Content-Type: multipart/mixed; boundary="-------------0201031058829" This is a multi-part message in MIME format. ---------------0201031058829 Content-Type: text/plain; name="02-2.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-2.comments" accepted for publication in CMP ---------------0201031058829 Content-Type: text/plain; name="02-2.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-2.keywords" Harper model, anomalous transport ---------------0201031058829 Content-Type: application/postscript; name="harper17.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="harper17.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: harper17.dvi %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips harper17 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 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b([Her])f(\(namely)-8 b(,)27 b(for)f(an)m(y)h Fw(\017)h(>)g Fx(0,)f(there)g(is)f Fw(c)h(>)h Fx(0)e(suc)m(h)i(that)e Fr(j)p Fw(\022)12 b Fr(\000)d Fw(p=q)t Fr(j)28 b(\025)g Fw(c=q)3068 1403 y FB(2+)p Fv(\017)3217 1439 y Fx(for)e(all)e Fw(p=q)31 b Fr(2)d Fs(Q)12 b Fx(\),)-118 1560 y(and)33 b(for)f(the)h(class)f(of)g(mo)s (dels)g(men)m(tioned)g(ab)s(o)m(v)m(e,)i(the)f(follo)m(wing)c (inequalit)m(y)j(holds)g(for)g(all)f(0)c Fw(<)h(q)j(<)d Fx(1)1441 1753 y Fw(\014)6 b Fx(\()p Fw(q)t Fx(\))43 b Fr(\025)i Fw(D)1871 1768 y Fu(N)1943 1753 y Fx(\(1)22 b Fr(\000)g Fw(q)t Fx(\))32 b(.)1478 b(\(1\))-118 1947 y(This)33 b(result)f(can)h(b)s(e)g(reform)m(ulated)e(in)h(terms)h(of)f (t)m(w)m(o-dimensional)e(magnetic)h(op)s(erators)i(on)f(the)h(lattice) -118 2067 y(and)e(then)g(giv)m(es)h(an)e(impro)m(v)m(emen)m(t)h(of)f (the)i(general)e(Guarneri-Com)m(b)s(es-Last)f(lo)m(w)m(er)i(b)s(ound)h ([Gua)o(,)f(Com,)-118 2187 y(Las])i(b)m(y)g(a)f(factor)h(2.)43 b(More)33 b(precise)g(de\014nitions)f(and)h(statemen)m(ts)g(will)e(b)s (e)h(giv)m(en)h(in)f(Section)g(2.)28 2355 y(The)k(inequalit)m(y)d (\(1\))h(has)h(b)s(een)g(motiv)-5 b(ated)33 b(b)m(y)i(w)m(ork)g(b)m(y)g (Pi)m(\023)-46 b(ec)m(hon)35 b([Pie],)g(who)g(ga)m(v)m(e)g(heuristic)f (argu-)-118 2475 y(men)m(ts)g(and)g(n)m(umerical)e(supp)s(ort)i(for)g Fw(\014)6 b Fx(\()p Fw(q)t Fx(\))28 b(=)i Fw(D)1719 2490 y Fu(N)1791 2475 y Fx(\(1)22 b Fr(\000)i Fw(q)t Fx(\))33 b(for)h Fw(q)f(>)d Fx(0,)j(v)-5 b(alid)32 b(for)h(the)i(Harp)s(er)e(mo) s(del)f(and)-118 2596 y(the)g(Fib)s(onacci)e(c)m(hain)i(\(for)f(the)h (latter)f(case,)i(a)e(p)s(erturbativ)m(e)h(argumen)m(t)g(w)m(as)g(also) f(giv)m(en\).)43 b(It)32 b(w)m(as)h(theo-)-118 2716 y(retically)26 b(and)i(n)m(umerically)d(demonstrated)k(b)m(y)f(Man)m(tica)g([Man])g (that)f(the)i(same)e Fm(exact)h Fx(relation)e(b)s(et)m(w)m(een)-118 2837 y(sp)s(ectral)36 b(and)g(transp)s(ort)h(exp)s(onen)m(ts)h(is)e (also)f(v)-5 b(alid)35 b(for)g(the)i(Jacobi)f(matrices)f(asso)s(ciated) h(with)g(a)g(Julia)-118 2957 y(set.)45 b(This)33 b(result)g(w)m(as)h (rigorously)d(pro)m(v)m(en)j(in)e([GSB1,)h(BSB].)45 b(F)-8 b(or)32 b(the)h(latter)f(op)s(erators,)h(the)g(DOS)f(and)-118 3077 y(the)h(lo)s(cal)d(densit)m(y)k(of)e(states)h(\(LDOS\))f (coincide.)28 3245 y(Numerous)g(w)m(orks)h([Gua,)f(Com)o(,)g(Las,)g (GSB2,)g(GSB3)o(,)g(BGT])g(yield)f(lo)m(w)m(er)h(b)s(ounds)g(on)g(the)g (quan)m(tum)-118 3365 y(di\013usion)g(of)g(a)g(giv)m(en)h(w)m(a)m(v)m (e)i(pac)m(k)m(et)f(in)e(terms)h(of)f(the)i(fractal)d(prop)s(erties)i (of)f(the)h(corresp)s(onding)g(LDOS.)-118 3486 y(These)50 b(rigorous)e(lo)m(w)m(er)h(b)s(ounds)g(are)g(t)m(ypically)f(not)g (optimal)e(as)j(sho)m(wn)h(b)m(y)g(n)m(umerical)d(sim)m(ulations)-118 3606 y([GM,)30 b(KKK)m(G].)42 b(Better)31 b(lo)m(w)m(er)g(b)s(ounds)g (are)f(obtained)g(if)f(the)i(b)s(eha)m(viour)f(of)g(generalized)g (eigenfunctions)-118 3726 y(is)g(tak)m(en)i(in)m(to)e(accoun)m(t)h ([KKK)m(G].)43 b(Kiselev)30 b(and)h(Last)g(ha)m(v)m(e)h(pro)m(v)m(en)g (general)f(rigorous)e(b)s(ounds)j(in)e(terms)-118 3847 y(of)i(upp)s(er)h(b)s(ounds)g(for)f(the)h(algebraic)e(deca)m(y)j(of)e (the)h(eigenfunctions)g([KL].)28 4014 y(Ho)m(w)m(ev)m(er,)e(in)26 b(most)h(mo)s(dels)f(used)j(in)d(solid)g(state)h(ph)m(ysics,)j(the)e (Hamiltonian)c(is)i(a)h(co)m(v)-5 b(arian)m(t)27 b(strongly)-118 4135 y(con)m(tin)m(uous)45 b(family)d(of)i(self-adjoin)m(t)f(op)s (erators)h([Bel)o(])h(indexed)g(b)m(y)h(a)e(v)-5 b(ariable)42 b(whic)m(h)j(represen)m(ts)i(the)-118 4255 y(phase)41 b(or)f(the)g(con\014guration)g(of)g(disorder.)66 b(The)41 b(measure)f(class)h(of)f(the)g(singular)f(part)h(of)f(the)i(LDOS)-118 4376 y(ma)m(y)g(sensitiv)m(ely)g(dep)s(end)i(on)e(the)g(phase)i([DS)o (].)70 b(In)41 b(addition,)h(the)g(m)m(ultifractal)37 b(dimensions)k(are)g(not)-118 4496 y(ev)m(en)29 b(measure)e(class)h(in) m(v)-5 b(arian)m(ts)26 b([SBB)q(])h(\(unlik)m(e)g(the)h(Hausdor\013)f (and)h(pac)m(king)f(dimensions\).)41 b(This)28 b(raises)-118 4616 y(concerns)g(ab)s(out)e(the)g(practical)f(relev)-5 b(ance)27 b(of)f(b)s(ounds)h(based)g(on)f(m)m(ultifractal)d(dimensions) j(of)f(the)i(LDOS)-118 4737 y(in)38 b(this)i(con)m(text.)65 b(The)40 b(b)s(ound)g(\(1\))f(has)g(a)g(threefold)g(adv)-5 b(an)m(tage:)58 b(\(i\))38 b(it)g(in)m(v)m(olv)m(es)i(the)g(DOS,)f (whic)m(h)h(is)-118 4857 y(phase-a)m(v)m(eraged;)49 b(\(ii\))40 b(it)h(do)s(es)h(not)g(require)h(information)c(ab)s(out)j (eigenfunctions;)k(\(iii\))40 b(the)i(exp)s(onen)m(t)-118 4978 y(of)36 b(phase-a)m(v)m(eraged)i(transp)s(ort)e(is)g(the)h(one)f (that)g(determines)h(the)g(lo)m(w)f(temp)s(erature)g(b)s(eha)m(viour)g (of)g(the)-118 5098 y(conductivit)m(y)d([SBB].)28 5266 y(The)g(presen)m(t)h(form)m(ulation)29 b(uses)34 b(the)e(C)1536 5229 y Fu(\003)1576 5266 y Fx(-algebraic)e(framew)m(ork)i(in)m(tro)s (duced)g(b)m(y)h(one)f(of)g(the)g(authors)-118 5386 y(for)j(the)h (study)g(of)f(homogeneous)h(mo)s(dels)e(of)h(solid)f(state)i(ph)m (ysics.)54 b(While)34 b(referring)h(to)g([Bel,)g(SBB)q(])g(for)1865 5635 y(2)p eop %%Page: 3 3 3 2 bop -118 22 a Fx(motiv)-5 b(ations)35 b(and)j(details,)g(in)g(the)g (op)s(ening)f(Section)h(2)f(w)m(e)i(brie\015y)f(recall)f(some)h(of)f (the)h(basic)g(notions.)-118 142 y(A)32 b(precise)h(statemen)m(t)g(of)f (our)g(main)f(results)i(is)f(also)f(giv)m(en)i(in)f(Section)g(2,)g (along)f(with)h(an)g(outline)f(of)h(the)-118 263 y(logical)26 b(structure)k(of)e(their)h(pro)s(ofs.)42 b(In)29 b(the)g(subsequen)m(t) j(sections)e(w)m(e)g(presen)m(t)g(more)f(results)g(and)g(pro)s(ofs.) -118 430 y Fl(Ac)m(kno)m(wledgemen)m(ts:)39 b Fx(W)-8 b(e)28 b(w)m(ould)f(lik)m(e)f(to)h(thank)g(B.)g(Simon,)g(R.)g(Seiler)f (and)h(S.)g(Jitormisk)-5 b(a)m(y)m(a)26 b(for)g(v)m(ery)-118 551 y(useful)37 b(commen)m(ts.)58 b(The)38 b(w)m(ork)g(of)f(H.)g(S.-B.) g(w)m(as)h(supp)s(orted)g(b)m(y)h(NSF)e(Gran)m(t)g(DMS-0070755)e(and)i (the)-118 671 y(DF)m(G)30 b(Gran)m(t)h(SCHU)i(1358/1-1.)41 b(J.B.)31 b(w)m(an)m(ts)i(to)e(thank)h(the)g(Institut)g(Univ)m (ersitaire)e(de)i(F)-8 b(rance)32 b(and)f(the)-118 792 y(MSRI)i(at)f(Berk)m(eley)i(for)e(pro)m(viding)g(supp)s(ort)h(while)e (this)i(w)m(ork)g(w)m(as)h(in)d(progress.)-118 1242 y Fy(2)161 b(Notations)54 b(and)f(results)-118 1461 y Fx(A)35 b(n)m(um)m(b)s(er)g Fw(\013)e Fr(2)f Fs(R)46 b Fx(is)35 b(of)f(Roth)h(t)m(yp)s(e)h(if)e(and)h(only)g(if,)f(for)h(an)m(y)h Fw(\017)c(>)g Fx(0,)j(there)h(is)e(a)h(constan)m(t)h Fw(c)3454 1476 y Fv(\017)3519 1461 y Fw(>)31 b Fx(0)k(suc)m(h)-118 1582 y(that)d(for)g(all)f(rational)f(n)m(um)m(b)s(er)j Fw(p=q)j Fx(the)d(follo)m(wing)d(inequalit)m(y)h(holds)1514 1808 y Fq(\014)1514 1868 y(\014)1514 1928 y(\014)1514 1987 y(\014)1547 1953 y Fw(\013)23 b Fr(\000)1741 1885 y Fw(p)p 1741 1930 49 4 v 1742 2021 a(q)1800 1808 y Fq(\014)1800 1868 y(\014)1800 1928 y(\014)1800 1987 y(\014)1889 1953 y Fr(\025)2079 1885 y Fw(c)2121 1900 y Fv(\017)p 2032 1930 170 4 v 2032 2021 a Fw(q)2079 1992 y FB(2+)p Fv(\017)2239 1953 y Fw(:)1507 b Fx(\(2\))-118 2228 y(Most)25 b(prop)s(erties)g(of)f (n)m(um)m(b)s(ers)i(of)e(Roth)g(t)m(yp)s(e)i(can)f(b)s(e)g(found)g(in)f ([Her].)41 b(They)27 b(form)c(a)i(set)g(of)f(full)f(Leb)s(esgue)-118 2348 y(measure)34 b(con)m(taining)e(all)f(algebraic)g(n)m(um)m(b)s(ers) k(\(Roth's)e(theorem\).)46 b Fw(\022)32 b(>)d Fx(0)k(will)e(b)s(e)i (called)f(a)h(Roth)g(angle)-118 2469 y(if)e Fw(\022)s(=)p Fx(2)p Fw(\031)36 b Fx(is)d(a)f(n)m(um)m(b)s(er)h(of)f(Roth)g(t)m(yp)s (e.)28 2636 y(The)k Fm(r)-5 b(otation)36 b(algebr)-5 b(a)34 b Fr(A)1012 2651 y Fv(\022)1085 2636 y Fx([Rie])g(is)g(the)h (smallest)e Fw(C)2040 2600 y Fu(\003)2080 2636 y Fx(-algebra)g (generated)i(b)m(y)g(t)m(w)m(o)h(unitaries)d Fw(U)45 b Fx(and)-118 2757 y Fw(V)21 b Fx(,)49 b(suc)m(h)f(that)d Fw(U)10 b(V)72 b Fx(=)50 b Fw(e)870 2721 y Fv({)12 b(\022)961 2757 y Fw(V)22 b(U)10 b Fx(.)83 b(It)46 b(is)f(con)m(v)m(enien)m(t)j (to)d(set)h Fw(W)2341 2772 y Fv(\022)2380 2757 y Fx(\()p Fl(m)p Fx(\))k(=)g Fw(e)2771 2721 y 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b(and)g(\(9\).)51 b(This)36 b(crucial)e(impro)m(v)m(emen)m(t)h(follo)m(ws)f(from)g(an)h (almost-sure)f(estimate)h(on)g(the)g(gro)m(wth)h(of)-118 3605 y(the)i(generalized)g(eigenfunctions)g(in)f(the)i(W)-8 b(eyl)38 b(represen)m(tation)h(\()p Fi(cf.)60 b Fx(Prop)s(osition)37 b(4)h(b)s(elo)m(w\))g(whic)m(h)g(in)-118 3725 y(turn)33 b(follo)m(ws)f(from)g(n)m(um)m(b)s(er-theoretic)i(estimates.)45 b(As)34 b(in)e([KL],)i(a)f(con)m(trol)f(on)i(the)f(asymptotics)g(of)g (the)-118 3845 y(generalized)g(eigenfunctions)h(then)g(leads)g(to)f(an) h(impro)m(v)m(ed)f(lo)m(w)m(er)h(b)s(ound)g(on)g(the)g(di\013usion)e (co)s(e\016cien)m(ts)-118 3966 y(\(here)h(b)m(y)h(a)e(factor)g(2)g(at)h Fw(q)e Fx(=)d(0\).)-118 4194 y Fl(Remark)37 b(2)49 b Fx(The)32 b(b)s(ound)g(\(8\))f(is)g(of)g(practical)f(in)m(terest)i(esp) s(ecially)f(if)f Fw(H)39 b Fx(is)31 b(in)m(v)-5 b(arian)m(t)30 b(under)i(some)g(sym-)-118 4315 y(metry)-8 b(.)63 b(Non-symmetric)38 b(Hamiltonians)e(ma)m(y)j(lead)f(to)h(ballistic)d(motion)h(and)i (absolutely)g(con)m(tin)m(uous)-118 4435 y(sp)s(ectral)28 b(measures)h(\(as)f(it)g(is)f(generically)g(the)i(case)g(for)f(the)h (non-critical)c(Harp)s(er)j(Hamiltonian,)e(see)j([Jit])-118 4555 y(and)34 b(references)h(therein\).)46 b(In)34 b(this)f(situation,) g(the)h(b)s(ound)f(b)s(ecomes)h(trivial)d(b)s(ecause)k(b)s(oth)e(sides) h(in)f(\(9\))-118 4676 y(are)f(equal)h(to)f(1.)-118 4904 y Fl(Remark)37 b(3)49 b Fx(Numerical)39 b(results)i([TK)q(,)g(RP])g(as) g(w)m(ell)f(as)h(the)h(Thouless)g(prop)s(ert)m(y)f([RP])g(supp)s(ort)h (that)-118 5024 y Fw(D)-37 5039 y Fu(N)35 5024 y Fx(\()p Fr(\000)p Fx(1\))37 b(=)397 4985 y FB(1)p 397 5001 V 397 5059 a(2)480 5024 y Fx(in)h(the)g(case)h(of)f(the)g(critical)e (Harp)s(er)i(Hamiltonian)c Fw(H)2589 5039 y FB(4)2667 5024 y Fx(for)j(Diophan)m(tine)g Fw(\022)s(=)p Fx(\(2)p Fw(\031)t Fx(\).)60 b(Ac-)-118 5145 y(cording)32 b(to)g(\(9\),)g(one)h (th)m(us)h(exp)s(ects)g Fw(\014)1328 5160 y Fn(1D)1408 5145 y Fx(\()p Fw(H)1527 5160 y FB(4)1567 5145 y Fw(;)17 b Fs(R)t Fx(;)g(2\))33 b Fr(\025)1955 5106 y FB(1)p 1955 5122 V 1955 5179 a(2)2001 5145 y Fx(.)1865 5635 y(5)p eop %%Page: 6 6 6 5 bop -118 28 a Fl(Remark)37 b(4)49 b Fx(Numerical)33 b(sim)m(ulations)g(b)m(y)j(Pi)m(\023)-46 b(ec)m(hon)36 b([Pie])f(for)g(the)g(Harp)s(er)g(mo)s(del)f(with)h(some)g(strongly) -118 148 y(incommensurate)28 b Fw(\022)s(=)p Fx(\(2)p Fw(\031)t Fx(\))g(indicate)g(that)g(\(9\))g(ma)m(y)h(actually)e(b)s(e)i (an)g(exact)g(estimate.)42 b(Piec)m(hon)29 b(also)f(ga)m(v)m(e)-118 268 y(a)43 b(p)s(erturbativ)m(e)h(argumen)m(t)f(supp)s(orting)g(the)h (equalit)m(y)f Fw(\014)2111 283 y Fn(1D)2191 268 y Fx(\()p Fw(H)8 b Fx(;)17 b Fw(q)t Fx(\))46 b(=)g Fw(D)2696 283 y Fu(N)2768 268 y Fx(\(1)29 b Fr(\000)h Fw(q)t Fx(\))43 b(in)g(the)h(case)g(of)f(the)-118 389 y(Fib)s(onacci)38 b(Hamiltonian,)f(and)j(v)m(eri\014ed)h(it)d(n)m(umerically)-8 b(.)63 b(The)41 b(tec)m(hniques)g(of)e(the)h(presen)m(t)i(article)c(do) -118 509 y(not)32 b(apply)h(to)f(the)h(Fib)s(onacci)e(mo)s(del)f(whic)m (h)k(has)f(no)f(phase-space)i(symmetry)-8 b(.)-118 720 y Fl(Remark)37 b(5)49 b Fx(Our)34 b(pro)s(of)f(forces)i Fw(q)f Fr(2)d Fx(\(0)p Fw(;)17 b Fx(1\))33 b(\(see)i(Lemma)e(3\).)48 b(If)34 b Fw(D)2463 679 y FB(+)2460 747 y Fu(N)2532 720 y Fx(\(\001;)17 b Fw(q)t Fx(\))30 b(=)g Fw(D)3000 679 y Fu(\000)2997 747 y(N)3069 720 y Fx(\(\001;)17 b Fw(q)t Fx(\))34 b(for)f(all)f Fw(q)j Fr(6)p Fx(=)30 b(1,)-118 840 y(the)k(large)f(deviation)g(tec)m(hnique)j(of)e([GSB3)o(])g(leads)g (to)g(\(8\))g(for)f(all)f Fw(q)i(>)d Fx(0)i(\(if)g Fw(H)38 b Fr(2)31 b(C)3090 804 y Fu(1)3165 840 y Fx(\()p Fr(A)3283 855 y Fv(\022)3321 840 y Fx(\)\))j(and)g(\(9\))g(for)-118 961 y(all)d Fw(q)h Fr(2)d Fx(\(0)p Fw(;)17 b Fx(2].)43 b(Numerical)32 b(results)h([TK,)g(RP])g(suggest)h(that)f(the)g(upp)s (er)h(and)f(lo)m(w)m(er)g(fractal)e(dimensions)-118 1081 y(indeed)f(coincide)e(for)h(Diophan)m(tine)f Fw(\022)s(=)p Fx(\(2)p Fw(\031)t Fx(\).)42 b(This)29 b(is)g(hardly)g(to)g(b)s(e)g (exp)s(ected)j(for)c(Liouville)f Fw(\022)s(=)p Fx(\(2)p Fw(\031)t Fx(\):)41 b(the)-118 1201 y(study)34 b(in)e([Las])g(can)h(b)s (e)g(tak)m(en)h(as)e(an)h(indicator)e(for)h(suc)m(h)i Fi(bad)e Fx(scaling)g(b)s(eha)m(vior.)-118 1412 y Fl(Remark)37 b(6)49 b Fx(Tw)m(o-sided)33 b(time)e(a)m(v)m(erages)j(are)f(used)g(for) g(tec)m(hnical)f(con)m(v)m(enience.)28 1623 y(Imp)s(ortan)m(t)47 b(in)m(termediate)e(steps)k(of)e(the)g(pro)s(of)g(are)g(summarized)f(b) s(elo)m(w.)87 b(Asso)s(ciated)47 b(with)g(the)-118 1743 y(symmetry)42 b Fw(S)47 b Fx(there)42 b(is)f(a)h(harmonic)e(oscillator) f(Hamiltonian)f Fh(H)2416 1758 y Fv(S)2524 1743 y Fx(in)m(v)-5 b(arian)m(t)41 b(under)46 b Fq(b)-59 b Fw(\021)3276 1758 y Fv(S)3368 1743 y Fx(with)42 b(ground)-118 1863 y(state)32 b Fw(\036)178 1878 y Fv(S)273 1863 y Fr(2)c(S)7 b Fx(\()p Fs(R)f Fx(\),)37 b(see)c(Section)f(3.3.)42 b(In)32 b(the)g(case)h(of)e Fw(S)1997 1878 y FB(4)2068 1863 y Fx(\(relev)-5 b(an)m(t)32 b(to)f(the)h(critical)d(Harp)s(er)j(mo)s(del\))e(this)-118 1984 y(is)d(the)i(con)m(v)m(en)m(tional)f(harmonic)e(oscillator)f (hamiltonian)f Fh(H)2143 1999 y Fv(S)h FB(4)2278 1984 y Fx(=)i(\()p Fw(P)2496 1948 y FB(2)2548 1984 y Fx(+)13 b Fw(Q)2714 1948 y FB(2)2753 1984 y Fx(\))p Fw(=)p Fx(2,)28 b(and)g Fw(\036)3187 1999 y Fv(S)3282 1984 y Fx(is)g(the)g(gaussian) -118 2104 y(state.)44 b(Let)33 b Fw(\032)384 2119 y Fv(S)467 2104 y Fx(b)s(e)g(the)g(sp)s(ectral)f(measure)h(of)f Fw(H)1706 2119 y Fv(W)1814 2104 y Fx(=)c Fw(\031)1973 2119 y Fv(W)2054 2104 y Fx(\()p Fw(H)8 b Fx(\))32 b(with)g(resp)s(ect)i (to)e Fw(\036)2984 2119 y Fv(S)3051 2104 y Fx(.)