Content-Type: multipart/mixed; boundary="-------------0306030922779" This is a multi-part message in MIME format. ---------------0306030922779 Content-Type: text/plain; name="03-255.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-255.keywords" ground state ---------------0306030922779 Content-Type: application/postscript; name="gdstate.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="gdstate.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.86 p1.5d Copyright 1996-2001 ASCII Corp.(www-ptex@ascii.co.jp) %%based on dvipsk 5.86 Copyright 1999 Radical Eye Software (www.radicaleye.com) %%Title: gdstate.dvi %%Pages: 34 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips gdstate.dvi -o gdstate.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.06.03:1309 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind 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bop Fy 440 975 a(The) 52 b(ground) f(state) i(problem) d(for) i(a) h (quan) l(tum) 555 1183 y(Hamiltonian) e(mo) t(del) h(describing) f (friction) p Fx 1451 1479 a(Lauren) m(t) 38 b(Bruneau) 647 1778 y(Departmen) m(t) f(of) h(Mathematical) e(Metho) s(ds) i(in) g(Ph) m(ysics) 1408 1927 y(W) -9 b(arsa) m(w) 37 b(Univ) m(ersit) m(y) 990 2077 y(Ho\273a) h(74,) f(00-682,) g(W) -9 b(arsza) m(w) m(a,) 36 b(P) m(oland) 1563 2309 y(June) i(3,) f(2003) p Fw 257 2684 a(Abstract:) p Fv 38 w(In) 23 b(this) g(pap) s(er,) i(w) m(e) f (consider) f(the) g(quan) m(tum) g(v) m(ersion) g(of) g(the) g (hamiltonian) 257 2804 y(mo) s(del) 30 b(describing) h(friction) f(in) m (tro) s(duced) i(in) e([4].) 43 b(This) 32 b(mo) s(del) e(consists) i (of) f(a) h(particle) 257 2925 y(whic) m(h) 27 b(in) m(teracts) f(with) g(a) f(b) s(osonic) h(reserv) m(oir) g(represen) m(ting) h(a) f (homogeneous) g(medium) 257 3045 y(through) 35 b(whic) m(h) h(the) f (particle) f(mo) m(v) m(es.) 51 b(W) -8 b(e) 36 b(sho) m(w) g(that) f (if) e(the) j(particle) d(is) i(con\034ned,) 257 3166 y(then) 30 b(the) g(Hamiltonian) 25 b(admits) j(a) h(ground) g(state) g (if) f(and) h(only) g(if) f(a) h(suitable) f(infrared) 257 3286 y(condition) 43 b(is) g(satis\034ed.) 79 b(The) 45 b(latter) d(is) i(violated) f(in) g(the) h(case) h(of) f(linear) e (friction,) 257 3406 y(but) 33 b(satis\034ed) g(when) h(the) e (friction) f(force) i(is) f(prop) s(ortional) d(to) j(a) h(higher) e(p) s(o) m(w) m(er) j(of) e(the) 257 3527 y(particle) g(sp) s(eed.) p Fu 257 3860 a(1) 156 b(In) l(tro) t(duction) p Fv 257 4079 a(In) 50 b([4]) g(w) m(e) h(in) m(tro) s(duced) f(a) f(classical) f (Hamiltonian) e(mo) s(del) i(of) i(a) f(particle) f(mo) m(ving) 257 4199 y(through) 40 b(a) g(homogeneous) h(dissipativ) m(e) e(medium) g (at) h(zero) g(temp) s(erature) g(in) g(suc) m(h) h(a) 257 4319 y(w) m(a) m(y) k(that) e(the) g(particle) f(exp) s(eriences) j(an) e(e\033ectiv) m(e) p Ft 44 w(line) -5 b(ar) p Fv 54 w(friction) 41 b(force) i(prop) s(or-) 257 4440 y(tional) f(to) h(its) h(v) m(elo) s (cit) m(y) -8 b(.) 76 b(The) 45 b(medium) d(consists) j(at) e(eac) m(h) i(p) s(oin) m(t) e(in) g(the) h(space) h(of) 257 4560 y(a) 37 b(vibration) e(\034eld) h(with) g(whic) m(h) h(the) g(particle) f(exc) m(hanges) i(energy) g(and) f(momen) m(tum.) 257 4680 y(More) c(precisely) g(the) g(Hamiltonian) c(is) j(giv) m(en) g(b) m(y) p Fs 358 4954 a(H) p Fr 8 w(\() p Fs(q) t(;) 17 b(p;) g(\036;) g(\031) p Fr 4 w(\)) 81 b(=) p Fs 1117 4887 a(p) p Fq 1166 4851 a(2) p 1117 4931 89 4 v Fr 1137 5022 a(2) 1238 4954 y(+) p Fs 22 w(V) p Fr 21 w(\() p Fs(q) p Fr 4 w(\)) 22 b(+) 1667 4887 y(1) p 1667 4931 49 4 v 1667 5022 a(2) p Fp 1743 4818 a(Z) p Fo 1798 5044 a(R) p Fn 1846 5025 a(d) p Fs 1903 4954 a(dx) p Fp 2026 4818 a(Z) p Fo 2081 5044 a(R) p Fn 2129 5025 a(n) p Fs 2192 4954 a(dy) e(c) p Fq 2353 4913 a(2) p Fm 2392 4954 a(jr) p Fl 2503 4969 a(y) p Fs 2544 4954 a(\036) p Fr(\() p Fs(x;) d(y) p Fr 4 w(\)) p Fm(j) p Fq 2857 4913 a(2) p Fr 2917 4954 a(+) p Fm 22 w(j) p Fs(\031) p Fr 4 w(\() p Fs(x;) g(y) p Fr 4 w(\)) p Fm(j) p Fq 3357 4913 a(2) p Fr 1498 5213 a(+) p Fp 1591 5077 a(Z) p Fo 1645 5303 a(R) p Fn 1693 5284 a(d) p Fs 1750 5213 a(dx) p Fp 1873 5077 a(Z) p Fo 1928 5303 a(R) p Fn 1976 5284 a(n) p Fs 2040 5213 a(dy) i(\032) p Fq 2208 5228 a(1) p Fr 2248 5213 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(q) p Fr 4 w(\)) p Fs(\032) p Fq 2598 5228 a(2) p Fr 2637 5213 a(\() p Fs(y) p Fr 4 w(\)) p Fs(\036) p Fr(\() p Fs(x;) e(y) p Fr 4 w(\)) p Fs(;) p Fv 218 w(\(1.1\)) 1852 5637 y(1) p 90 rotate dyy eop %%Page: 2 2 2 1 bop Fv 257 573 a(where) p Fs 29 w(V) p Fv 49 w(is) 27 b(an) g(external) h(p) s(oten) m(tial,) p Fs 27 w(c) p Fv 27 w(represen) m(ts) i(the) e(sp) s(eed) g(of) f(the) h(w) m(a) m(v) m(e) h(propaga-) 257 693 y(tion) h(in) g(the) h(\020mem) m(branes\021) 38 b(and) 30 b(the) h(functions) p Fs 31 w(\032) p Fq 2140 708 a(1) p Fv 2210 693 a(and) p Fs 31 w(\032) p Fq 2448 708 a(2) p Fv 2518 693 a(determine) g(the) g(coupling) 257 814 y(b) s(et) m(w) m(een) g(the) e(particle) e(and) h(the) h(\034eld) f (and) h(are) f(smo) s(oth) g(radial) e(functions) i(with) g(com-) 257 934 y(pact) 33 b(supp) s(ort.) 404 1054 y(W) -8 b(e) 33 b(studied) g(the) g(asymptotic) f(b) s(eha) m(viour) h(of) f(the) h (particle) e(motion) g(for) h(t) m(w) m(o) i(cat-) 257 1175 y(egories) g(of) f(p) s(oten) m(tials:) 45 b(linear) 32 b(ones) j(\(whic) m(h) f(means) g(constan) m(t) g(external) g(force\)) g (and) 257 1295 y(con\034ning) 40 b(ones.) 65 b(W) -8 b(e) 40 b(pro) m(v) m(ed) h(that) e(under) i(suitable) d(assumptions) i (\(on) f(the) h(initial) 257 1416 y(conditions\),) h(for) p Fs 39 w(c) p Fv 40 w(su\036cien) m(tly) g(large) d(and,) k(most) d(imp) s(ortan) m(tly) -8 b(,) p Fs 40 w(n) p Fr 40 w(=) 40 b(3) p Fs(;) p Fv 40 w(the) g(par-) 257 1536 y(ticle) g(b) s(eha) m(v) m (es) i(asymptotically) c(as) j(if) e(its) h(motion) f(w) m(as) i(go) m (v) m(erned) h(b) m(y) f(the) g(e\033ectiv) m(e) 257 1656 y(equation) p Fr 1322 1777 a(\177) p Fs -56 w(q) p Fr 4 w(\() p Fs(t) p Fr(\)) 22 b(+) p Fs 22 w(\015) p Fr 23 w(_) p Fs -45 w(q) p Fr 4 w(\() p Fs(t) p Fr(\)) 27 b(=) p Fm 28 w(\000r) p Fs(V) p Fr 22 w(\() p Fs(q) p Fr 4 w(\() p Fs(t) p Fr(\)\)) p Fs(;) p Fv 257 1951 a(where) 32 b(the) f(friction) d(co) s(e\036cien) m(t) p Fs 31 w(\015) p Fv 36 w(is) i(non) g(negativ) m(e) h(and) f(is) g(explicit) f(in) h (terms) g(of) g(the) 257 2071 y(parameters) j(of) f(the) h(mo) s(del:) p Fs 993 2334 a(\015) p Fr 33 w(:=) p Fs 1229 2267 a(\031) p 1218 2311 82 4 v 1218 2403 a(c) p Fq 1260 2374 a(3) p Fm 1309 2334 a(j) p Fr 9 w(^) p Fs -58 w(\032) p Fq 1387 2349 a(2) p Fr 1426 2334 a(\(0\)) p Fm(j) p Fq 1579 2293 a(2) p Fp 1635 2199 a(Z) p Fo 1690 2424 a(R) p Fn 1738 2405 a(n) p Fs 1801 2334 a(d\030) p Fp 1916 2199 a(Z) p Fo 1971 2424 a(R) p Fn 2019 2405 a(d) p Fk(\000) p Fj(1) p Fs 2155 2334 a(d\021) p Fm 20 w(j) p Fr 9 w(^) p Fs -58 w(\032) p Fq 2352 2349 a(1) p Fr 2391 2334 a(\() p Fm(j) p Fs(\030) p Fm 5 w(j) p Fs(;) 17 b(\021) p Fr 4 w(\)) p Fm(j) p Fq 2695 2293 a(2) p Fs 2732 2334 a(:) p Fv 536 w(\(1.2\)) 257 2601 y(If) p Fs 40 w(V) p Fr 61 w(=) p Fm 39 w(\000) p Fs(F) p Fm 41 w(\001) p Fs 26 w(q) t(;) p Fv 39 w(whic) m(h) 40 b(means) g(that) f(w) m(e) h(apply) f(a) g (constan) m(t) i(external) e(force) p Fs 39 w(F) p Fv 54 w(to) 257 2721 y(the) k(particle,) g(then) g(this) f(particle) f (reac) m(hes) i(exp) s(onen) m(tially) e(fast) h(\(with) g(rate) p Fs 42 w(\015) p Fv 5 w(\)) g(an) 257 2842 y(asymptotic) 33 b(v) m(elo) s(cit) m(y) p Fs 33 w(v) p Fr 4 w(\() p Fs(F) p Fr 14 w(\)) 28 b(=) p Fl 1467 2802 a(F) p 1467 2819 55 4 v 1474 2876 a(\015) p Fv 1564 2842 a(whic) m(h) 34 b(is) f(prop) s(ortional) d(to) j(the) g(applied) f(force) i(\(at) 257 2972 y(least) 43 b(for) f(small) f(forces\).) 76 b(This) 43 b(is,) i(in) d(particular,) i(at) f(the) g(origin) e(of) i(Ohm's) g(la) m(w.) 257 3092 y(On) 33 b(the) g(other) f(hand,) h(if) p Fs 31 w(V) p Fv 54 w(is) f(con\034ning,) h(the) g(particle) e(stops) i (at) f(one) h(of) f(the) h(critical) 257 3213 y(p) s(oin) m(ts) 27 b(of) f(the) h(p) s(oten) m(tial,) f(the) h(con) m(v) m(ergence) j (rate) c(b) s(eing) g(still) f(exp) s(onen) m(tial) h(\(but) h(with) 257 3333 y(rate) p Fl 468 3289 a(\015) p 468 3310 41 4 v Fq 470 3367 a(2) p Fv 551 3333 a(as) 32 b(exp) s(ected) j(from) c(the) i (e\033ectiv) m(e) g(equation\).) 404 3453 y(In) k([4]) g(w) m(e) h (mostly) f(concen) m(trated) h(on) f(linear) f(friction.) 55 b(This) 38 b(is) e(wh) m(y) j(the) p Fs 37 w(n) p Fr 36 w(=) c(3) p Fv 257 3574 a(assumption) k(w) m(as) h(required.) 64 b(Ho) m(w) m(ev) m(er,) 44 b(for) 39 b(other) g(v) -5 b(alues) 39 b(of) p Fs 39 w(n) p Fr 17 w(\() p Fs(>) p Fr 39 w(3\)) p Fv(,) i(our) e(mo) s(del) 257 3694 y(still) g(describ) s (es) j(friction.) 66 b(Indeed,) 45 b(the) c(reaction) f(force) h(of) f (the) i(en) m(vironmen) m(t) f(on) f(a) 257 3814 y(particle) 33 b(mo) m(ving) f(with) h(v) m(elo) s(cit) m(y) p Fs 34 w(v) p Fv 37 w(tak) m(es) i(the) f(form) p Fm 33 w(\000) p Fs(\015) p Fm 5 w(j) p Fs(v) p Fm 4 w(j) p Fl 2514 3778 a(n) p Fi(\000) p Fq(3) p Fs 2650 3814 a(v) p Fv 38 w(\(at) f(least) g (for) h(small) p Fs 257 3935 a(v) p Fv 34 w(and) d(where) p Fs 31 w(\015) p Fv 36 w(is) f(de\034ned) h(in) f(\(1.2\)\).) 42 b(One) 31 b(can) f(therefore) h(see) h(that) e(w) m(e) h(ha) m(v) m(e) h (linear) 257 4055 y(friction) i(when) p Fs 37 w(n) p Fr 33 w(=) e(3) p Fs(;) p Fv 36 w(and) j(otherwise) h(a) f(friction) f (force) i(whic) m(h) g(is) f(prop) s(ortional) e(to) 257 4176 y(some) g(other) f(p) s(o) m(w) m(er) i(of) e(the) h(v) m(elo) s (cit) m(y) f(of) g(the) h(particle.) 404 4296 y(Suc) m(h) 39 b(mo) s(dels,) g(where) h(a) e(small) e(system) k(in) m(teracts) e (with) g(a) p Ft 39 w(lar) -5 b(ge) p Fv 45 w(en) m(vironmen) m(t,) 257 4416 y(are) 34 b(called) e(op) s(en) h(systems.) 47 b(The) 34 b(reason) f(for) g(studying) h(those) f(mo) s(dels) f(is) h(usually) g (to) 257 4537 y(ha) m(v) m(e) 26 b(a) e(Hamiltonian) c(description) k (of) f(dissipativ) m(e) h(phenomena.) 41 b(There) 25 b(exist) g(sev) m(eral) 257 4657 y(mec) m(hanisms) 38 b(leading) e(to) h(dissipation.) 57 b(Among) 36 b(them,) j(t) m(w) m(o) f(imp) s(ortan) m(t,) f(and) h(v) m(ery) 257 4778 y(di\033eren) m(t,) 26 b(mec) m(hanisms) d(are) g(radiation) e(damping) h(and) h(friction) f (\(whic) m(h) i(can) f(b) s(e) h(linear) 257 4898 y(or) 32 b(not\).) 43 b(As) 32 b(far) f(as) h(radiation) d(damping) h(is) h (concerned,) j(there) e(exist) g(man) m(y) f(mo) s(dels,) 257 5018 y(whic) m(h) 47 b(are) f(more) f(or) h(less) g(related) f(to) h (electromagnetism.) 82 b(One) 46 b(example) f(is) h(the) 1852 5637 y(2) p 90 rotate dyy eop %%Page: 3 3 3 2 bop Fv 257 573 a(\020classical) 31 b(Nelson) h(mo) s(del\021) p Fs 396 801 a(H) p Fh 477 816 a(nels) p Fr 599 801 a(\() p Fs(q) t(;) 17 b(p;) g(\036;) g(\031) p Fr 4 w(\)) 82 b(=) p Fs 1271 734 a(p) p Fq 1320 698 a(2) p 1271 778 89 4 v Fr 1290 869 a(2) 1391 801 y(+) p Fs 22 w(V) p Fr 22 w(\() p Fs(q) p Fr 4 w(\)) 22 b(+) 1821 734 y(1) p 1821 778 49 4 v 1821 869 a(2) p Fp 1896 665 a(Z) p Fo 1951 891 a(R) p Fn 1999 872 a(d) p Fs 2056 801 a(dx) p Fp 2179 720 a(\000) p Fm 2224 801 a(jr) p Fs(\036) p Fr(\() p Fs(x) p Fr(\)) p Fm(j) p Fq 2552 760 a(2) p Fr 2613 801 a(+) p Fm 22 w(j) p Fs(\031) p Fr 4 w(\() p Fs(x) p Fr(\)) p Fm(j) p Fq 2957 760 a(2) p Fp 2996 720 a(\001) p Fr 2431 1060 a(+) p Fp 2524 924 a(Z) p Fo 2579 1150 a(R) p Fn 2627 1131 a(d) p Fs 2684 1060 a(dx\032) p Fr(\() p Fs(x) p Fm 23 w(\000) p Fs 22 w(q) p Fr 4 w(\)) p Fs(\036) p Fr(\() p Fs(x) p Fr(\)) p Fs(;) p Fv 257 1306 a(whic) m(h) 35 b(has) g(b) s(een) h(studied) f(in) e([16]) i(\(except) h(for) e(the) h(kinetic) f(energy) h(of) g(the) g(particle) 257 1439 y(whic) m(h) f(w) m(as) p Fp 726 1352 a(p) p 826 1352 258 4 v Fs 826 1439 a(p) p Fq 875 1410 a(2) p Fr 936 1439 a(+) 22 b(1) p Fv 34 w(instead) 33 b(of) p Fl 1576 1395 a(p) p Fj 1612 1372 a(2) p 1576 1416 71 4 v Fq 1594 1473 a(2) p Fv 1656 1439 a(\).) 47 b(This) 34 b(mo) s(del) e(describ) s(es) i(a) g(particle) e(in) m(teract-) 257 1559 y(ing) 26 b(with) g(a) g(scalar) g(radiation) e(\034eld,) k(and) f (exhibits) f(radiation) e(damping.) 40 b(Concerning) 257 1680 y(friction,) f(although) f(there) h(exist) g(v) -5 b(arious) 38 b(Hamiltonian) e(mo) s(dels) h(in) h(the) i(literature,) 257 1800 y(ours) 34 b(is) f(the) h(only) f(one) g(w) m(e) i(are) e(a) m (w) m(are) h(of) f(that) g(describ) s(es) i(the) f(friction) d(pro) s (duced) j(b) m(y) 257 1920 y(the) 40 b(motion) e(of) h(the) h(particle) e(through) h(a) h(homogeneous) f(medium.) 63 b(In) 40 b(particular,) 257 2041 y(the) 26 b(coupling) e(b) s(et) m(w) m(een) k (the) e(medium) d(and) j(the) g(particle) e(is) h(translationally) d (in) m(v) -5 b(arian) m(t) 257 2161 y(and) 32 b(hence) g(non-linear) e (in) g(the) i(particle) e(v) -5 b(ariable.) 41 b(This) 32 b(means) f(that) g(no) g(dip) s(ole) f(ap-) 257 2282 y(pro) m(ximation) e(is) h(assumed) h(and) g(is) f(essen) m(tial) g (for) g(a) g(correct) i(treatmen) m(t) e(of) g(a) g(constan) m(t) 257 2402 y(external) 43 b(force) g(\034eld.) 73 b(Despite) 43 b(the) g(formal) d(similarit) m(y) f(b) s(et) m(w) m(een) 45 b(our) d(mo) s(del) f(and) 257 2522 y(the) 36 b(classical) e(Nelson) h (mo) s(del,) g(w) m(e) h(w) m(an) m(t) g(to) f(stress) i(once) f(again) e(that) h(they) i(describ) s(e) 257 2643 y(ph) m(ysically) 45 b(totally) f(di\033eren) m(t) i(phenomena.) 83 b(This) 46 b(is) f(re\035ected) j(in) c(mathematical) 257 2763 y(di\033erences) 34 b(that) e(will) f(b) s(ecome) h(apparen) m(t) h(b) s(elo) m(w.) 404 2883 y(Our) 39 b(goal) e(in) h(this) h(pap) s(er) g(is) f(to) h(b) s (egin) f(the) i(study) g(of) e(the) i(quan) m(tum) f(v) m(ersion) h(of) 257 3004 y(the) c(mo) s(del) d(\(1.1\).) 51 b(Since) 35 b(the) g(sp) s(eed) i(of) d(the) i(w) m(a) m(v) m(e) h(propagation) c (will) g(not) i(pla) m(y) g(an) m(y) 257 3124 y(role) 30 b(in) f(our) h(pap) s(er,) h(w) m(e) h(tak) m(e) f(it) e(equal) h(to) g (1.) 43 b(The) 31 b(quan) m(tum) g(Hamiltonian) 26 b(can) 31 b(then) 257 3245 y(b) s(e) i(written) f(as) h(follo) m(ws) p Fs 394 3462 a(H) p Fr 91 w(=) 83 b(\() p Fm(\000) p Fr(\001) 22 b(+) p Fs 22 w(V) p Fr 22 w(\)) p Fm 22 w(\012) p Fr 23 w(1) -22 b(l) 21 b(+) h(1) -22 b(l) p Fm 20 w(\012) p Fp 1629 3326 a(Z) p Fs 1745 3462 a(dx) 17 b(dk) i(!) p Fr 4 w(\() p Fs(x;) e(k) p Fr 3 w(\)) p Fs(a) p Fi 2334 3421 a(\003) p Fr 2373 3462 a(\() p Fs(x;) g(k) p Fr 3 w(\)) p Fs(a) p Fr(\() p Fs(x;) g(k) p Fr 3 w(\)) 1505 3723 y(+) p Fp 1598 3588 a(Z) p Fs 1714 3723 a(dx) g(dk) 1968 3656 y(\032) p Fq 2018 3671 a(1) p Fr 2057 3656 a(\() p Fs(x) p Fm 23 w(\000) p Fs 22 w(Q) p Fr(\)) 9 b(^) p Fs -58 w(\032) p Fq 2437 3671 a(2) p Fr 2477 3656 a(\() p Fs(k) p Fr 3 w(\)) p 1968 3700 640 4 v Fp 2067 3720 a(p) p 2166 3720 343 4 v Fr 2166 3805 a(2) p Fs(!) p Fr 4 w(\() p Fs(x;) 17 b(k) p Fr 3 w(\)) p Fm 2639 3723 a(\012) p Fs 23 w(a) p Fi 2790 3682 a(\003) p Fr 2830 3723 a(\() p Fs(x;) g(k) p Fr 3 w(\)) 22 b(+) p Fs 22 w(h:c:;) p Fv 257 3986 a(where) p Fs 48 w(a) p Fv 47 w(and) p Fs 46 w(a) p Fi 905 3950 a(\003) p Fv 991 3986 a(are) 46 b(the) h(usual) f (annihilation) d(and) k(creation) e(op) s(erators) h(on) h(the) 257 4107 y(b) s(osonic) 34 b(F) -8 b(o) s(c) m(k) 35 b(space) p Fm 35 w(F) p Fr 10 w(\() p Fs(L) p Fq 1291 4070 a(2) p Fr 1330 4107 a(\() p Fg(R) p Fl 1434 4070 a(d) p Fq 1 w(+) p Fl(n) p Fs 1578 4107 a(;) 17 b(dx) g(dk) p Fr 3 w(\)\)) p Fs(;) p Fv 34 w(and) p Fs 35 w(!) p Fr 4 w(\() p Fs(x;) g(k) p Fr 3 w(\)) 30 b(=) p Fm 31 w(j) p Fs(k) p Fm 3 w(j) p Fv 33 w(is) k(the) h(b) s(osons) g(dis-) 257 4227 y(p) s(ersion) h(relation.) 53 b(In) 36 b(this) g(pap) s(er,) i(w) m(e) f(start) f(with) g(the) h(study) g(of) f(con\034ning) g(p) s (oten-) 257 4347 y(tials,) k(whic) m(h) h(are) f(less) g(di\036cult.) 64 b(More) 41 b(precisely) -8 b(,) 42 b(w) m(e) e(deal) g(with) f(the) h (question) h(of) 257 4468 y(existence) 31 b(of) e(a) f(ground) i (state,) g(whic) m(h) g(is) e(essen) m(tial) h(b) s(efore) g(studying) h (questions) g(suc) m(h) 257 4588 y(as) 36 b(scattering) e(theory) i(or) f(return) g(to) g(equilibrium) d(for) i(example.) 50 b(If) 35 b(a) g(Hamiltonian) 257 4709 y(is) e(b) s(ounded) h(from) e(b) s(elo) m(w,) h(w) m(e) h(sa) m(y) g(that) f(it) g(admits) e(a) i (ground) h(state) f(if) f(the) i(in\034m) m(um) 257 4829 y(of) j(its) f(sp) s(ectrum) h(is) g(an) f(eigen) m(v) -5 b(alue.) 56 b(W) -8 b(e) 37 b(call) e(ground) i(state) g(energy) h (this) f(in\034m) m(um) 257 4949 y(and) j(ground) f(state) g(an) m(y) h (corresp) s(onding) f(eigen) m(v) m(ector) h(if) e(it) g(exists.) 64 b(W) -8 b(e) 39 b(will) e(pro) m(v) m(e) 257 5070 y(that) g(suc) m(h) i (a) d(ground) h(state) h(exists) f(pro) m(vided) h(the) f(follo) m (wing) p Ft 34 w(infr) -5 b(ar) g(e) g(d) 38 b(c) -5 b(ondition) p Fv 44 w(is) 257 5190 y(satis\034ed:) p Fp 1398 5209 a(Z) p Fo 1454 5435 a(R) p Fn 1502 5416 a(n) p Fs 1565 5345 a(dk) p Fm 1696 5277 a(j) p Fr 9 w(^) p Fs -58 w(\032) p Fq 1774 5292 a(2) p Fr 1814 5277 a(\() p Fs(k) p Fr 3 w(\)) p Fm(j) p Fq 1972 5241 a(2) p 1696 5322 315 4 v Fm 1779 5413 a(j) p Fs(k) p Fm 3 w(j) p Fq 1889 5384 a(3) p Fs 2048 5345 a(<) p Fr 28 w(+) p Fm(1) p Fs(:) p Fv 1852 5637 a(3) p 90 rotate dyy eop %%Page: 4 4 4 3 bop Fv 257 573 a(This) 33 b(condition) e(will) g(b) s(e) h(used) i (to) e(con) m(trol) g(the) h(n) m(um) m(b) s(er) g(of) f(b) s(osons) i (whic) m(h) f(ha) m(v) m(e) h(lo) m(w) 257 693 y(energy) f(\(soft) f(b) s(osons\).) 44 b(Let) 32 b(us) g(supp) s(ose) i(that) p Fr 40 w(^) p Fs -58 w(\032) p Fq 2112 708 a(2) p Fr 2152 693 a(\(0\)) p Fm 27 w(6) p Fr(=) 28 b(0) p Fs(:) p Fv 32 w(Indeed,) 33 b(this) f(is) f(the) i(only) 257 814 y(in) m(teresting) 38 b(case) h(since) g(the) g(friction) d(co) s (e\036cien) m(t) p Fs 39 w(\015) p Fv 44 w(v) -5 b(anishes) 38 b(together) h(with) p Fr 47 w(^) p Fs -58 w(\032) p Fq 3331 829 a(2) p Fr 3371 814 a(\(0\)) p Fv 257 934 a(\(see) 46 b(\(1.2\)\).) 77 b(Then,) 49 b(there) 44 b(exists) h(a) f(ground) g (state) h(if) e(the) i(infrared) e(condition) g(is) 257 1054 y(satis\034ed) 31 b(\(Theorem) f(3.3\).) 42 b(One) 30 b(can) g(see) i(that) d(this) h(condition) e(is) i(ful\034lled) e(when) j(the) 257 1175 y(friction) k(is) g(non-linear.) 53 b(On) 36 b(the) h(other) f(hand,) i(for) e(linear) e(friction,) h(there) i(is) f (gener-) 257 1295 y(ically) h(no) i(ground) g(state) h(\(Prop) s (osition) d(3.4\).) 63 b(Th) m(us,) 42 b(w) m(e) e(ha) m(v) m(e) h(a) d (class) i(of) e(mo) s(dels,) 257 1416 y(dep) s(ending) h(on) g(a) f (parameter) p Fs 39 w(n;) p Fv 39 w(describing) g(friction) f (phenomena,) j(linear) d(or) i(pro-) 257 1536 y(p) s(ortional) 31 b(to) i(a) f(p) s(o) m(w) m(er) i(of) f(the) h(v) m(elo) s(cit) m(y) f (of) f(the) i(particle,) e(for) g(whic) m(h) i(w) m(e) g(are) f(able) g (to) 257 1656 y(sa) m(y) h(w) m(ether) g(they) f(admit) e(a) h(ground) h (state) g(or) f(not.) 404 1777 y(As) 27 b(in) e(the) i(classical) e (case,) j(our) f(mo) s(del) d(lo) s(oks) i(v) m(ery) i(similar) 23 b(to) j(the) g(Nelson) h(mo) s(del,) 257 1897 y(and) 37 b(more) e(generally) h(to) g(the) g(P) m(auli-Fierz) e(mo) s(dels) i (\(follo) m(wing) d(the) k(terminology) d(of) 257 2017 y([9]\),) c(in) f(whic) m(h) h(a) f(\(small\)) e(quan) m(tum) i(system) h(in) m(teracts) g(with) f(a) g(scalar) g(b) s(osonic) f(\034eld,) 257 2138 y(although) 38 b(they) i(lead) f(to) f(v) m(ery) j(di\033eren) m (t) e(dissipativ) m(e) f(phenomena.) 64 b(W) -8 b(e) 39 b(will) e(recall) 257 2258 y(some) 31 b(basic) h(facts) f(ab) s(out) g (F) -8 b(o) s(c) m(k) 31 b(spaces) i(and) e(describ) s(e) h(the) g (quan) m(tum) f(v) m(ersion) h(of) f(the) 257 2379 y(mo) s(del) g(in) h (Sect.) 44 b(2,) 33 b(while,) e(in) h(Sect.) 44 b(3,) 33 b(w) m(e) g(state) g(our) g(main) d(results.) 404 2499 y(T) -8 b(o) 34 b(pro) m(v) m(e) i(the) f(existence) h(of) e(a) h (ground) f(sate,) i(w) m(e) g(follw) d(the) i(standard) g(strategy:) 257 2619 y(w) m(e) i(\034rst) g(pro) m(v) m(e) g(the) f(result) g(for) f (coupling) g(to) g(a) h(massiv) m(e) g(\034eld) f(and) h(then) h(w) m (e) g(let) e(the) 257 2740 y(mass) 43 b(tend) g(to) f(zero.) 74 b(W) -8 b(e) 43 b(study) h(the) f(massiv) m(e) g(case) g(in) f(Sect.) 75 b(4) 42 b(along) f(the) i(lines) 257 2860 y(of) 35 b([5]-[6]-[14) o(]:) 49 b(w) m(e) 37 b(\034rst) f(constrain) f(the) h(mo) s(del) e(to) h(a) g (\034nite) g(b) s(o) m(x) h(\(|x|) p Fr 28 w(0) p Fs(:) p Fv 257 2107 a(Our) 41 b(pro) s(of) f(will) f(use) j (di\033eren) m(t) f(metho) s(ds) f(dev) m(elop) s(ed) i(in) e(the) h (literature) f([5]-[6) o(]-[9]-) 257 2228 y([13]-[14) o(].) 404 2348 y(Finally) -8 b(,) 25 b(w) m(e) j(w) m(ould) f(lik) m(e) f(to) g (emphasize) h(that) g(all) e(the) i(Hamiltonians) c(w) m(e) 28 b(will) d(deal) 257 2468 y(with) 33 b(ha) m(v) m(e) h(the) f(same) g (structure) h(as) f(\(2.11\)) f(and) h(so,) h(a) e(similar) e(result) j (to) f(the) i(one) f(of) 257 2589 y(Prop) s(osition) e(3.1) h(is) g(a) m (v) -5 b(ailable) 30 b(for) i(eac) m(h) i(of) e(them.) p Fu 257 2922 a(4) 156 b(Ground) 52 b(state) g(for) h(massiv) l(e) f(b) t (osons) p Fv 257 3141 a(Our) 34 b(goal) e(in) g(this) h(section) h(is) f (to) g(pro) m(v) m(e) i(a) e(\034rst) h(result) f(similar) d(to) j (Theorem) h(3.3) f(but) 257 3261 y(in) 40 b(the) g(case) h(of) f (massiv) m(e) g(b) s(osons) g(\(Theorem) h(4.7,) g(Sect.) 67 b(4.2\).) e(W) -8 b(e) 41 b(use) g(the) f(same) 257 3381 y(approac) m(h) 28 b(as) g(in) f([14]) h(and) g([5) o(].) 42 b(The) 29 b(idea) e(is) g(\034rst) i(to) e(consider) h(a) f(\034nite) h (b) s(o) m(x) g(\() p Fm(j) p Fs(x) p Fm(j) p Fs 27 w(<) g(L) p Fv(\)) 257 3502 y(and) 41 b(then) f(to) g(con) m(trol) g(the) g (remaining) e(part) i(as) p Fs 40 w(L) p Fv 41 w(go) s(es) g(to) g (in\034nit) m(y) -8 b(.) 66 b(W) -8 b(e) 40 b(will) e(see,) 257 3622 y(in) 33 b(Sect.) 47 b(4.2,) 33 b(that) h(the) f(\020cuto\033) 7 b(\021) 41 b(mo) s(del) 32 b(so) h(obtained) g(can) h(b) s(e) g (written) f(in) g(the) g(form) 257 3743 y(\(4.1\).) 43 b(W) -8 b(e) 33 b(therefore) g(\034rst) g(study) h(mo) s(dels) d(of) h (this) h(latter) e(t) m(yp) s(e) j(\(Theorem) e(4.1\).) p Ff 257 4031 a(4.1) 131 b(Discrete) 44 b(mo) t(dels) p Fw 257 4216 a(4.1.1) 113 b(Description) p Fv 257 4401 a(W) -8 b(e) 33 b(consider) g(Hamiltonians) c(of) j(te) h(form) p Fs 465 4644 a(H) p Fh 554 4603 a(d) p Fr 680 4644 a(:=) p Fs 83 w(H) p Fl 947 4659 a(p) p Fm 1009 4644 a(\012) p Fr 23 w(1) -22 b(l) 20 b(+) i(1) -22 b(l) p Fm 21 w(\012) p Fp 1461 4550 a(X) p Fl 1458 4769 a(l) p Fi 1 w(2) p Fo(Z) p Fn 1577 4750 a(d) p Fp 1624 4509 a(Z) p Fo 1679 4734 a(R) p Fn 1727 4715 a(n) p Fs 1790 4644 a(dk) 20 b(!) p Fl 1973 4659 a(m) p Fr 2039 4644 a(\() p Fs(k) p Fr 3 w(\)) p Fs(a) p Fi 2220 4603 a(\003) p Fl 2220 4669 a(l) p Fr 2260 4644 a(\() p Fs(k) p Fr 3 w(\)) p Fs(a) p Fl 2441 4659 a(l) p Fr 2467 4644 a(\() p Fs(k) p Fr 3 w(\)) 1451 4952 y(+) p Fp 1547 4858 a(X) p Fl 1544 5077 a(l) p Fi 1 w(2) p Fo(Z) p Fn 1663 5058 a(d) p Fp 1710 4817 a(Z) p Fo 1765 5042 a(R) p Fn 1813 5023 a(n) p Fs 1876 4952 a(dk) p Fr 20 w(\() p Fs(\014) p Fl 2091 4967 a(l) p Fr 2117 4952 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 23 w(a) p Fi 2420 4911 a(\003) p Fl 2420 4977 a(l) p Fr 2459 4952 a(\() p Fs(k) p Fr 3 w(\)) i(+) 2723 4926 y(\026) p Fs 2709 4952 a(\014) p Fl 2764 4967 a(l) p Fr 2790 4952 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 23 w(a) p Fl 3093 4967 a(l) p Fr 3119 4952 a(\() p Fs(k) p Fr 3 w(\)\)) 694 5211 y(=) p Fs 96 w(H) p Fh 955 5170 a(d) p Fq 947 5236 a(0) p Fr 1020 5211 a(+) p Fs 22 w(W) p Fh 1224 5170 a(d) p Fs 1267 5211 a(;) p Fv 2001 w(\(4.1\)) 1828 5637 y(10) p 90 rotate dyy eop %%Page: 11 11 11 10 bop Fv 257 573 a(on) 33 b(the) g(space) p Fm 1071 693 a(H) p Fh 1156 652 a(d) p Fr 1227 693 a(:=) p Fs 27 w(L) p Fq 1423 652 a(2) p Fr 1463 693 a(\() p Fg(R) p Fl 1567 652 a(d) p Fr 1613 693 a(\)) p Fm 23 w(\012) 22 b(F) p Fp 1871 613 a(\000) p Fs 1917 693 a(l) p Fq 1948 652 a(2) p Fr 1988 693 a(\() p Fg(Z) p Fl 2095 652 a(d) p Fr 2132 693 a(\)) p Fm 22 w(\012) p Fs 23 w(L) p Fq 2358 652 a(2) p Fr 2398 693 a(\() p Fg(R) p Fl 2502 652 a(n) p Fr 2555 693 a(\)) p Fp 2593 613 a(\001) p Fs 2655 693 a(;) p Fv 613 w(\(4.2\)) 257 857 y(and) 33 b(where) h(the) p Fs 33 w(\014) p Fl 952 872 a(l) p Fr 978 857 a(\() p Fs(k) p Fr 3 w(\)) p Fv 32 w(satisfy) p Fr 501 1176 a(\() p Fs(C) p Fl 609 1191 a(\014) p Fr 656 1176 a(\)) p Fs 98 w(\014) p Fl 847 1191 a(l) p Fr 873 1176 a(\() p Fs(k) p Fr 3 w(\)) 27 b(=) p Fs 28 w(\020) p Fl 1177 1191 a(l) p Fq 1294 1128 a(^) p Fl -42 w(\032) p Fj 1323 1137 a(2) p Fq 1358 1128 a(\() p Fl(k) p Fq 2 w(\)) p 1212 1153 315 4 v Fm 1212 1165 a(p) p 1295 1165 232 4 v Fq 1295 1231 a(2) p Fl(!) p Fn 1374 1239 a(m) p Fq 1433 1231 a(\() p Fl(k) p Fq 2 w(\)) p Fv 1569 1176 a(where) p Fs 33 w(\020) p Fl 1893 1191 a(l) p Fv 1951 1176 a(is) 32 b(a) g(m) m(ultiplication) c (op) s(erator) j(on) p Fs 501 1351 a(L) p Fq 567 1315 a(2) p Fr 607 1351 a(\() p Fg(R) p Fl 711 1315 a(d) p Fr 757 1351 a(\)) p Fv 33 w(suc) m(h) j(that) p Fr 32 w(sup) p Fl 1406 1375 a(l) p Fm 1449 1351 a(kj) p Fs(l) p Fm 2 w(j) p Fl 1586 1315 a(s) p Fs 1622 1351 a(\020) p Fl 1665 1366 a(l) p Fm 1690 1351 a(k) p Fs 28 w(<) p Fr 27 w(+) p Fm(1) p Fv 32 w(for) e(all) p Fs 31 w(s) c(>) p Fr 27 w(0) p Fv(,) p Fs 257 1670 a(a) p Fl 308 1685 a(l) p Fr 335 1670 a(\() p Fs(k) p Fr 3 w(\)) p Fv 46 w(and) p Fs 47 w(a) p Fi 766 1634 a(\003) p Fl 766 1696 a(l) p Fr 806 1670 a(\() p Fs(k) p Fr 3 w(\)) p Fv 47 w(are) 46 b(the) i(annihilation) 43 b(and) k(creation) f(op) s(erators) g(on) h (the) g(space) p Fm 257 1791 a(F) p Fp 355 1710 a(\000) p Fs 401 1791 a(l) p Fq 432 1754 a(2) p Fr 472 1791 a(\() p Fg(Z) p Fl 579 1754 a(d) p Fr 617 1791 a(\)) p Fm 22 w(\012) p Fs 22 w(L) p Fq 842 1754 a(2) p Fr 882 1791 a(\() p Fg(R) p Fl 986 1754 a(n) p Fr 1039 1791 a(\)) p Fp 1077 1710 a(\001) p Fs 1139 1791 a(;) p Fv 33 w(and) 32 b(for) p Fs 32 w(l) p Fr 30 w(=) c(\() p Fs(l) p Fq 1767 1806 a(1) p Fs 1806 1791 a(;) 17 b(:) g(:) g(:) f(;) h(l) p Fl 2054 1806 a(d) p Fr 2094 1791 a(\)) p Fm 28 w(2) p Fg 28 w(Z) p Fl 2323 1754 a(d) p Fs 2361 1791 a(;) p Fm 33 w(j) p Fs(l) p Fm 2 w(j) p Fr 27 w(:=) 27 b(sup) p Fl 2812 1814 a(i) p Fm 2857 1791 a(j) p Fs(l) p Fl 2914 1806 a(i) p Fm 2942 1791 a(j) p Fs(:) p Fv 404 1911 a(W) -8 b(e) 23 b(w) m(ould) h(lik) m(e) e(to) i(stress) h(that) e(one) g(can) h (consider) g(the) g(Hamiltonians) c(of) j(the) g(form) 257 2031 y(\(4.1\)) 33 