Content-Type: multipart/mixed; boundary="-------------9908280430917" This is a multi-part message in MIME format. ---------------9908280430917 Content-Type: text/plain; name="99-315.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-315.keywords" long range magnetic fields, tight binding limit, essential spectrum ---------------9908280430917 Content-Type: application/postscript; name="lmpf.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lmpf.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: lmpf.dvi %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: A4 %%EndComments %DVIPSCommandLine: dvips lmpf %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1999.08.28:1121 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N 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b(fact)h(that)g(the)g(\\one)f(w)n(ell")523 1013 y(p)r(erturbations)g(w)n(ere)f(tak)n(en)h(compactly)g(supp)r(orted)g (allo)n(w)n(ed)f(us)h(to)g(conclude)g(that)h(the)523 1112 y(Hamiltonian)f(with)h(an)f(in\014nite)g(n)n(um)n(b)r(er)g(of)h(w) n(ells)e(\(not)i(necessarily)d(disp)r(osed)i(in)h(a)e(p)r(e-)523 1212 y(rio)r(dic)e(lattice\))h(has)e(\(for)h(a)g(su\016cien)n(tly)h (large)d(in)n(ter)i(w)n(ell)g(distance\))h(essen)n(tial)e(sp)r(ectrum) 523 1312 y(near)32 b(the)i(discrete)f(eigen)n(v)-5 b(alues)32 b(of)h Fo(H)7 b Fp(,)35 b(and)e(is)g(lo)r(cated)g(in)g(mini-bands)g (whose)g(width)523 1411 y(presen)n(ts)27 b(a)g Fg(Gaussian)h Fp(deca)n(y)f(with)h(the)g(in)n(ter)f(w)n(ell)h(distance.)648 1511 y(In)33 b(this)g(pap)r(er)g(w)n(e)f(in)n(tend)i(to)f(study)g(the)g (case)g(in)g(whic)n(h)g(the)g(\\one)g(w)n(ell")f(p)r(ertur-)523 1611 y(bations)i(are)g(no)h(longer)f(compactly)g(supp)r(orted,)i(but)g (p)r(erio)r(dic.)59 b(W)-7 b(e)35 b(will)g(sho)n(w)f(that)523 1710 y(\(see)f(Theorem)g(2.1)f(for)h(the)h(precise)e(statemen)n(t\))i (the)g(essen)n(tial)e(sp)r(ectrum)i(whic)n(h)f(ap-)523 1810 y(p)r(ears)27 b(near)h(a)f(discrete)h(eigen)n(v)-5 b(alue)27 b(of)h Fo(H)35 b Fp(lo)r(oks)27 b(lik)n(e)h(a)f(mini-band)i (to)r(o,)f(and)f(its)i(width)523 1910 y(is)k(shrinking)f(\(at)g (least\))h Fg(faster)i(than)g(any)f(exp)l(onential)p Fp(.)53 b(This)33 b(together)f(with)h(an)f(im-)523 2009 y(pro)n(v)n(ed)h(resolv)n(en)n(t)g(estimate)i(for)f(the)h(\\one)e(w)n (ell")h(Hamiltonian)h(\(see)f(Lemma)g(2.1\))g(is)523 2109 y(what)27 b(w)n(e)g(add)h(new)f(to)g(the)h(already)e(kno)n(wn)h (results.)36 b(W)-7 b(e)28 b(w)n(an)n(t)e(to)i(stress)e(though)h(that) 523 2208 y(the)36 b(magnetic)g(p)r(erturbation)f(considered)g(here)g (is)g(the)i(magnetic)e(\014eld)h(whic)n(h)g(is)f(not)523 2308 y(alw)n(a)n(ys)30 b(a)h(v)n(ery)f(regular)g(p)r(erturbation)h(and) g(one)g(has)g(to)h(b)r(e)g(a)f(little)h(bit)g(more)f(careful)523 2408 y(with)d(it.)37 b(Nev)n(ertheless,)27 b(from)g(the)h(tec)n(hnical) f(p)r(oin)n(t)h(of)g(view,)f(our)g(w)n(ork)f(follo)n(ws)g(rather)523 2507 y(closely)h(the)h(geometric)e(p)r(erturbation)h(theory)g(dev)n (elop)r(ed)g(in)h([1].)648 2607 y(Let)f(us)h(commen)n(t)f(a)h(few)f (previous)g(results)g(whic)n(h)g(are)g(connected)h(to)f(ours.)648 2707 y(In)41 b([7])g(the)h(case)e(of)h(pure)g(magnetic)g(\014eld)h (Hamiltonians)e(is)h(considered.)77 b(T)-7 b(ak)n(e)523 2806 y Fo(H)592 2818 y Fj(a)632 2806 y Fp(\()p Fo(\025)p Fp(\))45 b(=)e(\()p Fm(\000)p Fo({)p Fm(r)26 b(\000)g Fo(\025)p Fn(a)p Fp(\))1333 2776 y Fl(2)1411 2806 y Fp(with)41 b Fo(\025)j Fm(2)f Fn(R)p Fp(,)g Fn(a)h Fm(2)f(C)2178 2776 y Fl(1)2229 2739 y Ff(\000)2267 2806 y Fn(R)2339 2776 y Fj(d)2377 2806 y Fo(;)14 b Fn(R)2486 2776 y Fj(d)2525 2739 y Ff(\001)2563 2806 y Fp(,)43 b Fo(d)g Fm(\025)g Fp(2)d(and)f Fn(B)44 b Fp(=)f Fo(d)p Fn(a)523 2906 y Fp(p)r(erio)r(dic)28 b(with)h(resp)r(ect)e(to)h Fn(Z)1476 2876 y Fj(d)1515 2906 y Fp(.)39 b(Let)28 b Fo(M)1807 2918 y Fj(B)1888 2906 y Fp(=)23 b Fm(f)p Fn(x)p Fm(j)p Fn(B)p Fp(\()p Fn(x)p Fp(\))i(=)e(0)p Fm(g)28 b Fp(and)f Fo(M)2740 2918 y Fj(a)2804 2906 y Fp(=)d Fm(f)p Fn(x)p Fm(j)p Fn(a)p Fp(\()p Fn(x)p Fp(\))h(=)e(0)p Fm(g)p Fp(.)523 3005 y(Under)34 b(the)g(condition)f(that)h Fm(j)p Fo(M)1590 3017 y Fj(B)1647 3005 y Fm(n)p Fo(M)1770 3017 y Fj(a)1809 3005 y Fm(j)f Fp(=)f(0,)j(it)f(is)g(sho)n(wn)e(that)i Fo(H)2753 3017 y Fj(a)2793 3005 y Fp(\()p Fo(\025)p Fp(\))h(con)n(v)n (erges)c(in)523 3105 y(the)e(norm)e(resolv)n(en)n(t)g(sense)g(as)h Fo(\025)c Fm(!)g(1)29 b Fp(to)f(the)g(Diric)n(hlet)g(Laplacian)f(on)h Fo(M)9 b Fp(,)28 b(therefore)523 3205 y(the)g(sp)r(ectrum)g(of)f(the)h (pure)g(magnetic)f(op)r(erator)f(will)h(ha)n(v)n(e)g(band)g(structure.) 648 3304 y(Starting)37 b(from)h(these)h(results,)h(Nak)-5 b(am)n(ura)37 b([14])h(studied)g(the)h(existence)f(of)g(op)r(en)523 3404 y(gaps)27 b(for)g(op)r(erators)e(of)j(the)g(follo)n(wing)e(t)n(yp) r(e:)1471 3587 y Fo(H)1540 3599 y Fj(a)1580 3587 y Fp(\()p Fo(\025)p Fp(\))e(=)e(\()p Fm(\000)p Fo({)p Fm(r)d(\000)f Fo(\025)p Fn(a)p Fp(\))2224 3552 y Fl(2)2281 3587 y Fp(+)g Fo(V)5 b(;)523 3769 y Fp(where)22 b Fo(V)40 b Fp(is)22 b(b)r(ounded.)36 b(He)22 b(also)f(considered)g(that)h(the)h(electric)e (p)r(oten)n(tial)h(and)g(the)g(mag-)523 3869 y(netic)32 b(\014eld)f(are)f(p)r(erio)r(dic)h(with)h(resp)r(ect)f(to)g(a)f (lattice)i(\000)c Fm(\032)h Fn(Z)2525 3839 y Fj(d)2564 3869 y Fp(,)j(with)g(b)r(ounded)f(funda-)523 3968 y(men)n(tal)25 b(domain)f(\012)1147 3980 y Fl(\000)1193 3968 y Fp(.)36 b(Let)25 b Fo(H)1474 3938 y Fj(D)1467 3989 y(a)1534 3968 y Fp(\()p Fo(\025)p Fp(\))h(b)r(e)f(the)h(restriction)e(of)g Fo(H)2478 3980 y Fj(a)2519 3968 y Fp(\()p Fo(\025)p Fp(\))i(to)f(\012) 2816 3980 y Fl(\000)2886 3968 y Fp(with)g(Diric)n(hlet)523 4068 y(b)r(oundary)32 b(conditions)f(imp)r(osed)i(on)f(its)g(fron)n (tier.)50 b(Denote)32 b(with)h(\006\()p Fo(\025)p Fp(\))h(the)e(sp)r (ectrum)523 4168 y(of)c Fo(H)694 4138 y Fj(D)687 4188 y(a)754 4168 y Fp(\()p Fo(\025)p Fp(\).)39 b(If)29 b(the)f(magnetic)g (\014eld)g(is)g(di\013eren)n(t)g(from)g(zero)f(on)h Fo(@)5 b Fp(\012)2701 4180 y Fl(\000)2746 4168 y Fp(,)28 b(it)h(is)f(pro)n(v)n (en)e(that)523 4267 y Fo(\033)s Fp(\()p Fo(H)674 4279 y Fj(a)715 4267 y Fp(\()p Fo(\025)p Fp(\)\))d(is)f(situated)g(near)f (\006\()p Fo(\025)p Fp(\))i(up)f(to)g(an)g(error)e(of)i(order)e Fm(O)r Fp(\(exp)15 b(\()p Fm(\000)p Fo(\013)2857 4197 y Fm(p)p 2926 4197 49 4 v 70 x Fo(\025)q Fp(\)\),)24 b(for)d(some)523 4367 y(p)r(ositiv)n(e)27 b Fo(\013)p Fp(.)648 4467 y(If)33 b(in)g(these)g(t)n(w)n(o)f(pap)r(ers)g(\\tigh)n (t-binding)h(limit")g(means)f(that)h(the)h(magnetic)e(\014eld)523 4566 y(outside)f(the)i(w)n(ells)e(is)g(v)n(ery)g(large,)g(in)h([5])f (\(as)g(in)h(our)f(case\))g(the)h(large)e(quan)n(tit)n(y)h(is)h(the)523 4666 y(distance)27 b(b)r(et)n(w)n(een)h(w)n(ells.)648 4765 y(More)d(precisely)-7 b(,)26 b(Daumer)g([5])h(considered)e(the)i (t)n(w)n(o)f(dimensional)g(case)g Fo(d)d Fp(=)f(2,)27 b(with)523 4865 y(a)h(p)r(erio)r(dic)f(lattice)h(\000)c(=)f Fn(Z)1387 4835 y Fl(2)1453 4865 y Fp(and)k(a)h(constan)n(t)f(magnetic)h (\014eld)g Fo(B)t Fp(.)38 b(The)28 b(Hamiltonian)f(is)1934 5147 y(2)p eop %%Page: 3 3 3 2 bop 523 614 a Fp(giv)n(en)27 b(b)n(y:)1568 714 y Fo(H)1644 680 y Fj(B)1637 735 y(R)1724 714 y Fp(=)c(\()p Fn(p)c Fm(\000)f Fn(a)p Fp(\))2077 680 y Fl(2)2133 714 y Fp(+)g Fo(V)2264 726 y Fj(R)2319 714 y Fo(;)523 847 y Fp(where)27 b Fn(a)h Fp(is)g(the)g(transv)n(ersal)d(gauge)h(whic)n(h) i(giv)n(es)e Fo(B)32 b Fp(and)1416 1013 y Fo(V)1464 1025 y Fj(R)1542 1013 y Fp(=)1632 935 y Ff(X)1630 1113 y Fj(\015)t Fk(2)p Fl(\000)1768 1013 y Fo(V)19 b Fp(\()p Fm(\001)g(\000)f Fo(R)q(\015)5 b Fp(\))p Fo(;)96 b(R)24 b(>)e Fp(0)p Fo(;)523 1253 y Fp(where)27 b Fo(V)47 b Fp(is)27 b(a)g(smo)r(oth)h(p)r(oten)n (tial)f(ob)r(eying)g(the)h(follo)n(wing)f(conditions:)904 1408 y Fm(8)p Fn(x)c Fm(2)g Fn(R)1174 1374 y Fl(2)1211 1408 y Fo(;)37 b Fm(8)p Fo(\013)23 b Fm(2)g Fn(N)1547 1374 y Fl(2)1584 1408 y Fo(;)37 b Fm(j)p Fo(D)1738 1374 y Fj(\013)1785 1408 y Fo(V)19 b Fm(j)p Fp(\()p Fn(x)p Fp(\))24 b Fm(\024)f Fp(C)2161 1420 y Fj(\013)2208 1408 y Fp(\(1)c(+)f Fm(j)p Fn(x)p Fm(j)p Fp(\))2512 1374 y Fk(\000)p Fj(\033)r Fk(\000j)p Fj(\013)p Fk(j)2743 1408 y Fo(;)c(\033)27 b(>)22 b Fp(2)p Fo(:)523 1563 y Fp(One)41 b(of)g(the)h(man)n(y)e(problems)h(considered)f(there)h(is)g(the)g (comparison)f(b)r(et)n(w)n(een)h(the)523 1662 y(negativ)n(e)25 b(sp)r(ectrum)g(of)h Fo(H)1378 1632 y Fj(B)1371 1685 y(R)1460 1662 y Fp(and)g(the)g(negativ)n(e)e(sp)r(ectrum)i(of)g(the)g (\\one)e(w)n(ell")h(op)r(erator)523 1762 y Fo(H)30 b Fp(=)22 b Fm(\000)p Fp(\001)d(+)e Fo(V)47 b Fp(for)27 b Fo(B)32 b Fp(small)27 b(enough)g(and)g Fo(R)h Fp(su\016cien)n(tly)g (large.)35 b(It)28 b(is)f(sho)n(wn)g(that)h(the)523 1862 y(sp)r(ectral)33 b(problem)h(can)f(b)r(e)i(reduced)e(to)h(that)g(of)g (a)g(p)r(erturb)r(ed)g(Harp)r(er)f(t)n(yp)r(e)h(pseudo-)523 1961 y(di\013eren)n(tial)39 b(op)r(erator)e(whic)n(h)h(can)h(b)r(e)g (handled)f(with)i(the)f(theory)f(dev)n(elop)r(ed)g(b)n(y)g(B.)523 2061 y(Hel\013er)28 b(and)f(J.)h(Sj\177)-42 b(ostrand)27 b(in)g([8],)h([9],...,)f([12].)648 2160 y(Amongst)18 b(other)h(things,)i(it)e(is)g(pro)n(v)n(en)f(that)h(if)h Fo(\025)j(<)g Fp(0)c(is)g(a)f(not)h(degenerate)f(eigen)n(v)-5 b(alue)523 2260 y(of)23 b Fo(H)7 b Fp(,)23 b(then)g(in)g(the)g(limit)g (\()p Fo(R)q(;)14 b(B)t Fp(\))23 b Fm(!)g Fp(\()p Fm(1)p Fo(;)14 b Fp(0\),)24 b Fo(H)2052 2230 y Fj(B)2045 2283 y(R)2131 2260 y Fp(has)e(a)g(nonempt)n(y)h(sp)r(ectral)f(band)g(near) 523 2360 y Fo(\025)28 b Fp(included)g(in)g([)p Fo(\025)19 b Fm(\000)f Fo(o)p Fp(\(1\))p Fo(;)c(\025)19 b Fp(+)f Fo(o)p Fp(\(1\)].)648 2459 y(The)27 b(ma)5 b(jor)27 b(di\013erence)h(b) r(et)n(w)n(een)g(Daumer's)f(w)n(ork)g(and)g(ours)g(is)h(that)g(the)g (constan)n(t)523 2559 y(magnetic)d(\014eld)i(is)e(treated)h(there)g(as) f(a)g(p)r(erturbation)h(to)f Fo(H)7 b Fp(,)27 b(whic)n(h)e(restricts)g (the)i(sp)r(ec-)523 2659 y(tral)22 b(study)h(to)f(the)h(negativ)n(e)f (part)g(of)h(the)g(sp)r(ectrum.)35 b(In)23 b(our)f(setting,)i(it)f(is)f (the)h(constan)n(t)523 2758 y(magnetic)35 b(\014eld)g(whic)n(h)g(is)g (resp)r(onsible)g(for)f(the)i(enhanced)f(lo)r(calisation)f(of)h(the)g (mini-)523 2858 y(bands.)47 b(Moreo)n(v)n(er,)29 b(b)r(ecause)i(our)f (one)g(w)n(ell)h(Hamiltonian)g(\(see)g(\(2.9\)\))g(ob)r(eys)f(\(1.4\),) i(it)523 2957 y(could)27 b(ha)n(v)n(e)g(p)r(ositiv)n(e)g(eigen)n(v)-5 b(alues)27 b(whic)n(h)g(generate)f(mini-bands)i(at)f(their)h(turn)g (and)f(in)523 3057 y(fact)h(w)n(e)f(mak)n(e)g(no)g(di\013erence)h(b)r (et)n(w)n(een)f(the)h(negativ)n(e)f(and)g(the)h(p)r(ositiv)n(e)f (regions.)523 3327 y Fq(2)135 b(Preliminaries)47 b(and)d(the)i(results) 523 3509 y Fp(As)30 b(already)e(said,)i(w)n(e)g(shall)f(consider)g (only)g(the)h(t)n(w)n(o)f(dimensional)h(case)e(\(i.e.)44 b(the)30 b(par-)523 3608 y(ticle)36 b(is)g(con\014ned)g(in)g(the)g (plane)g Fo(x)1672 3620 y Fl(3)1746 3608 y Fp(=)h(0)e(and)h(the)g (magnetic)g(\014eld)g(is)g(orthogonal)e(to)523 3708 y(that)f(plane\).) 