-118 2315 y Fl(Prop)s(osition)j(1)49 b Fm(L)-5 b(et)32 b Fw(\022)f(>)c Fx(2)p Fw(\031)t Fm(.)43 b(Ther)-5 b(e)31 b(ar)-5 b(e)31 b(two)g(p)-5 b(ositive)31 b(c)-5 b(onstants)31 b Fw(c)2549 2330 y Fu(\006)2639 2315 y Fm(such)g(that)h(for)f(any)g(Bor)-5 b(el)31 b(subset)-118 2435 y Fx(\001)d Fr(\032)g Fs(R)714 2556 y Fw(c)756 2571 y Fu(\000)832 2556 y Fr(N)15 b Fx(\(\001\))54 b Fr(\024)i Fw(\032)1323 2571 y Fv(S)1374 2556 y Fx(\(\001\))g(=)f Fr(h)p Fw(\036)1815 2571 y Fv(S)1882 2556 y Fr(j)p Fw(\037)1971 2571 y FB(\001)2034 2556 y Fx(\()p Fw(H)2153 2571 y Fv(W)2233 2556 y Fx(\))p Fr(j)p Fw(\036)2357 2571 y Fv(S)2424 2556 y Fr(i)g(\024)h Fw(c)2693 2571 y FB(+)2768 2556 y Fr(N)15 b Fx(\(\001\))i Fw(:)-118 2723 y Fm(In)34 b(p)-5 b(articular,)35 b Fr(N)49 b Fm(and)34 b Fw(\032)855 2738 y Fv(S)941 2723 y Fm(have)h(same)f(multifr)-5 b(actal)34 b(exp)-5 b(onents.)28 2934 y Fx(The)36 b(Hamiltonian)c Fh(H)867 2949 y Fv(S)969 2934 y Fx(will)g(b)s(e)k(used)g(to)f(study)h(transp)s(ort)f(in)f(phase) i(space.)52 b(Similarly)31 b(to)k(eqs.)h(\(5\))-118 3054 y(and)d(\(6\),)f(momen)m(ts)h(of)f(the)h(phase)h(space)g(distance)f (and)f(gro)m(wth)h(exp)s(onen)m(ts)i(thereof)e(can)g(b)s(e)g(de\014ned) h(in)-118 3175 y(the)f(W)-8 b(eyl)32 b(represen)m(tation)i(as)f(follo)m (ws:)522 3379 y Fw(M)616 3394 y Fv(W)697 3379 y Fx(\()p Fw(H)r(;)17 b Fx(\001;)g Fw(q)t(;)g(t)p Fx(\))55 b(=)g Fr(h)p Fw(\036)1434 3394 y Fv(S)1501 3379 y Fr(j)p Fw(\037)1590 3394 y FB(\001)1653 3379 y Fx(\()p Fw(H)1772 3394 y Fv(W)1852 3379 y Fx(\))17 b Fw(e)1952 3338 y Fv({tH)2058 3349 y Fp(W)2132 3379 y Fh(H)2204 3394 y Fv(S)2271 3338 y(q)r(=)p FB(2)2380 3379 y Fw(e)2425 3338 y Fu(\000)p Fv({tH)2586 3349 y Fp(W)2661 3379 y Fw(\037)2722 3394 y FB(\001)2785 3379 y Fx(\()p Fw(H)2904 3394 y Fv(W)2984 3379 y Fx(\))p Fr(j)p Fw(\036)3108 3394 y Fv(S)3175 3379 y Fr(i)g Fw(;)904 3639 y(\014)965 3598 y Fu(\006)959 3666 y Fv(W)1040 3639 y Fx(\()p Fw(H)r(;)g Fx(\001;)g Fw(q)t Fx(\))55 b(=)83 b(lim)1601 3701 y Fv(T)10 b Fu(!1)1770 3596 y(\006)1866 3572 y Fx(log\()p Fr(h)p Fw(M)2163 3587 y Fv(W)2244 3572 y Fx(\()p Fw(H)r(;)17 b Fx(\001;)g Fw(q)t(;)g Fr(\001)p Fx(\))p Fr(i)2730 3587 y Fv(T)2784 3572 y Fx(\))p 1866 3616 956 4 v 2176 3707 a Fw(q)j Fx(log\()p Fw(T)14 b Fx(\))2848 3639 y Fw(:)-118 3910 y Fl(Prop)s(osition)35 b(2)49 b Fm(L)-5 b(et)35 b Fw(\022)c(>)d Fx(2)p Fw(\031)38 b Fm(and)d Fw(H)g Fx(=)27 b Fw(H)1580 3874 y Fu(\003)1647 3910 y Fr(2)h(C)1799 3874 y FB(2)1839 3910 y Fx(\()p Fr(A)1957 3925 y Fv(\022)1995 3910 y Fx(\))p Fm(.)45 b(F)-7 b(or)34 b Fw(q)d Fr(2)e Fx(\(0)p Fw(;)17 b Fx(2])p Fm(,)1265 4114 y Fw(\014)1326 4073 y Fu(\006)1320 4141 y Fv(W)1400 4114 y Fx(\()p Fw(H)r(;)g Fx(\001;)g Fw(q)t Fx(\))55 b(=)g Fw(\014)2022 4073 y Fu(\006)2016 4139 y Fn(2D)2097 4114 y Fx(\()p Fw(H)r(;)17 b Fx(\001;)g Fw(q)t Fx(\))g Fw(:)-118 4325 y Fl(Prop)s(osition)35 b(3)49 b Fm(L)-5 b(et)38 b Fw(\022)e(>)d Fx(2)p Fw(\031)41 b Fm(and)d Fw(H)i Fx(=)33 b Fw(H)1610 4289 y Fu(\003)1681 4325 y Fr(2)h(C)1839 4289 y FB(2)1878 4325 y Fx(\()p Fr(A)1996 4340 y Fv(\022)2035 4325 y Fx(\))k Fm(b)-5 b(e)37 b(invariant)g(under)k Fq(b)-59 b Fw(\021)2986 4340 y Fv(S)3075 4325 y Fm(for)38 b(some)e(symmetry)-118 4445 y Fw(S)d Fr(2)28 b Fw(S)6 b(L)p Fx(\(2)p Fw(;)17 b Fs(Z)p Fx(\))p Fm(.)42 b(Then)947 4566 y Fw(\014)1008 4525 y Fu(\006)1002 4593 y Fv(W)1082 4566 y Fx(\()p Fw(H)r(;)17 b Fx(\001;)g Fw(q)t Fx(\))55 b Fr(\024)h Fw(\014)1706 4525 y Fu(\006)1700 4590 y Fn(1D)1780 4566 y Fx(\()p Fw(H)r(;)17 b Fx(\001;)g Fw(q)t Fx(\))g Fw(;)215 b(q)32 b Fr(2)c Fx(\(0)p Fw(;)17 b Fx(2])g Fw(:)28 4777 y Fx(Thanks)32 b(to)e(Prop)s(ositions)f(1,)h(2)g(and)h(3)f(and)g(since)h Fw(\022)i Fx(ma)m(y)d(b)s(e)h(replaced)f(b)m(y)h Fw(\022)21 b Fx(+)c(2)p Fw(\031)34 b Fx(without)c(c)m(hanging)-118 4897 y(the)j(1)p Fw(D)i Fx(and)e(2)p Fw(D)s Fx(-represen)m(tations,)f (the)h(Main)f(Theorem)h(is)f(a)h(direct)f(consequence)k(of)c(the)h (follo)m(wing:)-118 5108 y Fl(Theorem)k(1)49 b Fm(L)-5 b(et)39 b Fw(H)i Fx(=)34 b Fw(H)955 5071 y Fu(\003)1028 5108 y Fr(2)h(C)1187 5071 y FB(2)1227 5108 y Fx(\()p Fr(A)1345 5123 y Fv(\022)1383 5108 y Fx(\))k Fm(and)e Fw(\022)h(>)c Fx(2)p Fw(\031)42 b Fm(b)-5 b(e)38 b(a)g(R)-5 b(oth)39 b(angle.)54 b(Then,)38 b(for)h(any)f(Bor)-5 b(el)38 b(subset)-118 5228 y Fx(\001)28 b Fr(\032)g Fs(R)704 5348 y Fw(\014)765 5307 y Fu(\006)759 5375 y Fv(W)839 5348 y Fx(\()p Fw(H)r(;)17 b Fx(\001;)g Fw(q)t Fx(\))55 b Fr(\025)73 b Fw(D)1503 5307 y Fu(\006)1500 5373 y Fv(\032)1536 5384 y Fp(S)1585 5348 y Fx(\(\001;)17 b(1)k Fr(\000)i Fw(q)t Fx(\))17 b Fw(;)489 b Fr(8)55 b Fw(q)32 b Fr(2)c Fx(\(0)p Fw(;)17 b Fx(1\))g Fw(:)1865 5635 y Fx(6)p eop %%Page: 7 7 7 6 bop 28 22 a Fx(The)41 b(pro)s(of)e(of)g(Theorem)i(1)e(will)f (require)i(t)m(w)m(o)g(tec)m(hnical)g(steps)h(that)f(are)f(w)m(orth)i (b)s(eing)e(men)m(tioned)-118 142 y(here.)j(The)28 b(\014rst)f(one)g (requires)h(some)e(notations.)41 b(Giv)m(en)26 b(a)h(symmetry)g Fw(S)6 b Fx(,)27 b(let)g(\005)2887 157 y Fv(S)2964 142 y Fx(b)s(e)g(the)g(pro)5 b(jection)27 b(on)m(to)-118 263 y(the)36 b Fw(H)134 278 y Fv(W)214 263 y Fx(-cyclic)e(subspace)k Fr(H)1012 278 y Fv(S)1095 263 y Fr(\032)33 b(H)j Fx(of)f Fw(\036)1497 278 y Fv(S)1564 263 y Fx(.)52 b(Using)34 b(the)i(sp)s(ectral)f(theorem,)h(there)g(is)f(an)g(isomorphism)-118 383 y(b)s(et)m(w)m(een)d Fr(H)340 398 y Fv(S)421 383 y Fx(and)e Fw(L)674 347 y FB(2)714 383 y Fx(\()p Fs(R)5 b Fw(;)17 b(d\032)963 398 y Fv(S)1019 383 y Fx(\).)43 b(If)30 b(\()p Fw(\036)1318 332 y FB(\()p Fv(n)p FB(\))1318 410 y Fv(S)1420 383 y Fx(\))1458 398 y Fv(n)p Fu(2)p Ft(N)1629 383 y Fx(denotes)i(the)e(orthonormal)e(basis)i(of)g (eigenstates)h(of)e Fh(H)3719 398 y Fv(S)3816 383 y Fx(in)-118 519 y Fr(H)q Fx(,)k(let)f(\010)238 534 y Fv(n;S)351 519 y Fx(\()p Fw(E)6 b Fx(\))33 b(b)s(e)f(the)h(represen)m(tativ)m(e)i(of)d (\005)1649 534 y Fv(S)1716 519 y Fw(\036)1774 468 y FB(\()p Fv(n)p FB(\))1774 546 y Fv(S)1909 519 y Fx(in)f Fw(L)2088 483 y FB(2)2128 519 y Fx(\()p Fs(R)5 b Fw(;)17 b(d\032)2377 534 y Fv(S)2434 519 y Fx(\).)43 b(Then:)-118 748 y Fl(Prop)s(osition)35 b(4)49 b Fm(L)-5 b(et)37 b Fw(H)h Fx(=)30 b Fw(H)1076 711 y Fu(\003)1145 748 y Fr(2)g(C)1299 711 y FB(2)1339 748 y Fx(\()p Fr(A)1457 763 y Fv(\022)1496 748 y Fx(\))36 b Fm(and)f(let)i Fw(\022)i Fm(b)-5 b(e)36 b(a)g(R)-5 b(oth)36 b(angle.)48 b(Then)36 b(for)g(any)g Fw(\017)31 b(>)f Fx(0)36 b Fm(ther)-5 b(e)36 b(is)-118 868 y Fw(c)-76 883 y Fv(\017)-16 868 y Fw(>)28 b Fx(0)34 b Fm(such)h(that)177 1029 y Fu(1)141 1059 y Fq(X)146 1268 y Fv(n)p FB(=0)318 1154 y Fr(j)p Fx(\010)416 1169 y Fv(n;S)529 1154 y Fx(\()p Fw(E)6 b Fx(\))p Fr(j)711 1113 y FB(2)767 1154 y Fw(e)812 1113 y Fu(\000)p Fv(\016)14 b FB(\()p Fv(n)p FB(+1)p Fv(=)p FB(2\))1230 1154 y Fr(\024)56 b Fw(c)1405 1169 y Fv(\017)1454 1154 y Fw(\016)1501 1113 y Fu(\000)p FB(\(1)p Fv(=)p FB(2+)p Fv(\017)p FB(\))1821 1154 y Fw(;)253 b Fr(8)56 b Fx(0)27 b Fw(<)h(\016)j(<)d Fx(1)17 b Fw(;)215 b(\032)2928 1169 y Fv(S)3001 1154 y Fr(\000)23 b Fw(a:)17 b(e:)56 b(E)33 b Fr(2)28 b Fs(R)22 b Fw(:)-118 1469 y Fl(Remark)37 b(7)49 b Fx(This)29 b(result)h(is)f(uniform)f(\()p Fw(\032)1446 1484 y Fv(S)1497 1469 y Fx(-almost)f(surely\))k(with)e (resp)s(ect)i(to)e(the)h(sp)s(ectral)f(parameter)h Fw(E)-118 1603 y Fx(and)35 b(to)f Fw(\016)t Fx(.)49 b(In)35 b(particular,)f(in)m (tegrating)f(o)m(v)m(er)i Fw(E)41 b Fx(with)34 b(resp)s(ect)i(to)e Fw(\032)2474 1618 y Fv(S)2560 1603 y Fx(sho)m(ws)i(that)3056 1529 y Fq(P)3161 1555 y Fv(N)7 b Fu(\000)p FB(1)3161 1632 y Fv(n)p FB(=0)3351 1603 y Fr(k)p Fx(\005)3474 1618 y Fv(S)3542 1603 y Fw(\036)3600 1552 y FB(\()p Fv(n)p FB(\))3600 1630 y Fv(S)3701 1603 y Fr(k)3751 1567 y FB(2)3822 1603 y Fx(=)-118 1724 y Fw(O)s Fx(\()p Fw(N)86 1688 y FB(1)p Fv(=)p FB(2+)p Fv(\017)279 1724 y Fx(\).)55 b(This)36 b(is)g(p)s(ossible)g(b)s(ecause)i(of)e(the)h(follo)m(wing)c(complemen)m (tary)j(result)g(pro)m(v)m(ed)i(in)e(the)h(Ap-)-118 1844 y(p)s(endix:)-118 2072 y Fl(Prop)s(osition)e(5)49 b Fm(L)-5 b(et)35 b Fw(H)h Fx(=)27 b Fw(H)1069 2036 y Fu(\003)1136 2072 y Fr(2)h(A)1310 2087 y Fv(\022)1348 2072 y Fm(.)45 b(Then)34 b Fw(H)1758 2087 y Fv(W)1873 2072 y Fm(has)h(in\014nite)f (multiplicity)h(and)f(no)h(cyclic)f(ve)-5 b(ctor.)28 2301 y Fx(The)41 b(second)h(tec)m(hnical)e(result)g(concerns)i(the)e (so-called)f Fm(Mehler)j(kernel)d Fx(of)h(the)h(Hamiltonian)36 b Fh(H)3803 2316 y Fv(S)3871 2301 y Fx(,)-118 2421 y(notably)c(the)h (in)m(tegral)e(k)m(ernel)i(of)f(the)h(op)s(erator)f Fw(e)1764 2385 y Fu(\000)p Fv(t)12 b Fg(H)1906 2396 y Fp(S)1999 2421 y Fx(in)32 b(the)h Fw(Q)p Fx(-represen)m(tation:)1269 2641 y Fr(M)1389 2656 y Fv(S)1439 2641 y Fx(\()p Fw(t)p Fx(;)17 b Fw(x;)g(y)t Fx(\))54 b(=)i Fr(h)p Fw(x)p Fr(j)17 b Fw(e)2115 2600 y Fu(\000)p Fv(t)11 b Fg(H)2256 2611 y Fp(S)2333 2641 y Fr(j)p Fw(y)t Fr(i)17 b Fw(;)1228 b Fx(\(10\))-118 2869 y Fl(Prop)s(osition)35 b(6)49 b Fm(L)-5 b(et)35 b Fw(\022)j Fm(b)-5 b(e)35 b(a)g(R)-5 b(oth)35 b(angle.)44 b(Then,)34 b(for)g(al)5 b(l)35 b Fw(\017)28 b(>)f Fx(0)p Fm(,)279 3101 y Fx(sup)-42 3194 y FB(0)p Fu(\024)p Fv(x)p Fu(\024)p FB(2)p Fv(\031)r(\022)256 3175 y Fo(\000)p Fn(1)p Fp(=)p Fn(2)401 3194 y Fv(;)11 b FB(0)p Fu(\024)p Fv(y)r Fu(\024)p Fv(\022)649 3175 y Fn(1)p Fp(=)p Fn(2)813 3007 y Fq(X)790 3224 y Fv(m)p Fu(2)p Ft(Z)950 3205 y Fn(2)1012 3101 y Fr(jM)1160 3116 y Fv(S)1210 3101 y Fx(\()p Fw(t)p Fx(;)17 b Fw(x)22 b Fx(+)h(2)p Fw(\031)t(m)1696 3116 y FB(1)1735 3101 y Fw(\022)1783 3060 y Fu(\000)p FB(1)p Fv(=)p FB(2)1948 3101 y Fw(;)17 b(y)25 b Fx(+)d Fw(\022)2211 3060 y FB(1)p Fv(=)p FB(2)2321 3101 y Fw(m)2406 3116 y FB(2)2446 3101 y Fx(\))p Fr(j)55 b Fx(=)g Fr(O)s Fx(\()p Fw(t)2853 3060 y Fu(\000)p FB(1)p Fv(=)p FB(2)p Fu(\000)p Fv(\017)3102 3101 y Fx(\))17 b Fw(;)252 b(as)56 b(t)28 b Fr(#)f Fx(0)17 b Fw(:)-118 3651 y Fy(3)161 b(W)-13 b(eyl's)53 b(calculus)-118 3870 y Fx(This)40 b(c)m(hapter)i(b)s(egins)e(with)g(a)g(review)h(of)f(basic) g(facts)h(ab)s(out)f(W)-8 b(eyl)40 b(op)s(erators,)i(the)f(rotation)e (algebra)-118 3991 y(and)29 b(implemen)m(tation)c(of)j(symmetries)g (therein.)43 b(The)29 b(form)m(ulas)e(are)i(w)m(ell-kno)m(wn)g(\()p Fi(e.g.)42 b Fx([P)m(er)q(,)29 b(Bel94)o(])f(and)-118 4111 y(mainly)36 b(giv)m(en)j(in)e(order)i(to)f(\014x)h(notations,)g (but)f(for)g(the)h(con)m(v)m(enience)h(of)e(the)h(reader)g(their)f(pro) s(ofs)g(are)-118 4231 y(nev)m(ertheless)e(giv)m(en)d(in)g(the)h(App)s (endix.)46 b(The)35 b(c)m(hapter)f(also)f(con)m(tains)g(a)g(new)i(and)e (compact)g(solution)f(of)-118 4352 y(the)h(frame)f(problem)f(for)h (coheren)m(t)i(states)f(\(Section)g(3.4\).)-118 4641 y Ff(3.1)135 b(W)-11 b(eyl)45 b(op)t(erators)-118 4825 y Fx(Let)35 b Fr(H)g Fx(denote)g(the)g(Hilb)s(ert)e(space)j Fw(L)1331 4789 y FB(2)1371 4825 y Fx(\()p Fs(R)5 b Fx(\).)55 b(Giv)m(en)34 b(a)g(v)m(ector)i Fh(a)31 b Fx(=)g(\()p Fw(a)2534 4840 y FB(1)2573 4825 y Fw(;)17 b(a)2668 4840 y FB(2)2708 4825 y Fx(\))30 b Fr(2)i Fs(R)2940 4789 y FB(2)2985 4825 y Fx(,)j(the)g(asso)s(ciated)g(W)-8 b(eyl)-118 4946 y(op)s(erator)32 b(is)g(de\014ned)i(b)m(y:)56 5166 y Fh(W)p Fx(\()p Fh(a)p Fx(\))44 b(=)h Fw(e)496 5124 y Fv({)p FB(\()p Fv(a)583 5133 y Fn(1)618 5124 y Fv(P)10 b FB(+)p Fv(a)765 5133 y Fn(2)800 5124 y Fv(Q)p FB(\))1080 5166 y Fr(,)193 b Fh(W)p Fx(\()p Fh(a)p Fx(\))17 b Fw( )t Fx(\()p Fw(x)p Fx(\))44 b(=)g Fw(e)2027 5124 y Fv({a)2087 5133 y Fn(1)2123 5124 y Fv(a)2160 5133 y Fn(2)2195 5124 y Fv(=)p FB(2)2286 5166 y Fw(e)2331 5124 y Fv({a)2391 5133 y Fn(2)2426 5124 y Fv(x)2487 5166 y Fw( )t Fx(\()p Fw(x)22 b Fx(+)g Fw(a)2818 5181 y FB(1)2858 5166 y Fx(\))17 b Fw(;)182 b Fr(8)55 b Fw( )32 b Fr(2)c(H)18 b Fw(:)174 b Fx(\(11\))1865 5635 y(7)p eop %%Page: 8 8 8 7 bop -118 22 a Fx(The)33 b(W)-8 b(eyl)33 b(op)s(erators)f(are)h (unitaries,)f(strongly)g(con)m(tin)m(uous)h(with)f(resp)s(ect)i(to)e Fh(a)h Fx(and)f(satisfy)628 231 y Fh(W)p Fx(\()p Fh(a)p Fx(\))17 b Fh(W)p Fx(\()p Fh(b)p Fx(\))44 b(=)g Fw(e)1315 189 y Fv({)12 b Fg(a)p Fu(^)p Fg(b)p Fv(=)p FB(2)1569 231 y Fh(W)p Fx(\()p Fh(a)23 b Fx(+)f Fh(b)p Fx(\))17 b Fw(;)252 b Fh(a)22 b Fr(^)h Fh(b)44 b Fx(=)g Fw(a)2694 246 y FB(1)2734 231 y Fw(b)2775 246 y FB(2)2837 231 y Fr(\000)23 b Fw(a)2988 246 y FB(2)3027 231 y Fw(b)3068 246 y FB(1)3125 231 y Fw(:)572 b Fx(\(12\))-118 439 y(The)33 b(follo)m(wing)d(w)m(eak-in)m(tegral)i(iden)m(tities)g(are)g(v)m (eri\014ed)i(in)d(the)j(App)s(endix:)615 802 y Fr(h)p Fw( )t Fr(j)p Fh(W)p Fx(\()p Fh(a)p Fx(\))979 761 y Fu(\000)p FB(1)1089 802 y Fr(j)p Fw( )t Fr(i)27 b Fh(W)p Fx(\()p Fh(a)p Fx(\))60 b(=)1671 667 y Fq(Z)1727 892 y Ft(R)1775 873 y Fn(2)1857 735 y Fw(d)1908 699 y FB(2)1947 735 y Fh(b)p 1857 779 142 4 v 1873 870 a Fx(2)p Fw(\031)2024 802 y(e)2069 761 y Fv({)13 b Fg(a)p Fu(^)p Fg(b)2253 802 y Fh(W)p Fx(\()p Fh(b)p Fx(\))k Fr(j)p Fw( )t Fr(ih)p Fw( )t Fr(j)p Fh(W)p Fx(\()p Fh(b)p Fx(\))3000 761 y Fu(\000)p FB(1)3121 802 y Fw(;)576 b Fx(\(13\))626 1223 y Fh(W)p Fx(\()p Fh(b)p Fx(\))17 b Fr(j)p Fw( )t Fr(ih)p Fw( )t Fr(j)p Fh(W)p Fx(\()p Fh(b)p Fx(\))1373 1182 y Fu(\000)p FB(1)1522 1223 y Fx(=)1653 1087 y Fq(Z)1708 1313 y Ft(R)1756 1294 y Fn(2)1838 1156 y Fw(d)1889 1119 y FB(2)1928 1156 y Fh(a)p 1838 1200 140 4 v 1854 1291 a Fx(2)p Fw(\031)2005 1223 y(e)2050 1182 y Fv({)12 b Fg(b)p Fu(^)p Fg(a)2233 1223 y Fr(h)p Fw( )t Fr(j)p Fh(W)p Fx(\()p Fh(a)p Fx(\))2597 1182 y Fu(\000)p FB(1)2708 1223 y Fr(j)p Fw( )t Fr(i)27 b Fh(W)p Fx(\()p Fh(a)p Fx(\))g Fw(:)571 b Fx(\(14\))-118 1484 y(Applying)32 b(\(13\))g(to)g Fw(\036)g Fx(and)h(setting)f Fh(a)27 b Fx(=)h(0)k(leads)h(to)371 1847 y Fw(\036)44 b Fx(=)593 1712 y Fq(Z)648 1937 y Ft(R)696 1918 y Fn(2)778 1780 y Fw(d)829 1744 y FB(2)868 1780 y Fh(b)p 778 1824 142 4 v 795 1916 a Fx(2)p Fw(\031)973 1847 y Fr(h)p Fw( )t Fr(j)p Fh(W)p Fx(\()p Fh(b)p Fx(\))1338 1806 y Fu(\000)p FB(1)1432 1847 y Fr(j)p Fw(\036)p Fr(i)27 b Fh(W)p Fx(\()p Fh(b)p Fx(\))17 b Fw( )53 b Fx(,)228 b Fw(\036;)44 b( )k Fr(2)d(H)50 b Fx(,)228 b Fr(k)p Fw( )t Fr(k)27 b Fx(=)h(1)49 b(.)358 b(\(15\))-118 2112 y(In)28 b(particular,)g(an)m(y)g(non)h(zero) f(v)m(ector)h(in)f Fr(H)h Fx(is)e(cyclic)h(for)g(the)g(W)-8 b(eyl)28 b(algebra)f Fr(f)p Fh(W)p Fx(\()p Fh(a)p Fx(\))p Fr(j)p Fh(a)g Fr(2)h Fs(R)3339 2076 y FB(2)3385 2112 y Fr(g)p Fx(.)42 b(If)28 b Fw( )j Fr(2)d(H)q Fx(,)-118 2233 y(the)41 b(map)f Fh(a)i Fr(2)h Fs(R)550 2197 y FB(2)637 2233 y Fr(7!)f(h)p Fw( )t Fr(j)p Fh(W)p Fx(\()p Fh(a)p Fx(\))p Fr(j)p Fw( )t Fr(i)f(2)h Fs(C)67 b Fx(is)40 b(con)m(tin)m (uous,)k(tends)e(to)f(zero)g(at)g(in\014nit)m(y)f(and)h(b)s(elongs)g (to)-118 2353 y Fw(L)-52 2317 y FB(2)-12 2353 y Fx(\()p Fs(R)92 2317 y FB(2)137 2353 y Fx(\),)33 b(whereas)h Fw( )d Fr(2)d(S)7 b Fx(\()p Fs(R)f Fx(\))39 b(if)31 b(and)i(only)f(if)f (this)h(map)g(b)s(elongs)g(to)g Fr(S)7 b Fx(\()p Fs(R)2671 2317 y FB(2)2717 2353 y Fx(\).)