b(as) g(mo) s(dels) f(similar) e(to) j(ours,) g(but) h(with) e(only) h(a) g(discrete) h(set) f(of) g(\020mem) m(branes\021) 257 2152 y(\(situated) g(at) f(eac) m(h) p Fs 33 w(l) p Fm 30 w(2) p Fg 28 w(Z) p Fl 1230 2116 a(d) p Fv 1268 2152 a(\)) h(rather) f(than) h(a) f(con) m(tin) m(uous) h(one.) 404 2272 y(No) m(w,) 42 b(let) p Fs 39 w(E) p Fh 888 2236 a(d) p Fq 882 2297 a(0) p Fv 971 2272 a(denote) e(the) g(ground) f (state) h(energy) h(for) p Fs 39 w(H) p Fh 2614 2236 a(d) p Fs 2657 2272 a(:) p Fv 40 w(W) -8 b(e) 40 b(will) d(pro) m(v) m (e) k(the) 257 2392 y(follo) m(wing:) p Fw 257 2574 a(Theorem) g(4.1.) p Fs 43 w(\033) p Fl 1008 2589 a(ess) p Fr 1111 2574 a(\() p Fs(H) p Fb 1238 2538 a(d) p Fr 1278 2574 a(\)) p Fm 33 w(\032) p Fp 1459 2493 a(\002) p Fs 1501 2574 a(E) p Fb 1579 2538 a(d) p Fq 1573 2598 a(0) p Fr 1642 2574 a(+) p Fs 22 w(m;) p Fr 17 w(+) p Fm(1) p Fp 2045 2493 a(\002) p Fs 2102 2574 a(:) p Ft 38 w(In) c(p) -5 b(articular,) p Fs 38 w(H) p Fb 2865 2538 a(d) p Ft 2943 2574 a(has) 37 b(a) h(gr) -5 b(ound) 257 2694 y(state) p Fs 35 w(\036) p Fb 549 2658 a(d) p Fq 549 2719 a(0) p Ft 590 2694 a(.) p Fw 257 2950 a(4.1.2) 113 b(Cuto\033) 37 b(mo) s(dels) p Fv 257 3135 a(In) c(the) g(follo) m(wing,) p Fs 30 w(M) p Fv 43 w(will) e(b) s(e) h(a) h(non) f(negativ) m(e) h(n) m(um) m(b) s (er.) 44 b(On) p Fm 32 w(H) p Fh 2736 3099 a(d) p Fs 2780 3135 a(;) p Fv 32 w(w) m(e) 34 b(de\034ne) p Fs 567 3372 a(H) p Fh 656 3331 a(d) p Fr 699 3372 a(\() p Fs(M) p Fr 10 w(\)) 84 b(:=) p Fs 83 w(H) p Fh 1238 3331 a(d) p Fq 1230 3397 a(0) p Fr 1303 3372 a(+) p Fp 1424 3277 a(X) p Fi 1401 3493 a(j) p Fl(l) p Fi 1 w(j\024) p Fl(M) p Fp 1608 3236 a(Z) p Fo 1663 3462 a(R) p Fn 1711 3443 a(n) p Fs 1775 3372 a(dk) p Fr 19 w(\() p Fs(\014) p Fl 1989 3387 a(l) p Fr 2015 3372 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 23 w(a) p Fi 2318 3331 a(\003) p Fl 2318 3397 a(l) p Fr 2357 3372 a(\() p Fs(k) p Fr 3 w(\)) 23 b(+) 2622 3346 y(\026) p Fs 2608 3372 a(\014) p Fl 2663 3387 a(l) p Fr 2689 3372 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 22 w(a) p Fl 2991 3387 a(l) p Fr 3018 3372 a(\() p Fs(k) p Fr 3 w(\)\)) p Fv 109 w(\(4.3\)) p Fr 976 3640 a(=) p Fs 97 w(H) p Fh 1238 3599 a(d) p Fq 1230 3664 a(0) p Fr 1303 3640 a(+) p Fs 22 w(W) p Fh 1507 3599 a(d) p Fr 1550 3640 a(\() p Fs(M) p Fr 10 w(\)) p Fs(:) p Fv 257 3834 a(W) -8 b(e) 33 b(also) f(de\034ne) p Fr 451 4027 a(~) p Fs 426 4052 a(H) p Fh 515 4011 a(d) p Fr 558 4052 a(\() p Fs(M) p Fr 10 w(\)) 83 b(:=) p Fs 83 w(H) p Fl 1088 4067 a(p) p Fm 1150 4052 a(\012) p Fr 23 w(1) -22 b(l) 21 b(+) h(1) -22 b(l) p Fm 20 w(\012) p Fp 1622 3957 a(X) p Fi 1599 4173 a(j) p Fl(l) p Fi 1 w(j\024) p Fl(M) p Fp 1806 3916 a(Z) p Fo 1862 4142 a(R) p Fn 1910 4123 a(n) p Fs 1973 4052 a(dk) 19 b(!) p Fl 2155 4067 a(m) p Fr 2222 4052 a(\() p Fs(k) p Fr 3 w(\)) p Fs(a) p Fi 2403 4011 a(\003) p Fl 2403 4076 a(l) p Fr 2442 4052 a(\() p Fs(k) p Fr 3 w(\)) p Fs(a) p Fl 2623 4067 a(l) p Fr 2649 4052 a(\() p Fs(k) p Fr 3 w(\)) k(+) p Fs 22 w(W) p Fh 3006 4011 a(d) p Fr 3049 4052 a(\() p Fs(M) p Fr 10 w(\)) p Fv 66 w(\(4.4\)) p Fr 835 4324 a(=) 1033 4299 y(~) p Fs 1007 4324 a(H) p Fh 1096 4283 a(d) p Fq 1088 4349 a(0) p Fr 1139 4324 a(\() p Fs(M) p Fr 10 w(\)) g(+) p Fs 22 w(W) p Fh 1546 4283 a(d) p Fr 1589 4324 a(\() p Fs(M) p Fr 10 w(\)) p Fs(;) p Fv 257 4518 a(as) 33 b(an) g(op) s(erator) e(on) i(the) g(space) p Fm 1033 4712 a(H) p Fh 1118 4671 a(d) p Fl 1117 4737 a(M) p Fr 1224 4712 a(:=) p Fs 28 w(L) p Fq 1421 4671 a(2) p Fr 1461 4712 a(\() p Fg(R) p Fl 1564 4671 a(d) p Fr 1611 4712 a(\)) p Fm 22 w(\012) 23 b(F) p Fp 1869 4632 a(\000) p Fs 1914 4712 a(l) p Fq 1945 4671 a(2) p Fr 1985 4712 a(\(\003) p Fl 2091 4727 a(M) p Fr 2170 4712 a(\)) p Fm 22 w(\012) p Fs 22 w(L) p Fq 2395 4671 a(2) p Fr 2435 4712 a(\() p Fg(R) p Fl 2539 4671 a(n) p Fr 2592 4712 a(\)) p Fp 2630 4632 a(\001) p Fs 2692 4712 a(;) p Fv 576 w(\(4.5\)) 257 4907 y(where) p Fr 35 w(\003) p Fl 608 4922 a(M) p Fr 715 4907 a(=) p Fm 29 w(f) p Fs(l) p Fm 31 w(2) p Fg 29 w(Z) p Fl 1094 4870 a(d) p Fs 1132 4907 a(;) p Fm 17 w(j) p Fs(l) p Fm 2 w(j) 28 b(\024) p Fs 30 w(M) p Fm 10 w(g) p Fs(;) p Fv 34 w(so) 33 b(that) p Fs 33 w(l) p Fq 1976 4870 a(2) p Fr 2016 4907 a(\(\003) p Fl 2122 4922 a(M) p Fr 2201 4907 a(\)) p Fv 33 w(is) f(a) h(\034nite) g (dimensional) e(space.) 257 5027 y(Let) p Fs 45 w(E) p Fh 522 4991 a(d) p Fq 516 5052 a(0) p Fr 566 5027 a(\() p Fs(M) p Fr 10 w(\)) p Fv 45 w(\(resp.) p Fr 1133 5002 a(~) p Fs 1110 5027 a(E) p Fh 1188 4991 a(d) p Fq 1182 5052 a(0) p Fr 1232 5027 a(\() p Fs(M) p Fr 10 w(\)) p Fv(\)) 45 b(b) s(e) g(the) g(ground) g(state) g(energy) h(for) p Fs 44 w(H) p Fh 2988 4991 a(d) p Fr 3031 5027 a(\() p Fs(M) p Fr 10 w(\)) p Fv 45 w(\(resp.) p Fr 283 5122 a(~) p Fs 257 5147 a(H) p Fh 346 5111 a(d) p Fr 389 5147 a(\() p Fs(M) p Fr 10 w(\)) p Fv(\).) k(Our) 34 b(goal) f(is) g(to) h (get) h(informations) c(on) p Fs 34 w(H) p Fh 2273 5111 a(d) p Fv 2351 5147 a(from) i(the) h(ones) i(w) m(e) f(will) d(ha) m(v) m(e) 257 5268 y(on) p Fs 29 w(H) p Fh 478 5232 a(d) p Fr 520 5268 a(\() p Fs(M) p Fr 10 w(\)) p Fv 29 w(\(taking) c(the) g (limit) p Fs 25 w(M) p Fm 38 w(!) p Fr 28 w(+) p Fm(1) p Fv(\).) 41 b(Th) m(us,) 31 b(w) m(e) e(\034rst) g(pro) m(v) m(e) g(a) f (result) h(similar) 257 5388 y(to) k(Theorem) f(4.1,) h(but) f(for) p Fs 32 w(H) p Fh 1389 5352 a(d) p Fr 1432 5388 a(\() p Fs(M) p Fr 10 w(\)) p Fs(:) p Fv 1828 5637 a(11) p 90 rotate dyy eop %%Page: 12 12 12 11 bop Fw 257 573 a(Prop) s(osition) 23 b(4.2.) p Fs 34 w(\033) p Fl 1113 588 a(ess) p Fr 1215 573 a(\() p Fs(H) p Fb 1342 537 a(d) p Fr 1382 573 a(\() p Fs(M) p Fr 10 w(\)\)) p Fm 29 w(\032) p Fp 1734 492 a(\002) p Fs 1775 573 a(E) p Fb 1853 537 a(d) p Fq 1847 597 a(0) p Fr 1894 573 a(\() p Fs(M) p Fr 10 w(\)) g(+) p Fs 22 w(m;) p Fr 17 w(+) p Fm(1) p Fp 2500 492 a(\002) p Fs 2557 573 a(:) p Ft 25 w(In) i(p) -5 b(articular,) p Fs 26 w(H) p Fb 3283 537 a(d) p Fr 3323 573 a(\() p Fs(M) p Fr 10 w(\)) p Ft 257 707 a(has) 35 b(a) f(gr) -5 b(ound) 35 b(state) p Fs 35 w(\036) p Fb 1131 671 a(d) p Fq 1131 732 a(0) p Fr 1171 707 a(\() p Fs(M) p Fr 10 w(\)) p Fs(:) p Ft 36 w(Mor) -5 b(e) g(over,) p Fs 34 w(E) p Fb 1950 671 a(d) p Fq 1944 732 a(0) p Fr 1991 707 a(\() p Fs(M) p Fr 10 w(\)) 28 b(=) 2325 682 y(~) p Fs 2303 707 a(E) p Fb 2381 671 a(d) p Fq 2375 732 a(0) p Fr 2421 707 a(\() p Fs(M) p Fr 10 w(\)) p Fs(:) p Fw 257 924 a(Lemma) i(4.3.) p Fs 37 w(\033) p Fl 913 939 a(ess) p Fr 1016 924 a(\() 1079 899 y(~) p Fs 1054 924 a(H) p Fb 1143 888 a(d) p Fr 1183 924 a(\() p Fs(M) p Fr 10 w(\)\)) p Fm 28 w(\032) p Fp 1534 814 a(h) p Fr 1604 899 a(~) p Fs 1581 924 a(E) p Fb 1659 888 a(d) p Fq 1653 949 a(0) p Fr 1700 924 a(\() p Fs(M) p Fr 10 w(\)) 23 b(+) p Fs 22 w(m;) p Fr 17 w(+) p Fm(1) p Fp 2306 814 a(h) p Fs 2369 924 a(:) p Ft 29 w(In) 28 b(p) -5 b(articular,) p Fr 3042 899 a(~) p Fs 3017 924 a(H) p Fb 3106 888 a(d) p Fr 3146 924 a(\() p Fs(M) p Fr 10 w(\)) p Ft 30 w(has) 257 1090 y(a) 35 b(gr) -5 b(ound) 34 b(state) p Fr 911 1064 a(~) p Fs 899 1090 a(\036) p Fb 957 1054 a(d) p Fq 957 1115 a(0) p Fr 997 1090 a(\() p Fs(M) p Fr 10 w(\)) p Fs(:) p Fw 257 1280 a(Pro) s(of) 48 b(of) g(Lemma) f(4.3:) p Fv 61 w(The) c(set) p Fr 42 w(\003) p Fl 1796 1295 a(M) p Fv 1916 1280 a(is) e(\034nite.) 70 b(If) 41 b(its) g(cardinal) f(w) m (as) i(one,) i(w) m(e) 257 1400 y(w) m(ould) e(ha) m(v) m(e) i(exactly) e(the) h(mo) s(del) d(studied) i(in) g([9],) i(and) e(the) h(lemma) c (w) m(ould) j(corre-) 257 1520 y(sp) s(ond) 32 b(to) f(their) f (Theorem) i(4.1.) 42 b(Ha) m(ving) 31 b(\034nitely) f(man) m(y) h (elemen) m(ts) h(do) s(es) f(not) g(c) m(hange) 257 1641 y(an) m(ything) i(and) f(the) h(result) g(can) g(b) s(e) f(pro) m(v) m (en) i(the) f(same) g(w) m(a) m(y) -8 b(.) p Fc 904 w(2) p Fw 257 1881 a(Pro) s(of) 40 b(of) g(Prop) s(osition) e(4.2:) p Fv 48 w(The) e(prop) s(osition) d(follo) m(ws) h(immediately) d(from) j (the) 257 2002 y(preceding) h(lemma) c(using) j(an) g(iden) m (ti\034cation) e(b) s(et) m(w) m(een) p Fm 36 w(H) p Fh 2459 1966 a(d) p Fl 2458 2027 a(M) p Fv 2571 2002 a(and) i(some) g(subspace) i(of) p Fm 257 2122 a(H) p Fh 342 2086 a(d) p Fs 386 2122 a(;) p Fv 32 w([14].) 44 b(Indeed,) 34 b(one) e(can) h(write) p Fs 1315 2326 a(l) p Fq 1346 2285 a(2) p Fr 1386 2326 a(\() p Fg(Z) p Fl 1493 2285 a(d) p Fr 1531 2326 a(\)) p Fm 28 w(') p Fs 28 w(l) p Fq 1733 2285 a(2) p Fr 1772 2326 a(\(\003) p Fl 1878 2341 a(M) p Fr 1957 2326 a(\)) p Fm 22 w(\010) p Fs 23 w(l) p Fq 2148 2285 a(2) p Fr 2187 2326 a(\(\003) p Fl 2293 2285 a(c) 2293 2351 y(M) p Fr 2372 2326 a(\)) p Fs(;) p Fr 257 2530 a(\003) p Fl 325 2494 a(c) 325 2555 y(M) p Fv 437 2530 a(denoting) f(the) g(complemen) m(t) g(of) p Fr 32 w(\003) p Fl 1729 2545 a(M) p Fv 1840 2530 a(in) p Fg 32 w(Z) p Fl 2023 2494 a(d) p Fs 2061 2530 a(;) p Fv 33 w(so) g(one) h(has) p Fm 389 2734 a(F) p Fp 488 2654 a(\000) p Fs 533 2734 a(l) p Fq 564 2693 a(2) p Fr 604 2734 a(\() p Fg(Z) p Fl 711 2693 a(d) p Fr 749 2734 a(\)) p Fm 22 w(\012) p Fs 22 w(L) p Fq 974 2693 a(2) p Fr 1014 2734 a(\() p Fg(R) p Fl 1118 2693 a(n) p Fr 1171 2734 a(\)) p Fp 1209 2654 a(\001) p Fm 1282 2734 a(') c(F) p Fp 1486 2654 a(\000) p Fs 1531 2734 a(l) p Fq 1562 2693 a(2) p Fr 1602 2734 a(\(\003) p Fl 1708 2749 a(M) p Fr 1787 2734 a(\)) p Fm 22 w(\012) p Fs 22 w(L) p Fq 2012 2693 a(2) p Fr 2052 2734 a(\() p Fg(R) p Fl 2156 2693 a(n) p Fr 2209 2734 a(\)) p Fp 2247 2654 a(\001) p Fm 2315 2734 a(\012) 22 b(F) p Fp 2512 2654 a(\000) p Fs 2558 2734 a(l) p Fq 2589 2693 a(2) p Fr 2629 2734 a(\(\003) p Fl 2735 2693 a(c) 2735 2759 y(M) p Fr 2813 2734 a(\)) p Fm 23 w(\012) p Fs 22 w(L) p Fq 3039 2693 a(2) p Fr 3079 2734 a(\() p Fg(R) p Fl 3183 2693 a(n) p Fr 3236 2734 a(\)) p Fp 3274 2654 a(\001) p Fs 3336 2734 a(:) p Fv 257 2938 a(And) 33 b(\034nally) p Fm 1129 3059 a(H) p Fh 1214 3018 a(d) p Fm 1285 3059 a(') 28 b(H) p Fh 1475 3018 a(d) p Fl 1474 3083 a(M) p Fm 1575 3059 a(\012) 23 b(F) p Fp 1773 2978 a(\000) p Fs 1819 3059 a(l) p Fq 1850 3018 a(2) p Fr 1889 3059 a(\(\003) p Fl 1995 3018 a(c) 1995 3083 y(M) p Fr 2074 3059 a(\)) p Fm 22 w(\012) p Fs 23 w(L) p Fq 2300 3018 a(2) p Fr 2340 3059 a(\() p Fg(R) p Fl 2443 3018 a(n) p Fr 2496 3059 a(\)) p Fp 2534 2978 a(\001) p Fs 2597 3059 a(:) p Fv 257 3226 a(One) 50 b(can) f(then) h(iden) m(tify) p Fm 49 w(H) p Fh 1371 3190 a(d) p Fl 1370 3251 a(M) p Fv 1498 3226 a(with) p Fm 49 w(H) p Fh 1822 3190 a(d) p Fl 1821 3251 a(M) p Fm 1934 3226 a(\012) p Fr 34 w(\012) p Fl 2115 3190 a(c) 2115 3251 y(M) p Fv 2243 3226 a(where) p Fr 51 w(\012) p Fl 2612 3190 a(c) 2612 3251 y(M) p Fv 2740 3226 a(is) f(the) h(v) -5 b(acuum) 49 b(of) p Fm 257 3347 a(F) p Fr 26 w(\() p Fs(l) p Fq 424 3310 a(2) p Fr 464 3347 a(\(\003) p Fl 570 3310 a(c) 570 3372 y(M) p Fr 649 3347 a(\)) p Fm 22 w(\012) p Fs 22 w(L) p Fq 874 3310 a(2) p Fr 914 3347 a(\() p Fg(R) p Fl 1018 3310 a(n) p Fr 1071 3347 a(\)\)) p Fs 17 w(:) p Fv 32 w(W) -8 b(e) 33 b(can) g(rewrite) p Fm 32 w(H) p Fh 1985 3310 a(d) p Fv 2061 3347 a(as) p Fm 815 3638 a(H) p Fh 900 3597 a(d) p Fr 971 3638 a(=) p Fq 1088 3514 a(+) p Fi(1) p Fp 1075 3544 a(M) p Fl 1089 3754 a(j) p Fq 4 w(=0) p Fp 1242 3558 a(\000) p Fm 1288 3638 a(H) p Fh 1373 3597 a(d) p Fl 1372 3663 a(M) p Fm 1473 3638 a(\012) p Fl 1550 3597 a(j) 1550 3663 y(s) p Fp 1610 3558 a(\000) p Fs 1655 3638 a(l) p Fq 1686 3597 a(2) p Fr 1726 3638 a(\(\003) p Fl 1832 3597 a(c) 1832 3663 y(L) p Fr 1884 3638 a(\)) p Fm 22 w(\012) p Fs 22 w(L) p Fq 2109 3597 a(2) p Fr 2149 3638 a(\() p Fg(R) p Fl 2253 3597 a(n) p Fr 2306 3638 a(\)) p Fp 2344 3558 a(\001\001) p Fr 2463 3638 a(=) p Fq 2579 3514 a(+) p Fi(1) p Fp 2566 3544 a(M) p Fl 2580 3754 a(j) p Fq 4 w(=0) p Fm 2734 3638 a(H) p Fq 2819 3597 a(\() p Fl(j) p Fq 4 w(\)) p Fs 2910 3638 a(:) p Fv 257 3937 a(A) m(ctually) -8 b(,) 32 b(w) m(e) i(ha) m(v) m(e) p Fm 1063 4223 a(H) p Fh 1148 4182 a(d) p Fl 1147 4248 a(M) p Fr 1254 4223 a(=) p Fm 28 w(H) p Fq 1443 4182 a(\(0\)) p Fv 1635 4223 a(and) p Fr 97 w(\() p Fm(H) p Fh 2012 4182 a(d) p Fl 2011 4248 a(M) p Fr 2090 4223 a(\)) p Fi 2128 4182 a(?) p Fr 2215 4223 a(=) p Fq 2331 4098 a(+) p Fi(1) p Fp 2319 4128 a(M) p Fl 2333 4338 a(j) p Fq 4 w(=1) p Fm 2486 4223 a(H) p Fq 2571 4182 a(\() p Fl(j) p Fq 4 w(\)) p Fs 2662 4223 a(:) p Fv 257 4539 a(One) f(sees) h(that) f(the) p Fm 33 w(H) p Fq 1124 4503 a(\() p Fl(j) p Fq 4 w(\)) p Fv 1248 4539 a(are) f(in) m(v) -5 b(arian) m(ts) 32 b(for) p Fs 32 w(H) p Fh 2096 4503 a(d) p Fr 2139 4539 a(\() p Fs(M) p Fr 10 w(\)) p Fs(:) p Fv 33 w(But,) h(on) p Fm 32 w(H) p Fq 2820 4503 a(\() p Fl(j) p Fq 4 w(\)) p Fs 2912 4539 a(;) p Fv 32 w(one) g(has) p Fs 630 4791 a(H) p Fh 719 4750 a(d) p Fr 762 4791 a(\() p Fs(M) p Fr 10 w(\)) 84 b(=) 1211 4766 y(~) p Fs 1186 4791 a(H) p Fh 1275 4750 a(d) p Fr 1318 4791 a(\() p Fs(M) p Fr 10 w(\)) p Fm 22 w(\012) p Fr 23 w(1) -22 b(l) 21 b(+) h(1) -22 b(l) p Fm 21 w(\012) p Fp 1979 4697 a(X) p Fi 1969 4913 a(j) p Fl(l) p Fi 1 w(j) p Fl(>L) p Fp 2150 4656 a(Z) p Fo 2205 4881 a(R) p Fn 2253 4862 a(n) p Fs 2317 4791 a(dk) 19 b(!) p Fl 2499 4806 a(m) p Fr 2565 4791 a(\() p Fs(k) p Fr 3 w(\)) p Fs(a) p Fi 2746 4750 a(\003) p Fl 2746 4816 a(l) p Fr 2786 4791 a(\() p Fs(k) p Fr 3 w(\)) p Fs(a) p Fl 2967 4806 a(l) p Fr 2993 4791 a(\() p Fs(k) p Fr 3 w(\)) p Fm 1025 5064 a(\025) p Fr 1211 5039 a(~) p Fs 1186 5064 a(H) p Fh 1275 5023 a(d) p Fr 1318 5064 a(\() p Fs(M) p Fr 10 w(\)) p Fm 22 w(\012) p Fr 23 w(1) -22 b(l) 21 b(+) p Fs 22 w(mj;) p Fv 257 5268 a(and) 33 b(on) p Fm 32 w(H) p Fq 667 5232 a(\(0\)) p Fs 762 5268 a(;) 1397 5388 y(H) p Fh 1486 5347 a(d) p Fr 1529 5388 a(\() p Fs(M) p Fr 10 w(\)) 28 b(=) 1866 5363 y(~) p Fs 1841 5388 a(H) p Fh 1930 5347 a(d) p Fr 1972 5388 a(\() p Fs(M) p Fr 10 w(\)) p Fm 23 w(\012) p Fr 23 w(1) -22 b(l) p Fs -1 w(:) p Fv 1828 5637 a(12) p 90 rotate dyy eop %%Page: 13 13 13 12 bop Fv 257 573 a(Then,) 34 b(w) m(e) g(ha) m(v) m(e) p Fs 267 804 a(\033) p Fp 342 693 a(\020) p Fs 402 804 a(H) p Fh 491 763 a(d) p Fr 534 804 a(\() p Fs(M) p Fr 10 w(\)) p Fm(j) p Fi 742 827 a(H) p Fa 802 804 a(d) p Fn 802 850 a(M) p Fp 874 693 a(\021) p Fr 961 804 a(=) p Fs 27 w(\033) p Fp 1140 693 a(\020) p Fr 1225 779 a(~) p Fs 1199 804 a(H) p Fh 1288 763 a(d) p Fr 1331 804 a(\() p Fs(M) p Fr 10 w(\)) p Fp 1511 693 a(\021) p Fv 1686 804 a(and) p Fs 97 w(\033) p Fl 1995 819 a(ess) p Fp 2114 693 a(\020) p Fs 2174 804 a(H) p Fh 2263 763 a(d) p Fr 2306 804 a(\() p Fs(M) p Fr 10 w(\)) p Fm(j) p Fi 2514 827 a(H) p Fa 2574 804 a(d) p Fn 2574 850 a(M) p Fp 2646 693 a(\021) p Fr 2733 804 a(=) p Fs 27 w(\033) p Fl 2891 819 a(ess) p Fp 3011 693 a(\020) p Fr 3095 779 a(~) p Fs 3070 804 a(H) p Fh 3159 763 a(d) p Fr 3202 804 a(\() p Fs(M) p Fr 10 w(\)) p Fp 3382 693 a(\021) p Fs 3459 804 a(;) p Fv 257 1034 a(and) f(also) p Fs 488 1243 a(\033) p Fl 543 1258 a(ess) p Fp 663 1133 a(\020) p Fs 722 1243 a(H) p Fh 811 1202 a(d) p Fr 854 1243 a(\() p Fs(M) p Fr 10 w(\)) p Fm(j) p Fq 1062 1267 a(\() p Fi(H) p Fa 1149 1244 a(d) p Fn 1149 1290 a(M) p Fq 1217 1267 a(\)) p Fk 1244 1248 a(?) p Fp 1301 1133 a(\021) p Fm 1388 1243 a(\032) p Fs 28 w(\033) p Fp 1569 1133 a(\020) p Fs 1628 1243 a(H) p Fh 1717 1202 a(d) p Fr 1760 1243 a(\() p Fs(M) p Fr 10 w(\)) p Fm(j) p Fq 1968 1267 a(\() p Fi(H) p Fa 2055 1244 a(d) p Fn 2055 1290 a(M) p Fq 2124 1267 a(\)) p Fk 2151 1248 a(?) p Fp 2207 1133 a(\021) p Fm 2294 1243 a(\032) p Fp 2400 1133 a(h) p Fr 2469 1218 a(~) p Fs 2447 1243 a(E) p Fh 2525 1202 a(d) p Fq 2519 1268 a(0) p Fr 2568 1243 a(\() p Fs(M) p Fr 10 w(\)) 23 b(+) p Fs 22 w(m;) p Fr 17 w(+) p Fm(1) p Fp 3174 1133 a(h) p Fs 3237 1243 a(;) p Fv 257 1498 a(whic) m(h) k(ends) h(the) f (pro) s(of.) 40 b(Moreo) m(v) m(er,) 29 b(one) e(can) g(remark) f(that) p Fs 26 w(\036) p Fh 2585 1462 a(d) p Fq 2585 1522 a(0) p Fr 2628 1498 a(\() p Fs(M) p Fr 10 w(\)) j(=) 2953 1471 y(~) p Fs 2940 1498 a(\036) p Fh 2998 1462 a(d) p Fq 2998 1522 a(0) p Fr 3041 1498 a(\() p Fs(M) p Fr 10 w(\)) p Fm 10 w(\012) p Fr 10 w(\012) p Fl 3388 1462 a(c) 3388 1523 y(M) p Fs 3468 1498 a(:) p Fc 257 1618 a(2) p Fw 257 1997 a(4.1.3) 113 b(Remo) m(ving) 36 b(the) h(cuto\033) p Fv 257 2181 a(W) -8 b(e) 33 b(\034rst) g(pro) m(v) m(e) h(some) e(con) m (v) m(ergence) k(results) c(as) p Fs 33 w(M) p Fv 44 w(go) s(es) g(to) g(in\034nit) m(y) -8 b(.) p Fw 257 2375 a(Prop) s(osition) 36 b(4.4.) p Fs 42 w(H) p Fb 1168 2339 a(d) p Fr 1208 2375 a(\() p Fs(M) p Fr 10 w(\)) p Ft 35 w(c) -5 b(onver) g(ges) 34 b(to) p Fs 35 w(H) p Fb 2064 2339 a(d) p Ft 2139 2375 a(in) g(the) h(str) -5 b(ong) 35 b(r) -5 b(esolvent) 34 b(sens.) p Fw 257 2570 a(Pro) s(of) j(:) p Fv 38 w(W) -8 b(e) 33 b(ha) m(v) m(e) p Fs 294 2802 a(H) p Fh 383 2761 a(d) p Fm 448 2802 a(\000) p Fs 22 w(H) p Fh 636 2761 a(d) p Fr 679 2802 a(\() p Fs(M) p Fr 10 w(\)) c(=) p Fs 27 w(W) p Fh 1097 2761 a(d) p Fm 1162 2802 a(\000) p Fs 23 w(W) p Fh 1368 2761 a(d) p Fr 1411 2802 a(\() p Fs(M) p Fr 10 w(\)) f(=) p Fp 1746 2708 a(X) p Fi 1723 2923 a(j) p Fl(l) p Fi 1 w(j) p Fl(>M) p Fp 1930 2667 a(Z) p Fo 1986 2892 a(R) p Fn 2034 2873 a(n) p Fs 2097 2802 a(dk) 19 b(\014) p Fl 2273 2817 a(l) p Fr 2299 2802 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 23 w(a) p Fi 2602 2761 a(\003) p Fl 2602 2827 a(l) p Fr 2642 2802 a(\() p Fs(k) p Fr 3 w(\)) j(+) 2906 2776 y(\026) p Fs 2892 2802 a(\014) p Fl 2947 2817 a(l) p Fr 2973 2802 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 22 w(a) p Fl 3275 2817 a(l) p Fr 3302 2802 a(\() p Fs(k) p Fr 3 w(\)) p Fs(:) p Fv 257 3129 a(Let) p Fs 33 w( ) p Fm 32 w(2) p Fs 28 w(D) p Fr 3 w(\() p Fs(H) p Fh 832 3093 a(d) p Fq 824 3154 a(0) p Fr 874 3129 a(\)) p Fs(:) p Fv 33 w(Using) 32 b(condition) p Fr 31 w(\() p Fs(C) p Fl 1782 3144 a(\014) p Fr 1829 3129 a(\)) p Fs(;) p Fv 32 w(one) h(has) p Fm 257 3396 a(k) p Fp 347 3301 a(X) p Fi 324 3517 a(j) p Fl(l) p Fi 1 w(j) p Fl(>M) p Fp 531 3260 a(Z) p Fo 587 3486 a(R) p Fn 635 3467 a(n) p Fs 698 3396 a(dk) p Fr 833 3370 a(\026) p Fs 819 3396 a(\014) p Fl 874 3411 a(l) p Fr 900 3396 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 23 w(a) p Fl 1203 3411 a(l) p Fr 1229 3396 a(\() p Fs(k) p Fr 3 w(\)) p Fs( ) p Fm 4 w(k) 83 b(\024) p Fs 1785 3329 a(C) p Fr 7 w(\() p Fs(s) p Fr(\)) p 1729 3373 311 4 v 1729 3464 a(1) 22 b(+) p Fs 22 w(M) p Fl 2002 3436 a(s) p Fm 2050 3396 a(k) p Fp 2140 3301 a(X) p Fi 2117 3517 a(j) p Fl(l) p Fi 1 w(j) p Fl(>M) p Fp 2324 3260 a(Z) p Fo 2379 3486 a(R) p Fn 2427 3467 a(n) p Fs 2490 3396 a(dk) p Fr 20 w(1) -22 b(l) p Fm 20 w(\012) p Fs 23 w(a) p Fl 2838 3411 a(l) p Fr 2864 3396 a(\() p Fs(k) p Fr 3 w(\)) p Fs( ) p Fm 4 w(k) 1559 3719 y(\024) p Fs 1785 3652 a(C) p Fr 7 w(\() p Fs(s) p Fr(\)) p 1729 3696 V 1729 3788 a(1) 22 b(+) p Fs 22 w(M) p Fl 2002 3759 a(s) p Fm 2050 3719 a(k) p Fr(\(1) -22 b(l) p Fm 20 w(\012) p Fs 23 w(N) p Fh 2401 3678 a(d) p Fr 2445 3719 a(\)) p Fj 2493 3651 a(1) p 2493 3663 31 3 v 2493 3704 a(2) p Fs 2537 3719 a( ) p Fm 4 w(k) 60 b(\024) p Fs 2918 3652 a(C) p Fr 7 w(\() p Fs(s) p Fr(\)) p 2862 3696 311 4 v 2862 3788 a(1) 22 b(+) p Fs 22 w(M) p Fl 3135 3759 a(s) p Fm 3182 3719 a(k) p Fr(\() p Fs(H) p Fh 3359 3678 a(d) p Fq 3351 3744 a(0) p Fr 3402 3719 a(\)) p Fj 3450 3651 a(1) p 3450 3663 31 3 v 3450 3704 a(2) p Fs 3495 3719 a( ) p Fm 4 w(k) p Fs(:) p Fv 257 3967 a(Then,) 34 b(using) e(the) h(comm) m (utation) e(relations) g(\(2.1\),) h(w) m(e) h(ha) m(v) m(e) p Fm 266 4225 a(k) p Fp 356 4130 a(X) p Fi 333 4346 a(j) p Fl(l) p Fi 1 w(j) p Fl(>M) p Fp 540 4089 a(Z) p Fo 595 4314 a(R) p Fn 643 4296 a(n) p Fs 706 4225 a(dk) 19 b(\014) p Fl 882 4240 a(l) p Fr 909 4225 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 22 w(a) p Fi 1211 4183 a(\003) p Fl 1211 4249 a(l) p Fr 1251 4225 a(\() p Fs(k) p Fr 3 w(\)) p Fs( ) p Fm 4 w(k) p Fq 1498 4183 a(2) p Fr 1620 4225 a(=) p Fm 83 w(k) p Fp 1869 4130 a(X) p Fi 1846 4346 a(j) p Fl(l) p Fi 1 w(j) p Fl(>M) p Fp 2053 4089 a(Z) p Fo 2108 4314 a(R) p Fn 2156 4296 a(n) p Fs 2219 4225 a(dk) p Fr 2355 4198 a(\026) p Fs 2341 4225 a(\014) p Fl 2396 4240 a(l) p Fr 2422 4225 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 23 w(a) p Fl 2725 4240 a(l) p Fr 2751 4225 a(\() p Fs(k) p Fr 3 w(\)) p Fs( ) p Fm 4 w(k) p Fq 2998 4183 a(2) p Fr 2169 4610 a(+) p Fp 2262 4409 a(0) 2262 4589 y(@) 2372 4515 y(X) p Fi 2349 4731 a(j) p Fl(l) p Fi 1 w(j) p Fl(>M) p Fp 2556 4474 a(Z) p Fo 2612 4700 a(R) p Fn 2660 4681 a(n) p Fs 2723 4610 a(dk) p Fm 19 w(j) p Fs(\014) p Fl 2927 4625 a(l) p Fr 2953 4610 a(\() p Fs(k) p Fr 3 w(\)) p Fm(j) p Fq 3111 4569 a(2) p Fp 3150 4409 a(1) 3150 4589 y(A) p Fm 3254 4610 a(k) p Fs( ) p Fm 4 w(k) p Fq 3421 4569 a(2) p Fs 3460 4610 a(:) p Fv 257 4930 a(Finally) -8 b(,) 30 b(one) j(gets) p Fm 350 5284 a(k) p Fs(H) p Fh 489 5242 a(d) p Fs 532 5284 a( ) p Fm 26 w(\000) p Fs 23 w(H) p Fh 810 5242 a(d) p Fr 852 5284 a(\() p Fs(M) p Fr 10 w(\)) p Fs( ) p Fm 4 w(k) c(\024) p Fr 1324 5216 a(2) p Fs(C) p Fr 7 w(\() p Fs(s) p Fr(\)) p 1293 5261 311 4 v 1293 5352 a(1) 22 b(+) p Fs 22 w(M) p Fl 1566 5323 a(s) p Fm 1613 5284 a(k) p Fr(\() p Fs(H) p Fh 1790 5242 a(d) p Fq 1782 5308 a(0) p Fr 1833 5284 a(\)) p Fj 1881 5215 a(1) p 1881 5227 31 3 v 1881 5268 a(2) p Fs 1925 5284 a( ) p Fm 4 w(k) p Fr 22 w(+) p Fp 2162 5083 a(0) 2162 5263 y(@) 2273 5189 y(X) p Fi 2249 5405 a(j) p Fl(l) p Fi 1 w(j) p Fl(>M) p Fp 2457 5148 a(Z) p Fo 2512 5373 a(R) p Fn 2560 5355 a(n) p Fs 2623 5284 a(dk) p Fm 20 w(j) p Fs(\014) p Fl 2828 5299 a(l) p Fr 2853 5284 a(\() p Fs(k) p Fr 3 w(\)) p Fm(j) p Fq 3011 5242 a(2) p Fp 3050 5083 a(1) 3050 5263 y(A) p Fj 3148 5078 a(1) p 3148 5090 V 3148 5132 a(2) p Fm 3209 5284 a(k) p Fs( ) p Fm 4 w(k) p Fs(:) p Fv 1828 5637 a(13) p 90 rotate dyy eop %%Page: 14 14 14 13 bop Fv 257 573 a(Using) 30 b(condition) p Fr 28 w(\() p Fs(C) p Fl 1062 588 a(\014) p Fr 1109 573 a(\)) p Fs(;) p Fv 29 w(one) g(sho) m(ws) i(that) d(the) h(righ) m(t) f(hand) h (side) g(tends) g(to) g(zero) g(as) p Fs 29 w(M) p Fv 257 693 a(go) s(es) 38 b(to) f(in\034nit) m(y) -8 b(.) 58 b(So,) p Fs 39 w(H) p Fh 1240 657 a(d) p Fr 1283 693 a(\() p Fs(M) p Fr 10 w(\)) p Fv 38 w(con) m(v) m(erges) 40 b(strongly) d(to) p Fs 37 w(H) p Fh 2538 657 a(d) p Fv 2618 693 a(and) h(then) g(also) f(in) g(the) 257 814 y(strong) c(resolv) m(en) m(t) h(sens) f(\([18],) g(Theorem) g(VI) s(I) s(I.25\).) p Fc 1213 w(2) p Fw 257 1122 a(Prop) s(osition) j(4.5.) p Fs 41 w(E) p Fb 1156 1086 a(d) p Fq 1150 1147 a(0) p Fr 1197 1122 a(\() p Fs(M) p Fr 10 w(\)) p Ft 35 w(is) f(a) f(de) -5 b(cr) g(e) g(asing) 33 b(function) i(of) p Fs 34 w(M) p Ft 46 w(which) e(tends) i(to) p Fs 34 w(E) p Fb 3427 1086 a(d) p Fq 3421 1147 a(0) p Fs 3468 1122 a(:) p Fw 257 1311 a(Pro) s(of) c(:) p Fv 31 w(W) -8 b(e) 27 b(kno) m(w) h(that,) f(if) p Fs 26 w(\036) p Fh 1408 1274 a(d) p Fq 1408 1335 a(0) p Fr 1451 1311 a(\() p Fs(M) p Fr 10 w(\)) p Fv 27 w(is) g(a) f(ground) h(state) g(for) p Fs 26 w(H) p Fh 2616 1274 a(d) p Fr 2659 1311 a(\() p Fs(M) p Fr 10 w(\)) p Fs(;) p Fv 28 w(then) p Fs 27 w(\036) p Fh 3168 1274 a(d) p Fq 3168 1335 a(0) p Fr 3211 1311 a(\() p Fs(M) p Fr 10 w(\)) h(=) 270 1405 y(~) p Fs 257 1431 a(\036) p Fh 315 1395 a(d) p Fq 315 1456 a(0) p Fr 358 1431 a(\() p Fs(M) p Fr 10 w(\)) p Fm 23 w(\012) p Fr 23 w(\012) p Fl 731 1395 a(c) 731 1456 y(M) p Fs 810 1431 a(;) p Fv 33 w(and) 33 b(so) p Fm 1132 1633 a(8) p Fs(l) p Fm 30 w(2) p Fr 28 w(\003) p Fl 1408 1592 a(c) 1408 1658 y(M) p Fs 1487 1633 a(;) p Fm 17 w(8) p Fs(k) p Fm 31 w(2) p Fg 28 w(R) p Fl 1828 1592 a(n) p Fs 1881 1633 a(;) 17 b(a) p Fl 1976 1648 a(l) p Fr 2002 1633 a(\() p Fs(k) p Fr 3 w(\)) p Fs(\036) p Fh 2190 1592 a(d) p Fq 2190 1658 a(0) p Fr 2233 1633 a(\() p Fs(M) p Fr 10 w(\)) 28 b(=) g(0) p Fs(:) p Fv 257 1836 a(Let) p Fs 33 w(M) p Fi 536 1800 a(0) p Fs 588 1836 a(>) f(M) 5 b(;) 257 2038 y(E) p Fh 335 1997 a(d) p Fq 329 2063 a(0) p Fr 379 2038 a(\() p Fs(M) p Fi 521 1997 a(0) p Fr 545 2038 a(\)) p Fm 83 w(\024) 83 b(h) p Fs(\036) p Fh 923 1997 a(d) p Fq 923 2063 a(0) p Fr 966 2038 a(\() p Fs(M) p Fr 10 w(\);) p Fs 17 w(H) p Fh 1279 1997 a(d) p Fr 1322 2038 a(\() p Fs(M) p Fi 1464 1997 a(0) p Fr 1488 2038 a(\)) p Fs(\036) p Fh 1584 1997 a(d) p Fq 1584 2063 a(0) p Fr 1627 2038 a(\() p Fs(M) p Fr 10 w(\)) p Fm(i) 666 2183 y(\024) g(h) p Fs(\036) p Fh 923 2142 a(d) p Fq 923 2208 a(0) p Fr 966 2183 a(\() p Fs(M) p Fr 10 w(\);) p Fs 17 w(H) p Fh 1279 2142 a(d) p Fr 1322 2183 a(\() p Fs(M) p Fr 10 w(\)) p Fs(\036) p Fh 1560 2142 a(d) p Fq 1560 2208 a(0) p Fr 1604 2183 a(\() p Fs(M) p Fr 10 w(\)) p Fm(i) p Fp 826 2245 a(|) p 871 2245 409 12 v 409 w({z) p 1370 2245 V 409 w(}) p Fq 1186 2344 a(=) p Fl(E) p Fa 1297 2321 a(d) p Fj 1293 2365 a(0) p Fq 1334 2344 a(\() p Fl(M) p Fq 7 w(\)) p Fr 1840 2183 a(+) p Fm 17 w(h) p Fs(\036) p Fh 2030 2142 a(d) p Fq 2030 2208 a(0) p Fr 2072 2183 a(\() p Fs(M) p Fr 10 w(\);) 17 b(\() p Fs(W) p Fh 2440 2142 a(d) p Fr 2484 2183 a(\() p Fs(M) p Fi 2626 2142 a(0) p Fr 2650 2183 a(\)) p Fm 22 w(\000) p Fs 22 w(W) p Fh 2915 2142 a(d) p Fr 2958 2183 a(\() p Fs(M) p Fr 10 w(\)\)) p Fs(\036) p Fh 3234 2142 a(d) p Fq 3234 2208 a(0) p Fr 3278 2183 a(\() p Fs(M) p Fr 10 w(\)) p Fm(i) p Fp 1933 2245 a(|) p 1978 2245 693 12 v 693 w({z) p 2761 2245 V 693 w(}) p Fq 2670 2330 a(=0) p Fs 3514 2183 a(:) p Fv 257 2548 a(So) 30 b(the) g(function) p Fs 29 w(E) p Fh 1012 2511 a(d) p Fq 1006 2572 a(0) p Fr 1055 2548 a(\() p Fs(M) p Fr 10 w(\)) p Fv 30 w(decreases.) 