Content-Type: multipart/mixed; boundary="-------------0112050226516" This is a multi-part message in MIME format. ---------------0112050226516 Content-Type: text/plain; name="01-449.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-449.keywords" Schrodinger operator, propagation properties, anisotropic potential, compactification, C*-algebra ---------------0112050226516 Content-Type: application/postscript; name="Prop.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Prop.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.47 Copyright 1986-91 Radical Eye Software %%Title: Prop.dvi %%Pages: 15 1 %%BoundingBox: 0 0 596 843 %%EndComments %%BeginProcSet: tex.pro /TeXDict 200 dict def TeXDict begin /N /def load def /B{bind def}N /S /exch load def /X{S N}B /TR /translate load N /isls false N /vsize 10 N /@rigin{ isls{[0 1 -1 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale Resolution VResolution vsize neg mul TR matrix 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(careful)h(sp)r(ectral)g(analysis)e(of)j(the)f(Hamiltoni-)418 1912 y(ans)20 b(under)h(study)-7 b(.)35 b(W)-7 b(e)21 b(prop)r(ose)f(here)g (a)h(di\013eren)n(t)g(metho)r(d)g(for)f(obtaining)h(non-propagation)418 2011 y(prop)r(erties,)f(based)e(on)h(a)f(relativ)n(ely)g(recen)n(t)g(approac) n(h)f(to)i(sp)r(ectral)f(theory)g(in)i(the)f(framew)n(ork)418 2111 y(of)29 b(C*-algebras)e(and)i(without)h(in)n(v)n(oking)e(an)n(y)g (scattering)h(theory)-7 b(.)41 b(W)-7 b(e)30 b(shall)f(in)g(particular)418 2211 y(obtain)i(the)g(non-propagation)d(result)j(stated)g(in)g(the)g(second)f (example)h(ab)r(o)n(v)n(e.)45 b(Belo)n(w)30 b(w)n(e)418 2310 y(giv)n(e)d(a)g(brief)h(non-tec)n(hnical)e(description)h(of)h(this)g(metho)r (d.)518 2414 y(T)n(ypically)j(the)h(p)r(oten)n(tial)h Fr(V)51 b Fz(to)32 b(b)r(e)g(considered)f(is)h(an)g(elemen)n(t)g(of)g(a)g(C*-algebra) e Fq(A)i Fz(of)418 2514 y(b)r(ounded,)h(con)n(tin)n(uous)d(functions)i(on)f Fp(R)1739 2484 y Fo(n)1790 2514 y Fz(.)48 b(The)31 b(functions)h(in)f Fq(A)h Fz(are)e(c)n(haracterized)f(b)n(y)i(a)418 2613 y(sp)r(eci\014c)21 b(asymptotic)f(b)r(eha)n(viour)f(\(for)i(example)f(asymptotic)g(p)r(erio)r (dicit)n(y)g(in)h(certain)f(cones\).)418 2713 y(Then,)31 b(b)n(y)f(in)n(v)n (oking)f(the)h(Neumann)g(series)g(for)f(\()p Fr(H)e Fq(\000)20 b Fr(z)t Fz(\))2280 2683 y Fn(\000)p Fm(1)2399 2713 y Fz(\(whic)n(h)30 b(is)g(con)n(v)n(ergen)n(t)e(for)i Fq(=)p Fr(z)418 2813 y Fz(large)24 b(enough\),)h(one)g(\014nds)g(that)h(the)f(resolv)n(en)n(t)f(of)h(the)g(op)r (erator)f Fr(H)29 b Fz(=)23 b Fq(\000)p Fz(\001)14 b(+)g Fr(V)43 b Fz(b)r(elongs)24 b(to)418 2912 y(a)j(C*-algebra)d Fj(C)957 2924 y Fn(A)1041 2912 y Fz(generated)i(b)n(y)h(pro)r(ducts)f(of)h(elemen)n (ts)g(of)g Fq(A)g Fz(\(view)n(ed)g(as)f(m)n(ultiplication)418 3012 y(op)r(erators)f(in)i Fr(L)938 2982 y Fm(2)975 3012 y Fz(\()p Fp(R)1061 2982 y Fo(n)1112 3012 y Fz(\)\))g(and)g(suitable)g (functions)g(of)g(momen)n(tum.)36 b(W)-7 b(e)28 b(shall)e(sa)n(y)f(that)j Fr(H)33 b Fz(is)418 3112 y(a\016liated)25 b(to)g Fj(C)907 3124 y Fn(A)965 3112 y Fz(.)36 b(A)25 b(cen)n(tral)g(concept)g(is)g(that)g(of)h (the)f(sp)r(ectrum)h(of)f Fr(H)32 b Fz(relativ)n(e)24 b(to)h(an)g(ideal)418 3211 y Fj(K)f Fz(of)g Fj(C)639 3223 y Fn(A)720 3211 y Fz(of)f(the)h(form)f Fj(K)h Fz(=)e Fj(C)1358 3223 y Fn(K)1413 3211 y Fz(,)i(where)f Fq(K)i Fz(is)f(an)f(ideal)g(of)h Fq(A)g Fz(and)f Fj(C)2558 3223 y Fn(K)2636 3211 y Fz(is)h(de\014ned)g(similarly)e(to)418 3311 y Fj(C)469 3323 y Fn(A)556 3311 y Fz(\(just)30 b(replace)f Fq(A)h Fz(b)n(y)f Fq(K)i Fz(in)f(the)g(de\014nition)g(of)f Fj(C)2107 3323 y Fn(A)2164 3311 y Fz(\).)43 b(Let)30 b(us)f(denote)h(this)g (sp)r(ectrum)f(b)n(y)418 3410 y Fr(\033)468 3380 y Fn(K)523 3410 y Fz(\()p Fr(H)7 b Fz(\))26 b(\(a)e(precise)g(de\014nition)h(is)g(giv)n (en)f(in)h(Section)f(2\).)36 b(F)-7 b(or)24 b Fq(K)h Fz(=)d Fq(f)p Fz(0)p Fq(g)p Fz(,)i Fr(\033)2776 3380 y Fn(K)2832 3410 y Fz(\()p Fr(H)7 b Fz(\))25 b(is)f(the)h(usual)418 3510 y(sp)r(ectrum)32 b(of)f Fr(H)7 b Fz(;)33 b(for)e Fq(K)g Fz(=)e Fr(C)1393 3522 y Fm(0)1431 3510 y Fz(\()p Fp(R)1517 3480 y Fo(n)1568 3510 y Fz(\))j(\(the)g(space)f(of)g(con)n(tin)n(uous)f(functions)i(con)n(v)n (erging)d(to)418 3610 y(zero)e(at)g(in\014nit)n(y\),)i Fj(K)f Fz(will)g(b)r(e)g(the)g(ideal)f(of)h(compact)f(op)r(erators)f(and)h Fr(\033)2737 3580 y Fn(K)2793 3610 y Fz(\()p Fr(H)7 b Fz(\))28 b(the)g(essen)n(tial)418 3709 y(sp)r(ectrum)36 b Fr(\033)836 3721 y Fm(ess)923 3709 y Fz(\()p Fr(H)7 b Fz(\))36 b(of)g Fr(H)7 b Fz(.)62 b(F)-7 b(or)35 b(an)g(ideal)h Fq(K)h Fz(of)f Fq(A)h Fz(suc)n(h)e(that)h Fr(C)2598 3721 y Fm(0)2636 3709 y Fz(\()p Fp(R)2722 3679 y Fo(n)2773 3709 y Fz(\))h Fq(\032)g(K)q Fz(,)h Fr(\033)3119 3679 y Fn(K)3175 3709 y Fz(\()p Fr(H)7 b Fz(\))36 b(is)418 3809 y(a)e(subset)h(of)g Fr(\033)905 3821 y Fm(ess)991 3809 y Fz(\()p Fr(H)7 b Fz(\).)59 b(A)35 b(t)n(ypical)f(non-propagation)e (result)i(will)h(assert)e(that)i(scattering)418 3909 y(states)i(of)h Fr(H)44 b Fz(with)38 b(sp)r(ectral)f(supp)r(ort)h(disjoin)n(t)f(from)g Fr(\033)2282 3878 y Fn(K)2338 3909 y Fz(\()p Fr(H)7 b Fz(\))38 b(will)f(essen)n(tially)g(nev)n(er)g(b)r(e)418 4008 y(lo)r(calized)27 b(in)h(certain)f(spatial)g(domains)g Fr(W)39 b Fz(determined)28 b(b)n(y)g Fq(K)q Fz(.)518 4112 y(In)39 b(these)g(considerations)e(the)i (ideal)g Fq(K)h Fz(will)f(b)r(e)h(giv)n(en)e(in)h(terms)f(of)h(the)g (asymptotic)418 4212 y(b)r(eha)n(viour)17 b(of)i(the)g(elemen)n(ts)f(of)h Fq(A)f Fz(in)h(some)f(neigh)n(b)r(ourho)r(o)r(d)g(of)g(in\014nit)n(y)-7 b(,)21 b(and)d(the)h(theory)f(will)418 4311 y(apply)h(if)g(the)g(spatial)f (domains)g Fr(W)31 b Fz(asso)r(ciated)18 b(to)g Fq(K)j Fz(co)n(v)n(er)c(or)h (in)n(tersect)g(this)h(neigh)n(b)r(ourho)r(o)r(d)418 4411 y(of)30 b(in\014nit)n(y)-7 b(.)46 b(In)30 