-118 2640 y Ff(3.2)135 b(The)45 b(rotation)h(algebra)-118 2825 y Fx(The)40 b Fm(r)-5 b(otation)41 b(algebr)-5 b(a)38 b Fr(A)879 2840 y Fv(\022)917 2825 y Fx(,)j(its)e(represen)m(tations)h(\()p Fw(\031)1903 2840 y Fv(!)1953 2825 y Fx(\))1991 2840 y Fv(!)r Fu(2)p Ft(R)2137 2825 y Fx(,)h Fw(\031)2260 2840 y Fn(2D)2379 2825 y Fx(and)e Fw(\031)2630 2840 y Fv(W)2750 2825 y Fx(as)g(w)m(ell)g(as)g(the)g(tracial)e(state)-118 2946 y Fr(T)-64 2961 y Fv(\022)9 2946 y Fx(and)d Fr(\003)p Fx(-deriv)-5 b(ations)32 b Fw(\016)823 2961 y FB(1)863 2946 y Fw(;)17 b(\016)950 2961 y FB(2)1023 2946 y Fx(w)m(ere)35 b(de\014ned)g(in)e(Section)h(2.)47 b(Here)34 b(w)m(e)h(giv)m(e)f(some)f (complemen)m(ts,)h(further)-118 3066 y(de\014nitions)f(and)h(the)g (short)g(pro)s(of)f(of)h(Prop)s(osition)e(5.)46 b(The)35 b(trace)f(is)g(faithful)d(and)j(satis\014es)h(the)f(F)-8 b(ourier)-118 3186 y(form)m(ula:)909 3395 y Fw(A)45 b Fx(=)1149 3300 y Fq(X)1147 3517 y Fk(l)p Fu(2)p Ft(Z)1267 3498 y Fn(2)1329 3395 y Fw(a)1380 3410 y Fk(l)1423 3395 y Fw(W)1515 3410 y Fv(\022)1554 3395 y Fx(\()p Fl(l)p Fx(\))17 b Fw(;)211 b(a)1968 3410 y Fk(l)2039 3395 y Fx(=)44 b Fr(T)2213 3410 y Fv(\022)2252 3395 y Fx(\()p Fw(W)2382 3410 y Fv(\022)2422 3395 y Fx(\()p Fl(l)p Fx(\))2530 3354 y Fu(\000)p FB(1)2639 3395 y Fw(A)p Fx(\))50 b(.)897 b(\(16\))-118 3697 y(In)33 b(addition,)-12 3965 y Fr(T)42 3980 y Fv(\022)81 3965 y Fx(\()p Fw(A)p Fx(\))55 b(=)416 3829 y Fq(Z)516 3856 y FB(2)p Fv(\031)642 3897 y Fw(d!)p 642 3942 116 4 v 646 4033 a Fx(2)p Fw(\031)783 3965 y Fr(h)p Fw(m)p Fr(j)p Fw(\031)990 3980 y Fv(!)1041 3965 y Fx(\()p Fw(A)p Fx(\))p Fr(j)p Fw(m)p Fr(i)71 b Fx(=)56 b 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Fl(m)p Fx(\)\))54 b(=)i Fr(W)1464 646 y Fv(\022)1503 631 y Fx(\()p Fl(m)p Fx(\))f(:=)g Fh(W)p Fx(\()2028 541 y Fr(p)p 2111 541 49 4 v 90 x Fw(\022)s Fl(m)p Fx(\))32 b(,)228 b Fr(8)56 b Fl(m)27 b Fr(2)h Fs(Z)2973 590 y FB(2)3042 631 y Fx(.)655 b(\(20\))-118 805 y(It)35 b(can)g(b)s(e)h(seen)g(as)g(a)e(direct)h(in)m(tegral)f(of)h (1)p Fw(D)s Fx(-represen)m(tations)g(b)m(y)h(in)m(tro)s(ducing)e(the)h (family)e(\()p Fr(G)3550 820 y Fv(!)3601 805 y Fx(\))3639 820 y Fv(!)r Fu(2)p Ft(R)3819 805 y Fx(of)-118 925 y(transformations)e (from)g Fr(H)j Fx(in)m(to)e Fw(`)1172 889 y FB(2)1211 925 y Fx(\()p Fs(Z)p Fx(\))860 1156 y(\()p Fr(G)957 1171 y Fv(!)1008 1156 y Fw(\036)p Fx(\)\()p Fw(n)p Fx(\))44 b(=)g Fw(\022)1450 1115 y Fu(\000)p FB(1)p Fv(=)p FB(4)1632 1156 y Fw(\036)1723 1016 y Fq(\022)1806 1089 y Fw(!)26 b Fr(\000)c Fw(n\022)p 1806 1133 293 4 v 1887 1153 a Fr(p)p 1970 1153 49 4 v 85 x Fw(\022)2109 1016 y Fq(\023)2215 1156 y Fw(;)182 b Fr(8)56 b Fw(\036)27 b Fr(2)h(H)51 b Fx(.)848 b(\(21\))-118 1387 y(Then)34 b(a)e(direct)g(computation)f (\(giv)m(en)i(in)f(the)h(App)s(endix\))g(sho)m(ws)h(that:)516 1629 y Fr(h)p Fw(\036)p Fr(j)p Fw(\031)696 1644 y Fv(W)776 1629 y Fx(\()p Fw(A)p Fx(\))p Fr(j)p Fw(\036)p Fr(i)55 b Fx(=)1236 1494 y Fq(Z)1335 1520 y Fv(\022)1291 1719 y FB(0)1408 1629 y Fw(d!)19 b Fr(hG)1637 1644 y Fv(!)1688 1629 y Fw(\036)p Fr(j)p Fw(\031)1829 1644 y Fv(!)1879 1629 y Fx(\()p Fw(A)p Fx(\))p Fr(jG)2115 1644 y Fv(!)2165 1629 y Fw( )t Fr(i)49 b Fx(,)33 b Fw(A)27 b Fr(2)i(A)2655 1644 y Fv(\022)2726 1629 y Fx(,)j Fw(\036;)17 b( )31 b Fr(2)d(H)34 b Fx(.)504 b(\(22\))-118 1881 y(In)43 b(particular,)h Fr(k)p Fw(\036)p Fr(k)662 1845 y FB(2)764 1881 y Fx(=)902 1801 y Fq(R)968 1827 y Fv(\022)949 1916 y FB(0)1040 1881 y Fw(d!)20 b Fr(kG)1281 1896 y Fv(!)1331 1881 y Fw(\036)p Fr(k)1439 1845 y FB(2)1439 1912 y Fv(`)1468 1893 y Fn(2)1507 1881 y Fx(.)75 b(The)44 b(link)e(b)s(et)m(w)m(een)j Fw(\031)2465 1896 y Fv(W)2589 1881 y Fx(and)e Fw(\031)2844 1896 y Fn(2D)2968 1881 y Fx(will)e(b)s(e)i(established)g(in)-118 2001 y(Section)32 b(4.2.)28 2169 y(It)24 b(follo)m(ws)e(from)h(a)g (theorem)h(b)m(y)g(Rie\013el)f([Rie)o(])h(that)f(the)h(comm)m(utan)m(t) f(of)h Fw(\031)2766 2184 y Fv(W)2847 2169 y Fx(\()p Fr(A)2965 2184 y Fv(\022)3003 2169 y Fx(\))g(is)f(the)h(v)m(on)h(Neumann)-118 2289 y(algebra)36 b(generated)j(b)m(y)f Fw(\031)873 2304 y Fv(W)954 2289 y Fx(\()p Fr(A)1072 2304 y Fv(\022)1107 2286 y Fo(0)1133 2289 y Fx(\))f(where)i Fw(\022)1543 2253 y Fu(0)1566 2289 y Fw(=)p Fx(2)p Fw(\031)h Fx(=)c(2)p Fw(\031)t(=\022)k Fx(and)d Fw(\031)2362 2304 y Fv(W)2443 2289 y Fx(\()p Fw(W)2573 2304 y Fv(\022)2608 2286 y Fo(0)2635 2289 y Fx(\()p Fl(l)p Fx(\)\))51 b(=)i Fr(W)3059 2304 y Fv(\022)3094 2286 y Fo(0)3121 2289 y Fx(\()p Fl(l)p Fx(\).)k(The)38 b(follo)m(wing)-118 2410 y(result)32 b(is)g(pro)m(v)m(en)j(in)c(the)i(App)s(endix:)-118 2585 y Fl(Prop)s(osition)i(7)49 b Fx(\(The)34 b(generalized)e(P)m(oisson)h (summation)d(form)m(ula\):)380 2904 y Fw(T)451 2863 y Fv(\022)437 2929 y( )545 2904 y Fx(:=)705 2809 y Fq(X)703 3026 y Fk(l)p Fu(2)p Ft(Z)823 3007 y Fn(2)885 2904 y Fr(W)983 2919 y Fv(\022)1018 2900 y Fo(0)1045 2904 y Fx(\()p Fl(l)p Fx(\))17 b Fr(j)p Fw( )t Fr(i)g(h)p Fw( )t Fr(jW)1553 2919 y Fv(\022)1588 2900 y Fo(0)1612 2904 y Fx(\()p Fl(l)p Fx(\))1720 2863 y Fu(\000)p FB(1)1868 2904 y Fx(=)2039 2837 y Fw(\022)p 2009 2881 108 4 v 2009 2972 a Fx(2)p Fw(\031)2169 2809 y Fq(X)2144 3026 y Fk(m)p Fu(2)p Ft(Z)2309 3007 y Fn(2)2354 2904 y Fr(h)p Fw( )t Fr(jW)2586 2919 y Fv(\022)2625 2904 y Fx(\()p Fl(m)p Fx(\))2795 2863 y Fu(\000)p FB(1)2888 2904 y Fr(j)p Fw( )t Fr(i)g(W)3137 2919 y Fv(\022)3176 2904 y Fx(\()p Fl(m)p Fx(\))27 b Fw(:)324 b Fx(\(23\))28 3186 y(By)41 b(eq.)f(\(23\),)h Fw( )j Fr(2)d(S)7 b Fx(\()p Fs(R)f Fx(\))45 b(implies)38 b Fw(T)1471 3150 y Fv(\022)1457 3212 y( )1550 3186 y Fr(2)i(C)1714 3150 y Fu(1)1789 3186 y Fx(\()p Fr(A)1907 3201 y Fv(\022)1946 3186 y Fx(\).)65 b(It)40 b(follo)m(ws)e (immediately)f(from)i(eq.)h(\(23\))g(that,)-118 3318 y(giv)m(en)28 b Fw( )k Fr(2)c(S)7 b Fx(\()p Fs(R)f Fx(\),)34 b(there)29 b(is)f(a)g(p)s(ositiv)m(e)f(elemen)m(t)h(in)f Fr(A)1909 3333 y Fv(\022)1948 3318 y Fx(,)i(denoted)g Fw(F)2445 3282 y Fv(\022)2431 3344 y( )2483 3318 y Fx(,)g(suc)m(h)h (that)e Fw(T)3033 3282 y Fv(\022)3019 3344 y( )3099 3318 y Fx(=)g(\()p Fw(\022)s(=)p Fx(2)p Fw(\031)t Fx(\))f Fw(\031)3566 3333 y Fv(W)3663 3237 y Fq(\000)3709 3318 y Fw(F)3786 3282 y Fv(\022)3772 3344 y( )3825 3237 y Fq(\001)3871 3318 y Fx(.)-118 3438 y(Moreo)m(v)m(er)938 3612 y Fr(h)p Fw( )t Fr(j)17 b Fw(\031)1144 3627 y Fv(W)1224 3612 y Fx(\()p Fw(A)p Fx(\))g Fr(j)p Fw( )t Fr(i)54 b Fx(=)h Fr(T)1763 3627 y Fv(\022)1819 3531 y Fq(\000)1865 3612 y Fw(A)17 b(F)2032 3571 y Fv(\022)2018 3637 y( )2070 3531 y Fq(\001)2149 3612 y Fw(;)253 b Fr(8)p Fw(A)28 b Fr(2)g(A)2759 3627 y Fv(\022)2815 3612 y Fw(:)882 b Fx(\(24\))-118 3995 y Ff(3.3)135 b(Symmetries)-118 4179 y Fx(It)34 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y(to)i(the)h(App)s(endix:)1865 5635 y(9)p eop %%Page: 10 10 10 9 bop -118 22 a Fl(Prop)s(osition)35 b(8)49 b Fm(F)-7 b(or)34 b(any)h Fw(\024;)17 b(\025;)g(s)27 b Fr(2)h Fs(R)5 b Fm(,)41 b Fw(\025)27 b Fr(6)p Fx(=)h(0)p Fm(,)34 b(up)h(to)h(a)e (phase)-100 362 y Fr(F)-28 331 y Fe(\022)58 370 y Fn(1)86 b(0)55 429 y Fp(\024)d Fn(1)268 331 y Fe(\023)363 362 y Fx(=)27 b Fw(e)511 321 y Fu(\000)p Fv({)12 b(\024)g(Q)710 298 y Fn(2)744 321 y Fv(=)p FB(2)835 362 y Fw(;)88 b Fr(F)1022 333 y Fe(\022)1105 368 y Fp(\025)125 b Fn(0)1108 435 y(0)87 b Fp(\025)1262 414 y Fd(\000)p Fc(1)1380 333 y Fe(\023)1476 362 y Fx(=)27 b Fw(e)1624 321 y Fu(\000)p Fv({)d FB(ln\()p Fv(\025)p FB(\))13 b(\()p Fv(QP)d FB(+)p Fv(P)g(Q)p FB(\))p Fv(=)p FB(2)2314 362 y Fw(;)87 b Fr(F)2500 331 y Fe(\022)2584 370 y Fn(cos)11 b Fp(s)83 b Fo(\000)11 b Fn(sin)f Fp(s)2587 429 y Fn(sin)g Fp(s)113 b Fn(cos)11 b Fp(s)3004 331 y Fe(\023)3099 362 y Fx(=)28 b Fw(e)3248 321 y Fu(\000)p Fv({)12 b(s)f FB(\()p Fv(Q)3465 298 y Fn(2)3500 321 y FB(+)p Fv(P)3610 298 y Fn(2)3644 321 y Fu(\000)p FB(1\))p Fv(=)p FB(2)3853 362 y Fw(:)3724 535 y Fx(\(26\))-118 743 y Fm(Note)35 b(in)g(p)-5 b(articular)35 b(that)g Fr(F)953 758 y Fv(S)1020 743 y Fr(F)1092 758 y Fv(S)1139 739 y Fo(0)1192 743 y Fx(=)28 b Fw(z)21 b Fr(F)1434 758 y Fv(S)15 b(S)1539 739 y Fo(0)1600 743 y Fm(for)35 b Fw(z)d Fr(2)c Fs(C)37 b Fw(;)17 b Fr(j)p Fw(z)t Fr(j)33 b Fx(=)28 b(1)p Fm(.)44 b(F)-7 b(urthermor)i(e,)34 b(if)h Fx(0)27 b Fw(<)h(s)g(<)f(\031)t Fm(,)468 1012 y Fr(F)540 980 y Fe(\022)623 1020 y Fn(cos)12 b Fp(s)83 b Fo(\000)11 b Fn(sin)f Fp(s)627 1078 y Fn(sin)g Fp(s)112 b Fn(cos)12 b Fp(s)1043 980 y Fe(\023)1128 1012 y Fw(\036)p Fx(\()p Fw(x)p Fx(\))55 b(=)1503 877 y Fq(Z)1559 1102 y Ft(R)1817 945 y Fw(dy)p 1665 989 407 4 v 1665 1009 a Fr(p)p 1748 1009 324 4 v 84 x Fx(2)p Fw(\031)37 b Fx(sin)16 b Fw(s)2109 1012 y(e)2154 968 y Fv({)2177 976 y Fx(\()2215 968 y FB(cos)c Fv(s)g FB(\()p Fv(x)2433 945 y Fn(2)2468 968 y FB(+)p Fv(y)2560 945 y Fn(2)2594 968 y FB(\))p Fu(\000)p FB(2)p Fv(xy)2788 976 y Fx(\))2827 968 y Fv(=)p FB(2)g(sin)g Fv(s)3072 1012 y Fw(\036)p Fx(\()p Fw(y)t Fx(\))26 b Fw(:)413 b Fx(\(27\))28 1343 y(In)36 b(the)g(sp)s(ecial)e (case)i Fw(s)c Fx(=)g Fw(\031)t(=)p Fx(2,)k(namely)e(for)h(the)h (matrix)e Fw(S)2303 1358 y FB(4)2377 1343 y Fx(\(see)j(Section)e(2\),)h (this)e(giv)m(es)i(the)g(usual)-118 1463 y(F)-8 b(ourier)31 b(transform)1208 1708 y Fr(F)1280 1723 y Fv(S)1323 1732 y Fn(4)1378 1708 y Fw(\036)p Fx(\()p Fw(x)p Fx(\))56 b(=)1754 1572 y Fq(Z)1809 1798 y Ft(R)1949 1641 y Fw(dy)p 1904 1685 191 4 v 1904 1705 a Fr(p)p 1987 1705 108 4 v 82 x Fx(2)p Fw(\031)2122 1708 y(e)2167 1667 y Fu(\000)p Fv({xy)2343 1708 y Fw(\036)p Fx(\()p Fw(y)t Fx(\))17 b Fw(:)1151 b Fx(\(28\))-118 1978 y(F)-8 b(or)24 b(the)i(case)g(of)e (the)i(3-fold)d(and)i(6-fold)e(symmetries)i Fw(S)1953 1993 y FB(3)2017 1978 y Fx(and)g Fw(S)2259 1993 y FB(6)2299 1978 y Fx(,)h(acting)e(on)h(a)g(hexagonal)f(or)h(a)g(triangular)-118 2098 y(lattice)31 b(\(see)j(Section)e(2\),)g(eqs.)45 b(\(26\))32 b(and)h(\(27\))f(giv)m(e)-84 2368 y Fr(F)-12 2383 y Fv(S)31 2392 y Fn(3)86 2368 y Fw(\036)p Fx(\()p Fw(x)p Fx(\))56 b(=)f Fw(e)507 2326 y Fv({\031)r(=)p FB(12)716 2232 y Fq(Z)772 2457 y Ft(R)895 2300 y Fw(dy)p 850 2345 191 4 v 850 2365 a Fr(p)p 933 2365 108 4 v 82 x Fx(2)p Fw(\031)1068 2368 y(e)1113 2326 y Fu(\000)p Fv({x)p FB(\()p Fv(x)p FB(+2)p Fv(y)r FB(\))p Fv(=)p FB(2)1544 2368 y Fw(\036)p Fx(\()p Fw(y)t Fx(\))17 b Fw(;)181 b Fr(F)2027 2383 y Fv(S)2070 2392 y Fn(6)2124 2368 y Fw(\036)p Fx(\()p Fw(x)p Fx(\))56 b(=)f Fw(e)2545 2326 y Fu(\000)p Fv({\031)r(=)p FB(12)2809 2232 y Fq(Z)2865 2457 y Ft(R)2988 2300 y Fw(dy)p 2943 2345 191 4 v 2943 2365 a Fr(p)p 3026 2365 108 4 v 82 x Fx(2)p Fw(\031)3161 2368 y(e)3206 2326 y Fu(\000)p Fv({y)r FB(\(2)p Fv(x)p Fu(\000)p Fv(y)r FB(\))p Fv(=)p FB(2)3635 2368 y Fw(\036)p Fx(\()p Fw(y)t Fx(\))17 b Fw(:)3724 2543 y Fx(\(29\))28 2751 y(No)m(w)40 b(supp)s(ose)h(that)e Fw(S)45 b Fr(2)40 b Fw(S)6 b(L)p 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FB(1)2738 3081 y Fq(X)2744 3290 y Fv(n)p FB(=0)2899 3176 y Fw(S)2965 3135 y Fv(n)3028 3176 y Fr(j)p Fw(e)3101 3191 y FB(2)3141 3176 y Fr(ih)p Fw(e)3264 3191 y FB(2)3303 3176 y Fr(j)17 b Fx(\()p Fw(S)3452 3135 y Fv(t)3480 3176 y Fx(\))3518 3135 y Fv(n)3582 3176 y Fw(;)-118 3487 y Fx(where)26 b Fw(K)35 b Fx(=)27 b(\()p Fw(P)s(;)17 b(Q)p Fx(\))25 b(and)g Fr(f)p Fw(e)942 3502 y FB(1)981 3487 y Fw(;)17 b(e)1070 3502 y FB(2)1109 3487 y Fr(g)25 b Fx(is)f(the)i(canonical)d(basis)i(of)f Fs(R)2256 3451 y FB(2)2302 3487 y Fx(.)41 b(Note)25 b(that)f Fh(H)2873 3502 y Fv(S)2916 3511 y Fn(4)2982 3487 y Fx(=)k(\()p Fw(P)3201 3451 y FB(2)3246 3487 y Fx(+)6 b Fw(Q)3405 3451 y FB(2)3445 3487 y Fx(\))p Fw(=)p Fx(2.)40 b(There)-118 3608 y(is)30 b(0)e Fr(\024)g Fw(n)g Fr(\024)g Fw(r)22 b Fr(\000)d Fx(1)31 b(suc)m(h)h(that)f Fw(S)1087 3571 y Fv(n)1134 3608 y Fw(e)1179 3623 y FB(2)1238 3608 y Fr(^)19 b Fw(e)1368 3623 y FB(2)1435 3608 y Fr(6)p Fx(=)28 b(0,)j(so)g Fw(M)1858 3623 y Fv(S)1940 3608 y Fx(is)g(p)s(ositiv)m(e)f (de\014nite)h(and)h(can)f(b)s(e)g(diagonalized)e(b)m(y)-118 3728 y(a)j(rotation:)524 4018 y Fw(M)618 4033 y Fv(S)724 4018 y Fx(=)856 3878 y Fq(\022)970 3957 y Fx(cos)18 b Fw(\015)88 b Fr(\000)17 b Fx(sin)f Fw(\015)976 4078 y Fx(sin)g Fw(\015)135 b Fx(cos)17 b Fw(\015)1585 3878 y Fq(\023)g(\022)1789 3957 y Fw(\026)1848 3916 y FB(+)1848 3984 y Fv(S)2025 3957 y Fx(0)1824 4078 y(0)117 b Fw(\026)2049 4037 y Fu(\000)2049 4105 y Fv(S)2149 3878 y Fq(\023)17 b(\022)2354 3957 y Fx(cos)g Fw(\015)88 b Fr(\000)17 b Fx(sin)g Fw(\015)2360 4078 y Fx(sin)f Fw(\015)135 b Fx(cos)17 b Fw(\015)2968 3878 y Fq(\023)3042 3899 y Fu(\000)p FB(1)3185 4018 y Fx(.)-118 4258 y(Hence)44 b Fh(H)254 4273 y Fv(S)363 4258 y Fx(is)e(unitarily)e(equiv)-5 b(alen)m(t)42 b(to)g(the)h (harmonic)e(oscillator)f(Hamiltonian)f(\()p Fw(\026)3193 4216 y FB(+)3193 4285 y Fv(S)3251 4258 y Fw(P)3328 4221 y FB(2)3396 4258 y Fx(+)29 b Fw(\026)3560 4216 y Fu(\000)3560 4285 y Fv(S)3618 4258 y Fw(Q)3695 4221 y FB(2)3735 4258 y Fx(\))p Fw(=)p Fx(2.)-118 4378 y(Therefore,)88 4778 y Fh(H)160 4793 y Fv(S)283 4778 y Fx(=)55 b Fw(\026)543 4654 y Fu(1)506 4684 y Fq(X)511 4893 y Fv(n)p FB(=0)683 4638 y Fq(\022)756 4778 y Fw(n)23 b Fx(+)945 4711 y(1)p 945 4756 V 945 4847 a(2)1003 4638 y Fq(\023)1110 4778 y Fr(j)p Fw(\036)1196 4728 y FB(\()p Fv(n)p FB(\))1196 4806 y Fv(S)1297 4778 y Fr(ih)p Fw(\036)1433 4728 y FB(\()p Fv(n)p FB(\))1433 4806 y Fv(S)1534 4778 y Fr(j)17 b Fw(;)211 b(\026)55 b Fx(=)h(\()p Fw(\026)2160 4737 y FB(+)2160 4806 y Fv(S)2218 4778 y Fw(\026)2277 4737 y Fu(\000)2277 4806 y Fv(S)2336 4778 y Fx(\))2374 4737 y FB(1)p Fv(=)p FB(2)2517 4778 y Fx(,)227 b Fw(\025)56 b Fx(=)3015 4638 y Fq(\022)3098 4711 y Fw(\026)3157 4670 y FB(+)3157 4738 y Fv(S)p 3098 4756 118 4 v 3098 4849 a Fw(\026)3157 4808 y Fu(\000)3157 4876 y Fv(S)3226 4638 y Fq(\023)3299 4656 y FB(1)p Fv(=)p FB(4)3458 4778 y Fx(,)239 b(\(30\))-118 5112 y(where)34 b(the)f Fw(\036)390 5061 y FB(\()p Fv(n)p FB(\))390 5139 y Fv(S)524 5112 y Fx(are)f(the)h(eigenstates.)45 b(The)33 b(ground)g(state)g(is)f(denoted)h Fw(\036)2681 5127 y Fv(S)2776 5112 y Fr(\021)28 b Fw(\036)2939 5061 y FB(\(0\))2939 5139 y Fv(S)3033 5112 y Fx(.)1841 5635 y(10)p eop %%Page: 11 11 11 10 bop -118 22 a Fl(Prop)s(osition)35 b(9)49 b Fm(Up)35 b(to)g(a)g(phase,)f(the)h(gr)-5 b(ound)34 b(state)i(is)e(given)g(by)322 313 y Fw(\036)380 328 y Fv(S)447 313 y Fx(\()p Fw(x)p Fx(\))56 b(=)765 172 y Fq(\022)848 245 y Fr(<)p Fw(e)p Fx(\()p Fw(\033)1058 260 y Fv(S)1110 245 y Fx(\))p 848 290 300 4 v 969 381 a Fw(\031)1158 172 y Fq(\023)1231 194 y FB(1)p Fv(=)p FB(4)1385 313 y Fw(e)1430 271 y Fu(\000)p Fv(\033)1525 282 y Fp(S)1582 271 y Fv(x)1622 248 y Fn(2)1656 271 y Fv(=)p FB(2)1748 313 y Fw(;)253 b(\033)2083 328 y Fv(s)2175 313 y Fx(=)2316 158 y Fq(p)p 2416 158 118 4 v 87 x Fw(\026)2475 204 y Fu(\000)2475 272 y Fv(S)2567 245 y Fx(cos)17 b Fw(\015)27 b Fx(+)22 b Fw({)2921 158 y Fq(p)p 3021 158 V 87 x Fw(\026)3080 204 y FB(+)3080 272 y Fv(S)3155 245 y Fx(sin)17 b Fw(\015)p 2316 290 1032 4 v 2316 310 a Fq(p)p 2416 310 118 4 v 87 x Fw(\026)2475 356 y FB(+)2475 424 y Fv(S)2567 397 y Fx(cos)g Fw(\015)27 b Fx(+)22 b Fw({)2921 310 y Fq(p)p 3021 310 V 87 x Fw(\026)3080 356 y Fu(\000)3080 424 y Fv(S)3155 397 y Fx(sin)17 b Fw(\015)3393 313 y Fm(,)301 b Fx(\(31\))-118 603 y Fm(and)34 b(the)h(Mehler)g(kernel)f Fx(\(10\))g Fm(by)114 1038 y Fr(M)234 1053 y Fv(S)285 1038 y Fx(\()p Fw(t)p Fx(;)17 b Fw(x;)g(y)t Fx(\))54 b(=)1006 