51 b(Let)32 b Fo(B)t Fp(\()p Fn(x)p Fp(\))g Fm(2)f Fo(C)1525 3678 y Fl(1)1563 3708 y Fp(\()p Fn(R)1667 3678 y Fl(2)1704 3708 y Fp(\).)51 b(W)-7 b(e)33 b(shall)f(use)g(the)h(follo)n(wing)e (family)h(of)h(v)n(ector)523 3808 y(p)r(oten)n(tials)27 b(corresp)r(onding)f(to)h Fo(B)t Fp(\()p Fn(x)p Fp(\):)1097 4022 y Fn(a)p Fp(\()p Fn(x)p Fo(;)14 b Fn(x)1312 3987 y Fk(0)1336 4022 y Fp(\))24 b(=)1479 3909 y Ff(Z)1562 3929 y Fl(1)1526 4097 y(0)1614 4022 y Fo(ds)18 b(s)g Fn(B)p Fp(\()p Fn(x)1921 3987 y Fk(0)1964 4022 y Fp(+)g Fo(s)p Fp(\()p Fn(x)h Fm(\000)f Fn(x)2320 3987 y Fk(0)2344 4022 y Fp(\)\))h Fm(^)f Fp(\()p Fn(x)i Fm(\000)e Fn(x)2735 3987 y Fk(0)2758 4022 y Fp(\))p Fo(:)404 b Fp(\(2.1\))523 4223 y(F)-7 b(or)27 b Fn(x)722 4193 y Fk(0)769 4223 y Fp(=)22 b(0,)28 b(this)g(is)f(nothing)g(but)i(the)f(usual)f(transv)n (ersal)e(gauge)h(\(see)i(e.g.)36 b([17]\):)1295 4437 y Fn(a)p Fp(\()p Fn(x)p Fo(;)14 b Fp(0\))24 b Fm(\021)e Fn(a)p Fp(\()p Fn(x)p Fp(\))j(=)1917 4324 y Ff(Z)2000 4345 y Fl(1)1964 4513 y(0)2052 4437 y Fo(ds)18 b(s)g Fn(B)p Fp(\()p Fo(s)h Fn(x)p Fp(\))g Fm(^)g Fn(x)p Fo(:)602 b Fp(\(2.2\))648 4639 y(If)28 b(w)n(e)f(de\014ne)h(\(see)f([5])g(for)h (details\):)755 4853 y Fo(')p Fp(\()p Fn(x)p Fo(;)14 b Fn(x)978 4818 y Fk(0)1003 4853 y Fp(\))23 b(=)g Fm(\000)1225 4736 y Ff(\022)1285 4740 y(Z)1368 4760 y Fl(1)1331 4928 y(0)1419 4853 y Fo(dt)1506 4740 y Ff(Z)1589 4760 y Fl(1)1553 4928 y(0)1641 4853 y Fo(ds)18 b(s)g Fn(B)p Fp(\()p Fo(s)h(t)f Fp(\()p Fn(x)i Fm(\000)e Fn(x)2239 4818 y Fk(0)2262 4853 y Fp(\))h(+)f Fo(s)g Fn(x)2503 4818 y Fk(0)2527 4853 y Fp(\))2559 4736 y Ff(\023)2639 4853 y Fm(\001)g Fp(\()p Fn(x)i Fm(^)e Fn(x)2905 4818 y Fk(0)2929 4853 y Fp(\))p Fo(;)233 b Fp(\(2.3\))1934 5147 y(3)p eop %%Page: 4 4 4 3 bop 523 614 a Fp(then)28 b(one)f(can)h(see)f(that:)1424 814 y Fm(r)1493 826 y Fe(x)1537 814 y Fo(')p Fp(\()p Fn(x)p Fo(;)14 b Fn(x)1760 779 y Fk(0)1785 814 y Fp(\))23 b(=)g Fn(a)p Fp(\()p Fn(x)p Fp(\))d Fm(\000)e Fn(a)p Fp(\()p Fn(x)p Fo(;)c Fn(x)2406 779 y Fk(0)2431 814 y Fp(\))p Fo(:)731 b Fp(\(2.4\))648 949 y(If)28 b Fo(B)t Fp(\()p Fn(x)p Fp(\))c(=)e Fo(B)1086 961 y Fl(0)1151 949 y Fp(is)28 b(constan)n(t,)f(then:)1265 1148 y Fo(')1319 1160 y Fl(0)1357 1148 y Fp(\()p Fn(x)p Fo(;)14 b Fn(x)1526 1114 y Fk(0)1550 1148 y Fp(\))23 b(=)g Fm(\000)1768 1092 y Fp(1)p 1768 1129 42 4 v 1768 1205 a(2)1819 1148 y Fo(B)1882 1160 y Fl(0)1938 1148 y Fp(\()p Fo(x)2017 1160 y Fl(1)2073 1148 y Fo(x)2120 1114 y Fk(0)2120 1169 y Fl(2)2176 1148 y Fm(\000)18 b Fo(x)2306 1114 y Fk(0)2306 1169 y Fl(1)2362 1148 y Fo(x)2409 1160 y Fl(2)2447 1148 y Fp(\))1487 1348 y Fn(a)1533 1360 y Fl(0)1571 1348 y Fp(\()p Fn(x)p Fo(;)c Fn(x)1740 1314 y Fk(0)1764 1348 y Fp(\))24 b(=)1917 1292 y(1)p 1917 1329 V 1917 1405 a(2)1969 1348 y Fn(B)2037 1360 y Fl(0)2092 1348 y Fm(^)19 b Fp(\()p Fn(x)g Fm(\000)f Fn(x)2400 1314 y Fk(0)2424 1348 y Fp(\))p Fo(:)738 b Fp(\(2.5\))648 1532 y(The)21 b(Hamiltonian)h(of)g(a)f(particle)g(in)h (the)g(presence)f(of)h(a)f(magnetic)g(\014eld)h(and)g(a)f(scalar)523 1631 y(p)r(oten)n(tial)28 b Fo(V)46 b Fp(is)28 b(\(in)g(the)g(transv)n (ersal)d(gauge\):)1084 1800 y Fo(H)90 b Fp(=)83 b(\()p Fn(p)19 b Fm(\000)f Fn(a)p Fp(\()p Fn(x)p Fp(\)\))1771 1758 y Fl(2)1827 1800 y Fp(+)g Fo(V)h Fp(\()p Fn(x)p Fp(\))p Fo(;)1107 1974 y Fn(p)83 b Fp(=)1391 1857 y Ff(\022)1452 1974 y Fm(\000)p Fo({)1596 1918 y(@)p 1554 1955 134 4 v 1554 2031 a(@)5 b(x)1650 2043 y Fl(1)1697 1974 y Fo(;)14 b Fm(\000)p Fo({)1877 1918 y(@)p 1836 1955 V 1836 2031 a(@)5 b(x)1932 2043 y Fl(2)1978 1857 y Ff(\023)2053 1974 y Fo(;)1141 b Fp(\(2.6\))999 2216 y Fn(a)p Fp(\()p Fn(x)p Fp(\))84 b(=)1391 2099 y Ff(\022)1452 2216 y Fm(\000)p Fo(x)1564 2228 y Fl(2)1615 2103 y Ff(Z)1698 2124 y Fl(1)1661 2292 y(0)1749 2216 y Fo(ds)19 b(s)f(B)t Fp(\()p Fo(s)h Fn(x)p Fp(\))p Fo(;)14 b(x)2230 2228 y Fl(1)2282 2103 y Ff(Z)2365 2124 y Fl(1)2328 2292 y(0)2416 2216 y Fo(ds)19 b(s)f(B)t Fp(\()p Fo(s)h Fn(x)p Fp(\))2813 2099 y Ff(\023)2888 2216 y Fo(:)648 2426 y Fp(In)29 b(order)f(to)h(giv)n(e)f(our)h(\014rst) g(result)g(\(used)g(in)h(the)f(estimation)g(on)g(the)h(width)g(of)f (the)523 2525 y(mini-bands)22 b(whic)n(h)f(app)r(ear)g(in)h(the)g(m)n (ultiple)g(w)n(ell)g(case,)g(but)g(also)f(in)n(teresting)g(in)h(itself) 6 b(\),)523 2625 y(w)n(e)27 b(need)h(a)f(few)h(auxiliary)e(notations.) 648 2725 y(Let)34 b Fo(B)866 2737 y Fl(0)937 2725 y Fo(>)f Fp(0,)i Fo(B)1202 2695 y Fk(0)1225 2725 y Fp(\()p Fn(x)p Fp(\))g Fm(2)f(C)1512 2695 y Fl(1)1549 2725 y Fp(\()p Fn(R)1653 2695 y Fl(2)1690 2725 y Fp(\))g(and)g Fo(B)t Fp(\()p Fn(x)p Fp(\))h(=)e Fo(B)2301 2737 y Fl(0)2361 2725 y Fp(+)22 b Fo(B)2515 2695 y Fk(0)2539 2725 y Fp(\()p Fn(x)p Fp(\).)57 b(Let)34 b Fo(V)2936 2737 y Fl(1)3007 2725 y Fm(2)g Fn(L)3153 2695 y Fl(2)3191 2725 y Fp(\()p Fn(R)3295 2695 y Fl(2)3332 2725 y Fp(\),)523 2824 y Fo(V)571 2836 y Fl(2)632 2824 y Fm(2)23 b Fn(L)767 2794 y Fk(1)838 2824 y Fp(\()p Fn(R)942 2794 y Fl(2)979 2824 y Fp(\))28 b(and)f Fo(V)42 b Fp(=)23 b Fo(V)1426 2836 y Fl(1)1482 2824 y Fp(+)18 b Fo(V)1613 2836 y Fl(2)1651 2824 y Fp(.)648 2937 y(De\014ne)32 b Fn(a)p Fp(\()p Fn(x)p Fp(\))f(=)1194 2870 y Ff(R)1249 2891 y Fl(1)1233 2967 y(0)1300 2937 y Fo(ds)14 b(s)g Fn(B)p Fp(\()p Fo(s)p Fn(x)p Fp(\))22 b Fm(^)f Fn(x)p Fp(;)34 b(then)e Fo(H)k Fp(=)29 b(\()p Fn(p)22 b Fm(\000)e Fn(a)p Fp(\))2537 2907 y Fl(2)2596 2937 y Fp(+)h Fo(V)50 b Fp(is)32 b(essen)n(tially)e(self)523 3037 y(adjoin)n(t)d(on)h Fm(C)969 3007 y Fk(1)964 3057 y Fl(0)1039 3037 y Fp(\()p Fn(R)1143 3007 y Fl(2)1180 3037 y Fp(\).)648 3165 y(De\014ne)g Fm(H)975 3177 y Fj(\026)1042 3165 y Fp(:=)23 b Fn(L)1210 3135 y Fl(2)1262 3098 y Ff(\000)1300 3165 y Fn(R)1372 3135 y Fl(2)1409 3165 y Fo(;)14 b(dx)37 b Fp(exp)14 b(\(2)p Fo(\026)o Fm(h\001i)p Fp(\))1957 3098 y Ff(\001)1995 3165 y Fp(,)28 b(where)f Fm(h\001i)d Fp(=)2485 3098 y Ff(\000)2523 3165 y Fp(1)18 b(+)g Fm(j)g(\001)h(j)2772 3135 y Fl(2)2809 3098 y Ff(\001)2857 3093 y Fd(1)p 2857 3102 29 4 v 2857 3135 a(2)2927 3165 y Fp(and)27 b Fo(\026)d Fm(\025)e Fp(0.)648 3265 y(Then)27 b(our)g(\014rst)h(result)f(is)g(con) n(tained)g(in)h(the)g(follo)n(wing)f(lemma:)523 3412 y Fn(Lemma)j(2.1)40 b Fg(L)l(et)32 b Fo(z)k Fg(b)l(e)d(a)h(c)l(omplex)f (numb)l(er)f(such)h(that)g Fp(dist)p Fm(f)p Fo(z)t(;)14 b(\033)s Fp(\()p Fo(H)7 b Fp(\))p Fm(g)27 b Fp(=)i Fo(d)f(>)g Fp(0)33 b Fg(and)523 3512 y(let)d Fo(\037)692 3524 y Fj(n)737 3512 y Fp(\()p Fn(x)p Fp(\))p Fg(,)h Fo(n)23 b Fm(2)g Fn(N)30 b Fg(b)l(e)g(the)f(char)l(acteristic)j(function)e(of)g Fm(fj)p Fn(x)p Fm(j)23 b(\025)g Fo(n)p Fm(g)p Fg(.)648 3612 y(De\014ne)39 b Fo(b)956 3624 y Fj(n)1043 3612 y Fp(=)i(max)14 b(\()p Fm(jj)p Fo(\037)1448 3624 y Fj(n)1493 3612 y Fn(D)1566 3582 y Fj(\013)1614 3612 y Fo(B)1681 3582 y Fk(0)1704 3612 y Fm(jj)1750 3624 y Fk(1)1821 3612 y Fo(;)g Fm(j)p Fo(\013)p Fm(j)23 b(\024)g Fp(1\))40 b Fg(and)h Fo(v)2394 3624 y Fj(n)2481 3612 y Fp(=)h Fm(jj)p Fo(\037)2686 3624 y Fj(n)2731 3612 y Fo(V)2779 3624 y Fl(1)2816 3612 y Fm(jj)2862 3624 y Fl(2)2926 3612 y Fp(+)25 b Fm(jj)p Fo(\037)3114 3624 y Fj(n)3160 3612 y Fo(V)3208 3624 y Fl(2)3245 3612 y Fm(jj)3291 3624 y Fk(1)3362 3612 y Fg(,)523 3711 y(ob)l(eying)36 b(the)f(c)l(ondition)g Fp(lim)1449 3723 y Fj(n)p Fk(!1)1626 3711 y Fp(\()p Fo(b)1694 3723 y Fj(n)1761 3711 y Fp(+)22 b Fo(v)1888 3723 y Fj(n)1933 3711 y Fp(\))33 b(=)e(0)p Fg(.)53 b(T)-6 b(ake)36 b Fn(n)c Fm(2)g Fn(R)2665 3681 y Fl(2)2702 3711 y Fg(,)k Fm(j)p Fn(n)p Fm(j)c Fp(=)g(1)p Fg(;)37 b(then)d(for)523 3811 y(every)d(p)l(ositive)g(numb)l(er)e Fo(\026)g Fg(the)h(fol)t(lowing)j (op)l(er)l(ators:)1610 3971 y Fo(A)1672 3983 y Fl(1)1710 3971 y Fp(\()p Fo(z)t Fp(\))22 b(=)h(\()p Fo(H)j Fm(\000)18 b Fo(z)t Fp(\))2211 3929 y Fk(\000)p Fl(1)3217 3971 y Fp(\(2.7\))523 4131 y Fg(and)1375 4231 y Fo(A)1437 4243 y Fl(2)1474 4231 y Fp(\()p Fo(z)t Fp(\))23 b(=)1692 4138 y Ff(h)1731 4231 y Fn(n)18 b Fm(\001)h Fp(\()p Fn(p)g Fm(\000)f Fn(a)p Fp(\))c(\()q Fo(H)25 b Fm(\000)18 b Fo(z)t Fp(\))2407 4189 y Fk(\000)p Fl(1)2496 4138 y Ff(i)3217 4231 y Fp(\(2.8\))523 4391 y Fg(b)l(elong)30 b(to)g Fm(B)16 b Fp(\()p Fm(H)1046 4403 y Fj(\026)1091 4391 y Fp(\))30 b Fg(and)g(their)h(norms)e(ar)l(e)h(uniform)g(in)g Fp(0)23 b Fo(<)g(d)2524 4403 y Fl(0)2584 4391 y Fm(\024)g Fo(d)g Fm(\024)g Fo(d)2869 4403 y Fl(1)2929 4391 y Fo(<)g Fm(1)p Fg(.)648 4539 y Fp(F)-7 b(rom)29 b(no)n(w)g(on,)h(a)f(single)h(w)n(ell) f(will)h(consist)f(from)h(a)f(magnetic)h(\014eld)g Fo(B)t Fp(\()p Fn(x)p Fp(\))d(=)g Fo(B)3266 4551 y Fl(0)3323 4539 y Fp(+)523 4639 y Fo(B)590 4609 y Fk(0)613 4639 y Fp(\()p Fn(x)p Fp(\))38 b(and)e(a)h(scalar)d(p)r(oten)n(tial)j Fo(V)19 b Fp(,)39 b(ob)r(eying)d(the)g(follo)n(wing)g(condition)g(\(in) h(order)f(to)523 4738 y(insure)27 b(the)h(existence)f(of)h(\(2.10\)\):) 1080 4898 y Fm(j)p Fo(D)1174 4864 y Fj(\013)1221 4898 y Fo(B)1288 4864 y Fk(0)1312 4898 y Fm(j)p Fp(\()p Fm(\001)p Fp(\))19 b(+)f Fm(j)p Fo(V)h Fm(j)p Fp(\()p Fm(\001)p Fp(\))24 b Fm(\024)e Fp(const)h Fm(h\001i)2135 4864 y Fk(\000)p Fj(\014)2232 4898 y Fo(;)37 b(\014)28 b(>)22 b Fp(2)p Fo(;)37 b Fm(j)p Fo(\013)p Fm(j)23 b(\024)g Fp(1)p Fo(:)386 b Fp(\(2.9\))1934 5147 y(4)p eop %%Page: 5 5 5 4 bop 523 614 a Fp(If)24 b Fn(a)g Fp(giv)n(es)e Fo(B)27 b Fp(=)c Fo(B)1112 626 y Fl(0)1159 614 y Fp(+)10 b Fo(B)1301 584 y Fk(0)1324 614 y Fp(,)24 b(then)g(the)f(one)g(w)n(ell)g (Hamiltonian)h(will)f(b)r(e)h Fo(H)30 b Fp(=)22 b(\()p Fn(p)10 b Fm(\000)g Fn(a)p Fp(\))3175 584 y Fl(2)3223 614 y Fp(+)g Fo(V)18 b Fp(.)648 714 y(Denote)27 b(with)h(\000)23 b(=)g Fn(Z)1342 684 y Fl(2)1407 714 y Fp(and)k(de\014ne)h(a)f(dilated,) g(rectangular)f(lattice)h(\000)2920 726 y Fj(R)2998 714 y Fp(=)22 b Fm(f)p Fo(R)q(\015)5 b Fm(j)p Fo(\015)27 b Fm(2)523 814 y Fp(\000)p Fo(;)37 b(R)24 b(>)e Fp(0)p Fm(g)p Fp(.)36 b(The)28 b(p)r(erio)r(dic)f(p)r(oten)n(tials)h(will)f(b) r(e:)813 1086 y Fo(V)861 1098 y Fl(\000)902 1106 y Fc(R)954 1086 y Fp(\()p Fm(\001)p Fp(\))d(=)1155 1007 y Ff(X)1152 1185 y Fj(\015)t Fk(2)p Fl(\000)1290 1086 y Fo(V)19 b Fp(\()p Fm(\001)g(\000)f Fo(R)q(\015)5 b Fp(\))p Fo(;)97 b(B)1841 1098 y Fl(\000)1882 1106 y Fc(R)1933 1086 y Fp(\()p Fm(\001)p Fp(\))23 b(=)g Fo(B)2194 1098 y Fl(0)2250 1086 y Fp(+)2335 1007 y Ff(X)2333 1185 y Fj(\015)t Fk(2)p Fl(\000)2471 1086 y Fo(B)2538 1052 y Fk(0)2561 1086 y Fp(\()p Fm(\001)c(\000)f Fo(R)q(\015)5 b Fp(\))p Fo(:)290 b Fp(\(2.10\))648 1348 y(If)28 b Fn(a)777 1360 y Fl(\000)818 1368 y Fc(R)898 1348 y Fp(stands)g(for)g(the)h(\(transv)n(ersal\))e (gauge)g(whic)n(h)h(giv)n(es)f Fo(B)2658 1360 y Fl(\000)2699 1368 y Fc(R)2750 1348 y Fp(,)i(then)g(our)f(Hamil-)523 1448 y(tonian)f(will)h(b)r(e)g Fo(H)1120 1460 y Fl(\000)1161 1468 y Fc(R)1235 1448 y Fp(=)23 b(\()p Fn(p)c Fm(\000)f Fn(a)1556 1460 y Fl(\000)1597 1468 y Fc(R)1648 1448 y Fp(\))1680 1418 y Fl(2)1736 1448 y Fp(+)g Fo(V)1867 1460 y Fl(\000)1908 1468 y Fc(R)1960 1448 y Fp(.)648 1547 y(Let)j Fo(c)i(<)g(d)e Fp(suc)n(h)g(that)h([)p Fo(c;)14 b(d)p Fp(])6 b Fm(\\)g Fo(\033)1632 1559 y Fj(Landau)1869 1547 y Fp(\()p Fo(B)1964 1559 y Fl(0)2002 1547 y Fp(\))23 b(=)g Fm(;)e Fp(\(see)g(\(1.4\)\))g(and)g(let)h Fo(\033)s Fp(\()p Fo(H)7 b Fp(\))f Fm(\\)g Fp(\()p Fo(c;)14 b(d)p Fp(\))25 b(=)523 1647 y Fm(f)p Fo(E)626 1659 y Fl(1)663 1647 y Fm(g)p Fp(,)h(where)e Fo(E)1052 1659 y Fl(1)1115 1647 y Fp(is)h(a)g(discrete)g(eigen)n(v)-5 b(alue)25 b(of)g Fo(H)32 b Fp(\(for)25 b(simplicit)n(y)h(w)n(e're)e(considering)g (it)523 1747 y(not)j(degenerate;)f(see)g(Remark)g(4)g(at)h(the)g(end)g (of)g(the)g(pro)r(ofs)f(for)g(a)h(short)f(discussion)g(on)523 1846 y(the)i(degenerate)e(case\).)648 1946 y(Then)31 b(the)g(existence)f(of)h(highly)g(lo)r(calised)f(essen)n(tial)g(sp)r (ectrum)h(near)f Fo(E)3010 1958 y Fl(1)3079 1946 y Fp(is)g(stated)523 2045 y(in)e(the)g(follo)n(wing)e(theorem:)523 2203 y