45 b(With) 29 b(the) h(same) f(argumen) m(t,) h(one) g(pro) m(v) m(es) h (that) p Fs 257 2668 a(E) p Fh 335 2632 a(d) p Fq 329 2693 a(0) p Fr 379 2668 a(\() p Fs(M) p Fr 10 w(\)) p Fm 35 w(\025) p Fs 35 w(E) p Fh 784 2632 a(d) p Fq 778 2693 a(0) p Fs 827 2668 a(:) p Fv 36 w(Then) p Fs 38 w(E) p Fh 1227 2632 a(d) p Fq 1221 2693 a(0) p Fr 1270 2668 a(\() p Fs(M) p Fr 10 w(\)) p Fv 37 w(con) m(v) m(erges) 39 b(to) d(some) p Fs 36 w(E) p Fi 2371 2683 a(1) p Fm 2481 2668 a(\025) p Fs 34 w(E) p Fh 2670 2632 a(d) p Fq 2664 2693 a(0) p Fs 2714 2668 a(:) p Fv 36 w(But) p Fs 37 w(E) p Fl 3053 2632 a(d) p Fq 3047 2693 a(0) p Fm 3128 2668 a(2) p Fs 34 w(\033) p Fr 4 w(\() p Fs(H) p Fh 3414 2632 a(d) p Fr 3457 2668 a(\)) p Fv 257 2788 a(and) p Fs 34 w(H) p Fh 537 2752 a(d) p Fr 580 2788 a(\() p Fs(M) p Fr 10 w(\)) p Fv 34 w(con) m(v) m(erges) f(to) p Fs 33 w(H) p Fh 1440 2752 a(d) p Fv 1517 2788 a(in) d(the) i(strong) f (resolv) m(en) m(t) i(sens,) g(so) f(\([18) o(],) g(Theorem) 257 2909 y(VI) s(I) s(I.24\),) p Fm 939 3029 a(8) p Fs(M) 39 b(>) p Fr 27 w(0) p Fs(;) p Fm 17 w(9) p Fs(E) p Fr 6 w(\() p Fs(M) p Fr 10 w(\)) p Fm 28 w(2) p Fs 28 w(\033) p Fr 4 w(\() p Fs(H) p Fh 1944 2988 a(d) p Fr 1987 3029 a(\() p Fs(M) p Fr 10 w(\)\)) p Fs(=E) p Fr 6 w(\() p Fs(M) p Fr 10 w(\)) p Fm 29 w(!) p Fs 27 w(E) p Fl 2746 2988 a(d) p Fq 2740 3054 a(0) p Fs 2787 3029 a(:) p Fv 257 3196 a(Using) 34 b(the) h(fact) f(that) p Fs 34 w(E) p Fh 1188 3160 a(d) p Fq 1182 3221 a(0) p Fr 1231 3196 a(\() p Fs(M) p Fr 10 w(\)) p Fv 35 w(is) f(the) i(ground) f(state) h (energy) g(of) p Fs 34 w(H) p Fh 2803 3160 a(d) p Fr 2845 3196 a(\() p Fs(M) p Fr 10 w(\)) p Fs(;) p Fv 35 w(w) m(e) g(\034nally) 257 3316 y(get) p Fs 33 w(E) p Fi 492 3331 a(1) p Fr 595 3316 a(=) p Fs 27 w(E) p Fh 776 3280 a(d) p Fq 770 3341 a(0) p Fs 819 3316 a(:) p Fc 2575 w(2) p Fw 257 3625 a(Prop) s(osition) j(4.6.) p Ft 43 w(L) -5 b(et) p Fr 38 w(\001) p Ft 37 w(b) g(e) 36 b(an) h(interval) f(b) -5 b(ounde) g(d) 36 b(fr) -5 b(om) 37 b(ab) -5 b(ove.) 50 b(F) -7 b(or) 36 b(al) 5 b(l) p Fs 36 w(s) 32 b(>) p Fr 31 w(0) p Fs(;) p Ft 257 3745 a(ther) -5 b(e) 35 b(exists) p Fs 35 w(K) p Fr 7 w(\() p Fs(s;) p Fr 17 w(\001\)) p Fs 27 w(>) p Fr 28 w(0) p Ft 34 w(such) g(that) p Fm 904 4003 a(k) p Fs(\037) p Fq 1015 4018 a(\001) p Fr 1078 4003 a(\() p Fs(H) p Fb 1205 3962 a(d) p Fr 1245 4003 a(\)\() p Fs(W) p Fb 1427 3962 a(d) p Fm 1489 4003 a(\000) p Fs 22 w(W) p Fb 1694 3962 a(d) p Fr 1735 4003 a(\() p Fs(M) p Fr 10 w(\)\)) p Fs(\037) p Fq 2014 4018 a(\001) p Fr 2077 4003 a(\() p Fs(H) p Fb 2204 3962 a(d) p Fr 2244 4003 a(\)) p Fm(k) 28 b(\024) p Fs 2475 3935 a(K) p Fr 7 w(\() p Fs(s;) p Fr 17 w(\001\)) p 2475 3980 337 4 v 2488 4071 a(1) 22 b(+) p Fs 22 w(M) p Fl 2761 4042 a(s) p Fs 2822 4003 a(:) p Fw 257 4251 a(Pro) s(of) 37 b(:) p Fv 38 w(Let) p Fs 33 w(\036;) 17 b( ) p Fm 31 w(2) 28 b(H) p Fh 1185 4215 a(d) p Fs 1228 4251 a(:) p Fv 33 w(W) -8 b(e) 33 b(ha) m(v) m(e) p Fp 623 4369 a(\014) 623 4428 y(\014) p Fm 656 4453 a(h) p Fs(\036) p Fr(;) p Fs 17 w(\037) p Fq 858 4468 a(\001) p Fr 921 4453 a(\() p Fs(H) p Fh 1048 4412 a(d) p Fr 1091 4453 a(\)\() p Fs(W) p Fh 1273 4412 a(d) p Fm 1338 4453 a(\000) p Fs 22 w(W) p Fh 1543 4412 a(d) p Fr 1586 4453 a(\() p Fs(M) p Fr 10 w(\)\)) p Fs(\037) p Fq 1865 4468 a(\001) p Fr 1929 4453 a(\() p Fs(H) p Fh 2056 4412 a(d) p Fr 2099 4453 a(\)) p Fs( ) p Fm 4 w(i) p Fp 2243 4369 a(\014) 2243 4428 y(\014) p Fr 464 4657 a(=) p Fm 83 w(jh) p Fs(\036) p Fr(;) p Fs 17 w(\037) p Fq 853 4672 a(\001) p Fr 915 4657 a(\() p Fs(H) p Fh 1042 4616 a(d) p Fr 1085 4657 a(\)\() p Fp 1184 4562 a(X) p Fi 1161 4778 a(j) p Fl(l) p Fi 1 w(j) p Fl(>M) p Fp 1368 4521 a(Z) p Fo 1424 4747 a(R) p Fn 1472 4728 a(n) p Fs 1535 4657 a(dk) 19 b(\014) p Fl 1711 4672 a(l) p Fr 1737 4657 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 23 w(a) p Fi 2040 4616 a(\003) p Fl 2040 4682 a(l) p Fr 2080 4657 a(\() p Fs(k) p Fr 3 w(\)) j(+) 2344 4631 y(\026) p Fs 2330 4657 a(\014) p Fl 2385 4672 a(l) p Fr 2411 4657 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 23 w(a) p Fl 2714 4672 a(l) p Fr 2740 4657 a(\() p Fs(k) p Fr 3 w(\)\)) p Fs(\037) p Fq 2969 4672 a(\001) p Fr 3032 4657 a(\() p Fs(H) p Fh 3159 4616 a(d) p Fr 3202 4657 a(\)) p Fs( ) p Fm 4 w(ij) 463 4974 y(\024) 83 b(jh) p Fs(\037) p Fq 751 4989 a(\001) p Fr 814 4974 a(\() p Fs(H) p Fh 941 4932 a(d) p Fr 984 4974 a(\)) p Fs(\036) p Fr(;) 17 b(\() p Fp 1184 4879 a(X) p Fi 1162 5095 a(j) p Fl(l) p Fi 1 w(j) p Fl(>M) p Fp 1368 4838 a(Z) p Fo 1424 5064 a(R) p Fn 1472 5045 a(n) p Fs 1535 4974 a(dk) p Fr 1670 4947 a(\026) p Fs 1656 4974 a(\014) p Fl 1711 4989 a(l) p Fr 1737 4974 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 23 w(a) p Fl 2040 4989 a(l) p Fr 2066 4974 a(\() p Fs(k) p Fr 3 w(\)\)) p Fs(\037) p Fq 2295 4989 a(\001) p Fr 2358 4974 a(\() p Fs(H) p Fh 2485 4932 a(d) p Fr 2528 4974 a(\)) p Fs( ) p Fm 4 w(ij) p Fr 1208 5290 a(+) p Fm(jh) p Fr(\() p Fp 1412 5196 a(X) p Fi 1389 5411 a(j) p Fl(l) p Fi 1 w(j) p Fl(>M) p Fp 1596 5155 a(Z) p Fo 1651 5380 a(R) p Fn 1699 5361 a(n) p Fs 1763 5290 a(dk) p Fr 1898 5264 a(\026) p Fs 1884 5290 a(\014) p Fl 1939 5305 a(l) p Fr 1965 5290 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 23 w(a) p Fl 2268 5305 a(l) p Fr 2294 5290 a(\() p Fs(k) p Fr 3 w(\)\)) p Fs(\037) p Fq 2523 5305 a(\001) p Fr 2586 5290 a(\() p Fs(H) p Fh 2713 5249 a(d) p Fr 2756 5290 a(\)) p Fs(\036) p Fr(;) p Fs 17 w(\037) p Fq 2957 5305 a(\001) p Fr 3019 5290 a(\() p Fs(H) p Fh 3146 5249 a(d) p Fr 3189 5290 a(\)) p Fs( ) p Fm 4 w(ij) p Fv 1828 5637 a(14) p 90 rotate dyy eop %%Page: 15 15 15 14 bop Fm 340 609 a(\024) 84 b(k) p Fs(\036) p Fm(k) 21 b(\002) i(k) p Fr(\() p Fp 891 514 a(X) p Fi 868 730 a(j) p Fl(l) p Fi 1 w(j) p Fl(>M) p Fp 1075 473 a(Z) p Fo 1131 699 a(R) p Fn 1179 680 a(n) p Fs 1242 609 a(dk) p Fr 1377 583 a(\026) p Fs 1363 609 a(\014) p Fl 1418 624 a(l) p Fr 1444 609 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 23 w(a) p Fl 1747 624 a(l) p Fr 1773 609 a(\() p Fs(k) p Fr 3 w(\)\)) p Fs(\037) p Fq 2002 624 a(\001) p Fr 2065 609 a(\() p Fs(H) p Fh 2192 568 a(d) p Fr 2235 609 a(\)) p Fs( ) p Fm 4 w(k) p Fr 1281 925 a(+) p Fm(k) p Fs( ) p Fm 4 w(k) f(\002) g(k) p Fr(\() p Fp 1757 831 a(X) p Fi 1733 1047 a(j) p Fl(l) p Fi 1 w(j) p Fl(>M) p Fp 1941 790 a(Z) p Fo 1996 1015 a(R) p Fn 2044 996 a(n) p Fs 2107 925 a(dk) p Fr 2242 899 a(\026) p Fs 2228 925 a(\014) p Fl 2283 940 a(l) p Fr 2310 925 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 22 w(a) p Fl 2612 940 a(l) p Fr 2638 925 a(\() p Fs(k) p Fr 3 w(\)\)) p Fs(\037) p Fq 2867 940 a(\001) p Fr 2930 925 a(\() p Fs(H) p Fh 3057 884 a(d) p Fr 3100 925 a(\)) p Fs(\036) p Fm(k) 340 1249 y(\024) p Fs 567 1181 a(C) p Fr 7 w(\() p Fs(s) p Fr(\)) p 511 1226 311 4 v 511 1317 a(1) g(+) p Fs 22 w(M) p Fl 784 1288 a(s) p Fp 848 1138 a(\020) p Fm 907 1249 a(k) p Fs(\036) p Fm(k) g(\002) h(k) p Fr(\(1) -22 b(l) p Fm 20 w(\012) p Fs 23 w(N) p Fh 1538 1207 a(d) p Fr 1582 1249 a(\)) p Fj 1630 1180 a(1) p 1630 1192 31 3 v 1630 1233 a(2) p Fs 1674 1249 a(\037) p Fq 1735 1264 a(\001) p Fr 1798 1249 a(\() p Fs(H) p Fh 1925 1207 a(d) p Fr 1968 1249 a(\)) p Fs( ) p Fm 4 w(k) p Fr 22 w(+) p Fm 22 w(k) p Fs( ) p Fm 4 w(k) 22 b(\002) g(k) p Fr(\(1) -22 b(l) p Fm 21 w(\012) p Fs 23 w(N) p Fh 2883 1207 a(d) p Fr 2926 1249 a(\)) p Fj 2974 1180 a(1) p 2974 1192 V 2974 1233 a(2) p Fs 3019 1249 a(\037) p Fq 3080 1264 a(\001) p Fr 3143 1249 a(\() p Fs(H) p Fh 3270 1207 a(d) p Fr 3313 1249 a(\)) p Fs(\036) p Fm(k) p Fp 3459 1138 a(\021) p Fs 3534 1249 a(:) p Fv 257 1485 a(But) p Fr 45 w(\001) p Fv 44 w(is) 44 b(b) s(ounded) h(from) e(ab) s(o) m(v) m (e,) p Fr 48 w(1) -22 b(l) p Fm 29 w(\012) p Fs 30 w(N) p Fh 1947 1449 a(d) p Fm 2039 1485 a(\024) p Fq 2187 1446 a(1) p 2174 1462 63 4 v Fl 2174 1519 a(m) p Fs 2246 1485 a(H) p Fh 2335 1449 a(d) p Fq 2327 1510 a(0) p Fv 2422 1485 a(and) p Fs 44 w(W) p Fh 2729 1449 a(d) p Fv 2817 1485 a(is) 43 b(relativ) m(ely) p Fs 43 w(H) p Fh 3452 1449 a(d) p Fq 3444 1510 a(0) p Fv 257 1624 a(b) s(ounded,) 34 b(so) p Fr 32 w(\(1) -22 b(l) p Fm 21 w(\012) p Fs 23 w(N) p Fh 1104 1587 a(d) p Fr 1148 1624 a(\)) p Fj 1196 1560 a(1) p 1196 1572 31 3 v 1196 1613 a(2) p Fs 1240 1624 a(\037) p Fq 1301 1639 a(\001) p Fr 1364 1624 a(\() p Fs(H) p Fh 1491 1587 a(d) p Fr 1534 1624 a(\)) p Fv 33 w(is) 32 b(a) g(b) s(ounded) h(op) s(erator.) 43 b(Finally) -8 b(,) 30 b(one) i(has) p Fp 267 1806 a(\014) 267 1865 y(\014) p Fm 300 1890 a(h) p Fs(\036) p Fr(;) p Fs 17 w(\037) p Fq 502 1905 a(\001) p Fr 564 1890 a(\() p Fs(H) p Fh 691 1849 a(d) p Fr 734 1890 a(\)\() p Fs(W) p Fh 916 1849 a(d) p Fm 981 1890 a(\000) p Fs 23 w(W) p Fh 1187 1849 a(d) p Fr 1230 1890 a(\() p Fs(M) p Fr 10 w(\)\)) p Fs(\037) p Fq 1509 1905 a(\001) p Fr 1572 1890 a(\() p Fs(H) p Fh 1699 1849 a(d) p Fr 1742 1890 a(\)) p Fs( ) p Fm 4 w(i) p Fp 1886 1806 a(\014) 1886 1865 y(\014) p Fm 1947 1890 a(\024) p Fr 2062 1823 a(2) p Fs(C) p Fr 7 w(\() p Fs(s) p Fr(\)) p Fm(k) p Fr(\() p Fs(N) p Fh 2486 1787 a(d) p Fr 2529 1823 a(\)) p Fj 2577 1759 a(1) p 2577 1771 V 2577 1812 a(2) p Fs 2621 1823 a(\037) p Fq 2682 1838 a(\001) p Fr 2745 1823 a(\() p Fs(H) p Fh 2872 1787 a(d) p Fr 2915 1823 a(\)) p Fm(k) p 2062 1867 942 4 v Fr 2377 1959 a(1) 22 b(+) p Fs 22 w(M) p Fl 2650 1930 a(s) p Fm 3013 1890 a(k) p Fs(\036) p Fm(k) f(\002) i(k) p Fs( ) p Fm 4 w(k) p Fs(;) p Fv 257 2113 a(whic) m(h) 33 b(ends) h(the) f(pro) s(of.) p Fc 2244 w(2) p Fw 257 2354 a(Pro) s(of) k(of) h(Theorem) f(4.1:) p Fv 404 2475 a(W) -8 b(e) 45 b(use) h(the) f(metho) s(d) f(of) g([6].) 80 b(Giv) m(en) 45 b(an) g(op) s(erator) p Fs 44 w(A;) p Fr 45 w([) p Fs(A) p Fr(]) p Fi 2765 2490 a(\000) p Fv 2869 2475 a(will) d(denote) k(its) 257 2595 y(negativ) m(e) 34 b(part) f(and) h(T) -8 b(r) p Fr(\() p Fs(A) p Fr(\)) p Fv 34 w(its) 33 b(trace.) 46 b(T) -8 b(o) 34 b(pro) m(v) m(e) g(the) g (theorem,) g(it) e(su\036ces) k(to) d(sho) m(w) 257 2715 y(that,) g(for) f(all) p Fs 30 w(\017) c(>) p Fr 28 w(0) p Fs(;) p Fv 32 w(w) m(e) 34 b(ha) m(v) m(e) 1165 2906 y(T) -8 b(r) p Fp 1281 2825 a(\010) p Fr 1340 2906 a([) p Fs(H) p Fh 1456 2865 a(d) p Fm 1521 2906 a(\000) p Fs 22 w(E) p Fh 1698 2865 a(d) p Fq 1692 2930 a(0) p Fm 1764 2906 a(\000) p Fs 22 w(m) p Fr 23 w(+) p Fs 22 w(\017) p Fr(]) p Fi 2135 2921 a(\000) p Fp 2194 2825 a(\011) p Fs 2280 2906 a(>) p Fm 28 w(\0001) p Fs(:) p Fv 257 3096 a(Let) p Fs 33 w(\017) 28 b(>) p Fr 28 w(0) p Fs(;) p Fv 32 w(and) p Fr 33 w(\001) g(=]) p Fm 22 w(\000) 22 b(1) p Fs(;) 17 b(E) p Fh 1456 3060 a(d) p Fq 1450 3121 a(0) p Fr 1521 3096 a(+) p Fs 22 w(m) p Fm 23 w(\000) p Fs 22 w(\017) p Fr([) p Fs(:) p Fv 33 w(Then) p Fr 629 3287 a([) p Fs(H) p Fh 745 3246 a(d) p Fm 810 3287 a(\000) p Fs 23 w(E) p Fh 988 3246 a(d) p Fq 982 3311 a(0) p Fm 1053 3287 a(\000) p Fs 23 w(m) p Fr 22 w(+) p Fs 22 w(\017) p Fr(]) p Fi 1424 3302 a(\000) p Fr 1512 3287 a(=) p Fs 27 w(\037) p Fq 1676 3302 a(\001) p Fr 1739 3287 a(\() p Fs(H) p Fh 1866 3246 a(d) p Fr 1909 3287 a(\)\() p Fs(H) p Fh 2074 3246 a(d) p Fm 2139 3287 a(\000) p Fs 23 w(E) p Fh 2317 3246 a(d) p Fq 2311 3311 a(0) p Fm 2382 3287 a(\000) p Fs 23 w(m) p Fr 22 w(+) p Fs 22 w(\017) p Fr(\)) p Fs(\037) p Fq 2825 3302 a(\001) p Fr 2889 3287 a(\() p Fs(H) p Fh 3016 3246 a(d) p Fr 3058 3287 a(\)) p Fs(;) p Fv 257 3477 a(and) 33 b(so) 257 3668 y(T) -8 b(r) p Fp 374 3587 a(\010) p Fr 432 3668 a([) p Fs(H) p Fh 548 3627 a(d) p Fm 613 3668 a(\000) p Fs 23 w(E) p Fh 791 3627 a(d) p Fq 785 3693 a(0) p Fm 857 3668 a(\000) p Fs 22 w(m) p Fr 23 w(+) p Fs 22 w(\017) p Fr(]) p Fi 1228 3683 a(\000) p Fp 1287 3587 a(\011) p Fr 1428 3668 a(=) p Fv 83 w(T) g(r) p Fp 1704 3587 a(\010) p Fs 1762 3668 a(\037) p Fq 1823 3683 a(\001) p Fr 1886 3668 a(\() p Fs(H) p Fh 2013 3627 a(d) p Fr 2056 3668 a(\)\() p Fs(H) p Fh 2221 3627 a(d) p Fm 2286 3668 a(\000) p Fs 23 w(E) p Fh 2464 3627 a(d) p Fq 2458 3693 a(0) p Fm 2529 3668 a(\000) p Fs 23 w(m) p Fr 22 w(+) p Fs 22 w(\017) p Fr(\)) p Fs(\037) p Fq 2972 3683 a(\001) p Fr 3036 3668 a(\() p Fs(H) p Fh 3163 3627 a(d) p Fr 3205 3668 a(\)) p Fp 3243 3587 a(\011) p Fr 1428 3823 a(=) p Fv 83 w(T) g(r) p Fm(f) p Fs(\037) p Fq 1798 3838 a(\001) p Fr 1861 3823 a(\() p Fs(H) p Fh 1988 3782 a(d) p Fr 2031 3823 a(\)\() p Fs(H) p Fh 2196 3782 a(d) p Fr 2239 3823 a(\() p Fs(M) p Fr 10 w(\)) p Fm 23 w(\000) p Fs 22 w(E) p Fh 2619 3782 a(d) p Fq 2613 3847 a(0) p Fr 2663 3823 a(\() p Fs(M) p Fr 10 w(\)) p Fm 23 w(\000) p Fs 22 w(m) p Fr 23 w(+) p Fs 22 w(\017) p Fr 1782 3968 a(+) p Fs(W) p Fh 1964 3927 a(d) p Fm 2029 3968 a(\000) p Fs 23 w(W) p Fh 2235 3927 a(d) p Fr 2278 3968 a(\() p Fs(M) p Fr 10 w(\)) p Fm 23 w(\000) p Fs 22 w(E) p Fh 2658 3927 a(d) p Fq 2652 3993 a(0) p Fr 2724 3968 a(+) p Fs 22 w(E) p Fh 2900 3927 a(d) p Fq 2894 3993 a(0) p Fr 2943 3968 a(\() p Fs(M) p Fr 10 w(\)\)) p Fs(\037) p Fq 3222 3983 a(\001) p Fr 3286 3968 a(\() p Fs(H) p Fh 3413 3927 a(d) p Fr 3456 3968 a(\)) p Fm(g) p Fs(:) p Fv 257 4159 a(But) p Fs 576 4349 a(E) p Fh 654 4308 a(d) p Fq 648 4374 a(0) p Fr 697 4349 a(\() p Fs(M) p Fr 10 w(\)) p Fm 29 w(!) p Fs 27 w(E) p Fh 1111 4308 a(d) p Fq 1105 4374 a(0) p Fv 1252 4349 a(and) p Fm 97 w(k) p Fs(\037) p Fq 1617 4364 a(\001) p Fr 1680 4349 a(\() p Fs(H) p Fh 1807 4308 a(d) p Fr 1850 4349 a(\)\() p Fs(W) p Fh 2032 4308 a(d) p Fm 2097 4349 a(\000) p Fs 23 w(W) p Fh 2303 4308 a(d) p Fr 2346 4349 a(\() p Fs(M) p Fr 10 w(\)\)) p Fs(\037) p Fq 2625 4364 a(\001) p Fr 2688 4349 a(\() p Fs(H) p Fh 2815 4308 a(d) p Fr 2858 4349 a(\)) p Fm(k) 28 b(!) p Fr 27 w(0) p Fs(;) p Fv 257 4540 a(using) 33 b(Prop) s(ositions) e(4.5) h(and) h(4.6,) f(so,) h(for) p Fs 32 w(M) p Fv 43 w(large) e(enough,) i (w) m(e) h(ha) m(v) m(e) 467 4722 y(T) -8 b(r) p Fp 584 4641 a(\010) p Fr 642 4722 a([) p Fs(H) p Fh 758 4686 a(d) p Fm 823 4722 a(\000) p Fs 22 w(E) p Fh 1000 4686 a(d) p Fq 994 4747 a(0) p Fm 1066 4722 a(\000) p Fs 23 w(m) p Fr 22 w(+) p Fs 22 w(\017) p Fr(]) p Fi 1437 4737 a(\000) p Fp 1497 4641 a(\011) p Fm 1052 4843 a(\025) p Fv 28 w(T) g(r) p Fp 1274 4762 a(\010) p Fs 1332 4843 a(\037) p Fq 1393 4858 a(\001) p Fr 1456 4843 a(\() p Fs(H) p Fh 1583 4807 a(d) p Fr 1626 4843 a(\)\() p Fs(H) p Fh 1791 4807 a(d) p Fr 1834 4843 a(\() p Fs(M) p Fr 10 w(\)) p Fm 22 w(\000) p Fs 23 w(E) p Fh 2214 4807 a(d) p Fq 2208 4868 a(0) p Fr 2257 4843 a(\() p Fs(M) p Fr 10 w(\)) p Fm 23 w(\000) p Fs 23 w(m) p Fr 22 w(+) p Fl 2779 4804 a(\017) p 2775 4820 36 4 v Fq 2775 4877 a(2) p Fr 2821 4843 a(\)) p Fs(\037) p Fq 2920 4858 a(\001) p Fr 2983 4843 a(\() p Fs(H) p Fh 3110 4807 a(d) p Fr 3152 4843 a(\)) p Fp 3190 4762 a(\011) p Fm 1052 4964 a(\025) p Fv 28 w(T) g(r) p Fp 1274 4883 a(\010) p Fs 1332 4964 a(\037) p Fq 1393 4979 a(\001) p Fr 1456 4964 a(\() p Fs(H) p Fh 1583 4928 a(d) p Fr 1626 4964 a(\)[) p Fs(H) p Fh 1780 4928 a(d) p Fr 1823 4964 a(\() p Fs(M) p Fr 10 w(\)) p Fm 23 w(\000) p Fs 22 w(E) p Fh 2203 4928 a(d) p Fq 2197 4988 a(0) p Fr 2247 4964 a(\() p Fs(M) p Fr 10 w(\)) p Fm 22 w(\000) p Fs 23 w(m) p Fr 23 w(+) p Fl 2768 4925 a(\017) p 2765 4941 V Fq 2765 4998 a(2) p Fr 2810 4964 a(]) p Fi 2837 4979 a(\000) p Fs 2896 4964 a(\037) p Fq 2957 4979 a(\001) p Fr 3020 4964 a(\() p Fs(H) p Fh 3147 4928 a(d) p Fr 3190 4964 a(\)) p Fp 3228 4883 a(\011) p Fm 1052 5085 a(\025) p Fv 28 w(T) g(r) p Fp 1274 5004 a(\010) p Fr 1332 5085 a([) p Fs(H) p Fh 1448 5048 a(d) p Fr 1491 5085 a(\() p Fs(M) p Fr 10 w(\)) p Fm 23 w(\000) p Fs 22 w(E) p Fh 1871 5048 a(d) p Fq 1865 5109 a(0) p Fr 1915 5085 a(\() p Fs(M) p Fr 10 w(\)) p Fm 23 w(\000) p Fs 22 w(m) p Fr 23 w(+) p Fl 2436 5045 a(\017) p 2433 5062 V Fq 2433 5119 a(2) p Fr 2478 5085 a(]) p Fi 2505 5100 a(\000) p Fp 2564 5004 a(\011) p Fs 2650 5085 a(>) p Fm 27 w(\0001) p Fv 257 5268 a(where) 34 b(in) e(the) h(last) f(step) h(w) m(e) h(used) f(Prop) s(osition) e (4.2.) p Fc 1166 w(2) p Fv 1828 5637 a(15) p 90 rotate dyy eop %%Page: 16 16 16 15 bop Ff 257 573 a(4.2) 131 b(Con) l(tin) l(uous) 44 b(mo) t(dels) p Fv 257 758 a(In) 29 b(this) f(section,) h(w) m(e) h (are) e(in) m(terested) h(in) f(the) h(mo) s(del) d(in) m(tro) s(duced) j(in) e(Sect.) 43 b(2.2) 28 b(but) h(for) 257 878 y(massiv) m(e) k(b) s (osons,) p Ft 34 w(i.e.) p Fv 43 w(the) g(function) p Fs 32 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fv 32 w(is) g(replaced) g(b) m(y) p Fs 33 w(!) p Fl 2597 893 a(m) p Fr 2664 878 a(\() p Fs(k) p Fr 3 w(\)) p Fv 32 w(satisfying) p Fr 32 w(\() p Fs(H) p Fl 3380 893 a(!) p Fr 3430 878 a(\)) p Fs(:) p Fv 257 998 a(W) -8 b(e) 33 b(then) g(consider,) g(on) p Fm 33 w(H) p Fs 1 w(;) p Fv 33 w(the) f(follo) m(wing) e(Hamiltonian:) p Fs 587 1261 a(H) p Fl 668 1276 a(m) p Fr 817 1261 a(:=) p Fs 83 w(H) p Fl 1084 1276 a(p) p Fm 1146 1261 a(\012) p Fr 23 w(1) -22 b(l) 21 b(+) h(1) -22 b(l) p Fm 20 w(\012) p Fp 1595 1126 a(Z) p Fo 1650 1351 a(R) p Fn 1698 1332 a(d) p Fs 1755 1261 a(dx) p Fp 1895 1126 a(Z) p Fo 1950 1351 a(R) p Fn 1998 1332 a(n) p Fs 2061 1261 a(dk) 20 b(!) p Fl 2244 1276 a(m) p Fr 2310 1261 a(\() p Fs(k) p Fr 3 w(\)) p Fs(a) p Fi 2491 1220 a(\003) p Fr 2530 1261 a(\() p Fs(x;) d(k) p Fr 3 w(\)) p Fs(a) p Fr(\() p Fs(x;) g(k) p Fr 3 w(\)) 1198 1526 y(+) p Fp 1291 1391 a(Z) p Fo 1346 1616 a(R) p Fn 1394 1598 a(d) p Fs 1451 1526 a(dx) p Fp 1590 1391 a(Z) p Fo 1646 1616 a(R) p Fn 1694 1598 a(n) p Fs 1757 1526 a(dk) i(\032) p Fq 1928 1541 a(1) p Fr 1968 1526 a(\() p Fs(x) p Fm 23 w(\000) p Fs 22 w(Q) p Fr(\)) 2410 1459 y(^) p Fs -58 w(\032) p Fq 2451 1474 a(2) p Fr 2491 1459 a(\() p Fs(k) p Fr 3 w(\)) p 2308 1504 406 4 v Fp 2308 1524 a(p) p 2408 1524 307 4 v Fr 2408 1609 a(2) p Fs(!) p Fl 2518 1624 a(m) p Fr 2584 1609 a(\() p Fs(k) p Fr 3 w(\)) p Fm 2746 1526 a(\012) p Fs 23 w(a) p Fi 2897 1485 a(\003) p Fr 2937 1526 a(\() p Fs(x;) e(k) p Fr 3 w(\)) 1784 1822 y(+) p Fs(\032) p Fq 1910 1837 a(1) p Fr 1949 1822 a(\() p Fs(x) p Fm 23 w(\000) p Fs 22 w(Q) p Fr(\)) 2391 1729 y(\026) 2391 1755 y(^) p Fs -57 w(\032) p Fq 2433 1770 a(2) p Fr 2472 1755 a(\() p Fs(k) p Fr 3 w(\)) p 2289 1799 406 4 v Fp 2289 1819 a(p) p 2389 1819 307 4 v Fr 2389 1904 a(2) p Fs(!) p Fl 2499 1919 a(m) p Fr 2565 1904 a(\() p Fs(k) p Fr 3 w(\)) p Fm 2727 1822 a(\012) p Fs 23 w(a) p Fr(\() p Fs(x;) g(k) p Fr 3 w(\)) p Fv 188 w(\(4.6\)) p Fr 831 2052 a(=) p Fs 96 w(H) p Fq 1092 2011 a(0) p Fl 1084 2077 a(m) p Fr 1173 2052 a(+) p Fs 22 w(W) p Fl 1363 2067 a(m) p Fs 1430 2052 a(:) p Fv 404 2272 a(W) -8 b(e) 29 b(denote) h(b) m(y) p Fs 30 w(E) p Fl 1083 2287 a(m) p Fv 1179 2272 a(the) f(ground) g(state) h(energy) g(of) p Fs 28 w(H) p Fl 2402 2287 a(m) p Fs 2469 2272 a(:) p Fv 29 w(The) g(main) d(result) i(of) g(this) 257 2392 y(section) k(is) f(the) p Fw 257 2596 a(Theorem) 39 b(4.7.) p Fs 43 w(\033) p Fl 1006 2611 a(ess) p Fr 1109 2596 a(\() p Fs(H) p Fl 1228 2611 a(m) p Fr 1294 2596 a(\)) p Fm 31 w(\032) p Fr 31 w([) p Fs(E) p Fl 1570 2611 a(m) p Fr 1660 2596 a(+) p Fs 24 w(m;) p Fr 17 w(+) p Fm(1) p Fr([) p Fs(:) p Ft 36 w(In) d(p) -5 b(articular,) p Fs 36 w(H) p Fl 2842 2611 a(m) p Ft 2945 2596 a(has) 36 b(a) h(gr) -5 b(ound) 257 2716 y(state) p Fs 35 w(\036) p Fl 549 2731 a(m) p Fs 616 2716 a(:) p Fv 404 2920 a(The) 26 b(strategy) f(of) g (the) g(pro) s(of) f(is) h(v) m(ery) h(similar) c(to) j(the) g(one) h (of) e(the) i(previous) f(section.) 257 3040 y(Ho) m(w) m(ev) m(er,) 37 b(one) e(has) f(to) g(b) s(e) h(more) e(careful) h(with) g(the) g (estimates) g(when) h(remo) m(ving) f(the) 257 3160 y(cuto\033) 44 b(b) s(ecause) h(the) f(norm) e(of) p Fs 43 w(\032) p Fq 1530 3175 a(1) p Fr 1570 3160 a(\() p Fs(x) p Fm 30 w(\000) p Fs 30 w(Q) p Fr(\)) p Fv 44 w(as) h(an) h(op) s(erator) e(on) p Fs 44 w(L) p Fq 2852 3124 a(2) p Fr 2892 3160 a(\() p Fg(R) p Fl 2996 3124 a(d) p Fr 3042 3160 a(\)) p Fv 43 w(do) s(es) i(not) 257 3281 y(decrease) 24 b(with) p Fs 22 w(x;) p Fv 22 w(ev) m(en) g(w) m(orse,) h(it) c(do) s(es) i(not) e (dep) s(end) j(on) d(it.) 39 b(T) -8 b(o) 22 b(con) m(trol) g(this) f (problem,) 257 3401 y(w) m(e) 28 b(will) d(use) i(the) g(exp) s(onen) m (tial) f(deca) m(y) i(of) f(the) g(sp) s(ectral) f(pro) 5 b(jectors) 28 b(in) e(the) p Fs 27 w(Q) p Fv 27 w(v) -5 b(ariable,) 257 3521 y(whic) m(h) 33 b(will) e(b) s(e) h(obtained) g (via) g(the) h(Agmon) f(metho) s(d) g(\(see) h(Sect.) 44 b(4.2.2\)) p Fw 257 3781 a(4.2.1) 113 b(Cuto\033) 37 b(mo) s(dels) p Fv 257 3966 a(Let) p Fs 33 w(j) p Fv 39 w(b) s(e) 32 b(a) h(smo) s(oth) e(function) h(with) g(compact) g (supp) s(ort) h(on) p Fg 33 w(R) p Fl 2626 3930 a(d) p Fv 2705 3966 a(suc) m(h) h(that) p Fr 377 4186 a(0) p Fm 27 w(\024) p Fs 28 w(j) p Fr 6 w(\() p Fs(x) p Fr(\)) p Fm 28 w(\024) p Fr 28 w(1) p Fs(;) 114 b(j) p Fr 6 w(\() p Fs(x) p Fr(\)) 28 b(=) g(1) 65 b(for) p Fm 32 w(j) p Fr(x) p Fm(j) 27 b(\024) p Fr 28 w(1) p Fs(=) p Fr(2) p Fs(;) p Fr 114 w(and) 98 b(j\(x\)) 27 b(=) h(0) 65 b(for) p Fm 32 w(j) p Fr(x) p Fm(j) 27 b(\025) p Fr 28 w(3) p Fs(=) p Fr(4) p Fs(:) p Fv 257 4406 a(F) -8 b(or) 31 b(all) p Fs 29 w(L) d(>) p Fr 27 w(0) p Fs(;) p Fv 32 w(w) m(e) k(de\034ne) p Fs 32 w(j) p Fl 1332 4421 a(L) p Fr 1385 4406 a(\() p Fs(x) p Fr(\)) c(=) p Fs 27 w(j) p Fr 6 w(\() p Fl 1745 4367 a(x) p 1741 4383 48 4 v 1741 4440 a(L) p Fr 1799 4406 a(\)) p Fv 31 w(and) p Fr 2055 4384 a(\026) p Fs 2056 4406 a(j) p Fl 2096 4421 a(L) p Fr 2149 4406 a(\() p Fs(x) p Fr(\)) g(=) f(1) p Fm 20 w(\000) p Fs 20 w(j) p Fl 2617 4421 a(L) p Fr 2669 4406 a(\() p Fs(x) p Fr(\)) p Fs(:) p Fv 32 w(W) -8 b(e) 31 b(then) h(de\034ne) p Fs 414 4690 a(H) p Fl 495 4705 a(m) p Fr 561 4690 a(\() p Fs(L) p Fr(\)) 84 b(:=) p Fs 83 w(H) p Fq 1062 4649 a(0) p Fl 1054 4715 a(m) p Fr 1142 4690 a(+) p Fp 1240 4555 a(Z) p Fo 1296 4780 a(R) p Fn 1344 4761 a(d) p Fs 1400 4690 a(dx) p Fp 1540 4555 a(Z) p Fo 1595 4780 a(R) p Fn 1643 4761 a(n) p Fs 1706 4690 a(dk) 20 b(\032) p Fq 1878 4705 a(1) p Fr 1917 4690 a(\() p Fs(x) p Fm 23 w(\000) p Fs 23 w(Q) p Fr(\)) p Fs(j) p Fl 2288 4705 a(L) p Fr 2340 4690 a(\() p Fs(x) p Fr(\)) 2583 4623 y(^) p Fs -58 w(\032) p Fq 2624 4638 a(2) p Fr 2664 4623 a(\() p Fs(k) p Fr 3 w(\)) p 2481 4667 406 4 v Fp 2481 4687 a(p) p 2581 4687 307 4 v Fr 2581 4773 a(2) p Fs(!) p Fl 2691 4788 a(m) p Fr 2757 4773 a(\() p Fs(k) p Fr 3 w(\)) p Fm 2919 4690 a(\012) p Fs 23 w(a) p Fi 3070 4649 a(\003) p Fr 3110 4690 a(\() p Fs(x;) d(k) p Fr 3 w(\)) 1558 4986 y(+) p Fs(\032) p Fq 1684 5001 a(1) p Fr 1723 4986 a(\() p Fs(x) p Fm 23 w(\000) p Fs 23 w(Q) p Fr(\)) p Fs(j) p Fl 2094 5001 a(L) p Fr 2146 4986 a(\() p Fs(x) p Fr(\)) 2389 4892 y(\026) 2389 4919 y(^) p Fs -58 w(\032) p Fq 2430 4934 a(2) p Fr 2470 4919 a(\() p Fs(k) p Fr 3 w(\)) p 2287 4963 406 4 v Fp 2287 4983 a(p) p 2387 4983 307 4 v Fr 2387 5068 a(2) p Fs(!) p Fl 2497 5083 a(m) p Fr 2563 5068 a(\() p Fs(k) p Fr 3 w(\)) p Fm 2725 4986 a(\012) p Fs 23 w(a) p Fr(\() p Fs(x;) g(k) p Fr 3 w(\)) p Fv 190 w(\(4.7\)) p Fr 800 5216 a(=) p Fs 97 w(H) p Fq 1062 5175 a(0) p Fl 1054 5240 a(m) p Fr 1142 5216 a(+) p Fs 22 w(W) p Fl 1332 5231 a(m) p Fr 1399 5216 a(\() p Fs(L) p Fr(\)) p Fv 1828 5637 a(16) p 90 rotate dyy eop %%Page: 17 17 17 16 bop Fv 257 573 a(on) p Fm 51 w(H) p Fs 1 w(:) p Fv 50 w(Using) 50 b(the) h(de\034nition) e(of) p Fs 50 w(j) p Fl 1671 588 a(L) p Fs 1724 573 a(;) p Fv 50 w(one) i(can,) k(in) p Fs 49 w(W) p Fl 2449 588 a(m) p Fr 2516 573 a(\() p Fs(L) p Fr(\)) p Fs(;) p Fv 51 w(replace) p Fp 3084 493 a(R) p Fo 3131 607 a(R) p Fn 3179 589 a(d) p Fs 3236 573 a(dx) p Fv 50 w(b) m(y) p Fp 257 613 a(R) p Fq 304 728 a([) p Fi(\000) p Fl(L;L) p Fq(]) p Fn 515 709 a(d) p Fs 571 693 a(dx:) p Fv 33 w(Finally) -8 b(,) 30 b(w) m(e) j(de\034ne) p Fr 332 962 a(~) p Fs 307 987 a(H) p Fl 388 1002 a(m) p Fr 455 987 a(\() p Fs(L) p Fr(\)) 27 b(:=) p Fs 28 w(H) p Fl 836 1002 a(p) p Fm 882 987 a(\012) p Fr 6 w(1) -22 b(l) 5 b(+) h(1) -22 b(l) p Fm 4 w(\012) p Fp 1249 851 a(Z) p Fq 1305 1077 a([) p Fi(\000) p Fl(L;L) p Fq(]) p Fn 1516 1058 a(d) p Fs 1572 987 a(dx) p Fp 1711 851 a(Z) p Fo 1766 1077 a(R) p Fn 1814 1058 a(n) p Fs 1878 987 a(dk) 19 b(!) p Fl 2060 1002 a(m) p Fr 2126 987 a(\() p Fs(k) p Fr 3 w(\)) p Fs(a) p Fi 2307 946 a(\003) p Fr 2347 987 a(\() p Fs(x;) e(k) p Fr 3 w(\)) p Fs(a) p Fr(\() p Fs(x;) g(k) p Fr 3 w(\)) 6 b(+) p Fs 6 w(W) p Fl 3036 1002 a(m) p Fr 3103 987 a(\() p Fs(L) p Fr(\)) p Fv 50 w(\(4.8\)) 257 1291 y(on) p Fs 41 w(L) p Fq 467 1255 a(2) p Fr 507 1291 a(\() p Fg(R) p Fl 610 1255 a(d) p Fr 657 1291 a(\)) p Fm 27 w(\012) 28 b(F) p Fp 926 1211 a(\000) p Fs 971 1291 a(L) p Fq 1037 1255 a(2) p Fr 1077 1291 a(\([) p Fm(\000) p Fs(L;) 17 b(L) p Fr(]) p Fl 1422 1255 a(d) p Fr 1464 1291 a(\)) p Fm 22 w(\012) p Fs 22 w(L) p Fq 1689 1255 a(2) p Fr 1729 1291 a(\() p Fg(R) p Fl 1833 1255 a(n) p Fr 1886 1291 a(\)) p Fp 1924 1211 a(\001) p Fs 1986 1291 a(:) p Fv 41 w(W) -8 b(e) 41 b(denote) g(b) m(y) p Fs 41 w(E) p Fl 2767 1306 a(m) p Fr 2834 1291 a(\() p Fs(L) p Fr(\)) p Fv 41 w(and) p Fr 3237 1266 a(~) p Fs 3214 1291 a(E) p Fl 3286 1306 a(m) p Fr 3353 1291 a(\() p Fs(L) p Fr(\)) p Fv 257 1412 a(the) 33 b(ground) g(state) g(energies) g (of) f(those) h(t) m(w) m(o) g(op) s(erators) g(resp) s(ectiv) m(ely) -8 b(.) 404 1532 y(W) g(e) 43 b(ha) m(v) m(e) i(\020cut\021) 51 b(the) 43 b(Hamiltonian) p Fs 40 w(H) p Fl 1901 1547 a(m) p Fv 2011 1532 a(in) f(the) p Fs 44 w(x) p Fv 43 w(v) -5 b(ariable.) 74 b(W) -8 b(e) 44 b(are) f(no) m(w) h(in) 257 1653 y(a) e(\034nite) g(v) m(olume) f(b) s(o) m(x.) 73 b(If) 42 b(w) m(e) h(consider) f(the) g(v) -5 b(ariable) p Fs 41 w(p;) p Fv 42 w(conjugate) 42 b(to) p Fs 41 w(x;) p Fv 43 w(this) g(is) 257 1773 y(equiv) -5 b(alen) m(t) 49 b(to) f(\020discretizing\021) 54 b(the) 49 b(problem.) 