971 y Fw(e)1051 913 y Fu(\000)1116 880 y Fn(\()p Fp(x)p Fo(\000)p Fp(y)r Fn(\))1281 859 y(2)1325 880 y(tanh)10 b(\()p Fp(t\026)p Fn(\))1566 859 y Fo(\000)p Fn(1)1649 880 y(+\()p Fp(x)p Fn(+)p Fp(y)r Fn(\))1860 859 y(2)1903 880 y(tanh)h(\()p Fp(t\026)p Fn(\))p 1116 898 1030 3 v 1302 947 a(4\()p Fp(\025)1393 933 y Fn(2)1439 947 y(cos)g Fp(\015)1566 933 y Fn(2)1601 947 y(+)p Fp(\025)1685 933 y Fo(\000)p Fn(2)1778 947 y(sin)f Fp(\015)1899 933 y Fn(2)1934 947 y(\))p 786 1015 1593 4 v 786 1120 a Fw(\025)843 1035 y Fq(p)p 943 1035 1436 4 v 85 x Fx(2)p Fw(\031)20 b Fx(sinh)d(\()p Fw(t\026)p Fx(\))o(\()p Fw(\025)1522 1091 y FB(2)1578 1120 y Fx(cos)g Fw(\015)1781 1091 y FB(2)1843 1120 y Fx(+)22 b Fw(\025)1998 1091 y Fu(\000)p FB(2)2109 1120 y Fx(sin)16 b Fw(\015)2301 1091 y FB(2)2341 1120 y Fx(\))2444 1038 y Fw(e)2489 981 y Fv({)p FB(\()p Fv(x)2579 957 y Fn(2)2614 981 y Fu(\000)p Fv(y)2706 957 y Fn(2)2741 981 y FB(\))2863 947 y Fn(sin)10 b(\(2)p Fp(\015)s Fn(\))q(\()p Fp(\025)3124 926 y Fn(2)3158 947 y Fo(\000)p Fp(\025)3243 926 y Fo(\000)p Fn(2)3326 947 y(\))p 2778 965 656 3 v 2778 1015 a(4\()p Fp(\025)2869 1001 y Fn(2)2915 1015 y(cos)h Fp(\015)3042 1001 y Fn(2)3077 1015 y(+)p Fp(\025)3161 1001 y Fo(\000)p Fn(2)3254 1015 y(sin)f Fp(\015)3375 1001 y Fn(2)3410 1015 y(\))3465 1038 y Fw(:)232 b Fx(\(32\))28 1391 y(By)40 b(construction,)g Fr(F)853 1406 y Fv(S)904 1391 y Fh(H)976 1406 y Fv(S)1043 1391 y Fr(F)1125 1355 y Fu(\003)1115 1416 y Fv(S)1203 1391 y Fx(=)e Fh(H)1389 1406 y Fv(S)1457 1391 y Fx(,)i(so)f(that)g Fr(F)1940 1406 y Fv(S)1990 1391 y Fw(\036)2048 1406 y Fv(S)2153 1391 y Fx(=)f Fw(e)2312 1355 y Fv({\016)2366 1366 y Fp(S)2416 1391 y Fw(\036)2474 1406 y Fv(S)2580 1391 y Fx(for)g(some)g(phase)i Fw(\016)3306 1406 y Fv(S)3357 1391 y Fx(.)62 b(Th)m(us,)42 b(it)c(is)-118 1511 y(p)s(ossible)31 b(to)h(c)m(ho)s(ose)h(the)g(phase)g(of)f Fr(F)1298 1526 y Fv(S)1381 1511 y Fx(suc)m(h)h(that)f Fr(F)1883 1526 y Fv(S)1934 1511 y Fw(\036)1992 1526 y Fv(S)2086 1511 y Fx(=)c Fw(\036)2248 1526 y Fv(S)2315 1511 y Fx(.)43 b(Suc)m(h)34 b(is)e(the)g(case)h(for)f Fr(F)3313 1526 y Fv(S)3356 1536 y Fp(i)3418 1511 y Fx(in)g(eqs.)h(\(28\))-118 1632 y(and)g(\(29\).)28 1799 y(Recall)d(from)f(Section)i(2)g(that)f Fr(\006)p Fl(1)f Fr(6)p Fx(=)e Fw(S)34 b Fr(2)28 b Fw(S)6 b(L)p Fx(\(2)p Fw(;)17 b Fs(Z)p Fx(\))27 b(is)k(called)f(a)g(symmetry)h (of)g Fr(A)3123 1814 y Fv(\022)3192 1799 y Fx(if)f(sup)3427 1823 y Fv(n)p Fu(2)p Ft(Z)3582 1799 y Fr(k)p Fw(S)3698 1763 y Fv(n)3744 1799 y Fr(k)e Fw(<)-118 1920 y Fr(1)p Fx(.)67 b(Since)40 b(the)h(set)g(of)f Fw(M)52 b Fr(2)41 b Fw(S)6 b(L)p Fx(\(2)p Fw(;)17 b Fs(Z)p Fx(\))38 b(with)i Fr(k)p Fw(M)10 b Fr(k)41 b(\024)h Fw(c)e Fx(is)g(\014nite)g(\(for)g(an) m(y)h(0)g Fw(<)g(c)g(<)g Fr(1)p Fx(\),)h(and)e(since)-118 2040 y Fw(S)48 b Fr(6)p Fx(=)43 b Fr(\006)p Fl(1)p Fx(,)h(there)e(is)f (an)h(in)m(teger)f Fw(r)k Fr(2)e Fs(N)1418 2055 y Fu(\003)1505 2040 y Fx(suc)m(h)g(that)e Fw(S)2020 2004 y Fv(r)2101 2040 y Fx(=)h Fl(1)g Fx(and)f Fw(S)2581 2004 y Fv(n)2671 2040 y Fr(6)p Fx(=)h Fl(1)g Fx(for)f(0)h Fw(<)h(n)g(<)f(r)s Fx(.)70 b(So)41 b(the)-118 2160 y(t)m(w)m(o)i(eigen)m(v)-5 b(alues)43 b(are)f Fr(f)p Fw(e)858 2124 y Fu(\006)p Fv({')982 2132 y Fp(s)1020 2160 y Fr(g)p Fx(,)j(with)d Fw(r)s(')1485 2175 y Fv(s)1566 2160 y Fx(=)j(0)f(\(mo)s(d)32 b(2)p Fw(\031)t Fx(\))42 b(and)h Fw(')2490 2175 y Fv(s)2571 2160 y Fr(6)p Fx(=)i(0)p Fw(;)17 b(\031)t Fx(.)72 b(In)43 b(particular)e(T)-8 b(r\()p Fw(S)6 b Fx(\))45 b(=)-118 2281 y(2)17 b(cos)f Fw(')158 2296 y Fv(s)246 2281 y Fr(2)52 b Fs(Z)p Fx(,)47 b(implying)c Fw(r)54 b Fr(2)d(f)p Fx(3)p Fw(;)17 b Fx(4)p Fw(;)g Fx(6)p Fr(g)45 b Fx(and)h Fw(')1786 2296 y Fv(s)1874 2281 y Fr(2)52 b(f\006)p Fw(\031)t(=)p Fx(3)17 b Fw(;)g Fr(\006)p Fw(\031)t(=)p Fx(2)g Fw(;)g Fr(\006)p Fx(2)p Fw(\031)t(=)p Fx(3)p Fr(g)p Fx(.)82 b(An)m(y)47 b Fw(S)57 b Fr(2)52 b Fw(S)6 b(L)p Fx(\(2)p Fw(;)17 b Fs(Z)p Fx(\))-118 2401 y(de\014nes)38 b(a)e Fr(\003)p Fx(-automorphism)h Fq(b)-59 b Fw(\021)1070 2416 y Fv(S)1157 2401 y Fx(of)36 b Fr(A)1352 2416 y Fv(\022)1427 2401 y Fx(through)j Fq(b)-58 b Fw(\021)1847 2416 y Fv(S)1898 2401 y Fx(\()p Fw(W)2028 2416 y Fv(\022)2067 2401 y Fx(\()p Fl(m)p Fx(\)\))33 b(=)h Fw(W)2510 2416 y Fv(\022)2549 2401 y Fx(\()p Fw(S)6 b Fl(m)p Fx(\).)53 b(According)36 b(to)g(the)h(ab)s(o)m(v)m(e,)-118 2522 y Fw(\031)-63 2537 y Fv(W)18 2522 y Fx(\()t Fq(b)-59 b Fw(\021)104 2537 y Fv(S)155 2522 y Fx(\()p Fw(W)285 2537 y Fv(\022)324 2522 y Fx(\()p Fl(m)p Fx(\)\)\))26 b(=)i Fr(F)772 2537 y Fv(S)822 2522 y Fw(\031)877 2537 y Fv(W)958 2522 y Fx(\()p Fw(W)1088 2537 y Fv(\022)1127 2522 y Fx(\()p Fw(S)6 b Fl(m)p Fx(\)\))p Fr(F)1483 2480 y Fu(\000)p FB(1)1473 2549 y Fv(S)1576 2522 y Fx(.)-118 2857 y Ff(3.4)135 b Fb(\022)s Ff(-traces)46 b(and)e Fb(\022)s Ff(-frames)-118 3042 y Fl(De\014nition)36 b(1)49 b Fm(A)36 b(ve)-5 b(ctor)35 b Fw( )e Fr(2)c(H)36 b Fm(wil)5 b(l)35 b(b)-5 b(e)36 b(c)-5 b(al)5 b(le)-5 b(d)34 b Fw(\022)s Fx(-tracial)f Fm(if)i Fr(h)p Fw( )t Fr(jW)2506 3057 y Fv(\022)2545 3042 y Fx(\()p Fl(l)p Fx(\))17 b Fr(j)p Fw( )t Fr(i)27 b Fx(=)i Fr(T)2990 3057 y Fv(\022)3029 3042 y Fx(\()p Fw(W)3159 3057 y Fv(\022)3198 3042 y Fx(\()p Fl(l)p Fx(\)\))f(=)g Fw(\016)3519 3057 y Fk(l)p Fv(;)p FB(0)3637 3042 y Fm(for)35 b(al)5 b(l)-118 3162 y Fl(l)27 b Fr(2)h Fs(Z)104 3126 y FB(2)141 3162 y Fm(.)44 b(Equivalently,)35 b(the)g(family)f Fx(\()p Fr(W)1391 3177 y Fv(\022)1431 3162 y Fx(\()p Fl(l)p Fx(\))p Fw( )t Fx(\))1644 3179 y Fk(l)p Fu(2)p Ft(Z)1763 3160 y Fn(2)1830 3162 y Fm(is)h(orthonormal.)28 3385 y Fx(Using)i(the)h(comm)m(utation)d(rules)i(\(12\),)h(it)e(is)g(p) s(ossible)h(to)g(c)m(hec)m(k)i(that)e Fw( )k Fx(is)c Fw(\022)s Fx(-tracial)e(if)h(and)h(only)g(if)-118 3505 y Fh(W)p Fx(\()p Fh(a)p Fx(\))p Fw( )i Fx(is)34 b Fw(\022)s Fx(-tracial)e(for)i(an)m(y)i Fh(a)31 b Fr(2)h Fs(R)1283 3469 y FB(2)1328 3505 y Fx(.)50 b(It)35 b(also)f(follo)m(ws)f(from)h (eq.)i(\(23\))e(that)g Fw( )39 b Fx(is)34 b Fw(\022)s Fx(-tracial)f(if)g(and)i(only)-118 3626 y(if)c Fw(T)42 3590 y Fv(\022)28 3652 y( )109 3626 y Fx(=)c(\()p Fw(\022)s(=)p Fx(2)p Fw(\031)t Fx(\))p Fl(1)p Fx(.)43 b(Suc)m(h)34 b Fw(\022)s Fx(-tracial)c(states)k(exist)f(under)g(the)g(follo)m(wing)d (condition:)-118 3849 y Fl(Theorem)37 b(2)49 b Fm(Ther)-5 b(e)41 b(is)h(a)h Fw(\022)s Fm(-tr)-5 b(acial)42 b(ve)-5 b(ctor)42 b Fw( )j Fr(2)d(H)i Fm(if)e(and)f(only)h(if)h Fw(\022)h Fr(\025)e Fx(2)p Fw(\031)t Fm(.)67 b(If)42 b Fw(\022)i(>)e Fx(2)p Fw(\031)j Fm(ther)-5 b(e)43 b(is)f(a)-118 3969 y Fw(\022)s Fm(-tr)-5 b(acial)37 b(ve)-5 b(ctor)37 b(in)g Fr(S)7 b Fx(\()p Fs(R)f Fx(\))p Fm(.)58 b(F)-7 b(or)36 b Fw(\022)g Fr(\025)d Fx(2)p Fw(\031)t Fm(,)k(denote)g(by)h Fx(\005)2047 3984 y Fv( )2137 3969 y Fm(the)f(pr)-5 b(oje)g(ction)37 b(on)g(the)g(ortho)-5 b(c)g(omplement)36 b(of)-118 4100 y(the)h Fw( )t Fm(-cyclic)f(subsp)-5 b(ac)g(e)p 814 4013 398 4 v 36 w Fw(\031)869 4115 y Fv(W)950 4100 y Fx(\()p Fr(A)1068 4115 y Fv(\022)1107 4100 y Fx(\))p Fw( )35 b Fr(\032)d(H)q Fm(.)51 b(Ther)-5 b(e)36 b(is)h(a)g(pr)-5 b(oje)g(ction)36 b Fw(P)2511 4115 y Fv( )2595 4100 y Fr(2)c(A)2773 4115 y Fv(\022)2808 4096 y Fo(0)2870 4100 y Fm(satisfying)37 b Fw(\031)3361 4115 y Fv(W)3442 4100 y Fx(\()p Fw(P)3543 4115 y Fv( )3595 4100 y Fx(\))32 b(=)f(\005)3845 4115 y Fv( )-118 4220 y Fm(and)j Fr(T)125 4235 y Fv(\022)160 4216 y Fo(0)187 4220 y Fx(\()p Fw(P)288 4235 y Fv( )340 4220 y Fx(\))28 b(=)f(1)22 b Fr(\000)h Fx(2)p Fw(\031)t(=\022)s Fm(.)44 b(In)35 b(p)-5 b(articular,)34 b Fw( )39 b Fm(is)c(also)f Fr(A)2048 4235 y Fv(\022)2087 4220 y Fm(-cyclic)g(for)h Fw(\022)30 b Fx(=)e(2)p Fw(\031)t Fm(.)-118 4443 y Fl(Pro)s(of:)93 b Fx(If)32 b Fw( )37 b Fx(is)32 b Fw(\022)s Fx(-tracial,)e(then)k(\()p Fw(\022)s(=)p Fx(2)p Fw(\031)t Fx(\))27 b(=)g Fr(h)p Fw( )t Fr(j)p Fw(T)1823 4407 y Fv(\022)1809 4469 y( )1862 4443 y Fr(j)p Fw( )t Fr(i)g Fx(=)2126 4368 y Fq(P)2231 4472 y Fk(l)p Fu(2)p Ft(Z)2351 4453 y Fn(2)2417 4443 y Fr(jhW)2582 4458 y Fv(\022)2617 4439 y Fo(0)2643 4443 y Fx(\()p Fl(l)p Fx(\))p Fw( )20 b Fr(j)p Fw( )t Fr(ij)2996 4407 y FB(2)3062 4443 y Fr(\025)28 b(k)p Fw( )t Fr(k)3334 4407 y FB(2)3401 4443 y Fx(=)g(1.)28 4563 y(If)34 b Fw(\022)f(>)d Fx(2)p Fw(\031)t Fx(,)k(for)g(0)c Fw(<)g(")f(<)h Fx(min)16 b(\(2)p Fw(\031)t(;)h(\022)24 b Fr(\000)f Fx(2)p Fw(\031)t Fx(\),)34 b(let)f Fw(\036)h Fx(b)s(e)g(a)g Fw(C)2271 4527 y Fu(1)2379 4563 y Fx(function)g(on)g Fs(R)45 b Fx(suc)m(h)35 b(that)f(0)c Fr(\024)g Fw(\036)g Fr(\024)h Fx(1,)-118 4684 y(with)36 b(supp)s(ort)h(in)f([0)p Fw(;)17 b Fx(2)p Fw(\031)28 b Fx(+)c Fw(")p Fx(],)38 b(suc)m(h)g(that)e Fw(\036)e Fx(=)g(1)j(on)f([)p Fw(";)17 b Fx(2)p Fw(\031)t Fx(],)37 b(and)f Fw(\036)p Fx(\()p Fw(x)p Fx(\))2645 4648 y FB(2)2710 4684 y Fx(+)24 b Fw(\036)p Fx(\()p Fw(x)h Fx(+)g(2)p Fw(\031)t Fx(\))3233 4648 y FB(2)3307 4684 y Fx(=)34 b(1)i(whenev)m(er)-118 4804 y(0)27 b Fr(\024)h Fw(x)h Fr(\024)f Fw(")p Fx(.)42 b(Using)30 b(\(22\),)h Fw(\036)f Fx(is)g Fw(\022)s Fx(-tracial)e(\(after)i(normalization\),)d(and)j(b)s (elongs)g(to)g Fr(S)7 b Fx(\()p Fs(R)f Fx(\).)49 b(If)30 b Fw(\022)h Fx(=)d(2)p Fw(\031)t Fx(,)i(the)-118 4925 y(same)i(argumen)m(t)h(holds)f(with)g Fw(")27 b Fx(=)h(0.)43 b(Then)34 b Fw(\036)27 b Fr(2)h(H)q Fx(,)33 b(but)g(it)e(is)i(not)f (smo)s(oth)g(an)m(ymore.)28 5045 y(Let)c Fw( )j Fx(b)s(e)c Fw(\022)s Fx(-tracial.)39 b(Exc)m(hanging)28 b(the)g(r^)-49 b(oles)26 b(of)h Fw(\022)j Fx(and)e Fw(\022)2160 5009 y Fu(0)2183 5045 y Fx(,)g(the)g(P)m(oisson)g(summation)d(form)m(ula)g (implies)406 5299 y Fw(T)477 5258 y Fv(\022)463 5324 y( )516 5238 y Fu(0)584 5299 y Fx(=)729 5205 y Fq(X)704 5422 y Fk(m)p Fu(2)p Ft(Z)869 5403 y Fn(2)931 5299 y Fr(W)1029 5314 y Fv(\022)1068 5299 y Fx(\()p Fl(m)p Fx(\))p Fr(j)p Fw( )t Fr(ih)p Fw( )t Fr(jW)1604 5314 y Fv(\022)1642 5299 y Fx(\()p Fl(m)p Fx(\))1812 5258 y Fu(\000)p FB(1)1961 5299 y Fx(=)2102 5232 y(2)p Fw(\031)p 2102 5276 108 4 v 2132 5368 a(\022)2255 5205 y Fq(X)2253 5422 y Fk(l)p Fu(2)p Ft(Z)2373 5403 y Fn(2)2418 5299 y Fr(h)p Fw( )t Fr(jW)2650 5314 y Fv(\022)2685 5295 y Fo(0)2711 5299 y Fx(\()p Fl(l)p Fx(\))2819 5258 y Fu(\000)p FB(1)2913 5299 y Fr(j)p Fw( )t Fr(i)17 b(W)3162 5314 y Fv(\022)3197 5295 y Fo(0)3222 5299 y Fx(\()p Fl(l)p Fx(\))g Fw(:)1841 5635 y Fx(11)p eop %%Page: 12 12 12 11 bop -118 46 a Fx(Hence)33 b(\005)244 61 y Fv( )324 46 y Fx(=)28 b(1)20 b Fr(\000)h Fw(T)666 10 y Fv(\022)652 72 y( )705 -10 y Fu(0)760 46 y Fx(is)31 b(the)h(desired)g(orthonormal)e (pro)5 b(jection)31 b(whic)m(h,)i(due)f(to)f(the)h(r.h.s.,)h(is)e(the)i (W)-8 b(eyl)-118 166 y(represen)m(tativ)m(e)40 b(of)d(an)h(elemen)m(t)f Fw(P)1199 181 y Fv( )1288 166 y Fr(2)g(A)1471 181 y Fv(\022)1506 162 y Fo(0)1532 166 y Fx(.)59 b(Its)39 b(trace)f(is)f Fr(T)2174 181 y Fv(\022)2209 162 y Fo(0)2236 166 y Fx(\()p Fw(P)2337 181 y Fv( )2389 166 y Fx(\))g(=)f(1)25 b Fr(\000)i Fx(2)p Fw(\031)t(=\022)s Fx(.)58 b(If)38 b Fw(\022)i Fx(=)c(2)p Fw(\031)t Fx(,)j(since)f(the)-118 295 y(trace)33 b(is)f(faithful,)e Fw(T)663 259 y Fv(\022)649 321 y( )702 239 y Fu(0)753 295 y Fx(=)e Fl(1)p Fx(,)k(so)h(that)g Fw( )j Fx(is)c(cyclic.)2059 b Fa(2)-118 571 y Fl(De\014nition)36 b(2)49 b Fm(A)35 b(ve)-5 b(ctor)35 b Fw( )d Fr(2)c(H)36 b Fm(is)f(c)-5 b(al)5 b(le)-5 b(d)34 b(a)g Fw(\022)1720 535 y Fu(0)1744 571 y Fx(-frame)p Fm(,)f(if)i(ther)-5 b(e)35 b(ar)-5 b(e)35 b(c)-5 b(onstants)34 b Fx(0)28 b Fw(<)f(c)h(<)g(C)35 b(<)27 b Fr(1)35 b Fm(such)-118 691 y(that)g Fw(c)p Fl(1)28 b Fr(\024)g Fw(T)383 655 y Fv(\022)369 717 y( )450 691 y Fr(\024)g Fw(C)23 b Fl(1)p Fm(.)28 919 y Fx(This)34 b(de\014nition)e(is)h(in)g(accordance)i(with)e (the)h(literature)e(\([Sei)o(])i(and)f(references)j(therein\))d(where)i (the)-118 1040 y(complete)42 b(set)h(\()p Fr(W)602 1055 y Fv(\022)637 1036 y Fo(0)664 1040 y Fx(\()p Fl(l)p Fx(\))p Fw( )t Fx(\))877 1056 y Fk(l)p Fu(2)p Ft(Z)996 1038 y Fn(2)1071 1040 y Fx(is)f(called)g(a)g(frame.)73 b(The)44 b(principal)d(in)m(terest)i(of)f(frames)g(is)g(due)i(to)e(the)-118 1172 y(follo)m(wing:)48 b(an)m(y)37 b(v)m(ector)g Fw(\036)c Fr(2)i(H)i Fx(can)f(b)s(e)g(decomp)s(osed)h(as)g Fw(\036)c Fx(=)h Fw(T)2399 1136 y Fv(\022)2385 1198 y( )2438 1172 y Fx(\()p Fw(T)2547 1136 y Fv(\022)2533 1198 y( )2585 1172 y Fx(\))2623 1116 y Fu(\000)p FB(1)2718 1172 y Fw(\036)f Fx(=)2919 1097 y Fq(P)3024 1201 y Fk(l)p Fu(2)p Ft(Z)3144 1182 y Fn(2)3193 1172 y Fw(c)3235 1187 y Fk(l)3278 1172 y Fr(W)3376 1187 y Fv(\022)3411 1168 y Fo(0)3438 1172 y Fx(\()p Fl(l)p Fx(\))p Fw( )39 b Fx(where)-118 1318 y Fw(c)-76 1333 y Fk(l)-21 1318 y Fx(=)28 b Fr(h)p Fw( )t Fr(jW)315 1333 y Fv(\022)350 1314 y Fo(0)376 1318 y Fx(\()p Fl(l)p Fx(\))484 1282 y Fu(\003)523 1318 y Fx(\()p Fw(T)632 1282 y Fv(\022)618 1344 y( )670 1318 y Fx(\))708 1262 y Fu(\000)p FB(1)803 1318 y Fr(j)p Fw(\036)p Fr(i)p Fx(.)43 b(If)33 b Fw( )f Fr(2)d(S)7 b Fx(\()p Fs(R)f Fx(\))38 b(and)33 b Fw(\036)28 b Fr(2)h(S)7 b Fx(\()p Fs(R)f Fx(\),)39 b(then)33 b(\()p Fw(c)2483 1333 y Fk(l)2510 1318 y Fx(\))2548 1334 y Fk(l)p Fu(2)p Ft(Z)2668 1315 y Fn(2)2729 1318 y Fr(2)28 b Fh(s)p Fx(\()p Fs(Z)2974 1282 y FB(2)3011 1318 y Fx(\).)45 b(F)-8 b(urther)33 b(note)g(that,)-118 1471 y(if)41 b Fw( )46 b Fx(is)41 b(a)h Fw(\022)336 1434 y Fu(0)360 1471 y Fx(-frame,)h(then)956 1444 y(^)936 1471 y Fw( )48 b Fx(=)c(\()p Fw(\022)s(=)p Fx(2)p Fw(\031)t Fx(\))1448 1434 y FB(1)p Fv(=)p FB(2)1557 1471 y Fx(\()p Fw(T)1666 1434 y Fv(\022)1652 1497 y( )1705 1471 y Fx(\))1743 1415 y Fu(\000)p FB(1)p Fv(=)p FB(2)1908 1471 y Fw( )i Fx(is)c Fw(\022)s Fx(-tracial.)69 b(In)43 b(addition,)f(if)f Fw( )48 b Fr(2)c(S)7 b Fx(\()p Fs(R)f Fx(\))48 b(then)-98 1587 y(^)-118 1613 y Fw( )32 b Fr(2)c(S)7 b Fx(\()p Fs(R)e Fx(\).)28 1781 y(The)33 b(next)g(result)f(sho)m(ws)i(that)d(so-called)g (W)-8 b(eyl-Heisen)m(b)s(erg)32 b(or)g(Gab)s(or)f(lattices)g (constructed)i(with)f(a)-118 1901 y(gaussian)27 b(mother)f(state)h(are) g(frames)g(if)f(only)g(the)i(v)m(olume)e(of)h(the)g(c)m(hosen)i (phase-space)f(cell)e(is)h(su\016cien)m(tly)-118 2022 y(small.)41 b(This)33 b(w)m(as)g(pro)m(v)m(ed)h(in)e([Sei],)h(but)f (the)h(presen)m(t)i(pro)s(of)c(is)i(new)g(and)g(co)m(v)m(ers)h(more)e (general)g(cases.)28 2189 y(Supp)s(ose)42 b Fw(S)49 b Fr(2)43 b Fw(S)6 b(L)p Fx(\(2)p Fw(;)17 b Fs(R)t Fx(\))47 b(satis\014es)42 b Fw(S)1484 2153 y Fv(r)1564 2189 y Fx(=)h(1)e(for)f(some)h Fw(r)s Fx(.)70 b(Using)40 b(the)i(results)g(of) f(Section)g(3.3)f(and)-118 2310 y(eq.)33 b(\(11\),)f(it)g(is)g(p)s (ossible)g(to)g(compute)682 2591 y Fr(h)p Fw(\036)779 2606 y Fv(S)846 2591 y Fr(j)p Fh(W)p Fx(\()p Fh(a)p Fx(\))p Fr(j)p Fw(\036)1190 2606 y Fv(S)1256 2591 y Fr(i)55 b Fx(=)g Fw(e)1526 2550 y Fu(\000j)p Fg(a)11 b Fu(j)1667 2527 y Fn(2)1667 2573 y Fp(S)1711 2550 y Fv(=)p FB(4)1814 2591 y Fw(;)253 b Fr(j)p Fh(a)17 b Fr(j)2217 2550 y FB(2)2217 2616 y Fv(S)2322 2591 y Fx(=)2463 2524 y Fw(\026)2522 2483 y FB(+)2522 2551 y Fv(S)2597 2524 y Fw(a)2648 2488 y FB(2)2648 2548 y(1)2710 2524 y Fx(+)22 b Fw(\026)2867 2483 y Fu(\000)2867 2551 y Fv(S)2942 2524 y Fw(a)2993 2488 y FB(2)2993 2548 y(2)p 2463 2568 571 4 v 2719 2660 a Fw(\026)3071 2591 y(:)626 b Fx(\(33\))-118 2870 y Fl(Theorem)37 b(3)49 b Fm(F)-7 b(or)34 b Fw(\022)d(>)c Fx(2)p Fw(\031)t Fm(,)35 b Fw(\036)1054 2885 y Fv(S)1156 2870 y Fm(is)f(a)h Fw(\022)1393 2834 y Fu(0)1416 2870 y Fm(-fr)-5 b(ame)34 b(in)h Fr(S)7 b Fx(\()p Fs(R)f Fx(\))p Fm(.)