b(the)h(second)f(example)f(men)n(tioned)i (at)f(the)h(b)r(eginning)f Fq(K)i Fz(could)e(b)r(e)g(the)h(set)418 4510 y(of)24 b(functions)h(in)g Fq(A)f Fz(that)h(are)e(asymptotically)h(p)r 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b Fz(and)28 b Fr(W)39 b Fz(is)27 b(a)418 5112 y(subset)i(of)g Fp(R)824 5082 y Fo(n)904 5112 y Fz(whic)n(h)g(is)g(su\016cien)n(tly)g(close)f(to)h Fr(F)12 b Fz(.)41 b(In)29 b(the)h(cited)f(example)g(one)f(ma)n(y)g(think,)p eop %%Page: 3 3 bop 779 314 a Fx(PR)n(OP)-6 b(A)n(GA)g(TION)29 b(PR)n(OPER)-6 b(TIES)28 b(F)n(OR)g(SCHR)2210 297 y(\177)2200 314 y(ODINGER)h(OPERA)-6 b(TORS)354 b(3)418 513 y Fz(in)n(tuitiv)n(ely)-7 b(,)26 b(of)g Fr(F)37 b Fz(as)25 b(the)h(part)f(of)g(the)h(compacti\014cation)f(attac)n (hed)g(at)g(in\014nit)n(y)h(to)g(the)f(cone)418 613 y Fq(C)5 b Fz(.)518 713 y(In)37 b(these)g(considerations)f(the)h(b)r(eha)n(viour)f(of) i(v)-5 b(arious)36 b(ob)5 b(jects)36 b(under)h(translations)f(is)418 813 y(imp)r(ortan)n(t.)j(The)29 b(algebra)e Fq(A)i Fz(is)f(assumed)g(to)g(b)r (e)h(in)n(v)-5 b(arian)n(t)28 b(under)g(translations;)g(then)h(the)418 913 y(natural)35 b(action)h(of)g(the)g(translation)f(group)g(in)h Fp(R)2061 882 y Fo(n)2148 913 y Fz(has)g(a)f(con)n(tin)n(uous)g(extension)h (to)g(the)418 1012 y(compacti\014cation)24 b Fq(X)38 b Fz(of)25 b Fp(R)1281 982 y Fo(n)1358 1012 y Fz(and)g Fr(F)37 b Fz(m)n(ust)25 b(b)r(e)h(in)n(v)-5 b(arian)n(t)24 b(under)h(this)g(extension)g(in)g(order)f (to)418 1112 y(allo)n(w)g(the)h(application)f(of)g(a)h(result)f(from)g(the)h (theory)f(of)h(crossed)e(pro)r(ducts)h(of)h(C*-algebras)418 1211 y(to)j(determine)f(the)h(sp)r(ectrum)g Fr(\033)1463 1181 y Fn(K)1518 1211 y Fz(\()p Fr(H)7 b Fz(\).)518 1312 y(In)37 b(Section)g(2)g(w)n(e)g(presen)n(t)g(some)g(of)g(the)h(C*-algebraic)c (concepts)j(that)h(are)e(useful)i(in)418 1412 y(the)27 b(sp)r(ectral)e (theory)h(of)g(self-adjoin)n(t)f(op)r(erators.)35 b(In)26 b(Section)g(3)g(w)n (e)g(discuss)f(some)h(algebras)418 1511 y Fq(A)33 b Fz(of)f(con)n(tin)n(uous) f(functions)i(on)f Fp(R)1573 1481 y Fo(n)1624 1511 y Fz(,)i(the)f(asso)r (ciated)d(compacti\014cations)i(of)g Fp(R)3042 1481 y Fo(n)3125 1511 y Fz(and)h(the)418 1611 y(con)n(tin)n(uous)24 b(extension)h(of)h(the)f (translation)f(group)h(to)g(these)g(compacti\014cations.)35 b(Section)25 b(4)418 1711 y(con)n(tains)30 b(a)g(few)g(remarks)f(on)h (crossed)f(pro)r(duct)i(algebras,)e(and)h(in)h(Section)f(5)g(w)n(e)g(giv)n(e) g(the)418 1810 y(pro)r(of)21 b(of)g(an)g(abstract)f(theorem)g(on)h (non-propagation.)32 b(An)22 b(application)e(of)h(this)h(theorem)e(to)418 1910 y(Sc)n(hr\177)-42 b(odinger)23 b(Hamiltonians)h(that)g(are)g (asymptotically)f(p)r(erio)r(dic)i(in)f(sev)n(eral)f(cones)h(is)g(stud-)418 2010 y(ied)f(in)g(detail)g(in)g(Section)g(6,)g(and)g(in)g(Section)g(7)f(w)n (e)g(men)n(tion)h(other)f(classes)g(of)g(Hamiltonians)418 2109 y(that)28 b(can)f(b)r(e)h(treated)f(in)h(a)f(similar)g(w)n(a)n(y)-7 b(.)518 2210 y(W)g(e)39 b(refer)g(to)g(Chapter)f(8)h(of)g([1])g(for)f(a)h (presen)n(tation)f(of)h(the)h(algebraic)d(approac)n(h)g(to)418 2310 y(sp)r(ectral)d(theory)-7 b(,)35 b(and)f(w)n(e)g(use)h(the)f (terminology)f(and)h(results)g(of)g([10])g(for)g(the)g(theory)g(of)418 2409 y(C*-algebras.)58 b(The)36 b(algebras)e Fj(C)1489 2421 y Fn(A)1582 2409 y Fz(and)h Fj(C)1802 2421 y Fn(K)1893 2409 y Fz(men)n(tioned)h(ab)r(o)n(v)n(e)e(ha)n(v)n(e)g(the)j(structure)e(of)g(a) 418 2509 y(crossed)30 b(pro)r(duct;)j(Reference)e([4])g(con)n(tains)f(a)h (description)f(of)i(suc)n(h)e(algebras)g(that)h(is)g(w)n(ell)418 2608 y(adapted)c(to)h(applications)f(in)g(sp)r(ectral)g(theory)-7 b(.)518 2709 y(Finally)25 b(w)n(e)g(p)r(oin)n(t)h(out)f(that)h(v)-5 b(arious)24 b(generalizations)f(of)i(our)g(results)g(are)f(p)r(ossible)h (with)418 2809 y(almost)39 b(no)g(extra)g(e\013ort)g(\(cf.)73 b(also)39 b([1],)j([4],)g([8])d(and)h([9]\).)72 b(Lo)r(cal)39 b(singularities)f(of)i(the)418 2908 y(p)r(oten)n(tial)26 b(can)f(easily)g(b)r (e)i(tak)n(en)e(in)n(to)g(accoun)n(t.)36 b(The)26 b(kinetic)g(energy)f Fq(\000)p Fz(\001)g(can)h(b)r(e)g(replaced)418 3008 y(b)n(y)37 b Fr(h)p Fz(\()p Fr(P)12 b Fz(\),)40 b(an)e(arbitrary)d(con)n(tin)n(uous)h (function)i(of)f(momen)n(tum)h(satisfying)f 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b(of)f Fr(H)40 b Fz(if)34 b Fr(\021)s Fz(\()p Fr(H)7 b Fz(\))33 b(is)g(a)418 4360 y(non-compact)27 b(op)r(erator)e(whenev)n(er)i Fr(\021)f Fq(2)e Fr(C)1816 4372 y Fm(0)1853 4360 y Fz(\()p Fp(R)q Fz(\))34 b(and)27 b Fr(\021)s Fz(\()p Fr(\025)p Fz(\))d Fq(6)p Fz(=)f(0.)518 4460 y(If)36 b Fj(C)g Fz(is)f(a)g(C*-algebra)e(of)j(b)r(ounded)g(op)r(erators)e(in)i Fq(H)h Fz(suc)n(h)e(that)h Fr(\021)s Fz(\()p Fr(H)7 b Fz(\))37 b Fq(2)g Fj(C)f Fz(for)f(eac)n(h)418 4560 y Fr(\021)30 b Fq(2)c Fr(C)629 4572 y Fm(0)667 4560 y Fz(\()p Fp(R)p Fz(\),)36 b(then)30 b Fr(H)37 b Fz(is)29 b(said)g(to)g(b)r(e)h Fi(a\016liate)l(d)h Fz(to)f Fj(C)p Fz(.)42 b(A)30 b(su\016cien)n(t)g(condition)f(for)g Fr(H)36 b Fz(to)30 b(b)r(e)418 4660 y(a\016liated)35 b(to)f Fj(C)h Fz(is)f(the)h(requiremen)n(t)f(that)h(\()p Fr(H)30 b Fq(\000)23 b Fr(z)t Fz(\))2151 4629 y Fn(\000)p Fm(1)2275 4660 y Fq(2)35 b Fj(C)g Fz(for)f(some)g(complex)g(n)n(um)n(b)r(er)418 4759 y Fr(z)i(=)-52 b Fq(2)24 b Fr(\033)s Fz(\()p Fr(H)7 b Fz(\).)518 4860 y(The)24 b(preceding)g(situation)g(can)g(b)r(e)h(view)n(ed)f (as)g(a)g(sp)r(ecial)h(case)e(of)i(the)g(follo)n(wing)e(abstract)418 4960 y(de\014nition:)418 5112 y FB(De\014nition)28 b(1.)47 b Fh(\(a\))24 b Fz(An)g(observ)-5 b(able)23 b(a\016liated)g(to)g(a)h (C*-algebra)d Fj(C)i Fh(is)h(a)f Fq(\003)p Fh(-homomorphism)p eop %%Page: 4 4 bop 418 314 a Fx(4)646 b(W.O.)28 b(AMREIN,)g(M.)f(M)1824 297 y(\025)1815 314 y(ANTOIU)h(AND)g(R.)f(PURICE)418 513 y Fh(from)32 b(the)g(C*-algebra)d Fr(C)1249 525 y Fm(0)1287 