90 b(One) 49 b(has) g(to) f(note) h(that) f(here) i(the) 257 1893 y(v) -5 b(ariable) p Fs 31 w(p) p Fv 33 w(is) 32 b(discrete:) p Fs 44 w(p) p Fm 27 w(2) p Fg 28 w(Z) p Fl 1438 1857 a(d) p Fs 1476 1893 a(:) p Fv 33 w(If) p Fs 257 2155 a(a) p Fi 308 2114 a(\003) p Fl 308 2180 a(p) p Fr 348 2155 a(\() p Fs(k) p Fr 3 w(\)) c(=) 719 2088 y(1) p 619 2132 248 4 v 619 2243 a(\(2) p Fs(L) p Fr(\)) p Fn 820 2179 a(d) p 820 2191 33 3 v Fj 821 2232 a(2) p Fp 893 2019 a(Z) p Fq 948 2245 a([) p Fi(\000) p Fl(L;L) p Fq(]) p Fn 1159 2226 a(d) p Fs 1215 2155 a(dx) 17 b(e) p Fl 1383 2114 a(ipx) p Fs 1486 2155 a(a) p Fi 1537 2114 a(\003) p Fr 1577 2155 a(\() p Fs(x;) g(k) p Fr 3 w(\)) p Fs(;) 114 b(a) p Fl 1998 2170 a(p) p Fr 2038 2155 a(\() p Fs(k) p Fr 3 w(\)) 28 b(=) 2408 2088 y(1) p 2309 2132 248 4 v 2309 2243 a(\(2) p Fs(L) p Fr(\)) p Fn 2510 2179 a(d) p 2510 2191 33 3 v Fj 2511 2232 a(2) p Fp 2583 2019 a(Z) p Fq 2638 2245 a([) p Fi(\000) p Fl(L;L) p Fq(]) p Fn 2849 2226 a(d) p Fs 2905 2155 a(dx) 17 b(e) p Fi 3073 2114 a(\000) p Fl(ipx) p Fs 3231 2155 a(a) p Fr(\() p Fs(x;) g(k) p Fr 3 w(\)) p Fv 257 2443 a(and) p Fs 1077 2587 a(\014) p Fl 1132 2602 a(p) p Fr 1200 2587 a(=) 1412 2519 y(1) p 1313 2564 248 4 v 1313 2674 a(\(2) p Fs(L) p Fr(\)) p Fn 1514 2610 a(d) p 1514 2622 33 3 v Fj 1515 2664 a(2) p Fp 1587 2451 a(Z) p Fq 1642 2677 a([) p Fi(\000) p Fl(L;L) p Fq(]) p Fn 1853 2658 a(d) p Fs 1909 2587 a(dx) g(\032) p Fq 2082 2602 a(1) p Fr 2121 2587 a(\() p Fs(x) p Fm 23 w(\000) p Fs 22 w(Q) p Fr(\)) p Fs(j) p Fl 2491 2602 a(L) p Fr 2544 2587 a(\() p Fs(x) p Fr(\)) p Fv 257 2836 a(denote) 32 b(the) f(F) -8 b(ourier) 29 b(co) s(e\036cien) m(ts) i(of) p Fs 30 w(a) p Fi 1722 2800 a(\003) p Fr 1762 2836 a(\() p Fs(x;) 17 b(k) p Fr 3 w(\)) p Fs(;) g(a) p Fr(\() p Fs(x;) g(k) p Fr 3 w(\)) p Fv 31 w(and) p Fs 30 w(\032) p Fq 2583 2851 a(1) p Fr 2623 2836 a(\() p Fs(x) p Fm 18 w(\000) p Fs 19 w(Q) p Fr(\)) p Fs(j) p Fl 2985 2851 a(L) p Fr 3038 2836 a(\() p Fs(x) p Fr(\)) p Fv 30 w(resp) s(ec-) 257 2956 y(tiv) m(ely) -8 b(,) 33 b(the) f(problem) g(can) h(no) m(w) g(b) s(e) f(written) h(as) g (follo) m(ws) p Fr 284 3193 a(~) p Fs 258 3218 a(H) p Fl 339 3233 a(m) p Fr 406 3218 a(\() p Fs(L) p Fr(\)) 83 b(=) p Fs 83 w(H) p Fl 871 3233 a(p) p Fm 933 3218 a(\012) p Fr 22 w(1) -22 b(l) 21 b(+) h(1) -22 b(l) p Fm 21 w(\012) p Fp 1391 3123 a(X) p Fl 1382 3343 a(p) p Fi(2) p Fo(Z) p Fn 1515 3324 a(d) p Fp 1561 3082 a(Z) p Fo 1617 3308 a(R) p Fn 1665 3289 a(n) p Fs 1728 3218 a(dk) 19 b(!) p Fl 1910 3233 a(m) p Fr 1977 3218 a(\() p Fs(k) p Fr 3 w(\)) p Fs(a) p Fi 2158 3177 a(\003) p Fl 2158 3243 a(p) p Fr 2197 3218 a(\() p Fs(k) p Fr 3 w(\)) p Fs(a) p Fl 2378 3233 a(p) p Fr 2418 3218 a(\() p Fs(k) p Fr 3 w(\)) 985 3549 y(+) p Fp 1087 3454 a(X) p Fl 1078 3674 a(p) p Fi(2) p Fo(Z) p Fn 1210 3655 a(d) p Fp 1257 3413 a(Z) p Fo 1313 3639 a(R) p Fn 1361 3620 a(n) p Fs 1424 3549 a(dk) p Fr 19 w(\() p Fs(\014) p Fl 1638 3564 a(p) p Fr 1790 3481 a(^) p Fs -58 w(\032) p Fq 1831 3496 a(2) p Fr 1870 3481 a(\() p Fs(k) p Fr 3 w(\)) p 1688 3526 406 4 v Fp 1688 3546 a(p) p 1787 3546 307 4 v Fr 1787 3631 a(2) p Fs(!) p Fl 1897 3646 a(m) p Fr 1963 3631 a(\() p Fs(k) p Fr 3 w(\)) p Fm 2126 3549 a(\012) p Fs 22 w(a) p Fi 2276 3508 a(\003) p Fl 2276 3573 a(p) p Fr 2316 3549 a(\() p Fs(k) p Fr 3 w(\)) j(+) 2580 3523 y(\026) p Fs 2566 3549 a(\014) p Fl 2621 3564 a(p) p Fr 2773 3455 a(\026) 2773 3481 y(^) p Fs -58 w(\032) p Fq 2814 3496 a(2) p Fr 2854 3481 a(\() p Fs(k) p Fr 3 w(\)) p 2671 3526 406 4 v Fp 2671 3546 a(p) p 2770 3546 307 4 v Fr 2770 3631 a(2) p Fs(!) p Fl 2880 3646 a(m) p Fr 2947 3631 a(\() p Fs(k) p Fr 3 w(\)) p Fm 3109 3549 a(\012) p Fs 22 w(a) p Fl 3259 3564 a(p) p Fr 3299 3549 a(\() p Fs(k) p Fr 3 w(\)\)) p Fs(;) p Fv 257 3878 a(whic) m(h) 49 b(has) g(the) g(form) f(\(4.1\).) 90 b(If) 49 b(the) p Fs 49 w(\014) p Fl 1844 3893 a(p) p Fv 1932 3878 a(satisfy) p Fr 49 w(\() p Fs(C) p Fl 2361 3893 a(\014) p Fr 2407 3878 a(\)) p Fs(;) p Fv 49 w(w) m(e) h(will) c (then) j(ha) m(v) m(e) h(the) 257 3998 y(follo) m(wing) 30 b(result:) p Fw 257 4201 a(Prop) s(osition) 36 b(4.8.) p Fm 42 w(8) p Fs(L) 28 b(>) p Fr 28 w(0) p Fs(;) 17 b(\033) p Fl 1480 4216 a(ess) p Fr 1582 4201 a(\() 1645 4176 y(~) p Fs 1620 4201 a(H) p Fl 1701 4216 a(m) p Fr 1767 4201 a(\() p Fs(L) p Fr(\)\)) p Fm 28 w(\032) p Fr 28 w([) 2130 4176 y(~) p Fs 2107 4201 a(E) p Fl 2179 4216 a(m) p Fr 2246 4201 a(\() p Fs(L) p Fr(\)) 22 b(+) p Fs 22 w(m;) p Fr 17 w(+) p Fm(1) p Fr([) p Fs(:) p Fv 257 4403 a(Finally) -8 b(,) 38 b(a) h(splitting) e(of) p Fs 38 w(L) p Fq 1279 4367 a(2) p Fr 1319 4403 a(\() p Fg(R) p Fl 1423 4367 a(d) p Fr 1469 4403 a(\)) p Fv 39 w(in) m(to) p Fs 39 w(L) p Fq 1817 4367 a(2) p Fr 1856 4403 a(\([) p Fm(\000) p Fs(L;) 17 b(L) p Fr(]) p Fl 2201 4367 a(d) p Fr 2243 4403 a(\)) p Fm 27 w(\010) p Fs 27 w(L) p Fq 2478 4367 a(2) p Fr 2517 4403 a(\() p Fg(R) p Fl 2621 4367 a(d) p Fm 2668 4403 a(n) p Fr([) p Fm(\000) p Fs(L;) g(L) p Fr(]) p Fl 3025 4367 a(d) p Fr 3066 4403 a(\)) p Fv 39 w(together) 257 4524 y(with) 33 b(the) f(argumen) m(t) h(of) f(the) h (previous) g(section) f(will) e(lead) i(to) h(the) p Fw 257 4726 a(Prop) s(osition) 41 b(4.9.) p Fs 45 w(\033) p Fl 1142 4741 a(ess) p Fr 1244 4726 a(\() p Fs(H) p Fl 1363 4741 a(m) p Fr 1430 4726 a(\() p Fs(L) p Fr(\)\)) p Fm 36 w(\032) p Fr 36 w([) p Fs(E) p Fl 1858 4741 a(m) p Fr 1925 4726 a(\() p Fs(L) p Fr(\)) 26 b(+) p Fs 25 w(m;) p Fr 17 w(+) p Fm(1) p Fr([) p Fs(:) p Ft 40 w(In) 38 b(p) -5 b(articular,) p Fs 41 w(H) p Fl 3287 4741 a(m) p Fr 3353 4726 a(\() p Fs(L) p Fr(\)) p Ft 257 4846 a(has) 35 b(a) f(gr) -5 b(ound) 35 b(state) p Fs 35 w(\036) p Fl 1131 4861 a(m) p Fr 1197 4846 a(\() p Fs(L) p Fr(\)) p Fs(:) p Fv 404 5049 a(So,) 25 b(it) c(remains) h(to) h(c) m(hec) m(k) i(that) e(the) p Fs 23 w(\014) p Fl 1778 5064 a(p) p Fv 1841 5049 a(satisfy) f(the) i (condition) p Fr 21 w(\() p Fs(C) p Fl 2820 5064 a(\014) p Fr 2867 5049 a(\)) p Fs(:) p Fv 23 w(The) g(function) p Fs 257 5169 a(j) p Fl 297 5184 a(L) p Fv 380 5169 a(is) 29 b(zero) h(for) p Fm 29 w(j) p Fs(x) p Fm(j) p Fs 28 w(>) d(L) p Fv 31 w(and) p Fs 30 w(\032) p Fq 1400 5184 a(1) p Fv 1469 5169 a(has) j(compact) g(supp) s(ort) g(\(in) f(a) g(ball) f(of) i (radius) p Fs 29 w(R) p Fq 3273 5184 a(1) p Fv 3313 5169 a(\),) g(so) p Fm 947 5388 a(8j) p Fs(q) p Fm 4 w(j) p Fs 27 w(>) e(L) p Fr 22 w(+) p Fs 22 w(R) p Fq 1496 5403 a(1) p Fs 1536 5388 a(;) p Fm 17 w(8) p Fs(x) p Fm 28 w(2) p Fg 28 w(R) p Fl 1878 5347 a(d) p Fs 1925 5388 a(;) 33 b(\032) p Fq 2035 5403 a(1) p Fr 2075 5388 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(q) p Fr 4 w(\)) p Fs(j) p Fl 2415 5403 a(L) p Fr 2467 5388 a(\() p Fs(x) p Fr(\)) 28 b(=) g(0) p Fs(:) p Fv 1828 5637 a(17) p 90 rotate dyy eop %%Page: 18 18 18 17 bop Fv 257 573 a(Then,) 45 b(for) 40 b(all) p Fs 39 w(p) p Fv 41 w(in) p Fg 40 w(Z) p Fl 1132 537 a(d) p Fs 1170 573 a(;) h(\014) p Fl 1293 588 a(p) p Fv 1374 573 a(is) f(a) h(m) m(ultiplication) 36 b(op) s(erator) k(b) m(y) i(a) f (compactly) f(sup-) 257 693 y(p) s(orted) 32 b(function.) 42 b(Moreo) m(v) m(er,) 34 b(the) d(function) p Fs 31 w(\032) p Fq 2039 708 a(1) p Fr 2079 693 a(\() p Fs(x) p Fm 20 w(\000) p Fs 21 w(q) p Fr 4 w(\)) p Fs(j) p Fl 2415 708 a(L) p Fr 2467 693 a(\() p Fs(x) p Fr(\)) p Fv 32 w(is) p Fs 31 w(C) p Fi 2804 657 a(1) p Fs 2878 693 a(;) p Fv 32 w(so) h(its) e(F) -8 b(ourier) 257 814 y(co) s(e\036cien) m(ts) 32 b(deca) m(y) h(faster) e(than) f(an) m(y) i(p) s(o) m(w) m(er) g(of) p Fs 30 w(p:) p Fv 31 w(Those) g(t) m(w) m(o) g(facts) f(ensure) h(us) f (that) p Fr 257 934 a(sup) p Fl 404 958 a(p) p Fr 461 934 a(sup) p Fl 607 958 a(q) p Fp 662 849 a(\014) 662 909 y(\014) p Fs 695 934 a(\014) p Fl 750 949 a(p) p Fr 790 934 a(\() p Fs(q) p Fr 4 w(\)) p Fm(j) p Fs(p) p Fm(j) p Fl 1018 898 a(n) p Fp 1064 849 a(\014) 1064 909 y(\014) p Fs 1134 934 a(<) 36 b(C) p Fl 1316 949 a(n) p Fr 1363 934 a(\() p Fs(L) p Fr(\)) p Fs 37 w(<) p Fr 37 w(+) p Fm(1) p Fv 37 w(and) i(so) g(condition) p Fr 36 w(\() p Fs(C) p Fl 2729 949 a(\014) p Fr 2776 934 a(\)) p Fv 38 w(is) f(satis\034ed.) 60 b(T) -8 b(o) 257 1054 y(pro) m(v) m(e) 34 b(Theorem) f(4.7,) f(it) g(remains) f(to) h (con) m(trol) g(the) h(limit) p Fs 29 w(L) p Fm 28 w(!) p Fr 28 w(+) p Fm(1) p Fs(:) p Fw 257 1312 a(4.2.2) 113 b(Exp) s(onen) m(tial) 36 b(b) s(ounds) 257 1496 y(Prop) s(osition) d (4.10.) p Ft 39 w(L) -5 b(et) p Fr 33 w(\001) p Ft 33 w(b) g(e) 32 b(a) g(b) -5 b(ounde) g(d) 31 b(fr) -5 b(om) 32 b(ab) -5 b(ove) 32 b(interval.) 43 b(F) -7 b(or) 31 b(any) p Fs 32 w(\013) e(>) p Fr 27 w(0) p Fs(;) p Ft 257 1617 a(ther) -5 b(e) 35 b(exists) p Fs 35 w(M) p Fr 10 w(\() p Fs(\013) q(;) p Fr 17 w(\001\)) p Fs 28 w(>) p Fr 27 w(0) p Ft 35 w(such) f(that) 418 1805 y(-) p Fm 48 w(k) p Fr(\() p Fs(e) p Fl 634 1769 a(\013) p Fi(j) p Fl(Q) p Fi(j) p Fm 801 1805 a(\012) p Fr 22 w(1) -22 b(l) o(\)) p Fs(\037) p Fq 1053 1820 a(\001) p Fr 1116 1805 a(\() p Fs(H) p Fl 1235 1820 a(m) p Fr 1301 1805 a(\() p Fs(L) p Fr(\)\)) p Fm(k) 28 b(\024) p Fs 28 w(M) p Fr 10 w(\() p Fs(\013) q(;) p Fr 17 w(\001\)) p Fs(:) p Ft 418 2004 a(-) p Fm 48 w(k) p Fr(\() p Fs(e) p Fl 634 1967 a(\013) p Fi(j) p Fl(Q) p Fi(j) p Fm 801 2004 a(\012) p Fr 22 w(1) -22 b(l) o(\)) p Fs(\037) p Fq 1053 2019 a(\001) p Fr 1116 2004 a(\() p Fs(H) p Fl 1235 2019 a(m) p Fr 1301 2004 a(\)) p Fm(k) 28 b(\024) p Fs 28 w(M) p Fr 10 w(\() p Fs(\013) q(;) p Fr 17 w(\001\)) p Fs(:) p Ft 418 2202 a(-) p Fm 48 w(k) p Fr(\() p Fs(e) p Fl 634 2166 a(\013) p Fi(j) p Fl(Q) p Fi(j) p Fm 801 2202 a(\012) p Fr 22 w(1) -22 b(l) o(\)) p Fs(\037) p Fq 1053 2217 a(\001) p Fr 1116 2202 a(\() p Fs(H) p Fr 8 w(\)) p Fm(k) 27 b(\024) p Fs 28 w(M) p Fr 10 w(\() p Fs(\013) q(;) p Fr 17 w(\001\)) p Fs(:) p Fv 257 2391 a(This) k(b) s(ound) g(is) f(uniform) f(in) p Fs 29 w(L) p Fv 31 w(and) p Fs 31 w(m:) p Fv 31 w(The) i(pro) s(of) f (is) g(exactly) h(the) g(same) f(as) h(the) g(one) 257 2511 y(of) d(Theorem) h(I) s(I.1) g(of) f([5].) 42 b(The) 29 b(only) f(di\033erence) i(is) e(that) p Fs 28 w(\033) p Fl 2424 2526 a(ess) p Fr 2527 2511 a(\() p Fs(H) p Fl 2646 2526 a(p) p Fr 2685 2511 a(\)) g(=) p Fm 27 w(;) p Fs(;) p Fv 29 w(whic) m(h) h(mak) m(es) 257 2631 y(things) 36 b(easier) h(and) f(in) g(particular) f(one) i(do) s(es) g(not) f(need) h (an) m(y) h(condition) d(on) p Fs 36 w(\013) p Fv 37 w(or) h(on) 257 2752 y(the) d(suprem) m(um) g(of) f(the) h(in) m(terv) -5 b(al) p Fr 31 w(\001) p Fs(:) p Fv 404 2872 a(F) d(or) 31 b(an) m(y) p Fs 34 w(R) d(>) p Fr 28 w(0) p Fs(;) p Fv 32 w(w) m(e) 34 b(no) m(w) f(de\034ne) p Fs 897 3118 a(N) p Fr 10 w(\() p Fm(j) p Fs(x) p Fm(j) 28 b(\024) p Fs 28 w(R) p Fr 1 w(\)) f(:=) p Fp 1538 2982 a(Z) p Fi 1593 3207 a(j) p Fl(x) p Fi(j\024) p Fl(R) p Fs 1802 3118 a(dx) p Fp 1941 2982 a(Z) p Fo 1996 3207 a(R) p Fn 2044 3189 a(n) p Fs 2107 3118 a(dk) 20 b(a) p Fi 2280 3076 a(\003) p Fr 2319 3118 a(\() p Fs(x;) d(k) p Fr 3 w(\)) p Fs(a) p Fr(\() p Fs(x;) g(k) p Fr 3 w(\)) p Fs(;) p Fv 440 w(\(4.9\)) 257 3383 y(and) p Fs 897 3527 a(N) p Fr 10 w(\() p Fm(j) p Fs(x) p Fm(j) p Fs 28 w(>) 28 b(R) p Fr 1 w(\)) g(:=) p Fp 1537 3392 a(Z) p Fi 1593 3617 a(j) p Fl(x) p Fi(j) p Fl(>R) p Fs 1801 3527 a(dx) p Fp 1940 3392 a(Z) p Fo 1995 3617 a(R) p Fn 2043 3598 a(n) p Fs 2107 3527 a(dk) 19 b(a) p Fi 2279 3486 a(\003) p Fr 2319 3527 a(\() p Fs(x;) e(k) p Fr 3 w(\)) p Fs(a) p Fr(\() p Fs(x;) g(k) p Fr 3 w(\)) p Fs(:) p Fv 391 w(\(4.10\)) p Fs 257 3763 a(N) p Fr 10 w(\() p Fm(j) p Fs(x) p Fm(j) 41 b(\024) p Fs 42 w(R) p Fr 1 w(\)) p Fv 40 w(is) f(the) h(n) m(um) m(b) s (er) f(of) g(b) s(osons) h(inside) f(the) g(ball) f(cen) m(tered) j(at) e(the) g(origin) 257 3884 y(and) 31 b(of) g(radius) p Fs 30 w(R) p Fv 32 w(\(in) f(the) p Fs 31 w(x) p Fv 32 w(v) -5 b(ariable\),) 29 b(and) p Fs 31 w(N) p Fr 10 w(\() p Fm(j) p Fs(x) p Fm(j) p Fs 28 w(>) f(R) p Fr 1 w(\)) p Fv 30 w(is) j(the) g(n) m(um) m(b) s(er) g(of) g(b) s(osons) 257 4004 y(outside) j(this) g(ball.) 46 b(W) -8 b(e) 34 b(will) e(pro) m(v) m(e) j(that) f(the) h(n) m(um) m(b) s(er) f(of) g (these) h(\020far) e(a) m(w) m(a) m(y\021) 43 b(b) s(osons) 257 4124 y(deca) m(ys) 33 b(exp) s(onen) m(tially) d(fast) i(with) p Fs 30 w(R) q(:) p Fv 32 w(More) f(precisely) -8 b(,) 32 b(w) m(e) g(ha) m(v) m(e) h(the) e(follo) m(wing) e(esti-) 257 4245 y(mate:) p Fw 257 4433 a(Prop) s(osition) 36 b(4.11.) p Ft 42 w(F) -7 b(or) 33 b(any) p Fs 35 w(\013) c(>) p Fr 27 w(0) p Fs(;) p Ft 35 w(ther) -5 b(e) 35 b(exists) p Fs 34 w(C) p Fr 7 w(\() p Fs(\013) p Fr 1 w(\)) p Fs 27 w(>) p Fr 28 w(0) p Ft 34 w(such) g(that) p Fm 945 4636 a(h) p Fs(\036) p Fl 1042 4651 a(m) p Fr 1108 4636 a(\() p Fs(L) p Fr(\);) 17 b(1) -22 b(l) p Fm 21 w(\012) p Fs 22 w(N) p Fr 10 w(\() p Fm(j) p Fs(x) p Fm(j) p Fs 28 w(>) 28 b(R) p Fr 1 w(\)) p Fs(\036) p Fl 2009 4651 a(m) p Fr 2075 4636 a(\() p Fs(L) p Fr(\)) p Fm(i) g(\024) p Fs 28 w(C) p Fr 7 w(\() p Fs(\013) p Fr 1 w(\)) p Fs(e) p Fi 2650 4595 a(\000) p Fl(\013R) p Fv 3246 4636 a(\(4.11\)) p Ft 257 4838 a(uniformly) 35 b(in) p Fs 34 w(L:) p Fv 404 5027 a(The) h(idea) g(is) f(to) g(adapt) h(the) g(pro) s(of) f(of) g ([5].) 53 b(What) 36 b(is) g(new) g(in) f(our) h(mo) s(del) e(is) h (that) 257 5147 y(w) m(e) 46 b(need) g(an) e(explicit) f(con) m(trol) h (on) h(the) g(n) m(um) m(b) s(er) g(of) f(\020far) f(a) m(w) m(a) m (y\021) 54 b(b) s(osons) 45 b(in) f(the) p Fs 45 w(x) p Fv 257 5268 a(direction,) 39 b(ev) m(en) h(for) e(massiv) m(e) h(b) s (osons.) 62 b(F) -8 b(ot) 37 b(that) h(purp) s(ose,) j(w) m(e) f(use) f (the) g(follo) m(wing) 257 5388 y(lemma) 27 b(whic) m(h) i(comes) f (from) g(the) g(w) m(ell) g(kno) m(wn) i(pullthrough) d(form) m(ula) g (\(see) p Ft 29 w(e.g.) p Fv 41 w([13]\):) 1828 5637 y(18) p 90 rotate dyy eop %%Page: 19 19 19 18 bop Fw 257 573 a(Lemma) 25 b(4.12.) p Fm 34 w(k) p Fr(1) -22 b(l) p Fm -1 w(\012) p Fs 1 w(a) p Fr(\() p Fs(x;) 17 b(k) p Fr 3 w(\)) p Fs(\036) p Fl 1426 588 a(m) p Fr 1493 573 a(\() p Fs(L) p Fr(\)) p Fm(k) 28 b(\024) p Fq 1908 534 a(1) p 1828 550 196 4 v Fl 1828 607 a(!) p Fn 1872 615 a(m) p Fq 1931 607 a(\() p Fl(k) p Fq 2 w(\)) p Fm 2034 573 a(k) p Fs(\032) p Fq 2134 588 a(1) p Fr 2174 573 a(\() p Fs(x) p Fm 1 w(\000) p Fs 1 w(Q) p Fr(\)) p Fs(j) p Fl 2501 588 a(L) p Fr 2554 573 a(\() p Fs(x) p Fr(\)) p Fq 2776 525 a(^) p Fl -41 w(\032) p Fj 2806 534 a(2) p Fq 2841 525 a(\() p Fl(k) p Fq 2 w(\)) p 2695 550 315 4 v Fm 2695 562 a(p) p 2778 562 232 4 v Fq 2778 628 a(2) p Fl(!) p Fn 2857 636 a(m) p Fq 2916 628 a(\() p Fl(k) p Fq 2 w(\)) p Fm 3020 573 a(\012) p Fr 1 w(1) -22 b(l) p Fs -1 w(\036) p Fl 3210 588 a(m) p Fr 3276 573 a(\() p Fs(L) p Fr(\)) p Fm(k) p Fs(:) p Fw 257 798 a(Pro) s(of) 37 b(of) h(Prop) s(osition) d(4.11) j (:) p Fv 81 w(Let) p Fs 33 w(\013) 28 b(>) p Fr 27 w(0) p Fs(;) p Fm 501 996 a(h) p Fs(\036) p Fl 598 1011 a(m) p Fr 664 996 a(\() p Fs(L) p Fr(\);) 17 b(1) -22 b(l) p Fm 21 w(\012) p Fs 22 w(N) p Fr 10 w(\() p Fm(j) p Fs(x) p Fm(j) p Fs 28 w(>) 28 b(R) p Fr 1 w(\)) p Fs(\036) p Fl 1565 1011 a(m) p Fr 1631 996 a(\() p Fs(L) p Fr(\)) p Fm(i) p Fr 341 1189 a(=) p Fp 501 1054 a(Z) p Fi 556 1279 a(j) p Fl(x) p Fi(j) p Fl(>R) p Fs 764 1189 a(dx) p Fp 904 1054 a(Z) p Fo 959 1279 a(R) p Fn 1007 1260 a(n) p Fs 1070 1189 a(dk) p Fm 20 w(k) p Fr(1) -22 b(l) p Fm 20 w(\012) p Fs 23 w(a) p Fr(\() p Fs(x;) 17 b(k) p Fr 3 w(\)) p Fs(\036) p Fl 1755 1204 a(m) p Fr 1822 1189 a(\() p Fs(L) p Fr(\)) p Fm(k) p Fq 2014 1148 a(2) p Fm 340 1471 a(\024) p Fp 501 1336 a(Z) p Fi 556 1561 a(j) p Fl(x) p Fi(j) p Fl(>R) p Fs 764 1471 a(dx) p Fp 904 1336 a(Z) p Fo 959 1561 a(R) p Fn 1007 1542 a(n) p Fs 1070 1471 a(dk) p Fr 1306 1404 a(1) p 1202 1448 258 4 v Fs 1202 1540 a(!) p Fq 1267 1511 a(2) p Fl 1263 1564 a(m) p Fr 1329 1540 a(\() p Fs(k) p Fr 3 w(\)) p Fm 1469 1471 a(k) p Fs(\032) p Fq 1569 1486 a(1) p Fr 1608 1471 a(\() p Fs(x) p Fm 23 w(\000) p Fs 23 w(Q) p Fr(\)) p Fs(j) p Fl 1979 1486 a(L) p Fr 2031 1471 a(\() p Fs(x) p Fr(\)) 2274 1404 y(^) p Fs -58 w(\032) p Fq 2315 1419 a(2) p Fr 2355 1404 a(\() p Fs(k) p Fr 3 w(\)) p 2172 1448 406 4 v Fp 2172 1468 a(p) p 2272 1468 307 4 v Fr 2272 1553 a(2) p Fs(!) p Fl 2382 1568 a(m) p Fr 2448 1553 a(\() p Fs(k) p Fr 3 w(\)) p Fm 2610 1471 a(\012) p Fr 23 w(1) -22 b(l) p Fs -1 w(\036) p Fl 2822 1486 a(m) p Fr 2888 1471 a(\() p Fs(L) p Fr(\)) p Fm(k) p Fq 3080 1430 a(2) p Fm 340 1763 a(\024) p Fp 501 1628 a(Z) p Fi 556 1853 a(j) p Fl(x) p Fi(j) p Fl(>R) p Fs 764 1763 a(dx) p Fp 904 1628 a(Z) p Fo 959 1853 a(R) p Fn 1007 1834 a(n) p Fs 1070 1763 a(dk) p Fm 1202 1696 a(j) p Fr 9 w(^) p Fs -58 w(\032) p Fq 1280 1711 a(2) p Fr 1319 1696 a(\() p Fs(k) p Fr 3 w(\)) p Fm(j) p Fq 1477 1660 a(2) p 1202 1741 315 4 v Fr 1206 1832 a(2) p Fs(!) p Fq 1320 1803 a(3) p Fl 1316 1856 a(m) p Fr 1382 1832 a(\() p Fs(k) p Fr 3 w(\)) p Fm 1526 1763 a(k) p Fs(\032) p Fq 1626 1778 a(1) p Fr 1666 1763 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(Q) p Fr(\)) p Fs(j) p Fl 2036 1778 a(L) p Fr 2088 1763 a(\() p Fs(x) p Fr(\)) p Fs(e) p Fi 2264 1722 a(\000) p Fl(\013) p Fi(j) p Fl(Q) p Fi(j) p Fm 2464 1763 a(k) p Fq 2514 1722 a(2) p Fi 2514 1789 a(B) p Fq 2 w(\() p Fl(L) p Fj 2637 1770 a(2) p Fq 2672 1789 a(\)) p Fm 2726 1763 a(\002) 22 b(k) p Fs(e) p Fl 2920 1722 a(\013) p Fi(j) p Fl(Q) p Fi(j) p Fm 3087 1763 a(\012) p Fr 22 w(1) -22 b(l) p Fs -1 w(\036) p Fl 3298 1778 a(m) p Fr 3364 1763 a(\() p Fs(L) p Fr(\)) p Fm(k) p Fq 3556 1722 a(2) p Fs 3596 1763 a(:) p Fv 404 2024 a(The) 41 b(function) p Fr 48 w(^) p Fs -58 w(\032) p Fq 1051 2039 a(2) p Fv 1131 2024 a(is) f(a) g(Sc) m(h) m(w) m(artz) i(function) d (and) p Fs 41 w(!) p Fl 2396 2039 a(m) p Fv 2502 2024 a(is) h(b) s(ounded) h(from) d(b) s(elo) m(w) 257 2145 y(b) m(y) p Fs 36 w(m) 30 b(>) p Fr 31 w(0) p Fs(;) p Fv 34 w(so) 35 b(the) f(in) m(tegral) f(with) h(resp) s(ect) h(to) f (the) p Fs 35 w(k) p Fv 37 w(v) -5 b(ariable) 33 b(con) m(v) m(erges.) 50 b(No) m(w) 35 b(w) m(e) 257 2265 y(recall) d(that) h(the) g (function) p Fs 32 w(\032) p Fq 1330 2280 a(1) p Fv 1403 2265 a(has) h(compact) e(supp) s(ort) h(in) g(the) g(ball) e(of) i (radius) p Fs 32 w(R) p Fq 3281 2280 a(1) p Fs 3321 2265 a(;) p Fv 33 w(so,) 257 2386 y(for) f(an) m(y) i(giv) m(en) p Fs 32 w(x) p Fm 28 w(2) p Fg 28 w(R) p Fl 1088 2349 a(d) p Fs 1135 2386 a(;) p Fv 32 w(w) m(e) g(ha) m(v) m(e) p Fm 350 2583 a(k) p Fs(\032) p Fq 450 2598 a(1) p Fr 490 2583 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(Q) p Fr(\)) p Fs(e) p Fi 865 2542 a(\000) p Fl(\013) p Fi(j) p Fl(Q) p Fi(j) p Fm 1064 2583 a(k) p Fi 1114 2600 a(B) p Fq 2 w(\() p Fl(L) p Fj 1237 2581 a(2) p Fq 1272 2600 a(\)) p Fr 1387 2583 a(=) 92 b(sup) p Fl 1545 2673 a(q) p Fi 2 w(2) p Fo(R) p Fn 1674 2654 a(d) p Fm 1727 2583 a(j) p Fs(\032) p Fq 1805 2598 a(1) p Fr 1844 2583 a(\() p Fs(x) p Fm 23 w(\000) p Fs 23 w(q) p Fr 4 w(\)) p Fs(e) p Fi 2190 2542 a(\000) p Fl(\013) p Fi(j) p Fl(q) p Fi 2 w(j) p Fm 2367 2583 a(j) p Fr 1387 2820 a(=) 164 b(sup) p Fi 1545 2905 a(j) p Fl(q) p Fi 2 w(\000) p Fl(x) p Fi(j\024) p Fl(R) p Fj 1822 2914 a(1) p Fm 1872 2820 a(j) p Fs(\032) p Fq 1950 2835 a(1) p Fr 1990 2820 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(q) p Fr 4 w(\)) p Fs(e) p Fi 2335 2778 a(\000) p Fl(\013) p Fi(j) p Fl(q) p Fi 2 w(j) p Fm 2512 2820 a(j) 55 b(\024) 29 b(k) p Fs(\032) p Fq 2801 2835 a(1) p Fm 2840 2820 a(k) p Fi 2890 2835 a(1) p Fs 2965 2820 a(e) p Fl 3010 2778 a(\013R) p Fj 3108 2787 a(1) p Fs 3147 2820 a(e) p Fi 3192 2778 a(\000) p Fl(\013) p Fi(j) p Fl(x) p Fi(j) p Fs 3376 2820 a(:) p Fv 257 3086 a(Th) m(us) p Fp 270 3171 a(Z) p Fi 325 3397 a(j) p Fl(x) p Fi(j) p Fl(>R) p Fs 533 3307 a(dx) p Fm 17 w(k) p Fs(\032) p Fq 756 3322 a(1) p Fr 796 3307 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(Q) p Fr(\)) p Fs(e) p Fi 1171 3266 a(\000) p Fl(\013) p Fi(j) p Fl(Q) p Fi(j) p Fm 1370 3307 a(k) p Fq 1420 3266 a(2) p Fi 1420 3333 a(B) p Fq 2 w(\() p Fl(L) p Fj 1543 3314 a(2) p Fq 1578 3333 a(\)) p Fm 1637 3307 a(\024) g(k) p Fs(\032) p Fq 1843 3322 a(1) p Fm 1882 3307 a(k) p Fq 1932 3266 a(2) p Fi 1932 3331 a(1) p Fs 2007 3307 a(e) p Fq 2052 3266 a(2) p Fl(\013R) p Fj 2185 3275 a(1) p Fp 2241 3171 a(Z) p Fi 2296 3397 a(j) p Fl(x) p Fi(j) p Fl(>R) p Fs 2504 3307 a(dx) 17 b(e) p Fi 2672 3266 a(\000) p Fq(2) p Fl(\013) p Fi(j) p Fl(x) p Fi(j) p Fm 2919 3307 a(\024) p Fs 28 w(K) p Fr 7 w(\() p Fs(\013) p Fr 1 w(\)) p Fs(e) p Fi 3298 3266 a(\000) p Fl(\013R) p Fs 3456 3307 a(:) p Fv 257 3567 a(And) 33 b(so,) g(\034nally) -8 b(,) p Fm 528 3765 a(h) p Fs(\036) p Fl 625 3780 a(m) p Fr 691 3765 a(\() p Fs(L) p Fr(\);) 17 b(1) -22 b(l) p Fm 21 w(\012) p Fs 22 w(N) p Fr 10 w(\() p Fm(j) p Fs(x) p Fm(j) p Fs 28 w(>) 28 b(R) p Fr 1 w(\)) p Fs(\036) p Fl 1592 3780 a(m) p Fr 1658 3765 a(\() p Fs(L) p Fr(\)) p Fm(i) g(\024) p Fs 28 w(K) p Fi 2062 3724 a(0) p Fr 2085 3765 a(\() p Fs(\013) p Fr 1 w(\)) p Fs(e) p Fi 2269 3724 a(\000) p Fl(\013R) p Fm 2427 3765 a(k) p Fs(e) p Fl 2522 3724 a(\013) p Fi(j) p Fl(Q) p Fi(j) p Fm 2688 3765 a(\012) p Fr 23 w(1) -22 b(l) p Fs -1 w(\036) p Fl 2900 3780 a(m) p Fr 2966 3765 a(\() p Fs(L) p Fr(\)) p Fm(k) p Fq 3158 3724 a(2) p Fs 3198 3765 a(:) p Fv 404 3962 a(But,) 28 b(for) f(an) m(y) p Fs 28 w(L;) p Fv 28 w(w) m(e) h(ha) m(v) m(e) p Fs 29 w(E) p Fl 1494 3977 a(m) p Fr 1561 3962 a(\() p Fs(L) p Fr(\)) p Fm 27 w(\024) p Fs 29 w(E) p Fq 1914 3926 a(0) p Fl 1908 3987 a(p) p Fv 1980 3962 a(where) p Fs 29 w(E) p Fq 2335 3926 a(0) p Fl 2329 3987 a(p) p Fv 2402 3962 a(is) e(the) i(ground) f(state) h(energy) 257 4083 y(of) p Fs 32 w(H) p Fl 449 4098 a(p) p Fs 489 4083 a(:) p Fv 33 w(Indeed,) 34 b(if) p Fs 31 w( ) p Fq 1049 4046 a(0) p Fl 1045 4107 a(p) p Fv 1121 4083 a(is) e(the) h(ground) f(state) h(of) p Fs 32 w(H) p Fl 2148 4098 a(p) p Fs 2188 4083 a(;) p Fv 33 w(w) m(e) g(ha) m(v) m(e) p Fs 1014 4289 a(E) p Fl 1086 4304 a(m) p Fr 1152 4289 a(\() p Fs(L) p Fr(\)) p Fm 28 w(\024) 28 b(h) p Fs( ) p Fq 1533 4248 a(0) p Fl 1529 4314 a(p) p Fm 1595 4289 a(\012) p Fr 22 w(\012;) p Fs 17 w(H) p Fl 1889 4304 a(m) p Fr 1956 4289 a(\() p Fs(L) p Fr(\)) p Fs 28 w( ) p Fq 2193 4248 a(0) p Fl 2189 4314 a(p) p Fm 2254 4289 a(\012) p Fr 23 w(\012) p Fm(i) p Fr 28 w(=) p Fs 27 w(E) p Fq 2672 4248 a(0) p Fl 2666 4314 a(p) p Fs 2712 4289 a(:) p Fv 257 4487 a(T) -8 b(ak) m(e) 42 b(no) m(w) p Fr 41 w(\001) g(=]) p Fm 27 w(\000) 28 b(1) p Fs(;) 17 b(E) p Fq 1292 4451 a(0) p Fl 1286 4511 a(p) p Fr 1331 4487 a(]) p Fs(;) p Fv 41 w(one) 41 b(can) g(then) g(write) p Fs 40 w(\036) p Fl 2345 4502 a(m) p Fr 2411 4487 a(\() p Fs(L) p Fr(\)) h(=) p Fs 41 w(\037) p Fq 2773 4502 a(\001) p Fr 2836 4487 a(\() p Fs(H) p Fl 2955 4502 a(m) p Fr 3021 4487 a(\() p Fs(L) p Fr(\)\)) p Fs(\036) p Fl 3259 4502 a(m) p Fr 3326 4487 a(\() p Fs(L) p Fr(\)) p Fs(;) p Fv 257 4607 a(and) 33 b(so) p Fm 676 4805 a(k) p Fs(e) p Fl 771 4764 a(\013) p Fi(j) p Fl(Q) p Fi(j) p Fm 937 4805 a(\012) p Fr 23 w(1) -22 b(l) p Fs -2 w(\036) p Fl 1148 4820 a(m) p Fr 1215 4805 a(\() p Fs(L) p Fr(\)) p Fm(k) p Fq 1407 4764 a(2) p Fm 1529 4805 a(\024) 84 b(k) p Fs(e) p Fl 1785 4764 a(\013) p Fi(j) p Fl(Q) p Fi(j) p Fm 1951 4805 a(\012) p Fr 23 w(1) -22 b(l) p Fs 26 w(\037) p Fq 2193 4820 a(\001) p Fr 2256 4805 a(\() p Fs(H) p Fl 2375 4820 a(m) p Fr 2441 4805 a(\() p Fs(L) p Fr(\)\)) p Fm(k) p Fq 2671 4764 a(2) p Fm 2711 4805 a(k) p Fs(\036) p Fl 2819 4820 a(m) p Fr 2885 4805 a(\() p Fs(L) p Fr(\)) p Fm(k) 1529 4950 y(\024) p Fs 84 w(M) p Fr 10 w(\() p Fs(\013) q(;) p Fr 17 w(\001\)) p Fq 2058 4909 a(2) p Fs 2097 4950 a(;) p Fv 257 5147 a(whic) m(h) 33 b(ends) h(the) f(pro) s(of.) p Fc 2244 w(2) p Fv 404 5388 a(W) -8 b(e) 33 b(\034nally) e(giv) m(e) h (an) h(estimate) e(similar) f(to) i(the) h(one) g(of) f(Prop) s (osition) e(4.6.) 1828 5637 y(19) p 90 rotate dyy eop %%Page: 20 20 20 19 bop Fw 257 573 a(Prop) s(osition) 33 b(4.13.) p Ft 40 w(L) -5 b(et) p Fr 33 w(\001) p Ft 33 w(and) p Fs 32 w(\013) p Ft 34 w(b) g(e) 32 b(as) h(in) f(Pr) -5 b(op) g(osition) 32 b(4.10,) g(then) h(ther) -5 b(e) 33 b(exists) p Fs 257 693 a(K) p Fr 7 w(\() p Fs(\013) q(;) p Fr 17 w(\001\)) p Ft 35 w(such) i(that) p Fm 795 913 a(k) p Fs(\037) p Fq 906 928 a(\001) p Fr 969 913 a(\() p Fs(H) p Fl 1088 928 a(m) p Fr 1154 913 a(\)\() p Fs(W) p Fl 1322 928 a(m) p Fm 1411 913 a(\000) p Fs 22 w(W) p Fl 1602 928 a(m) p Fr 1669 913 a(\() p Fs(L) p Fr(\)\)) p Fs(\037) p Fq 1910 928 a(\001) p Fr 1973 913 a(\() p Fs(H) p Fl 2092 928 a(m) p Fr 2159 913 a(\)) p Fm(k) 27 b(\024) p Fs 28 w(K) p Fr 7 w(\() p Fs(\013) q(;) p Fr 17 w(\001\)) p Fs(e) p Fi 2778 872 a(\000) p Fl(\013L) p Fs 2931 913 a(:) p Fw 257 1133 a(Pro) s(of) g(:) p Fv 27 w(W) -8 b(e) 25 b(follo) m(w) c(the) j(sc) m(heme) h(of) e(the) i (pro) s(of) d(of) h(Prop) s(osition) f(4.10) h(using) h(estimates) 257 1254 y(similar) 30 b(to) i(the) h(ones) g(of) f(the) h(previous) g (prop) s(osition.) 