-118 3098 y Fl(Pro)s(of:)93 b Fx(The)33 b(pro)s(of)f(b)s(elo)m(w)h(is)f(giv)m(en) g(for)g Fw(\036)1569 3113 y FB(0)1653 3098 y Fr(\021)c Fw(\036)1816 3113 y Fv(S)1859 3122 y Fn(4)1897 3098 y Fx(,)33 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Fr(A)2284 3474 y Fv(\022)2323 3459 y Fx(.)82 b(W)-8 b(riting)44 b Fw(\022)53 b Fx(=)c(2)p Fw(\031)35 b Fx(+)c Fw(\016)49 b Fx(with)c Fw(\016)54 b(>)c Fx(0,)-118 3580 y(there)c(is)e(a)h Fr(\003)p Fx(-isomorphism)d(b)s(et)m(w)m(een)48 b Fr(A)1483 3595 y Fv(\022)1566 3580 y Fx(and)e(the)f(closed)h(subalgebra)f(of)f Fr(A)2954 3595 y FB(2)p Fv(\031)3067 3580 y Fr(\012)31 b(A)3255 3595 y Fv(\016)3338 3580 y Fx(generated)46 b(b)m(y)-118 3700 y(\()p Fw(W)12 3715 y FB(2)p Fv(\031)94 3700 y Fx(\()p Fl(m)p Fx(\))28 b Fr(\012)i Fw(W)491 3715 y Fv(\016)529 3700 y Fx(\()p Fl(m)p Fx(\)\))737 3717 y Fk(m)p Fu(2)p Ft(Z)901 3698 y Fn(2)934 3700 y Fx(.)73 b(It)42 b(is)g(enough)h(to)f (sho)m(w)i(that)2224 3675 y(^)2205 3700 y Fw(T)2262 3715 y FB(0)2346 3700 y Fx(=)2467 3625 y Fq(P)2572 3729 y Fk(m)2660 3700 y Fw(e)2705 3664 y Fu(\000)p Fv(\022)r Fu(j)p Fk(m)p Fu(j)2904 3641 y Fn(2)2937 3664 y Fv(=)p FB(4)3012 3700 y Fw(W)3104 3715 y FB(2)p Fv(\031)3186 3700 y Fx(\()p Fl(m)p Fx(\))28 b Fr(\012)h Fw(W)3582 3715 y Fv(\016)3620 3700 y Fx(\()p Fl(m)p Fx(\))42 b(is)-118 3820 y(b)s(ounded)30 b(from)e(b)s(elo)m(w)h(in)f Fr(A)967 3835 y FB(2)p Fv(\031)1065 3820 y Fr(\012)15 b(A)1237 3835 y Fv(\016)1275 3820 y Fx(.)42 b Fr(A)1424 3835 y FB(2)p Fv(\031)1535 3820 y Fx(is)29 b(ab)s(elian)f(and)h Fr(\003)p Fx(-isomorphic)e(to)i Fw(C)7 b Fx(\()p Fs(T)3019 3784 y FB(2)3061 3820 y Fx(\),)30 b(pro)m(vided)g Fw(W)3646 3835 y FB(2)p Fv(\031)3728 3820 y Fx(\()p Fl(m)p Fx(\))-118 3941 y(is)h(iden)m(ti\014ed)g(with)h(the)g(map)e Fw(\024)e Fx(=)g(\()p Fw(\024)1289 3956 y FB(1)1328 3941 y Fw(;)17 b(\024)1428 3956 y FB(2)1467 3941 y Fx(\))28 b Fr(2)g Fs(T)1690 3905 y FB(2)1761 3941 y Fr(7!)f Fx(\()p Fr(\000)p Fx(1\))2090 3905 y Fv(m)2152 3914 y Fn(1)2187 3905 y Fv(m)2249 3914 y Fn(2)2288 3941 y Fw(e)2333 3905 y Fv({\024)p Fu(\001)p Fk(m)2517 3941 y Fr(2)h Fs(C)20 b Fx(.)49 b(Hence)33 b(it)d(is)h(enough)h(to)f(sho)m(w)-118 4061 y(that)114 4036 y(^)95 4061 y Fw(T)152 4076 y FB(0)192 4061 y Fx(\()p Fw(\024)p Fx(\))g(=)461 3987 y Fq(P)566 4090 y Fk(m)638 4061 y Fx(\()p Fr(\000)p Fx(1\))840 4025 y Fv(m)902 4034 y Fn(1)937 4025 y Fv(m)999 4034 y Fn(2)1055 4061 y Fw(e)1100 4025 y Fu(\000)p Fv(\022)r Fu(j)p Fk(m)p Fu(j)1299 4002 y Fn(2)1332 4025 y Fv(=)p FB(4)12 b(+)p Fv({\024)p Fu(\001)p Fk(m)1641 4061 y Fw(W)1733 4076 y Fv(\016)1771 4061 y Fx(\()p Fl(m)p Fx(\))34 b(is)g(b)s(ounded)h(from)e(b)s(elo)m(w)h(in)g Fr(A)3181 4076 y Fv(\016)3253 4061 y Fx(uniformly)e(in)h Fw(\024)p Fx(.)-118 4182 y(Since)39 b(the)g(W)-8 b(eyl)39 b(represen)m(tation)g(is)g(faithful,)f Fw(W)1786 4197 y Fv(\016)1824 4182 y Fx(\()p Fl(m)p Fx(\))g(can)h(b)s(e)g(replaced)g (b)m(y)h Fr(W)2987 4197 y Fv(\016)3025 4182 y Fx(\()p Fl(m)p Fx(\).)61 b(Using)39 b(eq.)g(\(13\))-118 4302 y(with)32 b Fw( )g Fx(=)27 b Fw(\036)360 4317 y FB(0)449 4302 y Fx(and)32 b Fh(a)c Fx(=)819 4217 y Fr(p)p 902 4217 47 4 v 85 x Fw(\016)t Fl(m)p Fx(,)k(it)g(is)g(th)m(us)i(enough)f (to)f(sho)m(w)i(that)505 4550 y(~)486 4576 y Fw(T)543 4591 y FB(0)583 4576 y Fx(\()p Fw(\024)p Fx(\))55 b(=)901 4440 y Fq(Z)957 4666 y Ft(R)1005 4647 y Fn(2)1098 4508 y Fw(d)1149 4472 y FB(2)1188 4508 y Fh(b)p 1098 4553 142 4 v 1115 4644 a Fx(2)p Fw(\031)1266 4576 y Fx(\002\()p Fw(\024)1436 4591 y FB(1)1497 4576 y Fx(+)1595 4486 y Fr(p)p 1678 4486 47 4 v 90 x Fw(\016)t(b)1766 4591 y FB(2)1806 4576 y Fw(;)17 b(\024)1906 4591 y FB(2)1967 4576 y Fr(\000)2067 4486 y(p)p 2150 4486 V 90 x Fw(\016)t(b)2238 4591 y FB(1)2278 4576 y Fx(\))g Fh(W)p Fx(\()p Fh(b)p Fx(\))p Fr(j)p Fw(\036)2650 4591 y FB(0)2705 4576 y Fr(ih)p Fw(\036)2841 4591 y FB(0)2896 4576 y Fr(j)p Fh(W)p Fx(\()p Fh(b)p Fx(\))3155 4535 y Fu(\000)p FB(1)3266 4576 y Fw(;)-118 4843 y Fx(where)1030 5063 y(\002\()p Fw(\024)p Fx(\))55 b(=)1449 4968 y Fq(X)1424 5185 y Fk(m)p Fu(2)p Ft(Z)1589 5166 y Fn(2)1618 5063 y Fx(\()p Fr(\000)p Fx(1\))1820 5022 y Fv(m)1882 5031 y Fn(1)1917 5022 y Fv(m)1979 5031 y Fn(2)2035 5063 y Fw(e)2080 5022 y Fu(\000)p Fv(\031)r Fu(j)p Fk(m)p Fu(j)2287 4998 y Fn(2)2319 5022 y Fv(=)p FB(2)13 b(+)p Fv({)p FB(\()p Fv(\024)p Fu(\001)p Fk(m)d FB(\))2706 5063 y Fw(;)991 b Fx(\(34\))1841 5635 y(12)p eop %%Page: 13 13 13 12 bop -118 22 a Fx(is)30 b(b)s(ounded)i(from)e(b)s(elo)m(w.)42 b(Clearly)30 b(the)i(function)e(\002)h(is)f(2)p Fw(\031)t Fx(-p)s(erio)s(dic)f(in)h(b)s(oth)g(of)h(its)f(argumen)m(ts.)43 b(Hence,)-118 142 y(decomp)s(osing)31 b(the)i(in)m(tegral)d(in)m(to)h (a)h(sum)g(of)g(in)m(tegrals)f(o)m(v)m(er)i(the)f(shifted)g(unit)g (cell)e Fw(C)35 b Fx(=)28 b([0)p Fw(;)17 b Fx(2)p Fw(\031)t Fx(\))j Fr(\002)h Fx([0)p Fw(;)c Fx(2)p Fw(\031)t Fx(\))-118 263 y(and)33 b(using)f Fr(W)425 278 y Fv(\016)458 259 y Fo(0)485 263 y Fx(\()p Fh(a)p Fx(\))c(=)f Fh(W)p Fx(\(2)p Fw(\031)t Fh(a)p Fw(=)1091 178 y Fr(p)p 1174 178 47 4 v 85 x Fw(\016)t Fx(\))32 b(giv)m(es)350 531 y(~)330 557 y Fw(T)387 572 y FB(0)427 557 y Fx(\()p Fw(\024)p Fx(\))55 b(=)748 462 y Fq(X)746 679 y Fk(l)p Fu(2)p Ft(Z)866 660 y Fn(2)911 421 y Fq(Z)966 647 y Fv(C)1087 489 y Fw(d)1138 453 y FB(2)1177 489 y Fh(a)p 1079 534 155 4 v 1079 625 a Fx(2)p Fw(\031)t(\016)1261 557 y Fx(\002\()p Fh(a)p Fx(\))17 b Fr(W)1578 572 y Fv(\016)1611 553 y Fo(0)1654 416 y Fq(\022)1727 557 y Fl(l)22 b Fx(+)1889 489 y Fh(a)g Fx(+)j(^)-52 b Fw(\024)p 1889 534 227 4 v 1948 625 a Fx(2)p Fw(\031)2125 416 y Fq(\023)2215 557 y Fr(j)p Fw(\036)2301 572 y FB(0)2356 557 y Fr(ih)p Fw(\036)2492 572 y FB(0)2547 557 y Fr(jW)2673 572 y Fv(\016)2706 553 y Fo(0)2750 416 y Fq(\022)2824 557 y Fl(l)21 b Fx(+)2985 489 y Fh(a)h Fx(+)k(^)-53 b Fw(\024)p 2985 534 V 3044 625 a Fx(2)p Fw(\031)3221 416 y Fq(\023)3294 438 y Fu(\000)p FB(1)3422 557 y Fw(;)-118 876 y Fx(where)50 b(^)-53 b Fw(\024)50 b Fx(=)f(\()p Fw(\024)501 891 y FB(2)540 876 y Fw(;)17 b Fr(\000)p Fw(\024)717 891 y FB(1)757 876 y Fx(\).)81 b(The)46 b(P)m(oisson)g(summation)d(form)m(ula)g(applied)h(to)h(the)g (summation)f(o)m(v)m(er)i Fw(m)3858 891 y FB(1)-118 996 y Fx(in)e(\(34\))h(giv)m(es)h(a)f(sum)g(o)m(v)m(er)i(an)e(index)h Fw(n)1496 1011 y FB(1)1535 996 y Fx(.)82 b(Changing)45 b(summation)e(indexes)k Fw(n)3029 1011 y FB(2)3118 996 y Fx(=)i Fw(m)3328 1011 y FB(2)3399 996 y Fr(\000)31 b Fw(n)3565 1011 y FB(1)3650 996 y Fx(sho)m(ws)-118 1117 y(that)43 b(\002\()p Fw(\024)p Fx(\))k(=)481 1034 y Fr(p)p 564 1034 49 4 v 83 x Fx(2)16 b Fw(e)674 1080 y Fu(\000)p Fv(\024)770 1057 y Fn(2)770 1101 y(1)805 1080 y Fv(=)p FB(2)p Fv(\031)939 1117 y Fr(j)p Fw(f)11 b Fx(\()p Fw(\024)1120 1132 y FB(1)1188 1117 y Fx(+)30 b Fw({\024)1381 1132 y FB(2)1421 1117 y Fx(\))p Fr(j)1487 1080 y FB(2)1526 1117 y Fx(,)46 b(where)f Fw(f)54 b Fx(is)43 b(the)h(holomorphic)d(en)m (tire)j(function)f(giv)m(en)h(b)m(y)-118 1237 y Fw(f)11 b Fx(\()p Fw(z)t Fx(\))28 b(=)197 1162 y Fq(P)303 1266 y Fv(n)p Fu(2)p Ft(Z)474 1237 y Fw(e)519 1201 y Fu(\000)p Fv(\031)14 b(n)672 1177 y Fn(2)718 1201 y Fu(\000)p Fv(nz)856 1237 y Fx(.)40 b(It)23 b(can)g(b)s(e)g(c)m(hec)m(k)m(ed)j(that)c Fw(f)11 b Fx(\()p Fw(z)c Fx(+)r(2)p Fw({\031)t Fx(\))27 b(=)h Fw(f)11 b Fx(\()p Fw(z)t Fx(\))23 b(and)g(that)f Fw(f)11 b Fx(\()p Fw(z)c Fx(+)r(2)p Fw(\031)t Fx(\))27 b(=)h Fw(e)3532 1201 y Fv(z)s FB(+)p Fv(\031)3686 1237 y Fw(f)11 b Fx(\()p Fw(z)t Fx(\).)-118 1357 y(Moreo)m(v)m(er,)39 b(using)d(the)i(P)m(oisson)f(summation)d(form)m(ula,)i Fw(f)47 b Fx(do)s(es)37 b(not)g(v)-5 b(anish)36 b(on)h Fw(\015)5 b Fx(,)37 b(the)h(b)s(oundary)f(of)f Fw(C)-118 1478 y Fx(orien)m(ted)f(clo)s(c)m(kwise.)53 b(As)36 b(\002)g(has)g(no)f (p)s(oles,)h(the)g(n)m(um)m(b)s(er)g(of)f(zeros)h(of)f Fw(f)46 b Fx(within)35 b Fw(C)42 b Fx(coun)m(ted)37 b(with)e(their)-118 1598 y(m)m(ultiplicit)m(y)e(is)j(giv)m(en)g(b)m(y)909 1518 y Fq(H)956 1633 y Fv(\015)1017 1598 y Fw(d)-16 b(f)11 b(=)p Fx(2)p Fw({\031)t(f)g Fx(.)54 b(Using)36 b(the)h(p)s(erio)s (dicit)m(y)d(prop)s(erties)j(of)f Fw(f)11 b Fx(,)37 b(this)f(in)m (tegral)f(equals)-118 1718 y(1.)42 b(Moreo)m(v)m(er,)32 b(a)e(direct)f(calculation)f(sho)m(ws)j(that)f(the)g(unique)h(zero)f (with)f(m)m(ultiplicit)m(y)d(1)k(of)f Fw(f)41 b Fx(lies)29 b(at)g(the)-118 1839 y(cen)m(ter)36 b Fw(\031)t Fx(\(1)24 b(+)g Fw({)p Fx(\))35 b(of)g Fw(C)7 b Fx(.)51 b(Hence)37 b(there)f(is)e(a)h(constan)m(t)i Fw(c)1985 1854 y FB(1)2056 1839 y Fw(>)32 b Fx(0)j(suc)m(h)i(that)e Fr(j)p Fw(f)11 b Fx(\()p Fw(\031)27 b Fx(+)d Fw({\031)k Fx(+)c Fw(r)s(e)3298 1803 y Fv({')3371 1839 y Fx(\))p Fr(j)32 b(\025)g Fw(c)3620 1854 y FB(1)3660 1839 y Fw(r)3707 1803 y FB(2)3781 1839 y Fx(for)-118 1959 y(all)e Fw(')e Fr(2)g Fx([0)p Fw(;)17 b Fx(2)p Fw(\031)t Fx(\).)43 b(Let)32 b Fw(B)787 1974 y Fv(r)858 1959 y Fx(denote)h(the)g(ball)e(of)h(size)h Fw(r)i Fx(around)e Fw(\031)t Fx(\(1)21 b(+)h Fw({)p Fx(\).)44 b(Replacing)31 b(this)h(sho)m(ws)499 2237 y(~)480 2262 y Fw(T)537 2277 y FB(0)576 2262 y Fx(\()p Fw(\024)p Fx(\))56 b Fr(\025)906 2195 y Fw(c)948 2210 y FB(1)988 2195 y Fw(r)1035 2159 y FB(2)p 906 2239 168 4 v 967 2331 a Fw(\016)1101 2092 y Fq( )1179 2262 y Fl(1)23 b Fr(\000)1357 2127 y Fq(Z)1413 2352 y Fv(B)1466 2360 y Fp(r)1531 2195 y Fw(d)1582 2159 y FB(2)1621 2195 y Fh(a)p 1531 2239 140 4 v 1547 2331 a Fx(2)p Fw(\031)1698 2262 y Fr(W)1796 2277 y Fv(\016)1829 2258 y Fo(0)1873 2122 y Fq(\022)1956 2195 y Fh(a)f Fx(+)k(^)-53 b Fw(\024)p 1956 2239 227 4 v 2015 2331 a Fx(2)p Fw(\031)2192 2122 y Fq(\023)2310 2262 y Fw(T)2381 2221 y Fv(\016)2414 2198 y Fo(0)2367 2287 y Fv(\036)2409 2296 y Fn(0)2487 2262 y Fr(W)2585 2277 y Fv(\016)2618 2258 y Fo(0)2663 2122 y Fq(\022)2746 2195 y Fh(a)22 b Fx(+)k(^)-53 b Fw(\024)p 2746 2239 V 2805 2331 a Fx(2)p Fw(\031)2982 2122 y Fq(\023)3055 2144 y Fu(\000)p FB(1)3150 2092 y Fq(!)3273 2262 y Fw(:)-118 2540 y Fx(As)35 b Fw(T)99 2504 y Fv(\016)132 2481 y Fo(0)85 2566 y Fv(\036)127 2575 y Fn(0)208 2540 y Fr(\024)c Fw(c)358 2555 y FB(2)397 2540 y Fl(1)p Fx(,)534 2515 y(~)515 2540 y Fw(T)572 2555 y FB(0)611 2540 y Fx(\()p Fw(\024)p Fx(\))g Fr(\025)g Fl(1)17 b Fw(c)997 2555 y FB(1)1036 2540 y Fw(r)1083 2504 y FB(2)1122 2540 y Fx(\(1)23 b Fr(\000)h Fw(c)1375 2555 y FB(2)1414 2540 y Fw(r)1461 2504 y FB(2)1500 2540 y Fw(=)p Fx(2\))p Fw(=\016)t Fx(.)48 b(Cho)s(osing)33 b Fw(r)k Fx(small)32 b(enough,)2956 2515 y(~)2937 2540 y Fw(T)2994 2555 y FB(0)3034 2540 y Fx(\()p Fw(\024)p Fx(\))i(is)g(b)s(ounded)h(from)-118 2661 y(b)s(elo)m(w)d(b)m(y)i(a)e(p) s(ositiv)m(e)g(constan)m(t)h(uniformly)e(in)h Fw(\024)p Fx(.)2052 b Fa(2)-118 3041 y Fy(4)161 b(Comparison)52 b(theorems)-118 3289 y Ff(4.1)135 b(Pro)t(of)45 b(of)g(Prop)t(osition)h (1)-118 3474 y Fx(Let)41 b Fw(H)50 b Fx(=)43 b Fw(H)404 3437 y Fu(\003)485 3474 y Fr(2)g(A)674 3489 y Fv(\022)754 3474 y Fx(and)e(set)h Fw(H)1194 3489 y Fv(W)1317 3474 y Fx(=)h Fw(\031)1491 3489 y Fv(W)1571 3474 y Fx(\()p Fw(H)8 b Fx(\).)69 b(F)-8 b(or)41 b(normalized)e Fw(\036)j Fr(2)h(H)q Fx(,)h Fw(\032)2938 3489 y Fv(\036)3026 3474 y Fx(denotes)e(the)g(sp)s(ectral)-118 3594 y(measure)33 b(of)f Fw(H)454 3609 y Fv(W)567 3594 y Fx(relativ)m(e)g(to)g Fw(\036)p Fx(.)43 b(Prop)s(osition)31 b(1)h(is)g(a)h(corollary)d(of)i (the)h(follo)m(wing)d(result:)-118 3822 y Fl(Theorem)37 b(4)49 b Fm(F)-7 b(or)34 b Fw(\022)d Fr(\025)d Fx(2)p Fw(\031)t Fm(,)34 b(for)h(any)g(normalize)-5 b(d)33 b Fw(\022)1882 3786 y Fu(0)1905 3822 y Fm(-fr)-5 b(ame)35 b Fw(\036)27 b Fr(2)h(H)36 b Fm(and)e(any)h(Bor)-5 b(el)34 b(subset)h Fx(\001)g Fm(of)g Fs(R)5 b Fm(,)823 4014 y Fx(2)p Fw(\031)p 823 4058 108 4 v 853 4150 a(\022)941 4081 y Fr(k)p Fx(\()p Fw(T)1100 4040 y Fv(\022)1086 4106 y(\036)1139 4081 y Fx(\))1177 4020 y Fu(\000)p FB(1)1271 4081 y Fr(k)1321 4040 y Fu(\000)p FB(1)1432 4081 y Fr(N)15 b Fx(\(\001\))55 b Fr(\024)g Fw(\032)1923 4096 y Fv(\036)1970 4081 y Fx(\(\001\))g Fr(\024)2325 4014 y Fx(2)p Fw(\031)p 2325 4058 V 2355 4150 a(\022)2443 4081 y Fr(k)p Fw(T)2564 4040 y Fv(\022)2550 4106 y(\036)2602 4081 y Fr(k)17 b(N)e Fx(\(\001\))i Fw(:)757 b Fx(\(35\))-118 4341 y Fl(Pro)s(of:)93 b Fx(Eq.)33 b(\(24\))f(leads)h(to)1000 4561 y Fw(\032)1050 4576 y Fv(\036)1096 4561 y Fx(\(\001\))56 b(=)f Fr(T)1494 4576 y Fv(\022)1550 4480 y Fq(\000)1596 4561 y Fw(\037)1657 4576 y FB(\001)1720 4561 y Fx(\()p Fw(H)8 b Fx(\))17 b Fw(F)1979 4519 y Fv(\022)1965 4585 y(\036)2016 4480 y Fq(\001)2117 4561 y Fr(\024)56 b(k)p Fw(F)2377 4519 y Fv(\022)2363 4585 y(\036)2416 4561 y Fr(k)17 b(N)e Fx(\(\001\))i Fw(;)-118 4781 y Fx(and)33 b(to)758 4901 y Fr(N)15 b Fx(\(\001\))55 b(=)g Fr(T)1252 4916 y Fv(\022)1308 4790 y Fq(\020)1368 4901 y Fw(\037)1429 4916 y FB(\001)1492 4901 y Fx(\()p Fw(H)8 b Fx(\))17 b Fw(F)1751 4860 y Fv(\022)1737 4926 y(\036)1788 4901 y Fx(\()p Fw(F)1903 4860 y Fv(\022)1889 4926 y(\036)1942 4901 y Fx(\))1980 4840 y Fu(\000)p FB(1)2074 4790 y Fq(\021)2189 4901 y Fr(\024)56 b Fw(\032)2372 4916 y Fv(\036)2418 4901 y Fx(\(\001\))17 b Fr(k)p Fx(\()p Fw(F)2757 4860 y Fv(\022)2743 4926 y(\036)2796 4901 y Fx(\))2834 4840 y Fu(\000)p FB(1)2928 4901 y Fr(k)g Fw(:)-118 5110 y Fx(Since)33 b Fw(T)208 5074 y Fv(\022)194 5136 y(\036)274 5110 y Fx(=)28 b Fw(\022)s(=)p Fx(2)p Fw(\031)j(\031)665 5125 y Fv(W)746 5110 y Fx(\()p Fw(F)861 5074 y Fv(\022)847 5136 y(\036)899 5110 y Fx(\),)i(the)g(theorem)f(follo)m(ws.)1963 b Fa(2)1841 5635 y Fx(13)p eop %%Page: 14 14 14 13 bop -118 36 a Ff(4.2)135 b(Pro)t(of)45 b(of)g(Prop)t(osition)h(2) -118 221 y Fx(Let)d Fw(\022)k(>)e Fx(2)p Fw(\031)t Fx(.)73 b(The)44 b(ground)e(state)h Fw(\036)1346 236 y Fv(S)1439 221 y Fx(of)f Fh(H)1632 236 y Fv(S)1742 221 y Fx(is)g(a)g Fw(\022)1989 184 y Fu(0)2013 221 y Fx(-frame)f(according)h(to)g (Theorem)h(3.)73 b(Let)43 b Fw( )3726 236 y Fv(S)3822 221 y Fx(=)-118 354 y(\()p Fw(\022)s(=)p Fx(2)p Fw(\031)t Fx(\))163 318 y FB(1)p Fv(=)p FB(2)272 354 y Fx(\()p Fw(T)381 318 y Fv(\022)367 380 y(\036)409 391 y Fp(S)458 354 y Fx(\))496 298 y Fu(\000)p FB(1)p Fv(=)p FB(2)661 354 y Fw(\036)719 369 y Fv(S)805 354 y Fx(b)s(e)35 b(the)h(asso)s (ciated)f Fw(\022)s Fx(-tracial)e(v)m(ector.)54 b(F)-8 b(urther)35 b(set)h Fr(H)2900 369 y Fv(S)2983 354 y Fx(=)p 3092 267 445 4 v 33 w Fw(\031)3147 369 y Fv(W)3228 354 y Fx(\()p Fr(A)3346 369 y Fv(\022)3384 354 y Fx(\))p Fw( )3485 369 y Fv(S)3536 354 y Fx(.)52 b(In)36 b(this)-118 474 y(section,)29 b Fw(\031)286 489 y Fv(W)394 474 y Fx(denotes)g(the)e(restriction)g(of)g(the)h(W)-8 b(eyl)27 b(represen)m(tation)h(to)g Fr(H)2707 489 y Fv(S)2757 474 y Fx(.)42 b(A)28 b(unitary)f(transformation)-118 595 y Fr(U)38 b Fx(:)28 b Fr(H)121 610 y Fv(S)199 595 y Fr(!)f Fw(`)367 559 y FB(2)407 595 y Fx(\()p Fl(Z)513 559 y FB(2)553 595 y Fx(\))32 b(is)g(de\014ned)i(b)m(y)763 836 y(\()p Fr(U)10 b Fw(\036)p Fx(\)\()p Fl(l)p Fx(\))54 b(=)h Fr(h)p Fw( )1364 851 y Fv(S)1415 836 y Fr(jW)1541 851 y Fv(\022)1580 836 y Fx(\()p Fl(l)p Fx(\))1688 794 y Fu(\000)p FB(1)1782 836 y Fr(j)p Fw(\036)p Fr(i)31 b Fx(,)228 b Fw(\036)27 b Fr(2)i(H)2457 851 y Fv(S)2540 836 y Fx(,)88 b Fl(l)27 b Fr(2)h Fs(Z)2877 794 y FB(2)2946 836 y Fx(.)