513 y Fz(\()p Fp(R)p Fz(\))38 b Fh(to)32 b Fj(C)g Fh(\(i.e.)50 b(a)31 b(linear)g(mapping)h Fz(\010)e(:)g Fr(C)2676 525 y Fm(0)2714 513 y Fz(\()p Fp(R)p Fz(\))37 b Fq(!)30 b Fj(C)i Fh(satisfying)418 613 y Fz(\010\()p Fr(\030)t(\021)s Fz(\))24 b(=)f(\010\()p Fr(\030)t Fz(\)\010\()p Fr(\021)s Fz(\))29 b Fh(and)e Fz(\010\()p Fr(\021)s Fz(\))1428 582 y Fn(\003)1490 613 y Fz(=)c(\010\()p 1670 567 45 4 v Fr(\021)s Fz(\))28 b Fh(if)g Fr(\030)t(;)14 b(\021)27 b Fq(2)c Fr(C)2132 625 y Fm(0)2170 613 y Fz(\()p Fp(R)p Fz(\))p Fh(\).)518 713 y(\(b\))k(The)f(sp)r(ectrum)g Fr(\033)s Fz(\(\010\))h Fh(of)f(the)h(observ)-5 b(able)25 b Fz(\010)h Fh(is)g(de\014ned)g(as)g(the)g(set)g(of)g(real)f(n)n (um)n(b)r(ers)418 813 y Fr(\025)j Fh(suc)n(h)f(that)h Fz(\010\()p Fr(\021)s Fz(\))c Fq(6)p Fz(=)f(0)k Fh(whenev)n(er)g Fr(\021)s Fz(\()p Fr(\025)p Fz(\))d Fq(6)p Fz(=)e(0)p Fh(.)37 b Fr(\033)s Fz(\(\010\))29 b Fh(is)e(a)g(closed)g(subset)h(of)f Fp(R)p Fh(.)518 974 y Fz(No)n(w)d(let)h Fj(K)g Fz(b)r(e)h(a)e(\(closed,)h (self-adjoin)n(t,)g(bilateral\))f(ideal)g(in)h Fj(C)p Fz(.)36 b(W)-7 b(e)25 b(denote)g(b)n(y)3102 952 y(^)3097 974 y Fj(C)e Fq(\021)g Fj(C)p Fr(=)p Fj(K)418 1073 y Fz(the)31 b(asso)r(ciated)d(quotien)n (t)i(C*-algebra)e(and)i(b)n(y)g(\005)g(the)h(canonical)e Fq(\003)p Fz(-homomorphism)f(of)i Fj(C)418 1177 y Fz(on)n(to)620 1155 y(^)615 1177 y Fj(C)p Fz(.)68 b(If)39 b(\010)f(is)f(an)h(observ)-5 b(able)37 b(a\016liated)g(to)h Fj(C)p Fz(,)j(then)d(clearly)f(\005)26 b 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y Fz(\()p Fp(R)q Fz(\))34 b Fh(is)27 b(suc)n(h)g(that)h Fr(\021)s Fz(\()p Fr(\025)p Fz(\))c Fq(6)p Fz(=)f(0)p Fh(.)518 1884 y Fz(T)-7 b(o)36 b(motiv)-5 b(ate)36 b(the)h(presen)n(t)e(terminology)-7 b(,)37 b(let)g(us)f(consider)f(the)i(situation)f(in)n(tro)r(duced)418 1984 y(at)e(the)f(b)r(eginning,)i(where)e Fj(C)h Fz(is)f(a)g(C*-subalgebra)e (of)j Fr(B)t Fz(\()p Fq(H)q Fz(\))g(and)f(\010)2695 1954 y Fo(H)2792 1984 y Fz(is)g(the)h(observ)-5 b(able)418 2083 y(determined)23 b(b)n(y)g(a)f(self-adjoin)n(t)g(op)r(erator)f Fr(H)30 b Fz(a\016liated)23 b(to)f Fj(C)h Fz(\(so)f(\010)2569 2053 y Fo(H)2632 2083 y Fz(\()p Fr(\021)s Fz(\))i(=)f Fr(\021)s Fz(\()p Fr(H)7 b Fz(\)\).)36 b(Assume)418 2183 y(that)26 b Fj(C)g Fz(con)n(tains)f(the)i(ideal)e Fr(K)6 b Fz(\()p Fq(H)q Fz(\))26 b(of)g(all)g(compact)g(op)r(erators)e(in)i Fq(H)q Fz(.)36 b(Then)26 b Fr(\033)2964 2198 y Fo(K)t Fm(\()p Fn(H)p Fm(\))3138 2183 y Fz(\(\010)3230 2153 y Fo(H)3293 2183 y Fz(\))g(is)418 2283 y(just)i(the)g(essen)n(tial)f(sp)r(ectrum)h Fr(\033)1466 2295 y Fm(ess)1552 2283 y Fz(\()p Fr(H)7 b Fz(\))28 b(of)g(the)g(self-adjoin)n(t)f(op)r(erator)f Fr(H)7 b Fz(.)518 2384 y(No)n(w)21 b(let)i(us)f(observ)n(e)f(that,)i(if)g Fj(K)1532 2396 y Fm(1)1592 2384 y Fz(and)f Fj(K)1803 2396 y Fm(2)1863 2384 y Fz(are)f(t)n(w)n(o)h(ideals)g(in)g Fj(C)g Fz(satisfying)g Fj(K)2958 2396 y Fm(1)3019 2384 y Fq(\032)g Fj(K)3161 2396 y Fm(2)3199 2384 y Fz(,)h(then)418 2483 y Fr(\033)465 2495 y Fg(K)510 2503 y Fk(2)544 2483 y Fz(\(\010\))34 b Fq(\032)f Fr(\033)847 2495 y Fg(K)892 2503 y Fk(1)926 2483 y Fz(\(\010\))i Fq(\032)e Fr(\033)s Fz(\(\010\).)57 b(In)34 b(particular,)g(if)h Fr(H)40 b Fz(is)34 b(a)g(self-adjoin)n(t)f(op)r(erator)f(a\016liated)418 2583 y(to)38 b(a)f(C*-subalgebra)e Fj(C)j Fz(of)f Fr(B)t Fz(\()p Fq(H)q Fz(\))i(and)e(if)h Fj(K)g Fz(is)g(an)f(ideal)h(in)g Fj(C)f Fz(with)h Fr(K)6 b Fz(\()p Fq(H)q Fz(\))40 b Fq(\032)g Fj(K)p Fz(,)g(then)418 2682 y Fr(\033)465 2694 y Fg(K)512 2682 y Fz(\(\010)604 2652 y Fo(H)667 2682 y Fz(\))23 b Fq(\032)g Fr(\033)857 2694 y Fm(ess)944 2682 y Fz(\()p Fr(H)7 b Fz(\).)518 2783 y(It)32 b(seems)f(that)i(the)f Fj(K)p Fz(-essen)n(tial)f(sp)r(ectrum)h (for)f Fj(K)i Fz(di\013eren)n(t)f(from)f Fq(f)p Fz(0)p Fq(g)g Fz(and)h Fr(K)6 b Fz(\()p Fq(H)q Fz(\))32 b(has)418 2883 y(nev)n(er)27 b(b)r(een)h(used)f(b)r(efore.)518 2983 y(One)k(of)g(the)h(in)n(teresting)f (asp)r(ects)g(of)g(the)h(preceding)e(framew)n(ork)g(in)h(the)h(study)g(of)f (self-)418 3083 y(adjoin)n(t)g(op)r(erators)e(in)i(a)f(Hilb)r(ert)i(space)e Fq(H)i Fz(is)e(as)h(follo)n(ws.)45 b(Let)31 b Fj(C)g Fz(b)r(e)g(a)g (C*-subalgebra)d(of)418 3182 y Fr(B)t Fz(\()p Fq(H)q Fz(\))23 b(and)e(consider)g(a)g(class)g(\002)h(of)f(self-adjoin)n(t)h(op)r(erators)d Fr(H)29 b Fz(a\016liated)21 b(to)h Fj(C)g Fz(suc)n(h)f(that,)j(for)418 3282 y(some)g(ideal)g Fj(K)h Fz(of)g Fj(C)p Fz(,)g(the)g Fq(\003)p Fz(-homomorphisms)d(\005)12 b Fq(\016)g Fz(\010)2102 3252 y Fo(H)2190 3282 y Fz(do)25 b(not)f(dep)r(end)h(on)g Fr(H)7 b Fz(.)35 b(So)25 b(all)f(mem-)418 3382 y(b)r(ers)h Fr(H)32 b Fz(of)26 b(\002)f(ha)n(v)n(e)f(the)i(same)f Fj(K)p Fz(-essen)n(tial)f(sp)r (ectrum)i Fr(\033)2233 3394 y Fg(K)2279 3382 y Fz(.)37 b(In)25 b(some)g(situations)g(it)h(is)f(rather)418 3485 y(easy)k(to)h(determine)g Fr(\033)1144 3497 y Fg(K)1191 3485 y Fz(:)42 b(although)29 b(the)i(quotien)n(t)f(C*-algebra)2507 3464 y(^)2502 3485 y Fj(C)g Fz(will)g(not)g(b)r(e)h(iden)n(ti\014able)418 3585 y(with)23 b(a)g(subalgebra)e(of)h Fr(B)t Fz(\()p Fq(H)q Fz(\),)j(it)e(ma)n(y)f(b)r(e)h (p)r(ossible)f(to)h(sp)r(ecify)g(a)f(faithful)i(represen)n(tation)d(of)423 3672 y(^)418 3693 y Fj(C)26 b Fz(in)g(a)g(Hilb)r(ert)g(space)1187 3672 y(^)1163 3693 y Fq(H)h Fz(\(an)f(injectiv)n(e)g Fq(\003)p Fz(-homomorphism)e Fr(\031)i Fz(:)2512 3672 y(^)2507 3693 y Fj(C)d Fq(!)g Fr(B)t Fz(\()2811 3672 y(^)2786 3693 y Fq(H)r Fz(\)\))j(and)g(a)g(simple)418 3802 y(self-adjoin)n(t)33 b(op)r(erator)1213 3781 y(^)1191 3802 y Fr(H)40 b Fz(in)1427 3781 y(^)1403 3802 y Fq(H)34 b Fz(a\016liated)f(to)g Fr(\031)s Fz(\()2048 3780 y(^)2043 3802 y Fj(C)q Fz(\))g(suc)n(h)g(that)h Fr(\031)17 b Fz([\005\()p Fr(\021)s Fz(\()p Fr(H)7 b Fz(\)\)])34 b(=)e Fr(\021)s Fz(\()3188 3781 y(^)3166 3802 y Fr(H)7 b Fz(\))34 