Fn(Theorem)k(2.1)41 b Fg(Fix)f Fo(\026)i(>)g Fp(0)p Fg(.)69 b(Then)41 b(ther)l(e)f(exists)g(a)h(r)l(e)l(al)f(function)g Fo(E)2916 2215 y Fl(1)2954 2203 y Fp(\()p Fo(R)q Fp(\))g Fg(ob)l(eying)523 2303 y Fp(lim)638 2315 y Fj(R)p Fk(!1)839 2303 y Fo(E)900 2315 y Fl(1)938 2303 y Fp(\()p Fo(R)q Fp(\))23 b(=)f Fo(E)1237 2315 y Fl(1)1275 2303 y Fg(,)30 b(and)g Fo(R)q Fp(\()p Fo(\026)p Fp(\))24 b Fo(>)e Fp(0)30 b Fg(such)f(that,)i(for)f(every)h Fo(R)24 b(>)e(R)q Fp(\()p Fo(\026)p Fp(\))30 b Fg(one)g(has:)648 2403 y(i\))g Fo(\033)s Fp(\()p Fo(H)888 2415 y Fl(\000)929 2423 y Fc(R)981 2403 y Fp(\))18 b Fm(\\)h Fp([)p Fo(c;)14 b(d)p Fp(])23 b Fm(\032)g Fp([)p Fo(E)1462 2415 y Fl(1)1500 2403 y Fp(\()p Fo(R)q Fp(\))18 b Fm(\000)g Fp(2)c(exp)f(\()p Fm(\000)p Fo(\026R)q Fp(\))p Fo(;)h(E)2266 2415 y Fl(1)2304 2403 y Fp(\()p Fo(R)q Fp(\))k(+)g(2)c(exp)f(\()p Fm(\000)p Fo(\026R)q Fp(\)])648 2502 y Fg(and)648 2602 y(ii\))30 b Fp(dimRan)p Fo(P)1102 2614 y Fl(\000)1143 2622 y Fc(R)1195 2602 y Fp(\([)p Fo(c;)14 b(d)p Fp(]\))24 b(=)e Fm(1)30 b Fg(\(i.e.)40 b(the)30 b(b)l(and)g(is)g(not)f(empty\).)523 2875 y Fq(3)135 b(The)44 b(pro)t(ofs)523 3073 y Fb(3.1)112 b(An)38 b(impro)m(v)m(ed)e(resolv)m(en)m(t)g(estimate)523 3226 y Fp(Before)23 b(starting)f(the)i(pro)r(of)f(of)g(Lemma)g(2.1,)h (let)g(us)f(remark)f(the)i(follo)n(wing)e(prop)r(osition:)523 3384 y Fn(Prop)s(osition)30 b(3.1)41 b Fg(L)l(et)32 b Fo(K)38 b Fg(b)l(e)32 b(a)h(c)l(omp)l(act)g(in)g Fn(C)f Fg(and)h Fo(z)e Fm(2)e Fo(K)6 b Fg(.)46 b(L)l(et)32 b Fo(A)2840 3396 y Fl(1)2878 3384 y Fp(\()p Fo(z)t Fp(\))g Fg(and)h Fo(A)3243 3396 y Fl(2)3280 3384 y Fp(\()p Fo(z)t Fp(\))523 3484 y Fg(de\014ne)l(d)d(as)g(in)g(\(2.7\))h(and)f(\(2.8\).) 40 b(Then)30 b(ther)l(e)g(exists)g(a)g(c)l(onstant)f Fm(C)5 b Fp(\()p Fo(K)h Fp(\))29 b Fg(such)h(that:)1377 3657 y Fm(jj)p Fo(A)1485 3669 y Fl(2)1523 3657 y Fp(\()p Fo(z)t Fp(\))p Fm(jj)23 b(\024)g(C)5 b Fp(\()p Fo(K)h Fp(\))14 b(\()o(1)k(+)g Fm(jj)p Fo(A)2273 3669 y Fl(1)2311 3657 y Fp(\()p Fo(z)t Fp(\))p Fm(jj)p Fp(\))c Fo(:)648 3829 y Fg(Pr)l(o)l(of:)648 3929 y Fp(In)20 b(the)h(argumen)n(t)f(w)n(e) g(are)f(presen)n(ting)h(here,)h(app)r(ear)f(the)h(same)f(basic)f(ideas) h(as)g(in)h(the)523 4028 y(pro)r(of)30 b(of)g(Theorem)f(B.6.5.)44 b(from)30 b([16])f(\(whic)n(h)i(treats)e(atomic)h(Hamiltonians)g (without)523 4128 y(magnetic)c(\014elds)g(and)h(with)f(p)r(olynomial)g (w)n(eigh)n(ts\).)36 b(Nev)n(ertheless,)26 b(w)n(e)g(decided)g(to)g (giv)n(e)523 4228 y(a)g(rather)g(detailed)g(pro)r(of)g(in)h(order)e(to) h(ha)n(v)n(e)g(a)g(b)r(etter)h(con)n(trol)e(on)h(the)h(b)r(ounds)g (\(see)f(the)523 4327 y(reasoning)g(for)h(this)h(in)g(Remark)e(2)h(b)r (elo)n(w)h(the)g(pro)r(of)f(of)g(Lemma)h(2.1\).)648 4427 y(In)k(order)e(to)i(w)n(ork)f(with)h(b)r(ounded)h(w)n(eigh)n(ts)e(only) -7 b(,)32 b(w)n(e)g(shall)f(consider)g(that)i Fo(f)3210 4439 y Fj(m)3272 4427 y Fp(\()p Fn(x)p Fp(\))523 4527 y(is)c(a)g(sequence)g(of)g(some)g(smo)r(oth)g(and)g(b)r(ounded)h (functions)f(suc)n(h)g(that)h Fo(f)2901 4539 y Fj(m)2963 4527 y Fp(\()p Fn(x)p Fp(\))d(=)f Fm(h)p Fn(x)p Fm(i)k Fp(if)523 4626 y Fm(j)p Fn(x)p Fm(j)24 b(\024)e Fo(m=)p Fp(2)27 b(and)g(sup)1200 4646 y Fj(m)1277 4626 y Fp(max)p Fm(fjr)p Fo(f)1607 4638 y Fj(m)1669 4626 y Fm(j)p Fo(;)14 b Fm(j)p Fp(\001)p Fo(f)1862 4638 y Fj(m)1925 4626 y Fm(jg)23 b Fo(<)f Fp(2.)648 4726 y(Because)k Fo(b)1001 4738 y Fl(0)1056 4726 y Fp(+)18 b Fo(v)1179 4738 y Fl(0)1240 4726 y Fo(<)23 b Fm(1)p Fp(,)k(there)h(exists)f(a)g Fo(\025)d(<)e Fp(0)28 b(suc)n(h)f(that:)1568 4898 y Fo(\025)d(<)e Fm(\000)p Fp(1)c Fm(\000)g Fp(\(inf)j Fo(\033)s Fp(\()p Fo(H)7 b Fp(\)\))15 b Fo(;)875 b Fp(\(3.1\))1934 5147 y(5)p eop %%Page: 6 6 6 5 bop 523 614 a Fp(and)27 b(therefore)g Fo(R)q Fp(\()p Fo(\025)p Fp(\))d(=)f(\()p Fo(H)i Fm(\000)18 b Fo(\025)p Fp(\))1610 584 y Fk(\000)p Fl(1)1723 614 y Fo(>)23 b Fp(0.)648 731 y(Denote)30 b(with)h Fo(R)1191 701 y Fl(0)1228 731 y Fp(\()p Fo(\025)p Fp(\))e(=)1462 664 y Ff(\002)1496 731 y Fp(\()p Fn(p)19 b Fm(\000)f Fn(a)p Fp(\))1761 701 y Fl(2)1817 731 y Fm(\000)g Fo(\025)1948 664 y Ff(\003)1983 681 y Fk(\000)p Fl(1)2072 731 y Fp(;)33 b(then)e(for)f Fm(\000)p Fo(\025)g Fp(large)f(enough,)i(one)f(has)523 831 y(the)e(follo)n(wing)f(relations:)1152 1050 y Fm(jj)1212 958 y Ff(\020)1262 1050 y Fp(1)18 b(+)g(\()p Fo(R)1501 1016 y Fl(0)1538 1050 y Fp(\()p Fo(\025)p Fp(\)\))1692 994 y Fd(1)p 1693 1003 29 4 v 1693 1036 a(2)1736 1050 y Fo(V)h Fp(\()p Fo(R)1899 1016 y Fl(0)1936 1050 y Fp(\()p Fo(\025)p Fp(\)\))2090 994 y Fd(1)p 2091 1003 V 2091 1036 a(2)2133 958 y Ff(\021)2183 975 y Fk(\000)p Fl(1)2286 1050 y Fm(jj)2332 1065 y Fk(B)q Fl(\()p Fk(H)2460 1073 y Fd(0)2493 1065 y Fl(\))2546 1050 y Fo(<)j Fp(1)p Fo(=)p Fp(2)458 b(\(3.2\))523 1253 y(and)990 1374 y Fo(R)q Fp(\()p Fo(\025)p Fp(\))24 b(=)e Fo(R)1341 1340 y Fl(0)1378 1374 y Fp(\()p Fo(\025)p Fp(\))1500 1317 y Fd(1)p 1501 1326 V 1501 1360 a(2)1557 1282 y Ff(\020)1607 1374 y Fp(1)c(+)g(\()p Fo(R)1846 1340 y Fl(0)1883 1374 y Fp(\()p Fo(\025)p Fp(\)\))2037 1317 y Fd(1)p 2038 1326 V 2038 1360 a(2)2081 1374 y Fo(V)h Fp(\()p Fo(R)2244 1340 y Fl(0)2281 1374 y Fp(\()p Fo(\025)p Fp(\)\))2435 1317 y Fd(1)p 2436 1326 V 2436 1360 a(2)2479 1282 y Ff(\021)2528 1299 y Fk(\000)p Fl(1)2631 1374 y Fo(R)2695 1340 y Fl(0)2732 1374 y Fp(\()p Fo(\025)p Fp(\))2854 1317 y Fd(1)p 2855 1326 V 2855 1360 a(2)2897 1374 y Fo(:)297 b Fp(\(3.3\))648 1544 y(Under)19 b(the)h(same)e(condition)i(on)f Fm(\000)p Fo(\025)p Fp(,)i(it)f(is)f(kno)n(wn)f(from)h(the)h(usual)f (Com)n(b)r(es-Thomas)523 1643 y(theory)27 b(that)h Fo(R)q Fp(\()p Fo(\025)p Fp(\))23 b Fm(2)h(B)s Fp(\()p Fm(H)1402 1655 y Fj(\026)1446 1643 y Fp(\):)1242 1826 y Fm(jj)14 b Fp(exp\()p Fo(\026)p Fm(h\001i)p Fp(\))p Fo(R)q Fp(\()p Fo(\025)p Fp(\))g(exp)q(\()p Fm(\000)p Fo(\026)p Fm(h\001i)p Fp(\))p Fm(jj)2260 1841 y Fk(B)q Fl(\()p Fk(H)2388 1849 y Fd(0)2422 1841 y Fl(\))2475 1826 y Fo(<)22 b Fm(1)p Fo(:)549 b Fp(\(3.4\))648 2009 y(Then)21 b Fo(A)920 2021 y Fl(2)957 2009 y Fp(\()p Fo(z)t Fp(\))g(can)f(b)r(e)h(rewritten)g (\(as)f(an)h(op)r(erator)e(in)i Fm(B)s Fp(\()p Fm(H)2504 2021 y Fl(0)2540 2009 y Fp(\)\))h(in)f(the)g(follo)n(wing)f(form:)883 2191 y(exp\()p Fo(\026)p Fm(h\001i)p Fp(\))14 b([)q Fn(n)k Fm(\001)h Fp(\()p Fn(p)g Fm(\000)f Fn(a)p Fp(\)\()p Fo(R)q Fp(\()p Fo(\025)p Fp(\))i(+)e(\()p Fo(z)k Fm(\000)c Fo(\025)p Fp(\))p Fo(R)q Fp(\()p Fo(\025)p Fp(\))p Fo(R)q Fp(\()p Fo(z)t Fp(\)\)])d(exp\()p Fm(\000)p Fo(\026)p Fm(h\001i)p Fp(\))p Fo(;)190 b Fp(\(3.5\))523 2374 y(so)27 b(it)h(w)n(ould)f(b)r(e) h(su\016cien)n(t)g(pro)n(ving)e(that)i Fo(A)1954 2386 y Fl(2)1991 2374 y Fp(\()p Fo(\025)p Fp(\))h(is)e(b)r(ounded.)648 2474 y(Let)32 b Fo(\036;)14 b( )35 b Fm(2)c Fo(C)1127 2444 y Fk(1)1121 2494 y Fl(0)1212 2406 y Ff(\000)1250 2474 y Fn(R)1322 2444 y Fl(2)1359 2406 y Ff(\001)1429 2474 y Fp(with)i(their)g(supp)r(ort)f(included)h(in)f Fm(j)p Fn(x)p Fm(j)g Fo(<)f(M)40 b(<)30 b Fm(1)p Fp(.)52 b(If)33 b(one)523 2573 y(manages)26 b(to)i(pro)n(v)n(e)e(that:)1416 2756 y Fm(jh)p Fp(exp\()p Fo(\026)p Fm(h\001i)p Fp(\))p Fo(\036;)14 b Fp([)r Fn(n)k Fm(\001)h Fp(\()p Fn(p)g Fm(\000)f Fn(a)p Fp(\)])c Fo(R)q Fp(\()p Fo(\025)p Fp(\))p Fo( )s Fm(ij)24 b(\024)1268 2880 y(\024)83 b Fp(const)23 b(\(1)18 b(+)g Fm(jj)p Fo(A)1912 2892 y Fl(1)1949 2880 y Fp(\()p Fo(\025)p Fp(\))p Fm(jj)p Fp(\))25 b Fm(jj)p Fo(\036)p Fm(jj)2305 2892 y Fk(H)2362 2900 y Fd(0)2399 2880 y Fm(jj)p Fo( )s Fm(jj)2548 2892 y Fk(H)2605 2900 y Fc(\026)2649 2880 y Fo(;)545 b Fp(\(3.6\))523 3063 y(then)28 b(via)f(a)g(densit)n(y)h(argumen)n(t)e(the)i(pro)r(of)f(w)n (ould)g(b)r(e)h(\014nished.)648 3163 y(Let)20 b(us)g(put)h(the)g (scalar)e(pro)r(duct)h(from)g(\(3.6\))g(in)g(a)g(more)g(con)n(v)n (enien)n(t)f(form.)34 b(Pro)n(vided)523 3262 y Fo(M)e(<)22 b(m=)p Fp(2,)27 b(one)g(has:)1340 3445 y Fm(h)p Fp(exp)q(\()p Fo(\026)p Fm(h\001i)p Fp(\))p Fo(\036;)14 b Fp([)q Fn(n)19 b Fm(\001)f Fp(\()p Fn(p)h Fm(\000)f Fn(a)p Fp(\)])d Fo(R)q Fp(\()p Fo(\025)p Fp(\))p Fo( )s Fm(i)2492 3457 y Fk(H)2549 3465 y Fd(0)2609 3445 y Fp(=)1193 3570 y(=)82 b Fm(h)p Fp([)q Fn(n)18 b Fm(\001)h Fp(\()p Fn(p)g Fm(\000)f Fn(a)p Fp(\)])c(exp\()p Fo(\026f)2061 3582 y Fj(m)2124 3570 y Fp(\()p Fm(\001)p Fp(\)\))p Fo(\036;)g(R)q Fp(\()p Fo(\025)p Fp(\))p Fo( )s Fm(i)2594 3582 y Fk(H)2651 3590 y Fd(0)2713 3570 y Fp(=)1193 3694 y(=)82 b Fm(h)p Fp([)p Fn(n)19 b Fm(\001)g Fp(\()p Fn(p)g Fm(\000)f Fn(a)p Fp(\)])p Fo(\036;)c Fp(exp)q(\()p Fo(\026f)2134 3706 y Fj(m)2197 3694 y Fp(\()p Fm(\001)p Fp(\)\))p Fo(R)q Fp(\()p Fo(\025)p Fp(\))p Fo( )s Fm(i)20 b Fp(+)1193 3819 y(+)82 b Fo({)23 b(\026)g Fm(h)p Fp([)p Fn(n)c Fm(\001)g Fp(\()p Fm(r)p Fo(f)1774 3831 y Fj(m)1837 3819 y Fp(\()p Fm(\001)p Fp(\)\)])p Fo(\036;)14 b Fp(exp)q(\()p Fo(\026f)2316 3831 y Fj(m)2379 3819 y Fp(\()p Fm(\001)p Fp(\)\))p Fo(R)q Fp(\()p Fo(\025)p Fp(\))p Fo( )s Fm(i)p Fo(:)431 b Fp(\(3.7\))648 4001 y(The)33 b(second)g(term)h(of)f(the)h(last)g(equalit)n(y)f(is)g(easy)g (to)g(con)n(trol;)j(w)n(e)d(will)h(treat)f(here)523 4101 y(only)27 b(the)h(\014rst)g(one.)648 4201 y(Comm)n(uting)f(exp\()p Fo(\026f)1352 4213 y Fj(m)1415 4201 y Fp(\()p Fm(\001)p Fp(\)\))h(with)g Fo(R)q Fp(\()p Fo(\025)p Fp(\),)h(one)e(obtains:)1095 4383 y(exp)o(\()p Fo(\026f)1344 4395 y Fj(m)1408 4383 y Fp(\()p Fm(\001)p Fp(\)\))p Fo(R)q Fp(\()p Fo(\025)p Fp(\))p Fo( )g Fp(=)947 4508 y(=)83 b Fo(R)q Fp(\()p Fo(\025)p Fp(\))14 b(exp\()p Fo(\026)p Fm(h\001i)p Fp(\))p Fo( )23 b Fp(+)18 b Fo(R)q Fp(\()p Fo(\025)p Fp(\)[)p Fo(H)r(;)c Fp(exp\()p Fo(\026f)2330 4520 y Fj(m)2393 4508 y Fp(\()p Fm(\001)p Fp(\)\)])p Fo(R)q Fp(\()p Fo(\025)p Fp(\))p Fo( )28 b Fp(=)947 4632 y(=)83 b Fo(R)q Fp(\()p Fo(\025)p Fp(\))14 b(exp\()p Fo(\026)p Fm(h\001i)p Fp(\))p Fo( )23 b Fp(+)18 b Fo(R)q Fp(\()p Fo(\025)p Fp(\)\()p Fo(\026)2031 4598 y Fl(2)2069 4632 y Fm(j)p Fp(\()p Fm(r)p Fo(f)2234 4644 y Fj(m)2297 4632 y Fp(\()p Fm(\001)p Fp(\)\))p Fm(j)2439 4598 y Fl(2)2496 4632 y Fp(+)g Fo(\026)23 b Fp(\001)p Fo(f)2762 4644 y Fj(m)2825 4632 y Fp(\()p Fm(\001)p Fp(\)\))c Fm(\002)947 4757 y(\002)83 b Fp(exp)o(\()p Fo(\026f)1344 4769 y Fj(m)1408 4757 y Fp(\()p Fm(\001)p Fp(\)\))p Fo(R)q Fp(\()p Fo(\025)p Fp(\))p Fo( )23 b Fm(\000)18 b Fp(2)k Fo({)h(\026)g(R)q Fp(\()p Fo(\025)p Fp(\)[\()p Fn(p)d Fm(\000)e Fn(a)p Fp(\))h Fm(\001)f Fp(\()p Fm(r)p Fo(f)2717 4769 y Fj(m)2781 4757 y Fp(\()p Fm(\001)p Fp(\)\)])h Fm(\002)947 4881 y(\002)83 b Fp(exp)o(\()p Fo(\026f)1344 4893 y Fj(m)1408 4881 y Fp(\()p Fm(\001)p Fp(\)\))p Fo(R)q Fp(\()p Fo(\025)p Fp(\))p Fo( )s(:)1434 b Fp(\(3.8\))1934 5147 y(6)p eop %%Page: 7 7 7 6 bop 648 614 a Fp(With)28 b(the)g(help)g(of)f(the)h(follo)n(wing)f (t)n(w)n(o)g(inequalities:)1351 797 y Fm(jj)p Fn(n)18 b Fm(\001)h Fp(\()p Fn(p)g Fm(\000)f Fn(a)p Fp(\)\()p Fo(R)1871 763 y Fl(0)1909 797 y Fp(\()p Fo(\025)p Fp(\)\))2063 740 y Fd(1)p 2064 749 29 4 v 2064 783 a(2)2106 797 y Fm(jj)2152 812 y Fk(B)q Fl(\()p Fk(H)2280 820 y Fd(0)2313 812 y Fl(\))2366 797 y Fo(<)23 b Fm(1)p Fo(;)657 b Fp(\(3.9\))523 980 y(and)1189 1079 y(sup)1222 1145 y Fj(m)1328 1079 y Fm(jj)p Fp(\()p Fo(R)1470 1045 y Fl(0)1507 1079 y Fp(\()p Fo(\025)p Fp(\)\))1661 1023 y Fd(1)p 1663 1032 V 1663 1065 a(2)1705 1079 y Fp(\()p Fn(p)19 b Fm(\000)f Fn(a)p Fp(\))h Fm(\001)f Fp(\()p Fm(r)p Fo(f)2172 1091 y Fj(m)2236 1079 y Fp(\))p Fm(jj)2314 1094 y Fk(B)q Fl(\()p Fk(H)2442 1102 y