41 b(Let) p Fs 33 w(\036;) 17 b( ) p Fm 31 w(2) 28 b(H) p Fs 1 w(;) p Fm 513 1474 a(jh) p Fs(\036) p Fr(;) p Fs 17 w(\037) p Fq 743 1489 a(\001) p Fr 805 1474 a(\() p Fs(H) p Fl 924 1489 a(m) p Fr 990 1474 a(\)\() p Fs(W) p Fl 1158 1489 a(m) p Fm 1247 1474 a(\000) p Fs 22 w(W) p Fl 1438 1489 a(m) p Fr 1505 1474 a(\() p Fs(L) p Fr(\)\)) p Fs(\037) p Fq 1746 1489 a(\001) p Fr 1809 1474 a(\() p Fs(H) p Fl 1928 1489 a(m) p Fr 1995 1474 a(\)) p Fs( ) p Fm 4 w(ij) 352 1667 y(\024) 84 b(jh) p Fr(\() p Fs(e) p Fq 663 1626 a(2) p Fl(\013) p Fi(j) p Fl(Q) p Fi(j) p Fm 864 1667 a(\012) p Fr 23 w(1) -22 b(l) n(\)) p Fs(\037) p Fq 1116 1682 a(\001) p Fr 1179 1667 a(\() p Fs(H) p Fl 1298 1682 a(m) p Fr 1365 1667 a(\)) p Fs(\036) p Fr(;) p Fp 1505 1557 a(\020) 1580 1532 y(Z) p Fi 1636 1757 a(j) p Fl(x) p Fi(j) p Fl(>) p Fn 1781 1730 a(L) p 1779 1742 43 3 v Fj 1785 1783 a(2) p Fs 1852 1667 a(dx) p Fp 1992 1532 a(Z) p Fo 2047 1757 a(R) p Fn 2095 1738 a(n) p Fs 2158 1667 a(dk) 20 b(e) p Fi 2325 1626 a(\000) p Fq(2) p Fl(\013) p Fi(j) p Fl(Q) p Fi(j) p Fs 2559 1667 a(\032) p Fq 2609 1682 a(1) p Fr 2649 1667 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(Q) p Fr(\)) 2978 1645 y(\026) p Fs 2979 1667 a(j) p Fl 3019 1682 a(L) p Fr 3072 1667 a(\() p Fs(x) p Fr(\)) p Fm 1878 1967 a(\002) p Fr 2068 1873 a(\026) 2068 1899 y(^) p Fs -58 w(\032) p Fq 2109 1914 a(2) p Fr 2148 1899 a(\() p Fs(k) p Fr 3 w(\)) p 1965 1944 406 4 v Fp 1965 1964 a(p) p 2065 1964 307 4 v Fr 2065 2049 a(2) p Fs(!) p Fl 2175 2064 a(m) p Fr 2241 2049 a(\() p Fs(k) p Fr 3 w(\)) p Fm 2404 1967 a(\012) p Fs 22 w(a) p Fr(\() p Fs(x;) d(k) p Fr 3 w(\)) p Fp 2783 1856 a(\021) p Fs 2843 1967 a(\037) p Fq 2904 1982 a(\001) p Fr 2967 1967 a(\() p Fs(H) p Fl 3086 1982 a(m) p Fr 3152 1967 a(\)) p Fs( ) p Fm 4 w(ij) p Fr 708 2248 a(+) p Fm(jh) p Fr(\() p Fs(e) p Fq 934 2207 a(2) p Fl(\013) p Fi(j) p Fl(Q) p Fi(j) p Fm 1135 2248 a(\012) p Fr 23 w(1) -22 b(l) n(\)) p Fs(\037) p Fq 1387 2263 a(\001) p Fr 1450 2248 a(\() p Fs(H) p Fl 1569 2263 a(m) p Fr 1636 2248 a(\)) p Fs( ) p Fr 4 w(;) p Fp 1785 2137 a(\020) 1860 2112 y(Z) p Fi 1916 2338 a(j) p Fl(x) p Fi(j) p Fl(>) p Fn 2061 2311 a(L) p 2060 2323 43 3 v Fj 2066 2364 a(2) p Fs 2132 2248 a(dx) p Fp 2272 2112 a(Z) p Fo 2327 2338 a(R) p Fn 2375 2319 a(n) p Fs 2438 2248 a(dk) 20 b(e) p Fi 2605 2207 a(\000) p Fq(2) p Fl(\013) p Fi(j) p Fl(Q) p Fi(j) p Fs 2839 2248 a(\032) p Fq 2889 2263 a(1) p Fr 2929 2248 a(\() p Fs(x) p Fm 23 w(\000) p Fs 22 w(Q) p Fr(\)) 3258 2225 y(\026) p Fs 3259 2248 a(j) p Fl 3299 2263 a(L) p Fr 3352 2248 a(\() p Fs(x) p Fr(\)) p Fm 1976 2547 a(\002) p Fr 2165 2453 a(\026) 2165 2480 y(^) p Fs -58 w(\032) p Fq 2206 2495 a(2) p Fr 2246 2480 a(\() p Fs(k) p Fr 3 w(\)) p 2063 2524 406 4 v Fp 2063 2544 a(p) p 2163 2544 307 4 v Fr 2163 2629 a(2) p Fs(!) p Fl 2273 2644 a(m) p Fr 2339 2629 a(\() p Fs(k) p Fr 3 w(\)) p Fm 2501 2547 a(\012) p Fs 23 w(a) p Fr(\() p Fs(x;) d(k) p Fr 3 w(\)) p Fp 2881 2436 a(\021) p Fs 2940 2547 a(\037) p Fq 3001 2562 a(\001) p Fr 3065 2547 a(\() p Fs(H) p Fl 3184 2562 a(m) p Fr 3250 2547 a(\)) p Fs(\036) p Fm(ij) p Fs(:) p Fv 257 2836 a(W) -8 b(e) 33 b(consider) g(only) f(the) h(\034rst) g(term,) f(the) h(other) g(one) g (b) s(eing) e(similar.) p Fm 501 3099 a(jh) p Fr(\() p Fs(e) p Fq 651 3058 a(2) p Fl(\013) p Fi(j) p Fl(Q) p Fi(j) p Fm 852 3099 a(\012) p Fr 23 w(1) -22 b(l) n(\)) p Fs(\037) p Fq 1104 3114 a(\001) p Fr 1167 3099 a(\() p Fs(H) p Fl 1286 3114 a(m) p Fr 1353 3099 a(\)) p Fs(\036) p Fr(;) p Fp 1493 2988 a(\020) 1568 2963 y(Z) p Fi 1624 3189 a(j) p Fl(x) p Fi(j) p Fl(>) p Fn 1769 3162 a(L) p 1767 3174 43 3 v Fj 1773 3215 a(2) p Fs 1840 3099 a(dx) p Fp 1980 2963 a(Z) p Fo 2035 3189 a(R) p Fn 2083 3170 a(n) p Fs 2146 3099 a(dk) 20 b(e) p Fi 2313 3058 a(\000) p Fq(2) p Fl(\013) p Fi(j) p Fl(Q) p Fi(j) p Fs 2547 3099 a(\032) p Fq 2597 3114 a(1) p Fr 2637 3099 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(Q) p Fr(\)) 2966 3077 y(\026) p Fs 2967 3099 a(j) p Fl 3007 3114 a(L) p Fr 3060 3099 a(\() p Fs(x) p Fr(\)) p Fm 1866 3398 a(\002) p Fr 2056 3305 a(\026) 2056 3331 y(^) p Fs -58 w(\032) p Fq 2097 3346 a(2) p Fr 2136 3331 a(\() p Fs(k) p Fr 3 w(\)) p 1953 3375 406 4 v Fp 1953 3395 a(p) p 2053 3395 307 4 v Fr 2053 3480 a(2) p Fs(!) p Fl 2163 3495 a(m) p Fr 2229 3480 a(\() p Fs(k) p Fr 3 w(\)) p Fm 2391 3398 a(\012) p Fs 23 w(a) p Fr(\() p Fs(x;) d(k) p Fr 3 w(\)) p Fp 2771 3288 a(\021) p Fs 2831 3398 a(\037) p Fq 2892 3413 a(\001) p Fr 2955 3398 a(\() p Fs(H) p Fl 3074 3413 a(m) p Fr 3140 3398 a(\)) p Fs( ) p Fm 4 w(ij) 340 3680 y(\024) 84 b(k) p Fr(\() p Fs(e) p Fq 634 3638 a(2) p Fl(\013) p Fi(j) p Fl(Q) p Fi(j) p Fm 835 3680 a(\012) p Fr 23 w(1) -22 b(l) o(\)) p Fs(\037) p Fq 1088 3695 a(\001) p Fr 1151 3680 a(\() p Fs(H) p Fl 1270 3695 a(m) p Fr 1336 3680 a(\)) p Fs(\036) p Fm(k) 22 b(\002) g(k) p Fr(\() p Fp 1691 3544 a(Z) p Fi 1747 3770 a(j) p Fl(x) p Fi(j) p Fl(>) p Fn 1892 3742 a(L) p 1890 3754 43 3 v Fj 1896 3796 a(2) p Fs 1963 3680 a(dx) p Fp 2103 3544 a(Z) p Fo 2158 3770 a(R) p Fn 2206 3751 a(n) p Fs 2269 3680 a(dk) d(e) p Fi 2435 3638 a(\000) p Fq(2) p Fl(\013) p Fi(j) p Fl(Q) p Fi(j) p Fs 2670 3680 a(\032) p Fq 2720 3695 a(1) p Fr 2760 3680 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(Q) p Fr(\)) 3089 3657 y(\026) p Fs 3090 3680 a(j) p Fl 3130 3695 a(L) p Fr 3183 3680 a(\() p Fs(x) p Fr(\)) p Fm 1964 3979 a(\002) p Fr 2153 3885 a(\026) 2153 3911 y(^) p Fs -58 w(\032) p Fq 2194 3926 a(2) p Fr 2234 3911 a(\() p Fs(k) p Fr 3 w(\)) p 2051 3956 406 4 v Fp 2051 3976 a(p) p 2151 3976 307 4 v Fr 2151 4061 a(2) p Fs(!) p Fl 2261 4076 a(m) p Fr 2327 4061 a(\() p Fs(k) p Fr 3 w(\)) p Fm 2489 3979 a(\012) p Fs 23 w(a) p Fr(\() p Fs(x;) e(k) p Fr 3 w(\)\)) p Fs(\037) p Fq 2968 3994 a(\001) p Fr 3031 3979 a(\() p Fs(H) p Fl 3150 3994 a(m) p Fr 3216 3979 a(\)) p Fs( ) p Fm 4 w(k) 340 4332 y(\024) p Fs 84 w(M) p Fr 10 w(\(2) p Fs(\013) q(;) p Fr 17 w(\001\)) p Fm(k) p Fs(\036) p Fm(k) p Fp 1093 4162 a(") 1150 4196 y(Z) p Fi 1205 4422 a(j) p Fl(x) p Fi(j) p Fl(>) p Fn 1350 4395 a(L) p 1349 4407 43 3 v Fj 1355 4448 a(2) p Fs 1422 4332 a(dx) p Fp 1561 4196 a(Z) p Fo 1617 4422 a(R) p Fn 1665 4403 a(n) p Fs 1728 4332 a(dk) p Fm 19 w(k) p Fs(e) p Fi 1944 4291 a(\000) p Fq(2) p Fl(\013) p Fi(j) p Fl(Q) p Fi(j) p Fs 2179 4332 a(\032) p Fq 2229 4347 a(1) p Fr 2268 4332 a(\() p Fs(x) p Fm 23 w(\000) p Fs 23 w(Q) p Fr(\)) 2598 4309 y(\026) p Fs 2599 4332 a(j) p Fl 2639 4347 a(L) p Fr 2691 4332 a(\() p Fs(x) p Fr(\)) 2934 4238 y(\026) 2934 4264 y(^) p Fs -57 w(\032) p Fq 2976 4279 a(2) p Fr 3015 4264 a(\() p Fs(k) p Fr 3 w(\)) p 2832 4309 406 4 v Fp 2832 4329 a(p) p 2932 4329 307 4 v Fr 2932 4414 a(2) p Fs(!) p Fl 3042 4429 a(m) p Fr 3108 4414 a(\() p Fs(k) p Fr 3 w(\)) p Fm 3248 4332 a(k) p Fq 3298 4291 a(2) p Fi 3298 4358 a(B) p Fq 2 w(\() p Fl(L) p Fj 3421 4339 a(2) p Fq 3456 4358 a(\)) p Fp 3488 4162 a(#) p Fj 3556 4157 a(1) p 3556 4169 31 3 v 3556 4210 a(2) p Fm 1476 4697 a(\002) p Fp 1570 4527 a(") 1628 4561 y(Z) p Fi 1684 4787 a(j) p Fl(x) p Fi(j) p Fl(>) p Fn 1829 4760 a(L) p 1828 4772 43 3 v Fj 1834 4813 a(2) p Fs 1900 4697 a(dx) p Fp 2040 4561 a(Z) p Fo 2095 4787 a(R) p Fn 2143 4768 a(n) p Fs 2206 4697 a(dk) p Fm 20 w(k) p Fr(1) -22 b(l) p Fm 20 w(\012) p Fs 23 w(a) p Fr(\() p Fs(x;) 17 b(k) p Fr 3 w(\)) p Fs(\037) p Fq 2894 4712 a(\001) p Fr 2957 4697 a(\() p Fs(H) p Fl 3076 4712 a(m) p Fr 3143 4697 a(\)) p Fs( ) p Fm 4 w(k) p Fq 3298 4656 a(2) p Fp 3337 4527 a(#) p Fj 3405 4522 a(1) p 3405 4534 31 3 v 3405 4575 a(2) p Fm 340 4955 a(\024) p Fs 84 w(C) p Fr 7 w(\() p Fs(\013) q(;) p Fr 17 w(\001\)) p Fs(e) p Fi 887 4914 a(\000) p Fl(\013L) p Fm 1039 4955 a(k) p Fs(\036) p Fm(k) k(\002) i(k) p Fr(\(1) -22 b(l) p Fm 21 w(\012) p Fs 22 w(N) p Fr 10 w(\)) p Fj 1717 4887 a(1) p 1718 4899 V 1718 4940 a(2) p Fs 1762 4955 a(\037) p Fq 1823 4970 a(\001) p Fr 1886 4955 a(\() p Fs(H) p Fl 2005 4970 a(m) p Fr 2072 4955 a(\)) p Fs( ) p Fm 4 w(k) p Fs(:) p Fv 257 5175 a(The) 34 b(result) e(then) h(follo) m (ws) f(as) g(in) g(the) h(discrete) g(case.) p Fc 1190 w(2) p Fv 1828 5637 a(20) p 90 rotate dyy eop %%Page: 21 21 21 20 bop Fw 257 573 a(4.2.3) 113 b(Remo) m(ving) 36 b(the) h(cuto\033) 257 758 y(Prop) s(osition) f(4.14.) p Fs 42 w(H) p Fl 1216 773 a(m) p Fr 1282 758 a(\() p Fs(L) p Fr(\)) p Ft 35 w(c) -5 b(onver) g(ges) 34 b(to) p Fs 35 w(H) p Fl 2092 773 a(m) p Ft 2193 758 a(in) g(the) h(str) -5 b(ong) 35 b(r) -5 b(esolvent) 34 b(sens.) p Fw 257 939 a(Pro) s(of) 49 b(:) p Fv 50 w(As) 43 b(for) g(Prop) s(osition) e(4.4,) k(it) d(su\036ces) j(to) e(sho) m(w) h(that) p Fs 42 w(H) p Fl 2839 954 a(m) p Fr 2906 939 a(\() p Fs(L) p Fr(\)) p Fv 43 w(con) m(v) m(erges) 257 1060 y(strongly) 32 b(to) p Fs 33 w(H) p Fl 835 1075 a(m) p Fv 901 1060 a(.) 44 b(Let) p Fs 32 w( ) p Fm 32 w(2) p Fs 28 w(D) p Fr 3 w(\() p Fs(H) p Fq 1546 1023 a(0) p Fl 1538 1084 a(m) p Fr 1604 1060 a(\)) p Fs(;) p Fm 600 1254 a(k) p Fs(H) p Fl 731 1269 a(m) p Fs 797 1254 a( ) p Fm 26 w(\000) p Fs 23 w(H) p Fl 1067 1269 a(m) p Fr 1133 1254 a(\() p Fs(L) p Fr(\)) p Fs( ) p Fm 4 w(k) p Fq 1392 1213 a(2) p Fi 1392 1279 a(H) p Fr 441 1399 a(=) p Fm 83 w(k) p Fs(W) p Fl 742 1414 a(m) p Fs 808 1399 a( ) p Fm 26 w(\000) p Fs 23 w(W) p Fl 1089 1414 a(m) p Fr 1156 1399 a(\() p Fs(L) p Fr(\)) p Fs( ) p Fm 4 w(k) p Fq 1415 1358 a(2) p Fi 1415 1424 a(H) p Fr 441 1600 a(=) p Fp 600 1464 a(Z) p Fi 655 1690 a(j) p Fl(q) p Fi 2 w(j) p Fl(>) p Fn 794 1662 a(L) p 793 1674 43 3 v Fj 799 1716 a(2) p Fi 845 1690 a(\000) p Fl(R) p Fj 953 1699 a(1) p Fs 1009 1600 a(dq) p Fp 1123 1485 a(\015) 1123 1545 y(\015) 1123 1605 y(\015) 1178 1489 y(\020) 1254 1464 y(Z) p Fi 1310 1690 a(j) p Fl(x) p Fi(j) p Fl(>) p Fn 1455 1662 a(L) p 1453 1674 V Fj 1459 1716 a(2) p Fs 1526 1600 a(dx) p Fp 1666 1464 a(Z) p Fo 1721 1690 a(R) p Fn 1769 1671 a(n) p Fs 1832 1600 a(dk) 19 b(\032) p Fq 2003 1615 a(1) p Fr 2043 1600 a(\() p Fs(x) p Fm 23 w(\000) p Fs 22 w(q) p Fr 4 w(\)) 2342 1577 y(\026) p Fs 2343 1600 a(j) p Fl 2383 1615 a(L) p Fr 2436 1600 a(\() p Fs(x) p Fr(\)) 2679 1532 y(^) p Fs -58 w(\032) p Fq 2720 1547 a(2) p Fr 2760 1532 a(\() p Fs(k) p Fr 3 w(\)) p 2577 1577 406 4 v Fp 2577 1597 a(p) p 2677 1597 307 4 v Fr 2677 1682 a(2) p Fs(!) p Fl 2787 1697 a(m) p Fr 2853 1682 a(\() p Fs(k) p Fr 3 w(\)) p Fs 2993 1600 a(a) p Fi 3044 1558 a(\003) p Fr 3083 1600 a(\() p Fs(x;) e(k) p Fr 3 w(\)) 600 1899 y(+) p Fp 693 1763 a(Z) p Fo 748 1989 a(R) p Fn 796 1970 a(d) p Fs 853 1899 a(dq) p Fp 983 1763 a(Z) p Fi 1039 1989 a(j) p Fl(x) p Fi(j) p Fl(>) p Fn 1184 1962 a(L) p 1182 1974 43 3 v Fj 1188 2015 a(2) p Fs 1255 1899 a(dx) p Fp 1395 1763 a(Z) p Fo 1450 1989 a(R) p Fn 1498 1970 a(n) p Fs 1561 1899 a(dk) j(\032) p Fq 1733 1914 a(1) p Fr 1772 1899 a(\() p Fs(x) p Fm 23 w(\000) p Fs 22 w(q) p Fr 4 w(\)) 2071 1876 y(\026) p Fs 2072 1899 a(j) p Fl 2112 1914 a(L) p Fr 2165 1899 a(\() p Fs(x) p Fr(\)) 2408 1805 y(\026) 2408 1831 y(^) p Fs -58 w(\032) p Fq 2449 1846 a(2) p Fr 2489 1831 a(\() p Fs(k) p Fr 3 w(\)) p 2306 1876 406 4 v Fp 2306 1896 a(p) p 2406 1896 307 4 v Fr 2406 1981 a(2) p Fs(!) p Fl 2516 1996 a(m) p Fr 2582 1981 a(\() p Fs(k) p Fr 3 w(\)) p Fs 2722 1899 a(a) p Fr(\() p Fs(x;) d(k) p Fr 3 w(\)) p Fp 3002 1788 a(\021) p Fs 3061 1899 a( ) p Fr 4 w(\() p Fs(q) p Fr 4 w(\)) p Fp 3251 1784 a(\015) 3251 1844 y(\015) 3251 1904 y(\015) p Fq 3306 1810 a(2) p Fi 3306 1968 a(F) p Fs 3368 1899 a(:) p Fv 257 2166 a(With) 32 b(similar) d(computations) j(as) h(the) g (ones) g(of) f(Prop) s(osition) f(4.4,) h(w) m(e) h(get) p Fm 578 2403 a(k) p Fs(H) p Fl 709 2418 a(m) p Fs 775 2403 a( ) p Fm 26 w(\000) p Fs 23 w(H) p Fl 1045 2418 a(m) p Fr 1111 2403 a(\() p Fs(L) p Fr(\)) p Fs( ) p Fm 4 w(k) p Fq 1370 2362 a(2) p Fi 1370 2428 a(H) p Fm 1462 2403 a(\024) p Fs 28 w(C) p Fp 1661 2268 a(Z) p Fi 1716 2493 a(j) p Fl(q) p Fi 2 w(j) p Fl(>) p Fn 1855 2466 a(L) p 1854 2478 43 3 v Fj 1860 2519 a(2) p Fi 1906 2493 a(\000) p Fl(R) p Fj 2014 2502 a(1) p Fs 2070 2403 a(dq) p Fm 20 w(k) p Fs(N) p Fj 2332 2335 a(1) p 2332 2347 31 3 v 2332 2388 a(2) p Fs 2377 2403 a( ) p Fr 4 w(\() p Fs(q) p Fr 4 w(\)) p Fm(k) p Fq 2617 2362 a(2) p Fi 2617 2428 a(F) p Fr 2699 2403 a(+) p Fm 22 w(k) p Fs( ) p Fr 4 w(\() p Fs(q) p Fr 4 w(\)) p Fm(k) p Fq 3087 2362 a(2) p Fi 3087 2428 a(F) p Fs 3148 2403 a(:) p Fv 257 2698 a(But) p Fs 37 w(N) p Fj 553 2634 a(1) p 553 2646 V 553 2687 a(2) p Fs 598 2698 a( ) p Fr 4 w(\() p Fs(q) p Fr 4 w(\)) p Fv 36 w(and) p Fs 37 w( ) p Fr 4 w(\() p Fs(q) p Fr 4 w(\)) p Fv 36 w(b) s(elong) i(to) p Fs 37 w(L) p Fq 1749 2662 a(2) p Fr 1788 2698 a(\() p Fg(R) p Fl 1892 2662 a(d) p Fs 1939 2698 a(;) p Fm 17 w(F) p Fr 10 w(\)) p Fs(;) p Fv 35 w(so) i(the) g(righ) m(t-hand) e (side) i(tends) h(to) 257 2818 y(zero) 33 b(as) p Fs 33 w(L) p Fv 33 w(go) s(es) f(to) h(in\034nit) m(y) -8 b(.) p Fc 2082 w(2) p Fw 257 3120 a(Prop) s(osition) 36 b(4.15.) p Fs 42 w(E) p Fl 1207 3135 a(m) p Fr 1273 3120 a(\() p Fs(L) p Fr(\)) p Ft 35 w(c) -5 b(onver) g(ges) 34 b(to) p Fs 35 w(E) p Fl 2074 3135 a(m) p Ft 2176 3120 a(as) p Fs 34 w(L) p Ft 35 w(go) -5 b(es) 35 b(to) g(in\034nity.) p Fw 257 3302 a(Pro) s(of) i(:) p Fv 38 w(Remem) m(b) s(er) 32 b(that) p Fs 32 w(\036) p Fl 1393 3317 a(m) p Fr 1459 3302 a(\() p Fs(L) p Fr(\)) p Fv 33 w(is) g(a) h(ground) f(state) h(of) p Fs 32 w(H) p Fl 2575 3317 a(m) p Fr 2642 3302 a(\() p Fs(L) p Fr(\)) p Fs(:) p Fv 32 w(W) -8 b(e) 33 b(ha) m(v) m(e) p Fs 257 3496 a(E) p Fl 329 3511 a(m) p Fm 479 3496 a(\024) 84 b(h) p Fs(\036) p Fl 737 3511 a(m) p Fr 803 3496 a(\() p Fs(L) p Fr(\);) p Fs 17 w(H) p Fl 1070 3511 a(m) p Fs 1136 3496 a(\036) p Fl 1194 3511 a(m) p Fr 1260 3496 a(\() p Fs(L) p Fr(\)) p Fm(i) 479 3641 y(\024) p Fs 84 w(E) p Fl 712 3656 a(m) p Fr 778 3641 a(\() p Fs(L) p Fr(\)) 23 b(+) p Fm 22 w(h) p Fs(\036) p Fl 1138 3656 a(m) p Fr 1204 3641 a(\() p Fs(L) p Fr(\);) 17 b(\() p Fs(W) p Fl 1520 3656 a(m) p Fm 1609 3641 a(\000) p Fs 22 w(W) p Fl 1800 3656 a(m) p Fr 1867 3641 a(\() p Fs(L) p Fr(\)\)) p Fs(\036) p Fl 2105 3656 a(m) p Fr 2171 3641 a(\() p Fs(L) p Fr(\)) p Fm(i) 479 3835 y(\024) p Fs 84 w(E) p Fl 712 3850 a(m) p Fr 778 3835 a(\() p Fs(L) p Fr(\)) 23 b(+) f(2) p Fm(R) p Fs(e) p Fp 1219 3754 a(\000) p Fm 1265 3835 a(h) p Fs(\036) p Fl 1362 3850 a(m) p Fr 1428 3835 a(\() p Fs(L) p Fr(\);) p Fp 1614 3699 a(Z) p Fi 1669 3925 a(j) p Fl(x) p Fi(j) p Fl(>) p Fn 1814 3898 a(L) p 1813 3910 43 3 v Fj 1819 3951 a(2) p Fs 1886 3835 a(dx) p Fp 2025 3699 a(Z) p Fo 2081 3925 a(R) p Fn 2129 3906 a(n) p Fs 2192 3835 a(dk) d(\032) p Fq 2363 3850 a(1) p Fr 2403 3835 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(Q) p Fr(\)) 2732 3813 y(\026) p Fs 2733 3835 a(j) p Fl 2773 3850 a(L) p Fr 2826 3835 a(\() p Fs(x) p Fr(\)) p Fm 2005 4126 a(\002) p Fr 2194 4059 a(^) p Fs -58 w(\032) p Fq 2235 4074 a(2) p Fr 2275 4059 a(\() p Fs(k) p Fr 3 w(\)) p 2092 4104 406 4 v Fp 2092 4123 a(p) p 2192 4123 307 4 v Fr 2192 4209 a(2) p Fs(!) p Fl 2302 4224 a(m) p Fr 2368 4209 a(\() p Fs(k) p Fr 3 w(\)) p Fm 2530 4126 a(\012) p Fs 23 w(a) p Fr(\() p Fs(x;) e(k) p Fr 3 w(\)) p Fs(\036) p Fl 2968 4141 a(m) p Fr 3035 4126 a(\() p Fs(L) p Fr(\)) p Fm(i) p Fp 3216 4046 a(\001) p Fm 479 4408 a(\024) p Fs 84 w(E) p Fl 712 4423 a(m) p Fr 778 4408 a(\() p Fs(L) p Fr(\)) 23 b(+) f(2) p Fm(R) p Fs(e) p Fp 1219 4327 a(\000) p Fm 1265 4408 a(h) p Fs(e) p Fl 1349 4367 a(\013) p Fi(j) p Fl(Q) p Fi(j) p Fm 1515 4408 a(\012) p Fr 23 w(1) -22 b(l) p Fs -1 w(\036) p Fl 1727 4423 a(m) p Fr 1793 4408 a(\() p Fs(L) p Fr(\);) p Fp 1979 4272 a(Z) p Fi 2034 4498 a(j) p Fl(x) p Fi(j) p Fl(>) p Fn 2179 4471 a(L) p 2178 4483 43 3 v Fj 2184 4524 a(2) p Fs 2251 4408 a(dx) p Fp 2390 4272 a(Z) p Fo 2446 4498 a(R) p Fn 2494 4479 a(n) p Fs 2557 4408 a(dk) 19 b(e) p Fi 2723 4367 a(\000) p Fl(\013) p Fi(j) p Fl(Q) p Fi(j) p Fs 2922 4408 a(\032) p Fq 2972 4423 a(1) p Fr 3012 4408 a(\() p Fs(x) p Fm 23 w(\000) p Fs 22 w(Q) p Fr(\)) p Fm 2005 4699 a(\002) p Fr 2081 4677 a(\026) p Fs 2082 4699 a(j) p Fl 2122 4714 a(L) p Fr 2175 4699 a(\() p Fs(x) p Fr(\)) 2418 4632 y(^) p Fs -58 w(\032) p Fq 2459 4647 a(2) p Fr 2499 4632 a(\() p Fs(k) p Fr 3 w(\)) p 2316 4676 406 4 v Fp 2316 4696 a(p) p 2416 4696 307 4 v Fr 2416 4781 a(2) p Fs(!) p Fl 2526 4796 a(m) p Fr 2592 4781 a(\() p Fs(k) p Fr 3 w(\)) p Fm 2754 4699 a(\012) p Fs 23 w(a) p Fr(\() p Fs(x;) e(k) p Fr 3 w(\)) p Fs(\036) p Fl 3192 4714 a(m) p Fr 3258 4699 a(\() p Fs(L) p Fr(\)) p Fm(i) p Fp 3439 4618 a(\001) p Fs 3485 4699 a(:) p Fv 257 4963 a(Then,) 34 b(the) f(same) f(computation) f(as) i(in) f(Prop) s(osition) f(4.13) g (leads) i(to) p Fs 606 5200 a(E) p Fl 678 5215 a(m) p Fm 828 5200 a(\024) p Fs 83 w(E) p Fl 1060 5215 a(m) p Fr 1127 5200 a(\() p Fs(L) p Fr(\)) 22 b(+) p Fs 22 w(K) p Fr 7 w(\() p Fs(\013) p Fr 1 w(\)) p Fs(e) p Fi 1663 5159 a(\000) p Fn 1728 5132 a(\013L) p 1728 5144 83 3 v Fj 1754 5185 a(2) p Fm 1824 5200 a(h) p Fs(\036) p Fl 1921 5215 a(m) p Fr 1987 5200 a(\() p Fs(L) p Fr(\);) 17 b(1) -22 b(l) p Fm 21 w(\012) p Fs 23 w(N) p Fr 10 w(\() p Fm(j) p Fs(x) p Fm(j) p Fs 28 w(>) 2727 5133 y(L) p 2727 5177 67 4 v Fr 2736 5268 a(2) 2804 5200 y(\)) p Fs(\036) p Fl 2900 5215 a(m) p Fr 2966 5200 a(\() p Fs(L) p Fr(\)) p Fm(i) 828 5388 y(\024) p Fs 83 w(E) p Fl 1060 5403 a(m) p Fr 1127 5388 a(\() p Fs(L) p Fr(\)) 22 b(+) p Fs 22 w(C) p Fr 7 w(\() p Fs(\013) p Fr 1 w(\)) p Fs(e) p Fi 1650 5347 a(\000) p Fl(\013L) p Fs 1802 5388 a(:) p Fv 1828 5637 a(21) p 90 rotate dyy eop %%Page: 22 22 22 21 bop Fv 257 573 a(So,) 49 b(the) d(function) p Fs 44 w(E) p Fl 1083 588 a(m) p Fr 1150 573 a(\() p Fs(L) p Fr(\)) p Fv 46 w(is) f(b) s(ounded) h(from) e(b) s(elo) m(w) h(\(and) g (from) f(ab) s(o) m(v) m(e) i(b) m(y) p Fs 47 w(E) p Fq 3391 537 a(0) p Fl 3385 597 a(p) p Fv 3430 573 a(\).) 257 693 y(Then) 34 b(there) f(exists) h(a) e(sequence) p Fs 35 w(L) p Fl 1582 708 a(n) p Fv 1662 693 a(and) p Fs 32 w(E) p Fi 1923 708 a(1) p Fv 2031 693 a(suc) m(h) i(that) p Fr 1348 909 a(lim) p Fl 1296 969 a(n) p Fi(!) p Fq(+) p Fi(1) p Fs 1552 909 a(E) p Fl 1624 924 a(m) p Fr 1690 909 a(\() p Fs(L) p Fl 1794 924 a(n) p Fr 1842 909 a(\)) 27 b(=) p Fs 28 w(E) p Fi 2083 924 a(1) p Fm 2185 909 a(\025) p Fs 29 w(E) p Fl 2363 924 a(m) p Fs 2429 909 a(:) p Fv 257 1165 a(No) m(w,) p Fs 28 w(H) p Fl 582 1180 a(m) p Fr 649 1165 a(\() p Fs(L) p Fl 753 1180 a(n) p Fr 800 1165 a(\)) p Fv 26 w(con) m(v) m(erges) h(to) p Fs 25 w(H) p Fl 1487 1180 a(m) p Fv 1579 1165 a(in) d(the) h(strong) g (resolv) m(en) m(t) h(sens) g(and) p Fs 26 w(E) p Fl 2997 1180 a(m) p Fm 3092 1165 a(2) p Fs 28 w(\033) p Fr 4 w(\() p Fs(H) p Fl 3364 1180 a(m) p Fr 3430 1165 a(\)) p Fv(,) 257 1285 y(so,) 33 b(for) f(all) p Fs 31 w(n;) p Fv 32 w(there) i(exists) p Fs 33 w(E) p Fr 6 w(\() p Fs(L) p Fl 1507 1300 a(n) p Fr 1554 1285 a(\)) p Fm 28 w(2) p Fs 28 w(\033) p Fr 4 w(\() p Fs(H) p Fl 1892 1300 a(m) p Fr 1958 1285 a(\() p Fs(L) p Fl 2062 1300 a(n) p Fr 2110 1285 a(\)\)) p Fv 32 w(suc) m(h) g(that) p Fr 1518 1501 a(lim) p Fl 1466 1561 a(n) p Fi(!) p Fq(+) p Fi(1) p Fs 1722 1501 a(E) p Fr 6 w(\() p Fs(L) p Fl 1904 1516 a(n) p Fr 1951 1501 a(\)) 28 b(=) p Fs 27 w(E) p Fl 2192 1516 a(m) p Fs 2259 1501 a(:) p Fv 257 1757 a(But) p Fs 27 w(E) p Fr 6 w(\() p Fs(L) p Fl 627 1772 a(n) p Fr 674 1757 a(\)) p Fv 27 w(is) e(bigger) f(than) p Fs 27 w(E) p Fl 1411 1772 a(m) p Fr 1478 1757 a(\() p Fs(L) p Fl 1582 1772 a(n) p Fr 1629 1757 a(\)) p Fv 26 w(for) h(all) p Fs 25 w(n;) p Fv 27 w(so) g(\034nally) p Fs 26 w(E) p Fl 2553 1772 a(m) p Fr 2647 1757 a(=) p Fs 28 w(E) p Fi 2823 1772 a(1) p Fs 2898 1757 a(:) p Fv 26 w(The) i(function) p Fs 257 1877 a(E) p Fl 329 1892 a(m) p Fr 396 1877 a(\() p Fs(L) p Fr(\)) p Fv 32 w(is) j(then) i(b) s(ounded) f(with) g(only) f (one) h(accum) m(ulating) e(p) s(oin) m(t) p Fs 31 w(E) p Fl 2823 1892 a(m) p Fv 2889 1877 a(,) i(whic) m(h) g(pro) m(v) m(es) 257 1997 y(that) h(the) g(function) f(con) m(v) m(erges) i(to) f(this) f(p) s(oin) m(t.) p Fc 1406 w(2) p Fw 257 2238 a(Pro) s(of) h(of) h(Theorem) f(4.7) h(:) p Fv 75 w(The) c(pro) s(of) e(is) h(iden) m(tical) e(to) i (the) g(one) h(of) e(Theorem) i(4.1.) p Fw 257 2438 a(Remark) k(4.1.) p Ft 40 w(A) n(nother) e(way) g(to) h(pr) -5 b(ove) 32 b(our) h(r) -5 b(esults) 32 b(c) -5 b(onc) g(erning) 31 b(the) i(massive) e(c) -5 b(ase) 257 2559 y(would) 37 b(b) -5 b(e) 36 b(to) h(use) g(the) f(ide) -5 b(as) 36 b(of) g([9]-[15) o(].) 50 b(The) 37 b(ide) -5 b(a) 36 b(is) g(to) h(pr) -5 b(ove) 36 b(that) p Fs 37 w(E) p Fl 3027 2574 a(m) p Ft 3130 2559 a(is) h(not) g(in) 257 2679 y(the) 30 b(essential) f(sp) -5 b(e) g(ctrum) 30 b(using) f(the) h(W) -7 b(eyl) 30 b(criterion.) 42 b(F) -7 b(or) 29 b(that) i(purp) -5 b(ose,) 30 b(one) f(pr) -5 b(oves) 257 2799 y(that,) 35 b(given) f(a) h(norme) -5 b(d) 34 b(se) -5 b(quenc) g(e) p Fs 34 w( ) p Fl 1634 2814 a(j) p Ft 1706 2799 a(tending) 34 b(we) -5 b(akly) 34 b(to) h(zer) -5 b(o,) p Fr 1235 3015 a(lim) 17 b(inf) p Fl 1284 3076 a(j) p Fi 4 w(!1) p Fm 1505 3015 a(h) p Fs( ) p Fl 1607 3030 a(j) p Fr 1644 3015 a(;) g(\() p Fs(H) p Fl 1807 3030 a(m) p Fm 1895 3015 a(\000) p Fs 23 w(E) p Fl 2067 3030 a(m) p Fr 2134 3015 a(\)) p Fs( ) p Fl 2235 3030 a(j) p Fm 2272 3015 a(i) p Fs 27 w(>) p Fr 27 w(0) p Fs(:) p Fv 729 w(\(4.12\)) p Ft 257 3272 a(The) 31 b(philosophy) f(is) h(that,) h(if) p Fs 31 w( ) p Fl 1406 3287 a(j) p Ft 1475 3272 a(tends) f(we) -5 b(akly) 30 b(to) i(zer) -5 b(o,) 31 b(it) h(must) f(\020esc) -5 b(ap) g(e) 31 b(to) g(in\034nity\021) 257 3392 y(in) g(some) e(way.) 44 b(In) 29 b(our) i(mo) -5 b(del,) 31 b(if) f(it) h(esc) -5 b(ap) g(es) 29 b(in) i(the) f(p) -5 b(article) 30 b(p) -5 b(art,) 32 b(with) e(the) h(numb) -5 b(er) 257 3513 y(of) 31 b(b) -5 b(osons) 31 b(or) g(with) g(their) g(momentum) g(in) f(the) p Fs 32 w(y) p Ft 34 w(dir) -5 b(e) g(ction) 31 b(\(that) g(is) g(when) p Fs 31 w(k) p Ft 34 w(tends) g(to) 257 3633 y(in\034nity\),) 39 b(then) f(the) g(ener) -5 b(gy) 37 b(gr) -5 b(ows) 38 b(ne) -5 b(c) g(essarily) 37 b(and) g(\(4.12\)) g(is) h(c) -5 b(ertainly) 38 b(satis\034e) -5 b(d.) 257 3754 y(Now,) 29 b(if) f(it) g(esc) -5 b(ap) g(es) 27 b(with) g(far) h(away) f(b) -5 b(osons,) 28 b(either) g(in) f(\020sp) -5 b(ac) g(e\021) 34 b(\(that) 28 b(is) g(in) f(the) p Fs 28 w(x) p Ft 28 w(or) p Fs 28 w(y) p Ft 257 3874 a(dir) -5 b(e) g(ction\)) 33 b(or) h(in) f(\020momentum) f(in) i(the) p Fs 33 w(x) p Ft 34 w(dir) -5 b(e) g(ction) -10 b(\021,) 33 b(the) h(ide) -5 b(a) 33 b(is) g(that) h(those) f(b) -5 b(osons) 257 3994 y(do) 37 b(not) h(inter) -5 b(act) 37 b(with) g(the) g(p) -5 b(article) 37 b(and) g(so) g(e) -5 b(ach) 36 b(of) h(them) g(has) g(an) g(ener) -5 b(gy) 37 b(at) g(le) -5 b(ast) p Fs 257 4115 a(m:) p Ft 30 w(A) 29 b(W) -7 b(eyl) 29 b(se) -5 b(quenc) g(e) 28 b(c) -5 b(an) 29 b(then) g(exist) g(only) g(for) p Fs 28 w(E) p Fm 34 w(\025) p Fs 28 w(E) p Fl 2370 4130 a(m) p Fr 2446 4115 a(+) p Fs 9 w(m:) p Ft 30 w(A) h(pr) -5 b(e) g(cise) 28 b(writing) h(of) 257 4235 y(such) 35 b(a) g(pr) -5 b(o) g(of) 35 b(would) g(imply) f(a) h(c) -5 b(ontr) g(ol) 35 b(on) g(the) g(momentum) g(of) f(the) i(b) -5 b(osons) 34 b(in) h(the) p Fs 35 w(x) p Ft 257 4355 a(dir) -5 b(e) g(ction,) 29 b(which) f(is) g(the) h(new) f(element) g(of) g(our) h (mo) -5 b(del.) 42 b(In) 28 b(our) h(pr) -5 b(o) g(of,) 29 b(such) f(a) h(c) -5 b(ontr) g(ol) 257 4476 y(alr) g(e) g(ady) 41 b(exists) g(but) h(app) -5 b(e) g(ars) 41 b(in) g(a) g(hidden) g(way) g (in) g(Pr) -5 b(op) g(osition) 41 b(4.6.) 64 b(Final) 5 b(ly,) 42 b(we) 257 4596 y(would) e(like) g(to) h(emphasize) d(that) j (writing) f(a) g(pr) -5 b(o) g(of) 40 b(using) g(this) h(other) f (metho) -5 b(d) 40 b(would) 257 4717 y(not) 35 b(b) -5 b(e) 35 b(much) f(shorter.) p Fu 257 5049 a(5) 156 b(Pro) t(of) 53 b(of) f(the) g(main) g(results) p Fv 257 5268 a(The) 43 b(goal) d(of) i(this) f(section) h(is) f(to) h(pro) m(v) m(e) h(the) f (results) g(of) g(Sect.) 72 b(3.) f(W) -8 b(e) 42 b(start) g(with) 257 5388 y(Theorem) 31 b(3.3.) 43 b(W) -8 b(e) 31 b(adapt) f(the) h(metho) s (d) f(of) g([13].) 43 b(W) -8 b(e) 31 b(will) d(insist) i(on) g(the) h (di\033erences) 1828 5637 y(22) p 90 rotate dyy eop %%Page: 23 23 23 22 bop Fv 257 573 a(with) 39 b(this) f(pap) s(er.) 63 b(The) 40 b(idea) e(is) g(to) h(approac) m(h) g(\(in) f(a) g(w) m(a) m (y) j(whic) m(h) e(has) g(to) g(b) s(e) g(made) 257 693 y(precise\)) p Fs 40 w(H) p Fv 46 w(with) g(Hamiltonians) c(for) k (whic) m(h) g(w) m(e) h(kno) m(w) g(that) f(they) g(ha) m(v) m(e) i(a) d (ground) 257 814 y(state) f(and) g(then) g(to) f(obtain) f(the) i(same) f(result) h(for) p Fs 36 w(H) r(:) p Fv 36 w(More) g(precisely) -8 b(,) 38 b(w) m(e) f(will) d(use) 257 934 y(the) f(follo) m(wing) d (lemma:) p Fw 257 1136 a(Lemma) i(5.1.) p Ft 38 w(\([3],) f(L) -5 b(emma) 30 b(4.9\)) g(L) -5 b(et) p Fs 31 w(H) r(;) 17 b(H) p Fl 1985 1151 a(n) p Fr 2031 1136 a(\() p Fs(n) p Fm 28 w(2) p Fg 28 w(N) p Fr 9 w(\)) p Ft 37 w(b) -5 b(e) 30 b(selfadjoint) g(op) -5 b(er) g(ators) 30 b(on) 257 1256 y(a) 35 b(Hilb) -5 b(ert) 35 b(sp) -5 b(ac) g(e) p Fm 34 w(H) p Fs 1 w(:) p Ft 35 w(W) e(e) 35 b(supp) -5 b(ose) 34 b(that) 343 1458 y(\(i\)) p Fm 48 w(8) p Fs(n) p Fm 28 w(2) p Fg 29 w(N) p Fs 9 w(;) 17 b(H) p Fl 928 1473 a(n) p Ft 1015 1458 a(has) 34 b(a) h(gr) -5 b(ound) 34 b(state) p Fs 35 w( ) p Fl 1893 1473 a(n) p Ft 1976 1458 a(with) g(gr) -5 b(ound) 35 b(state) g(ener) -5 b(gy) p Fs 34 w(E) p Fl 3123 1473 a(n) p Fs 3171 1458 a(;) p Ft 313 1661 a(\(ii\)) p Fs 48 w(H) p Fl 582 1676 a(n) p Ft 664 1661 a(tends) 34 b(to) p Fs 35 w(H) p Ft 43 w(in) g(the) h(str) -5 b(ong) 35 b(r) -5 b(esolvent) 34 b(sens,) 283 1864 y(\(iii\)) p Fr 48 w(lim) p Fl 637 1879 a(n) p Fi(!) p Fq(+) p Fi(1) p Fs 896 1864 a(E) p Fl 968 1879 a(n) p Fr 1043 1864 a(=) p Fs 28 w(E) 6 b(;) p Ft 298 2067 a(\(iv\)) p Fv 48 w(w) p Fm(\000) p Fr 17 w(lim) p Fl 801 2082 a(n) p Fi(!) p Fq(+) p Fi(1) p Fs 1061 2067 a( ) p Fl 1124 2082 a(n) p Fr 1199 2067 a(=) p Fs 27 w( ) p Fm 32 w(6) p Fr(=) 28 b(0) p Fs(:) p Ft 257 2269 a(Then) p Fs 34 w( ) p Ft 39 w(is) 35 b(a) f(gr) -5 b(ound) 35 b(state) g(of) p Fs 35 w(H) p Ft 42 w(with) g(gr) -5 b(ound) 34 b(state) h(ener) -5 b(gy) p Fs 35 w(E) 6 b(:) p Ff 257 2557 a(5.1) 131 b(Infrared) 45 b(cuto\033) p Fv 257 2742 a(W) -8 b(e) 25 b(denote) h(b) m(y) p Fs 25 w(\037) p Fl 912 2757 a(\033) p Fi 2 w(\024) p Fl(!) p Fq 2 w(\() p Fl(k) p Fq 2 w(\)) p Fv 1178 2742 a(the) f(caracteristic) f(function) g(of) h(the) g(set) p Fm 25 w(f) p Fs(k) p Fm 31 w(2) p Fg 28 w(R) p Fl 2952 2706 a(n) p Fm 3005 2742 a(j) p Fs(\033) p Fm 31 w(\024) p Fs 28 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fm(g) p Fs(:) p Fv 257 2862 a(F) -8 b(or) 32 b(an) m(y) p Fs 33 w(\033) g(>) p Fr 28 w(0) p Fs(;) p Fv 32 w(w) m(e) i(then) f (de\034ne) p Fs 320 3130 a(H) p Fl 409 3089 a(\033) p Fr 539 3130 a(:=) p Fs 83 w(H) p Fq 806 3145 a(0) p Fr 867 3130 a(+) p Fp 965 2994 a(Z) p Fo 1020 3220 a(R) p Fn 1068 3201 a(d) p Fs 1125 3130 a(dx) p Fp 1248 2994 a(Z) p Fo 1303 3220 a(R) p Fn 1351 3201 a(n) p Fs 1415 3130 a(dk) 19 b(\032) p Fq 1586 3145 a(1) p Fr 1626 3130 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(Q) p Fr(\)) 2036 3062 y(^) p Fs -58 w(\032) p Fq 2077 3077 a(2) p Fr 2117 3062 a(\() p Fs(k) p Fr 3 w(\)) p 1966 3107 343 4 v Fp 1966 3127 a(p) p 2066 3127 244 4 v Fr 2066 3212 a(2) p Fs(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fs 2319 3130 a(\037) p Fl 2380 3145 a(\033) p Fi 2 w(\024) p Fl(!) p Fq 2 w(\() p Fl(k) p Fq 2 w(\)) p Fr 2621 3130 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 23 w(a) p Fi 2924 3089 a(\003) p Fr 2963 3130 a(\() p Fs(x;) e(k) p Fr 3 w(\)) 1700 3426 y(+) p Fs(\032) p Fq 1826 3441 a(1) p Fr 1865 3426 a(\() p Fs(x) p Fm 23 w(\000) p Fs 23 w(Q) p Fr(\)) p Fi 2196 3384 a(\003) p Fr 2316 3332 a(\026) 2316 3358 y(^) p Fs -58 w(\032) p Fq 2357 3373 a(2) p Fr 2396 3358 a(\() p Fs(k) p Fr 3 w(\)) p 2245 3403 343 4 v Fp 2245 3423 a(p) p 2345 3423 244 4 v Fr 2345 3508 a(2) p Fs(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fs 2598 3426 a(\037) p Fl 2659 3441 a(\033) p Fi 2 w(\024) p Fl(!) p Fq 2 w(\() p Fl(k) p Fq 2 w(\)) p Fr 2900 3426 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fs 23 w(a) p Fr(\() p Fs(x;) g(k) p Fr 3 w(\)) 552 3646 y(=) p Fs 97 w(H) p Fq 806 3661 a(0) p Fr 867 3646 a(+) p Fs 22 w(H) p Fl 1046 3661 a(I) 5 b(;\033) p Fs 1148 3646 a(;) p Fv 2120 w(\(5.1\)) 257 3864 y(where) p Fs 38 w(H) p Fq 624 3879 a(0) p Fv 700 3864 a(is) 36 b(the) h(free) g(Hamiltonian) 32 b(de\034ned) 38 b(in) e(\(2.9\).) 