b(for)418 3901 y(all)f Fr(\021)j Fq(2)c Fr(C)762 3913 y Fm(0)800 3901 y Fz(\()p Fp(R)p Fz(\))40 b(and)33 b(all)g Fr(H)39 b Fq(2)32 b Fz(\002.)54 b(Then)33 b Fr(\033)1852 3913 y Fg(K)1932 3901 y Fz(is)g(just)h(the)f(sp)r(ectrum)h(of)f(the)h(\(presumably)418 4009 y(simple\))28 b(op)r(erator)1066 3988 y(^)1044 4009 y Fr(H)7 b Fz(.)37 b(Examples)26 b(will)i(b)r(e)g(considered)f(in)h(Sections)f (6)g(and)g(7.)518 4110 y(The)g(follo)n(wing)g(result)g(will)h(b)r(e)g(used)g (in)f(Section)h(5:)418 4262 y FB(Lemma)33 b(1.)47 b Fh(Let)29 b Fj(K)h Fh(b)r(e)f(an)g(ideal)g(in)g(a)g(C*-algebra)d Fj(C)j Fh(and)g Fz(\010)g Fh(an)g(observ)-5 b(able)27 b(a\016liated)i(to)418 4361 y Fj(C)p Fh(.)37 b(If)28 b Fr(\021)e Fq(2)e Fr(C)817 4373 y Fm(0)854 4361 y Fz(\()p Fp(R)p Fz(\))34 b Fh(is)28 b(suc)n(h)f(that)h Fr(\021)s Fz(\()p Fr(\026)p Fz(\))c(=)f(0)k Fh(for)g(all)g Fr(\026)c Fq(2)h Fr(\033)2237 4373 y Fg(K)2283 4361 y Fz(\(\010\))p Fh(,)k(then)h Fz(\010\()p Fr(\021)s Fz(\))23 b Fq(2)h Fj(K)p Fh(.)418 4514 y Fi(Pr)l(o)l(of.)44 b Fz(\(i\))29 b(Let)f Fr(\025)c Fq(2)g Fp(R)h Fq(n)18 b Fr(\033)1281 4526 y Fg(K)1328 4514 y Fz(\(\010\).)38 b(There)28 b(are)f(a)g(n)n(um)n(b)r(er)h Fr(")23 b(>)g Fz(0)28 b(and)g(a)f(function)i Fr(\022)d Fq(2)e Fr(C)3245 4526 y Fm(0)3282 4514 y Fz(\()p Fp(R)q Fz(\))418 4613 y(suc)n(h)18 b(that)h Fq(j)p Fr(\022)r Fz(\()p Fr(\026)p Fz(\))p Fq(j)24 b Fr(>)f(")18 b Fz(for)g(all)h Fr(\026)k Fq(2)g Fz(\()p Fr(\025)p Fq(\000)p Fr(";)14 b(\025)p Fz(+)p Fr(")p Fz(\))21 b(and)d(\010\()p Fr(\022)r Fz(\))24 b Fq(2)f Fj(K)p Fz(.)35 b(No)n(w)18 b(let)h Fr(\030)27 b Fq(2)d Fr(C)2963 4625 y Fm(0)3000 4613 y Fz(\()p Fp(R)p Fz(\))h(b)r(e)19 b(suc)n(h)418 4713 y(that)24 b(supp)p Fr(\030)j Fq(\032)22 b Fz(\()p 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Fz(has)d(compact)f(supp)r(ort)h(in)h Fp(R)14 b Fq(n)g Fr(\033)2911 5124 y Fg(K)2962 5112 y Fz(\(\010\).)37 b(Cho)r(ose)p eop %%Page: 5 5 bop 779 314 a Fx(PR)n(OP)-6 b(A)n(GA)g(TION)29 b(PR)n(OPER)-6 b(TIES)28 b(F)n(OR)g(SCHR)2210 297 y(\177)2200 314 y(ODINGER)h(OPERA)-6 b(TORS)354 b(5)418 513 y Fz(a)29 b(\014nite)g(collection)f(of)h(n)n(um)n(b)r (ers)g Fr(\025)1553 525 y Fm(1)1590 513 y Fr(;)14 b(:)g(:)g(:)g(;)g(\025)1823 525 y Fo(M)1922 513 y Fq(2)26 b Fz(supp)p Fr(\021)32 b Fz(suc)n(h)c(that)i (supp)p Fr(\021)e Fq(\032)d([)3002 525 y Fo(k)3043 513 y Fq(V)3094 525 y Fo(\025)3133 534 y Ff(k)3203 513 y Fz(and)j(a)418 613 y(corresp)r(onding)22 b(partition)i(of)h(unit)n(y)f(on)g(supp)p Fr(\021)s Fz(,)h(i.e.)36 b(a)24 b(collection)g(of)g(functions)h Fr(\030)3027 625 y Fo(k)3092 613 y Fz(in)g Fr(C)3245 625 y Fm(0)3282 613 y Fz(\()p Fp(R)q Fz(\))418 723 y(suc)n(h)h(that)h(supp)p Fr(\030)990 735 y Fo(k)1054 723 y Fq(\032)c(V)1193 735 y Fo(\025)1232 744 y Ff(k)1299 723 y Fz(and)1459 661 y Fe(P)1547 681 y Fo(M)1547 748 y(k)q Fm(=1)1685 723 y Fr(\030)1721 735 y Fo(k)1762 723 y Fz(\()p Fr(\025)p Fz(\))i(=)d(1)k(for)g(all)g Fr(\025)d Fq(2)h Fz(supp)p Fr(\021)s Fz(.)37 b(Since)26 b(\010\()p Fr(\030)3062 735 y Fo(k)3104 723 y Fz(\))d Fq(2)g Fj(K)k Fz(b)n(y)418 838 y(\(i\),)h(w)n(e)g(get)f(\010\()p Fr(\021)s Fz(\))d(=)1097 775 y Fe(P)1184 796 y Fo(M)1184 863 y(k)q Fm(=1)1323 838 y Fz(\010\()p Fr(\021)s Fz(\)\010\()p Fr(\030)1619 850 y Fo(k)1661 838 y Fz(\))f Fq(2)h Fj(K)p Fz(.)83 b Fd(\003)1281 1056 y FA(3.)46 b(Some)30 b(Abelian)g(C*-Algebras)518 1210 y Fz(If)39 b Fr(Y)58 b Fz(is)39 b(a)g(lo)r(cally)g(compact,)j(Hausdor\013)c(space,)k(w)n(e)c (denote)i(b)n(y)e Fr(C)2787 1222 y Fo(b)2821 1210 y Fz(\()p Fr(Y)19 b Fz(\))40 b(the)f(ab)r(elian)418 1309 y(C*-algebra)30 b(of)i(all)g(b)r(ounded,)h(con)n(tin)n(uous)e(complex)h(functions)h (de\014ned)f(on)g Fr(Y)19 b Fz(.)50 b(If)33 b Fr(G)f Fz(is)g(a)418 1409 y(closed)27 b(subset)h(of)g Fr(Y)18 b Fz(,)28 b(w)n(e)g(set)g Fr(C)1448 1379 y Fo(G)1504 1409 y Fz(\()p Fr(Y)19 b Fz(\))k(=)g Fq(f)p Fr(')h Fq(2)f Fr(C)2003 1421 y Fo(b)2037 1409 y Fz(\()p Fr(Y)c Fz(\))24 b Fq(j)f Fr(')p Fz(\()p Fr(y)s Fz(\))h(=)f(0)p Fr(;)41 b Fq(8)p Fr(y)25 b Fq(2)f Fr(G)p Fq(g)p Fz(.)38 b(Certain)27 b(C*-)418 1508 y(subalgebras)20 b(of)h Fr(C)1008 1520 y Fo(b)1042 1508 y Fz(\()p Fr(Y)e Fz(\))i(will)h(b)r(e)g(imp)r(ortan)n(t)f(further)g(on,) i(in)f(particular)e(the)i(algebras)d Fr(C)3232 1478 y Fo(u)3226 1532 y(b)3276 1508 y Fz(\()p Fr(Y)g Fz(\))418 1608 y(and)32 b Fr(C)643 1620 y Fm(0)680 1608 y Fz(\()p Fr(Y)19 b Fz(\))33 b(consisting)d(resp)r(ectiv)n(ely)h(of)h(all)g(b)r(ounded,)h(uniformly)f(con) n(tin)n(uous)f(functions)418 1708 y(and)h(of)h(all)f(con)n(tin)n(uous)f (functions)i(v)-5 b(anishing)32 b(at)g(in\014nit)n(y)-7 b(.)52 b(In)33 b(fact)f Fr(C)2727 1720 y Fm(0)2765 1708 y Fz(\()p Fr(Y)19 b Fz(\))32 b(is)h(an)f(ideal)g(of)418 1807 y Fr(C)477 1819 y Fo(b)511 1807 y Fz(\()p Fr(Y)19 b Fz(\).)518 1911 y(Throughout)28 b(this)i(pap)r(er)e(w)n(e)h(set)h Fr(X)i Fz(=)25 b Fp(R)1880 1881 y Fo(n)1931 1911 y Fz(.)43 b(Let)29 b Fr(Y)48 b Fz(b)r(e)30 b(as)e(ab)r(o)n(v)n(e)g(and)h(assume)g(that)h Fr(X)418 2010 y Fz(acts)19 b(on)h Fr(Y)39 b Fz(as)19 b(a)g(group)g(of)h(homeomorphisms:)31 b(so)20 b(if)g Fr(\013)2146 2022 y Fo(x)2208 2010 y Fz(denotes)f(the)i (homeomorphism)d(in)i Fr(Y)418 2110 y Fz(asso)r(ciated)k(to)g(the)i(elemen)n (t)f Fr(x)e Fq(2)g Fr(X)7 b Fz(,)25 b(w)n(e)g(ha)n(v)n(e)e Fr(\013)1987 2122 y Fo(x)2042 2110 y Fq(\016)13 b Fr(\013)2150 2122 y Fo(x)2188 2106 y Fc(0)2237 2110 y Fz(=)23 b Fr(\013)2378 2122 y Fo(x)p Fm(+)p Fo(x)2505 2106 y Fc(0)2531 2110 y Fz(.)36 b(The)25 b(mapping)f Fr(X)19 b Fq(\002)13 b Fr(Y)42 b Fq(3)418 2210 y Fz(\()p Fr(x;)14 b(y)s Fz(\))36 b Fq(7!)g Fr(\013)818 2222 y Fo(x)860 2210 y Fz(\()p Fr(y)s Fz(\))f Fq(2)h Fr(Y)54 b Fz(is)35 b(assumed)f(con)n(tin)n(uous.)58 b(Then)36 b Fr(\013)f Fz(induces)g(a)g(represen)n(tation)e(of)418 2309 y(the)28 b(group)e Fr(X)34 b Fz(b)n(y)27 