-118 994 y(Then)34 b Fr(U)10 b Fw(\031)264 1009 y Fv(W)345 994 y Fx(\()p Fw(A)p Fx(\))p Fr(U)566 958 y Fu(\003)633 994 y Fx(=)28 b Fw(\031)792 1009 y Fn(2D)872 994 y Fx(\()p Fw(A)p Fx(\))33 b(for)f(all)e Fw(A)e Fr(2)g(A)1613 1009 y Fv(\022)1652 994 y Fx(.)43 b(Moreo)m(v)m(er)34 b Fr(U)k Fx(:)28 b Fr(S)7 b Fx(\()p Fs(R)f Fx(\))28 b Fr(\\)22 b Fh(H)2703 1009 y Fv(S)2798 994 y Fr(!)27 b Fh(s)p Fx(\()p Fs(Z)3076 958 y FB(2)3113 994 y Fx(\).)44 b(As)49 b Fr(U)10 b Fw( )3517 1009 y Fv(S)3596 994 y Fx(=)28 b Fr(j)p Fl(0)p Fr(i)p Fx(,)451 1235 y Fw(M)545 1250 y Fv(W)626 1235 y Fx(\()p Fw(H)r(;)17 b Fx(\001;)g Fw(q)t(;)g(t)p Fx(\))55 b(=)g Fr(h)p Fl(0)p Fr(j)p Fw(\037)1450 1250 y FB(\001)1513 1235 y Fx(\()p Fw(H)1632 1250 y Fn(2D)1712 1235 y Fx(\))17 b Fw(e)1812 1194 y Fv({H)1893 1205 y Fn(2D)1973 1194 y Fv(t)2002 1235 y Fx(\()p Fr(U)10 b Fh(H)2184 1250 y Fv(S)2252 1235 y Fr(U)2324 1194 y Fu(\003)2363 1235 y Fx(\))2401 1194 y Fv(q)r(=)p FB(2)2510 1235 y Fw(e)2555 1194 y Fu(\000)p Fv({H)2691 1205 y Fn(2D)2771 1194 y Fv(t)2801 1235 y Fw(\037)2862 1250 y FB(\001)2925 1235 y Fx(\()p Fw(H)3044 1250 y Fn(2D)3124 1235 y Fx(\))p Fr(j)p Fl(0)p Fr(i)17 b Fw(:)-118 1394 y Fx(Recall)31 b(that)h Fh(H)459 1409 y Fv(S)559 1394 y Fx(is)g(a)g(p)s(olynomial)d(of)j(second)i(degree)g (in)e Fw(Q)g Fx(and)h Fw(P)14 b Fx(.)43 b(F)-8 b(rom)31 b(\(19\))h(follo)m(ws)637 1634 y Fr(U)10 b Fw(Q)p Fr(U)858 1593 y Fu(\003)953 1634 y Fx(=)55 b Fr(\000)p Fw(\022)1209 1593 y FB(1)p Fv(=)p FB(2)1320 1634 y Fw(X)1401 1649 y FB(1)1462 1634 y Fx(+)22 b Fw(A)1633 1649 y FB(1)1706 1634 y Fx(,)227 b Fr(U)10 b Fw(P)k Fr(U)2181 1593 y Fu(\003)2276 1634 y Fx(=)55 b Fr(\000)p Fw(\022)2532 1593 y Fu(\000)p FB(1)p Fv(=)p FB(2)2698 1634 y Fw(X)2779 1649 y FB(2)2840 1634 y Fx(+)22 b Fw(A)3011 1649 y FB(2)3083 1634 y Fx(,)-118 1793 y(where)42 b Fr(h)p Fl(l)p Fr(j)p Fw(A)344 1808 y FB(1)382 1793 y Fr(j)p Fl(m)p Fr(i)d Fx(=)i Fr(h)p Fw( )801 1808 y Fv(S)852 1793 y Fr(jW)978 1808 y Fv(\022)1017 1793 y Fx(\()p Fl(l)27 b Fr(\000)h Fl(m)p Fx(\))p Fr(j)p Fw(Q )1519 1808 y Fv(S)1569 1793 y Fr(i)40 b Fx(and)h Fr(h)p Fl(l)p Fr(j)p Fw(A)2018 1808 y FB(2)2056 1793 y Fr(j)p Fl(m)p Fr(i)f Fx(=)h Fr(h)p Fw( )2476 1808 y Fv(S)2527 1793 y Fr(jW)2653 1808 y Fv(\022)2692 1793 y Fx(\()p Fl(l)26 b Fr(\000)i Fl(m)p Fx(\))p Fr(j)p Fw(P)14 b( )3193 1808 y Fv(S)3243 1793 y Fr(i)p Fx(.)66 b(Because)42 b Fw( )3819 1808 y Fv(S)3871 1793 y Fx(,)-118 1913 y Fw(Q )22 1928 y Fv(S)112 1913 y Fx(and)d Fw(P)14 b( )448 1928 y Fv(S)537 1913 y Fx(are)39 b(in)f Fr(S)7 b Fx(\()p Fs(R)f Fx(\),)46 b Fw(A)1182 1928 y FB(1)1261 1913 y Fx(and)39 b Fw(A)1530 1928 y FB(2)1608 1913 y Fx(are)g(b)s(ounded)g(op) s(erators.)62 b(Using)39 b(the)g(standard)g(op)s(erator)-118 2034 y(inequalities)22 b Fr(j)p Fw(AB)5 b Fr(j)27 b(\024)i(k)p Fw(A)p Fr(k)17 b(j)p Fw(B)5 b Fr(j)23 b Fx(and)h Fr(j)p Fw(A)5 b Fx(+)g Fw(B)g Fr(j)27 b(\024)h Fx(2\()p Fr(j)p Fw(A)p Fr(j)5 b Fx(+)g Fr(j)p Fw(B)g Fr(j)p Fx(\))23 b(and)h(the)h(comm)m(utation)d(relation)g([)p Fw(X)3563 2049 y FB(1)3603 2034 y Fw(;)17 b(X)3728 2049 y FB(2)3767 2034 y Fx(])28 b(=)-118 2154 y(0,)f(it)e(is)g(no)m(w)h(p)s(ossible)f (to)h(deduce)h Fw(M)1248 2169 y Fv(W)1329 2154 y Fx(\()p Fw(H)r(;)17 b Fx(\001;)g Fw(q)t(;)g(t)p Fx(\))27 b Fr(\024)h Fw(c)1957 2169 y FB(1)1997 2154 y Fw(M)2091 2169 y Fn(2D)2171 2154 y Fx(\()p Fw(H)r(;)17 b Fx(\001;)g Fw(q)t(;)g(t)p Fx(\))8 b(+)g Fw(c)2759 2169 y FB(2)2824 2154 y Fx(for)26 b(t)m(w)m(o)g(p)s(ositiv)m(e)f(constan)m(ts)-118 2275 y Fw(c)-76 2290 y FB(1)3 2275 y Fx(and)40 b Fw(c)242 2290 y FB(2)281 2275 y Fx(.)65 b(An)39 b(inequalit)m(y)g Fw(M)1093 2290 y Fn(2D)1174 2275 y Fx(\()p Fw(H)r(;)17 b Fx(\001;)g Fw(q)t(;)g(t)p Fx(\))39 b Fr(\024)h Fw(c)1826 2290 y FB(1)1865 2275 y Fw(M)1959 2290 y Fv(W)2040 2275 y Fx(\()p Fw(H)r(;)17 b Fx(\001;)g Fw(q)t(;)g(t)p Fx(\))27 b(+)f Fw(c)2665 2290 y FB(2)2744 2275 y Fx(is)40 b(obtained)f (similarly)-8 b(.)60 b(This)-118 2395 y(implies)30 b(Prop)s(osition)h (2.)-118 2678 y Ff(4.3)135 b(Pro)t(of)45 b(of)g(Prop)t(osition)h(3)-118 2863 y Fl(Lemma)37 b(1)49 b Fm(L)-5 b(et)43 b Fw(Y)617 2878 y FB(1)656 2863 y Fw(;)17 b(:)g(:)g(:)33 b(;)17 b(Y)949 2878 y Fv(N)1058 2863 y Fm(b)-5 b(e)43 b(selfadjoint)f(op)-5 b(er)g(ators)43 b(on)g Fr(H)h Fm(with)f(c)-5 b(ommon)41 b(domain)h(which)h(satisfy)-118 2983 y Fx([)p Fw(Y)-34 2998 y Fv(m)32 2983 y Fw(;)17 b(Y)133 2998 y Fv(n)179 2983 y Fx(])28 b(=)g Fw({)17 b(c)428 2998 y Fv(m;n)556 2983 y Fl(1)p Fm(.)45 b(Then,)34 b(if)h Fw(c)27 b Fx(=)h(max)1421 2998 y Fv(m;n)1566 2983 y Fx(\()p Fr(j)p Fw(c)1674 2998 y Fv(m;n)1803 2983 y Fr(j)p Fx(\))f Fw(>)h Fx(0)34 b Fm(and)h(if)f Fx(0)28 b Fr(\024)g Fw(\013)g Fr(\024)g Fx(1)p Fm(,)678 3197 y Fx(1)p 658 3242 89 4 v 658 3333 a Fw(N)830 3140 y Fv(N)790 3170 y Fq(X)795 3379 y Fv(n)p FB(=1)967 3265 y Fw(Y)1045 3224 y FB(2)p Fv(\013)1024 3289 y(n)1185 3265 y Fr(\024)1318 3094 y Fq( )1437 3140 y Fv(N)1397 3170 y Fq(X)1402 3379 y Fv(n)p FB(=1)1574 3265 y Fw(Y)1652 3224 y FB(2)1631 3289 y Fv(n)1692 3094 y Fq(!)1771 3111 y Fv(\013)1876 3265 y Fr(\024)2049 3140 y Fv(N)2008 3170 y Fq(X)2014 3379 y Fv(n)p FB(=1)2185 3265 y Fw(Y)2264 3224 y FB(2)p Fv(\013)2242 3289 y(n)2398 3265 y Fx(+)50 b(2)p Fw(N)10 b Fx(\()p Fw(N)33 b Fr(\000)23 b Fx(1\))p Fw(c)3039 3224 y Fv(\013)3104 3265 y Fw(:)593 b Fx(\(36\))-118 3533 y Fl(Pro)s(of:)93 b Fx(F)-8 b(or)32 b Fw(\013)c Fx(=)g(0)p Fw(;)17 b Fx(1)31 b(b)s(oth)i(inequalities)d (are)j(trivial.)41 b(F)-8 b(or)32 b(0)27 b Fw(<)h(\013)g(<)g Fx(1)k(the)h(follo)m(wing)d(iden)m(tit)m(y)i(holds)1151 3866 y Fw(A)1224 3825 y Fv(\013)1329 3866 y Fx(=)1470 3799 y(sin)17 b(\()p Fw(\031)t(\013)p Fx(\))p 1470 3844 334 4 v 1608 3935 a Fw(\031)1847 3731 y Fq(Z)1947 3757 y Fu(1)1903 3956 y FB(0)2120 3799 y Fw(dv)p 2076 3844 191 4 v 2076 3935 a(v)2127 3906 y FB(1)p Fu(\000)p Fv(\013)2388 3799 y Fw(A)p 2303 3844 245 4 v 2303 3935 a(v)26 b Fx(+)c Fw(A)2585 3866 y(;)1112 b Fx(\(37\))-118 4121 y(for)36 b(a)h(p)s(ositiv)m(e)f(op)s(erator)g Fw(A)p Fx(.)56 b(If)37 b Fw(A)e Fx(=)1360 4047 y Fq(P)1465 4073 y Fv(N)1465 4151 y(n)p FB(=1)1619 4121 y Fw(Y)1697 4085 y FB(2)1676 4146 y Fv(n)1773 4121 y Fx(then)i(the)h(left-hand)d(inequalit)m(y)h(in) g(\(36\))g(follo)m(ws)g(from)-118 4242 y Fw(Y)-40 4206 y FB(2)-61 4266 y Fv(n)27 4242 y Fr(\024)28 b Fw(A)i Fx(and)g(from)f(the)h(op)s(erator)g(monotonicit)m(y)e(of)h Fw(A=)p Fx(\()p Fw(v)20 b Fx(+)d Fw(A)p Fx(\))28 b(=)f Fl(1)17 b Fr(\000)g Fw(v)t(=)p Fx(\()p Fw(v)j Fx(+)c Fw(A)p Fx(\).)43 b(On)30 b(the)g(other)g(hand)937 4450 y Fw(A)p 852 4495 V 852 4586 a(v)c Fx(+)c Fw(A)1161 4518 y Fx(=)1333 4393 y Fv(N)1292 4423 y Fq(X)1298 4632 y Fv(n)p FB(=1)1470 4377 y Fq(\022)1543 4518 y Fw(Y)1600 4533 y Fv(n)1754 4450 y Fx(1)p 1656 4495 V 1656 4586 a Fw(v)k Fx(+)c Fw(A)1910 4518 y(Y)1967 4533 y Fv(n)2053 4518 y Fx(+)38 b Fw(Y)2224 4533 y Fv(n)2288 4377 y Fq(\024)2340 4518 y Fw(Y)2397 4533 y Fv(n)2444 4518 y Fw(;)2595 4450 y Fx(1)p 2497 4495 V 2497 4586 a Fw(v)26 b Fx(+)c Fw(A)2751 4377 y Fq(\025\023)2910 4518 y Fw(:)-118 4793 y Fx(The)47 b(\014rst)f(term)f(of)g(eac)m(h)i(summand)e(is)g(b)s(ounded)h(b)m(y)h Fw(Y)2120 4756 y FB(2)2099 4817 y Fv(n)2160 4793 y Fw(=)p Fx(\()p Fw(v)34 b Fx(+)d Fw(Y)2513 4756 y FB(2)2492 4817 y Fv(n)2553 4793 y Fx(\).)82 b(Noting)45 b Fw(Y)3093 4808 y Fv(n)3156 4793 y Fx([)p Fw(Y)3240 4808 y Fv(n)3287 4793 y Fw(;)17 b Fx(\()p Fw(v)25 b Fx(+)d Fw(A)p Fx(\))3650 4756 y Fu(\000)p FB(1)3745 4793 y Fx(])50 b(=)-118 4913 y Fw(Y)-61 4928 y Fv(n)-15 4913 y Fx(\()p Fw(v)29 b Fx(+)24 b Fw(A)p Fx(\))310 4877 y Fu(\000)p FB(1)421 4913 y Fx([)p Fw(A;)17 b(Y)622 4928 y Fv(n)668 4913 y Fx(])g(\()p Fw(v)28 b Fx(+)c Fw(A)p Fx(\))1036 4877 y Fu(\000)p FB(1)1131 4913 y Fx(,)36 b(and)g(using)g(the)g(comm)m(utation)e(rules)i(for)f (the)i Fw(Y)3032 4928 y Fv(n)3078 4913 y Fx('s,)h(the)e(second)h(term) -118 5033 y(in)32 b(the)h(r.h.s.)44 b(is)32 b(estimated)g(b)m(y)685 5124 y Fq(\015)685 5184 y(\015)685 5243 y(\015)685 5303 y(\015)685 5363 y(\015)740 5298 y Fr(\000)p Fx(2)p Fw({)914 5204 y Fq(X)924 5413 y Fv(m;n)1091 5298 y Fw(c)1133 5313 y Fv(m;n)1279 5298 y Fw(Y)1336 5313 y Fv(n)1490 5231 y Fx(1)p 1392 5275 V 1392 5367 a Fw(v)26 b Fx(+)c Fw(A)1646 5298 y(Y)1703 5313 y Fv(m)1877 5231 y Fx(1)p 1779 5275 V 1779 5367 a Fw(v)k Fx(+)c Fw(A)2033 5124 y Fq(\015)2033 5184 y(\015)2033 5243 y(\015)2033 5303 y(\015)2033 5363 y(\015)2144 5298 y Fr(\024)56 b Fx(2)2454 5231 y(1)p 2352 5275 253 4 v 2352 5367 a Fw(v)26 b Fx(+)c Fw(c)2565 5382 y FB(0)2648 5204 y Fq(X)2657 5413 y Fv(m;n)2825 5298 y Fr(j)p Fw(c)2895 5313 y Fv(m;n)3023 5298 y Fr(j)17 b Fw(;)1841 5635 y Fx(14)p eop %%Page: 15 15 15 14 bop -118 32 a Fx(where)35 b Fw(c)207 47 y FB(0)280 32 y Fx(is)e(the)h(in\014m)m(um)f(of)g(the)h(sp)s(ectrum)g(of)g Fw(A)p Fx(.)47 b(In)34 b(the)g(latter)f(inequalit)m(y)f Fw(Y)2991 -4 y FB(2)2969 56 y Fv(n)3060 32 y Fr(\024)e Fw(A)k Fx(has)g(b)s(een)g(used.)-118 152 y(By)43 b(de\014nition,)i (there)e(are)g Fw(m;)17 b(n)43 b Fx(suc)m(h)h(that)f Fw(c)1675 167 y Fv(m;n)1849 152 y Fx(=)i Fw(c)g(>)g Fx(0)d(so)h(that)g Fw(Y)2699 116 y FB(2)2678 177 y Fv(n)2768 152 y Fx(+)29 b Fw(Y)2951 116 y FB(2)2930 177 y Fv(m)3041 152 y Fx(=)45 b(\()p Fw(Y)3257 167 y Fv(m)3352 152 y Fr(\000)30 b Fw({Y)3547 167 y Fv(n)3594 152 y Fx(\)\()p Fw(Y)3727 167 y Fv(m)3822 152 y Fx(+)-118 273 y Fw({Y)-30 288 y Fv(n)17 273 y Fx(\))e(+)h Fw(c)p Fl(1)44 b Fr(\025)h Fw(c)p Fl(1)p Fx(.)73 b(Hence)44 b Fw(c)992 288 y FB(0)1075 273 y Fr(\025)h Fw(c)p Fx(.)73 b(In)m(tegrating)42 b(o)m(v)m(er)h Fw(v)t Fx(,)i(using)d(the)h(eq.)g (\(37\),)h(and)f(remarking)e(that)-118 318 y Fq(P)-13 422 y Fv(m;n)133 393 y Fr(j)p Fw(c)203 408 y Fv(m;n)331 393 y Fr(j)28 b(\024)g Fw(N)10 b Fx(\()p Fw(N)33 b Fr(\000)23 b Fx(1\))p Fw(c)48 b Fx(giv)m(es)33 b(the)g(result.)2144 b Fa(2)28 572 y Fx(If)44 b Fw(S)53 b Fr(2)48 b Fw(S)6 b(L)p Fx(\(2)p Fw(;)17 b Fs(Z)p Fx(\))41 b(is)j(a)f(symmetry)i(suc)m(h) g(that)f Fw(S)1965 536 y Fv(r)2050 572 y Fx(=)j(1,)g(the)e(op)s (erators)e Fw(Y)2975 587 y Fv(n)3069 572 y Fx(=)k Fr(F)3274 536 y Fv(n)3264 597 y(S)3321 572 y Fw(Q)p Fr(F)3480 531 y Fu(\000)p Fv(n)3470 599 y(S)3626 572 y Fx(satisfy)-118 692 y(the)c(h)m(yp)s(othesis)g(of)f(Lemma)f(1,)k(b)s(ecause)e (calculating)d(the)j(deriv)-5 b(ativ)m(e)42 b(of)g(\(25\))f(at)h Fh(a)j Fx(=)f(0)e(sho)m(ws)i(that)-118 813 y(eac)m(h)37 b Fw(Y)162 828 y Fv(n)245 813 y Fx(is)f(linear)e(in)i Fw(P)49 b Fx(and)37 b Fw(Q)p Fx(.)54 b(Clearly)36 b Fh(H)1618 828 y Fv(S)1719 813 y Fx(=)d(1)p Fw(=)p Fx(\(2)p Fw(r)s Fx(\))2130 738 y Fq(P)2236 764 y Fv(r)2236 842 y(n)p FB(=1)2389 813 y Fw(Y)2468 776 y FB(2)2446 837 y Fv(n)2507 813 y Fx(.)54 b(If)37 b Fw(H)k Fr(2)34 b(A)2992 828 y Fv(\022)3067 813 y Fx(is)i Fw(S)6 b Fx(-in)m(v)-5 b(arian)m(t,)35 b(then)-118 933 y Fh(H)-46 948 y Fv(S)21 933 y Fx(\()p Fw(t)p Fx(\))28 b(=)f(1)p Fw(=)p Fx(\(2)p Fw(r)s Fx(\))565 858 y Fq(P)671 885 y Fv(r)671 962 y(n)p FB(=1)824 933 y Fr(F)906 897 y Fv(n)896 958 y(S)953 933 y Fw(Q)1030 897 y FB(2)1070 933 y Fx(\()p Fw(t)p Fx(\))p Fr(F)1263 892 y Fu(\000)p Fv(n)1253 960 y(S)1364 933 y Fx(,)j(where)h Fw(A)p Fx(\()p Fw(t)p Fx(\))d(=)g Fw(e)2061 897 y Fv({t)13 b(H)2180 908 y Fp(W)2253 933 y Fw(Ae)2371 897 y Fu(\000)p Fv({t)g(H)2545 908 y Fp(W)2648 933 y Fx(whenev)m(er)33 b Fw(A)d Fx(is)f(an)h(op)s(erator)f(on)-118 1053 y Fr(H)q Fx(.)43 b(Therefore,)34 b(if)e(0)27 b Fr(\024)h Fw(q)k Fr(\024)c Fx(2,)k(the)h(inequalit)m(y)f(\(36\))g(leads)g(to)g(\(with)h Fw(\037)2577 1068 y FB(\001)2667 1053 y Fx(=)28 b Fw(\037)2832 1068 y FB(\001)2895 1053 y Fx(\()p Fw(H)3014 1068 y Fv(W)3094 1053 y Fx(\)\))131 1290 y Fr(h)p Fw(\036)228 1305 y Fv(S)295 1290 y Fr(j)17 b Fw(\037)401 1305 y FB(\001)463 1290 y Fh(H)535 1305 y Fv(S)602 1290 y Fx(\()p Fw(t)p Fx(\))713 1249 y Fv(q)r(=)p FB(2)839 1290 y Fw(\037)900 1305 y FB(\001)963 1290 y Fr(j)p Fw(\036)1049 1305 y Fv(S)1116 1290 y Fr(i)55 b(\024)g Fw(r)s Fx(\(2)p Fw(r)s Fx(\))1561 1249 y Fu(\000)p Fv(q)r(=)p FB(2)1741 1290 y Fr(h)p Fw(\036)1838 1305 y Fv(S)1904 1290 y Fr(j)17 b Fw(\037)2010 1305 y FB(\001)2073 1290 y Fr(j)p Fw(Q)p Fx(\()p Fw(t)p Fx(\))p Fr(j)2317 1249 y Fv(q)2355 1290 y Fw(\037)2416 1305 y FB(\001)2495 1290 y Fr(j)p Fw(\036)2581 1305 y Fv(S)2648 1290 y Fr(i)38 b Fx(+)h(2)p Fw(r)s Fx(\()p Fw(r)24 b Fr(\000)f Fx(1\))3246 1179 y Fq(\020)3341 1223 y Fw(c)p 3315 1267 96 4 v 3315 1358 a Fx(2)p Fw(r)3420 1179 y Fq(\021)3480 1202 y Fv(q)r(=)p FB(2)3621 1290 y Fw(;)-118 1501 y Fx(where)33 b Fr(F)235 1516 y Fv(s)272 1501 y Fw(\036)330 1516 y Fv(S)424 1501 y Fx(=)28 b Fw(\036)586 1516 y Fv(S)685 1501 y Fx(has)33 b(b)s(een)f(used.)45 b(Prop)s(osition)30 b(3)i(is)g(then)g(a)g(direct)g(consequence)j(of)d (the)g(de\014nitions)-118 1622 y(of)g(the)h(exp)s(onen)m(ts)h Fw(\014)680 1585 y Fu(\006)674 1646 y Fn(1D)755 1622 y Fx(\()p Fw(H)r(;)17 b Fx(\001;)g Fw(q)t Fx(\))p Fw(;)33 b(\014)1251 1580 y Fu(\006)1245 1649 y Fv(W)1325 1622 y Fx(\()p Fw(H)r(;)17 b Fx(\001;)g Fw(q)t Fx(\))32 b(and)h(of)f(the)h (follo)m(wing)d(lemma:)-118 1811 y Fl(Lemma)37 b(2)49 b Fm(L)-5 b(et)32 b Fw(\036)c Fr(2)g(S)7 b Fx(\()p Fs(R)f Fx(\))p Fm(,)38 b Fw(\022)31 b Fr(\025)d Fx(2)p Fw(\031)36 b Fm(and)c Fw(q)f Fr(\025)d Fx(0)p Fm(.)44 b(Then,)32 b(ther)-5 b(e)32 b(ar)-5 b(e)32 b(two)g(p)-5 b(ositive)32 b(c)-5 b(onstants)31 b Fw(c)3497 1826 y FB(0)3537 1811 y Fw(;)i(c)3639 1826 y FB(1)3711 1811 y Fm(such)-118 1931 y(that,)i(for)g(any)g(element)f Fw(B)e Fr(2)d(A)1096 1946 y Fv(\022)654 2176 y Fr(h)p Fw(\036)p Fr(j)p Fw(B)858 2135 y Fu(\003)853 2201 y Fv(W)950 2176 y Fr(j)p Fw(Q)p Fr(j)1083 2135 y Fv(q)1137 2176 y Fw(B)1211 2191 y Fv(W)1308 2176 y Fr(j)p Fw(\036)p Fr(i)55 b(\024)h Fw(c)1663 2191 y FB(0)1746 2041 y Fq(Z)1846 2067 y FB(2)p Fv(\031)1802 2266 y FB(0)1983 2109 y Fw(d!)p 1983 2153 116 4 v 1987 2245 a Fx(2)p Fw(\031)2135 2176 y Fr(h)p Fx(0)p Fr(j)p Fw(B)2330 2135 y Fu(\003)2325 2201 y Fv(!)2392 2176 y Fr(j)2444 2151 y Fq(b)2420 2176 y Fw(X)7 b Fr(j)2536 2135 y Fv(q)2590 2176 y Fw(B)2664 2191 y Fv(!)2731 2176 y Fr(j)p Fx(0)p Fr(i)38 b Fx(+)h Fw(c)3042 2191 y FB(1)3098 2176 y Fw(;)-118 2415 y Fm(wher)-5 b(e)34 b Fw(B)231 2430 y Fv(W)339 2415 y Fx(=)28 b Fw(\031)498 2430 y Fv(W)579 2415 y Fx(\()p Fw(B)5 b Fx(\))35 b Fm(and)f Fw(B)1032 2430 y Fv(!)1110 2415 y Fx(=)28 b Fw(\031)1269 2430 y Fv(!)1319 2415 y Fx(\()p Fw(B)5 b Fx(\))p Fm(.)