55 b(W) -8 b(e) 37 b(w) m(an) m(t) g(to) f(use) i(Lemma) 257 3985 y(5.1) 32 b(with) p Fs 33 w(H) p Fv 40 w(and) p Fs 32 w(H) p Fl 1036 3949 a(\033) p Fn 1076 3957 a(n) p Fv 1155 3985 a(where) p Fs 34 w(\033) p Fl 1492 4000 a(n) p Fv 1572 3985 a(is) g(some) g(sequence) k(going) 31 b(to) h(zero.) 404 4105 y(W) -8 b(e) 33 b(consider) f(a) h(function) p Fr 39 w(~) p Fs -56 w(!) p Fl 1476 4120 a(\033) p Fr 1522 4105 a(\() p Fs(k) p Fr 3 w(\)) p Fv 33 w(satisfying) p Fp 1170 4237 a(8) 1170 4327 y(<) 1170 4506 y(:) p Fm 1300 4320 a(r) p Fr 8 w(~) p Fs -57 w(!) p Fl 1444 4335 a(\033) p Fm 1518 4320 a(2) p Fs 28 w(L) p Fi 1678 4284 a(1) p Fr 1753 4320 a(\() p Fg(R) p Fl 1857 4284 a(n) p Fr 1910 4320 a(\)) p Fs(;) p Fr 1308 4441 a(~) p Fs -57 w(!) p Fl 1361 4456 a(\033) p Fr 1407 4441 a(\() p Fs(k) p Fr 3 w(\)) 28 b(=) p Fs 28 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fv 96 w(si) p Fs 98 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fm 26 w(\025) p Fs 29 w(\033) n(;) p Fr 1300 4561 a(inf) i(~) p Fs -56 w(!) p Fl 1496 4576 a(\033) p Fr 1542 4561 a(\() p Fs(k) p Fr 3 w(\)) p Fm 28 w(\025) p Fl 1815 4522 a(\033) p 1815 4538 43 4 v Fq 1819 4595 a(2) p Fs 1895 4561 a(>) p Fr 28 w(0) p Fs(;) p Fv 257 4772 a(and) j(w) m(e) h(de\034ne) p Fr 1158 4867 a(~) p Fs 1133 4892 a(H) p Fl 1222 4851 a(\033) p Fr 1296 4892 a(=) p Fs 27 w(H) p Fl 1480 4907 a(p) p Fm 1542 4892 a(\012) p Fr 23 w(1) -22 b(l) 20 b(+) i(1) -22 b(l) p Fm 21 w(\012) p Fv 23 w(d) p Fr(\000\() 8 b(~) p Fs -57 w(!) p Fl 2205 4907 a(\033) p Fr 2252 4892 a(\)) 22 b(+) p Fs 22 w(H) p Fl 2491 4907 a(I) 5 b(;\033) p Fs 2593 4892 a(:) p Fv 675 w(\(5.2\)) 257 5066 y(Then) 34 b(w) m(e) g(ha) m(v) m(e) f(the) g(follo) m(wing) d(result:) p Fw 257 5268 a(Prop) s(osition) 39 b(5.2.) p Ft 43 w(F) -7 b(or) 36 b(any) p Fs 37 w(\033) f(>) p Fr 32 w(0) p Fs(;) 17 b(H) p Fl 1836 5232 a(\033) p Ft 1919 5268 a(has) 37 b(a) f(gr) -5 b(ound) 37 b(state) p Fs 37 w( ) p Fl 2806 5283 a(\033) p Fs 2854 5268 a(:) p Ft 37 w(W) -7 b(e) 37 b(denote) f(by) p Fs 257 5388 a(E) p Fl 329 5403 a(\033) p Ft 411 5388 a(its) f(gr) -5 b(ound) 35 b(state) g(ener) -5 b(gy.) p Fv 1828 5637 a(23) p 90 rotate dyy eop %%Page: 24 24 24 23 bop Fv 257 573 a(T) -8 b(o) 33 b(pro) m(v) m(e) h(this) e(result) g(w) m(e) i(use) f(the) g(follo) m(wing) d(lemma:) p Fw 257 776 a(Lemma) 46 b(5.3.) p Ft 46 w(\([13) o(],) e(L) -5 b(emma) 41 b(3.2\)) p Fs 42 w(H) p Fl 1809 740 a(\033) p Ft 1897 776 a(has) g(a) h(gr) -5 b(ound) 42 b(state) g(if) g(and) f (only) h(if) p Fr 3385 751 a(~) p Fs 3360 776 a(H) p Fl 3449 740 a(\033) p Ft 257 897 a(has) 35 b(one.) p Fw 257 1100 a(Pro) s(of) 40 b(of) g(Prop) s(osition) e(5.2) i(:) p Fv 87 w(A) m(ccording) 35 b(to) f(the) h(previous) g(lemma,) e(it) h (su\036ces) 257 1220 y(to) i(sho) m(w) h(that) p Fr 865 1195 a(~) p Fs 840 1220 a(H) p Fl 929 1184 a(\033) p Fv 1011 1220 a(has) f(a) g(ground) g(state.) 53 b(But) p Fr 2116 1195 a(~) p Fs 2090 1220 a(H) p Fl 2179 1184 a(\033) p Fv 2262 1220 a(is) 35 b(a) h(Hamiltonian) c(of) j(the) h (form) 257 1341 y(studied) d(in) f(Sect.) 44 b(4.2,) 32 b(so,) h(according) f(to) g(Theorem) h(4.7,) f(it) g(has) h(a) f (ground) h(state.) p Fc 57 w(2) p Fw 257 1665 a(Prop) s(osition) j (5.4.) p Fs 42 w(H) p Fl 1168 1628 a(\033) p Ft 1249 1665 a(tends) e(to) p Fs 35 w(H) p Ft 43 w(in) g(the) h(norm) f(r) -5 b(esolvent) 34 b(sens.) p Fw 257 1868 a(Pro) s(of) 29 b(:) p Fv 28 w(W) -8 b(e) 26 b(use) g(Lemma) d(A.2) i(of) g([13) o(]) g (whic) m(h) h(sa) m(ys) g(that) f(it) f(su\036ces) j(to) d(sho) m(w) i (that) p Fs 25 w(Q) p Fl 3448 1832 a(\033) p Fv 257 1988 a(con) m(v) m(erges) 36 b(to) p Fs 33 w(Q) p Fv 34 w(in) d(the) h(top) s (ology) e(of) p Fm 33 w(D) p Fr 3 w(\() p Fs(Q) p Fr(\)) p Fs(;) p Fv 33 w(where) p Fs 35 w(Q) p Fl 2377 1952 a(\033) p Fv 2458 1988 a(and) p Fs 33 w(Q) p Fv 34 w(are) i(the) g(quadratic) 257 2109 y(forms) h(asso) s(ciated) g(to) p Fs 35 w(H) p Fl 1207 2073 a(\033) p Fv 1289 2109 a(and) p Fs 36 w(H) r(:) p Fv 35 w(But,) i(with) e(a) g(similar) d(computation) i(to) h(the) h (one) 257 2229 y(of) c(Lemma) g(3.2,) g(one) h(has) p Fm 441 2536 a(j) p Fs(Q) p Fr(\() p Fs(u;) 17 b(v) p Fr 4 w(\)) p Fm 21 w(\000) p Fs 22 w(Q) p Fl 970 2495 a(\033) p Fr 1017 2536 a(\() p Fs(u;) g(v) p Fr 4 w(\)) p Fm(j) 82 b(\024) p Fp 1514 2395 a(\022) 1588 2400 y(Z) p Fo 1643 2626 a(R) p Fn 1691 2607 a(d) p Fs 1748 2536 a(dx) p Fp 1887 2400 a(Z) p Fl 1943 2626 a(!) p Fq 2 w(\() p Fl(k) p Fq 2 w(\)) p Fi(\024) p Fl(\033) p Fs 2201 2536 a(dk) 2332 2468 y(\032) p Fq 2382 2483 a(1) p Fr 2422 2468 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(q) p Fr 4 w(\)) p Fq 2722 2432 a(2) p Fm 2761 2468 a(j) p Fr 9 w(^) p Fs -58 w(\032) p Fq 2839 2483 a(2) p Fr 2879 2468 a(\() p Fs(k) p Fr 3 w(\)) p Fm(j) p Fq 3037 2432 a(2) p 2332 2513 744 4 v Fr 2563 2604 a(2) p Fs(!) p Fq 2677 2575 a(2) p Fr 2715 2604 a(\() p Fs(k) p Fr 3 w(\)) p Fp 3086 2395 a(\023) p Fj 3169 2389 a(1) p 3169 2401 31 3 v 3169 2442 a(2) p Fm 2100 2750 a(\002) p Fr(\() p Fs(Q) p Fr(\() p Fs(u;) 17 b(u) p Fr(\)) p Fm(k) p Fs(v) p Fm 4 w(k) p Fr 20 w(+) p Fs 22 w(Q) p Fr(\() p Fs(v) t(;) g(v) p Fr 4 w(\)) p Fm(k) p Fs(u) p Fm(k) p Fr(\)) p Fs(:) p Fw 257 2970 a(Corollary) 38 b(5.5.) p Fr 41 w(lim) p Fl 1108 2985 a(\033) p Fi 2 w(!) p Fq(0) p Fs 1277 2970 a(E) p Fl 1349 2985 a(\033) p Fr 1424 2970 a(=) p Fs 28 w(E) p Fq 1600 2985 a(0) p Fs 1639 2970 a(:) p Fw 257 3174 a(Remark) f(5.1.) p Ft 42 w(As) e(in) g(the) f(massive) g(c) -5 b(ase,) 34 b(one) g(has) p Fs 35 w(E) p Fl 2360 3189 a(\033) p Fm 2434 3174 a(\024) p Fs 29 w(E) p Fq 2618 3137 a(0) p Fl 2612 3198 a(p) p Ft 2692 3174 a(for) h(al) 5 b(l) p Fs 34 w(\033) 32 b(>) p Fr 27 w(0) p Fs(:) p Fv 404 3377 a(Using) k(Prop) s(ositions) f(5.2) h(and) h(5.4) f(together) h (with) f(Corollary) f(5.5,) i(one) g(can) g(see) 257 3497 y(that) 30 b(the) h(op) s(erators) p Fs 30 w(H) p Fl 1150 3461 a(\033) p Fv 1227 3497 a(and) p Fs 30 w(H) p Fv 38 w(satisfy) g(assumptions) p Fr 30 w(\() p Fs(i) p Fr(\)) p Fm 17 w(\000) p Fr 19 w(\() p Fs(ii) p Fr(\)) p Fm 18 w(\000) p Fr 18 w(\() p Fs(iii) p Fr(\)) p Fv 31 w(of) f(Lemma) 257 3618 y(5.1.) 43 b(So,) 33 b(it) e(remains) h(to) g (c) m(hec) m(k) j(condition) p Fr 31 w(\() p Fs(iv) p Fr 4 w(\)) p Fv 32 w(and) e(Theorem) g(3.3) f(will) e(b) s(e) j(pro) m (v) m(en.) p Ff 257 3907 a(5.2) 131 b(Uniform) 44 b(estimates) p Fw 257 4091 a(Lemma) 37 b(5.6.) p Ft 42 w(Ther) -5 b(e) 34 b(exists) p Fs 35 w(C) p Fq 1487 4106 a(1) p Fs 1554 4091 a(>) p Fr 27 w(0) p Ft 35 w(such) g(that) i(for) e(al) 5 b(l) p Fs 35 w(\033) 31 b(>) p Fr 28 w(0) p Fs(;) p Fm 1511 4311 a(h) p Fs( ) p Fl 1613 4326 a(\033) p Fr 1660 4311 a(;) p Fs 17 w(H) p Fq 1785 4326 a(0) p Fs 1824 4311 a( ) p Fl 1887 4326 a(\033) p Fm 1934 4311 a(i) c(\024) p Fs 29 w(C) p Fq 2176 4326 a(1) p Fs 2215 4311 a(:) p Fv 404 4531 a(This) 39 b(inequalit) m(y) e(comes) i(from) f(the) h (fact) g(that) p Fs 39 w(H) p Fl 2294 4546 a(I) 5 b(;\033) p Fv 2434 4531 a(is) 39 b(relativ) m(ely) p Fs 37 w(H) p Fq 3051 4546 a(0) p Fv 3129 4531 a(b) s(ounded) 257 4652 y(with) 24 b(in\034nitesimal) e(b) s(ound,) k(uniformly) c(with) i (resp) s(ect) i(to) p Fs 24 w(\033) 32 b(>) p Fr 27 w(0) p Fs(:) p Fv 24 w(Of) 25 b(course,) i(w) m(e) e(need) 257 4772 y(an) 35 b(estimate) g(on) g(the) g(n) m(um) m(b) s(er) h(of) e (soft) i(b) s(osons,) g(estimate) e(whic) m(h) i(uses) g(the) g (infrared) 257 4892 y(condition) 31 b(\(IR\).) p Fw 257 5096 a(Lemma) 37 b(5.7.) p Ft 42 w(Ther) -5 b(e) 34 b(exists) p Fs 35 w(C) p Fq 1487 5111 a(2) p Fs 1554 5096 a(>) p Fr 27 w(0) p Ft 35 w(such) g(that) i(for) e(al) 5 b(l) p Fs 35 w(\033) 31 b(>) p Fr 28 w(0) p Fs(;) p Fm 1439 5316 a(h) p Fs( ) p Fl 1541 5331 a(\033) p Fr 1588 5316 a(;) 17 b(1) -22 b(l) p Fm 20 w(\012) p Fs 23 w(N) 10 b( ) p Fl 1958 5331 a(\033) p Fm 2006 5316 a(i) 27 b(\024) p Fs 28 w(C) p Fq 2247 5331 a(2) p Fs 2287 5316 a(:) p Fv 1828 5637 a(24) p 90 rotate dyy eop %%Page: 25 25 25 24 bop Fw 257 573 a(Pro) s(of) 37 b(:) p Fv 38 w(As) c(in) f(Lemma) f (4.12,) h(one) h(can) f(sho) m(w) i(that) p Fm 424 832 a(k) p Fr(1) -22 b(l) p Fm 21 w(\012) p Fs 23 w(a) p Fr(\() p Fs(x;) 17 b(k) p Fr 3 w(\)) p Fs( ) p Fl 993 847 a(\033) p Fm 1040 832 a(k) 28 b(\024) p Fr 1306 764 a(1) p 1233 809 195 4 v Fs 1233 900 a(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fm 1437 832 a(k) p Fs(\032) p Fq 1537 847 a(1) p Fr 1577 832 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(Q) p Fr(\)) 2019 764 y(^) p Fs -58 w(\032) p Fq 2060 779 a(2) p Fr 2100 764 a(\() p Fs(k) p Fr 3 w(\)) p 1917 809 406 4 v Fp 1917 829 a(p) p 2017 829 307 4 v Fr 2017 914 a(2) p Fs(!) p Fl 2127 929 a(m) p Fr 2193 914 a(\() p Fs(k) p Fr 3 w(\)) p Fs 2333 832 a(\037) p Fl 2394 847 a(!) p Fq 2 w(\() p Fl(k) p Fq 2 w(\)) p Fi(\025) p Fl(\033) p Fr 2635 832 a(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\012) p Fr 23 w(1) -22 b(l) p Fs -1 w( ) p Fl 3004 847 a(\033) p Fm 3051 832 a(k) p Fs(:) p Fv 167 w(\(5.3\)) 257 1110 y(Th) m(us,) p Fm 341 1362 a(h) p Fs( ) p Fl 443 1377 a(\033) p Fr 490 1362 a(;) 17 b(1) -22 b(l) p Fm 21 w(\012) p Fs 22 w(N) 10 b( ) p Fl 860 1377 a(\033) p Fm 908 1362 a(i) p Fr 84 w(=) p Fp 1190 1227 a(Z) p Fo 1246 1452 a(R) p Fn 1294 1433 a(d) p Fs 1351 1362 a(dx) p Fp 1490 1227 a(Z) p Fo 1545 1452 a(R) p Fn 1593 1433 a(n) p Fs 1656 1362 a(dk) p Fm 20 w(k) p Fr(1) -22 b(l) p Fm 20 w(\012) p Fs 23 w(a) p Fr(\() p Fs(x;) 17 b(k) p Fr 3 w(\)) p Fs( ) p Fl 2346 1377 a(\033) p Fm 2394 1362 a(k) p Fq 2444 1321 a(2) p Fi 2444 1387 a(H) p Fm 1030 1627 a(\024) p Fp 1190 1492 a(Z) p Fo 1246 1717 a(R) p Fn 1294 1698 a(d) p Fs 1351 1627 a(dx) p Fp 1490 1492 a(Z) p Fl 1545 1717 a(!) p Fq 2 w(\() p Fl(k) p Fq 2 w(\)) p Fi(\025) p Fl(\033) p Fs 1803 1627 a(dk) p Fr 2027 1560 a(1) p 1935 1604 234 4 v Fs 1935 1696 a(!) p Fq 2000 1667 a(2) p Fr 2038 1696 a(\() p Fs(k) p Fr 3 w(\)) p Fm 2178 1627 a(k) p Fs(\032) p Fq 2278 1642 a(1) p Fr 2318 1627 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(Q) p Fr(\)) 2728 1560 y(^) p Fs -57 w(\032) p Fq 2770 1575 a(2) p Fr 2809 1560 a(\() p Fs(k) p Fr 3 w(\)) p 2658 1604 343 4 v Fp 2658 1624 a(p) p 2758 1624 244 4 v Fr 2758 1710 a(2) p Fs(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fm 3033 1627 a(\012) p Fr 23 w(1) -22 b(l) p Fs -2 w( ) p Fl 3249 1642 a(\033) p Fm 3297 1627 a(k) p Fq 3347 1586 a(2) p Fi 3347 1652 a(H) p Fm 1030 1920 a(\024) p Fp 1190 1784 a(Z) p Fo 1246 2009 a(R) p Fn 1294 1991 a(d) p Fs 1351 1920 a(dq) p Fp 1465 1784 a(Z) p Fo 1520 2009 a(R) p Fn 1568 1991 a(d) p Fs 1625 1920 a(dx) p Fp 1764 1784 a(Z) p Fl 1820 2009 a(!) p Fq 2 w(\() p Fl(k) p Fq 2 w(\)) p Fi(\025) p Fl(\033) p Fs 2078 1920 a(dk) p Fm 2209 1852 a(j) p Fr 9 w(^) p Fs -58 w(\032) p Fq 2287 1867 a(2) p Fr 2326 1852 a(\() p Fs(k) p Fr 3 w(\)) p Fm(j) p Fq 2484 1816 a(2) p 2209 1897 315 4 v Fr 2225 1988 a(2) p Fs(!) p Fq 2339 1959 a(3) p Fr 2377 1988 a(\() p Fs(k) p Fr 3 w(\)) p Fm 2533 1920 a(j) p Fs(\032) p Fq 2611 1935 a(1) p Fr 2651 1920 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(q) p Fr 4 w(\)) p Fm(j) p Fq 2979 1878 a(2) p Fm 3018 1920 a(k) p Fs( ) p Fl 3131 1935 a(\033) p Fr 3178 1920 a(\() p Fs(q) p Fr 4 w(\)) p Fm(k) p Fq 3351 1878 a(2) p Fi 3351 1944 a(F) p Fm 1030 2206 a(\024) 83 b(k) p Fs(\032) p Fq 1290 2221 a(1) p Fm 1330 2206 a(k) p Fq 1380 2165 a(2) 1380 2230 y(2) p Fp 1436 2065 a(\022) 1509 2070 y(Z) p Fo 1564 2296 a(R) p Fn 1612 2277 a(n) p Fs 1676 2206 a(dk) p Fm 1807 2138 a(j) p Fr 9 w(^) p Fs -58 w(\032) p Fq 1885 2153 a(2) p Fr 1924 2138 a(\() p Fs(k) p Fr 3 w(\)) p Fm(j) p Fq 2082 2102 a(2) p 1807 2183 V Fr 1823 2274 a(2) p Fs(!) p Fq 1937 2245 a(3) p Fr 1976 2274 a(\() p Fs(k) p Fr 3 w(\)) p Fp 2131 2065 a(\023) 2221 2070 y(Z) p Fo 2277 2296 a(R) p Fn 2325 2277 a(d) p Fs 2382 2206 a(dq) p Fm 20 w(k) p Fs( ) p Fl 2609 2221 a(\033) p Fr 2656 2206 a(\() p Fs(q) p Fr 4 w(\)) p Fm(k) p Fq 2829 2165 a(2) p Fi 2829 2230 a(F) p Fm 2917 2206 a(\024) p Fs 28 w(C) p Fq 3092 2221 a(2) p Fs 3132 2206 a(:) p Fc 3421 2442 a(2) p Fv 404 2683 a(W) -8 b(e) 33 b(ha) m(v) m(e) g (obtained) f(a) g(con) m(trol) g(on) h(the) g(total) d(n) m(um) m(b) s (er) j(of) f(b) s(osons.) 44 b(Ho) m(w) m(ev) m(er,) 35 b(w) m(e) 257 2803 y(will) k(also) i(need) h(some) f(con) m(trol) f (\(uniform) g(with) h(resp) s(ect) h(to) p Fs 41 w(\033) p Fv 4 w(\)) f(on) g(the) h(n) m(um) m(b) s(er) g(of) 257 2923 y(\020far) 37 b(a) m(w) m(a) m(y) i(b) s(osons\021,) g(that) f(is) f(on) h(the) g(follo) m(wing) d(quan) m(tities:) p Fm 54 w(h) p Fs( ) p Fl 2729 2938 a(\033) p Fr 2776 2923 a(;) p Fs 17 w(N) p Fr 10 w(\() p Fm(j) p Fs(x) p Fm(j) p Fs 37 w(>) h(R) p Fr 1 w(\)) p Fs( ) p Fl 3382 2938 a(\033) p Fm 3429 2923 a(i) p Fv(,) p Fm 257 3044 a(h) p Fs( ) p Fl 359 3059 a(\033) p Fr 406 3044 a(;) p Fs 17 w(N) p Fr 10 w(\() p Fm(j) p Fs(y) p Fm 4 w(j) p Fs 27 w(>) 27 b(S) p Fr 6 w(\)) p Fs( ) p Fl 981 3059 a(\033) p Fm 1028 3044 a(i) p Fv 33 w(and) p Fm 32 w(h) p Fs( ) p Fl 1391 3059 a(\033) p Fr 1438 3044 a(;) p Fs 17 w(N) p Fr 10 w(\() p Fm(j) p Fs(p) p Fm(j) p Fs 27 w(>) h(P) p Fr 14 w(\)) p Fs( ) p Fl 2022 3059 a(\033) p Fm 2069 3044 a(i) p Fv 32 w(where) p Fs 911 3301 a(N) p Fr 10 w(\() p Fm(j) p Fs(x) p Fm(j) p Fs 28 w(>) f(R) p Fr 1 w(\)) h(=) p Fp 1524 3166 a(Z) p Fi 1579 3391 a(j) p Fl(x) p Fi(j) p Fl(>R) p Fs 1787 3301 a(dx) p Fp 1927 3166 a(Z) p Fo 1982 3391 a(R) p Fn 2030 3372 a(n) p Fs 2093 3301 a(dk) 19 b(a) p Fi 2265 3260 a(\003) p Fr 2305 3301 a(\() p Fs(x;) e(k) p Fr 3 w(\)) p Fs(a) p Fr(\() p Fs(x;) g(k) p Fr 3 w(\)) p Fs(;) 927 3641 y(N) p Fr 10 w(\() p Fm(j) p Fs(y) p Fm 4 w(j) p Fs 27 w(>) 27 b(S) p Fr 6 w(\)) g(=) p Fp 1526 3505 a(Z) p Fi 1581 3731 a(j) p Fl(y) p Fi 2 w(j) p Fl(>S) p Fs 1780 3641 a(dx) p Fp 1919 3505 a(Z) p Fo 1975 3731 a(R) p Fn 2023 3712 a(n) p Fs 2086 3641 a(dy) p Fr 21 w(~) p Fs -51 w(a) p Fi 2255 3600 a(\003) p Fr 2295 3641 a(\() p Fs(x;) 17 b(y) p Fr 4 w(\)) q(~) p Fs -50 w(a) p Fr -1 w(\() p Fs(x;) g(y) p Fr 4 w(\)) p Fs(;) 925 3941 y(N) p Fr 10 w(\() p Fm(j) p Fs(p) p Fm(j) p Fs 27 w(>) 28 b(P) p Fr 14 w(\)) f(=) p Fp 1532 3805 a(Z) p Fi 1588 4031 a(j) p Fl(p) p Fi(j) p Fl(>P) p Fs 1793 3941 a(dp) p Fp 1926 3805 a(Z) p Fo 1981 4031 a(R) p Fn 2029 4012 a(n) p Fs 2092 3941 a(dk) p Fr 21 w(^) p Fs -50 w(a) p Fi 2265 3899 a(\003) p Fr 2304 3941 a(\() p Fs(p;) 17 b(k) p Fr 3 w(\)) q(^) p Fs -50 w(a) p Fr(\() p Fs(p;) g(k) p Fr 3 w(\)) p Fs(:) p Fv 257 4173 a(The) 36 b(op) s(erators) p Fr 35 w(~) p Fs -50 w(a) p Fv 34 w(and) p Fr 36 w(~) p Fs -50 w(a) p Fi 1221 4137 a(\003) p Fv 1295 4173 a(come) e(from) p Fs 33 w(a) p Fv 35 w(and) p Fs 34 w(a) p Fi 2106 4137 a(\003) p Fv 2180 4173 a(via) g(a) g(partial) e(F) -8 b(ourier) 33 b(transform) 257 4294 y(in) e(the) p Fs 32 w(k) p Fv 34 w(v) -5 b(ariable,) 30 b(and) h(the) h(op) s(erators) p Fr 33 w(^) p Fs -51 w(a) p Fv 32 w(and) p Fr 33 w(^) p Fs -51 w(a) p Fi 2121 4258 a(\003) p Fv 2192 4294 a(via) f(a) g (partial) e(F) -8 b(ourier) 30 b(transform) 257 4414 y(in) i(the) p Fs 33 w(x) p Fv 33 w(v) -5 b(ariable.) 42 b(W) -8 b(e) 33 b(then) g(pro) m(v) m(e) h(a) e(result) g(similar) d (to) k(Prop) s(osition) d(4.11:) p Fw 257 4608 a(Lemma) 37 b(5.8.) p Ft 42 w(F) -7 b(or) 34 b(any) p Fs 35 w(\013) 28 b(>) p Fr 28 w(0) p Fs(;) p Ft 34 w(ther) -5 b(e) 35 b(exists) p Fs 35 w(C) p Fr 7 w(\() p Fs(\013) p Fr 1 w(\)) p Fs 27 w(>) p Fr 27 w(0) p Ft 35 w(such) g(that) p Fm 1088 4818 a(h) p Fs( ) p Fl 1190 4833 a(\033) p Fr 1237 4818 a(;) 17 b(1) -22 b(l) p Fm 20 w(\012) p Fs 23 w(N) p Fr 10 w(\() p Fm(j) p Fs(x) p Fm(j) p Fs 28 w(>) 27 b(R) p Fr 1 w(\)) p Fs( ) p Fl 2000 4833 a(\033) p Fm 2048 4818 a(i) g(\024) p Fs 28 w(C) p Fr 7 w(\() p Fs(\013) p Fr 1 w(\)) p Fs(e) p Fi 2480 4777 a(\000) p Fl(\013R) p Fs 2638 4818 a(:) p Fv 404 5027 a(The) 41 b(pro) s(of) f(of) g(this) g(lemma) f(is) h(exactly) h(the) g(same) g (to) f(the) h(one) g(of) f(Prop) s(osition) 257 5147 y(4.11.) 76 b(This) 43 b(lemma) e(giv) m(es) j(us) g(a) f(con) m(trol) g (on) g(the) h(n) m(um) m(b) s(er) g(of) f(\020far) f(a) m(w) m(a) m (y\021) 53 b(b) s(osons) 257 5268 y(in) 40 b(the) p Fs 42 w(x) p Fv 41 w(direction.) 67 b(Similarly) 37 b(one) k(can) h(con) m (trol) e(the) h(n) m(um) m(b) s(er) g(of) g(b) s(osons) g(whose) 257 5388 y(momen) m(tum) 31 b(in) h(the) p Fs 33 w(x) p Fv 33 w(direction) f(is) i(large:) 1828 5637 y(25) p 90 rotate dyy eop %%Page: 26 26 26 25 bop Fw 257 573 a(Lemma) 37 b(5.9.) p Ft 42 w(F) -7 b(or) 34 b(any) p Fs 35 w(s) 27 b(>) p Fr 28 w(0) p Fs(;) p Ft 35 w(ther) -5 b(e) 35 b(exists) p Fs 34 w(C) p Fr 7 w(\() p Fs(s) p Fr(\)) p Fs 27 w(>) p Fr 28 w(0) p Ft 35 w(such) f(that) p Fm 1149 848 a(h) p Fs( ) p Fl 1251 863 a(\033) p Fr 1298 848 a(;) 17 b(1) -22 b(l) p Fm 20 w(\012) p Fs 23 w(N) p Fr 10 w(\() p Fm(j) p Fs(p) p Fm(j) p Fs 27 w(>) 28 b(P) p Fr 14 w(\)) p Fs( ) p Fl 2057 863 a(\033) p Fm 2103 848 a(i) g(\024) p Fs 2327 780 a(C) p Fr 7 w(\() p Fs(s) p Fr(\)) p 2285 825 283 4 v 2285 916 a(1) 22 b(+) p Fs 22 w(P) p Fl 2531 887 a(s) p Fs 2577 848 a(:) p Fw 257 1106 a(Pro) s(of) 49 b(:) p Fv 49 w(Using) 42 b(\(5.3\)) g(and) h(a) f(computation) f (similar) f(to) i(the) h(one) g(of) f(Prop) s(osition) 257 1227 y(4.11,) 32 b(one) h(gets) p Fm 463 1500 a(h) p Fs( ) p Fl 565 1515 a(\033) p Fr 612 1500 a(;) 17 b(1) -22 b(l) p Fm 21 w(\012) p Fs 22 w(N) p Fr 10 w(\() p Fm(j) p Fs(p) p Fm(j) p Fs 28 w(>) 27 b(P) p Fr 14 w(\)) p Fs( ) p Fl 1371 1515 a(\033) p Fm 1418 1500 a(i) g(\024) p Fp 1589 1360 a(\022) 1663 1365 y(Z) p Fs 1779 1500 a(dk) p Fm 1910 1433 a(j) p Fr 9 w(^) p Fs -58 w(\032) p Fq 1988 1448 a(2) p Fr 2028 1433 a(\() p Fs(k) p Fr 3 w(\)) p Fm(j) p Fq 2186 1397 a(2) p 1910 1478 315 4 v Fr 1926 1569 a(2) p Fs(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fq 2170 1540 a(3) p Fp 2235 1360 a(\023) p Fm 2330 1500 a(\002) p Fp 2430 1360 a(\022) 2503 1365 y(Z) p Fi 2559 1590 a(j) p Fl(p) p Fi(j) p Fl(>P) p Fs 2764 1500 a(dp) p Fm(j) p Fr 9 w(^) p Fs -58 w(\032) p Fq 2942 1515 a(1) p Fr 2981 1500 a(\() p Fs(p) p Fr(\)) p Fm(j) p Fq 3134 1459 a(2) p Fp 3173 1360 a(\023) p Fs 3262 1500 a(;) p Fv 257 1784 a(and) 33 b(the) g(result) f(follo) m(ws) g(.) p Fc 2187 w(2) p Fv 404 2024 a(Finally) -8 b(,) 41 b(to) g(con) m(trol) p Fs 40 w(N) p Fr 10 w(\() p Fm(j) p Fs(y) p Fm 4 w(j) p Fs 42 w(>) i(S) p Fr 6 w(\)) p Fs(;) p Fv 41 w(w) m(e) f(use) g(the) g (follo) m(wing) d(result) i(noting) f(that) 257 2145 y(d) p Fr 1 w(\000\(1) p Fm 21 w(\000) p Fs 23 w(F) p Fl 644 2160 a(S) p Fr 695 2145 a(\() p Fs(y) p Fr 4 w(\)\)) p Fm 27 w(\024) p Fs 28 w(N) p Fr 10 w(\() p Fm(j) p Fs(y) p Fm 4 w(j) p Fs 26 w(>) p Fl 1367 2105 a(S) p 1367 2122 47 4 v Fq 1373 2179 a(2) p Fr 1423 2145 a(\)) p Fs(:) p Fw 257 2348 a(Lemma) d(5.10.) p Ft 42 w(L) -5 b(et) p Fs 35 w(F) p Fm 42 w(2) p Fs 28 w(C) p Fi 1370 2312 a(1) p Fq 1363 2373 a(0) p Fr 1445 2348 a(\() p Fg(R) p Fl 1548 2312 a(n) p Fr 1601 2348 a(\)) p Ft 35 w(such) 35 b(that) p Fr 371 2568 a(0) p Fm 27 w(\024) p Fs 28 w(F) p Fr 14 w(\() p Fs(y) p Fr 4 w(\)) p Fm 27 w(\024) p Fr 28 w(1) p Fs(;) 116 b(F) p Fr 14 w(\() p Fs(y) p Fr 4 w(\)) 26 b(=) i(1) 69 b(for) p Fm 35 w(j) p Fr(y) p Fm 1 w(j) 27 b(\024) p Fr 29 w(1) p Fs(=) p Fr(2) p Fs(;) p Fr 115 w(and) 100 b(F\(y\)) 28 b(=) f(0) 70 b(for) p Fm 34 w(j) p Fr(y) p Fm 1 w(j) 28 b(\025) p Fr 28 w(1) p Fs(:) p Ft 257 2798 a(L) -5 b(et) p Fs 36 w(F) p Fl 489 2813 a(S) p Fr 539 2798 a(\() p Fs(y) p Fr 4 w(\)) 27 b(=) p Fs 28 w(F) p Fr 14 w(\() p Fi 923 2751 a(j) p Fl(y) p Fi 2 w(j) p 923 2775 77 4 v Fl 937 2833 a(S) p Fr 1009 2798 a(\)) p Fs(:) p Ft 35 w(Then) p Fr 1235 3018 a(lim) p Fl 1097 3080 a(\033) p Fi 2 w(!) p Fq(0) p Fl(;S) p Fi 4 w(!) p Fq(+) p Fi(1) p Fm 1508 3018 a(h) p Fs( ) p Fl 1610 3033 a(\033) p Fr 1657 3018 a(;) p Ft 17 w(d) p Fr(\000\(1) p Fm 22 w(\000) p Fs 22 w(F) p Fl 2083 3033 a(S) p Fr 2134 3018 a(\() p Fs(y) p Fr 4 w(\)\)) p Fs( ) p Fl 2363 3033 a(\033) p Fm 2409 3018 a(i) p Fr 28 w(=) g(0) p Fs(:) p Fw 257 3286 a(Pro) s(of) e(:) p Fv 26 w(There) e(is) f(a) f (similar) e(result) j(in) f([13]) h(\(Lemma) f(4.5\),) i(and) f(w) m(e) h(essen) m(tially) f(follo) m(w) 257 3406 y(its) 33 b(pro) s(of.) 46 b(The) 35 b(main) d(di\033erence) i(is) f(that) g(the) i(norm) d(of) p Fs 33 w(\032) p Fq 2472 3421 a(1) p Fr 2512 3406 a(\() p Fs(x) p Fm 23 w(\000) p Fs 24 w(Q) p Fr(\)) p Fv 34 w(as) h(an) h(op) s (erator) 257 3526 y(on) p Fs 33 w(L) p Fq 459 3490 a(2) p Fr 499 3526 a(\() p Fg(R) p Fl 603 3490 a(d) p Fr 649 3526 a(\)) p Fv 33 w(do) s(es) f(not) f(dep) s(end) i(on) p Fs 33 w(x) p Fv 33 w(and) e(is) h(therefore) g(not) f(square) i(in) m (tegrable) e(with) 257 3647 y(resp) s(ect) 27 b(to) f(this) f(v) -5 b(ariable.) 39 b(As) 27 b(in) e(Sect.) 42 b(4.2.2,) 26 b(to) g(con) m(trol) f(this) g(problem,) h(w) m(e) h(will) c(use) 257 3767 y(the) g(exp) s(onen) m(tial) e(deca) m(y) j(of) d(the) i(sp) s (ectral) f(pro) 5 b(jectors) 23 b(in) e(the) p Fs 22 w(Q) p Fv 23 w(v) -5 b(ariable) 20 b(\(Prop) s(osition) 257 3888 y(4.10\).) 43 b(First,) 32 b(one) h(easily) e(sees) j(that) 717 4163 y(d) p Fr(\000\(1) p Fm 22 w(\000) p Fs 22 w(F) p Fl 1103 4178 a(S) p Fr 1154 4163 a(\() p Fs(y) p Fr 4 w(\)\)) 27 b(=) p Fp 1450 4027 a(Z) p Fs 1567 4163 a(dx) 17 b(dk) i(a) p Fi 1862 4121 a(\003) p Fr 1901 4163 a(\() p Fs(x;) e(k) p Fr 3 w(\)\(1) p Fm 22 w(\000) p Fs 23 w(F) p Fr 14 w(\() p Fm 2464 4095 a(j) p Fs(D) p Fl 2573 4110 a(k) p Fm 2615 4095 a(j) p 2464 4140 180 4 v Fs 2520 4231 a(S) p Fr 2653 4163 a(\)\)) p Fs(a) p Fr(\() p Fs(x;) g(k) p Fr 3 w(\)) p Fs(:) p Fv 259 w(\(5.4\)) 257 4425 y(W) -8 b(e) 33 b(recall) e(that) i(for) f(an) m(y) p Fs 33 w(\033) p Fv 36 w(one) h(has) p Fs 569 4694 a(a) p Fr(\() p Fs(x;) 17 b(k) p Fr 3 w(\)) p Fs( ) p Fl 912 4709 a(\033) p Fr 988 4694 a(=) 27 b(\() p Fs(E) p Fl 1201 4709 a(\033) p Fm 1270 4694 a(\000) p Fs 23 w(H) p Fl 1459 4653 a(\033) p Fm 1527 4694 a(\000) p Fs 23 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)\)) p Fi 1860 4653 a(\000) p Fq(1) p Fs 1964 4627 a(\032) p Fq 2014 4642 a(1) p Fr 2053 4627 a(\() p Fs(x) p Fm 23 w(\000) p Fs 22 w(Q) p Fr(\)) 9 b(^) p Fs -58 w(\032) p Fq 2433 4642 a(2) p Fr 2473 4627 a(\() p Fs(k) p Fr 3 w(\)) p 1964 4671 640 4 v Fp 2112 4691 a(p) p 2212 4691 244 4 v Fr 2212 4777 a(2) p Fs(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fs 2613 4694 a(\037) p Fl 2674 4710 a(\033) p Fi 2 w(\024) p Fl(!) p Fq 2 w(\() p Fl(k) p Fq 2 w(\)) p Fr 2916 4694 a(\() p Fs(k) p Fr 3 w(\)) p Fs( ) p Fl 3109 4709 a(\033) p Fs 3156 4694 a(:) p Fv 257 4990 a(Then) 34 b(one) f(can) g(pro) m(v) m(e) g(\([13],) g(Prop) f (4.4\)) g(that) p Fr 665 5264 a(lim) p Fl 658 5324 a(\033) p Fi 2 w(!) p Fq(0) p Fs 824 5264 a(a) p Fr(\() p Fs(x;) 17 b(k) p Fr 3 w(\)) p Fs( ) p Fl 1167 5279 a(\033) p Fm 1236 5264 a(\000) p Fr 23 w(\() p Fs(E) p Fq 1446 5279 a(0) p Fm 1508 5264 a(\000) p Fs 22 w(H) p Fm 30 w(\000) p Fs 23 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)\)) p Fi 2051 5223 a(\000) p Fq(1) p Fs 2154 5197 a(\032) p Fq 2204 5212 a(1) p Fr 2244 5197 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(Q) p Fr(\)) 9 b(^) p Fs -58 w(\032) p Fq 2624 5212 a(2) p Fr 2664 5197 a(\() p Fs(k) p Fr 3 w(\)) p 2154 5242 640 4 v Fp 2303 5261 a(p) p 2402 5261 244 4 v Fr 2402 5347 a(2) p Fs(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fs 2804 5264 a( ) p Fl 2867 5279 a(\033) p Fr 2942 5264 a(=) 27 b(0) p Fv 1828 5637 a(26) p 90 rotate dyy eop %%Page: 27 27 27 26 bop Fv 257 573 a(in) p Fs 32 w(L) p Fq 437 537 a(2) p Fr 477 573 a(\() p Fg(R) p Fl 581 537 a(d) p Fq(+) p Fl(n) p Fs 725 573 a(;) 17 b(dx) g(dk) p Fr 3 w(;) p Fm 17 w(H) p Fr 1 w(\)) p Fs(:) p Fv 32 w(Using) 32 b(this) g(together) h(with) f(\(5.4\),) g(w) m(e) h(then) g(ha) m(v) m(e) p Fm 299 774 a(h) p Fs( ) p Fl 401 789 a(\033) p Fr 448 774 a(;) p Fv 17 w(d) p Fr(\000\(1) p Fm 22 w(\000) p Fs 22 w(F) p Fl 878 789 a(S) p Fr 929 774 a(\() p Fs(y) p Fr 4 w(\)\)) p Fs( ) p Fl 1158 789 a(\033) p Fm 1204 774 a(i) p Fr 494 908 a(=) p Fp 597 827 a(R) p Fs 681 908 a(dx) 17 b(dk) p Fm 19 w(h) p Fr(\() p Fs(E) p Fq 1074 923 a(0) p Fm 1135 908 a(\000) p Fs 23 w(H) p Fm 29 w(\000) p Fs 23 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)\)) p Fi 1678 871 a(\000) p Fq(1) p Fl 1781 860 a(\032) p Fj 1817 869 a(1) p Fq 1852 860 a(\() p Fl(x) p Fi(\000) p Fl(Q) p Fq(\)) 6 b(^) p Fl -41 w(\032) p Fj 2093 869 a(2) p Fq 2128 860 a(\() p Fl(k) p Fq 2 w(\)) p 1781 885 440 4 v Fm 1872 897 a(p) p 1955 897 175 4 v Fq 1955 963 a(2) p Fl(!) p Fq 2 w(\() p Fl(k) p Fq 2 w(\)) p Fs 2231 908 a( ) p Fl 2294 923 a(\033) p Fr 2342 908 a(;) 1079 1090 y(\(1) p Fm 22 w(\000) p Fs 23 w(F) p Fr 14 w(\() p Fi 1413 1042 a(j) p Fl(D) p Fn 1491 1054 a(k) p Fi 1528 1042 a(j) p 1413 1067 136 4 v Fl 1456 1124 a(S) p Fr 1557 1090 a(\)\)\() p Fs(E) p Fq 1743 1105 a(0) p Fm 1805 1090 a(\000) p Fs 23 w(H) p Fm 29 w(\000) p Fs 23 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)\)) p Fi 2348 1053 a(\000) p Fq(1) p Fl 2451 1042 a(\032) p Fj 2487 1051 a(1) p Fq 2522 1042 a(\() p Fl(x) p Fi(\000) p Fl(Q) p Fq(\)) g(^) p Fl -41 w(\032) p Fj 2763 1051 a(2) p Fq 2798 1042 a(\() p Fl(k) p Fq 2 w(\)) p 2451 1067 440 4 v Fm 2542 1079 a(p) p 2625 1079 175 4 v Fq 2625 1145 a(2) p Fl(!) p Fq 2 w(\() p Fl(k) p Fq 2 w(\)) p Fs 2901 1090 a( ) p Fl 2964 1105 a(\033) p Fm 3011 1090 a(i) p Fr 22 w(+) p Fs 22 w(o) p Fr(\() p Fs(\033) p Fq 3314 1053 a(0) p Fr 3354 1090 a(\)) p Fm 494 1272 a(\024) 28 b(k) p Fr(\() p Fs(E) p Fq 759 1287 a(0) p Fm 821 1272 a(\000) p Fs 22 w(H) p Fm 30 w(\000) p Fs 23 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)\)) p Fi 1364 1235 a(\000) p Fq(1) p Fl 1467 1224 a(\032) p Fj 1503 1233 a(1) p Fq 1538 1224 a(\() p Fl(x) p Fi(\000) p Fl(Q) p Fq(\)) 6 b(^) p Fl -41 w(\032) p Fj 1779 1233 a(2) p Fq 1813 1224 a(\() p Fl(k) p Fq 2 w(\)) p 1467 1249 440 4 v Fm 1558 1261 a(p) p 1641 1261 175 4 v Fq 1641 1327 a(2) p Fl(!) p Fq 2 w(\() p Fl(k) p Fq 2 w(\)) p Fs 1917 1272 a( ) p Fl 1980 1287 a(\033) p Fm 2027 1272 a(k) p Fl 2077 1291 a(L) p Fj 2125 1272 a(2) p Fq 2160 1291 a(\() p Fo(R) p Fn 2235 1272 a(d) p Fj(+) p Fn(n) p Fq 2356 1291 a(;) p Fi(H) p Fq(\)) p Fm 689 1454 a(\002k) p Fr(\(1) p Fm 22 w(\000) p Fs 23 w(F) p Fr 14 w(\() p Fi 1150 1406 a(j) p Fl(D) p Fn 1228 1418 a(k) p Fi 1265 1406 a(j) p 1150 1431 136 4 v Fl 1194 1488 a(S) p Fr 1295 1454 a(\)\)\() p Fs(E) p Fq 1481 1469 a(0) p Fm 1542 1454 a(\000) p Fs 23 w(H) p Fm 29 w(\000) p Fs 23 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)\)) p Fi 2085 1417 a(\000) p Fq(1) p Fl 2189 1406 a(\032) p Fj 2225 1415 a(1) p Fq 2259 1406 a(\() p Fl(x) p Fi(\000) p Fl(Q) p Fq(\)) g(^) p Fl -41 w(\032) p Fj 2500 1415 a(2) p Fq 2535 1406 a(\() p Fl(k) p Fq 2 w(\)) p 2189 1431 440 4 v Fm 2279 1443 a(p) p 2363 1443 175 4 v Fq 2363 1509 a(2) p Fl(!) p Fq 2 w(\() p Fl(k) p Fq 2 w(\)) p Fs 2638 1454 a( ) p Fl 2701 1469 a(\033) p Fm 2749 1454 a(k) p Fl 2799 1473 a(L) p Fj 2847 1454 a(2) p Fq 2881 1473 a(\() p Fo(R) p Fn 2956 1454 a(d) p Fj(+) p Fn(n) p Fq 3078 1473 a(;) p Fi(H) p Fq(\)) p Fr 3211 1454 a(+) p Fs 22 w(o) p Fr(\() p Fs(\033) p Fq 3453 1417 a(0) p Fr 3492 1454 a(\)) p Fm 494 1647 a(\024) 28 b(k) p Fr(\() p Fs(E) p Fq 759 1662 a(0) p Fm 821 1647 a(\000) p Fs 22 w(H) p Fm 30 w(\000) p Fs 23 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)\)) p Fi 1364 1610 a(\000) p Fq(1) p Fl 1467 1599 a(\032) p Fj 1503 1608 a(1) p Fq 1538 1599 a(\() p Fl(x) p Fi(\000) p Fl(Q) p Fq(\)) p Fl(e) p Fk 1776 1576 a(\000) p Fn(\013) p Fk(j) p Fn(Q) p Fk(j) p Fq 1960 1599 a(^) p Fl -41 w(\032) p Fj 1990 1608 a(2) p Fq 2024 1599 a(\() p Fl(k) p Fq 2 w(\)) p 1467 1624 651 4 v Fm 1663 1636 a(p) p 1746 1636 175 4 v Fq 1746 1702 a(2) p Fl(!) p Fq 2 w(\() p Fl(k) p Fq 2 w(\)) p Fm 2128 1647 