b Fq(\003)p Fz(-automorphisms)f(of)h Fr(C)1819 2321 y Fo(b)1853 2309 y Fz(\()p Fr(Y)19 b Fz(\))28 b(as)f(w)n(ell)g(as)g(of)g(v)-5 b(arious)26 b(C*-subalgebras)f(of)418 2409 y Fr(C)477 2421 y Fo(b)511 2409 y Fz(\()p Fr(Y)19 b Fz(\):)40 b(for)28 b Fr(')e Fq(2)f Fr(C)1052 2421 y Fo(b)1086 2409 y Fz(\()p Fr(Y)19 b Fz(\))29 b(and)g Fr(x)c Fq(2)h Fr(X)7 b Fz(,)29 b(de\014ne)g Fr(a)1975 2421 y Fo(x)2017 2409 y Fz(\()p Fr(')p Fz(\))d Fq(2)f Fr(C)2300 2421 y Fo(b)2334 2409 y Fz(\()p Fr(Y)19 b Fz(\))29 b(b)n(y)g([)p Fr(a)2678 2421 y Fo(x)2720 2409 y Fz(\()p Fr(')p Fz(\)]\()p Fr(y)s Fz(\))d(=)f Fr(')p Fz(\()p Fr(\013)3224 2421 y Fo(x)3266 2409 y Fz(\()p Fr(y)s Fz(\)\))418 2509 y(\()p Fr(y)45 b Fq(2)d Fr(Y)18 b Fz(\).)71 b(On)38 b Fr(C)1039 2478 y Fo(u)1033 2532 y(b)1083 2509 y Fz(\()p Fr(X)7 b Fz(\))39 b(the)g(represen)n(tation)e(is)h(con)n(tin)n(uous.)69 b(W)-7 b(e)39 b(observ)n(e)e(that)i(a)g(C*-)418 2610 y(subalgebra)d(of)h(the) g(form)g Fr(C)1371 2580 y Fo(G)1428 2610 y Fz(\()p Fr(Y)18 b 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Fr(i)h Fz(from)g Fr(X)41 b Fz(to)35 b(a)418 3112 y(dense)30 b(subset)g(of)g Fq(X)42 b Fz(\(see)30 b(e.g.)44 b Fq(x)o Fz(8.1)30 b(of)f([6]\).)44 b(F)-7 b(or)30 b Fr(x)d Fq(2)g Fr(X)7 b Fz(,)30 b(the)h(c)n(haracter)d Fr(i)p Fz(\()p Fr(x)p Fz(\))i(is)g(giv)n(en)f(b)n(y) 418 3212 y(the)j(form)n(ula)g([)p Fr(i)p Fz(\()p Fr(x)p Fz(\)]\()p Fr(')p Fz(\))g(=)d Fr(')p Fz(\()p Fr(x)p Fz(\),)35 b(for)c Fr(')g Fq(2)g(A)p Fz(.)50 b(W)-7 b(e)33 b(write)e Fq(Z)38 b Fz(=)30 b Fq(X)j(n)21 b Fr(i)p Fz(\()p Fr(X)7 b Fz(\))32 b(and)g(call)f(it)i (the)418 3311 y Fi(fr)l(ontier)g Fz(of)f Fr(X)38 b Fz(in)32 b Fq(X)12 b Fz(.)50 b(By)32 b(the)g(Gelfand)g(Theorem,)h Fq(A)f Fz(is)g(isomorphic)f(to)g(the)i(C*-algebra)418 3411 y Fr(C)6 b Fz(\()p Fq(X)12 b Fz(\))24 b(of)f(con)n(tin)n(uous)e(functions)i(on)g (\012\()p Fq(A)p Fz(\).)36 b(W)-7 b(e)23 b(shall)f(use)h(the)g(notation)f Fq(G)28 b Fz(:)23 b Fr(C)6 b Fz([\012\()p Fq(A)p Fz(\)])25 b Fq(!)e(A)418 3511 y Fz(for)j(the)g(in)n(v)n(erse)f(of)h(the)h(Gelfand)f (isomorphism.)36 b(The)26 b(C*-subalgebra)d Fr(C)2799 3481 y Fn(Z)2856 3511 y Fz(\()p Fq(X)12 b Fz(\))28 b(\(consisting)418 3610 y(of)g(con)n(tin)n(uous)e(functions)i(on)g Fq(X)40 b Fz(that)28 b(v)-5 b(anish)27 b(on)g(the)h(fron)n(tier)f Fq(Z)35 b Fz(of)27 b Fq(X)12 b Fz(\))29 b(can)e(b)r(e)h(naturally)418 3710 y(iden)n(ti\014ed)h (with)g Fr(C)1030 3722 y Fm(0)1067 3710 y Fz(\()p Fr(X)7 b Fz(\),)29 b(more)f(precisely)f Fr(C)1867 3722 y Fm(0)1905 3710 y Fz(\()p Fr(X)7 b Fz(\))24 b(=)g Fq(G)5 b Fr(C)2277 3680 y Fn(Z)2334 3710 y Fz([\012\()p Fq(A)p Fz(\)].)41 b(There)27 b(is)i(a)f(one-to-one)418 3810 y(corresp)r(ondence)23 b(b)r(et)n(w)n(een)h (\(self-adjoin)n(t,)h(closed\))f(ideals)g Fq(K)i Fz(of)f Fq(A)g Fz(and)f(closed)g(subsets)g Fr(G)h Fz(of)418 3909 y Fq(X)12 b Fz(,)27 b(giv)n(en)f(b)n(y)g Fq(K)e Fz(=)f Fq(G)5 b Fr(C)1163 3879 y Fo(G)1219 3909 y Fz(\()p Fq(X)12 b Fz(\))28 b(\(Theorem)d(3.4.1.)36 b(of)26 b([7]\).)36 b(In)27 b(particular)e(eac)n(h)g(closed)h(subset)418 4009 y Fr(F)43 b Fz(of)30 b(the)h(fron)n(tier)e Fq(Z)38 b Fz(determines)30 b(an)g(ideal)g Fq(K)1959 3979 y Fo(F)2045 4009 y Fz(in)h Fq(A)p Fz(,)h(viz.)45 b Fq(K)2502 3979 y Fo(F)2585 4009 y Fz(=)28 b Fq(G)5 b Fr(C)2797 3979 y Fo(F)2852 4009 y Fz(\()p Fq(X)12 b Fz(\).)47 b(It)31 b(is)f(clear)418 4108 y(that)e(suc)n(h)f(an)h(ideal)f (con)n(tains)g Fr(C)1484 4120 y Fm(0)1521 4108 y Fz(\()p Fr(X)7 b Fz(\).)518 4212 y(Supp)r(ose)23 b(no)n(w)f(that)i(the)f(C*-algebra)e Fq(A)i Fz(considered)f(ab)r(o)n(v)n(e)g(is)h(con)n(tained)f(in)h Fr(C)3066 4182 y Fo(u)3060 4235 y(b)3110 4212 y Fz(\()p Fr(X)7 b Fz(\))23 b(and)418 4312 y(in)n(v)-5 b(arian)n(t)17 b(under)i(translations,) g(i.e.)33 b(suc)n(h)19 b(that)f Fr(a)1985 4324 y Fo(x)2027 4312 y Fq(A)23 b(\032)g(A)c Fz(for)f(all)g Fr(x)23 b Fq(2)h Fr(X)7 b Fz(,)19 b(with)g([)p Fr(a)3027 4324 y Fo(x)3069 4312 y Fz(\()p Fr(')p Fz(\)]\()p Fr(y)s Fz(\))24 b(=)418 4411 y Fr(')p Fz(\()p Fr(x)d Fz(+)e Fr(y)s Fz(\).)41 b(Since)29 b Fq(A)d(\032)f Fr(C)1261 4381 y Fo(u)1255 4435 y(b)1304 4411 y Fz(\()p Fr(X)7 b Fz(\),)30 b(the)f(mapping)g Fr(x)d Fq(7!)f Fr(a)2209 4423 y Fo(x)2251 4411 y Fz(\()p Fr(')p Fz(\))30 b(is)f(norm)f(con)n (tin)n(uous)g(for)h(eac)n(h)418 4511 y Fr(')34 b Fq(2)g(A)p Fz(.)57 b(F)-7 b(urthermore)32 b(the)j(action)e(of)h Fr(X)40 b Fz(on)34 b(itself)h(\(giv)n(en)e(as)g Fr(\013)2589 4523 y Fo(x)2631 4511 y Fz(\()p Fr(y)s Fz(\))h(=)f Fr(x)24 b Fz(+)e Fr(y)s Fz(\))34 b(induces)418 4610 y(a)39 b(con)n(tin)n(uous)f(represen)n (tation)f Fr(\032)j Fz(of)f Fr(X)45 b Fz(b)n(y)39 b(homeomorphisms)f(of)h (the)h(c)n(haracter)d(space)418 4710 y Fq(X)47 b Fz(=)35 b(\012\()p Fq(A)p Fz(\):)52 b(for)34 b Fr(\034)45 b Fq(2)35 b(X)47 b Fz(the)35 b(c)n(haracter)e Fr(\032)1863 4722 y Fo(x)1905 4710 y Fr(\034)44 b Fz(is)35 b(de\014ned)g(as)f([)p Fr(\032)2544 4722 y Fo(x)2586 4710 y Fr(\034)9 b Fz(]\()p Fr(')p Fz(\))36 b(=)f Fr(\034)9 b Fz([)p Fr(a)3020 4722 y Fo(x)3062 4710 y Fz(\()p Fr(')p Fz(\)].)60 b(F)-7 b(or)418 4810 y Fr(y)26 b Fq(2)d Fr(X)7 b Fz(,)27 b(set)h Fr(\034)855 4822 y Fo(y)918 4810 y Fz(=)23 b Fr(i)p Fz(\()p Fr(y)s Fz(\);)k(then)h Fr(\032)1425 4822 y Fo(x)1467 4810 y Fr(\034)1503 4822 y Fo(y)1566 4810 y Fz(=)23 b Fr(\034)1690 4822 y Fo(x)p Fm(+)p Fo(y)1847 4810 y Fz(\()p Fr(x)g Fq(2)h Fr(X)7 b Fz(\).)518 4913 y(W)-7 b(e)26 b(end)g(this)g(section)f(with)h(a)f (result)h(whic)n(h)f(will)h(b)r(e)g(useful)h(in)f(the)g(examples)f(presen)n (ted)418 5013 y(further)e(on.)35 b(Let)22 b Fr(\034)33 b Fq(2)24 b(X)35 b Fz(b)r(e)23 b(a)f(c)n(haracter)f(of)h Fq(A)p Fz(.)36 b(A)23 b(neigh)n(b)r(ourho)r(o)r(d)e(base)h(of)h Fr(\034)32 b Fz(in)23 b Fq(X)35 b Fz(is)23 b(giv)n(en)418 5112 y(b)n(y)h(the)g (collection)f Fq(fV)1127 5124 y Fn(F)6 b Fo(;")1235 