-118 2604 y Fl(Pro)s(of:)93 b Fx(De\014nition)31 b(\(21\))h(and)g(iden)m(tit)m(y) h(\(22\))f(of)g(Section)g(3.2)g(lead)g(to)153 2867 y Fr(h)p Fw(\036)p Fr(j)p Fw(B)357 2826 y Fu(\003)352 2891 y Fv(W)449 2867 y Fr(j)p Fw(Q)p Fr(j)582 2826 y Fv(q)636 2867 y Fw(B)710 2882 y Fv(W)807 2867 y Fr(j)p Fw(\036)p Fr(i)55 b Fx(=)g Fw(\022)1166 2826 y FB(\()p Fv(q)r Fu(\000)p FB(1\))p Fv(=)p FB(2)1453 2731 y Fq(Z)1553 2758 y Fv(\022)1508 2957 y FB(0)1625 2867 y Fw(d!)1811 2772 y Fq(X)1773 2985 y Fv(n;n)1879 2966 y Fo(0)1901 2985 y Fu(2)p Ft(Z)p 2025 2710 534 4 v 2025 2867 a Fw(\036)2100 2726 y Fq(\022)2183 2799 y Fw(!)25 b Fr(\000)e Fw(n\022)p 2183 2844 293 4 v 2264 2864 a Fr(p)p 2347 2864 49 4 v 85 x Fw(\022)2485 2726 y Fq(\023)2586 2867 y Fw(\036)2661 2726 y Fq(\022)2744 2799 y Fw(!)j Fr(\000)c Fw(n)2988 2763 y Fu(0)3012 2799 y Fw(\022)p 2744 2844 316 4 v 2836 2864 a Fr(p)p 2919 2864 49 4 v 85 x Fw(\022)3070 2726 y Fq(\023)3176 2867 y Fr(h)p Fw(n)p Fr(j)p Fw(K)3384 2882 y Fv(!)3434 2867 y Fr(j)p Fw(n)3520 2826 y Fu(0)3544 2867 y Fr(i)17 b Fw(;)-118 3178 y Fx(with)37 b Fw(K)192 3193 y Fv(!)278 3178 y Fx(=)e Fw(B)468 3142 y Fu(\003)463 3203 y Fv(!)514 3178 y Fr(j)p Fx(\()p Fw(!)t(=\022)s Fx(\))24 b Fr(\000)931 3153 y Fq(b)907 3178 y Fw(X)7 b Fr(j)1023 3142 y Fv(q)1061 3178 y Fw(B)1135 3193 y Fv(!)1186 3178 y Fx(.)57 b(Since)37 b Fw(K)1612 3193 y Fv(!)1700 3178 y Fx(is)g(a)g(p)s(ositiv)m(e)f(op)s (erator,)i(the)g(Sc)m(h)m(w)m(arz)h(inequalit)m(y)d(giv)m(es)-118 3299 y Fr(jh)p Fw(n)p Fr(j)p Fw(K)118 3314 y Fv(!)168 3299 y Fr(j)p Fw(n)254 3263 y Fu(0)277 3299 y Fr(ij)27 b(\024)h Fx(\()p Fr(h)p Fw(n)p Fr(j)p Fw(K)722 3314 y Fv(!)772 3299 y Fr(j)p Fw(n)p Fr(i)13 b Fx(+)g Fr(h)p Fw(n)1096 3263 y Fu(0)1119 3299 y Fr(j)p Fw(K)1230 3314 y Fv(!)1281 3299 y Fr(j)p Fw(n)1367 3263 y Fu(0)1390 3299 y Fr(i)p Fx(\))p Fw(=)p Fx(2.)41 b(Both)28 b(terms)g(can)h(b)s(e)f (b)s(ounded)h(similarly)-8 b(.)38 b(The)30 b(co)m(v)-5 b(ariance)-118 3419 y(prop)s(ert)m(y)37 b(of)f Fw(\031)454 3434 y Fv(!)542 3419 y Fx(\(see)h(Section)g(3.2\))f(giv)m(es)h Fr(h)p Fw(n)p Fr(j)p Fw(K)1737 3434 y Fv(!)1787 3419 y Fr(j)p Fw(n)p Fr(i)d Fx(=)g Fr(h)p Fx(0)p Fr(j)p Fw(K)2255 3434 y Fv(!)r Fu(\000)p Fv(n\022)2438 3419 y Fr(j)p Fx(0)p Fr(i)p Fx(.)54 b(Since)37 b Fw(\036)d Fr(2)h(S)7 b Fx(\()p Fs(R)f Fx(\),)43 b(summing)35 b(up)-118 3540 y(o)m(v)m(er)f Fw(n)149 3503 y Fu(0)205 3540 y Fx(\014rst,)f(then)g(o)m(v)m(er)h Fw(n)p Fx(,)e(there)i(are)e(constan)m(ts)i Fw(C)r(;)f(c)1999 3555 y FB(1)2071 3540 y Fx(suc)m(h)h(that)-99 3768 y Fr(h)p Fw(\036)p Fr(j)p Fw(B)105 3727 y Fu(\003)100 3793 y Fv(W)196 3768 y Fr(j)p Fw(Q)p Fr(j)329 3727 y Fv(q)384 3768 y Fw(B)458 3783 y Fv(W)555 3768 y Fr(j)p Fw(\036)p Fr(i)43 b(\024)i Fw(C)939 3633 y Fq(Z)994 3858 y Ft(R)1079 3768 y Fw(dx)17 b Fr(j)p Fw(\036)p Fx(\()p Fw(x)p Fx(\))p Fr(j)g(h)p Fx(0)p Fr(j)g Fw(K)1680 3797 y Fv(x)1720 3740 y Fu(p)p 1776 3740 35 3 v 1776 3797 a Fv(\022)1832 3768 y Fr(j)p Fx(0)p Fr(i)43 b(\024)i Fw(C)2207 3633 y Fq(Z)2262 3858 y Ft(R)2347 3768 y Fw(dx)17 b Fr(j)p Fw(\036)p Fx(\()p Fw(x)p Fx(\))p Fr(j)g(h)p Fx(0)p Fr(j)g Fw(B)2944 3727 y Fu(\003)2939 3806 y Fv(x)2979 3749 y Fu(p)p 3035 3749 V 3035 3806 a Fv(\022)3075 3768 y Fr(j)3127 3743 y Fq(b)3103 3768 y Fw(X)7 b Fr(j)3219 3727 y Fv(q)3257 3768 y Fw(B)3331 3797 y Fv(x)3371 3740 y Fu(p)p 3429 3740 V 3429 3797 a Fv(\022)3485 3768 y Fr(j)p Fx(0)p Fr(i)38 b Fx(+)h Fw(c)3796 3783 y FB(1)3852 3768 y Fw(;)-118 4029 y Fx(where)31 b(the)g(inequalit)m(y)e Fr(j)p Fw(x)17 b Fr(\000)997 4004 y Fq(b)972 4029 y Fw(X)8 b Fr(j)1089 3993 y Fv(q)1154 4029 y Fr(\024)29 b Fw(C)1330 4044 y Fv(q)1367 4029 y Fx(\()p Fr(j)p Fw(x)p Fr(j)1516 3993 y Fv(q)1571 4029 y Fx(+)18 b Fr(j)1717 4004 y Fq(b)1693 4029 y Fw(X)7 b Fr(j)1809 3993 y Fv(q)1847 4029 y Fx(\),)30 b(v)-5 b(alid)29 b(for)g Fw(q)j Fr(\025)c Fx(0)i(and)g(some)g(suitable)g (constan)m(t)h Fw(C)3833 4044 y Fv(q)3871 4029 y Fx(,)-118 4149 y(has)g(b)s(een)g(used.)44 b(Thanks)32 b(to)f(the)g(p)s(erio)s (dicit)m(y)e(of)h Fw(\031)1824 4164 y Fv(!)1875 4149 y Fx(,)h(the)g(r.h.s.)43 b(of)30 b(the)i(latter)d(estimate)h(can)h(b)s (e)g(written)-118 4270 y(as)439 4430 y(r.h.s.)45 b Fr(\024)817 4295 y Fq(Z)917 4321 y FB(2)p Fv(\031)873 4520 y FB(0)1050 4363 y Fw(d!)p 1042 4407 132 4 v 1042 4427 a Fr(p)p 1125 4427 49 4 v 85 x Fw(\022)1200 4430 y Fr(h)p Fx(0)p Fr(j)17 b Fw(B)1412 4389 y Fu(\003)1407 4455 y Fv(!)1456 4430 y Fr(j)1509 4405 y Fq(b)1484 4430 y Fw(X)8 b Fr(j)1601 4389 y Fv(q)1639 4430 y Fw(B)1713 4445 y Fv(!)1780 4430 y Fr(j)p Fx(0)p Fr(i)93 b Fx(sup)1928 4509 y FB(0)p Fv()27 b Fx(0)32 b(one)h(has)550 1267 y Fr(h)p Fw(M)683 1282 y Fv(q)721 1267 y Fx(\()p Fw(H)r(;)17 b Fx(\001;)g Fw(T)d Fx(\))p Fr(i)1158 1219 y Fv(g)1158 1299 y(T)1269 1267 y Fr(\025)55 b Fw(\016)1448 1226 y Fu(\000)p Fv(q)r(=)p FB(2)1628 1186 y Fq(\000)1674 1267 y Fr(k)p Fw( )1787 1282 y FB(\001)1850 1267 y Fr(k)1900 1226 y FB(2)1978 1267 y Fr(\000)2095 1186 y Fq(\012)2142 1267 y Fr(h)p Fw( )2244 1282 y FB(\001)2307 1267 y Fx(\()p Fw(t)p Fx(\))p Fr(j)p Fw(e)2491 1226 y Fu(\000)p Fv(\016)14 b Fg(H)2641 1237 y Fp(S)2718 1267 y Fr(j)p Fw( )2809 1282 y FB(\001)2872 1267 y Fx(\()p Fw(t)p Fx(\))p Fr(i)3022 1186 y Fq(\013)3069 1206 y Fv(g)3069 1306 y(T)3124 1186 y Fq(\001)3203 1267 y Fw(:)-118 1457 y Fx(F)-8 b(or)30 b(\001)136 1472 y FB(1)203 1457 y Fr(\032)e Fx(\001,)k(\001)529 1420 y Fv(c)529 1481 y FB(1)599 1457 y Fx(will)d(denote)i(the)g(complemen)m(t) f(\001)19 b Fr(n)f Fx(\001)2052 1472 y FB(1)2092 1457 y Fx(.)42 b(The)32 b(decomp)s(osition)d(of)h Fw( )3173 1472 y FB(\001)3267 1457 y Fx(in)m(to)g Fw( )3526 1472 y FB(\001)3585 1481 y Fn(1)3642 1457 y Fx(+)19 b Fw( )3800 1472 y FB(\001)3859 1449 y Fp(c)3859 1493 y Fn(1)-118 1577 y Fx(giv)m(es)33 b(rise)f(to)g(the)i(follo)m(wing)29 b(lo)m(w)m(er)k(b)s(ound)355 1767 y Fr(h)o Fw(M)487 1782 y Fv(q)526 1767 y Fx(\()p Fw(H)r(;)17 b Fx(\001;)g Fw(T)d Fx(\))p Fr(i)963 1718 y Fv(g)963 1799 y(T)1073 1767 y Fr(\025)56 b Fw(\016)1253 1725 y Fu(\000)p Fv(q)r(=)p FB(2)1433 1686 y Fq(\000)1479 1767 y Fr(k)p Fw( )1592 1782 y FB(\001)1651 1791 y Fn(1)1690 1767 y Fr(k)1740 1725 y FB(2)1818 1767 y Fr(\000)39 b Fw(A)2007 1782 y FB(\001)2066 1791 y Fn(1)2100 1782 y Fv(;)p FB(\001)2179 1791 y Fn(1)2218 1767 y Fx(\()p Fw(T)8 b(;)17 b(\016)t Fx(\))38 b Fr(\000)h Fx(2)17 b Fr(<)p Fw(e)g(A)2877 1782 y FB(\001)2936 1791 y Fn(1)2970 1782 y Fv(;)p FB(\001)3049 1759 y Fp(c)3049 1803 y Fn(1)3087 1767 y Fx(\()p Fw(T)8 b(;)17 b(\016)t Fx(\))3319 1686 y Fq(\001)3398 1767 y Fw(;)-118 1967 y Fx(where)33 b Fw(A)236 1982 y FB(\001)295 1991 y Fn(1)329 1982 y Fv(;)p FB(\001)408 1991 y Fn(2)447 1967 y Fx(\()p Fw(T)8 b(;)17 b(\016)t Fx(\))27 b(:=)837 1886 y Fq(\012)884 1967 y Fr(h)p Fw( )986 1982 y FB(\001)1045 1991 y Fn(1)1083 1967 y Fx(\()p Fw(t)p Fx(\))p Fr(j)p Fw(e)1267 1931 y Fu(\000)p Fv(\016)15 b Fg(H)1418 1942 y Fp(S)1494 1967 y Fr(j)p Fw( )1585 1982 y FB(\001)1644 1991 y Fn(2)1683 1967 y Fx(\()p Fw(t)p Fx(\))p Fr(i)1833 1886 y Fq(\013)1880 1909 y Fv(g)1880 2006 y(T)1935 1967 y Fx(.)43 b(Using)32 b(the)g(sp)s(ectral)f(decomp)s(osition)f(of)i Fh(H)3636 1982 y Fv(S)3735 1967 y Fx(\(see)-118 2088 y(eq.)h(\(30\))f(in)g(Section)h(3.3\),)f(it)f(is)h(easy)i(to)e(get)64 2353 y Fw(A)137 2368 y FB(\001)196 2377 y Fn(1)231 2368 y Fv(;)p FB(\001)310 2377 y Fn(2)348 2353 y Fx(\()p Fw(T)8 b(;)17 b(\016)t Fx(\))55 b(=)766 2217 y Fq(Z)822 2443 y FB(\001)881 2452 y Fn(1)953 2353 y Fw(d\032)1054 2368 y Fv(S)1104 2353 y Fx(\()p Fw(E)6 b Fx(\))1292 2217 y Fq(Z)1347 2443 y FB(\001)1406 2452 y Fn(2)1478 2353 y Fw(d\032)1579 2368 y Fv(S)1629 2353 y Fx(\()p Fw(E)1745 2312 y Fu(0)1769 2353 y Fx(\))17 b Fw(e)1869 2312 y Fu(\000)p FB(\()p Fv(E)t Fu(\000)p Fv(E)2118 2288 y Fo(0)2139 2312 y FB(\))2166 2288 y Fn(2)2201 2312 y Fv(T)2252 2288 y Fn(2)2360 2229 y Fu(1)2324 2258 y Fq(X)2329 2468 y Fv(n)p FB(=0)2501 2353 y Fx(\010)2571 2368 y Fv(n;S)2685 2353 y Fx(\()p Fw(E)6 b Fx(\))p 2856 2266 361 4 v 17 w(\010)2926 2368 y Fv(n;S)3039 2353 y Fx(\()p Fw(E)3155 2324 y Fu(0)3178 2353 y Fx(\))17 b Fw(e)3278 2312 y Fu(\000)p Fv(\016)r(\026)p FB(\()p Fv(n)p FB(+1)p Fv(=)p FB(2\))3688 2353 y Fw(:)-118 2634 y Fx(The)34 b(Sc)m(h)m(w)m(arz)h(inequalit)m(y)d(2)17 b Fr(jh)p Fw( )1109 2649 y FB(1)1147 2634 y Fr(j)p Fw( )1238 2649 y FB(2)1278 2634 y Fr(ij)28 b(\024)g(k)p Fw( )1591 2649 y FB(1)1631 2634 y Fr(k)1681 2598 y FB(2)1743 2634 y Fx(+)22 b Fr(k)p Fw( )1954 2649 y FB(2)1994 2634 y Fr(k)2044 2598 y FB(2)2132 2634 y Fx(applied)32 b(to)h(the)g(sum)g(on)g (the)g(r.h.s.,)h(together)-118 2754 y(with)e(Prop)s(osition)f(4,)h (lead)g(to)489 2987 y Fr(j)p Fw(A)590 3002 y FB(\001)649 3011 y Fn(1)683 3002 y Fv(;)p FB(\001)762 3011 y Fn(2)800 2987 y Fx(\()p Fw(T)8 b(;)17 b(\016)t Fx(\))p Fr(j)55 b(\024)h Fw(c)1290 3002 y Fv(\017)1339 2987 y Fw(\016)1386 2946 y Fu(\000)p FB(\(1)p Fv(=)p FB(2+)p Fv(\017)p FB(\))1723 2851 y Fq(Z)1778 3077 y FB(\001)1837 3086 y Fn(1)1909 2987 y Fw(d\032)2010 3002 y Fv(S)2061 2987 y Fx(\()p Fw(E)6 b Fx(\))2248 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Fx(\))17 b Fw(e)2544 3444 y Fu(\000)p FB(\()p Fv(E)t Fu(\000)p Fv(E)2793 3421 y Fo(0)2814 3444 y FB(\))2841 3421 y Fn(2)2876 3444 y Fv(T)2927 3421 y Fn(2)3010 3485 y Fr(\024)28 b Fw(T)3186 3444 y Fu(\000)p Fv(\013)3290 3345 y Fq(\033)3408 3485 y Fw(:)-118 3726 y Fx(By)33 b(de\014nition)f(of)g Fw(\032)630 3741 y Fv(S)713 3726 y Fx(it)g(follo)m(ws)f(then)i(that)34 3916 y Fr(h)o Fw(M)166 3931 y Fv(q)205 3916 y Fx(\()p Fw(H)r(;)17 b Fx(\001;)g Fw(T)d Fx(\))p Fr(i)642 3868 y Fv(g)642 3948 y(T)752 3916 y Fr(\025)56 b Fw(\016)932 3875 y Fu(\000)p Fv(q)r(=)p FB(2)1112 3916 y Fw(\032)1162 3931 y Fv(S)1213 3916 y Fx(\(\001\()p Fw(\013)q(;)17 b(T)d Fx(\)\))1657 3835 y Fq(\000)1702 3916 y Fx(1)22 b Fr(\000)h Fw(c)1915 3931 y Fv(\017)1947 3916 y Fw(\016)1994 3875 y Fu(\000)p FB(\(1)p Fv(=)p FB(2+)p Fv(\017)p FB(\))2298 3916 y Fw(T)2369 3875 y Fu(\000)p Fv(\013)2473 3835 y Fq(\001)2574 3916 y Fr(\025)56 b Fw(c)17 b(T)2837 3875 y Fv(q)r(\013=)p FB(\(1+2)p Fv(\017)p FB(\))3191 3916 y Fw(\032)3241 3931 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1105 y Fv(k)r FB(=1)p Fv(:::)o(K)3289 1090 y Fx(of)g(length)g(1)p Fw(=T)14 b Fx(.)-118 1210 y(Then)45 b Fw(K)54 b Fr(\024)48 b Fw(T)30 b Fr(j)p Fx(supp)q(\()p Fw(\032)p Fx(\))p Fr(j)44 b Fx(\(where)h Fr(j)p Fw(A)p Fr(j)f Fx(denotes)h(the)g(diameter)d(of)i Fw(A)p Fx(\).)78 b(If)44 b Fw(a)2888 1225 y Fv(k)2978 1210 y Fx(=)j(inf)6 b Fr(f)p Fw(\032)p Fx(\()p Fw(B)3436 1163 y Fv(g)3431 1237 y(T)3486 1210 y Fx(\()p Fw(E)g Fx(\)\))p Fr(j)p Fw(E)53 b Fr(2)-118 1331 y Fw(A)-45 1346 y Fv(k)21 1331 y Fr(\\)23 b Fx(\012)180 1346 y FB(0)220 1331 y Fr(g)p Fx(,)34 b(then)g Fw(a)605 1346 y Fv(k)678 1331 y Fr(\024)d Fw(T)857 1295 y Fu(\000)p Fv(\024)990 1331 y Fx(b)m(y)j(de\014nition)f(of)h(\012)1744 1346 y FB(0)1784 1331 y Fx(.)47 b(Moreo)m(v)m(er)35 b Fw(\032)p Fx(\()p Fw(B)2454 1284 y Fv(g)2449 1358 y(T)2505 1331 y Fx(\()p Fw(E)6 b Fx(\)\))29 b Fr(\025)2834 1250 y Fq(R)2881 1365 y Fv(A)2934 1377 y Fp(k)2972 1365 y Fu(\\)p FB(\012)3070 1374 y Fn(0)3125 1331 y Fw(d\032)p Fx(\()p Fw(E)3342 1295 y 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FB(\012)246 2005 y Fn(0)301 1906 y Fw(d\032)p Fx(\()p Fw(E)6 b Fx(\))28 b Fw(\032)p Fx(\()p Fw(B)751 1859 y Fv(g)746 1933 y(T)801 1906 y Fx(\()p Fw(E)6 b Fx(\)\))993 1865 y Fv(p)p Fu(\000)p FB(1)1178 1906 y Fr(\024)1318 1811 y Fq(X)1311 2023 y Fv(k)r Fu(\024)p Fv(K)1485 1906 y Fw(\032)p Fx(\()p Fw(A)1646 1921 y Fv(k)1711 1906 y Fr(\\)23 b Fx(\012)1870 1921 y FB(0)1910 1906 y Fx(\))p Fw(a)1999 1859 y Fv(p)p Fu(\000)p FB(1)1999 1934 y Fv(k)2184 1906 y Fr(\024)56 b Fw(e)2386 1811 y Fq(X)2379 2023 y Fv(k)r Fu(\024)p Fv(K)2553 1906 y Fw(a)2604 1859 y Fv(p)2604 1934 y(k)2702 1906 y Fr(\024)g Fw(e)17 b(T)2968 1865 y FB(1)p Fu(\000)p Fv(\024p)3155 1906 y Fr(j)p Fx(supp\()p Fw(\032)p Fx(\))p Fr(j)33 b Fx(.)-118 2185 y(Hence)k(c)m(ho)s(osing)e Fw(\024)f Fx(=)f(2)p Fw(=p)p Fx(,)j(for)f(example,)h(pro)m(vides)h(a)f(sub)s(dominan)m(t)f (con)m(tribution)f(in)h(\(38\))h(suc)m(h)h(that)-118 2305 y(\(38\))32 b(ful\014lls)e(the)j(desired)g(b)s(ound.)2651 b Fa(2)-118 2588 y Ff(5.2)135 b(Pro)t(of)45 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Fv(n)2061 5386 y Fx(\()p Fw(\022)s(\021)t(=)p Fx(2)p Fw(\031)t(;)17 b(E)6 b Fx(\):)1841 5635 y(17)p eop %%Page: 18 18 18 17 bop -118 32 a Fl(Lemma)37 b(4)49 b Fm(L)-5 b(et)39 b Fw( )f Fr(2)c(S)7 b Fx(\()p Fs(R)f Fx(\))p Fm(.)61 b(Then)38 b(the)g(r)-5 b(epr)g(esentative)38 b(in)g Fw(L)2294 -4 y FB(2)2334 32 y Fx(\()p Fs(R)5 b Fw(;)17 b(\032)2532 47 y Fv(S)2589 32 y Fx(\))38 b Fm(of)g(the)h(pr)-5 b(oje)g(ction)37 b(of)h Fw( )43 b Fm(on)38 b(the)-118 152 y Fw(H)-37 167 y Fv(W)43 152 y Fm(-cyclic)d(c)-5 b(omp)g(onent)33 b(of)i Fw(\036)1008 167 y Fv(S)1110 152 y Fm(is)f(given)h(by)707 388 y Fx(~)687 414 y Fw( )t Fx(\()p Fw(E)6 b Fx(\))55 b(=)1094 279 y Fq(Z)1194 305 y FB(2)p Fv(\031)1150 504 y FB(0)1293 414 y Fw(d\027)1392 429 y Fv(E)1451 414 y Fx(\()p Fw(\021)t Fx(\))1623 320 y Fq(X)1628 532 y Fv(n)p Fu(2)p Ft(Z)1804 414 y Fx(~)-53 b Fw(g)1847 429 y Fv(n)1894 414 y Fx(\()p Fw(\021)t(;)17 b(E)6 b Fx(\))27 b Fw( )2255 334 y Fq(\000)2300 414 y Fx(\()p Fw(\021)f Fr(\000)d Fx(2)p Fw(\031)t(n)p 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Fx(The)45 b(last)f(sum)g(on)g(the)h(r.h.s.)79 b(of)44 b(this)g(iden)m(tit)m(y)g(reconstructs)i(the)f(Mehler)f(k)m(ernel)h(of) f(eq.)h(\(32\))f(with)-118 3218 y Fw(t)28 b Fx(=)f Fw(\016)t(=\026)p Fx(.)43 b(It)33 b(will)d(b)s(e)j(con)m(v)m(enien)m(t)h(to)e(de\014ne) 564 3469 y Fw(G)641 3484 y Fv(\016)679 3469 y Fx(\()p Fw(E)6 b Fx(;)17 b Fw(x)p Fx(\))55 b(=)1118 3334 y Fq(Z)1234 3469 y Fw(d\027)1333 3484 y Fv(E)1393 3469 y Fx(\()p Fw(\021)1483 3428 y Fu(0)1506 3469 y Fx(\))1561 3375 y Fq(X)1590 3587 y Fv(m)1652 3568 y Fo(0)1749 3385 y Fq(\014)1749 3444 y(\014)1782 3469 y Fr(M)1902 3484 y Fv(S)1952 3469 y Fx(\()p Fw(\016)t(=\026)p Fx(;)17 b Fw(x;)g Fx(\()p Fw(\021)25 b Fr(\000)e Fx(2)p Fw(\031)t(m)2692 3428 y Fu(0)2715 3469 y Fx(\))17 b Fw(\022)2818 3428 y FB(1)p Fv(=)p FB(2)2928 3469 y Fw(=)p Fx(2)p Fw(\031)t Fx(\))3123 3385 y Fq(\014)3123 3444 y(\014)3189 3469 y Fw(:)508 b Fx(\(42\))-118 3757 y(Since)23 b(the)h(Mehler)g(k)m(ernel) g(deca)m(ys)h(fastly)-8 b(,)25 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3629 y Fv(n)3629 3614 y Fx(+)10 b Fw(s)3761 3629 y Fv(n)p Fu(\000)p FB(1)-118 3735 y Fx(if)27 b Fw(s)h Fx(=)g Fw(p;)17 b(q)t Fx(.)41 b(It)29 b(can)g(b)s(e)g(pro)m(v)m(ed)h(\(see)g([Her])f(Prop.)43 b(7.8.3\))28 b(that)g Fw(\013)i Fx(is)e(a)g(n)m(um)m(b)s(er)h(of)g (Roth)f(t)m(yp)s(e)i(\(see)f(eq.)h(\(2\))-118 3855 y(in)i(Section)g (2\))g(if)g(and)g(only)h(if)1039 3780 y Fq(P)1144 3807 y Fu(1)1144 3884 y Fv(n)p FB(=1)1298 3855 y Fw(a)1349 3870 y Fv(n)p FB(+1)1486 3855 y Fw(=q)1582 3819 y Fv(\017)1578 3880 y(n)1653 3855 y Fw(<)27 b Fr(1)33 b Fx(for)f(all)e Fw(\017)e(>)g Fx(0.)28 4023 y(The)39 b(pro)s(of)d(of)h(Prop)s(osition)f (6)h(relies)g(up)s(on)g(the)h(so-called)e Fm(Denjoy-Koksma)h(ine)-5 b(quality)38 b Fx([Her].)58 b(Let)-118 4143 y Fw(')35 b Fx(b)s(e)g(a)f(p)s(erio)s(dic)f(function)i(on)f Fs(R)46 b Fx(with)34 b(p)s(erio)s(d)g(1,)h(of)g(b)s(ounded)g(total)f(v)-5 b(ariation)32 b(V)-8 b(ar\()p Fw(')p Fx(\))34 b(o)m(v)m(er)i(a)f(p)s (erio)s(d)-118 4263 y(in)m(terv)-5 b(al.)42 b(Then)34 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1134 y Fx(\))35 b(=)394 1060 y Fq(P)499 1163 y Fv(k)r(;m)641 1134 y Fw(e)686 1098 y Fu(\000)p Fv(a)12 b(F)835 1110 y Fp(\016)869 1098 y FB(\()p Fv(x)936 1107 y Fn(0)970 1098 y FB(+)p Fv(k)r(;y)1119 1107 y Fn(0)1153 1098 y FB(+)p Fv(m\013)p FB(\))1384 1134 y Fx(then)37 b Fw(S)42 b Fx(is)37 b(p)s(erio)s(dic)e(of)h(p)s(erio)s(d)f(1)i(in)e Fw(x)2873 1149 y FB(0)2950 1134 y Fx(and)i(of)f(p)s(erio)s(d)f Fw(\013)j Fx(in)e Fw(y)3832 1149 y FB(0)3871 1134 y Fx(.)