Fd(0)2474 1094 y Fl(\))2528 1079 y Fo(<)k Fm(1)p Fo(;)454 b Fp(\(3.10\))523 1274 y(after)27 b(using)h(\(3.3\))f(and)g (taking)g(the)h(suprem)n(um)g(o)n(v)n(er)d Fo(m)j Fp(in)g(\(3.8\),)f (one)g(obtains)g(\(3.6\))p Fo(:)3333 1286 y Fa(2)648 1374 y Fp(In)35 b(the)h(pro)r(of)f(of)g(Lemma)g(2.1,)i(a)e(cen)n(tral)f (role)h(will)g(b)r(e)h(pla)n(y)n(ed)f(b)n(y)g(the)g(follo)n(wing)523 1473 y(lemma)21 b(\(pro)n(v)n(ed)g(in)g([3])g(in)h(a)f(sligh)n(tly)g (di\013eren)n(t)h(form)f(as)f(Corollary)f(3.1\),)k(whic)n(h)e(con)n (trols)523 1573 y(the)28 b(\\asymptotic)f(Hamiltonians":)523 1739 y Fn(Lemma)j(3.1)40 b Fg(L)l(et)28 b Fo(z)j Fg(b)l(e)e(a)f(c)l (omplex)h(numb)l(er,)f(with)h Fp(dist\()p Fo(z)t(;)14 b(\033)2513 1751 y Fj(L)2563 1739 y Fp(\()p Fo(B)2658 1751 y Fl(0)2696 1739 y Fp(\)\))23 b(=)g Fo(d)g(>)g Fp(0)p Fg(.)37 b(De\014ne:)1216 1921 y Fo(B)1279 1933 y Fj(n)1324 1921 y Fp(\()p Fn(x)p Fp(\))24 b(=)e Fo(B)1612 1933 y Fl(0)1668 1921 y Fp(+)c Fo(B)1818 1887 y Fk(0)1814 1942 y Fj(n)1859 1921 y Fp(\()p Fn(x)p Fp(\))p Fo(;)100 b(V)2144 1933 y Fj(n)2213 1921 y Fp(=)22 b Fo(V)2348 1933 y Fl(1)p Fj(;n)2465 1921 y Fp(+)c Fo(V)2596 1933 y Fl(2)p Fj(;n)523 2104 y Fg(and)817 2287 y Fo(b)853 2299 y Fj(n)921 2287 y Fp(=)23 b(max)13 b(\()24 b(sup)1209 2360 y Fe(x)p Fk(2)p Fe(R)1350 2344 y Fd(2)1396 2287 y Fm(j)p Fo(D)1490 2253 y Fj(\013)1537 2287 y Fo(B)1604 2253 y Fk(0)1600 2307 y Fj(n)1645 2287 y Fp(\()p Fn(x)p Fp(\))p Fm(j)p Fo(;)38 b Fm(j)p Fo(\013)p Fm(j)24 b(\024)f Fp(1\))o(;)37 b Fo(v)2227 2299 y Fj(n)2296 2287 y Fp(=)23 b Fm(jj)p Fo(V)2478 2299 y Fl(1)p Fj(;n)2576 2287 y Fm(jj)2622 2299 y Fl(2)2678 2287 y Fp(+)18 b Fm(jj)p Fo(V)2855 2299 y Fl(2)p Fj(;n)2954 2287 y Fm(jj)3000 2299 y Fk(1)3070 2287 y Fo(;)523 2522 y Fg(with)1644 2621 y Fp(lim)1615 2671 y Fj(n)p Fk(!1)1788 2621 y Fp(\()p Fo(b)1856 2633 y Fj(n)1920 2621 y Fp(+)g Fo(v)2043 2633 y Fj(n)2088 2621 y Fp(\))23 b(=)g(0)p Fo(:)523 2795 y Fg(De\014ne:)1560 2895 y Fo(H)1629 2907 y Fj(n)1697 2895 y Fp(=)g(\()p Fn(p)18 b Fm(\000)g Fn(a)2017 2907 y Fj(n)2063 2895 y Fp(\))2095 2861 y Fl(2)2151 2895 y Fp(+)g Fo(V)2282 2907 y Fj(n)2328 2895 y Fo(:)523 3044 y Fg(Fix)32 b Fo(\026)c(>)f Fp(0)p Fg(.)45 b(Then)33 b(ther)l(e)f(exists)g Fo(N)9 b Fp(\()p Fo(\026;)14 b(d)p Fp(\))28 b Fo(>)f Fp(0)k Fg(such)h(that,)i(for)f(any)f Fo(n)27 b Fm(\025)g Fo(N)9 b Fp(\()p Fo(\026;)14 b(d)p Fp(\))p Fg(,)34 b(one)523 3144 y(has)d Fo(z)k(=)-51 b Fm(2)23 b Fo(\033)s Fp(\()p Fo(H)966 3156 y Fj(n)1012 3144 y Fp(\))30 b Fg(and)766 3335 y Fm(jj)14 b Fp(exp\()p Fo(\026)24 b(<)e Fm(\001)h Fo(>)p Fp(\))14 b(\()p Fo(H)1404 3347 y Fj(n)1468 3335 y Fm(\000)k Fo(z)t Fp(\))1626 3294 y Fk(\000)p Fl(1)1728 3335 y Fp(exp\()p Fm(\000)p Fo(\026)23 b(<)g Fm(\001)g Fo(>)p Fp(\))p Fm(jj)2302 3350 y Fk(B)q Fl(\()p Fk(H)2430 3358 y Fd(0)2462 3350 y Fl(\))2516 3335 y Fm(\024)f Fp(C\()p Fo(\026;)14 b(d)p Fp(\))p Fo(:)2880 3347 y Fa(2)3175 3335 y Fp(\(3.11\))648 3518 y Fg(Pr)l(o)l(of)30 b(of)h(L)l(emma)f(2.1:)648 3618 y Fp(The)d(main)g(idea)g(of)g(the)h (pro)r(of)f(consists)f(in)i(constructing)e(an)h(\\almost)g(in)n(v)n (erse")e(op-)523 3717 y(erator)c(for)h Fo(H)16 b Fm(\000)9 b Fo(z)t Fp(,)22 b(using)g(the)i(fact)e(that)h(after)g(\\gauging")d(a)n (w)n(a)n(y)h(some)h(magnetic)g(phases)523 3817 y(and)30 b(going)f(su\016cien)n(tly)h(far)f(from)h(the)g(origin,)g(the)g (particle)g(\\feels")f(only)g(smaller)h(and)523 3916 y(smaller)d(magnetic)f(and)i(scalar)d(p)r(erturbations.)37 b(In)27 b(that)h(region,)e(its)i(dynamics)f(should)523 4016 y(b)r(e)21 b(v)n(ery)d(close)i(to)g(that)g(of)g(a)g(particle)f (whic)n(h)h(is)g(placed)g(in)g(a)g Fg(glob)l(al)t(ly)i Fp(small)e(p)r(erturbation,)523 4116 y(for)28 b(whic)n(h)h(w)n(e)f(ha)n (v)n(e)g(the)h(informations)e(giv)n(en)h(in)h(Lemma)g(3.1.)39 b(The)29 b(\\almost)e(in)n(v)n(erse")523 4215 y(op)r(erator)i(will)i(b) r(e)g(constructed)f(via)g(a)h(\\cut)f(and)h(paste")f(tec)n(hnique)g (whic)n(h)h(tak)n(es)f(in)n(to)523 4315 y(accoun)n(t)e(that)h(for)f (long)g(range)f(magnetic)h(\014elds)h(the)g(gauge)e(transformations)g (w)n(e)h(need)523 4415 y(can)j(b)r(e)i(p)r(erformed)e(only)g(lo)r (cally)-7 b(.)49 b(Firstly)-7 b(,)33 b(w)n(e)e(need)h(a)f(quadratic)g (partition)g(of)h(unit)n(y)523 4514 y(constituted)23 b(from)e(\014v)n(e)h(cut-o\013)g(functions)h(with)f(con)n(v)n(ex)f (supp)r(orts)h(and)g(of)g(the)g(follo)n(wing)523 4614 y(t)n(yp)r(e:)1449 4713 y Fo(g)k Fm(2)d Fo(C)1658 4679 y Fk(1)1729 4713 y Fp(\()p Fn(R)1833 4679 y Fl(2)1870 4713 y Fp(\))c Fo(;)32 b Fp(0)23 b Fm(\024)f Fo(g)s Fp(\()p Fn(x)p Fp(\))i Fm(\024)f Fp(1)p Fo(:)713 b Fp(\(3.12\))1934 5147 y(7)p eop %%Page: 8 8 8 7 bop 523 614 a Fp(The)39 b(origin)e(\(where)i(the)g(p)r (erturbations)f(are)f(cen)n(tred\))i(is)g(co)n(v)n(ered)d(b)n(y)j(a)f (compactly)523 714 y(supp)r(orted)27 b(function:)1161 942 y Fo(g)1201 954 y Fl(0)1238 942 y Fp(\()p Fn(x)p Fp(\))d(=)1463 825 y Ff(\032)1567 892 y Fp(1)82 b(if)i Fm(j)p Fo(x)1893 904 y Fj(i)1921 892 y Fm(j)23 b(\024)2065 859 y Fl(1)p 2065 873 34 4 v 2065 920 a(2)2108 892 y Fo(;)37 b(i)22 b Fm(2)i(f)p Fp(1)p Fo(;)14 b Fp(2)p Fm(g)1567 992 y Fp(0)82 b(if)i Fm(j)p Fo(x)1893 1004 y Fl(1)1931 992 y Fm(j)23 b(\025)2074 959 y Fl(3)p 2074 973 V 2074 1020 a(2)2141 992 y Fp(or)f Fm(j)p Fo(x)2308 1004 y Fl(2)2345 992 y Fm(j)h(\025)2489 959 y Fl(3)p 2489 973 V 2489 1020 a(2)3175 942 y Fp(\(3.13\))523 1170 y(The)34 b(other)g(four)g(are)f (not)i(compactly)e(supp)r(orted)i(and)f(are)f(co)n(v)n(ering)f(the)j (rest)f(of)g(the)523 1270 y(plane.)i(The)27 b(righ)n(t)f(hand)g(side)h (is)f(co)n(v)n(ered)f(b)n(y)h Fo(g)2051 1282 y Fl(1)2115 1270 y Fp(whic)n(h)g(is)h(equal)f(to)g(one)g(if)h Fo(x)3025 1282 y Fl(1)3086 1270 y Fm(\025)3184 1237 y Fl(3)p 3184 1251 V 3184 1299 a(2)3254 1270 y Fp(and)523 1370 y Fm(j)p Fo(x)593 1382 y Fl(2)631 1370 y Fm(j)33 b(\024)794 1337 y Fl(1)p 794 1351 V 794 1398 a(2)837 1370 y Fp(,)i(and)e(equal)g(to)g (zero)f(if)i Fo(x)1704 1382 y Fl(1)1774 1370 y Fm(\024)1882 1337 y Fl(1)p 1882 1351 V 1882 1398 a(2)1958 1370 y Fp(or)f Fm(j)p Fo(x)2136 1382 y Fl(2)2173 1370 y Fm(j)g(\025)2336 1337 y Fl(3)p 2336 1351 V 2336 1398 a(2)2379 1370 y Fo(:)h Fp(The)f(upp)r(er)h(side)f(is)g(co)n(v)n(ered)523 1469 y(b)n(y)h Fo(g)685 1481 y Fl(2)757 1469 y Fp(whic)n(h)g(is)g(equal)g (to)g(one)g(if)h Fo(x)1714 1481 y Fl(2)1786 1469 y Fm(\025)1895 1437 y Fl(3)p 1895 1451 V 1895 1498 a(2)1973 1469 y Fp(and)f(equal)g (to)g(zero)g(if)h Fo(x)2788 1481 y Fl(2)2860 1469 y Fm(\024)2969 1437 y Fl(1)p 2969 1451 V 2969 1498 a(2)3012 1469 y Fp(.)57 b(The)35 b(left)523 1569 y(and)29 b(do)n(wn)f(sides)g(are)g(co)n(v)n (ered)e(b)n(y)j Fo(g)1701 1581 y Fl(3)1738 1569 y Fp(\()p Fn(x)p Fp(\))c(:=)g Fo(g)2030 1581 y Fl(1)2067 1569 y Fp(\()p Fm(\000)p Fn(x)p Fp(\))k(and)f Fo(g)2477 1581 y Fl(4)2514 1569 y Fp(\()p Fn(x)p Fp(\))e(:=)e Fo(g)2806 1581 y Fl(2)2843 1569 y Fp(\()p Fm(\000)p Fn(x)p Fp(\).)41 b(The)29 b(last)523 1669 y(condition)e(imp)r(osed)h(on)f(these)h (functions)g(is:)1747 1822 y Fl(4)1703 1847 y Ff(X)1706 2024 y Fj(j)s Fl(=0)1837 1926 y Fo(g)1880 1892 y Fl(2)1877 1947 y Fj(j)1917 1926 y Fp(\()p Fn(x)p Fp(\))c(=)f(1)p Fo(:)967 b Fp(\(3.14\))648 2159 y(In)23 b(order)f(to)i(de\014ne)f(the)h (asymptotic)f(p)r(oten)n(tials,)h(let)g(us)f(c)n(ho)r(ose)g(another)f (four)h(func-)523 2259 y(tions)37 b(~)-44 b Fo(g)774 2271 y Fj(j)808 2259 y Fp(,)37 b Fo(j)j Fm(2)35 b(f)p Fp(1)p Fo(;)14 b(:)g(:)g(:)f(;)h Fp(4)p Fm(g)34 b Fp(suc)n(h)g(that)h (the)g(origin)f(is)g(a)n(w)n(a)n(y)f(from)h(their)h(supp)r(orts)g(and) 526 2359 y(~)-45 b Fo(g)563 2371 y Fj(j)598 2359 y Fo(g)638 2371 y Fj(j)695 2359 y Fp(=)23 b Fo(g)823 2371 y Fj(j)857 2359 y Fp(.)648 2458 y(Let)k(us)h(no)n(w)f(\\dilate")f(the)i (partition:)951 2661 y Fo(g)991 2673 y Fj(j;n)1082 2661 y Fp(\()p Fn(x)p Fp(\))c(=)f Fo(g)1348 2673 y Fj(j)1396 2569 y Ff(\020)1456 2605 y Fn(x)p 1456 2642 51 4 v 1456 2718 a Fo(n)1516 2569 y Ff(\021)1580 2661 y Fp(;)99 b(~)-44 b Fo(g)1740 2673 y Fj(j;n)1831 2661 y Fp(\()p Fn(x)p Fp(\))24 b(=)i(~)-45 b Fo(g)2097 2673 y Fj(j)2145 2569 y Ff(\020)2205 2605 y Fn(x)p 2205 2642 V 2205 2718 a Fo(n)2265 2569 y Ff(\021)2329 2661 y Fp(;)97 b Fo(n)23 b Fm(2)g(f)p Fp(1)p Fo(;)14 b Fp(2)p Fo(;)g(:)g(:)g(:)n Fm(g)p Fo(:)215 b Fp(\(3.15\))523 2867 y(F)-7 b(rom)28 b(these)g(de\014nitions)g(one)f(can)h(see)g(that)g(all)g(the)g(deriv)-5 b(ativ)n(es)27 b(of)h Fo(g)2785 2879 y Fj(j;n)2905 2867 y Fp(con)n(v)n(erge)d(uni-)523 2967 y(formly)i(to)h(zero)e(when)i Fo(n)f Fp(go)r(es)g(to)h(in\014nit)n(y)-7 b(.)648 3066 y(Let)27 b(us)h(de\014ne)g(the)g(asymptotic)f(p)r(oten)n(tials:)1134 3249 y Fo(B)1197 3261 y Fj(j;n)1372 3249 y Fp(=)83 b Fo(B)1583 3261 y Fl(0)1638 3249 y Fp(+)21 b(~)-45 b Fo(g)1761 3261 y Fj(j;n)1853 3249 y Fo(B)1920 3215 y Fk(0)1944 3249 y Fo(;)97 b(V)2112 3261 y Fj(j;n)2227 3249 y Fp(=)26 b(~)-45 b Fo(g)2355 3261 y Fj(j;n)2447 3249 y Fo(V)5 b(;)1036 3436 y Fn(a)1082 3448 y Fj(j;n)1174 3436 y Fp(\()p Fn(x)p Fp(\))84 b(=)1520 3323 y Ff(Z)1603 3343 y Fl(1)1566 3512 y(0)1654 3436 y Fo(ds)23 b(s)g Fn(B)1889 3448 y Fj(j;n)1981 3436 y Fp(\()p Fo(s)p Fn(x)p Fp(\))c Fm(^)g Fn(x)p Fo(;)98 b(j)28 b Fm(2)23 b(f)p Fp(1)p Fo(;)14 b(:)g(:)g(:)f Fp(4)p Fm(g)p Fo(:)299 b Fp(\(3.16\))523 3661 y(T)-7 b(ak)n(e)22 b(no)n(w)g(four)h(p)r(oin)n(ts)g(situated)g(in) g(the)g(supp)r(ort)g(of)g(the)g(functions)g(whic)n(h)g(are)f(far)h (from)523 3760 y(the)32 b(origin.)47 b(They)31 b(will)h(b)r(e)g(useful) f(in)h(order)e(to)h(write)h(do)n(wn)f(the)g(needed)h(lo)r(cal)f(gauge) 523 3860 y(transformations:)1373 3892 y Ff(\010)1421 3959 y Fn(x)1471 3925 y Fj(j)1507 3892 y Ff(\011)1555 3993 y Fj(j)s Fk(2f)p Fl(1)p Fj(;:::;)p Fl(4)p Fk(g)1882 3959 y Fo(;)97 b Fn(x)2052 3925 y Fj(j)2110 3959 y Fm(2)23 b Fp(supp)h Fo(g)2423 3971 y Fj(j;n)2514 3959 y Fo(:)523 4128 y Fp(F)-7 b(rom)27 b(\(2.4\))h(and)f(\(2.3\),)g(one)h(can)f(see)g (that:)1001 4310 y Fn(a)p Fp(\()p Fn(x)p Fp(\))85 b(=)d Fn(a)p Fp(\()p Fn(x)p Fo(;)14 b Fn(x)1608 4276 y Fj(j)1645 4310 y Fp(\))k(+)g Fm(r)1847 4322 y Fe(x)1891 4310 y Fo(')p Fp(\()p Fn(x)p Fo(;)c Fn(x)2114 4276 y Fj(j)2151 4310 y Fp(\))83 b(and)909 4435 y Fn(a)955 4447 y Fj(j;n)1048 4435 y Fp(\()p Fn(x)p Fp(\))h(=)e Fn(a)1439 4447 y Fj(j;n)1532 4435 y Fp(\()p Fn(x)p Fo(;)14 b Fn(x)1701 4400 y Fj(j)1737 4435 y Fp(\))19 b(+)f Fm(r)1940 4447 y Fe(x)1984 4435 y Fo(')2038 4447 y Fj(j;n)2130 4435 y Fp(\()p Fn(x)p Fo(;)c Fn(x)2299 4400 y Fj(j)2335 4435 y Fp(\))p Fo(;)97 b(j)28 b Fm(2)c(f)p Fp(1)p Fo(;)14 b(:)g(:)g(:)e(;)i Fp(4)p Fm(g)p Fo(;)173 b Fp(\(3.17\))523 4617 y(and,)28 b(for)f Fn(x)c Fm(2)g Fp(supp)h Fo(g)1221 4629 y Fj(j;n)1312 4617 y Fo(;)k Fp(one)f(can)h(easily)e(sho)n(w)h(that:)1559 4800 y Fn(a)p Fp(\()p Fn(x)p Fo(;)14 b Fn(x)1774 4766 y Fj(j)1810 4800 y Fp(\))23 b(=)g Fn(a)1999 4812 y Fj(j;n)2091 4800 y Fp(\()p Fn(x)p Fo(;)14 b Fn(x)2260 4766 y Fj(j)2296 4800 y Fp(\))p Fo(:)1934 5147 y Fp(8)p eop %%Page: 9 9 9 8 bop 523 614 a Fp(With)29 b(the)e(help)h(of)g(the)g(follo)n(wing)e (de\014nition:)1288 795 y Fo(\016)s(')1382 807 y Fj(j;n)1475 795 y Fp(\()p Fn(x)p Fo(;)14 b Fn(x)1644 760 y Fj(j)1680 795 y Fp(\))23 b(=)f Fo(')p Fp(\()p Fn(x)p Fo(;)14 b Fn(x)2045 760 y Fj(j)2082 795 y Fp(\))k Fm(\000)h Fo(')2270 807 y Fj(j;n)2362 795 y Fp(\()p Fn(x)p Fo(;)14 b Fn(x)2531 760 y Fj(j)2567 795 y Fp(\))p Fo(;)523 975 y Fp(one)26 b(can)h(see)f(\(at)h(least)f(at)h(a)f(formal)g(lev)n(el\))h(ho)n(w)f (the)h(lo)r(cal)f(gauge)g(transformations)e(are)523 1074 y(acting:)694 1255 y(\()p Fn(p)19 b Fm(\000)f Fn(a)p Fp(\))c(exp\()p Fo({)24 b(\016)s(')1277 1267 y Fj(j;n)1369 1255 y Fp(\()p Fm(\001)p Fo(;)14 b Fn(x)1511 1220 y Fj(j)1547 1255 y Fp(\)\))p Fo(g)1651 1267 y Fj(j;n)1766 1255 y Fp(=)23 b(exp\()p Fo({)g(\016)s(')2157 1267 y Fj(j;n)2249 1255 y Fp(\()p Fm(\001)p Fo(;)14 b Fn(x)2391 1220 y Fj(j)2427 1255 y Fp(\)\)\()p Fn(p)19 b Fm(\000)f Fn(a)2724 1267 y Fj(j;n)2817 1255 y Fp(\))p Fo(g)2889 1267 y Fj(j;n)2981 1255 y Fo(:)171 b Fp(\(3.18\))523 1435 y(A)n(t)31 b(this)h(p)r(oin)n(t) f(w)n(e)g(are)f(able)h(to)g(de\014ne)g(the)h(\\asymptotic)e (Hamiltonians")g(\(they)i(ha)n(v)n(e)523 1534 y Fg(glob)l(al)t(ly)g Fp(small)e(p)r(erturbations,)g(uniformly)g(in)h Fo(n)p Fp(;)g(see)f(the)g(conditions)g(imp)r(osed)g(on)g Fo(B)3364 1504 y Fk(0)523 1634 y Fp(and)d Fo(V)19 b Fp(\):)1184 1734 y Fo(H)1253 1746 y Fj(j;n)1368 1734 y Fp(=)k(\()p Fn(p)18 b Fm(\000)g Fn(a)1688 1746 y Fj(j;n)1781 1734 y Fp(\))1813 1699 y Fl(2)1869 1734 y Fp(+)g Fo(V)2000 1746 y Fj(j;n)2093 1734 y Fo(;)97 b(j)28 b Fm(2)23 b(f)p Fp(1)p Fo(;)14 b(:)g(:)g(:)f(;)h Fp(4)p Fm(g)p Fo(:)648 1882 y Fp(T)-7 b(ak)n(e)25 b Fo(z)30 b Fp(as)c(stated.)36 b(F)-7 b(or)26 b Fo(n)1496 1894 y Fl(0)1559 1882 y Fp(large)f(enough)h (and)g Fo(n)d Fm(\025)g Fo(n)2417 1894 y Fl(0)2454 1882 y Fp(,)k(Lemma)f(3.1)f(implies)i(that:)1561 2100 y(dist\()p Fo(z)t(;)14 b(\033)s Fp(\()p Fo(H)1958 2112 y Fj(j;n)2051 2100 y Fp(\)\))23 b Fm(\025)2236 2044 y Fo(d)2279 2056 y Fl(0)p 2236 2081 81 4 v 2255 2157 a Fp(2)2326 2100 y Fo(:)826 b Fp(\(3.19\))523 2304 y(The)28 b(op)r(erator)e(whose)g (role)h(is)h(to)f(appro)n(ximate)f(the)i(resolv)n(en)n(t)e(reads)g(as:) 626 2484 y Fo(S)5 b Fp(\()p Fo(z)t Fp(\))83 b(=)f Fo(g)1059 2496 y Fl(0)p Fj(;n)1157 2484 y Fp(\()p Fo(H)25 b Fm(\000)18 b Fo(z)t Fp(\))1441 2450 y Fk(\000)p Fl(1)1530 2484 y Fo(g)1570 2496 y Fl(0)p Fj(;n)1686 2484 y Fp(+)1424 b(\(3.20\))872 2690 y(+)1063 2587 y Fl(4)1019 2612 y Ff(X)1022 2788 y Fj(j)s Fl(=1)1153 2690 y Fp(exp\()p Fo({)23 b(\016)s(')1456 2702 y Fj(j;n)1548 2690 y Fp(\()p Fm(\001)p Fo(;)14 b Fn(x)1690 2656 y Fj(j)1726 2690 y Fp(\)\))p Fo(g)1830 2702 y Fj(j;n)1922 2690 y Fp(\(\()p Fo(H)2055 2702 y Fj(j;n)2167 2690 y Fm(\000)k Fo(z)t Fp(\))2325 2656 y Fk(\000)p Fl(1)2413 2690 y Fo(g)2453 2702 y Fj(j;n)2559 2690 y Fp(exp\()p Fm(\000)p Fo({)k(\016)s(')2926 2702 y Fj(j;n)3019 2690 y Fp(\()p Fm(\001)p Fo(;)14 b Fn(x)3161 2656 y Fj(j)3197 2690 y Fp(\)\))p Fo(:)648 2964 y Fp(The)35 b(essen)n(tial)f(self-adjoin)n(tness)g(of)g Fo(H)42 b Fp(and)35 b Fm(f)p Fo(H)2239 2976 y Fj(j;n)2331 2964 y Fm(g)f Fp(on)h(the)g(same)f(core)g Fo(C)3143 2934 y Fk(1)3137 2985 y Fl(0)3214 2964 y Fp(\()p Fn(R)3318 2934 y Fl(2)3355 2964 y Fp(\))523 3064 y(and)27 b(the)h(use)g(of)f(\(3.18\)) g(enable)h(us)f(to)h(conclude)f(that)1457 3244 y(\()p Fo(H)f Fm(\000)18 b Fo(z)t Fp(\))p Fo(S)5 b Fp(\()p Fo(z)t Fp(\))22 b(=)g(1)c(+)g Fo(T)2206 3256 y Fl(0)2261 3244 y Fp(+)h Fo(T)2394 3256 y Fl(1)2430 3244 y Fo(;)722 b Fp(\(3.21\))523 3424 y(where)27 b Fo(T)812 3436 y Fl(0)877 3424 y Fp(and)g Fo(T)1087 3436 y Fl(1)1151 3424 y Fp(are)g(giv)n(en)g (b)n(y:)1444 3605 y Fo(T)1493 3617 y Fl(0)1553 3605 y Fp(=)c([)p Fo(H)r(;)14 b(g)1812 3617 y Fl(0)p Fj(;n)1909 3605 y Fp(]\()p Fo(H)26 b Fm(\000)18 b Fo(z)t Fp(\))2217 3570 y Fk(\000)p Fl(1)2306 3605 y Fo(g)2346 3617 y Fl(0)p Fj(;n)2443 3605 y Fo(;)709 b Fp(\(3.22\))706 3867 y Fo(T)755 3879 y Fl(1)815 3867 y Fp(=)946 3763 y Fl(4)903 3788 y Ff(X)905 3965 y Fj(j)s Fl(=1)1036 3867 y Fp(exp\()p Fo({)23 b(\016)s(')1339 3879 y Fj(j;n)1432 3867 y Fp(\)[)p Fo(H)1556 3879 y Fj(j;n)1648 3867 y Fo(;)14 b(g)1725 3879 y Fj(j;n)1817 3867 y Fp(]\(\()p Fo(H)1973 3879 y Fj(j;n)2084 3867 y Fm(\000)k Fo(z)t Fp(\))2242 3832 y Fk(\000)p Fl(1)2331 3867 y Fo(g)2371 3879 y Fj(j;n)2477 3867 y Fp(exp)o(\()p Fm(\000)p Fo({)23 b(\016)s(')2844 3879 y Fj(j;n)2937 3867 y Fp(\))p Fo(;)183 b Fp(\(3.23\))523 4103 y(and)27 b([)p Fo(H)776 4115 y Fj(j;n)869 4103 y Fo(;)14 b(g)946 4115 y Fj(j;n)1037 4103 y Fp(])23 b(=)g Fm(\000)p Fp(2)p Fo({)p Fp(\()p Fm(r)p Fo(g)1446 4115 y Fj(j;n)1537 4103 y Fp(\))c Fm(\001)f Fp(\()p Fn(p)h Fm(\000)f Fn(a)1862 4115 y Fj(j;n)1955 4103 y Fp(\))h Fm(\000)f Fp(\(\001)p Fo(g)2230 4115 y Fj(j;n)2322 4103 y Fp(\))p Fo(:)648 4203 y Fp(Fix)23 b Fo(\026)g(>)g Fp(0.)35 b(Prop)r(osition)21 b(3.1,)j(Lemma)f(3.1,)g(the)h(compact)f(supp)r(ort) g(of)g Fo(g)2994 4215 y Fl(0)p Fj(;n)3115 4203 y Fp(and)g(the)523 4302 y(fact)k(that)g(the)g(deriv)-5 b(ativ)n(es)26 b(of)h Fo(g)1554 4314 y Fj(j;n)1672 4302 y Fp(b)r(eha)n(v)n(e)f(at)h(least)f (lik)n(e)g(1)p Fo(=n)p Fp(,)g(assure)g(the)h(existence)f(of)523 4402 y Fo(n)p Fp(\()p Fo(\026;)14 b(d)p Fp(\))28 b(suc)n(h)g(that,)f (for)h Fo(n)22 b Fp(=)h Fo(n)p Fp(\()p Fo(\026;)14 b(d)p Fp(\))19 b(+)f(1,)27 b(one)h(has:)1196 4621 y Fm(jj)14 b Fp(exp\()p Fo(\026)23 b(<)g Fm(\001)g Fo(>)p Fp(\))p Fo(T)1768 4633 y Fl(1)1818 4621 y Fp(exp\()p Fm(\000)p Fo(\026)g(<)g Fm(\001)g Fo(>)p Fp(\))p Fm(jj)18 b Fp(+)g Fm(jj)p Fo(T)2588 4633 y Fl(0)2625 4621 y Fm(jj)23 b(\024)2792 4565 y Fp(1)p 2792 4602 42 4 v 2792 4678 a(4)2844 4621 y Fo(;)1196 4774 y Fm(jj)14 b Fp(exp\()p Fo(\026)23 b(<)g Fm(\001)g Fo(>)p Fp(\))p Fo(T)1768 4786 y Fl(0)1805 4774 y Fm(jj)g Fo(<)f Fm(1)p Fo(;)1196 4898 y Fm(jj)14 b Fp(exp\()p Fo(\026)23 b(<)g Fm(\001)g Fo(>)p Fp(\))p Fo(S)5 b Fp(\()p Fo(z)t Fp(\))14 b(exp)o(\()p Fm(\000)p Fo(\026)23 b(<)g Fm(\001)g Fo(>)p Fp(\))p Fm(jj)g(\024)f Fp(C\()p Fo(\026;)14 b(d)p Fp(\))p Fo(:)319 b Fp(\(3.24\))1934 5147 y(9)p eop %%Page: 10 10 10 9 bop 523 614 a Fp(Then)28 b(w)n(e)f(can)g(p)r(erform)g(the)h(follo) n(wing)f(computations:)1092 810 y(\(1)18 b(+)g Fo(T)1316 822 y Fl(1)1372 810 y Fp(+)g Fo(T)1504 822 y Fl(0)1540 810 y Fp(\))1572 776 y Fk(\000)p Fl(1)1685 810 y Fp(=)k(\(1)d(+)f Fo(T)1997 822 y Fl(1)2033 810 y Fp(\))2065 776 y Fk(\000)p Fl(1)2169 743 y Ff(\002)2203 810 y Fp(1)g(+)g Fo(T)2395 822 y Fl(0)2432 810 y Fp(\(1)g(+)g Fo(T)2656 822 y Fl(1)2693 810 y Fp(\))2725 776 y Fk(\000)p Fl(1)2814 743 y Ff(\003)2849 759 y Fk(\000)p Fl(1)2961 810 y Fp(=)944 968 y(=)83 b(\(1)18 b(+)g Fo(T)1316 980 y Fl(1)1353 968 y Fp(\))1385 934 y Fk(\000)p Fl(1)1488 876 y Ff(n)1544 968 y Fp(1)g Fm(\000)g Fo(T)1736 980 y Fl(0)1772 968 y Fp(\(1)h(+)f Fo(T)1997 980 y Fl(1)2033 968 y Fp(\))2065 934 y Fk(\000)p Fl(1)2169 901 y Ff(\002)2203 968 y Fp(1)g(+)g Fo(T)2395 980 y Fl(0)2432 968 y Fp(\(1)g(+)g Fo(T)2656 980 y Fl(1)2693 968 y Fp(\))2725 934 y Fk(\000)p Fl(1)2814 901 y Ff(\003)2849 917 y Fk(\000)p Fl(1)2938 876 y Ff(o)3007 968 y Fo(:)145 b Fp(\(3.25\))523 1181 y(If)29 b Fo(z)j Fp(has)c(a)g(small)g(imaginary)f(part,)i(then)g (w)n(e)f(kno)n(w)g(that)h(\()p Fo(H)d Fm(\000)19 b Fo(z)t Fp(\))2709 1151 y Fk(\000)p Fl(1)2826 1181 y Fp(exists)28 b(and)h(from)523 1281 y(\(3.21\))e(it)h(should)f(b)r(e)h(giv)n(en)f(b)n (y)g(the)h(follo)n(wing)f(relation:)737 1463 y(\()p Fo(H)e Fm(\000)18 b Fo(z)t Fp(\))1021 1429 y Fk(\000)p Fl(1)1193 1463 y Fp(=)82 b Fo(S)5 b Fp(\()p Fo(z)t Fp(\)\(1)18 b(+)g Fo(T)1727 1475 y Fl(1)1782 1463 y Fp(+)g Fo(T)1914 1475 y Fl(0)1951 1463 y Fp(\))1983 1429 y Fk(\000)p Fl(1)2096 1463 y Fp(=)1193 1588 y(=)82 b Fo(S)5 b Fp(\()p Fo(z)t Fp(\)\(1)18 b(+)g Fo(T)1727 1600 y Fl(1)1764 1588 y Fp(\))1796 1554 y Fk(\000)p Fl(1)1904 1588 y Fm(\000)1206 b Fp(\(3.26\))1193 1731 y Fm(\000)82 b Fo(S)5 b Fp(\()p Fo(z)t Fp(\)\(1)18 b(+)g Fo(T)1727 1743 y Fl(1)1764 1731 y Fp(\))1796 1696 y Fk(\000)p Fl(1)1885 1731 y Fo(T)1934 1743 y Fl(0)1971 1731 y Fp(\(1)g(+)g Fo(T)2195 1743 y Fl(1)2232 1731 y Fp(\))2264 1696 y Fk(\000)p Fl(1)2367 1663 y Ff(\002)2402 1731 y Fp(1)g(+)g Fo(T)2594 1743 y Fl(0)2631 1731 y Fp(\(1)g(+)g Fo(T)2855 1743 y Fl(1)2892 1731 y Fp(\))2924 1696 y Fk(\000)p Fl(1)3013 1663 y Ff(\003)3048 1679 y Fk(\000)p Fl(1)3151 1731 y Fo(:)523 1913 y Fp(Because)25 b(the)h(righ)n(t)f(hand)h(side)f (of)h(\(3.26\))f(is)h(b)r(ounded)g(for)f(all)h Fo(z)j Fp(ob)r(eying)c(0)e Fo(<)f(d)3108 1925 y Fl(0)3169 1913 y Fm(\024)g Fo(d)i Fm(\024)523 2013 y Fo(d)566 2025 y Fl(1)627 2013 y Fo(<)f Fm(1)p Fp(,)28 b(it)g(follo)n(ws)f(through)g (analytic)g(con)n(tin)n(uation)g(that)h(\(3.26\))f(is)h(true)g(on)f (the)h(real)523 2112 y(axis)f(to)r(o.)36 b(Using)28 b(no)n(w)f (\(3.24\))g(in)g(\(3.26\),)g(the)h(pro)r(of)f(is)h(\014nished)p Fo(:)2619 2124 y Fa(2)648 2212 y Fg(R)l(emark)h(1:)648 2312 y Fp(F)-7 b(or)27 b Fo(\013)c Fm(2)g Fn(R)p Fp(,)28 b(de\014ne)g(the)g(follo)n(wing)e(analytic)h(family)h(of)f(op)r (erators:)1221 2494 y Fo(H)7 b Fp(\()p Fo(\013)p Fp(\))24 b(=)f(exp\()p Fo({\013)g(<)g Fm(\001)g Fo(>)p Fp(\))p Fo(H)d Fp(exp\()p Fm(\000)p Fo({\013)j(<)g Fm(\001)g Fo(>)p Fp(\))p Fo(:)648 2677 y Fp(Lemma)f(2.1)h(implies)g(in)g (particular)f(that,)i(if)g Fo(z)i Fm(2)d Fo(\032)p Fp(\()p Fo(H)7 b Fp(\),)25 b(then)e(\()p Fo(H)7 b Fp(\()p Fo(\013)p Fp(\))i Fm(\000)g Fo(z)t Fp(\))3029 2647 y Fk(\000)p Fl(1)3142 2677 y Fp(admits)523 2777 y(an)27 b(en)n(tire)f(extension)h (to)g(the)g(whole)g(complex)f(plane)h(and)g Fo(\032)p Fp(\()p Fo(H)7 b Fp(\))23 b Fm(\032)g Fo(\032)p Fp(\()p Fo(H)7 b Fp(\()p Fo(\013)p Fp(\)\),)28 b(therefore)523 2876 y(the)33 b(essen)n(tial)f(sp)r(ectrum)h(of)g Fo(H)7 b Fp(\()p Fo(\013)p Fp(\))34 b(is)e(not)h(mo)n(ving)f(to)n(w)n(ards)f (its)i(discrete)g(eigen)n(v)-5 b(alues)523 2976 y(\(from)22 b(the)g(usual)g(analytic)f(p)r(erturbation)g(theory)g(w)n(e)h(kno)n(w)f (that)h(its)g(discrete)f(sp)r(ectrum)523 3075 y(is)k(indep)r(enden)n(t) h(of)e Fo(\013)p Fp(\).)37 b(Moreo)n(v)n(er,)23 b(the)i(pro)5 b(jector)24 b Fo(P)12 b Fp(\()p Fo(\013)p Fp(\))25 b(\(written)h(with)f (the)h(Riesz)e(for-)523 3175 y(m)n(ula\))j(asso)r(ciated)f(with)i(a)e (discrete)h(eigen)n(v)-5 b(alue)26 b(of)h Fo(H)7 b Fp(\()p Fo(\013)p Fp(\))28 b(admits)f(an)g(en)n(tire)g(extension)523 3275 y(to)r(o,)h(whic)n(h)g(\(via)f(O'Connor's)g(Lemma\))h(implies)g (that)g(the)g(discrete)g(eigenfunctions)f(of)523 3374 y Fo(H)33 b Fp(deca)n(y)25 b(faster)h(than)g(an)n(y)f(exp)r(onen)n (tial)h(\(this)h(result)f(w)n(as)f(obtained)h(in)g(a)g(simpler)g(w)n(a) n(y)523 3474 y(in)i([3]\).)648 3574 y Fg(R)l(emark)h(2:)648 3673 y Fp(Supp)r(ose)41 b(that)h(one)g(has)f(a)g(family)h(of)f(scalar)f (and)i(magnetic)f(p)r(erturbations)g Fo(V)3346 3685 y Fj(k)523 3773 y Fp(and)35 b Fo(B)759 3743 y Fk(0)782 3785 y Fj(k)858 3773 y Fp(suc)n(h)g(that)g(\(see)f(the)i(form)n (ulation)e(of)g(Lemma)h(2.1\))f(lim)2713 3785 y Fj(n)p Fk(!1)2890 3773 y Fm(f)p Fp(sup)3057 3793 y Fj(k)3098 3773 y Fp(\()p Fo(b)3166 3785 y Fj(n;k)3267 3773 y Fp(\))24 b(+)523 3872 y(sup)648 3893 y Fj(k)689 3872 y Fp(\()p Fo(v)761 3884 y Fj(n;k)863 3872 y Fp(\))p Fm(g)31 b Fp(=)f(0)p Fo(;)i Fp(and)g(inf)1427 3884 y Fj(k)1468 3872 y Fm(f)p Fp(dist\()p Fo(z)t(;)14 b(\033)s Fp(\()p Fo(H)1907 3884 y Fj(k)1948 3872 y Fp(\)\))p Fm(g)31 b(\025)g Fo(d)g(>)f Fp(0.)51 b(Then)33 b(it)f(can)g(b)r(e)h(seen)f(that)523 3972 y(the)e(c)n(hoices)f(for)g Fo(\025)i Fp(in)f(\(3.1\))f(and)h (\(3.2\),)g(the)g(b)r(ounds)g(in)g(\(3.4\),)h(\(3.2\),)f(\(3.3\),)g (\(3.9\))g(and)523 4072 y(\(3.10\),)i(the)g(c)n(hoice)e(of)h Fo(N)9 b Fp(\()p Fo(\026;)14 b(d)p Fp(\))32 b(and)g(the)g(b)r(ound)f (in)h(\(3.11\))f(can)g(b)r(e)h(made)f(uniform)g(in)523 4171 y Fo(k)s Fp(,)d(and)f(so)g(are)g(the)h(b)r(ounds)f(on)h(the)g (norms)e(whic)n(h)i(app)r(ear)f(in)g(Lemma)h(2.1.)523 4404 y Fb(3.2)112 b(P)m(erio)s(dic)36 b(w)m(ells)f(in)i(t)m(w)m(o)g (dimensions)523 4557 y Fg(Pr)l(o)l(of)31 b(of)g(The)l(or)l(em)f(2.1:) 648 4657 y Fp(i\).)35 b(The)21 b(strategy)e(in)j(\014nding)f(the)g(sp)r (ectrum)g(lo)r(cation)g(of)g(the)g(\\whole")f(Hamiltonian)523 4765 y Fo(H)592 4777 y Fl(\000)633 4785 y Fc(R)705 4765 y Fp(in)i(the)f(giv)n(en)f(in)n(terv)-5 b(al)20 b([)p Fo(c;)14 b(d)p Fp(])22 b(consists)e(in)h(constructing)f(its)h(resolv)n (en)n(t)f(\()p Fo(H)3030 4777 y Fl(\000)3071 4785 y Fc(R)3141 4765 y Fm(\000)e Fo(z)t Fp(\))3298 4723 y Fk(\000)p Fl(1)523 4865 y Fp(for)40 b(all)h Fo(z)48 b Fm(2)d Fp([)p Fo(c;)14 b(d)p Fp(])41 b(situated)g(outside)f(of)h(an)f(exp)r(onen)n(tially)g (small)g(strip)h(cen)n(tred)f(in)1914 5147 y(10)p eop %%Page: 11 11 11 10 bop 523 614 a Fo(E)584 626 y Fl(1)622 614 y Fp(\()p Fo(R)q Fp(\),)30 b(where)e Fo(E)1105 626 y Fl(1)1143 614 y Fp(\()p Fo(R)q Fp(\))h(will)h(turn)f(out)g(to)g(b)r(e)h(an)f (eigen)n(v)-5 b(alue)28 b(of)h(an)g(\\almost)f(one)h(w)n(ell")523 714 y(Hamiltonian)e Fo(H)1069 726 y Fj(R)1124 714 y Fp(.)37 b(So)27 b(one)g(has)g(to)h(mak)n(e)f(the)h(follo)n(wing)e(t)n(w)n(o)h (steps:)648 814 y(a\))g(Guess)g Fo(H)1059 826 y Fj(R)1142 814 y Fp(and)g Fo(E)1364 826 y Fl(1)1401 814 y Fp(\()p Fo(R)q Fp(\),)648 913 y(and)648 1013 y(b\))g(construct)g(a)g(b)r (ounded)g(op)r(erator)f Fo(S)1911 1025 y Fj(R)1965 1013 y Fp(\()p Fo(z)t Fp(\))h(suc)n(h)g(that,)g(as)g(so)r(on)f(as)h Fo(R)d Fm(\025)e Fo(R)q Fp(\()p Fo(\026)p Fp(\))28 b(one)523 1112 y(can)f(write:)1206 1295 y(\()p Fo(H)1307 1307 y Fl(\000)1348 1315 y Fc(R)1418 1295 y Fm(\000)18 b Fo(z)t Fp(\))p Fo(S)1627 1307 y Fj(R)1681 1295 y Fp(\()p Fo(z)t Fp(\))83 b(=)g(1)17 b(+)h Fo(T)2210 1307 y Fj(R)2264 1295 y Fp(\()p Fo(z)t Fp(\))p Fo(;)781 b Fp(\(3.27\))1486 1468 y Fm(jj)p Fo(T)1581 1480 y Fj(R)1635 1468 y Fp(\()p Fo(z)t Fp(\))p Fm(jj)83 b Fo(<)2045 1411 y Fp(exp)14 b(\()p Fm(\000)p Fo(\026R)q Fp(\))p 2028 1448 417 4 v 2028 1525 a Fm(j)p Fo(z)22 b Fm(\000)c Fo(E)2256 1537 y Fl(1)2294 1525 y Fp(\()p Fo(R)q Fp(\))p Fm(j)2474 1468 y Fp(+)g(1)p Fo(=)p Fp(2)p Fo(:)469 b Fp(\(3.28\))648 1694 y(Let)33 b(us)g(in)n(tro)r(duce)g(some)f(cut-o\013)h(functions)h (whic)n(h)f(will)g(help)g(to)g(write)g(do)n(wn)g(the)523 1794 y(needed)c(op)r(erators.)39 b(Let)29 b Fo(j)h Fm(2)c(f)p Fp(1)p Fo(;)14 b Fp(2)p Fo(;)g Fp(3)p Fm(g)p Fp(,)27 b(and)i(let)g Fo(g)2158 1806 y Fj(j)2222 1794 y Fp(b)r(e)g(of)g(the)g (same)f(kind)i(as)e(in)h(\(3.12\))523 1894 y(and)f(with)h(con)n(v)n(ex) d(supp)r(orts)i(included)h(in)f Fm(j)p Fn(x)p Fm(j)d(\024)e Fp(7)p Fo(=)p Fp(8.)38 b(Cho)r(ose)27 b Fo(g)2643 1906 y Fl(1)2680 1894 y Fp(\()p Fn(x)p Fp(\))e(=)e(1)28 b(if)h Fm(j)p Fn(x)p Fm(j)24 b(\024)g Fp(1)p Fo(=)p Fp(4)523 1993 y(and)32 b(equal)g(to)g(zero)g(if)h Fm(j)p Fn(x)p Fm(j)e(\025)g Fp(3)p Fo(=)p Fp(4.)50 b(T)-7 b(ak)n(e)31 b(then)i Fo(g)2142 2005 y Fl(2)2211 1993 y Fp(and)g Fo(g)2418 2005 y Fl(3)2487 1993 y Fp(suc)n(h)f(that)h Fo(g)2904 2005 y Fl(2)2941 1993 y Fo(g)2981 2005 y Fl(3)3049 1993 y Fp(=)d Fo(g)3184 2005 y Fl(2)3254 1993 y Fp(and)523 2093 y Fo(g)563 2105 y Fl(1)600 2093 y Fo(g)640 2105 y Fl(2)712 2093 y Fp(=)k Fo(g)851 2105 y Fl(1)888 2093 y Fp(.)58 b(If)35 b(supp\()p Fo(@)5 b(g)1351 2105 y Fl(2)1388 2093 y Fp(\))35 b(denotes)g(the)g(supp)r(ort)f(of)h(at)f(least)h(one)f (deriv)-5 b(ativ)n(e)34 b(of)g Fo(g)3327 2105 y Fl(2)3364 2093 y Fp(,)523 2192 y(then)28 b(w)n(e)f(ask)g(that:)1370 2292 y Fo(\016)f Fp(=)c(dist)q(\(supp\()p Fo(@)5 b(g)1979 2304 y Fl(2)2016 2292 y Fp(\))p Fo(;)14 b Fp(supp)p Fo(g)2296 2304 y Fl(1)2333 2292 y Fp(\))23 b Fo(>)g Fp(0)p Fo(:)634 b Fp(\(3.29\))648 2441 y(De\014ne)28 b(then)g Fo(g)1134 2453 y Fj(j;\015)t(;R)1316 2441 y Fp(=)23 b Fo(g)1444 2453 y Fj(j)1478 2441 y Fp(\(\()p Fm(\001)c(\000)f Fo(R)q(\015)5 b Fp(\))p Fo(=R)q Fp(\).)36 b(The)28 b(\014nal)g(requiremen)n(t)e(will) i(b)r(e:)1641 2565 y Ff(X)1638 2743 y Fj(\015)t Fk(2)p Fl(\000)1777 2644 y Fo(g)1817 2656 y Fl(1)p Fj(;\015)t(;R)1982 2644 y Fp(\()p Fn(x)p Fp(\))c(=)e(1)p Fo(:)903 b Fp(\(3.30\))523 2912 y Fg(R)l(emark)30 b(3)p Fp(:)648 3011 y(It)37 b(is)g(v)n(ery)f (imp)r(ortan)n(t)g(to)h(see)g(that,)j(for)c(an)n(y)h Fo(R)i(>)f Fp(0)f(and)g(an)n(y)f Fo(\015)5 b Fp(,)39 b(the)f(function)523 3111 y Fo(g)563 3123 y Fl(2)p Fj(;\015)t(;R)750 3111 y Fp(has)21 b(at)g(most)g(8)g(neigh)n(b)r(ours)g(whose)f(supp)r (orts)h(are)g(not)g(disjoin)n(t)h(from)f(supp)p Fo(g)3199 3123 y Fl(2)p Fj(;\015)t(;R)3364 3111 y Fp(.)523 3210 y(Moreo)n(v)n(er,)715 3393 y(dist)q(\(supp\()p Fo(@)5 b(g)1174 3405 y Fl(2)p Fj(;\015)t(;R)1339 3393 y Fp(\))p Fo(;)14 b Fp(supp)p Fo(g)1619 3405 y Fl(1)p Fj(;\015)t(;R)1784 3393 y Fp(\))24 b(=)e Fo(\016)s(R)q(;)180 b(g)2274 3405 y Fl(2)p Fj(;\015)t(;R)2439 3393 y Fo(g)2479 3405 y Fl(1)p Fj(;\015)t(;R)2667 3393 y Fp(=)23 b Fo(g)2795 3405 y Fl(1)p Fj(;\015)t(;R)2960 3393 y Fo(:)192 b Fp(\(3.31\))648 3576 y(Let)27 b(us)h(in)n(tro)r(duce)f(the)h(\\almost)f(one)g(w)n(ell)g (p)r(oten)n(tials":)659 3853 y Fo(B)722 3865 y Fj(R)776 3853 y Fp(\()p Fn(x)p Fp(\))84 b(=)f Fo(B)1185 3865 y Fl(0)1241 3853 y Fp(+)1324 3686 y Ff(2)1324 3836 y(4)1381 3774 y(X)1379 3953 y Fj(\015)t Fk(2)p Fl(\000)1517 3853 y Fo(B)1584 3819 y Fk(0)1608 3853 y Fp(\()p Fn(x)19 b Fm(\000)f Fo(R)q(\015)5 b Fp(\))1936 3686 y Ff(3)1936 3836 y(5)2004 3853 y Fo(g)2044 3865 y Fl(3)p Fj(;)p Fl(0)p Fj(;R)2204 3853 y Fp(\()p Fn(x)p Fp(\))24 b(=)768 b(\(3.32\))974 4185 y(=)83 b Fo(B)t Fp(\()p Fn(x)p Fp(\))20 b(+)e(\(1)g Fm(\000)g Fo(g)1621 4197 y Fl(3)p Fj(;)p Fl(0)p Fj(;R)1781 4185 y Fp(\()p Fn(x)p Fp(\)\))p Fo(B)1994 4151 y Fk(0)2018 4185 y Fp(\()p Fn(x)p Fp(\))i(+)2235 4019 y Ff(2)2235 4168 y(4)2291 4107 y(X)2290 4285 y Fj(\015)t Fk(6)p Fl(=0)2426 4185 y Fo(B)2493 4151 y Fk(0)2517 4185 y Fp(\()p Fn(x)f Fm(\000)f Fo(R)q(\015)5 b Fp(\))2845 4019 y Ff(3)2845 4168 y(5)2914 4185 y Fo(g)2954 4197 y Fl(3)p Fj(;)p Fl(0)p Fj(;R)3113 4185 y Fp(\()p Fn(x)p Fp(\))p Fo(;)690 4551 y(V)738 4563 y Fj(R)793 4551 y Fp(\()p Fn(x)p Fp(\))84 b(=)e Fo(V)1186 4563 y Fl(\000)1227 4571 y Fc(R)1279 4551 y Fp(\()p Fn(x)p Fp(\))p Fo(g)1433 4563 y Fl(3)p Fj(;)p Fl(0)p Fj(;R)1594 4551 y Fp(\()p Fn(x)p Fp(\))24 b(=)1378 b(\(3.33\))991 4778 y(=)82 b Fo(V)19 b Fp(\()p Fn(x)p Fp(\))h(+)e(\(1)g Fm(\000)g Fo(g)1637 4790 y Fl(3)p Fj(;)p Fl(0)p Fj(;R)1797 4778 y Fp(\()p Fn(x)p Fp(\)\))p Fo(V)i Fp(\()p Fn(x)p Fp(\))f(+)2227 4611 y Ff(2)2227 4760 y(4)2284 4699 y(X)2283 4878 y Fj(\015)t Fk(6)p Fl(=0)2419 4778 y Fo(V)g Fp(\()p Fn(x)g Fm(\000)f Fo(R)q(\015)5 b Fp(\))2814 4611 y Ff(3)2814 4760 y(5)2883 4778 y Fo(g)2923 4790 y Fl(3)p Fj(;)p Fl(0)p Fj(;R)3082 4778 y Fp(\()p Fn(x)p Fp(\))p Fo(:)1914 5147 y Fp(11)p eop %%Page: 12 12 12 11 bop 523 614 a Fp(If)35 b Fn(a)659 626 y Fj(R)749 614 y Fp(is)f(the)h(transv)n(ersal)d(gauge)h(corresp)r(onding)g(to)h Fo(B)2371 626 y Fj(R)2426 614 y Fp(\()p Fn(x)p Fp(\),)j(then)e(the)g (\\almost)f(one)523 714 y(w)n(ell")27 b(Hamiltonian)g(will)h(b)r(e)g Fo(H)1549 726 y Fj(R)1627 714 y Fp(=)22 b(\()p Fn(p)d Fm(\000)f Fn(a)1947 726 y Fj(R)2002 714 y Fp(\))2034 684 y Fl(2)2090 714 y Fp(+)g Fo(V)2221 726 y Fj(R)2276 714 y Fp(.)648 814 y(F)-7 b(rom)27 b(the)h(ab)r(o)n(v)n(e)e (de\014nitions)i(and)f(using)g(\(2.9\),)h(one)f(can)g(see)g(that:)1077 996 y(lim)1043 1050 y Fj(R)p Fk(!1)1226 996 y Fm(f)c Fp(sup)1268 1070 y Fe(x)p Fk(2)p Fe(R)1409 1053 y Fd(2)1440 996 y Fp([)p Fm(j)p Fo(D)1557 962 y Fj(\013)1605 996 y Fp(\()p Fo(B)g Fm(\000)18 b Fo(B)1869 1008 y Fj(R)1923 996 y Fp(\))p Fm(j)p Fp(\()p Fn(x)p Fp(\))i(+)e Fm(j)p Fo(V)37 b Fm(\000)18 b Fo(V)2434 1008 y Fj(R)2489 996 y Fm(j)p Fp(\()p Fn(x)p Fp(\)])p Fm(g)24 b Fp(=)e(0)p Fo(:)308 b Fp(\(3.34\))648 1236 y(The)25 b(stabilit)n(y)h(of)g(sp)r (ectrum)g(for)f(suc)n(h)g(Hamiltonians)h(\(see)f([15])g(for)g(a)h(more) f(general)523 1336 y(discussion\))37 b(implies)g(that)h(for)f Fo(R)j Fm(\025)e Fo(R)1841 1348 y Fl(0)1879 1336 y Fp(,)i(in)d([)p Fo(c;)14 b(d)p Fp(])38 b(liv)n(es)e(only)h(one)g(not)h(degenerated)523 1435 y(eigen)n(v)-5 b(alue)27 b(of)g Fo(H)1083 1447 y Fj(R)1165 1435 y Fp(\(denoted)h(with)g Fo(E)1761 1447 y Fl(1)1798 1435 y Fp(\()p Fo(R)q Fp(\)\))g(whic)n(h)g(con)n(v)n(erges) d(to)i Fo(E)2758 1447 y Fl(1)2823 1435 y Fp(when)h Fo(R)g Fp(go)r(es)e(to)523 1535 y(in\014nit)n(y)-7 b(.)648 1634 y(A)n(t)32 b(this)g(p)r(oin)n(t)g(w)n(e)g(ha)n(v)n(e)f(to)h(establish)f (the)i(link)f(b)r(et)n(w)n(een)g Fo(H)2644 1646 y Fj(R)2730 1634 y Fp(and)g Fo(H)2965 1646 y Fl(\000)3006 1654 y Fc(R)3057 1634 y Fp(.)50 b(Denote)523 1734 y(with)30 b Fo(t)744 1746 y Fj(\015)t(R)866 1734 y Fp(the)f(usual)g(translation)f (\(\()p Fo(t)1745 1746 y Fj(\015)t(R)1838 1734 y Fo(f)9 b Fp(\)\()p Fn(x)p Fp(\))26 b(=)f Fo(f)9 b Fp(\()p Fn(x)20 b Fm(\000)f Fo(R)q(\015)5 b Fp(\)\);)30 b(from)e(\(2.1\))h(and)g (\(2.10\))523 1834 y(one)e(obtains)g(that)h Fn(a)1192 1846 y Fl(\000)1233 1854 y Fc(R)1285 1834 y Fp(\()p Fn(x)p Fo(;)14 b(\015)5 b(R)q Fp(\))23 b(=)g Fn(a)1705 1846 y Fl(\000)1746 1854 y Fc(R)1797 1834 y Fp(\()p Fn(x)c Fm(\000)f Fo(\015)5 b(R)q Fp(\))28 b(and)1454 2016 y Fn(a)1500 2028 y Fl(\000)1541 2036 y Fc(R)1592 2016 y Fp(\()p Fn(x)p Fp(\))p Fo(g)1746 2028 y Fl(2)p Fj(;)p Fl(0)p Fj(;R)1930 2016 y Fp(=)23 b Fn(a)2064 2028 y Fj(R)2119 2016 y Fp(\()p Fn(x)p Fp(\))p Fo(g)2273 2028 y Fl(2)p Fj(;)p Fl(0)p Fj(;R)2433 2016 y Fo(:)719 b Fp(\(3.35\))523 2199 y(One)27 b(can)h(write)f(do)n(wn)g(a)g(gauge)f(phase)h Fo(')1871 2211 y Fj(\015)1942 2199 y Fp(suc)n(h)g(that:)1085 2382 y Fo(H)1154 2394 y Fl(\000)1195 2402 y Fc(R)1260 2382 y Fp(exp)14 b(\()p Fo({')1514 2394 y Fj(\015)1557 2382 y Fp(\))p Fo(g)1629 2394 y Fl(2)p Fj(;\015)t(;R)1794 2382 y Fo(t)1824 2394 y Fj(\015)t(R)1940 2382 y Fp(=)22 b(exp)14 b(\()p Fo({')2281 2394 y Fj(\015)2324 2382 y Fp(\))p Fo(t)2386 2394 y Fj(\015)t(R)2479 2382 y Fo(H)2548 2394 y Fj(R)2603 2382 y Fo(g)2643 2394 y Fl(2)p Fj(;)p Fl(0)p Fj(;R)2802 2382 y Fo(:)350 b Fp(\(3.36\))648 2564 y(The)27 b(resolv)n(en)n(t's)f(appro)n(ximation)g(will)h(then)i(b)r(e:) 654 2759 y Fo(S)705 2771 y Fj(R)759 2759 y Fp(\()p Fo(z)t Fp(\))23 b(=)979 2680 y Ff(X)977 2858 y Fj(\015)t Fk(2)p Fl(\000)1115 2759 y Fo(g)1155 2771 y Fl(2)p Fj(;\015)t(;R)1334 2759 y Fp(exp)13 b(\()p Fo({')1587 2771 y Fj(\015)1630 2759 y Fp(\))q Fo(t)1693 2771 y Fj(\015)t(R)1799 2759 y Fp(\()q Fo(H)1901 2771 y Fj(R)1974 2759 y Fm(\000)18 b Fo(z)t Fp(\))2131 2717 y Fk(\000)p Fl(1)2234 2759 y Fo(t)2264 2771 y Fk(\000)p Fj(\015)t(R)2423 2759 y Fp(exp)13 b(\()p Fm(\000)p Fo({')2741 2771 y Fj(\015)2784 2759 y Fp(\))p Fo(g)2856 2771 y Fl(1)p Fj(;\015)t(;R)3021 2759 y Fo(;)131 b Fp(\(3.37\))523 3031 y(where)23 b(the)g(in\014nite)h (sum)f(should)g(b)r(e)h(understo)r(o)r(d)f(as)f(a)h(strong)f(limit.)36 b(Using)23 b(\(3.31\))f(\(see)523 3131 y(Remark)31 b(3\))h(and)f(the)i (fact)f(that)g Fm(f)p Fo(g)1693 3143 y Fl(1)p Fj(;\015)t(;R)1857 3131 y Fm(g)g Fp(is)f(a)h(lo)r(cally)f(\014nite)h(partition)g(of)g (unit)n(y)-7 b(,)33 b(one)523 3231 y(obtains:)1388 3330 y Fm(jj)p Fo(S)1485 3342 y Fj(R)1540 3330 y Fp(\()p Fo(z)t Fp(\))p Fm(jj)23 b(\024)g Fp(const)o Fm(j)p Fo(E)2077 3342 y Fl(1)2115 3330 y Fp(\()p Fo(R)q Fp(\))18 b Fm(\000)g Fo(z)t Fm(j)2410 3296 y Fk(\000)p Fl(1)2499 3330 y Fo(;)653 b Fp(\(3.38\))523 3480 y(where)24 b(the)g(ab)r(o)n(v)n(e)f(constan)n(t) h(do)r(es)g(not)g(dep)r(end)h(on)f Fo(R)q