5112 y Fz(\()p Fr(\034)j Fz(\))p Fq(g)p Fz(,)26 b(where)d Fr(")h Fz(v)-5 b(aries)23 b(o)n(v)n(er)f(\(0)p Fr(;)14 b Fq(1)p Fz(\))24 b(and)g Fq(F)32 b Fz(o)n(v)n(er)22 b(all)i(\014nite)g(families)p eop %%Page: 6 6 bop 418 314 a Fx(6)646 b(W.O.)28 b(AMREIN,)g(M.)f(M)1824 297 y(\025)1815 314 y(ANTOIU)h(AND)g(R.)f(PURICE)418 513 y Fq(f)p Fr(')514 525 y Fm(1)551 513 y Fr(;)14 b(:)g(:)g(:)g(;)g(')790 525 y Fo(m)853 513 y Fq(g)31 b Fz(of)h(elemen)n(ts)g(of)g Fq(A)g Fz(and)g(where)f Fq(V)2027 525 y Fn(F)6 b Fo(;")2135 513 y Fz(\()p Fr(\034)j Fz(\))32 b(=)e Fq(f)p Fr(\034)2458 483 y Fn(0)2511 513 y Fq(2)h(X)42 b(j)30 b(j)p Fr(\034)2819 483 y Fn(0)2843 513 y Fz(\()p Fr(')2929 525 y Fo(i)2958 513 y Fz(\))21 b Fq(\000)g Fr(\034)9 b Fz(\()p Fr(')3228 525 y Fo(i)3257 513 y Fz(\))p Fq(j)30 b Fr(<)418 613 y(")d Fz(for)h(eac)n(h)e Fr(')852 625 y Fo(i)903 613 y Fq(2)e(F)8 b(g)p Fz(.)418 757 y FB(Lemma)34 b(2.)47 b Fh(Let)30 b Fq(A)g Fh(b)r(e)h(a)e(unital)h(C*-subalgebra)d(of)j Fr(C)2260 769 y Fo(b)2294 757 y Fz(\()p Fr(X)7 b Fz(\))p Fh(.)43 b(Let)30 b Fr(F)42 b Fh(b)r(e)30 b(a)g(closed)f(subset)418 857 y(of)g Fz(\012\()p Fq(A)p Fz(\))h Fh(and)f Fq(W)36 b Fh(a)28 b(neigh)n(b)r(ourho)r(o)r(d)g(of)h Fr(F)12 b Fh(.)41 b(Then)30 b(there)e(exist)h Fr(")c(>)g Fz(0)k Fh(and)g(a)f(\014nite)i(family)418 957 y Fq(F)h Fz(=)23 b Fq(f)p Fr(')693 969 y Fm(1)730 957 y Fr(;)14 b(:)g(:)g(:)f(;)h(')968 969 y Fo(m)1032 957 y Fq(g)27 b Fh(of)g(elemen)n(ts)h(of)f Fq(A)h Fh(suc)n(h)g(that)g Fr(F)35 b Fq(\032)22 b([)2321 969 y Fo(\034)7 b Fn(2)p Fo(F)2459 957 y Fq(V)2510 969 y Fn(F)f Fo(;")2618 957 y Fz(\()p Fr(\034)j Fz(\))25 b Fq(\032)d(W)7 b Fh(.)418 1101 y Fi(Pr)l(o)l(of.)44 b Fz(Let)30 b Fr(\034)36 b Fq(2)26 b Fr(F)12 b Fz(.)42 b(Then)30 b Fq(W)36 b Fz(is)30 b(a)f(neigh)n(b)r(ourho)r(o)r(d)f(of)h Fr(\034)9 b Fz(,)31 b(hence)f(there)f(are)f(a)h(\014nite)h(fam-)418 1201 y(ily)35 b Fq(F)8 b Fz(\()p Fr(\034)h Fz(\))37 b(of)e(elemen)n(ts)g(of)g Fq(A)h Fz(and)f(a)g(n)n(um)n(b)r(er)g Fr(")p Fz(\()p Fr(\034)9 b Fz(\))37 b Fr(>)e Fz(0)g(suc)n(h)g(that)h Fq(V)2762 1216 y Fn(F)6 b Fm(\()p Fo(\034)h Fm(\))p Fo(;")p Fm(\()p Fo(\034)g Fm(\))3049 1201 y Fz(\()p Fr(\034)i Fz(\))37 b Fq(\032)e(W)7 b Fz(.)418 1301 y(Since)30 b Fr(F)42 b Fz(is)30 b(compact,)h(there)e(are)g(a) h(\014nite)h(n)n(um)n(b)r(er)e(of)i(p)r(oin)n(ts)e Fr(\034)2508 1313 y Fm(1)2546 1301 y Fr(;)14 b(:)g(:)g(:)g(;)g(\034)2767 1313 y Fo(M)2871 1301 y Fz(in)30 b Fr(F)42 b Fz(suc)n(h)30 b(that)418 1400 y Fr(F)41 b Fq(\032)29 b([)661 1370 y Fo(M)661 1422 y(j)s Fm(=1)780 1400 y Fq(V)831 1415 y Fn(F)6 b Fm(\()p Fo(\034)942 1423 y Ff(j)972 1415 y Fm(\))p Fo(;")p Fm(\()p Fo(\034)1106 1423 y Ff(j)1137 1415 y Fm(\))p Fo(=)p Fm(2)1234 1400 y Fz(\()p Fr(\034)1302 1412 y Fo(j)1338 1400 y Fz(\).)48 b(Let)32 b Fq(F)37 b Fz(=)29 b Fq([)1840 1370 y Fo(M)1840 1422 y(j)s Fm(=1)1959 1400 y Fq(F)8 b Fz(\()p Fr(\034)2095 1412 y Fo(j)2130 1400 y Fz(\))32 b(and)f Fr(")e Fz(=)2531 1368 y Fm(1)p 2531 1382 34 4 v 2531 1429 a(2)2588 1400 y Fz(min)p Fq(f)p Fr(")p Fz(\()p Fr(\034)2875 1412 y Fm(1)2912 1400 y Fz(\))p Fr(;)14 b(:)g(:)g(:)g(;)g(")p Fz(\()p Fr(\034)3236 1412 y Fo(M)3310 1400 y Fz(\))p Fq(g)p Fz(.)418 1500 y(The)39 b(result)f(of)h(the)g(lemma)f(is)h(true)f(if)h(w)n(e)g(can)f(sho)n(w)g(that,) j(for)d(eac)n(h)g Fr(\034)51 b Fq(2)42 b Fr(F)12 b Fz(,)42 b(there)c(is)418 1600 y Fr(j)54 b Fq(2)49 b(f)p Fz(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(M)9 b Fq(g)42 b Fz(suc)n(h)h(that)g Fq(V)1501 1612 y Fn(F)6 b Fo(;")1609 1600 y Fz(\()p Fr(\034)j Fz(\))50 b Fq(\032)f(V)1933 1615 y Fn(F)6 b Fm(\()p Fo(\034)2044 1623 y Ff(j)2074 1615 y Fm(\))p Fo(;")p Fm(\()p Fo(\034)2208 1623 y Ff(j)2238 1615 y Fm(\))2268 1600 y Fz(\()p Fr(\034)2336 1612 y Fo(j)2372 1600 y Fz(\).)84 b(Since)43 b(clearly)f Fq(V)3075 1612 y Fn(F)6 b Fo(;")3183 1600 y Fz(\()p Fr(\034)j Fz(\))50 b Fq(\032)418 1702 y(V)469 1717 y Fn(F)6 b Fm(\()p Fo(\034)580 1725 y Ff(j)610 1717 y Fm(\))p Fo(;")691 1702 y Fz(\()p Fr(\034)j Fz(\))53 b Fq(\032)f(V)1021 1717 y Fn(F)6 b Fm(\()p Fo(\034)1132 1725 y Ff(j)1162 1717 y Fm(\))p Fo(;")p Fm(\()p Fo(\034)1296 1725 y Ff(j)1326 1717 y Fm(\))p Fo(=)p Fm(2)1424 1702 y Fz(\()p Fr(\034)j Fz(\))46 b(for)e(eac)n(h)g Fr(j)5 b Fz(,)50 b(it)45 b(is)g(enough)f(to)h(sho)n(w)f(that)h(for)g(some)418 1803 y Fr(j)28 b Fq(2)c(f)p Fz(1)p Fr(;)14 b(:)g(:)g(:)e(;)i(M)9 b Fq(g)27 b Fz(one)g(has)g Fq(V)1336 1818 y Fn(F)6 b Fm(\()p Fo(\034)1447 1826 y Ff(j)1477 1818 y Fm(\))p Fo(;")p Fm(\()p Fo(\034)1611 1826 y Ff(j)1642 1818 y Fm(\))p Fo(=)p Fm(2)1739 1803 y Fz(\()p Fr(\034)j Fz(\))24 b Fq(\032)f(V)2011 1818 y Fn(F)6 b Fm(\()p Fo(\034)2122 1826 y Ff(j)2152 1818 y Fm(\))p Fo(;")p Fm(\()p Fo(\034)2286 1826 y Ff(j)2316 1818 y Fm(\))2346 1803 y Fz(\()p Fr(\034)2414 1815 y Fo(j)2450 1803 y Fz(\).)518 1905 y(T)-7 b(o)34 b(pro)n(v)n(e)f(this)i(last)g(inclusion,)h(observ)n(e)d (that)i Fr(\034)45 b Fz(b)r(elongs)34 b(to)h Fq(V)2626 1920 y Fn(F)6 b Fm(\()p Fo(\034)2737 1928 y Ff(j)2766 1920 y Fm(\))p Fo(;")p Fm(\()p Fo(\034)2900 1928 y Ff(j)2931 1920 y Fm(\))p Fo(=)p Fm(2)3028 1905 y Fz(\()p Fr(\034)3096 1917 y Fo(j)3132 1905 y Fz(\))35 b(for)f(at)418 2007 y(least)29 b(one)h(v)-5 b(alue)29 b(of)h Fr(j)5 b Fz(.)43 b(Cho)r(ose)28 b(one)i(of)f(these)h(v)-5 b(alues)29 b(of)h Fr(j)k Fz(and)c(let)g Fr(\034)2688 1977 y Fn(0)2738 2007 y Fq(2)d(V)2871 2022 y Fn(F)6 b Fm(\()p Fo(\034)2982 2030 y Ff(j)3012 2022 y Fm(\))p Fo(;")p Fm(\()p Fo(\034)3146 2030 y Ff(j)3176 2022 y Fm(\))p Fo(=)p Fm(2)3274 2007 y Fz(\()p Fr(\034)j Fz(\).)418 2107 y(By)27 b(the)h(triangle)f(inequalit)n(y)g(one)g (has)g(for)h(eac)n(h)e Fr(')e Fq(2)f(A)p Fz(:)1031 2286 y Fq(j)p Fr(\034)1090 2298 y Fo(j)1125 2286 y Fz(\()p Fr(')p Fz(\))d Fq(\000)e Fr(\034)1391 2251 y Fn(0)1415 2286 y Fz(\()p Fr(')p Fz(\))p Fq(j)24 