-118 1255 y(Therefore,)d(it)f(is)g(enough)g(to)g(assume)h(0)28 b Fr(\024)g Fw(x)1574 1270 y FB(0)1641 1255 y Fw(<)g Fx(1)k(and)h(0)27 b Fr(\024)h Fw(y)2245 1270 y FB(0)2312 1255 y Fw(<)f Fx(1)32 b(\(since)h(0)28 b Fw(<)f(\013)h(<)g Fx(1\).)43 b(F)-8 b(or)32 b(0)27 b Fw(<)h(\033)j(<)d Fx(1)-118 1375 y(and)34 b(for)f Fw(j)j Fr(2)30 b Fs(N)9 b Fx(,)40 b(let)33 b Fr(L)739 1390 y Fv(j)809 1375 y Fx(b)s(e)h(the)g(set)g(of)g(p)s(oin)m(ts)f(\()p Fw(x;)17 b(y)t Fx(\))29 b Fr(2)h(L)j Fx(for)g(whic)m(h)i Fw(j)2603 1339 y FB(2)2642 1375 y Fw(\016)2689 1339 y Fu(\000)p Fv(\033)2821 1375 y Fr(\024)30 b Fw(F)2991 1390 y Fv(\016)3029 1375 y Fx(\()p Fw(x;)17 b(y)t Fx(\))29 b Fw(<)g Fx(\()p Fw(j)g Fx(+)23 b(1\))3683 1339 y FB(2)3722 1375 y Fw(\016)3769 1339 y Fu(\000)p Fv(\033)3871 1375 y Fx(.)-118 1496 y(Th)m(us)1236 1758 y Fw(S)6 b Fx(\()p Fw(x)1395 1773 y FB(0)1435 1758 y Fw(;)17 b(y)1527 1773 y FB(0)1565 1758 y Fx(\))56 b Fr(\024)1828 1634 y Fu(1)1791 1664 y Fq(X)1802 1874 y Fv(j)t FB(=0)1952 1758 y Fw(e)1997 1717 y Fu(\000)p Fv(aj)2122 1694 y Fn(2)2156 1717 y Fv(\016)2189 1694 y Fo(\000)p Fp(\033)2312 1758 y Fr(jL)2409 1773 y Fv(j)2444 1758 y Fr(j)28 b Fw(;)1197 b Fx(\(43\))-118 2071 y(where)40 b Fr(j)p Fw(A)p Fr(j)e Fx(denotes)i(the)f(n)m(um)m(b)s(er)g(of)f(p)s (oin)m(ts)h(in)f Fw(A)p Fx(.)62 b Fr(L)1999 2086 y Fv(j)2073 2071 y Fx(is)39 b(con)m(tained)f(in)g(an)h(elliptic)d(cro)m(wn)j(with)g (axis)-118 2191 y(along)31 b(the)i(t)m(w)m(o)g(diagonals)e Fw(x)d Fx(=)g Fr(\006)p Fw(y)t Fx(.)43 b(In)33 b(particular,)37 2403 y(\()p Fw(x;)17 b(y)t Fx(\))27 b Fr(2)h(L)454 2418 y Fv(j)573 2403 y Fr(\))83 b Fx(max)o Fr(fj)p Fw(x)p Fr(j)p Fw(;)17 b Fr(j)p Fw(y)t Fr(jg)42 b(\024)j Fx(\()p Fw(j)28 b Fx(+)22 b(1\))17 b Fw(\016)1819 2362 y Fu(\000)p FB(\(1+)p Fv(\033)r FB(\))p Fv(=)p FB(2)2219 2403 y Fx(and)83 b Fr(j)p Fw(x)22 b Fr(\000)h Fw(y)t Fr(j)43 b(\024)i Fx(\()p Fw(j)28 b Fx(+)22 b(1\))17 b Fw(\016)3264 2362 y FB(\(1)p Fu(\000)p Fv(\033)r FB(\))p Fv(=)p FB(2)3542 2403 y Fw(:)155 b Fx(\(44\))-118 2615 y(If)38 b Fw(j)44 b Fr(\025)39 b Fx(1,)h(the)f(n)m(um)m(b)s(er)g(of)f(p)s(oin)m(ts)g(con) m(tained)h(in)f Fr(L)1889 2630 y Fv(j)1963 2615 y Fx(can)h(b)s(e)g (estimated)f(b)m(y)i(coun)m(ting)e(the)h(n)m(um)m(b)s(er)g(of)-118 2735 y(rectangular)e(cells)g(of)g(sizes)i(\(1)p Fw(;)17 b(\013)q Fx(\))37 b(cen)m(tered)i(at)f(p)s(oin)m(ts)f(of)g Fr(L)h Fx(and)g(meeting)e(the)i(elliptic)d(cro)m(wn.)60 b(Since)-118 2856 y(this)35 b(cro)m(wn)i(is)e(included)g(inside)g(the)h (square)h(max)p Fr(fj)p Fw(x)p Fr(j)p Fw(;)17 b Fr(j)p Fw(y)t Fr(jg)30 b(\024)k Fx(\()p Fw(j)c Fx(+)24 b(1\))p Fw(\016)2646 2820 y Fu(\000)p FB(\(1+)p Fv(\033)r FB(\))p Fv(=)p FB(2)2999 2856 y Fx(it)34 b(is)h(enough)i(to)e(coun)m(t)-118 2976 y(suc)m(h)45 b(cells)d(meeting)g(this)h(square.)77 b(Suc)m(h)44 b(cells)f(are)g(all)e(included)i(inside)g(the)h(square)g Fw(C)53 b Fx(=)45 b Fr(f)p Fx(\()p Fw(x;)17 b(y)t Fx(\))45 b Fr(2)-118 3097 y Fs(R)-52 3060 y FB(2)10 3097 y Fr(j)33 b Fx(max)o Fr(fj)p Fw(x)p Fr(j)p Fw(;)17 b Fr(j)p Fw(y)t Fr(jg)26 b(\024)i Fx(\()p Fw(j)c Fx(+)18 b(2\))p Fw(\016)1076 3060 y Fu(\000)p FB(\(1+)p Fv(\033)r FB(\))p Fv(=)p FB(2)1393 3097 y Fr(g)31 b Fx(\(since)g Fw(\016)h Fr(\024)c Fx(1\).)42 b(Hence)32 b(the)f(n)m(um)m(b)s(er)h(of)e(suc)m(h)i(cells)e(is)g (certainly)-118 3217 y(dominated)h(b)m(y)j(the)f(ratio)e(of)h(the)h (area)f(of)g Fw(C)40 b Fx(to)32 b(the)h(area)f(of)g(eac)m(h)i(cell,)d (namely)1358 3495 y Fr(jL)1455 3510 y Fv(j)1491 3495 y Fr(j)55 b(\024)1716 3427 y Fx(\()p Fw(j)29 b Fx(+)22 b(2\))p 1716 3472 291 4 v 1831 3563 a Fw(\013)2017 3379 y FB(2)2084 3495 y Fw(\016)2131 3454 y Fu(\000)p FB(\(1+)p Fv(\033)r FB(\))2395 3495 y Fw(:)-118 3738 y Fx(Therefore,)48 b(the)d(part)f(of)g(the)h(sum)f(in)f(\(43\))h(coming)f(from)g Fw(j)53 b Fr(\025)48 b Fx(1)c(con)m(v)m(erges)j(to)d(zero)g(as)h Fw(\016)51 b Fr(#)d Fx(0.)78 b(In)-118 3858 y(particular,)33 b(it)g(is)g(b)s(ounded)i(b)m(y)g(a)f(constan)m(t)h Fw(c)1616 3873 y FB(1)1690 3858 y Fx(that)f(is)f(indep)s(enden)m(t)i(of)f(\()p Fw(x)2762 3873 y FB(0)2802 3858 y Fw(;)17 b(y)2894 3873 y FB(0)2933 3858 y Fx(\).)48 b(Th)m(us,)36 b(it)d(is)g(su\016cien)m(t) -118 3978 y(to)f(consider)h(the)g(term)f Fw(j)i Fx(=)27 b(0)33 b(only)-8 b(.)28 4146 y(Let)37 b Fw(')g Fx(b)s(e)g(the)h (function)e(on)h Fs(R)48 b Fx(de\014ned)38 b(b)m(y)g Fw(')p Fx(\()p Fw(x)p Fx(\))71 b(=)2145 4071 y Fq(P)2250 4175 y Fv(k)r Fu(2)p Ft(Z)2401 4146 y Fw(\037)2462 4161 y Fv(I)2502 4146 y Fx(\()p Fw(x)25 b Fx(+)g Fw(y)2769 4161 y FB(0)2833 4146 y Fr(\000)h Fw(x)2991 4161 y FB(0)3056 4146 y Fx(+)f Fw(k)s Fx(\))37 b(where)h Fw(I)45 b Fx(is)36 b(the)-118 4266 y(in)m(terv)-5 b(al)38 b Fw(I)47 b Fx(=)38 b([)p Fr(\000)p Fw(\016)596 4230 y FB(\(1)p Fu(\000)p Fv(\033)r FB(\))p Fv(=)p FB(2)860 4266 y Fw(;)17 b(\016)951 4230 y FB(\(1)p Fu(\000)p Fv(\033)r FB(\))p Fv(=)p FB(2)1213 4266 y Fx(])39 b Fr(\032)g Fs(R)5 b Fx(.)70 b(It)39 b(is)g(a)f(p)s (erio)s(dic)g(function)h(of)g(p)s(erio)s(d)f(1)h(with)f(V)-8 b(ar\()p Fw(')p Fx(\))39 b(=)g(2.)-118 4387 y(Moreo)m(v)m(er,)34 b(using)e(\(44\))g(it)g(can)h(b)s(e)f(c)m(hec)m(k)m(ed)k(easily)c(that) 503 4691 y Fw(S)6 b Fx(\()p Fw(x)662 4706 y FB(0)702 4691 y Fw(;)17 b(y)794 4706 y FB(0)832 4691 y Fx(\))56 b Fr(\024)f Fw(c)1100 4706 y FB(1)1162 4691 y Fx(+)1304 4597 y Fq(X)1260 4812 y Fu(j)p Fv(m)p Fu(j)p Fv()d(n)33 b Fx(so)f(that)h Fw(X)563 5116 y Fv(n)637 5101 y Fx(=)28 b Fr(;)k Fx(for)g(all)f Fw(n)d Fr(\025)g Fx(1.)28 5221 y(If)33 b Fw(H)207 5236 y Fv(W)320 5221 y Fx(had)f(a)h(cyclic)f(v)m(ector,)i(its)e(sp)s(ectrum)h(w)m(ould) f(b)s(e)h(simple.)1365 b Fa(2)1841 5635 y Fx(23)p eop %%Page: 24 24 24 23 bop -118 52 a Fy(References)-118 272 y Fx([BSB])105 b(J.-M.)44 b(Barbaroux,)j(H.)c(Sc)m(h)m(ulz-Baldes,)k Fi(Anomalous)c(transp)s(ort)g(in)g(presence)j(of)d(self-similar)233 392 y(sp)s(ectra)p Fx(,)34 b(Annales)f(I.H.P)-8 b(.)33 b(Ph)m(ys.)h(th)m(\023)-46 b(eo.)33 b Fl(71)p Fx(,)g(539-559)e (\(1999\).)-118 595 y([BGT])81 b(J.M.)44 b(Barbaroux,)i(F.)d(Germinet,) i(S.)e(Tc)m(heremc)m(han)m(tsev,)50 b Fi(Nonlinear)41 b(v)-5 b(ariation)41 b(of)i(di\013usion)233 716 y(exp)s(onen)m(ts)35 b(in)d(quan)m(tum)h(dynamics)p Fx(,)f(C.R.)h(Acad.)g(Sci.)f(P)m(aris)h Fl(330)p Fx(,)g(s)m(\023)-46 b(erie)32 b(I,)h(409-414)e(\(2000\).)-118 919 y([Bel])157 b(J.)40 b(Bellissard,)f Fi(K-theory)g(of)g(C)1438 883 y Fu(\003)1478 919 y Fi(-algebras)f(in)g(solid)g(state)i(ph)m (ysics)p Fx(,)i(in)c Fi(Statistical)f(Mec)m(hanics)233 1039 y(and)43 b(Field)e(Theory:)64 b(Mathematical)41 b(Asp)s(ects)p Fx(,)46 b Fi(Lecture)e(Notes)f(in)f(Ph)m(ysics)i Fl(257)p Fx(,)h(edited)e(b)m(y)233 1160 y(T.)e(Dorlas,)g(M.)g (Hugenholtz,)i(M.)e(Winnink,)h(99-156)c(\(Springer-V)-8 b(erlag,)41 b(Berlin,)g(1986\);)i(and)233 1280 y Fi(Gap)38 b(lab)s(elling)d(theorems)k(for)f(Sc)m(hr\177)-49 b(odinger)39 b(op)s(erators)p Fx(,)h(538-630,)e(in)g Fi(F)-8 b(rom)37 b(Num)m(b)s(er)h(Theory)233 1401 y(to)33 b(Ph)m(ysics)p Fx(,)h(\(Springer,)e(Berlin,)g(1992\).)-118 1604 y([Bel94])59 b(J.)30 b(Bellissard,)f Fi(Lipshitz)f(con)m(tin)m(uit)m(y)i(of)e(gap)i (b)s(oundaries)f(for)g(Hofstadter-lik)m(e)f(sp)s(ectra)p Fx(,)j(Comm.)233 1724 y(Math.)i(Ph)m(ys.)i Fl(160)p Fx(,)e(599-613)d (\(1994\).)-118 1928 y([Com])96 b(J.-M.)39 b(Com)m(b)s(es,)i(in)c (Di\013eren)m(tial)f(Equations)j(with)f(Applications)f(to)h (Mathematical)f(Ph)m(ysics,)233 2048 y(Ames)c(W.F.,)g(Harell)e(E.M.,)i (Hero)s(d)g(J.V.)g(Eds,)h(Academic)e(Press,)i(Boston)f(\(1993\).)-118 2252 y([Co])178 b(A.)33 b(Connes,)h Fi(Noncomm)m(utativ)m(e)e(Geometry) p Fx(,)g(Academic)g(Press,)j(London,)d(\(1994\).)-118 2455 y([DS])168 b(P)-8 b(.)32 b(Deift,)d(B.)i(Simon,)f Fi(Almost)g(p)s(erio)s(dic)f(Sc)m(hrdinger)i(op)s(erators.)g(I)s(I)s (I.)g(The)h(absolutely)e(con)m(tin)m(u-)233 2575 y(ous)j(sp)s(ectrum)g (in)f(one)h(dimension)p Fx(,)e(Comm.)h(Math.)h(Ph)m(ys.)h Fl(90)p Fx(,)f(389-411)d(\(1983\).)-118 2779 y([Gua])117 b(I.)41 b(Guarneri,)h Fi(Sp)s(ectral)e(prop)s(erties)h(of)f(quan)m(tum) h(di\013usion)f(on)g(discrete)i(lattices)p Fx(,)g(Europh)m(ys.)233 2899 y(Lett.)32 b Fl(10)p Fx(,)h(95-100)d(\(1989\);)h Fi(On)h(an)f(estimate)g(concerning)h(quan)m(tum)g(di\013usion)f(in)g (the)h(presence)233 3020 y(of)h(a)f(fractal)f(sp)s(ectrum)p Fx(,)i(Europh)m(ys.)i(Lett.)d Fl(21)p Fx(,)h(729-733)e(\(1993\).)-118 3223 y([GM])131 b(I.)39 b(Guarneri,)g(G.)f(Man)m(tica,)i Fi(Multifractal)c(Energy)j(Sp)s(ectra)g(and)g(Their)f(Dynamical)e (Implica-)233 3343 y(tions)p Fx(,)d(Ph)m(ys.)h(Rev.)f(Lett.)g Fl(73)p Fx(,)g(3379-3383)d(\(1994\).)-118 3547 y([GSB1])48 b(I.)24 b(Guarneri,)h(H.)f(Sc)m(h)m(ulz-Baldes,)i Fi(Upp)s(er)e(b)s (ounds)h(for)e(quan)m(tum)h(dynamics)f(go)m(v)m(erned)j(b)m(y)e(Jacobi) 233 3667 y(matrices)32 b(with)g(self-similar)d(sp)s(ectra)p Fx(,)k(Rev.)h(Math.)e(Ph)m(ys.)j Fl(11)p Fx(,)e(1249-1268)d(\(1999\).) -118 3871 y([GSB2])48 b(I.)37 b(Guarneri,)f(H.)h(Sc)m(h)m(ulz-Baldes,)g Fi(Lo)m(w)m(er)h(b)s(ounds)e(on)h(w)m(a)m(v)m(e)h(pac)m(k)m(et)g (propagation)c(b)m(y)j(pac)m(king)233 3991 y(dimensions)32 b(of)g(sp)s(ectral)h(measures)p Fx(,)g(Elect.)g(J.)g(Math.)f(Ph)m(ys.)j Fl(5)p Fx(,)d(\(1999\).)-118 4194 y([GSB3])48 b(I.)39 b(Guarneri,)h(H.)f(Sc)m(h)m(ulz-Baldes,)i Fi(In)m(termitten)m(t)d(lo)m (w)m(er)h(b)s(ound)g(on)g(quan)m(tum)g(di\013usion)p Fx(,)g(Lett.)233 4315 y(Math.)33 b(Ph)m(ys.)i Fl(49)p Fx(,)d(317-324)f(\(1999\).)-118 4518 y([Har])137 b(P)-8 b(.)44 b(G.)e(Harp)s(er,)k Fi(Single)c(Band)h(Motion)f(of)h(Conduction) g(Electrons)g(in)g(a)g(Uniform)e(Magnetic)233 4638 y(Field)p Fx(,)31 b(Pro)s(c.)i(Ph)m(ys.)i(So)s(c.)d(Lond.)h(A)g Fl(68)p Fx(,)f(874-878,)f(\(1955\).)-118 4842 y([Her])143 b(M.)35 b(R.)f(Herman,)g Fi(Sur)g(la)f(conjugaison)h(di\013)m(\023)-46 b(eren)m(tiable)32 b(des)k(di\013)m(\023)-46 b(eomorphismes)32 b(du)i(cercle)h(\022)-49 b(a)34 b(des)233 4962 y(rotations)p Fx(,)e(Publications)f(I.H.E.S.)j Fl(49)p Fx(,)f(5-233)e(\(1979\).)-118 5166 y([Jit])181 b(S.)33 b(Jitomirsk)-5 b(a)m(y)m(a,)32 b Fi(Metal-Insulator)f(T)-8 b(ransition)32 b(for)g(the)h(Almost)f (Mathieu)g(Op)s(erator)p Fx(,)h(Annals)233 5286 y(of)g(Math.)f Fl(150)p Fx(,)h(1159-1175)d(\(1999\).)1841 5635 y(24)p eop %%Page: 25 25 25 24 bop -118 28 a Fx([Las])149 b(Y.)48 b(Last,)j Fi(Quan)m(tum)c (Dynamics)f(and)h(decomp)s(osition)f(of)g(singular)g(con)m(tin)m(uous)i (sp)s(ectra)p Fx(,)j(J.)233 148 y(F)-8 b(unct.)33 b(Anal.)f Fl(142)p Fx(,)h(402-445)e(\(1996\).)-118 351 y([KKK)m(G])48 b(R.)31 b(Ketzmeric)m(k,)g(K.)g(Kruse,)g(S.)g(Kraut,)g(and)g(T.)g (Geisel,)f Fi(What)h(determines)f(the)h(spreading)g(of)233 472 y(a)i(w)m(a)m(v)m(e)h(pac)m(k)m(et?)p Fx(,)h(Ph)m(ys.)f(Rev.)f (Lett.)g Fl(79)p Fx(,)g(1959-1962)d(\(1997\).)-118 675 y([KL])160 b(A.)37 b(Kiselev,)g(Y.)f(Last,)i Fi(Solutions,)e(sp)s (ectrum,)i(and)e(dynamics)g(for)g(Sc)m(hr\177)-49 b(odinger)37 b(op)s(erators)f(on)233 796 y(in\014nite)c(domains)p Fx(,)f(Duk)m(e)j(Math.)f(J.)f Fl(102)p Fx(,)h(125-150)e(\(2000\).)-118 999 y([Man])105 b(G.)46 b(Man)m(tica,)k Fi(Quan)m(tum)c(in)m (termittency)g(in)g(almost)e(p)s(erio)s(dic)h(systems)j(deriv)m(ed)f (from)e(their)233 1119 y(sp)s(ectral)e(prop)s(erties)p Fx(,)i(Ph)m(ysica)f Fl(D)49 b(103)p Fx(,)d(576-589,)d(\(1997\);)k Fi(W)-8 b(a)m(v)m(e)44 b(Propagation)d(in)h(Almost-)233 1240 y(P)m(erio)s(dic)32 b(Structures)p Fx(,)i(Ph)m(ysica)g Fl(D)j(109)p Fx(,)c(113-127)e(\(1997\).)-118 1443 y([P)m(er])153 b(A.)32 b(P)m(erelomo)m(v,)g Fi(Generalized)f(Coheren)m(t)j(States)e (and)g(Their)g(Applications)p Fx(,)e(\(Springer,)i(Berlin,)233 1563 y(1986\).)-118 1767 y([Pie])160 b(F.)28 b(Pi)m(\023)-46 b(ec)m(hon,)29 b Fi(Anomalous)e(Di\013usion)f(Prop)s(erties)i(of)f(W)-8 b(a)m(v)m(e)29 b(P)m(ac)m(k)m(ets)h(on)e(Quasip)s(erio)s(dic)d(Chains)p Fx(,)233 1887 y(Ph)m(ys.)35 b(Rev.)e(Lett.)g Fl(76)p Fx(,)g(4372-4375)d(\(1996\).)-118 2091 y([Ram])94 b(R.)32 b(Rammal,)c Fi(Landau)j(lev)m(el)g(sp)s(ectrum)h(of)f(Blo)s(c)m(h)f (electron)i(in)e(a)h(honeycom)m(b)h(lattice)p Fx(,)e(J.)i(Ph)m(ys.)233 2211 y(F)-8 b(rance)33 b Fl(46)p Fx(,)g(1345-1354,)d(\(1985\).)-118 2414 y([Rie])154 b(M.)34 b(A.)g(Rie\013el,)e Fi(C)917 2378 y Fu(\003)957 2414 y Fi(-algebras)g(asso)s(ciated)i(with)f (irrational)d(rotations)p Fx(,)i(P)m(ac.)j(J.)e(Math.)h Fl(93)p Fx(,)g(415-)233 2535 y(429)e(\(1981\).)-118 2738 y([RP])159 b(A.)31 b(R)s(\177)-51 b(udinger,)30 b(F.)g(Pi)m(\023)-46 b(ec)m(hon,)32 b Fi(Hofstadter)f(rules)g(and)g(generalized)f (dimensions)g(of)g(the)h(sp)s(ectrum)233 2859 y(of)i(Harp)s(er's)g (equation)p Fx(,)f(J.)h(Ph)m(ys.)h Fl(A)j(30)p Fx(,)c(117-128)e (\(1997\).)-118 3062 y([Sak])143 b(S.)33 b(Sak)-5 b(ai,)32 b Fi(C)702 3026 y Fu(\003)742 3062 y Fi(-algebras)f(and)i(W)1444 3026 y Fu(\003)1483 3062 y Fi(-algebras)p Fx(,)f(\(Springer,)g(Berlin,) g(1971\).)-118 3265 y([SBB])105 b(H.)25 b(Sc)m(h)m(ulz-Baldes,)i(J.)d (Bellissard,)h Fi(Anomalous)e(transp)s(ort:)39 b(a)24 b(mathematical)e(framew)m(ork)p Fx(,)k(Rev.)233 3386 y(Math.)33 b(Ph)m(ys.)i Fl(10)p Fx(,)d(1-46)g(\(1998\).)-118 3589 y([Sei])172 b(K.)30 b(Seip,)g Fi(Densit)m(y)g(theorems)g(for)f (sampling)e(and)j(in)m(terp)s(olation)d(in)h(the)i(Bargmann-F)-8 b(o)s(c)m(k)28 b(space)233 3710 y Fx(I,)33 b(J.)g(reine)f(angew)m(a.)i (Math.)f Fl(429)p Fx(,)f(91-106,)f(\(1992\).)-118 3913 y([TK])151 b(C.)39 b(T)-8 b(ang,)39 b(M.)f(Kohmoto,)g Fi(Global)d(scaling)i(prop)s(erties)h(of)f(the)h(sp)s(ectrum)g(for)g(a) f(quasip)s(erio)s(dic)233 4033 y(Sc)m(hrdinger)c(equation)p Fx(,)g(Ph)m(ys.)h(Rev.)g Fl(B)j(34)p Fx(,)32 b(2041-2044)f(\(1986\).) 1841 5635 y(25)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0201031058829--