a(k) p Fl 2178 1666 a(L) p Fj 2226 1647 a(2) p Fq 2260 1666 a(\() p Fo(R) p Fn 2335 1647 a(d) p Fj(+) p Fn(n) p Fq 2457 1666 a(;) p Fi(B) p Fq 2 w(\() p Fi(H) p Fq(\)\)) p Fm 2693 1647 a(\002) 23 b(k) p Fs(e) p Fl 2888 1610 a(\013) p Fi(j) p Fl(Q) p Fi(j) p Fs 3032 1647 a( ) p Fl 3095 1662 a(\033) p Fm 3142 1647 a(k) p Fi 3192 1662 a(H) p Fm 689 1840 a(\002k) p Fr(\(1) p Fm 22 w(\000) p Fs 23 w(F) p Fr 14 w(\() p Fi 1150 1792 a(j) p Fl(D) p Fn 1228 1804 a(k) p Fi 1265 1792 a(j) p 1150 1817 136 4 v Fl 1194 1874 a(S) p Fr 1295 1840 a(\)\)\() p Fs(E) p Fq 1481 1855 a(0) p Fm 1542 1840 a(\000) p Fs 23 w(H) p Fm 29 w(\000) p Fs 23 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)\)) p Fi 2085 1803 a(\000) p Fq(1) p Fl 2189 1792 a(\032) p Fj 2225 1801 a(1) p Fq 2259 1792 a(\() p Fl(x) p Fi(\000) p Fl(Q) p Fq(\)) p Fl(e) p Fk 2497 1769 a(\000) p Fn(\013) p Fk(j) p Fn(Q) p Fk(j) p Fq 2681 1792 a(^) p Fl -41 w(\032) p Fj 2711 1801 a(2) p Fq 2746 1792 a(\() p Fl(k) p Fq 2 w(\)) p 2189 1817 651 4 v Fm 2385 1829 a(p) p 2468 1829 175 4 v Fq 2468 1895 a(2) p Fl(!) p Fq 2 w(\() p Fl(k) p Fq 2 w(\)) p Fm 2849 1840 a(k) p Fl 2899 1859 a(L) p Fj 2947 1840 a(2) p Fq 2982 1859 a(\() p Fo(R) p Fn 3057 1840 a(d) p Fj(+) p Fn(n) p Fq 3178 1859 a(;) p Fi(B) p Fq 2 w(\() p Fi(H) p Fq(\)\)) p Fm 884 2010 a(\002k) p Fs(e) p Fl 1056 1974 a(\013) p Fi(j) p Fl(Q) p Fi(j) p Fs 1201 2010 a( ) p Fl 1264 2025 a(\033) p Fm 1311 2010 a(k) p Fi 1361 2025 a(H) p Fr 1447 2010 a(+) p Fs 22 w(o) p Fr(\() p Fs(\033) p Fq 1689 1974 a(0) p Fr 1729 2010 a(\)) p Fs(:) p Fv 257 2247 a(W) -8 b(e) 28 b(c) m(hec) m(k) h (that) p Fr 27 w(\() p Fs(E) p Fq 993 2262 a(0) p Fm 1044 2247 a(\000) p Fs 11 w(H) p Fm 19 w(\000) p Fs 11 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)\)) p Fi 1553 2211 a(\000) p Fq(1) p Fl 1657 2199 a(\032) p Fj 1693 2208 a(1) p Fq 1728 2199 a(\() p Fl(x) p Fi(\000) p Fl(Q) p Fq(\)) p Fl(e) p Fk 1966 2176 a(\000) p Fn(\013) p Fk(j) p Fn(Q) p Fk(j) p Fq 2150 2199 a(^) p Fl -41 w(\032) p Fj 2180 2208 a(2) p Fq 2214 2199 a(\() p Fl(k) p Fq 2 w(\)) p 1657 2224 651 4 v Fm 1853 2236 a(p) p 1936 2236 175 4 v Fq 1936 2302 a(2) p Fl(!) p Fq 2 w(\() p Fl(k) p Fq 2 w(\)) p Fv 2345 2247 a(b) s(elongs) d(to) p Fs 27 w(L) p Fq 2869 2211 a(2) p Fr 2909 2247 a(\() p Fg(R) p Fl 3013 2211 a(d) p Fq(+) p Fl(n) p Fr 3157 2247 a(;) p Fm 17 w(B) p Fr 3 w(\() p Fm(H) p Fr 1 w(\)\)) p Fs(;) p Fv 257 2419 a(using) 31 b(the) g(fact) g(that) p Fm 31 w(k) p Fr(\() p Fs(E) p Fq 1238 2434 a(0) p Fm 1296 2419 a(\000) p Fs 20 w(H) p Fm 26 w(\000) p Fs 20 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)\)) p Fi 1830 2383 a(\000) p Fq(1) p Fm 1923 2419 a(k) d(\024) p Fs 28 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fi 2301 2383 a(\000) p Fq(1) p Fv 2425 2419 a(and) j(condition) f(\(IR\).) h(Th) m(us) p Fr 311 2689 a(lim) p Fl 257 2751 a(S) p Fi 4 w(!) p Fq(+) p Fi(1) p Fm 517 2689 a(k) p Fr(\(1) p Fm(\000) p Fs(F) p Fr 14 w(\() p Fm 856 2622 a(j) p Fs(D) p Fl 965 2637 a(k) p Fm 1007 2622 a(j) p 856 2666 180 4 v Fs 912 2758 a(S) p Fr 1045 2689 a(\)\)\() p Fs(E) p Fq 1231 2704 a(0) p Fm 1270 2689 a(\000) p Fs(H) p Fm 8 w(\000) p Fs(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)\)) p Fi 1746 2648 a(\000) p Fq(1) p Fs 1851 2622 a(\032) p Fq 1901 2637 a(1) p Fr 1941 2622 a(\() p Fs(x) p Fm 22 w(\000) p Fs 23 w(Q) p Fr(\)) p Fs(e) p Fi 2316 2586 a(\000) p Fl(\013) p Fi(j) p Fl(Q) p Fi(j) p Fr 2524 2622 a(^) p Fs -58 w(\032) p Fq 2565 2637 a(2) p Fr 2605 2622 a(\() p Fs(k) p Fr 3 w(\)) p 1851 2666 885 4 v Fp 2122 2686 a(p) p 2221 2686 244 4 v Fr 2221 2771 a(2) p Fs(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fm 2745 2689 a(k) p Fl 2795 2708 a(L) p Fj 2843 2689 a(2) p Fq 2877 2708 a(\() p Fo(R) p Fn 2952 2689 a(d) p Fj(+) p Fn(n) p Fq 3074 2708 a(;) p Fi(B) p Fq 2 w(\() p Fi(H) p Fq(\)\)) p Fr 3316 2689 a(=) c(0) p Fs(:) p Fv 257 2980 a(Moreo) m(v) m(er) p Fm 30 w(k) p Fs(e) p Fl 776 2944 a(\013) p Fi(j) p Fl(Q) p Fi(j) p Fs 920 2980 a( ) p Fl 983 2995 a(\033) p Fm 1030 2980 a(k) p Fi 1080 2995 a(H) p Fv 1172 2980 a(is) h(uniformly) e(b) s (ounded) i(\(w.r.t) p Fs 29 w(\033) p Fv 4 w(\),) h(whic) m(h) f(can) g (b) s(e) h(pro) m(v) m(en) g(as) 257 3100 y(for) p Fm 32 w(k) p Fs(e) p Fl 501 3064 a(\013) p Fi(j) p Fl(Q) p Fi(j) p Fs 646 3100 a( ) p Fl 709 3115 a(m) p Fr 776 3100 a(\() p Fs(L) p Fr(\)) p Fm(k) p Fi 968 3115 a(H) p Fv 1064 3100 a(\(see) 34 b(Sect.) 44 b(4.2\),) 32 b(and) h(the) g (result) f(follo) m(ws.) p Fc 745 w(2) p Ff 257 3507 a(5.3) 131 b(Pro) t(of) 43 b(of) h(Theorem) f(3.3) p Fv 257 3692 a(W) -8 b(e) 46 b(ha) m(v) m(e) h(seen) g(that) f(the) g (only) f(thing) g(w) m(e) h(had) g(to) g(c) m(hec) m(k) h(w) m(as) g (condition) p Fr 44 w(\() p Fs(iv) p Fr 4 w(\)) p Fv 46 w(of) 257 3812 y(Lemma) 27 b(5.1.) 42 b(The) 29 b(unit) e(ball) f (of) p Fm 28 w(H) p Fv 29 w(is) h(w) m(eakly) i(compact,) g(so) f (there) h(exists) g(a) f(sequence) p Fs 257 3933 a(\033) p Fl 312 3948 a(n) p Fm 391 3933 a(!) p Fr 30 w(0) p Fv 35 w(and) p Fs 34 w( ) p Fm 35 w(2) j(H) p Fv 36 w(suc) m(h) 36 b(that) p Fs 34 w( ) p Fl 1609 3948 a(\033) p Fn 1649 3956 a(n) p Fv 1731 3933 a(con) m(v) m(erges) g(w) m(eakly) g(to) p Fs 34 w( ) t(:) p Fv 34 w(It) f(then) g(su\036ces) i(to) 257 4053 y(pro) m(v) m(e) j(that) p Fs 38 w( ) p Fm 42 w(6) p Fr(=) e(0) p Fs(:) p Fv 38 w(The) i(idea) e(is) g(to) g(\034nd) h(a) f (compact) g(op) s(erator) p Fs 38 w(K) p Fv 46 w(suc) m(h) h(that) g (for) 257 4173 y(an) m(y) p Fs 34 w(n) p Fv 32 w(large) 32 b(enough) h(one) f(has) h(suc) m(h) h(an) f(estimate:) p Fm 1513 4378 a(k) p Fs(K) 7 b( ) p Fl 1716 4393 a(\033) p Fn 1756 4401 a(n) p Fm 1803 4378 a(k) 28 b(\025) p Fs 28 w(\016) j(>) p Fr 28 w(0) p Fs(:) p Fv 1055 w(\(5.5\)) 257 4582 y(This) 38 b(will) d(ensure) k(that) p Fs 37 w( ) p Fv 42 w(is) e(non) g(zero.) 59 b(Indeed,) p Fs 40 w(K) p Fv 45 w(is) 37 b(compact,) h(so) p Fs 38 w(K) 7 b( ) p Fl 3142 4597 a(\033) p Fn 3182 4605 a(n) p Fv 3267 4582 a(tends) 257 4702 y(strongly) 35 b(to) p Fs 35 w(K) 7 b( ) t(:) p Fv 36 w(If) p Fs 36 w( ) p Fv 39 w(w) m(as) 36 b(zero) g(then) p Fm 36 w(k) p Fs(K) 7 b( ) p Fl 2009 4717 a(\033) p Fn 2049 4725 a(n) p Fm 2097 4702 a(k) p Fv 35 w(w) m(ould) 35 b(go) g(to) g(zero,) i(whic) m(h) f(en) m(ters) 257 4823 y(in) c(con) m(tradiction) f(with) i(\(5.5\).) 404 4943 y(Let) e(us) g(then) g(tak) m(e) p Fs 32 w(F) p Fm 41 w(2) p Fs 28 w(C) p Fi 1405 4907 a(1) p Fq 1398 4968 a(0) p Fr 1480 4943 a(\() p Fg(R) p Fl 1584 4907 a(n) p Fr 1637 4943 a(\)) p Fv 30 w(and) p Fs 31 w(G) p Fm 28 w(2) p Fs 28 w(C) p Fi 2169 4907 a(1) p Fq 2162 4968 a(0) p Fr 2243 4943 a(\() p Fg(R) p Fl 2347 4907 a(d) p Fr 2394 4943 a(\)) p Fv 30 w(satisfying) f(the) h(conditions) 257 5063 y(of) 45 b(Lemma) f(5.10.) 82 b(Remem) m(b) s(ering) 44 b(that) p Fs 45 w(p) p Fv 46 w(is) h(the) g(v) -5 b(ariable) 44 b(conjugate) i(to) p Fs 45 w(x;) p Ft 46 w(i.e.) p Fs 257 5184 a(p) p Fr 28 w(=) p Fm 28 w(\000) p Fs(i) p Fm(r) p Fl 631 5199 a(x) p Fv 708 5184 a(on) p Fs 32 w(L) p Fq 909 5148 a(2) p Fr 949 5184 a(\() p Fg(R) p Fl 1053 5148 a(d) p Fq(+) p Fl(n) p Fs 1197 5184 a(;) 17 b(dx) g(dk) p Fr 3 w(\)) p Fs(;) p Fv 32 w(one) 33 b(has) f(the) h (follo) m(wing) d(inequalities:) p Fr 785 5388 a(\(1) p Fm 22 w(\000) p Fr 22 w(\000\() p Fs(F) p Fl 1155 5403 a(S) p Fr 1206 5388 a(\() p Fs(y) p Fr 4 w(\)\)\)) p Fq 1410 5347 a(2) p Fm 1476 5388 a(\024) p Fr 28 w(\(1) p Fm 22 w(\000) p Fr 23 w(\000\() p Fs(F) p Fl 1952 5403 a(S) p Fr 2003 5388 a(\() p Fs(y) p Fr 4 w(\)\)\)) p Fm 26 w(\024) p Fv 28 w(d) p Fr(\000\(1) p Fm 22 w(\000) p Fs 23 w(F) p Fl 2725 5403 a(S) p Fr 2776 5388 a(\() p Fs(y) p Fr 4 w(\)\)) p Fs(;) p Fv 326 w(\(5.6\)) 1828 5637 y(27) p 90 rotate dyy eop %%Page: 28 28 28 27 bop Fr 330 609 a(\(1) p Fm 22 w(\000) p Fr 22 w(\000\() p Fs(G) p Fl 714 624 a(R) p Fr 772 609 a(\() p Fs(x) p Fr(\)\)\)) p Fq 979 568 a(2) p Fm 1046 609 a(\024) p Fr 28 w(\(1) p Fm 22 w(\000) p Fr 23 w(\000\() p Fs(G) p Fl 1536 624 a(R) p Fr 1593 609 a(\() p Fs(x) p Fr(\)\)\)) p Fm 28 w(\024) p Fv 28 w(d) p Fr 1 w(\000\(1) p Fm 21 w(\000) p Fs 23 w(G) p Fl 2334 624 a(R) p Fr 2392 609 a(\() p Fs(x) p Fr(\)\)) p Fm 28 w(\024) p Fs 28 w(N) p Fr 10 w(\() p Fm(j) p Fs(x) p Fm(j) p Fs 28 w(>) 3072 541 y(R) p 3072 586 76 4 v Fr 3085 677 a(2) 3157 609 y(\)) p Fs(;) p Fv 73 w(\(5.7\)) p Fr 341 867 a(\(1) p Fm 22 w(\000) p Fr 22 w(\000\() p Fs(G) p Fl 725 882 a(P) p Fr 784 867 a(\() p Fs(p) p Fr(\)\)\)) p Fq 985 826 a(2) p Fm 1052 867 a(\024) p Fr 28 w(\(1) p Fm 22 w(\000) p Fr 22 w(\000\() p Fs(G) p Fl 1541 882 a(P) p Fr 1600 867 a(\() p Fs(p) p Fr(\)\)\)) p Fm 27 w(\024) p Fv 28 w(d) p Fr(\000\(1) p Fm 22 w(\000) p Fs 23 w(G) p Fl 2334 882 a(P) p Fr 2392 867 a(\() p Fs(p) p Fr(\)\)) p Fm 28 w(\024) p Fs 28 w(N) p Fr 10 w(\() p Fm(j) p Fs(p) p Fm(j) p Fs 28 w(>) 3060 800 y(P) p 3060 844 77 4 v Fr 3074 935 a(2) 3146 867 y(\)) p Fs(:) p Fv 84 w(\(5.8\)) 257 1064 y(Finally) -8 b(,) 32 b(let) p Fs 34 w(\037) p Fr(\() p Fs(s) p Fm 31 w(\024) p Fs 31 w(s) p Fq 1083 1079 a(0) p Fr 1122 1064 a(\)) p Fv 35 w(b) s(e) i(a) g(function) g(with) g(supp) s (ort) h(in) p Fm 33 w(fj) p Fs(s) p Fm(j) 30 b(\024) p Fs 31 w(s) p Fq 2834 1079 a(0) p Fm 2874 1064 a(g) p Fv 34 w(and) k(equal) h(to) p Fr 257 1185 a(1) p Fv 33 w(in) p Fm 31 w(fj) p Fs(s) p Fm(j) 27 b(\024) p Fl 746 1145 a(s) p Fj 779 1154 a(0) p 746 1162 68 4 v Fq 762 1219 a(2) p Fm 824 1185 a(g) p Fs(:) p Fv 32 w(F) -8 b(or) 32 b(an) m(y) h(non) g(negativ) m(e) p Fs 32 w(\022) s(;) 17 b(P) s(;) g(R) p Fv 33 w(and) p Fs 33 w(S;) p Fv 33 w(w) m(e) 33 b(de\034ne) p Fs 396 1386 a(K) p Fr 7 w(\() p Fs(\022) s(;) 17 b(P) s(;) g(R) q(;) g(S) p Fr 6 w(\)) 27 b(:=) p Fs 27 w(\037) p Fr(\() p Fs(N) p Fm 38 w(\024) p Fs 29 w(\022) p Fr 3 w(\)) p Fs(\037) p Fr(\() p Fs(H) p Fq 1693 1401 a(0) p Fm 1760 1386 a(\024) p Fs 28 w(\022) p Fr 3 w(\)\000\() p Fs(F) p Fl 2113 1401 a(S) p Fr 2164 1386 a(\() p Fs(y) p Fr 4 w(\)\)\000\() p Fs(G) p Fl 2506 1401 a(R) p Fr 2562 1386 a(\() p Fs(x) p Fr(\)\)\000\() p Fs(G) p Fl 2907 1401 a(P) p Fr 2966 1386 a(\() p Fs(p) p Fr(\)\)) p Fs(:) p Fv 139 w(\(5.9\)) 257 1588 y(The) 37 b(assumptions) e(on) p Fs 36 w(F) s(;) 17 b(G;) g(\037) p Fv 34 w(and) p Fs 36 w(!) p Fv 39 w(ensure) 37 b(that) p Fs 35 w(K) p Fr 7 w(\() p Fs(\022) s(;) 17 b(P) s(;) g(R) q(;) g(S) p Fr 6 w(\)) p Fv 35 w(is) 35 b(compact) g(for) 257 1708 y(an) m(y) p Fs 34 w(\022) s(;) 17 b(P) s(;) g(R) p Fv 32 w(and) p Fs 33 w(S:) p Fv 404 1829 a(Using) 36 b(Lemmas) g(5.6) h (and) g(5.7,) h(there) g(exists) p Fs 38 w(\022) p Fq 2192 1844 a(0) p Fs 2267 1829 a(>) p Fr 36 w(0) p Fv 36 w(suc) m(h) h(that,) f(for) f(all) p Fs 35 w(n;) p Fv 37 w(one) 257 1949 y(has:) p Fm 547 2170 a(k) p Fr(\(1) p Fm 22 w(\000) p Fs 22 w(\037) p Fr(\() p Fs(N) p Fm 39 w(\024) p Fs 28 w(\022) p Fr 3 w(\)\)) p Fs( ) p Fl 1313 2185 a(\033) p Fn 1353 2193 a(n) p Fm 1400 2170 a(k) 28 b(\024) p Fr 1617 2103 a(1) p 1593 2147 98 4 v 1593 2239 a(10) p Fs 1700 2170 a(;) p Fm 17 w(k) p Fr(\(1) p Fm 22 w(\000) p Fs 22 w(\037) p Fr(\() p Fs(H) p Fq 2182 2185 a(0) p Fm 2249 2170 a(\024) p Fs 29 w(\022) p Fr 3 w(\)\)) p Fs( ) p Fl 2542 2185 a(\033) p Fn 2582 2193 a(n) p Fm 2629 2170 a(k) f(\024) p Fr 2846 2103 a(1) p 2822 2147 V 2822 2239 a(10) p Fs 2929 2170 a(:) p Fv 290 w(\(5.10\)) 257 2403 y(Lik) m(ewise,) f(using) e(Lemmas) e(5.8) h (and) h(5.9) f(together) h(with) f(inequalities) f(\(5.7\)) h(and) h (\(5.8\),) 257 2523 y(there) 34 b(exist) p Fs 32 w(R) p Fq 811 2538 a(0) p Fs 851 2523 a(;) 17 b(P) p Fq 958 2538 a(0) p Fs 1025 2523 a(>) p Fr 27 w(0) p Fv 33 w(suc) m(h) 34 b(that,) e(for) g(all) p Fs 30 w(n;) p Fv 33 w(one) h(has:) p Fm 570 2764 a(k) p Fr(\(1) p Fm 21 w(\000) p Fr 23 w(\000\() p Fs(G) p Fl 1004 2779 a(R) p Fr 1061 2764 a(\() p Fs(x) p Fr(\)\)\)) p Fs( ) p Fl 1331 2779 a(\033) p Fn 1371 2787 a(n) p Fm 1419 2764 a(k) 27 b(\024) p Fr 1636 2696 a(1) p 1612 2741 V 1612 2832 a(10) p Fs 1719 2764 a(;) p Fm 17 w(k) p Fr(\(1) p Fm 21 w(\000) p Fr 23 w(\000\() p Fs(G) p Fl 2197 2779 a(P) p Fr 2256 2764 a(\() p Fs(p) p Fr(\)\)\)) p Fs( ) p Fl 2520 2779 a(\033) p Fn 2560 2787 a(n) p Fm 2607 2764 a(k) g(\024) p Fr 2824 2696 a(1) p 2799 2741 V 2799 2832 a(10) p Fs 2907 2764 a(:) p Fv 312 w(\(5.11\)) 257 2996 y(Finally) -8 b(,) 37 b(using) h(Lemma) f (5.10) h(and) g(\(5.6\),) h(there) g(exist) p Fs 38 w(S) p Fq 2455 3011 a(0) p Fs 2532 2996 a(>) p Fr 38 w(0) p Fv 37 w(and) p Fs 39 w(n) p Fq 2986 3011 a(0) p Fv 3064 2996 a(suc) m(h) g(that,) 257 3117 y(for) 32 b(all) p Fs 31 w(n) p Fm 28 w(\025) p Fs 28 w(n) p Fq 791 3132 a(0) p Fs 831 3117 a(;) p Fv 32 w(one) h(has:) p Fm 1300 3357 a(k) p Fr(\(1) p Fm 22 w(\000) p Fr 23 w(\000\() p Fs(F) p Fl 1721 3372 a(S) p Fr 1772 3357 a(\() p Fs(y) p Fr 4 w(\)\)\)) p Fs( ) p Fl 2039 3372 a(\033) p Fn 2079 3380 a(n) p Fm 2125 3357 a(k) 27 b(\024) p Fr 2342 3290 a(1) p 2317 3334 V 2317 3426 a(10) p Fs 2425 3357 a(:) p Fv 794 w(\(5.12\)) 257 3584 y(Then,) 34 b(for) e(an) m(y) p Fs 33 w(n) p Fm 28 w(\025) p Fs 28 w(n) p Fq 1121 3599 a(0) p Fr 1189 3584 a(:) p Fm 257 3786 a(k) p Fs( ) p Fl 370 3801 a(\033) p Fn 410 3809 a(n) p Fm 458 3786 a(k) 82 b(\024) i(k) p Fr(\(1) p Fm 22 w(\000) p Fs 22 w(\037) p Fr(\() p Fs(N) p Fm 38 w(\024) p Fs 29 w(\022) p Fq 1375 3801 a(0) p Fr 1415 3786 a(\)\)) p Fs( ) p Fl 1554 3801 a(\033) p Fn 1594 3809 a(n) p Fm 1641 3786 a(k) p Fr 22 w(+) p Fm 22 w(k) p Fs(\037) p Fr(\() p Fs(N) p Fm 38 w(\024) p Fs 28 w(\022) p Fq 2226 3801 a(0) p Fr 2266 3786 a(\)\(1) p Fm 22 w(\000) p Fs 22 w(\037) p Fr(\() p Fs(H) p Fq 2692 3801 a(0) p Fm 2759 3786 a(\024) p Fs 28 w(\022) p Fq 2909 3801 a(0) p Fr 2949 3786 a(\)\)) p Fs( ) p Fl 3088 3801 a(\033) p Fn 3128 3809 a(n) p Fm 3176 3786 a(k) p Fr 751 3931 a(+) p Fm(k) p Fs(\037) p Fr(\() p Fs(N) p Fm 38 w(\024) p Fs 28 w(\022) p Fq 1242 3946 a(0) p Fr 1282 3931 a(\)) p Fs(\037) p Fr(\() p Fs(H) p Fq 1500 3946 a(0) p Fm 1567 3931 a(\024) p Fs 28 w(\022) p Fq 1717 3946 a(0) p Fr 1757 3931 a(\)\(1) p Fm 22 w(\000) p Fr 22 w(\000\() p Fs(G) p Fl 2179 3946 a(R) p Fj 2232 3955 a(0) p Fr 2271 3931 a(\() p Fs(x) p Fr(\)\)\)) p Fs( ) p Fl 2541 3946 a(\033) p Fn 2581 3954 a(n) p Fm 2629 3931 a(k) p Fr 751 4076 a(+) p Fm(k) p Fs(\037) p Fr(\() p Fs(N) p Fm 38 w(\024) p Fs 28 w(\022) p Fq 1242 4091 a(0) p Fr 1282 4076 a(\)) p Fs(\037) p Fr(\() p Fs(H) p Fq 1500 4091 a(0) p Fm 1567 4076 a(\024) p Fs 28 w(\022) p Fq 1717 4091 a(0) p Fr 1757 4076 a(\)\000\() p Fs(G) p Fl 1971 4091 a(R) p Fj 2024 4100 a(0) p Fr 2063 4076 a(\() p Fs(x) p Fr(\)\)\(1) p Fm 22 w(\000) p Fr 22 w(\000\() p Fs(G) p Fl 2616 4091 a(P) p Fj 2661 4100 a(0) p Fr 2700 4076 a(\() p Fs(p) p Fr(\)\)\)) p Fs( ) p Fl 2964 4091 a(\033) p Fn 3004 4099 a(n) p Fm 3051 4076 a(k) p Fr 751 4221 a(+) p Fm(k) p Fs(\037) p Fr(\() p Fs(N) p Fm 38 w(\024) p Fs 28 w(\022) p Fq 1242 4236 a(0) p Fr 1282 4221 a(\)) p Fs(\037) p Fr(\() p Fs(H) p Fq 1500 4236 a(0) p Fm 1567 4221 a(\024) p Fs 28 w(\022) p Fq 1717 4236 a(0) p Fr 1757 4221 a(\)\000\() p Fs(G) p Fl 1971 4236 a(R) p Fj 2024 4245 a(0) p Fr 2063 4221 a(\() p Fs(x) p Fr(\)\)\000\() p Fs(G) p Fl 2408 4236 a(P) p Fj 2453 4245 a(0) p Fr 2491 4221 a(\() p Fs(p) p Fr(\)\)\(1) p Fm 22 w(\000) p Fr 22 w(\000\() p Fs(F) p Fl 3024 4236 a(S) p Fj 3067 4245 a(0) p Fr 3106 4221 a(\() p Fs(y) p Fr 4 w(\)\)\)) p Fs( ) p Fl 3373 4236 a(\033) p Fn 3413 4244 a(n) p Fm 3459 4221 a(k) p Fr 751 4367 a(+) p Fm(k) p Fs(K) p Fr 7 w(\() p Fs(\022) p Fq 1050 4382 a(0) p Fs 1090 4367 a(;) 17 b(P) p Fq 1197 4382 a(0) p Fs 1236 4367 a(;) g(R) p Fq 1354 4382 a(0) p Fs 1393 4367 a(;) g(S) p Fq 1497 4382 a(0) p Fr 1536 4367 a(\)) p Fs( ) p Fl 1637 4382 a(\033) p Fn 1677 4390 a(n) p Fm 1724 4367 a(k) 590 4556 y(\024) p Fr 761 4489 a(1) p 761 4534 49 4 v 761 4625 a(2) 842 4556 y(+) p Fm 22 w(k) p Fs(K) p Fr 7 w(\() p Fs(\022) p Fq 1163 4571 a(0) p Fs 1203 4556 a(;) g(P) p Fq 1310 4571 a(0) p Fs 1349 4556 a(;) g(R) p Fq 1467 4571 a(0) p Fs 1506 4556 a(;) g(S) p Fq 1610 4571 a(0) p Fr 1649 4556 a(\)) p Fs( ) p Fl 1750 4571 a(\033) p Fn 1790 4579 a(n) p Fm 1837 4556 a(k) p Fs(:) p Fv 257 4789 a(But) p Fm 33 w(k) p Fs( ) p Fl 564 4804 a(\033) p Fn 604 4812 a(n) p Fm 651 4789 a(k) p Fr 28 w(=) 27 b(1) p Fv 33 w(for) 32 b(all) p Fs 30 w(n;) p Fv 33 w(th) m(us) p Fm 1288 5035 a(k) p Fs(K) p Fr 7 w(\() p Fs(\022) p Fq 1511 5050 a(0) p Fs 1551 5035 a(;) 17 b(P) p Fq 1658 5050 a(0) p Fs 1697 5035 a(;) g(R) p Fq 1815 5050 a(0) p Fs 1854 5035 a(;) g(S) p Fq 1958 5050 a(0) p Fr 1998 5035 a(\)) p Fs( ) p Fl 2099 5050 a(\033) p Fn 2139 5058 a(n) p Fm 2186 5035 a(k) 27 b(\025) p Fr 2379 4968 a(1) p 2379 5012 V 2379 5103 a(2) p Fs 2437 5035 a(;) p Fv 257 5268 a(for) 32 b(an) m(y) p Fs 34 w(n) p Fm 27 w(\025) p Fs 29 w(n) p Fq 840 5283 a(0) p Fs 879 5268 a(;) p Fv 33 w(whic) m(h) h(is) f(an) g(estimate) g(of) g(the) h(form) e (\(5.5\).) p Fc 842 w(2) p Fv 1828 5637 a(28) p 90 rotate dyy eop %%Page: 29 29 29 28 bop Ff 257 573 a(5.4) 131 b(Pro) t(of) 43 b(of) h(Prop) t (osition) g(3.4) p Fv 257 758 a(The) 29 b(idea) e(of) g(the) h(pro) s (of) f(is) g(adapted) i(from) d([10].) 41 b(Once) 29 b(again,) e(one) h(of) f(the) i(main) d(to) s(ols) 257 878 y(is) 40 b(the) g(pullthrough) f(form) m(ula,) h(whic) m(h) g (comes) g(from) f(the) h(comm) m(utator) f(b) s(et) m(w) m(een) p Fs 42 w(H) p Fv 257 998 a(and) 33 b(annihilation) c(op) s(erators) p Fr 450 1240 a([) p Fs(H) r(;) p Fr 17 w(1) -22 b(l) p Fm 21 w(\012) p Fs 23 w(a) p Fr(\() p Fs(x;) 17 b(k) p Fr 3 w(\)]) 28 b(=) p Fm 27 w(\000) p Fs(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)1) -22 b(l) p Fm 21 w(\012) p Fs 23 w(a) p Fr(\() p Fs(x;) 17 b(k) p Fr 3 w(\)) p Fm 22 w(\000) p Fs 23 w(\032) p Fq 2118 1255 a(1) p Fr 2157 1240 a(\() p Fs(x) p Fm 23 w(\000) p Fs 23 w(Q) p Fr(\)) 2568 1173 y(^) p Fs -58 w(\032) p Fq 2609 1188 a(2) p Fr 2649 1173 a(\() p Fs(k) p Fr 3 w(\)) p 2498 1217 343 4 v Fp 2498 1237 a(p) p 2597 1237 244 4 v Fr 2597 1322 a(2) p Fs(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fm 2873 1240 a(\012) p Fr 22 w(1) -22 b(l) p Fs -1 w(:) p Fv 193 w(\(5.13\)) 257 1502 y(In) 41 b(order) g(to) f(get) h(our) f(result) h(w) m(e) g(will) e (need) i(to) g(use) g(this) f(form) m(ula) f(taking) h(in) m(to) g(ac-) 257 1622 y(coun) m(t) g(the) f(mem) m(branes) g(alltogether,) g(whic) m (h,) i(on) e(a) g(formal) d(lev) m(el,) k(means) f(that) g(w) m(e) 257 1743 y(will) d(in) m(tegrate) i(the) g(previous) g(form) m(ula) e(o) m (v) m(er) j(the) g(\020) p Fs 8 w(x) p Fv(-space\021.) 60 b(It) 38 b(is) f(therefore) i(more) 257 1863 y(con) m(v) m(enien) m(t) e (to) d(lo) s(ok) g(at) g(the) h(Hamiltonian) c(not) k(in) f(the) p Fr 35 w(\() p Fs(x;) 17 b(k) p Fr 3 w(\)) p Fv 34 w(v) -5 b(ariables) 34 b(but) h(in) f(the) p Fr 257 1983 a(\() p Fs(p;) 17 b(k) p Fr 3 w(\)) p Fv 29 w(v) -5 b(ariables,) 29 b(where) p Fs 31 w(p) p Fv 29 w(is) g(the) h(v) -5 b(ariable) 28 b(conjugate) h(to) p Fs 30 w(x) p Ft 29 w(via) p Fv 37 w(F) -8 b(ourier) 28 b(transform,) 257 2104 y(and) e(then) g(consider) g (the) g(v) -5 b(alue) p Fs 25 w(p) p Fr 28 w(=) 27 b(0) p Fv(.) 41 b(In) 26 b(suc) m(h) h(v) -5 b(ariables,) 26 b(the) g(pullthrough) e(form) m(ula) 257 2224 y(just) 33 b(b) s(ecomes) p Fr 450 2466 a([) p Fs(H) r(;) p Fr 17 w(1) -22 b(l) p Fm 21 w(\012) p Fr 24 w(^) p Fs -51 w(a) p Fr 1 w(\() p Fs(p;) 17 b(k) p Fr 3 w(\)]) 27 b(=) p Fm 27 w(\000) p Fs(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)1) -22 b(l) p Fm 21 w(\012) p Fr 24 w(^) p Fs -50 w(a) p Fr(\() p Fs(p;) 17 b(k) p Fr 3 w(\)) p Fm 22 w(\000) p Fr 31 w(^) p Fs -58 w(\032) p Fq 2104 2481 a(1) p Fr 2144 2466 a(\() p Fs(p) p Fr(\)) p Fs(e) p Fi 2314 2425 a(\000) p Fl(ipQ) p Fr 2569 2399 a(^) p Fs -58 w(\032) p Fq 2610 2414 a(2) p Fr 2649 2399 a(\() p Fs(k) p Fr 3 w(\)) p 2498 2443 343 4 v Fp 2498 2463 a(p) p 2598 2463 244 4 v Fr 2598 2548 a(2) p Fs(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fm 2873 2466 a(\012) p Fr 23 w(1) -22 b(l) p Fs -2 w(:) p Fv 193 w(\(5.14\)) 404 2728 y(Supp) s(ose) 38 b(no) m(w) h(that) p Fr 37 w(\011) p Fm 37 w(2) e(H) p Fv 38 w(satis\034es) p Fs 39 w(H) p Fr 8 w(\011) f(=) p Fs 36 w(E) p Fq 2304 2743 a(0) p Fr 2344 2728 a(\011) p Fs(;) p Fv 38 w(where) p Fs 38 w(E) p Fq 2843 2743 a(0) p Fv 2921 2728 a(is) h(the) h(ground) 257 2848 y(state) h(energy) g(of) p Fs 38 w(H) r(:) p Fv 39 w(W) -8 b(e) 38 b(will) e(sho) m(w) k(that) p Fr 38 w(\011) d(=) g(0) p Fs(:) p Fv 38 w(W) -8 b(e) 39 b(apply) f(equation) g (\(5.14\)) f(on) 257 2968 y(suc) m(h) d(a) f(v) m(ector.) 44 b(One) 33 b(then) g(gets) g(the) g(follo) m(wing) d(equalit) m(y) p Fr 518 3242 a(1) -22 b(l) p Fm 21 w(\012) p Fr 24 w(^) p Fs -50 w(a) p Fr(\() p Fs(p;) 17 b(k) p Fr 3 w(\)) 32 b(\011) c(=) p Fm 27 w(\000) p Fr(\() p Fs(H) p Fr 30 w(+) p Fs 22 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\000) p Fs 22 w(E) p Fq 1919 3257 a(0) p Fr 1959 3242 a(\)) p Fi 1997 3201 a(\000) p Fq(1) p Fp 2108 3072 a( ) p Fr 2206 3175 a(^) p Fs -58 w(\032) p Fq 2247 3190 a(1) p Fr 2286 3175 a(\() p Fs(p) p Fr(\)) p Fs(e) p Fi 2456 3139 a(\000) p Fl(ipQ) p Fr 2640 3175 a(^) p Fs -58 w(\032) p Fq 2681 3190 a(2) p Fr 2720 3175 a(\() p Fs(k) p Fr 3 w(\)) p 2197 3219 654 4 v Fp 2352 3239 a(p) p 2452 3239 244 4 v Fr 2452 3325 a(2) p Fs(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fm 2882 3242 a(\012) p Fr 23 w(1) -22 b(l) p Fp 3036 3072 a(!) p Fr 3131 3242 a(\011) p Fs(:) p Fv 257 3522 a(W) -8 b(e) 27 b(denote) f(with) g(an) g(exp) s(onen) m(t) p Fr 27 w(\() p Fs(m) p Fr(\)) p Fv 26 w(the) g(comp) s(onen) m(t) g (of) g(a) f(v) m(ector) i(in) e(the) p Fs 27 w(m) p Fv(-particle) 257 3642 y(sector.) 45 b(W) -8 b(e) 33 b(ha) m(v) m(e,) g(for) f(an) m(y) p Fs 34 w(m) p Fv(,) p Fr 401 3835 a(\(1) -22 b(l) p Fm 21 w(\012) p Fr 24 w(^) p Fs -51 w(a) p Fr 1 w(\() p Fs(p;) 17 b(k) p Fr 3 w(\)) 32 b(\011\)) p Fq 1035 3794 a(\() p Fl(m) p Fq(\)) p Fr 1156 3835 a(\() p Fs(p) p Fq 1243 3850 a(1) p Fs 1282 3835 a(;) 17 b(k) p Fq 1377 3850 a(1) p Fs 1416 3835 a(;) g(:) g(:) g(:) f(;) h(p) p Fl 1684 3850 a(m) p Fs 1750 3835 a(;) g(k) p Fl 1845 3850 a(m) p Fr 1911 3835 a(\)) 28 b(=) f(\011) p Fq 2156 3794 a(\() p Fl(m) p Fq(+1\)) p Fr 2368 3835 a(\() p Fs(p;) 17 b(k) s(;) g(p) p Fq 2646 3850 a(1) p Fs 2685 3835 a(;) g(k) p Fq 2780 3850 a(1) p Fs 2819 3835 a(;) g(:) g(:) g(:) e (;) i(p) p Fl 3086 3850 a(m) p Fs 3153 3835 a(;) g(k) p Fl 3248 3850 a(m) p Fr 3314 3835 a(\)) p Fv 257 4027 a(and) 32 b(the) h(righ) m(thand) e(side) h(is) f(square) i(in) m (tegrable) e(with) g(resp) s(ect) i(to) f(all) e(its) h(argumen) m(ts) 257 4148 y(b) s(ecause) p Fr 34 w(\011) p Fm 28 w(2) d(H) p Fs 1 w(:) p Fv 33 w(Therefore,) 33 b(for) f(all) p Fs 31 w(m;) p Fr 530 4445 a(\010) p Fq 600 4404 a(\() p Fl(m) p Fq(\)) p Fr 721 4445 a(\() p Fs(p;) 17 b(k) p Fr 3 w(\)) 28 b(:=) p Fp 1102 4275 a( ) p Fm 1181 4445 a(\000) p Fr(\() p Fs(H) p Fr 30 w(+) p Fs 22 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fm 22 w(\000) p Fs 22 w(E) p Fq 1893 4460 a(0) p Fr 1933 4445 a(\)) p Fi 1971 4404 a(\000) p Fq(1) p Fr 2084 4378 a(^) p Fs -58 w(\032) p Fq 2125 4393 a(1) p Fr 2165 4378 a(\() p Fs(p) p Fr(\)) p Fs(e) p Fi 2335 4342 a(\000) p Fl(ipQ) p Fr 2518 4378 a(^) p Fs -58 w(\032) p Fq 2559 4393 a(2) p Fr 2599 4378 a(\() p Fs(k) p Fr 3 w(\)) p 2075 4422 654 4 v Fp 2231 4442 a(p) p 2330 4442 244 4 v Fr 2330 4528 a(2) p Fs(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fm 2761 4445 a(\012) p Fr 23 w(1) -22 b(l) 31 b(\011) p Fp 3023 4275 a(!) p Fq 3102 4297 a(\() p Fl(m) p Fq(\)) p Fv 257 4725 a(is) d(square) i(in) m(tegrable) d(with) h(resp) s(ect) i(to) p Fr 28 w(\() p Fs(p;) 17 b(k) p Fr 3 w(\)) p Fs(:) p Fv 28 w(On) 29 b(the) g(other) f(hand,) i(it) d(is) h(a) g(con) m(tin) m (u-) 257 4845 y(ous) j(function) f(on) p Fg 30 w(R) p Fl 1008 4809 a(d) p Fm 1072 4845 a(\002) p Fr 18 w(\() p Fg(R) p Fl 1271 4809 a(n) p Fm 1341 4845 a(\000) 18 b(f) p Fr(0) p Fm(g) p Fr(\)) p Fs(:) p Fv 30 w(Then,) 32 b(for) d(an) m(y) p Fs 31 w(p) p Fq 2337 4860 a(0) p Fm 2404 4845 a(2) p Fg 28 w(R) p Fl 2564 4809 a(d) p Fs 2611 4845 a(;) p Fr 33 w(\010) p Fq 2741 4809 a(\() p Fl(m) p Fq(\)) p Fr 2863 4845 a(\() p Fs(p) p Fq 2950 4860 a(0) p Fs 2989 4845 a(;) 17 b(k) p Fr 3 w(\)) p Fv 30 w(is) 30 b(a) g(w) m(ell) 257 4966 y(de\034ned) i(function) e(of) p Fs 30 w(k) p Fv 33 w(and) h(it) e(is) h(square) i(in) m(tegrable.) 41 b(As) 31 b(w) m(e) h(ha) m(v) m(e) f(said) f(previously) -8 b(,) 257 5086 y(w) m(e) 34 b(consider) f(the) g(v) -5 b(alue) p Fs 32 w(p) p Fq 1250 5101 a(0) p Fr 1317 5086 a(=) 27 b(0) p Fs(:) p Fv 33 w(But) p Fr 1264 5328 a(\010) p Fq 1334 5287 a(\() p Fl(m) p Fq(\)) p Fr 1456 5328 a(\(0) p Fs(;) 17 b(k) p Fr 3 w(\)) 27 b(=) 1829 5261 y(^) p Fs -58 w(\032) p Fq 1870 5276 a(1) p Fr 1910 5261 a(\(0\)) 9 b(^) p Fs -58 w(\032) p Fq 2085 5276 a(2) p Fr 2124 5261 a(\() p Fs(k) p Fr 3 w(\)) p 1820 5305 435 4 v Fm 1847 5331 a(p) p 1930 5331 49 4 v Fr 1930 5413 a(2) p Fs -1 w(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fj 2183 5349 a(3) p 2183 5361 31 3 v 2183 5402 a(2) p Fr 2264 5328 a(\011) p Fq 2340 5287 a(\() p Fl(m) p Fq(\)) p Fs 2461 5328 a(;) p Fv 1828 5637 a(29) p 90 rotate dyy eop %%Page: 30 30 30 29 bop Fv 257 573 a(whic) m(h) 45 b(is) e(not) g(square) i(in) m (tegrable) e(if) f(the) i(infrared) f(condition) f(is) h(violated,) i (unless) p Fr 266 693 a(^) p Fs -58 w(\032) p Fq 307 708 a(1) p Fr 347 693 a(\(0\)\011) p Fq 548 657 a(\() p Fl(m) p Fq(\)) p Fr 697 693 a(=) 27 b(0) p Fs(:) p Fv 29 w(By) i(assumption,) p Fr 37 w(^) p Fs -58 w(\032) p Fq 1643 708 a(1) p Fr 1683 693 a(\(0\)) p Fm 27 w(6) p Fr(=) f(0) p Fs(;) p Fv 28 w(so) p Fr 29 w(\011) p Fq 2235 657 a(\() p Fl(m) p Fq(\)) p Fr 2384 693 a(=) f(0) p Fv 28 w(for) h(all) p Fs 27 w(m) p Fv 28 w(whic) m(h) h(means) 257 814 y(that) p Fr 33 w(\011) e(=) h(0) p Fs(:) p Fc 2669 w(2) p Fu 257 1262 a(6) 156 b(A) 34 b(classical) k(in) l(terpretation) g (of) d(the) g(infrared) i(prob-) 490 1444 y(lem) p Fv 257 1663 a(In) 30 b(this) f(section,) h(w) m(e) g(w) m(ould) g(lik) m (e) e(to) h(sa) m(y) h(a) f(few) h(w) m(ords) g(ab) s(out) f(the) h (infrared) e(problem.) 