b(\024)e(j)p Fr(\034)1726 2298 y Fo(j)1762 2286 y Fz(\()p Fr(')p Fz(\))d Fq(\000)f Fr(\034)9 b Fz(\()p Fr(')p Fz(\))p Fq(j)20 b Fz(+)e Fq(j)p Fr(\034)9 b Fz(\()p Fr(')p Fz(\))20 b Fq(\000)e Fr(\034)2605 2251 y Fn(0)2629 2286 y Fz(\()p Fr(')p Fz(\))p Fq(j)p Fr(:)418 2464 y Fz(F)-7 b(or)28 b(ev)n(ery)e Fr(')f Fq(2)f(F)8 b Fz(\()p Fr(\034)1081 2476 y Fo(j)1116 2464 y Fz(\))29 b(eac)n(h)e(term)h(on)g(the)g(r.h.s.)38 b(is)28 b(less)g(than)g Fr(")p Fz(\()p Fr(\034)2580 2476 y Fo(j)2615 2464 y Fz(\))p Fr(=)p Fz(2.)38 b(Hence)28 b Fr(\034)3084 2434 y Fn(0)3137 2464 y Fz(b)r(elongs)418 2564 y(to)g Fq(V)571 2579 y Fn(F)6 b Fm(\()p Fo(\034)682 2587 y Ff(j)711 2579 y Fm(\))p Fo(;")p Fm(\()p Fo(\034)845 2587 y Ff(j)876 2579 y Fm(\))906 2564 y Fz(\()p Fr(\034)974 2576 y Fo(j)1010 2564 y Fz(\).)83 b Fd(\003)1080 2756 y FA(4.)45 b(Some)31 b(Cr)n(ossed)g(Pr)n (oduct)f(C*-Algebras)518 2905 y Fz(W)-7 b(e)30 b(consider)f(some)g (C*-subalgebras)f(of)h(the)i(space)e Fr(B)t Fz(\()p Fq(H)q Fz(\))i(of)f(all)f(b)r(ounded,)i(linear)f(op-)418 3005 y(erators)f(in)h(the)h (Hilb)r(ert)h(space)d Fq(H)g Fz(=)f Fr(L)1707 2975 y Fm(2)1743 3005 y Fz(\()p Fr(X)7 b Fz(\).)46 b(If)31 b Fr(')d Fz(:)g Fr(X)35 b Fq(!)27 b Fp(C)52 b Fz(is)30 b(a)h(b)r(ounded,)g(measurable)418 3105 y(function,)44 b(w)n(e)39 b(denote)h(b)n(y)g Fr(')p Fz(\()p Fr(Q)p Fz(\))h(the)f(op)r(erator)e(of)i(m)n(ultiplication)g(b)n(y)g Fr(')g Fz(in)g Fq(H)h Fz(and)f(b)n(y)418 3204 y Fr(')p Fz(\()p Fr(P)12 b Fz(\))26 b(the)f(op)r(erator)e Fj(F)1150 3174 y Fn(\003)1187 3204 y Fr(')p Fz(\()p Fr(Q)p Fz(\))p Fj(F)j Fz(\(the)f(op)r(erator)e(of)h(m)n (ultiplication)h(b)n(y)g Fr(')g Fz(in)g(the)g(momen)n(tum)418 3304 y(space\),)32 b(where)f Fj(F)g Fz(is)g(the)g(F)-7 b(ourier)31 b(transformation.)46 b(A)31 b(C*-subalgebra)e Fq(A)j Fz(of)f Fr(C)3063 3274 y Fo(u)3057 3327 y(b)3107 3304 y Fz(\()p Fr(X)7 b Fz(\))31 b(will)418 3403 y(b)r(e)26 b(iden)n(ti\014ed)g(with)f(the)h (subalgebra)e(of)h Fr(B)t Fz(\()p Fq(H)q Fz(\))h(consisting)e(of)i(all)f(m)n (ultiplication)g(op)r(erators)418 3503 y Fr(')p Fz(\()p Fr(Q)p Fz(\))j(with)g Fr(')c Fq(2)f(A)p Fz(.)518 3603 y(If)28 b Fq(A)f Fz(is)h(a)f(C*-subalgebra)e(of)i Fr(C)1551 3573 y Fo(b)1545 3623 y(u)1589 3603 y Fz(\()p Fr(X)7 b Fz(\),)27 b(w)n(e)g(write)g Fj(C)2164 3615 y Fn(A)2249 3603 y Fz(for)g(the)h(norm)f(closure)f(in)i Fr(B)t Fz(\()p Fq(H)q Fz(\))g(of)418 3702 y(the)j(set)g(of)g(\014nite)g(sums) g(of)g(the)g(form)f Fr(')1719 3714 y Fm(1)1757 3702 y Fz(\()p Fr(Q)p Fz(\))p Fr( )1941 3714 y Fm(1)1978 3702 y Fz(\()p Fr(P)12 b Fz(\))21 b(+)g Fq(\001)14 b(\001)g(\001)20 b Fz(+)g Fr(')2470 3714 y Fo(N)2533 3702 y Fz(\()p Fr(Q)p Fz(\))p Fr( )2717 3714 y Fo(N)2781 3702 y Fz(\()p Fr(P)12 b Fz(\))31 b(with)g Fr(')3187 3714 y Fo(k)3257 3702 y Fq(2)e(A)418 3802 y Fz(and)f Fr( )634 3814 y Fo(k)698 3802 y Fq(2)c Fr(C)836 3814 y Fm(0)873 3802 y Fz(\()p Fr(X)7 b Fz(\).)38 b(W)-7 b(e)28 b(men)n(tion)g(the)g(fact)g(that,) g(if)g Fq(A)c Fz(=)f Fr(C)2361 3814 y Fm(0)2398 3802 y Fz(\()p Fr(X)7 b Fz(\),)28 b(then)g Fj(C)2829 3814 y Fn(A)2915 3802 y Fz(is)f(the)i(ideal)e(of)418 3902 y(all)g(compact)h(op)r(erators)d(in)j Fr(L)1387 3871 y Fm(2)1424 3902 y Fz(\()p Fr(X)7 b Fz(\).)518 4001 y(If)30 b Fq(A)h Fz(is)f(in)n(v)-5 b(arian)n(t)29 b(under)h (translations,)f(then)i Fj(C)2097 4013 y Fn(A)2184 4001 y Fz(is)f(a)g (C*-algebra)e(isomorphic)h(to)h(the)418 4101 y(crossed)i(pro)r(duct)h (algebra)e Fq(A)22 b Fp(o)g Fr(X)39 b Fz(de\014ned)34 b(in)f(terms)g(of)g (the)g(action)f Fr(a)2784 4113 y Fo(x)2859 4101 y Fz(of)h Fr(X)40 b Fz(on)32 b Fq(A)p Fz(.)54 b(In)418 4201 y(the)30 b(pro)r(of)f(of)h(Lemma)f (6)g(w)n(e)g(shall)h(use)f(the)h(follo)n(wing)f(result)g(from)g(the)h(theory) f(of)g(crossed)418 4300 y(pro)r(ducts:)k(If)21 b Fq(K)g Fz(is)f(an)g(ideal)g (in)h Fq(A)f Fz(that)h(is)f(in)n(v)-5 b(arian)n(t)19 b(under)h(translations,) g(then)h(the)f(quotien)n(t)418 4400 y(C*-algebra)f Fj(C)883 4412 y Fn(A)941 4400 y Fr(=)p Fj(C)1034 4412 y Fn(K)1110 4400 y Fz(is)j(isomorphic)e(to)i([)p Fq(A)p Fr(=)p Fq(K)q Fz(])7 b Fp(o)g Fr(X)g Fz(.)34 b(The)22 b(p)r(oin)n(t)g(is)g(that)g(the)g(general)f (theory)418 4499 y(allo)n(ws)33 b(us)g(to)h(de\014ne)g(the)h(crossed)d(pro)r (duct)i([)p Fq(A)p Fr(=)p Fq(K)q Fz(])23 b Fp(o)f Fr(X)40 b Fz(only)34 b(b)n(y)f(using)h(the)g(con)n(tin)n(uous)418 4599 y(action)28 b(of)h Fr(X)35 b Fz(b)n(y)28 b Fq(\003)p Fz(-automorphisms)f(of)i Fq(A)p Fr(=)p Fq(K)h Fz(\(the)f(quotien)n(t)g(action\);)g(the)g(fact)g(that)g Fq(A)p Fr(=)p Fq(K)418 4699 y Fz(is)f(not)f(a)g(C*-subalgebra)e(of)j Fr(B)t Fz(\()p Fq(H)q Fz(\))g(do)r(es)f(not)h(matter.)518 4798 y FB(Remark.)63 b Fz(If)36 b Fr(V)57 b Fq(2)38 b(A)p Fz(,)h(where)d Fq(A)h Fz(is)f(a)g(unital)g(C*-subalgebra)e(of)i Fr(C)2848 4768 y Fo(u)2842 4822 y(b)2892 4798 y Fz(\()p Fr(X)7 b Fz(\),)39 b(then)e(the)418 4898 y(self-adjoin)n(t)28 b(op)r(erator)f Fr(H)32 b Fz(=)25 b Fq(\000)p Fz(\001)19 b(+)g Fr(V)47 b Fz(is)29 b(a\016liated)g(to)f Fj(C)2286 4910 y Fn(A)2344 4898 y Fz(.)40 b(This)29 b(is)g(easily)f(seen)g(from)h(the)418 5013 y(fact)k(that)f(the)h (Neumann)g(series)e([)p Fr(H)e Fq(\000)21 b Fr(z)t Fz(])1800 4983 y Fn(\000)p Fm(1)1919 5013 y Fz(=)2015 4950 y Fe(P)2103 4971 y Fn(1)2103 5038 y Fo(k)q Fm(=0)2227 5013 y Fz(\()p Fr(P)2324 4983 y Fm(2)2383 5013 y Fq(\000)h Fr(z)t Fz(\))2545 4983 y Fn(\000)p Fm(1)2647 4945 y Fe(\000)2685 5013 y Fq(\000)p Fr(V)d Fz(\()p Fr(Q)p Fz(\)[)p Fr(P)3035 4983 y Fm(2)3091 5013 y Fq(\000)f Fr(z)t Fz(])3240 4983 y Fn(\000)p Fm(1)3328 4945 y Fe(\001)3366 4963 y Fo(k)418 5112 y Fz(con)n(v)n(erges)25 b(in)j(the)g(norm)f(of)h Fr(B)t Fz(\()p Fq(H)q Fz(\))g(if)g Fq(=)p Fr(z)j Fz(is)c(su\016cien)n(tly)h (large.)p eop %%Page: 7 7 bop 779 314 a Fx(PR)n(OP)-6 b(A)n(GA)g(TION)29 b(PR)n(OPER)-6 b(TIES)28 b(F)n(OR)g(SCHR)2210 297 