Fp(.)35 b(After)25 b(a)f(similar)g(reasoning)e(with)523 3579 y(that)28 b(one)f(made)h(in)f (obtaining)g(\(3.21\),)g(one)g(has)h(the)g(follo)n(wing)e(t)n(w)n(o)h (equalities:)753 3762 y(\()p Fo(H)854 3774 y Fl(\000)895 3782 y Fc(R)965 3762 y Fm(\000)18 b Fo(z)t Fp(\))p Fo(S)1174 3774 y Fj(R)1228 3762 y Fp(\()p Fo(z)t Fp(\))23 b(=)g(1)18 b(+)g Fo(T)1638 3774 y Fj(R)1692 3762 y Fp(\()p Fo(z)t Fp(\))p Fo(;)1353 b Fp(\(3.39\))753 3903 y Fo(T)802 3915 y Fj(R)856 3903 y Fp(\()p Fo(z)t Fp(\))23 b(=)1076 3824 y Ff(X)1074 4002 y Fj(\015)t Fk(2)p Fl(\000)1212 3903 y Fp(exp)13 b(\()p Fo({')1465 3915 y Fj(\015)1508 3903 y Fp(\))q Fo(t)1571 3915 y Fj(\015)t(R)1664 3903 y Fp([)p Fo(H)1756 3915 y Fj(R)1810 3903 y Fo(;)h(g)1887 3915 y Fl(2)p Fj(;)p Fl(0)p Fj(;R)2047 3903 y Fp(])g(\()p Fo(H)2185 3915 y Fj(R)2258 3903 y Fm(\000)k Fo(z)t Fp(\))2415 3861 y Fk(\000)p Fl(1)2518 3903 y Fo(g)2558 3915 y Fl(1)p Fj(;)p Fl(0)p Fj(;R)2718 3903 y Fo(t)2748 3915 y Fk(\000)p Fj(\015)t(R)2907 3903 y Fp(exp)13 b(\()p Fm(\000)p Fo({')3225 3915 y Fj(\015)3268 3903 y Fp(\))p Fo(:)648 4175 y Fp(The)27 b(op)r(erator)f Fo(T)1202 4187 y Fj(R)1256 4175 y Fp(\()p Fo(z)t Fp(\))i(is)f(also)g(b)r(ounded:)1069 4367 y Fm(jj)p Fo(T)1164 4379 y Fj(R)1218 4367 y Fp(\()p Fo(z)t Fp(\))p Fm(jj)c(\024)g Fp(const)o Fm(jj)p Fp([)p Fo(H)1809 4379 y Fj(R)1864 4367 y Fo(;)14 b(g)1941 4379 y Fl(2)p Fj(;)p Fl(0)p Fj(;R)2101 4367 y Fp(])g(\()p Fo(H)2239 4379 y Fj(R)2312 4367 y Fm(\000)k Fo(z)t Fp(\))2469 4325 y Fk(\000)p Fl(1)2572 4367 y Fo(g)2612 4379 y Fl(1)p Fj(;)p Fl(0)p Fj(;R)2772 4367 y Fm(jj)p Fo(;)334 b Fp(\(3.40\))523 4549 y(where)32 b([)p Fo(H)860 4561 y Fj(R)914 4549 y Fo(;)14 b(g)991 4561 y Fl(2)p Fj(;)p Fl(0)p Fj(;R)1151 4549 y Fp(])31 b(=)g Fm(\000)p Fp(2)p Fo({)p Fp(\()p Fm(r)p Fo(g)1576 4561 y Fl(2)p Fj(;)p Fl(0)p Fj(;R)1735 4549 y Fp(\))22 b Fm(\001)f Fp(\()p Fn(p)h Fm(\000)g Fn(a)2073 4561 y Fj(R)2128 4549 y Fp(\))f Fm(\000)h Fp(\001)p Fo(g)2377 4561 y Fl(2)p Fj(;)p Fl(0)p Fj(;R)2537 4549 y Fp(.)51 b(Because)31 b Fo(z)36 b Fp(is)c(situated)523 4649 y(in)j(the)g(in)n(terv)-5 b(al)34 b([)p Fo(c;)14 b(d)p Fp(])35 b(whic)n(h)g(con)n(tains)f(only)g Fo(E)2109 4661 y Fl(1)2147 4649 y Fp(,)j(one)d(can)g(\014nd)i(a)e(con)n(tour)g Fm(C)39 b Fp(in)c(the)523 4749 y(complex)g(plane)g(whic)n(h)g (surrounds)e Fo(E)1782 4761 y Fl(1)1855 4749 y Fp(suc)n(h)i(that)g (dist)p Fm(f)p Fo(\033)s Fp(\()p Fo(H)7 b Fp(\))p Fo(;)14 b Fm(C)5 b(g)35 b(\025)h Fo(d)f(>)h Fp(0;)i(due)d(to)1914 5147 y(12)p eop %%Page: 13 13 13 12 bop 523 614 a Fp(the)28 b(sp)r(ectrum)h(stabilit)n(y)e(of)h Fo(H)1517 626 y Fj(R)1572 614 y Fp(,)g(for)g Fo(R)c(>)f(R)1989 626 y Fl(0)2055 614 y Fp(one)k(has)h(dist)p Fm(f)p Fo(\033)s Fp(\()p Fo(H)2683 626 y Fj(R)2738 614 y Fp(\))p Fo(;)14 b Fm(C)5 b(g)23 b(\025)g Fo(d=)p Fp(2.)38 b(Riesz)523 714 y(form)n(ula)27 b(giv)n(es)f(at)i(once)f(that:)1421 928 y Fo(P)1474 940 y Fj(R)1612 928 y Fp(=)82 b Fm(\000)1872 872 y Fp(1)p 1834 909 119 4 v 1834 985 a(2)p Fo(\031)s({)1976 815 y Ff(Z)2022 1004 y Fk(C)2079 928 y Fo(d\030)t Fp(\()p Fo(H)2263 940 y Fj(R)2337 928 y Fm(\000)18 b Fo(\030)t Fp(\))2492 894 y Fk(\000)p Fl(1)2581 928 y Fo(;)792 1150 y Fp(\(1)h Fm(\000)f Fo(P)1021 1162 y Fj(R)1075 1150 y Fp(\)\()p Fo(H)1208 1162 y Fj(R)1282 1150 y Fm(\000)g Fo(z)t Fp(\))1440 1116 y Fk(\000)p Fl(1)1612 1150 y Fp(=)1808 1094 y(1)p 1769 1131 V 1769 1207 a(2)p Fo(\031)s({)1911 1037 y Ff(Z)1958 1226 y Fk(C)2015 1150 y Fo(d\030)2179 1094 y Fp(1)p 2108 1131 184 4 v 2108 1207 a Fo(\030)k Fm(\000)c Fo(z)2302 1150 y Fp(\()p Fo(H)2403 1162 y Fj(R)2476 1150 y Fm(\000)g Fo(\030)t Fp(\))2631 1116 y Fk(\000)p Fl(1)2720 1150 y Fo(;)1108 1369 y Fp(\()p Fo(H)1209 1381 y Fj(R)1282 1369 y Fm(\000)g Fo(z)t Fp(\))1440 1335 y Fk(\000)p Fl(1)1612 1369 y Fp(=)1934 1313 y(1)p 1769 1350 371 4 v 1769 1426 a Fo(E)1830 1438 y Fl(1)1868 1426 y Fp(\()p Fo(R)q Fp(\))g Fm(\000)g Fo(z)2150 1369 y(P)2203 1381 y Fj(R)2276 1369 y Fp(+)g(\(1)g Fm(\000)g Fo(P)2587 1381 y Fj(R)2642 1369 y Fp(\)\()p Fo(H)2775 1381 y Fj(R)2848 1369 y Fm(\000)g Fo(z)t Fp(\))3006 1335 y Fk(\000)p Fl(1)3095 1369 y Fo(:)57 b Fp(\(3.41\))648 1594 y(T)-7 b(ak)n(e)26 b(\(see)i(\(3.29\))e(for)h(the)h(de\014nition)g(of)f Fo(\016)s Fp(\))35 b(~)-49 b Fo(\026)23 b(>)g Fp(2)p Fo(\026=\016)s Fp(.)36 b(F)-7 b(rom)27 b(\(3.31\),)g(Lemma)g(2.1,)523 1694 y(Remark)g(2)g(and)g(\(3.41\))g(one)g(can)h(write:)965 1879 y Fm(jj)p Fo(T)1060 1891 y Fj(R)1114 1879 y Fp(\()p Fo(z)t Fp(\))p Fm(jj)83 b(\024)g Fp(const)p Fm(jj)p Fp([)p Fo(H)1826 1891 y Fj(R)1880 1879 y Fo(;)14 b(g)1957 1891 y Fl(2)p Fj(;)p Fl(0)p Fj(;R)2117 1879 y Fp(])g(\()p Fo(H)2255 1891 y Fj(R)2328 1879 y Fm(\000)k Fo(z)t Fp(\))2486 1837 y Fk(\000)p Fl(1)2588 1879 y Fo(g)2628 1891 y Fl(1)p Fj(;)p Fl(0)p Fj(;R)2788 1879 y Fm(jj)23 b(\024)1350 2043 y(\024)1622 1987 y Fp(const)p 1508 2024 417 4 v 1508 2100 a Fm(j)p Fo(E)1592 2112 y Fl(1)1630 2100 y Fp(\()p Fo(R)q Fp(\))18 b Fm(\000)g Fo(z)t Fm(j)1948 2043 y Fp(exp)c(\()p Fm(\000)7 b Fp(~)-49 b Fo(\026\016)s(R)q Fp(\))18 b(+)g(1)p Fo(=)p Fp(2)p Fo(;)553 b Fp(\(3.42\))523 2269 y(where)19 b(the)i(ab)r(o)n(v)n(e)d(constan)n(t)h(dep)r(ends)h(on) g(ev)n(erything)e(but)j Fo(R)q Fp(;)h(tak)n(e)d(no)n(w)g Fo(R)q Fp(\()p Fo(\026)p Fp(\))i(suc)n(h)e(that)523 2368 y(const)13 b(exp)h(\()p Fm(\000)7 b Fp(~)-49 b Fo(\026\016)s(R)q Fp(\()p Fo(\026)p Fp(\))p Fo(=)p Fp(2\))24 b Fo(<)g Fp(1;)k(then)h(for) f Fo(R)d Fm(\025)f Fo(R)q Fp(\()p Fo(\026)p Fp(\))k(and)h Fm(j)p Fo(E)2502 2380 y Fl(1)2539 2368 y Fp(\()p Fo(R)q Fp(\))19 b Fm(\000)g Fo(z)t Fm(j)24 b(\025)f Fp(2)14 b(exp)f(\()p Fm(\000)p Fo(\026R)q Fp(\))523 2468 y(one)27 b(\014nally)h(obtains)f Fm(jj)p Fo(T)1313 2480 y Fj(R)1367 2468 y Fp(\()p Fo(z)t Fp(\))p Fm(jj)c Fo(<)f Fp(1)28 b(and)f(the)h(pro)r(of)f(of)h(i\))g(is)f(\014nished.)648 2568 y(ii\).)56 b(First)34 b(of)g(all,)h(b)r(ecause)e Fo(H)1643 2580 y Fl(\000)1684 2588 y Fc(R)1770 2568 y Fp(comm)n(utes)g(with)i(the)f(magnetic)f(translations)g(it)523 2667 y(can)23 b(not)g(ha)n(v)n(e)g(discrete)g(sp)r(ectrum,)h(therefore) f(the)g(only)g(thing)h(w)n(e)f(should)g(c)n(hec)n(k)g(is)g(that)523 2767 y(its)34 b(sp)r(ectral)f(pro)5 b(jection)33 b(corresp)r(onding)e (to)j([)p Fo(c;)14 b(d)p Fp(])34 b(is)g(not)f(zero.)54 b(F)-7 b(or)33 b Fo(R)h Fm(\025)f Fo(R)q Fp(\()p Fo(\026)p Fp(\),)j(one)523 2866 y(has:)1272 2988 y Fo(P)1325 3000 y Fl(\000)1366 3008 y Fc(R)1418 2988 y Fp(\([)p Fo(c;)14 b(d)p Fp(]\))24 b(=)e Fm(\000)1868 2932 y Fp(1)p 1830 2969 119 4 v 1830 3045 a(2)p Fo(\031)s({)1972 2875 y Ff(Z)2018 3063 y Fk(C)2075 2988 y Fo(d\030)t Fp(\()p Fo(H)2259 3000 y Fl(\000)2300 3008 y Fc(R)2370 2988 y Fm(\000)c Fo(\030)t Fp(\))2525 2953 y Fk(\000)p Fl(1)2615 2988 y Fo(;)537 b Fp(\(3.43\))523 3180 y(and)27 b(using)h(the)g (expression)e(of)h Fo(S)1590 3192 y Fj(R)1645 3180 y Fp(\()p Fo(z)t Fp(\),)g(one)h(obtains)f(\()p Fo(R)h Fp(k)n(ept)g (su\016cien)n(tly)f(large\):)585 3373 y Fm(jj)p Fo(P)684 3385 y Fl(\000)725 3393 y Fc(R)777 3373 y Fp(\([)p Fo(c;)14 b(d)p Fp(]\))19 b Fm(\000)1107 3295 y Ff(X)1105 3473 y Fj(\015)t Fk(2)p Fl(\000)1243 3373 y Fo(g)1283 3385 y Fl(2)p Fj(;\015)t(;R)1462 3373 y Fp(exp)14 b(\()p Fo({')1716 3385 y Fj(\015)1759 3373 y Fp(\))p Fo(t)1821 3385 y Fj(\015)t(R)1914 3373 y Fo(P)1967 3385 y Fj(R)2021 3373 y Fo(t)2051 3385 y Fk(\000)p Fj(\015)t(R)2210 3373 y Fp(exp)g(\()p Fm(\000)p Fo({')2529 3385 y Fj(\015)2571 3373 y Fp(\))q Fo(g)2644 3385 y Fl(1)p Fj(;\015)t(;R)2809 3373 y Fm(jj)23 b Fo(<)f Fp(1)p Fo(=)p Fp(2)p Fo(:)61 b Fp(\(3.44\))523 3635 y(Here)29 b Fo(P)774 3647 y Fj(R)859 3635 y Fp(stands)g(for)g(the)h(one)f (dimensional)g(pro)5 b(jector)28 b(of)h Fo(H)2532 3647 y Fj(R)2616 3635 y Fp(whic)n(h)h(corresp)r(onds)e(to)523 3735 y Fo(E)584 3747 y Fl(1)622 3735 y Fp(\()p Fo(R)q Fp(\).)37 b(Denote)28 b(with)1302 3714 y(~)1284 3735 y Fo(P)12 b Fp(\()p Fo(R)q Fp(\))27 b(the)h(sum)g(whic)n(h)g(app)r (ears)e(in)i(\(3.44\):)655 3907 y(~)636 3928 y Fo(P)12 b Fp(\()p Fo(R)q Fp(\))24 b(=)e Fo(g)980 3940 y Fl(2)p Fj(;)p Fl(0)p Fj(;R)1140 3928 y Fo(P)1193 3940 y Fj(R)1248 3928 y Fo(g)1288 3940 y Fl(1)p Fj(;)p Fl(0)p Fj(;R)1466 3928 y Fp(+)1550 3849 y Ff(X)1549 4028 y Fj(\015)t Fk(6)p Fl(=0)1685 3928 y Fo(g)1725 3940 y Fl(2)p Fj(;\015)t(;R)1904 3928 y Fp(exp)14 b(\()p Fo({')2158 3940 y Fj(\015)2201 3928 y Fp(\))p Fo(t)2263 3940 y Fj(\015)t(R)2356 3928 y Fo(P)2409 3940 y Fj(R)2464 3928 y Fo(t)2494 3940 y Fk(\000)p Fj(\015)t(R)2652 3928 y Fp(exp)g(\()p Fm(\000)p Fo({')2971 3940 y Fj(\015)3013 3928 y Fp(\))q Fo(g)3086 3940 y Fl(1)p Fj(;\015)t(;R)3251 3928 y Fo(;)3175 4118 y Fp(\(3.45\))523 4218 y(where)34 b(w)n(e)g(used)g(the)g(fact)h(that)f Fo(')1655 4230 y Fl(0)1727 4218 y Fp(=)g(0)g(\(see)g(\(2.3\)\).)57 b(Denote)34 b(also)f(with)i Fo( )3072 4230 y Fl(1)3109 4218 y Fp(\()p Fo(R)q Fp(\))g(the)523 4317 y(normalised)29 b(eigenfunction)h(of)g Fo(H)1624 4329 y Fj(R)1678 4317 y Fp(.)44 b(If)30 b Fo(P)42 b Fp(\()p Fo( )2011 4329 y Fl(1)2049 4317 y Fp(\))30 b(denotes)f(the)i(pro)5 b(jector)28 b(\(eigen)n(v)n(ector\))523 4417 y(of)g Fo(H)34 b Fp(corresp)r(onding) 26 b(to)h Fo(E)1417 4429 y Fl(1)1455 4417 y Fp(,)g(then)i(w)n(e)e(kno)n (w)g(that)1620 4594 y(lim)1586 4647 y Fj(R)p Fk(!1)1783 4594 y Fm(jj)p Fo(P)1882 4606 y Fj(R)1955 4594 y Fm(\000)18 b Fo(P)12 b Fm(jj)23 b Fp(=)g(0)p Fo(:)850 b Fp(\(3.46\))523 4799 y(F)-7 b(rom)25 b(no)n(w)h(on,)g Fo(\017)f Fp(will)h(mean)g(a)f (generic,)g(in\014nitesimally)h(small)g(p)r(ositiv)n(e)f(quan)n(tit)n (y)-7 b(.)36 b(F)-7 b(or)523 4898 y Fo(R)586 4910 y Fl(0)649 4898 y Fo(>)26 b Fp(0,)j(denote)h(with)g Fo(\037)1347 4910 y Fl(0)1413 4898 y Fp(the)g(c)n(haracteristic)d(function)j(of)g Fm(fj)p Fn(x)p Fm(j)c(\025)f Fo(R)2811 4910 y Fl(0)2849 4898 y Fm(g)p Fp(;)30 b(pro)n(vided)e Fo(R)3350 4910 y Fl(0)1914 5147 y Fp(13)p eop %%Page: 14 14 14 13 bop 523 614 a Fp(su\016cien)n(tly)28 b(large,)e(one)h(has:)1619 792 y(sup)1589 861 y Fj(R)p Fk(\025)p Fj(R)1741 869 y Fd(0)1788 792 y Fm(jj)p Fo(P)1887 804 y Fj(R)1960 792 y Fm(\000)18 b Fo(P)12 b Fm(jj)23 b(\024)g Fo(\017:)853 b Fp(\(3.47\))523 1029 y(If)31 b Fo(U)9 b Fp(\()p Fo(R)q Fp(\))30 b(denotes)g(the)g(Nagy)f(unitary)h(whic)n(h)g(in)n(tert)n (wines)g Fo(P)2517 1041 y Fj(R)2602 1029 y Fp(and)g Fo(P)12 b Fp(,)30 b(then)h Fo( )3130 1041 y Fl(1)3167 1029 y Fp(\()p Fo(R)q Fp(\))d(=)523 1128 y Fo(U)9 b Fp(\()p Fo(R)q Fp(\))p Fo( )771 1140 y Fl(1)840 1128 y Fp(and)32 b(sup)1131 1148 y Fj(R)p Fk(\025)p Fj(R)1283 1156 y Fd(0)1333 1128 y Fm(jj)p Fo(U)9 b Fp(\()p Fo(R)q Fp(\))22 b Fm(\000)f Fp(1)p Fm(jj)29 b(\024)h Fo(\017)p Fp(.)49 b(Due)33 b(to)e(the)i(exp)r (onen)n(tial)e(lo)r(calisation)f(of)523 1228 y Fo( )577 1240 y Fl(1)642 1228 y Fp(near)d(the)h(origin,)e(one)h(has)h Fm(jj)p Fo(\037)1627 1240 y Fl(0)1664 1228 y Fo( )1718 1240 y Fl(1)1755 1228 y Fm(jj)23 b(\024)g Fo(\017)p Fp(,)k(therefore:) 1381 1405 y Fm(h)p Fo( )1467 1417 y Fl(1)1505 1405 y Fp(\()p Fo(R)q Fp(\))p Fo(;)14 b Fp(\(1)k Fm(\000)g Fo(\037)1897 1417 y Fl(0)1934 1405 y Fp(\))p Fo( )2020 1417 y Fl(1)2058 1405 y Fp(\()p Fo(R)q Fp(\))p Fm(i)24 b(\025)e Fp(1)c Fm(\000)g Fo(\017:)523 1582 y Fp(Moreo)n(v)n(er,)25 b(\(1)18 b Fm(\000)g Fo(\037)1138 1594 y Fl(0)1175 1582 y Fp(\))p Fo( )1261 1594 y Fl(1)1299 1582 y Fp(\()p Fo(R)q Fp(\))28 b(is)f(compactly)g(supp)r(orted.)37 b(It)28 b(follo)n(ws)f(that:)971 1759 y Fo( )1025 1771 y Fl(1)1062 1759 y