257 1784 y(W) -8 b(e) 37 b(kno) m(w) h(that) e (this) g(condition) f(is) h(necessary) i(and) f(su\036cien) m(t) g(for) f(the) h(existence) h(of) 257 1904 y(a) e(ground) f(state) h(in) f(the) h(case) h(of) e(the) h(Nelson) g(mo) s(del,) f(and) g(su\036cien) m(t) i (and) f(\020almost\021) 257 2025 y(necessary) f(in) d(our) g(mo) s (del.) 404 2145 y(On) e(the) g(other) g(side,) g(in) g([2) o(],) h(the) f(author) g(sho) m(ws) h(that) f(if) f(w) m(e) i(consider) f(another) p Ft 30 w(ad) 257 2265 y(ho) -5 b(c) p Fv 36 w(represen) m(tation) 32 b(of) e(the) h(canonical) e(comm) m(utation) g(relations) g(the) i (Nelson) g(mo) s(del) 257 2386 y(without) 46 b(infrared) f(condition) g (has) h(a) g(ground) g(state.) 84 b(In) 47 b(some) e(sens,) 51 b(this) 46 b(repre-) 257 2506 y(sen) m(tation) 41 b(regularizes) f(the) h(infrared) f(singularit) m(y) f(and) i(is) f(of) g(course) i(not) f (unitarily) 257 2627 y(equiv) -5 b(alen) m(t) 32 b(to) g(the) h(F) -8 b(o) s(c) m(k) 32 b(one.) 43 b(One) 33 b(could) e(think) h(that) g(the) h(same) f(approac) m(h) g(should) 257 2747 y(w) m(ork) d(in) d(our) h (case.) 43 b(Ho) m(w) m(ev) m(er,) 31 b(it) 26 b(turns) i(out) g(that) f (this) g(is) g(not) g(true.) 42 b(T) -8 b(o) 27 b(explain) g(wh) m(y) -8 b(,) 257 2867 y(w) m(e) 31 b(will) c(brie\035y) j(explain) f(the) h(ph) m(ysical) f(origin) e(of) j(this) f(represen) m(tation.) 43 b(It) 30 b(will) d(allo) m(w) 257 2988 y(us) 35 b(to) e(see) i(that) f (this) g(pro) s(cedure) g(can) g(apply) g(to) g(our) f(mo) s(del) f (but) j(do) s(es) f(not) g(ha) m(v) m(e) h(the) 257 3108 y(same) e(regularising) d(e\033ect.) 404 3228 y(W) -8 b(e) 43 b(w) m(ould) g(lik) m(e) g(to) f(explain) h(the) g(idea) g (whic) m(h) g(is) g(b) s(ehind) g(this) g(c) m(hange) h(of) e(rep-) 257 3349 y(resen) m(tation) c(coming) d(bac) m(k) k(to) e(classical) e(mec) m(hanics) j([8) o(].) 58 b(It) 37 b(will) e(then) j(allo) m(w) e(us) i (to) 257 3469 y(sho) m(w) 33 b(the) f(di\033erence) g(b) s(et) m(w) m (een) i(the) e(Nelson) g(mo) s(del) e(and) h(ours.) 44 b(W) -8 b(e) 32 b(th) m(us) h(consider) e(a) 257 3590 y(classical) g(Hamiltonian) e(of) j(the) h(form) p Fs 1066 3800 a(H) p Fr 35 w(=) 1296 3732 y(1) p 1296 3777 49 4 v 1296 3868 a(2) p Fp 1371 3664 a(Z) p Fl 1427 3890 a(A) p Fs 1500 3800 a(d\026) p Fr(\() p Fs(\013) p Fr 1 w(\)\() p Fs(!) p Fr 4 w(\() p Fs(\013) p Fr 1 w(\)) p Fq 1991 3759 a(2) p Fs 2028 3800 a(\036) p Fr(\() p Fs(\013) p Fr 1 w(\)) p Fq 2225 3759 a(2) p Fr 2286 3800 a(+) p Fs 22 w(\031) p Fr 4 w(\() p Fs(\013) p Fr 1 w(\)) p Fq 2582 3759 a(2) p Fr 2621 3800 a(\)) p Fs(;) p Fv 257 4039 a(where) p Fs 27 w(!) p Fr 4 w(\() p Fs(\013) p Fr 1 w(\)) p Fv 25 w(is) 26 b(some) f(almost) f(ev) m(erywhere) 29 b(non) d(negativ) m(e) g(function.) 41 b(W) -8 b(e) 26 b(will) e(simply) 257 4160 y(write) p Fs 33 w(X) p Fv 40 w(instead) 32 b(of) p Fr 32 w(\() p Fs(\036;) 17 b(\031) p Fr 4 w(\)) p Fs(:) p Fv 32 w(The) 34 b(Hamiltonian) 28 b(\035o) m(w) 34 b(can) e(b) s(e) h(written) g(as) p Fs 419 4404 a(X) p Fl 500 4419 a(t) p Fr 557 4404 a(=) 28 b(\010) p Fl 731 4419 a(t) p Fs 761 4404 a(X) p Fq 842 4419 a(0) p Fr 909 4404 a(=) g(cos\() p Fs(!) t(t) p Fr(\)) p Fs(X) p Fq 1400 4419 a(0) p Fm 1461 4404 a(\000) p Fr 23 w(sin\() p Fs(!) t(t) p Fr(\)) p Fs(J) 9 b(X) p Fq 2001 4419 a(0) p Fs 2040 4404 a(;) p Fr 114 w(where) 66 b(J) 28 b(=) p Fp 2677 4264 a(\022) p Fr 2807 4343 a(0) p Fm 83 w(\000) p Fs(!) p Fi 3081 4307 a(\000) p Fq(1) p Fs 2791 4464 a(!) p Fr 274 w(0) p Fp 3216 4264 a(\023) p Fs 3306 4404 a(:) p Fv 257 4654 a(One) 33 b(can) g(see) h(that) p Fs 32 w(J) p Fq 1074 4617 a(2) p Fr 1141 4654 a(=) p Fm 28 w(\000) p Fr(1) -22 b(l) p Fs -1 w(:) p Fv 404 4774 a(No) m(w,) 33 b(for) f(an) m(y) p Fs 33 w(s) p Fm 28 w(2) p Fg 28 w(R) p Fs 5 w(;) p Fv 38 w(w) m(e) i(de\034ne) p Fm 1256 4961 a(D) p Fr 3 w(\() p Fs(!) p Fl 1439 4919 a(s) p Fr 1474 4961 a(\)) 28 b(:=) p Fm 28 w(f) p Fs(\036) p Fm 27 w(2) p Fs 28 w(L) p Fq 1966 4919 a(2) p Fm 2006 4961 a(j) p Fs(!) p Fl 2099 4919 a(s) p Fs 2135 4961 a(\036) p Fm 27 w(2) p Fs 28 w(L) p Fq 2380 4919 a(2) p Fm 2420 4961 a(g) p Fs(;) p Fv 257 5147 a(and) p Fr 42 w([) p Fm(D) p Fr 3 w(\() p Fs(!) p Fl 666 5111 a(s) p Fr 702 5147 a(\)]) p Fv 42 w(its) 42 b(closure) g(for) f(the) h(norm) p Fm 41 w(k) p Fs(\036) p Fm(k) p Fl 2047 5162 a(s) p Fr 2127 5147 a(=) p Fm 44 w(k) p Fs(!) p Fl 2362 5111 a(s) p Fs 2398 5147 a(\036) p Fm(k) p Fl 2506 5164 a(L) p Fj 2554 5145 a(2) p Fs 2592 5147 a(:) p Fv 42 w(Giv) m(en) p Fs 42 w(s;) 17 b(r) p Fm 46 w(2) p Fg 44 w(R) p Fs 5 w(;) p Fv 48 w(w) m(e) 257 5268 y(de\034ne) p Fm 1324 5388 a(H) p Fl 1408 5403 a(s;r) p Fr 1526 5388 a(:=) 28 b([) p Fm(D) p Fr 3 w(\() p Fs(!) p Fl 1867 5347 a(s) p Fr 1903 5388 a(\)]) p Fm 22 w(\002) p Fr 22 w([) p Fm(D) p Fr 3 w(\() p Fs(!) p Fl 2299 5347 a(r) p Fr 2336 5388 a(\)]) p Fs(:) p Fv 1828 5637 a(30) p 90 rotate dyy eop %%Page: 31 31 31 30 bop Fv 257 573 a(The) 37 b(op) s(erator) p Fs 35 w(J) p Fv 45 w(is) f(w) m(ell) f(de\034ned) i(on) p Fm 36 w(H) p Fl 1821 588 a(s;r) p Fv 1947 573 a(\(as) f(a) f(b) s(ounded) i (op) s(erator\)) e(if) g(and) h(only) 257 693 y(if) p Fs 32 w(r) p Fr 30 w(=) p Fs 28 w(s) p Fm 22 w(\000) p Fr 22 w(1) p Fs(:) p Fv 33 w(On) c(the) h(other) g(side,) g(the) g (symplectic) f(form) p Fs 845 950 a(\033) p Fr 4 w(\() p Fs(X) p Fq 1023 965 a(1) p Fs 1062 950 a(;) 17 b(X) p Fq 1187 965 a(2) p Fr 1226 950 a(\)) 28 b(=) p Fp 1395 814 a(Z) p Fl 1451 1040 a(A) p Fs 1524 950 a(d\026) p Fr(\() p Fs(\013) p Fr 1 w(\)\() p Fs(\036) p Fq 1869 965 a(1) p Fr 1907 950 a(\() p Fs(\013) p Fr 1 w(\)) p Fs(\031) p Fq 2101 965 a(2) p Fr 2141 950 a(\() p Fs(\013) p Fr 1 w(\)) p Fm 21 w(\000) p Fs 23 w(\036) p Fq 2459 965 a(2) p Fr 2498 950 a(\() p Fs(\013) p Fr 1 w(\)) p Fs(\031) p Fq 2692 965 a(1) p Fr 2731 950 a(\() p Fs(\013) p Fr 1 w(\)\)) p Fv 257 1210 a(is) 39 b(meaningfull) c(only) k(on) g (spaces) h(of) f(the) g(form) p Fm 38 w(H) p Fl 2177 1225 a(s;) p Fi(\000) p Fl(s) p Fs 2321 1210 a(:) p Fv 39 w(So) p Fs 38 w(J) p Fv 48 w(and) p Fs 39 w(\033) p Fv 43 w(are) g(b) s(oth) g(w) m(ell) 257 1330 y(de\034ned) 34 b(only) e(on) p Fm 1338 1451 a(H) p Fr 29 w(=) 27 b([) p Fm(D) p Fr 3 w(\() p Fs(!) p Fj 1774 1382 a(1) p 1773 1394 31 3 v 1773 1435 a(2) p Fr 1818 1451 a(\)]) p Fm 22 w(\002) p Fr 22 w([) p Fm(D) p Fr 3 w(\() p Fs(!) p Fi 2214 1409 a(\000) p Fj 2279 1382 a(1) p 2278 1394 V 2278 1435 a(2) p Fr 2323 1451 a(\)]) p Fs(:) p Fv 257 1622 a(On) p Fm 33 w(H) p Fv 1 w(,) 33 b(w) m(e) g(consider) g(the) g (follo) m(wing) d(complex) i(structure) p Fm 33 w(h) p Fr 33 w(;) p Fm 50 w(i) p Fv 33 w(de\034ned) i(as) p Fm 1088 1836 a(h) p Fs(X) p Fq 1208 1851 a(1) p Fr 1248 1836 a(;) p Fs 17 w(X) p Fq 1373 1851 a(2) p Fm 1412 1836 a(i) p Fr 27 w(=) p Fs 28 w(\033) p Fr 4 w(\() p Fs(X) p Fq 1760 1851 a(1) p Fs 1799 1836 a(;) 17 b(J) 9 b(X) p Fq 1987 1851 a(2) p Fr 2026 1836 a(\)) 22 b(+) p Fs 22 w(i\033) p Fr 4 w(\() p Fs(X) p Fq 2395 1851 a(1) p Fs 2435 1836 a(;) 17 b(X) p Fq 2560 1851 a(2) p Fr 2599 1836 a(\)) p Fs(:) p Fv 257 2049 a(One) 33 b(can) g(then) g(iden) m (tify) p Fm 32 w(H) p Fv 34 w(and) p Fs 32 w(L) p Fq 1592 2013 a(2) p Fr 1632 2049 a(\() p Fs(A;) 17 b(d\026;) p Fg 17 w(C) p Fr 19 w(\)) p Ft 38 w(via) p Fv 40 w(the) 33 b(follo) m(wing) d(isometry:) p Fr 672 2308 a(\() p Fs(\036;) 17 b(\031) p Fr 4 w(\)) p Fm 27 w(2) 28 b(H) g(!) p Fp 1270 2218 a(p) p 1369 2218 203 4 v Fs 1369 2308 a(!) p Fr 4 w(\() p Fs(\013) p Fr 1 w(\)) p Fs -1 w(\036) p Fr(\() p Fs(\013) p Fr 1 w(\)) 22 b(+) p Fs 2033 2241 a(i) p 1899 2285 303 4 v Fp 1899 2305 a(p) p 1998 2305 203 4 v Fs 1998 2391 a(!) p Fr 4 w(\() p Fs(\013) p Fr 1 w(\)) p Fs 2211 2308 a(\031) p Fr 4 w(\() p Fs(\013) p Fr 1 w(\)) p Fm 27 w(2) p Fs 28 w(L) p Fq 2596 2267 a(2) p Fr 2636 2308 a(\() p Fs(A;) 17 b(d\026;) p Fg 17 w(C) p Fr 19 w(\)) p Fs(:) p Fv 257 2591 a(and) 33 b(if) e(w) m(e) j(de\034ne) p Fs 634 2867 a(a) p Fr(\() p Fs(\013) p Fr 1 w(\)) 27 b(:=) 1033 2800 y(1) p 992 2844 132 4 v Fm 992 2864 a(p) p 1075 2864 49 4 v Fr 1075 2946 a(2) p Fp 1150 2697 a( ) 1229 2777 y(p) p 1329 2777 203 4 v Fs 1329 2867 a(!) p Fr 4 w(\() p Fs(\013) p Fr 1 w(\)) p Fs -2 w(\036) p Fr(\() p Fs(\013) p Fr 1 w(\)) 22 b(+) p Fs 1992 2800 a(i) p 1858 2844 303 4 v Fp 1858 2864 a(p) p 1957 2864 203 4 v Fs 1957 2949 a(!) p Fr 4 w(\() p Fs(\013) p Fr 1 w(\)) p Fs 2170 2867 a(\031) p Fr 4 w(\() p Fs(\013) p Fr 1 w(\)) p Fp 2368 2697 a(!) p Fm 2474 2867 a(2) p Fs 28 w(L) p Fq 2634 2826 a(2) p Fr 2674 2867 a(\() p Fs(A;) 17 b(d\026;) p Fg 17 w(C) p Fr 19 w(\)) p Fs(;) p Fv 257 3161 a(one) 33 b(can) g(rewrite) p Fs 32 w(H) p Fv 41 w(as) p Fs 1254 3398 a(H) p Fr 35 w(=) p Fp 1473 3263 a(Z) p Fl 1529 3488 a(A) p Fs 1602 3398 a(!) p Fr 4 w(\() p Fs(\013) p Fr 1 w(\)) p Fs(a) p Fi 1857 3357 a(\003) p Fr 1896 3398 a(\() p Fs(\013) p Fr 1 w(\)) p Fs(a) p Fr(\() p Fs(\013) p Fr 1 w(\)) p Fs(d\026) p Fr(\() p Fs(\013) p Fr 1 w(\)) p Fs(:) p Fv 257 3664 a(The) j(P) m(oisson) f(brac) m(k) m(et) i(asso) s (ciated) e(to) p Fs 34 w(\033) p Fv 39 w(is) p Fm 35 w(f) p Fs(\036) p Fr(\() p Fs(\013) p Fr 1 w(\)) p Fs(;) 17 b(\031) p Fr 4 w(\() p Fs(\013) p Fr 1 w(\)) p Fm(g) p Fr 29 w(=) p Fs 32 w(\016) p Fl 2663 3679 a(\026) p Fr 2710 3664 a(\() p Fs(\013) p Fm 24 w(\000) p Fs 24 w(\013) p Fi 2998 3628 a(0) p Fr 3021 3664 a(\)) p Fs(;) p Fv 35 w(where) p Fs 36 w(\016) p Fl 3448 3679 a(\026) p Fv 257 3785 a(is) 32 b(de\034ned) i(b) m(y) p Fp 1170 3812 a(Z) p Fl 1225 4038 a(A) p Fs 1299 3948 a(f) p Fr 11 w(\() p Fs(\013) p Fi 1459 3907 a(0) p Fr 1482 3948 a(\)) p Fs(\016) p Fl 1563 3963 a(\026) p Fr 1609 3948 a(\() p Fs(\013) p Fm 23 w(\000) p Fs 23 w(\013) p Fi 1895 3907 a(0) p Fr 1918 3948 a(\)) p Fs(d\026) p Fr(\() p Fs(\013) p Fi 2167 3907 a(0) p Fr 2189 3948 a(\)) 28 b(=) p Fs 27 w(f) p Fr 11 w(\() p Fs(\013) p Fr 1 w(\)) p Fs(:) p Fv 257 4172 a(It) 35 b(is) f(easy) h(to) f(see) h(that) p Fm 35 w(f) p Fs(a) p Fr(\() p Fs(\013) p Fr 1 w(\)) p Fs(;) 17 b(a) p Fi 1510 4136 a(\003) p Fr 1548 4172 a(\() p Fs(\013) p Fi 1649 4136 a(0) p Fr 1672 4172 a(\)) p Fm(g) p Fr 31 w(=) p Fm 30 w(\000) p Fs(i\016) p Fl 2050 4187 a(\026) p Fr 2098 4172 a(\() p Fs(\013) p Fm 24 w(\000) p Fs 24 w(\013) p Fi 2386 4136 a(0) p Fr 2409 4172 a(\)) p Fs(:) p Fv 34 w(Those) 36 b(relations) c(are) j(the) 257 4292 y(classical) i(equiv) -5 b(alen) m(t) 38 b(of) f(the) i(relations) d(\(2.1\).) 60 b(T) -8 b(o) 38 b(write) f(the) i(quan) m(tum) f(v) m (ersion) h(of) 257 4413 y(this) 33 b(mo) s(del,) e(one) h(then) h (consider) g(the) g(F) -8 b(o) s(c) m(k) 33 b(space) g(o) m(v) m(er) p Fs 34 w(L) p Fq 2480 4377 a(2) p Fr 2520 4413 a(\() p Fs(A;) 17 b(d\026;) p Fg 17 w(C) p Fr 19 w(\)) p Fs(:) p Fv 404 4533 a(Consider) 44 b(no) m(w) g(the) g(Nelson) f(mo) s(del,) p Ft 45 w(i.e.) p Fs 76 w(A) p Fr 47 w(=) p Fg 46 w(R) p Fl 2393 4497 a(d) p Fv 2483 4533 a(and) p Fs 44 w(d\026) p Fr(\() p Fs(\013) p Fr 1 w(\)) i(=) p Fs 46 w(dk) s(:) p Fv 44 w(If) e(w) m(e) 257 4654 y(consider) 36 b(a) e(particle) g(whic) m (h) h(in) m(teracts) g(with) g(this) f(\034eld) h(and) g(whic) m(h) g (is) g(on) g(the) g(other) 257 4774 y(hand) 46 b(submitted) f(to) g(a) h (con\034ning) f(p) s(oten) m(tial) p Fs 44 w(V) p Fv 67 w(suc) m(h) i(that) p Fr 45 w(min) p Fs 15 w(V) p Fr 71 w(=) p Fs 50 w(V) p Fr 22 w(\(0\)) p Fs(;) p Fv 45 w(the) 257 4894 y(equilibrium) 37 b(p) s(oin) m(t) i(of) h(the) g (system) h(whic) m(h) g(correpsonds) g(to) f(the) h(minim) m(um) 36 b(of) k(the) 257 5015 y(energy) e(can) f(b) s(e) g(written) g(as) p Fr 37 w(\() p Fs(q) p Fi 1444 5030 a(\003) p Fs 1484 5015 a(;) 17 b(\036) p Fi 1586 5030 a(\003) p Fs 1625 5015 a(;) g(p) p Fi 1718 5030 a(\003) p Fs 1757 5015 a(;) g(\031) p Fi 1856 5030 a(\003) p Fr 1895 5015 a(\)) 35 b(=) g(\(0) p Fs(;) p Fm 17 w(\000) p Fq 2326 4971 a(^) p Fl -42 w(\032) p 2297 4992 81 4 v 2297 5049 a(!) p Fj 2343 5030 a(2) p Fs 2388 5015 a(;) p Fr 17 w(0) p Fs(;) p Fr 17 w(0\)) p Fv 35 w(and) p Fr 37 w(\() p Fs(\036) p Fi 2937 5030 a(\003) p Fs 2976 5015 a(;) 17 b(\031) p Fi 3075 5030 a(\003) p Fr 3115 5015 a(\)) p Fv 37 w(is) 36 b(in) p Fm 36 w(H) p Fv 257 5147 a(if) 30 b(and) i(only) e(if) p Fq 904 5104 a(^) p Fl -41 w(\032) p 844 5124 144 4 v 844 5185 a(!) p Fj 890 5166 a(3) p Fn(=) p Fj(2) p Fm 1025 5147 a(2) p Fs 28 w(L) p Fq 1185 5111 a(2) p Fr 1225 5147 a(\() p Fg(R) p Fl 1329 5111 a(d) p Fr 1375 5147 a(\)) p Fs(;) p Fv 31 w(whic) m(h) i(is) e(exactly) i(the) f (condition) f(\(IR\).) h(One) g(should) 257 5268 y(remind) d(that,) h (for) f(this) g(mo) s(del,) g(this) g(condition) f(is) h(necessary) j (and) e(su\036cien) m(t) h(to) e(ha) m(v) m(e) 257 5388 y(a) 36 b(ground) f(state.) 52 b(In) 36 b(other) g(w) m(ords,) h(the) f (minim) m(um) p Fr 32 w(\() p Fs(q) p Fi 2344 5403 a(\003) p Fs 2383 5388 a(;) 17 b(\036) p Fi 2485 5403 a(\003) p Fs 2524 5388 a(;) g(p) p Fi 2617 5403 a(\003) p Fs 2656 5388 a(;) g(\031) p Fi 2755 5403 a(\003) p Fr 2795 5388 a(\)) p Fv 35 w(of) 35 b(the) h(classical) 1828 5637 y(31) p 90 rotate dyy eop %%Page: 32 32 32 31 bop Fv 257 573 a(Hamiltonian) 24 b(b) s(elongs) j(to) p Fm 27 w(H) p Fv 28 w(if) g(and) g(only) g(if) f(\(IR\)) i(is) f (satis\034ed.) 42 b(In) 28 b(this) f(w) m(a) m(y) -8 b(,) 29 b(one) f(can) 257 693 y(sa) m(y) 37 b(that) f(the) h(condition) d(to) i(ha) m(v) m(e) h(a) f(ground) g(state) g(is) g(the) g(same) g (on) g(b) s(oth) g(classical) 257 814 y(and) d(quan) m(tum) g(lev) m (el.) 404 934 y(The) 28 b(represen) m(tation) g(considered) g(in) f ([2]) g(corresp) s(onds,) j(on) e(the) g(classical) d(lev) m(el,) k(to) 257 1054 y(the) e(a\036ne) g(space) g(\020) p Fr 997 1029 a(~) p Fm 968 1054 a(H) p Fr 28 w(=) p Fm 28 w(H) p Fr 10 w(+) 9 b(\() p Fq 1440 1011 a(^) p Fl -41 w(\032) p 1411 1031 81 4 v 1411 1089 a(!) p Fj 1457 1070 a(2) p Fs 1502 1054 a(;) p Fr 17 w(0\)) p Fv(\021,) 27 b(more) e(precisely) -8 b(,) 28 b(one) e(considers) h(the) g(follo) m(wing) 257 1175 y(symplectic) 32 b(transformation:) p Fr 1208 1425 a(~) p Fs -57 w(q) p Fr 32 w(=) p Fs 27 w(q) t(;) p Fr 1482 1399 a(~) p Fs 1469 1425 a(\036) p Fr 27 w(=) p Fs 28 w(\036) p Fr 22 w(+) 1881 1358 y(^) p Fs -57 w(\032) p 1846 1402 104 4 v 1846 1493 a(!) p Fq 1911 1465 a(2) p Fs 1960 1425 a(;) p Fr 24 w(~) p Fs -56 w(p) p Fr 27 w(=) p Fs 27 w(p;) p Fr 22 w(~) p Fs -54 w(\031) p Fr 32 w(=) p Fs 27 w(\031) t(:) p Fv 257 1680 a(Here,) p Fr 537 1654 a(~) p Fs 525 1680 a(\036) p Fv 40 w(represen) m(ts) 43 b(the) e(di\033erence) g(b) s(et) m(w) m(een) i(the) e(\034eld) f(and) h (its) f(equilibrium) d(p) s(o-) 257 1800 y(sition.) 86 b(W) -8 b(e) 47 b(already) g(note) g(that) p Fr 47 w(0) p Fs 64 w(=) p Fm -61 w(2) p Fr 1852 1775 a(~) p Fm 1823 1800 a(H) p Fv 48 w(if) f(and) h(only) g(if) e(\(IR\)) i(is) g(not) g (satis\034ed.) 257 1921 y(One) c(then) g(sees) i(that) d(the) h(t) m(w) m(o) g(equilibrium) d(p) s(oin) m(ts,) k(b) s(efore) f(\() p Fs(\036) p Fi 2764 1936 a(\003) p Fr 2848 1921 a(=) i(0) p Fv(\)) d(and) g(after) 257 2041 y(\() p Fs(\036) p Fi 353 2056 a(\003) p Fr 420 2041 a(=) p Fm 28 w(\000) p Fq 640 1997 a(^) p Fl -42 w(\032) p 611 2018 81 4 v 611 2075 a(!) p Fj 657 2057 a(2) p Fv 702 2041 a(\)) 28 b(ha) m(ving) f (turned) i(on) f(the) g(in) m(teraction) f(with) h(the) g(particle,) g (are) g(not) g(in) f(the) 257 2161 y(same) h(space.) 43 b(This) 29 b(is) e(this) h(phenomenon) g(whic) m(h,) i(on) e(the) g (quan) m(tum) h(lev) m(el,) f(expresses) 257 2282 y(that) 42 b(the) g(ground) g(state) g(exists) g(but) g(in) f(another) h(represen) m(tation,) j(non-equiv) -5 b(alen) m(t) 257 2402 y(to) 37 b(the) g(F) -8 b(o) s(c) m(k) 37 b(one.) 56 b(One) 37 b(sometimes) f(reads) h(that) g(the) g(\020ground) f(state) h(is) f (not) h(in) f(the) 257 2523 y(F) -8 b(o) s(c) m(k) 45 b(space\021) 54 b(\(within) 43 b(the) j(con) m(text) g(of) e(\020V) -8 b(an) 44 b(Ho) m(v) m(e) i(Hamiltonians\021) j(for) c(example) 257 2643 y([7]-[12) o(]-[20]\).) 404 2763 y(In) 32 b(those) h(new) h(v) -5 b(ariables,) 31 b(the) i(Hamiltonian) 28 b(of) k(the) h(whole) f (system) h(then) g(writes) p Fr 631 2987 a(~) p Fs 605 3013 a(H) p Fr 8 w(\() 7 b(~) p Fs -56 w(q) t(;) p Fr 835 2986 a(~) p Fs 823 3013 a(\036) o(;) p Fr 25 w(~) p Fs -57 w(p;) p Fr 21 w(~) p Fs -53 w(\031) p Fr 3 w(\)) 83 b(=) 1365 2945 y(1) p 1365 2990 49 4 v 1365 3081 a(2) p Fp 1440 2877 a(Z) p Fr 1540 3013 a(\() p Fs(!) p Fq 1643 2971 a(2) p Fr 1694 2986 a(~) p Fs 1682 3013 a(\036) p Fq 1740 2971 a(2) p Fr 1801 3013 a(+) 27 b(~) p Fs -54 w(\031) p Fq 1958 2971 a(2) p Fr 1997 3013 a(\)) c(+) p Fp 2156 2877 a(Z) p Fr 2281 3013 a(^) p Fs -58 w(\032) p Fr(\() p Fs(k) p Fr 3 w(\)\() p Fs(e) p Fi 2535 2971 a(\000) p Fl(ik) r(q) p Fm 2713 3013 a(\000) p Fr 22 w(1\)) 2912 2986 y(~) p Fs 2899 3013 a(\036) p Fr 1355 3281 a(+) 1449 3214 y(~) p Fs -57 w(p) p Fq 1490 3178 a(2) p 1441 3258 89 4 v Fr 1461 3350 a(2) 1561 3281 y(+) p Fs 22 w(V) p Fr 22 w(\() 7 b(~) p Fs -56 w(q) p Fr 4 w(\)) p Fm 17 w(\000) p Fp 1972 3146 a(Z) p Fm 2098 3214 a(j) p Fr 9 w(^) p Fs -58 w(\032) p Fr(\() p Fs(k) p Fr 3 w(\)) p Fm(j) p Fq 2334 3178 a(2) p Fs 2373 3214 a(e) p Fi 2418 3178 a(\000) p Fl(ik) r(q) p 2098 3258 476 4 v Fs 2284 3350 a(!) p Fq 2349 3321 a(2) p Fp 1877 3404 a(|) p 1922 3404 264 12 v 264 w({z) p 2276 3404 V 264 w(}) p Fq 1976 3495 a(=) p Fl(W) p Fq 10 w(\() e(~) p Fl -40 w(q) p Fq 2 w(\)) p Fh 24 w(b) r(ounded) p Fr 2600 3281 a(+) p Fp 2693 3146 a(Z) p Fm 2819 3214 a(j) p Fr 9 w(^) p Fs -58 w(\032) p Fr(\() p Fs(k) p Fr 3 w(\)) p Fm(j) p Fq 3055 3178 a(2) p 2819 3258 276 4 v Fr 2880 3350 a(2) p Fs(!) p Fq 2994 3321 a(2) p Fp 2692 3404 a(|) p 2737 3404 116 12 v 116 w({z) p 2943 3404 V 116 w(}) p Fq 2740 3489 a(=) p Fh(constan) n(t) p Fs 3120 3281 a(:) p Fv 148 w(\(6.1\)) 257 3685 y(If) 23 b(w) m(e) h(w) m(an) m (t) g(to) e(study) i(the) g(system) g(near) f(the) g(new) h (equilibriun,) e(one) i(then) f(has) g(to) g(c) m(hose) 257 3805 y(the) 32 b(phase) h(space) g(suc) m(h) g(that) p Fr 1397 3779 a(~) p Fs 1384 3805 a(\036) p Fr 28 w(=) 27 b(0) p Fv 32 w(b) s(elongs) k(to) g(it.) 43 b(It) 31 b(is) h(then) g(natural) e(to) i(study) p Fr 3432 3780 a(~) p Fs 3407 3805 a(H) p Fv 257 3925 a(not) f(on) p Fr 591 3900 a(~) p Fm 562 3925 a(H) p Fv 32 w(but) g(on) p Fm 30 w(H) p Fs 1 w(:) p Fv 31 w(In) g(other) f(w) m(ords,) i(to) e (mak) m(e) h(sure) g(that) g(the) g(equilibrium) c(p) s(oin) m(t) 257 4046 y(for) 32 b(the) g(in) m(teracting) f(system) h(is) g(in) f(the) h (phase) h(space,) g(one) f(has) g(to) g(consider) g(another) 257 4166 y(space.) 45 b(If) 32 b(one) h(do) s(es) g(so,) g(condition) e (\(IR\)) h(is) g(then) i(satis\034ed) f(ev) m(en) h(for) p Fs 32 w(d) p Fr 27 w(=) 28 b(3) p Fv(:) p Fr 1350 4368 a(^) p Fs -58 w(\032) p Fr(\() p Fs(k) p Fr 3 w(\)\() p Fs(e) p Fi 1604 4331 a(\000) p Fl(ik) r(q) p Fm 1782 4368 a(\000) p Fr 23 w(1\)) p 1341 4412 628 4 v Fs 1595 4520 a(!) p Fj 1670 4456 a(3) p 1670 4468 31 3 v 1670 4509 a(2) p Fm 2006 4435 a(2) p Fs 28 w(L) p Fq 2166 4394 a(2) p Fr 2206 4435 a(\() p Fg(R) p Fl 2310 4394 a(d) p Fr 2356 4435 a(\)) p Fs(:) p Fv 257 4706 a(Then,) 53 b(if) 47 b(one) h(quan) m(tizes) p Fr 1327 4680 a(~) p Fs 1301 4706 a(H) p Fv 8 w(,) k(one) c(obtains) f(a) h(mo) s(del) e(in) i(whic) m(h) g(a) g(ground) g(state) 257 4826 y(exists.) c(This) 32 b(is) f(precisely) g(what) h(Arai) e(do) s(es) i(in) f([2) o(],) h(but) g(without) f(explaining) e(it) i(this) 257 4946 y(w) m(a) m(y) -8 b(.) 51 b(Ho) m(w) m(ev) m(er,) 38 b(the) d(same) f(transformation) f (in) h(our) g(mo) s(del) f(do) s(es) i(not) g(mak) m(e) g(things) 257 5067 y(\020b) s(etter.\021) 51 b(Indeed,) 34 b(condition) d(\(IR\)) i (b) s(ecomes,) g(after) f(the) h(same) f(transform:) p Fr 1031 5261 a([) p Fs(\032) p Fq 1108 5276 a(1) p Fr 1148 5261 a(\() p Fs(x) p Fm 23 w(\000) p Fs 22 w(q) p Fr 4 w(\)) p Fm 22 w(\000) p Fs 23 w(\032) p Fq 1620 5276 a(1) p Fr 1660 5261 a(\() p Fs(x) p Fr(\)]) 9 b(^) p Fs -58 w(\032) p Fq 1868 5276 a(2) p Fr 1908 5261 a(\() p Fs(k) p Fr 3 w(\)) p 1031 5305 1007 4 v Fs 1410 5413 a(!) p Fr 4 w(\() p Fs(k) p Fr 3 w(\)) p Fj 1615 5349 a(3) p 1614 5361 31 3 v 1614 5402 a(2) p Fm 2075 5328 a(2) p Fs 28 w(L) p Fq 2235 5287 a(2) p Fr 2275 5328 a(\() p Fg(R) p Fl 2379 5287 a(d) p Fm 2448 5328 a(\002) p Fg 22 w(R) p Fl 2613 5287 a(n) p Fr 2666 5328 a(\)) p Fs(;) p Fv 1828 5637 a(32) p 90 rotate dyy eop %%Page: 33 33 33 32 bop Fv 257 573 a(whic) m(h) 33 b(is) f(still) f(not) h (satis\034ed) h(if) p Fs 31 w(n) p Fr 28 w(=) 28 b(3) p Fv 32 w(unless) p Fr 42 w(^) p Fs -58 w(\032) p Fq 2071 588 a(2) p Fr 2111 573 a(\(0\)) f(=) h(0) p Fs(:) p Fw 257 693 a(A) m(c) m(kno) m(wledgmen) m(ts:) p Fv 50 w(P) m(art) 37 b(of) f(this) g(w) m(ork) i(w) m(as) f(supp) s(orted) h(b) m(y) f(the) g (P) m(ostdo) s(ctoral) 257 814 y(T) -8 b(raining) 32 b(Program) g(HPRN-CT-2002-0277.) 44 b(The) 35 b(author) e(wishes) i(to) e(thank) h(S.) f(De) 257 934 y(Bi\350vre) g(for) f(man) m(y) h(enjo) m (y) m(able) g(discussions) g(and) g(useful) f(commen) m(ts.) p Fu 257 1267 a(References) p Fv 257 1486 a([1]) 342 b(Ara\357) 35 b(A.,) h(On) g(a) f(mo) s(del) f(of) h(a) h(harmonic) e(oscillator) f (coupled) i(to) h(a) f(quan-) 702 1606 y(tized,) 42 b(massless,) g (scalar) d(\034eld,) j(I,) f(Journal) e(of) g(Math.) i(Ph) m(ys.) p Fw 42 w(22) p Fv(,) h(2539-) 702 1727 y(2548) 31 b(\(1981\).) 257 1930 y([2]) 342 b(Ara\357) 38 b(A.,) i(Ground) e(state) h(of) f(the) h (massless) g(Nelson) g(mo) s(del) d(without) i(in-) 702 2050 y(frared) f(cuto\033) g(in) f(a) g(Non-F) -8 b(o) s(c) m(k) 37 b(represen) m(tation,) i(Rev.) e(Math.) g(Ph) m(ys) p Fw 39 w(13) p Fv(,) 702 2171 y(1075-1094) 30 b(\(2001\).) 257 2374 y([3]) 342 b(Ara\357) 28 b(A.,) j(Hirok) -5 b(a) m(w) m(a) 28 b(M.,) j(On) e(the) h(existence) h(and) e(uniqueness) i(of) e(ground) 702 2494 y(states) h(of) f(a) g(generalized) g(spin-b) s(oson) g(mo) s (del,) g(J.) h(F) -8 b(unc.) 29 b(Anal.) p Fw 29 w(151) p Fv(,) h(455-) 702 2615 y(503) h(\(1997\).) 257 2818 y([4]) 342 b(Bruneau) 43 b(L.,) j(De) e(Bi\350vre) f(S.,) j(A) d(Hamiltonian) c (mo) s(del) i(for) h(linear) g(fric-) 702 2939 y(tion) 25 b(in) h(a) g(homogeneous) h(medium,) f(Comm.) g(Math.) h(Ph) m(ys.) p Fw 28 w(229) p Fv(,) h(511-542) 702 3059 y(\(2002\).) 257 3262 y([5]) 342 b(Bac) m(h) 24 b(V.,) i(F) -8 b(r\366hlic) m(h) 22 b(J.,) k(Sigal) c(I,) i(Quan) m(tum) g(electro) s(dynamics) f(of) g (con\034ned) 702 3383 y(non-relativistic) 29 b(particles,) j(A) m (dv.Math.) p Fw 34 w(137) p Fv(,) h(299-395) d(\(1998\).) 257 3586 y([6]) 342 b(Bac) m(h) 25 b(V.,) h(F) -8 b(r\366hlic) m(h) 23 b(J.,) j(Sigal) d(I,) h(Sp) s(ectral) g(analysis) g(for) g(systems) i (of) e(atoms) 702 3707 y(and) 46 b(molecules) f(coupled) h(to) f(the) i (quan) m(tized) g(radiation) c(\034eld,) 49 b(Comm.) 702 3827 y(Math.) 33 b(Ph) m(ys.) p Fw 34 w(207) p Fv(,) g(249-290) d (\(1999\).) 257 4030 y([7]) 342 b(Derezi\253ski) 29 b(J.,) h(V) -8 b(an) 28 b(Ho) m(v) m(e) i(Hamiltonians) c(-) i(exactly) h(solv) -5 b(able) 27 b(mo) s(dels) h(of) 702 4151 y(the) 33 b(infrared) e(and) i (ultra) m(violet) d(problem,) i(preprin) m(t.) 257 4354 y([8]) 342 b(De) 36 b(Bi\350vre) g(S.,) p Ft 37 w(private) i(c) -5 b(ommunic) g(ation) p Fv 42 w(and) 36 b(Classical) e(and) i(quan) m (tum) 702 4475 y(linear) 31 b(oscillator) e(\034elds,) k(in) f (preparation.) 257 4678 y([9]) 342 b(Derezi\253ski) 45 b(J.,) j(G\351rard) c(C.,) k(Asymptotic) c(completeness) h(in) f(quan) m (tum) 702 4798 y(\034eld) 34 b(theory) -8 b(.) 34 b(Massiv) m(e) i(P) m (auli-Fierz) c(Hamiltonians,) f(Rev.) k(Math.) f(Ph) m(ys.) p Fw 702 4919 a(11) p Fv(,) e(383-450) f(\(1999\).) 257 5122 y([10]) 293 b(Derezi\253ski) 39 b(J.,) i(G\351rard) d(C.,) j (Scattering) e(theory) g(of) f(infrared) g(div) m(ergen) m(t) 702 5243 y(P) m(auli-Fierz) 30 b(Hamiltonians,) f(in) j(preparation.) 1828 5637 y(33) p 90 rotate dyy eop %%Page: 34 34 34 33 bop Fv 257 573 a([11]) 293 b(Derezi\253ski) 34 b(J.,) f(Jak\262i\242) h(V.,) f(Sp) s(ectral) g(theory) h(of) f(P) m (auli-Fierz) d(op) s(erators,) 702 693 y(J.) i(F) -8 b(unc.) 33 b(Anal.) p Fw 32 w(180) p Fv(,) g(243-327) d(\(2001\).) 257 897 y([12]) 293 b(F) -8 b(riedric) m(hs) 23 b(K.O.,) j(Mathematical) c (asp) s(ects) j(of) f(quan) m(tum) g(theory) h(of) f(\034elds,) 702 1017 y(New-Y) -8 b(ork) 33 b(\(1953\).) 257 1220 y([13]) 293 b(G\351rard) 37 b(C.,) i(On) f(the) g(existence) h(of) e(ground) h (states) g(for) f(massless) h(P) m(auli-) 702 1341 y(Fierz) 32 b(Hamiltonians,) d(Ann.) k(Henri.) f(P) m(oincar\351.) p Fw 33 w(1) p Fv(,) g(443-459) f(\(2000\).) 257 1544 y([14]) 293 b(Glimm) 26 b(J.,) 31 b(Ja\033e) f(A.,) h(The) p Fs 31 w(\025) p Fr(\() p Fs(\036) p Fq 1898 1508 a(4) p Fr 1937 1544 a(\)) p Fq 1975 1559 a(2) p Fv 2045 1544 a(quan) m(tum) f (\034eld) g(theory) h(without) e(cut-) 702 1665 y(o\033s) 46 b(I) s(I.) g(The) h(\034eld) e(op) s(erators) h(and) g(the) h(appro) m (ximate) d(v) -5 b(acuum,) 50 b(Ann.) 702 1785 y(Math.) p Fw 33 w(91) p Fv(,) 32 b(362-401) f(\(1970\).) 257 1988 y([15]) 293 b(Griesemer) 67 b(M.,) 78 b(Lieb) 68 b(E.H.,) 78 b(Loss) 69 b(M.,) 78 b(Ground) 69 b(states) g(in) f(non-) 702 2109 y(relativistic) 28 b(quan) m(tum) i(electro) s(dynamics,) h(In) m (v) m(en) m(t.) h(Math.) p Fw 31 w(145) p Fv(,) f(557-595) 702 2229 y(\(2001\).) 257 2433 y([16]) 293 b(K) m(omec) m(h) 44 b(A.,) k(Kunze) d(M.,) j(Sp) s(ohn) c(H.,) k(Long-time) 42 b(asymptotics) i(for) f(a) 702 2553 y(classical) 30 b(particle) h(in) m (teracting) f(with) i(a) g(scalar) f(w) m(a) m(v) m(e) j(\034eld,) e (Comm.) e(P) m(ar-) 702 2673 y(tial) g(Di\033eren) m(tial) h(Equation) p Fw 32 w(22) p Fv(,) i(307-335) d(\(1997\).) 257 2877 y([17]) 293 b(L\366rinczi) 46 b(J.,) 51 b(Minlos) 46 b(R.A.,) 52 b(Sp) s(ohn) 47 b(H.,) 52 b(The) c(infrared) f(b) s(eha) m (viour) g(in) 702 2997 y(Nelson's) 30 b(mo) s(del) f(of) g(a) h(quan) m (tum) g(particle) f(coupled) h(to) g(a) g(massless) g(scalar) 702 3117 y(\034eld,) i(Ann.) h(Henri.) f(P) m(oincar\351.) p Fw 33 w(3) p Fv(,) h(269-295) d(\(2002\).) 257 3321 y([18]) 293 b(Reed) 26 b(M.,) h(Simon) d(B.,) j(Metho) s(ds) g(of) e(mo) s(dern) g (mathematical) d(ph) m(ysics) 27 b(\(v) m(ol) 702 3441 y(1\),) 32 b(A) m(cademic) g(Press,) i(London) f(\(1976\).) 257 3645 y([19]) 293 b(Reed) 26 b(M.,) h(Simon) d(B.,) j(Metho) s(ds) g(of) e(mo) s(dern) g(mathematical) d(ph) m(ysics) 27 b(\(v) m(ol) 702 3765 y(2\),) 32 b(A) m(cademic) g(Press,) i(London) f(\(1976\).) 257 3968 y([20]) 293 b(V) -8 b(an) 32 b(Ho) m(v) m(e) i(L.,) e(Les) h (di\036cult\351s) g(de) g(div) m(ergences) h(p) s(our) e(un) h(mo) s (d\350le) e(parti-) 702 4089 y(culier) g(de) i(c) m(hamp) g(quan) m (ti\034\351,) g(Ph) m(ysica) p Fw 33 w(18) p Fv(,) g(145-152) e (\(1952\).) 1828 5637 y(34) p 90 rotate dyy eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0306030922779--