Fp(\()p Fo(R)q Fp(\))d(=)e(\()p Fm(h)p Fo( )1419 1771 y Fl(1)1457 1759 y Fp(\()p Fo(R)q Fp(\))p Fo(;)14 b Fp(\(1)k Fm(\000)h Fo(\037)1850 1771 y Fl(0)1887 1759 y Fp(\))p Fo( )1973 1771 y Fl(1)2010 1759 y Fp(\()p Fo(R)q Fp(\))p Fm(i)p Fp(\))2202 1725 y Fk(\000)p Fl(1)2292 1759 y Fo(P)2345 1771 y Fj(R)2400 1759 y Fp(\(1)f Fm(\000)g Fo(\037)2627 1771 y Fl(0)2664 1759 y Fp(\))p Fo( )2750 1771 y Fl(1)2788 1759 y Fp(\()p Fo(R)q Fp(\))p Fo(:)236 b Fp(\(3.48\))523 1936 y(Therefore:)1397 2036 y(sup)1337 2110 y Fj(R)p Fk(\025)p Fj(R)p Fl(\()p Fj(\026)p Fl(\))1596 2036 y Fm(jj)14 b Fp(exp)g(\()p Fo(\026)p Fm(h\001i)p Fp(\))q Fo( )2053 2048 y Fl(1)2090 2036 y Fp(\()p Fo(R)q Fp(\))p Fm(jj)24 b(\024)e Fp(C\()p Fo(\026)p Fp(\))p Fo(;)603 b Fp(\(3.49\))523 2250 y(whic)n(h)29 b(means)f(that)h Fo( )1252 2262 y Fl(1)1290 2250 y Fp(\()p Fo(R)q Fp(\))g(has)f(an)h (uniform)f(exp)r(onen)n(tial)h(lo)r(calisation)e(in)i Fo(R)d Fm(\025)f Fo(R)q Fp(\()p Fo(\026)p Fp(\).)523 2349 y(Then)f Fm(jj)801 2328 y Fp(~)782 2349 y Fo(P)12 b Fp(\()p Fo(R)q Fp(\))p Fo( )1029 2361 y Fl(1)1066 2349 y Fp(\()p Fo(R)q Fp(\))p Fm(jj)24 b Fo(>)e Fp(1)p Fo(=)p Fp(2)h(for)g(su\016cien)n(tly)g(large)f Fo(R)q Fp(.)36 b(This)23 b(and)h(\(3.44\))f(\014nally)g(imply)523 2449 y(that)28 b Fm(jj)p Fo(P)802 2461 y Fl(\000)843 2469 y Fc(R)895 2449 y Fp(\([)p Fo(c;)14 b(d)p Fp(]\))p Fo( )1175 2461 y Fl(1)1213 2449 y Fp(\()p Fo(R)q Fp(\))p Fm(jj)23 b Fo(>)g Fp(0)k(and)g(the)h(pro)r(of)f(is)h(\014nished)p Fo(:)2472 2461 y Fa(2)648 2548 y Fg(R)l(emark)h(4)p Fp(:)648 2648 y(If)21 b Fo(E)785 2660 y Fl(1)844 2648 y Fp(is)g(degenerated)g (of)g(order)f Fo(n)p Fp(,)i(\(sa)n(y)f Fo(n)i Fp(=)f(2\),)h(then)f(w)n (e)f(will)g(sho)n(w)f(that)i(the)g(ab)r(o)n(v)n(e)523 2748 y(argumen)n(t)27 b(can)h(b)r(e)h(rep)r(eated)f(in)h(order)e(to)h (obtain)h(that)f(for)g Fo(R)d Fm(\025)f Fo(R)q Fp(\()p Fo(\026)p Fp(\))29 b(and)f Fo(k)g Fm(2)c(f)p Fp(1)p Fo(;)14 b Fp(2)p Fm(g)523 2847 y Fp(one)27 b(has:)1396 2947 y(sup)1336 3021 y Fj(R)p Fk(\025)p Fj(R)p Fl(\()p Fj(\026)p Fl(\))1594 2947 y Fm(jj)14 b Fp(exp)g(\()p Fo(\026)p Fm(h\001i)p Fp(\))q Fo( )2051 2959 y Fj(k)2092 2947 y Fp(\()p Fo(R)q Fp(\))p Fm(jj)23 b(\024)g Fp(C\()p Fo(\026)p Fp(\))p Fo(:)601 b Fp(\(3.50\))523 3160 y(First)28 b(of)f(all,)g Fo( )1012 3172 y Fl(1)p Fj(;)p Fl(2)1103 3160 y Fp(\()p Fo(R)q Fp(\))h(are)e(normalised)h(linear)f(com)n(binations)h(of)g Fo(U)9 b Fp(\()p Fo(R)q Fp(\))p Fo( )2896 3172 y Fl(1)p Fj(;)p Fl(2)2987 3160 y Fp(,)27 b(therefore:)1575 3338 y(sup)1545 3407 y Fj(R)p Fk(\025)p Fj(R)1697 3415 y Fd(0)1744 3338 y Fm(jj)p Fo(\037)1842 3350 y Fl(0)1879 3338 y Fo( )1933 3350 y Fl(1)p Fj(;)p Fl(2)2023 3338 y Fp(\()p Fo(R)q Fp(\))p Fm(jj)d(\024)e Fo(\017:)810 b Fp(\(3.51\))523 3575 y(The)27 b(second)f(step)h(is)g(to)g(express)f Fo( )1662 3587 y Fl(1)p Fj(;)p Fl(2)1752 3575 y Fp(\()p Fo(R)q Fp(\))h(with)g(the)h(help)f(of)g Fo(P)2564 3587 y Fj(R)2618 3575 y Fp(\(1)17 b Fm(\000)g Fo(\037)2843 3587 y Fl(0)2880 3575 y Fp(\))p Fo( )2966 3587 y Fl(1)p Fj(;)p Fl(2)3056 3575 y Fp(\()p Fo(R)q Fp(\).)37 b(The)523 3674 y(relation)27 b(b)r(et)n(w)n(een)g(them)h(is)g(giv)n(en)f(b)n(y:)896 3851 y Fo(P)949 3863 y Fj(R)1004 3851 y Fp(\(1)18 b Fm(\000)g Fo(\037)1231 3863 y Fl(0)1268 3851 y Fp(\))p Fo( )1354 3863 y Fl(1)p Fj(;)p Fl(2)1444 3851 y Fp(\()p Fo(R)q Fp(\))84 b(=)e Fo( )1857 3863 y Fl(1)1895 3851 y Fp(\()p Fo(R)q Fp(\))p Fm(h)p Fo( )2109 3863 y Fl(1)2146 3851 y Fp(\()p Fo(R)q Fp(\))p Fo(;)14 b Fp(\(1)19 b Fm(\000)f Fo(\037)2539 3863 y Fl(0)2576 3851 y Fp(\))p Fo( )2662 3863 y Fl(1)p Fj(;)p Fl(2)2753 3851 y Fp(\()p Fo(R)q Fp(\))p Fm(i)h Fp(+)1656 3976 y(+)82 b Fo( )1857 3988 y Fl(2)1895 3976 y Fp(\()p Fo(R)q Fp(\))p Fm(h)p Fo( )2109 3988 y Fl(2)2146 3976 y Fp(\()p Fo(R)q Fp(\))p Fo(;)14 b Fp(\(1)19 b Fm(\000)f Fo(\037)2539 3988 y Fl(0)2576 3976 y Fp(\))p Fo( )2662 3988 y Fl(1)p Fj(;)p Fl(2)2753 3976 y Fp(\()p Fo(R)q Fp(\))p Fm(i)p Fo(:)239 b Fp(\(3.52\))523 4153 y(Using)27 b(\(3.51\),)g(one)h(can)f(in)n(v)n(ert)g(the)g(ab)r(o)n (v)n(e)g(matrix)g(and)g(\(3.50\))g(follo)n(ws)g(easily)-7 b(.)648 4253 y(Then)28 b Fo(H)934 4265 y Fj(R)1016 4253 y Fp(has)g(t)n(w)n(o)f(eigen)n(v)-5 b(alues)27 b Fo(E)1813 4265 y Fl(1)1851 4253 y Fp(\()p Fo(R)q Fp(\))h(and)g Fo(E)2230 4265 y Fl(2)2268 4253 y Fp(\()p Fo(R)q Fp(\))g(near)f Fo(E)2670 4265 y Fl(1)2708 4253 y Fp(;)h(lo)r(oking)f(at)h(\(3.41\),) 523 4352 y(one)f(realizes)g(that)g(the)h(only)g(thing)f(that)h(c)n (hanges)e(is:)1003 4573 y Fo(P)1056 4585 y Fj(R)1111 4573 y Fp(\()p Fo(H)1212 4585 y Fj(R)1285 4573 y Fm(\000)18 b Fo(z)t Fp(\))1443 4539 y Fk(\000)p Fl(1)1555 4573 y Fp(=)1653 4517 y Fm(j)p Fo( )1730 4529 y Fl(1)1767 4517 y Fp(\()p Fo(R)q Fp(\))p Fm(ih)p Fo( )2013 4529 y Fl(1)2051 4517 y Fp(\()p Fo(R)q Fp(\))p Fm(j)p 1653 4554 550 4 v 1742 4630 a Fo(E)1803 4642 y Fl(1)1841 4630 y Fp(\()p Fo(R)q Fp(\))h Fm(\000)f Fo(z)2231 4573 y Fp(+)2324 4517 y Fm(j)p Fo( )2401 4529 y Fl(2)2438 4517 y Fp(\()p Fo(R)q Fp(\))p Fm(ih)p Fo( )2684 4529 y Fl(2)2723 4517 y Fp(\()p Fo(R)q Fp(\))p Fm(j)p 2324 4554 V 2414 4630 a Fo(E)2475 4642 y Fl(2)2512 4630 y Fp(\()p Fo(R)q Fp(\))h Fm(\000)f Fo(z)2884 4573 y(:)268 b Fp(\(3.53\))523 4799 y(Using)33 b(\(3.50\),)h(one)f(concludes)f(that)i(the)f(sp)r(ectrum)h(of)f Fo(H)2441 4811 y Fl(\000)2482 4819 y Fc(R)2566 4799 y Fp(in)h(that)f(region)f(consists)523 4898 y(from)21 b(t)n(w)n(o)g (mini-bands)h(\(of)g(the)g(same)f(t)n(yp)r(e)h(as)f(in)h(i\)\))g(lo)r (calised)f(near)g Fo(E)2794 4910 y Fl(1)2832 4898 y Fp(\()p Fo(R)q Fp(\))h(and)f Fo(E)3198 4910 y Fl(2)3236 4898 y Fp(\()p Fo(R)q Fp(\).)1914 5147 y(14)p eop %%Page: 15 15 15 14 bop 648 614 a Fn(Ac)m(kno)m(wledgemen)m(ts.)648 714 y Fp(This)32 b(pap)r(er)g(w)n(as)g(conceiv)n(ed)g(during)g(a)g (visit)h(to)g(Erwin)e(Sc)n(hro)r(dinger)h(Institute)h(in)523 814 y(Vienna)41 b(at)g(the)h(in)n(vitation)f(of)g(Professor)d(T.)k (Ho\013mann-Ostenhof.)77 b(The)41 b(\014nancial)523 913 y(supp)r(ort)27 b(is)h(hereb)n(y)f(gratefully)g(ac)n(kno)n(wledged.)648 1013 y(W)-7 b(e)31 b(also)f(thank)h(the)h(referee)e(for)h(v)-5 b(aluable)31 b(suggestions)e(and)i(for)g(dra)n(wing)e(our)i(at-)523 1112 y(ten)n(tion)d(to)n(w)n(ard)e(Daumer's)h(w)n(ork)f([5].)523 1387 y Fq(References)523 1569 y Fp([1])290 b(Briet,)26 b(P)-7 b(.,)25 b(Com)n(b)r(es,)g(J.)g(M.,)h(Duclos,)f(P)-7 b(.:)35 b(Sp)r(ectral)25 b(stabilit)n(y)g(under)g(tunnel-)901 1669 y(ing.)j(Comm)n(un.)f(Math.)h(Ph)n(ys.)f Fn(126)f Fp(\(1989\),)h(133.)523 1835 y([2])290 b(Cycon,)34 b(H.L.,)g(F)-7 b(ro)r(ese,)33 b(R.G.,)i(Kirsc)n(h,)e(W.,)h(Simon,)g(B.:)47 b(Sc)n(hr\177)-42 b(odinger)31 b(op-)901 1934 y(erators)24 b(with)j(application)e(to)h(quan)n(tum)g(mec)n(hanics)f(and)h(global)f (geometry)-7 b(,)901 2034 y(Berlin,)27 b(Heidelb)r(erg,)h(New)g(Y)-7 b(ork,)27 b(Springer-V)-7 b(erlag,)25 b(1987.)523 2200 y([3])290 b(Cornean,)36 b(H.D.,)h(Nenciu,)h(G.:)52 b(On)34 b(eigenfunction)h(deca)n(y)f(of)h(t)n(w)n(o)f(dimen-)901 2299 y(sional)g(magnetic)h(Sc)n(hr\177)-42 b(odinger)33 b(op)r(erators.)g(Comm)n(un.)i(Math.)g(Ph)n(ys.)f Fn(192)901 2399 y Fp(\(1998\),)27 b(671.)523 2565 y([4])290 b(Cornean,)31 b(H.D.,)i(Nenciu,)g(G.:)44 b(Tw)n(o)31 b(dimensional)f(magnetic)h(Sc)n (hr\177)-42 b(odinger)901 2665 y(op)r(erators:)43 b(width)33 b(of)f(mini-bands)f(in)h(the)g(tigh)n(t)g(binding)g(appro)n(ximation.) 901 2764 y(Preprin)n(t)40 b(E.S.I.)g Fo(n)1538 2734 y Fj(o)1616 2764 y Fp(608,)i(Vienna,)j(\(1998\).)39 b(T)-7 b(o)40 b(app)r(ear)g(in)h(Ann.)g(I.H.P)-7 b(.)901 2864 y(\(Ph)n(ysique)27 b(Th)n(\023)-39 b(eorique\))523 3030 y([5])290 b(Daumer,)71 b(F.:)108 b(Equation)61 b(de)i(Sc)n(hr\177)-42 b(odinger)60 b(a)n(v)n(ec)i(c)n(hamp)e(\023)-39 b(el)n(\023)g(ectrique) 901 3130 y(p)n(\023)g(erio)r(dique)34 b(et)h(c)n(hamp)g(magn)n(\023)-39 b(etique)33 b(constan)n(t)h(dans)h(l'appro)n(ximation)e(du)901 3229 y(tigh)n(t-binding.)28 b(Comm.)f(in)h(P)-7 b(.D.E.)28 b Fn(18)f Fo(n)2252 3199 y Fj(o)2316 3229 y Fp(5,)g(6)h(\(1993\),)e (1021.)523 3395 y([6])290 b(Hel\013er,)31 b(B.:)41 b(On)30 b(sp)r(ectral)f(theory)h(for)f(Sc)n(hr\177)-42 b(odinger)28 b(op)r(erators)g(with)i(mag-)901 3495 y(netic)f(p)r(oten)n(tials.)g (Adv)-5 b(anced)29 b(Studies)g(in)h(Pure)e(Mathematics.)g Fn(23)h Fp(\(1994\),)901 3595 y(113.)523 3761 y([7])290 b(Hemp)r(el,)37 b(R.,)g(Herbst)d(I.:)51 b(Strong)33 b(magnetic)h (\014eds,)j(Diric)n(hlet)d(b)r(oundaries)901 3860 y(and)28 b(sp)r(ectral)f(gaps.)f(Preprin)n(t)h(E.S.I.)g Fo(n)2198 3830 y Fj(o)2263 3860 y Fp(74,)f(Vienna,)i(\(1994\).)523 4026 y([8])290 b(Hel\013er,)28 b(B.,)g(Sj\177)-42 b(ostrand,)27 b(J.:)37 b(Multiple)28 b(w)n(ells)f(in)h(the)g(semi-classical)e(limit)i Fn(I)p Fp(.)901 4126 y(Comm.)g(in)g(P)-7 b(.D.E.)27 b Fn(9)g Fp(\(1984\),)g(337.)523 4292 y([9])290 b(Hel\013er,)43 b(B.,)f(Sj\177)-42 b(ostrand,)41 b(J.:)60 b(Puits)39 b(m)n(ultiples)h(en)f(limite)h(semi-classique)901 4392 y Fn(I)s(I)p Fp(.)25 b(In)n(teraction)f(mol)n(\023)-39 b(eculaire.)23 b(Sym)n(\023)-39 b(etries.)24 b(P)n(erturbations.)f (Ann.)j(I.H.P)-7 b(.)25 b Fn(42)901 4491 y Fp(\(1985\),)i(127.)523 4657 y([10])248 b(Hel\013er,)38 b(B.,)g(Sj\177)-42 b(ostrand,)37 b(J.:)53 b(Multiple)37 b(w)n(ells)e(in)h(the)g(semi-classical)e(limit) 901 4757 y Fn(I)s(I)s(I)p Fp(.)28 b(Math.)g(Nac)n(hr.)e Fn(124)h Fp(\(1985\),)g(263.)1914 5147 y(15)p eop %%Page: 16 16 16 15 bop 523 614 a Fp([11])248 b(Hel\013er,)24 b(B.,)g(Sj\177)-42 b(ostrand,)23 b(J.:)35 b(E\013et)23 b(tunnel)g(p)r(our)g(l')n(\023)-39 b(equation)21 b(de)i(Sc)n(hr\177)-42 b(odinger)901 714 y(a)n(v)n(ec)26 b(c)n(hamp)i(magn)n(\023)-39 b(etique.)26 b(Ann.)i(l')n(\023)-39 b(ecole)26 b(de)i(Pise.)f Fn(14)g Fp(\(1987\),)f(625.)523 880 y([12])248 b(Hel\013er,)53 b(B.,)g(Sj\177)-42 b(ostrand,)53 b(J.:)77 b(Equation)47 b(de)h(Sc)n(hr\177)-42 b(odinger)46 b(a)n(v)n(ec)h(c)n(hamp)901 980 y(magn)n(\023)-39 b(etique)38 b(et)g(\023)-39 b(equation)38 b(de)i(Harp)r(er,)i(Lect.)d(notes)h(in)f(Ph)n(ysics,)j Fo(n)3205 950 y Fj(o)3281 980 y Fp(34,)901 1079 y(Springer)27 b(V)-7 b(erlag,)26 b(118-197,)f(1988)523 1245 y([13])248 b(Iw)n(atsuk)-5 b(a,)24 b(A.:)35 b(The)23 b(essen)n(tial)g(sp)r(ectrum) g(of)h(t)n(w)n(o)e(dimensional)h(Sc)n(hr\177)-42 b(odinger)901 1345 y(op)r(erators)30 b(with)j(p)r(erturb)r(ed)f(magnetic)g(\014elds.) g(J.)g(Math.)g(Ky)n(oto)e(Univ.)j Fn(23)901 1445 y Fp(\(1983\),)27 b(475.)523 1611 y([14])248 b(Nak)-5 b(am)n(ura,)25 b(S.:)36 b(Band)26 b(sp)r(ectrum)g(for)f(Sc)n(hr\177)-42 b(odinger)24 b(op)r(erators)f(with)k(strong)901 1710 y(p)r(erio)r(dic)35 b(magnetic)g(\014elds.)g(Op)r(erator)f(Theory:)51 b(Adv)-5 b(ances)35 b(and)g(Applica-)901 1810 y(tions)28 b Fn(78)p Fp(,)f(\(1995\),)f(261.)523 1976 y([15])248 b(Nenciu,)44 b(G.:)63 b(Dynamics)40 b(of)g(band)h(electrons)e(in)i(electric)f(and)g (magnetic)901 2076 y(\014elds:)d(rigorous)24 b(justi\014cation)j(of)g (the)g(e\013ectiv)n(e)g(Hamiltonians.)f(Rev.)h(Mo)r(d.)901 2175 y(Ph)n(ys.)g Fn(63)g Fp(\(1991\),)f(91.)523 2341 y([16])248 b(Simon,)32 b(B.:)44 b(Sc)n(hr\177)-42 b(odinger)29 b(semigroups.)h(Bull.)h(Amer.)g(Math.)g(So)r(c.)g(\(N.)h(S.\))901 2441 y Fn(7)c Fp(\(1982\),)e(447.)523 2607 y([17])248 b(Thaller,)50 b(B.:)72 b(The)46 b(Dirac)f(equation,)50 b(Berlin,)f(Heidelb)r(erg,)h(New)c(Y)-7 b(ork,)901 2707 y(Springer-V)g(erlag,)25 b(1992)1914 5147 y(16)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9908280430917--