y(\177)2200 314 y(ODINGER)h(OPERA)-6 b(TORS)354 b(7)1244 513 y FA(5.)45 b(A)31 b(Non-pr)n(op)-6 b(a)n(ga)g(tion)31 b(Theorem)518 662 y Fz(F)-7 b(or)27 b(our)g(principal)g (theorem)g(and)g(its)h(corollary)d(w)n(e)i(consider)g(the)h(follo)n(wing)418 803 y FB(F)-8 b(ramew)m(ork.)48 b Fq(A)30 b Fh(is)f(a)g(unital)g (C*-subalgebra)e(of)i Fr(C)2158 772 y Fo(u)2152 826 y(b)2202 803 y Fz(\()p Fr(X)7 b Fz(\))p Fh(,)29 b(in)n(v)-5 b(arian)n(t)28 b(under)h(translations)418 902 y(and)35 b(suc)n(h)f(that)h Fr(C)1027 914 y Fm(0)1064 902 y Fz(\()p Fr(X)7 b Fz(\))35 b Fq(\032)g(A)p Fh(,)h(and)f Fj(C)1684 914 y Fn(A)1776 902 y Fh(is)g(the)g(asso)r(ciated)e(C*-subalgebra)f(of)j Fr(B)t Fz(\()p Fq(H)q Fz(\))g Fh(in-)418 1002 y(tro)r(duced)f(in)h Fq(x)p Fh(4)f(\(with)i Fq(H)f Fz(=)f Fr(L)1480 972 y Fm(2)1517 1002 y Fz(\()p Fr(X)7 b Fz(\))p Fh(\).)58 b Fq(X)47 b Fz(=)34 b(\012\()p Fq(A)p Fz(\))h Fh(is)g(the)g(c)n(haracter)d(space)i(of)g Fq(A)p Fh(,)j Fr(F)46 b Fh(a)418 1101 y(translation)26 b(in)n(v)-5 b(arian)n(t,)26 b(closed)h(subset)g(of)g Fq(Z)j Fz(=)23 b Fq(X)30 b(n)17 b Fr(i)p Fz(\()p Fr(X)7 b Fz(\))27 b Fh(and)g Fr(C)2550 1071 y Fo(F)2605 1101 y Fz(\()p Fq(X)12 b Fz(\))29 b Fh(the)e(ideal)g(in)h Fr(C)6 b Fz(\()p Fq(X)12 b Fz(\))418 1201 y Fh(determined)29 b(b)n(y)g Fr(F)12 b Fh(.)41 b(W)-7 b(e)29 b(set)g Fq(K)1438 1171 y Fo(F)1519 1201 y Fq(\021)c(G)5 b Fr(C)1728 1171 y Fo(F)1784 1201 y Fz(\()p Fq(X)12 b Fz(\))p Fh(,)30 b(whic)n(h)f(is)g(a)f(translation)g (in)n(v)-5 b(arian)n(t)28 b(ideal)h(in)418 1301 y Fq(A)p Fh(.)36 b(Then)25 b Fj(K)812 1271 y Fo(F)891 1301 y Fq(\021)e Fj(C)1030 1317 y Fn(K)1081 1300 y Ff(F)1157 1301 y Fh(is)h(an)h(ideal)f(in)h Fj(C)1690 1313 y Fn(A)1772 1301 y Fh(that)g(con)n(tains)f(all)h(the)g (compact)f(op)r(erators)f(in)i Fq(H)q Fh(.)518 1441 y Fz(W)-7 b(e)31 b(shall)f(w)n(ork)f(with)j(families)c Fr(W)-82 b(W)1557 1437 y(W)1678 1441 y Fz(of)31 b(subsets)f(of)h Fr(X)37 b Fz(suc)n(h)30 b(that)h(their)g(images)f(through)418 1541 y Fr(i)j Fz(in)g Fq(X)45 b Fz(are)32 b(close)h(to)g Fr(F)12 b Fz(.)51 b Fr(W)-83 b(W)1274 1537 y(W)1397 1541 y Fz(will)34 b(ha)n(v)n(e)d(the)j(structure)e(of) h(a)g(\014lter)g(base,)g(i.e.)54 b(a)32 b(non-v)n(oid)418 1640 y(collection)22 b(of)h(non-v)n(oid)e(subsets)i(of)f Fr(X)29 b Fz(suc)n(h)23 b(that)g(for)f(an)n(y)g Fr(W)2388 1652 y Fm(1)2425 1640 y Fr(;)14 b(W)2540 1652 y Fm(2)2601 1640 y Fq(2)21 b Fr(W)-82 b(W)2671 1636 y(W)2784 1640 y Fz(there)22 b(is)h(a)f Fr(W)35 b Fq(2)21 b Fr(W)-82 b(W)3317 1636 y(W)418 1740 y Fz(with)24 b Fr(W)35 b Fq(\032)22 b Fr(W)881 1752 y Fm(1)928 1740 y Fq(\\)9 b Fr(W)1070 1752 y Fm(2)1109 1740 y Fz(.)35 b(If)21 b Fr(W)-82 b(W)1237 1736 y(W)1350 1740 y Fz(is)23 b(a)g(\014lter)g(base)f(in)i Fr(X)7 b Fz(,)23 b(then)g(the)h(family)f Fq(f)p Fr(i)p Fz(\()p Fr(W)12 b Fz(\))23 b Fq(j)g Fr(W)35 b Fq(2)21 b Fr(W)-82 b(W)3132 1736 y(W)3222 1740 y Fq(g)22 b Fz(is)h(a)418 1839 y(\014lter)i(base)g(in)h Fq(X)38 b Fz(and)25 b(w)n(e)g(sa)n(y)f(that)g Fr(W)-82 b(W)1579 1835 y(W)1694 1839 y Fz(is)26 b Fi(adjac)l(ent)i(to)e Fr(F)37 b Fz(if)26 b(all)f(cluster)g(p)r(oin)n(ts)h(in)f Fq(X)38 b Fz(of)26 b(this)418 1943 y(family)31 b Fq(f)p Fr(i)p Fz(\()p Fr(W)12 b Fz(\))29 b Fq(j)g Fr(W)42 b Fq(2)27 b Fr(W)-82 b(W)1177 1939 y(W)1267 1943 y Fq(g)31 b Fz(b)r(elong)f(to)i Fr(F)12 b Fz(,)32 b(i.e.)48 b(if)32 b Fq(\\)2123 1957 y Fo(W)9 b Fn(2)n Fo(W)-65 b(W)2232 1954 y(W)p 2308 1870 184 4 v 2308 1943 a Fr(i)p Fz(\()p Fr(W)12 b Fz(\))29 b Fq(\032)g Fr(F)12 b Fz(,)32 b(where)f(the)g(closures)418 2042 y(are)c(tak)n(en)g(in)i Fq(X)12 b Fz(.)38 b(W)-7 b(e)28 b(observ)n(e)e(that)j(the)f(set)g(of)g(these)g (cluster)f(p)r(oin)n(ts)h(is)g(non-empt)n(y)f(since)418 2142 y Fq(X)45 b Fz(is)32 b(a)g(compact)g(space.)51 b(In)33 b(the)f(ma)5 b(jorit)n(y)32 b(of)g(situations)g(considered)f(further)i(on)f(it)h(will)418 2242 y(su\016ce)h(to)g(tak)n(e)g(for)e Fr(W)-82 b(W)1095 2238 y(W)1219 2242 y Fz(the)35 b(family)f Fq(f)p Fr(W)46 b Fz(=)33 b Fr(i)1922 2211 y Fn(\000)p Fm(1)2011 2242 y Fz([)p Fq(W)d(\\)23 b Fr(i)p Fz(\()p Fr(X)7 b Fz(\))34 b Fq(j)g(W)41 b(2)35 b Fj(W)p Fz(])p Fq(g)p Fz(,)g(where)f Fj(W)g Fz(is)g(a)418 2341 y(neigh)n(b)r(ourho)r (o)r(d)c(base)h(of)g Fr(F)43 b Fz(in)32 b Fq(X)43 b Fz(\(since)32 b Fr(i)p Fz(\()p Fr(X)7 b Fz(\))31 b(is)g(dense)g(in)g Fq(X)12 b Fz(,)33 b(eac)n(h)d(of)i(these)f(sets)g Fr(W)43 b Fz(is)418 2441 y(non-v)n(oid\).)518 2540 y(If)32 b Fr(W)-82 b(W)599 2536 y(W)723 2540 y Fz(is)34 b(a)f(\014lter)h(base)g(adjacen)n(t)f(to)h Fr(F)46 b Fz(and)34 b Fr(')g Fq(2)g Fr(C)6 b Fz(\()p Fq(X)12 b Fz(\))36 b(is)e(suc)n(h)f(that)i Fr(')p Fq(j)2958 2552 y Fo(F)3047 2540 y Fz(=)e(0,)i(then)418 2640 y(giv)n(en)30 b(an)n(y)f Fr(")e(>)h Fz(0,)i(there)g(is)h(some)e Fr(W)40 b Fq(2)26 b Fr(W)-82 b(W)1755 2636 y(W)1876 2640 y Fz(suc)n(h)30 b(that)g Fq(j)e Fr(')p Fz(\()p Fr(\034)9 b Fz(\))29 b Fq(j)p Fr(<)e(")k Fz(for)e(all)h Fr(\034)38 b Fq(2)28 b Fr(i)p Fz(\()p Fr(W)12 b Fz(\).)45 b(In)418 2740 y(the)28 b(sequel)f(w)n(e)h(shall)f(denote)g(b)n(y) h Fr(\037)1560 2752 y Fo(W)1663 2740 y Fz(the)g(c)n(haracteristic)d(function) j(of)g Fr(W)12 b Fz(.)418 2880 y FB(Theorem.)47 b Fh(Let)26 b Fq(A)f Fh(and)h Fr(F)37 b Fh(b)r(e)26 b(as)f(in)g(the)h(F)-7 b(ramew)n(ork)24 b(and)h(let)f Fr(W)-82 b(W)2497 2876 y(W)2613 2880 y Fh(a)25 b(\014lter)g(base)g(in)h Fr(X)32 b Fh(that)418 2980 y(is)c(adjacen)n(t)f(to)h Fr(F)12 b Fh(.)37 b(Let)28 b Fr(H)35 b Fh(b)r(e)28 b(a)f(self-adjoin)n(t)h(op)r(erator)e(in)i Fq(H)g Fh(a\016liated)g(to)g Fj(C)2949 2992 y Fn(A)3006 2980 y Fh(.)37 b(Let)28 b Fr(")23 b(>)g Fz(0)418 3079 y Fh(and)28 b Fr(\021)e Fq(2)d Fr(C)784 3091 y Fm(0)822 3079 y Fz(\()p Fp(R)p Fz(\))34 b Fh(with)28 b(supp)p Fr(\021)22 b Fq(\\)d Fr(\033)1518 3096 y Fg(K)1563 3079 y Ff(F)13 b Fz(\(\010)1704 3049 y Fo(H)1767 3079 y Fz(\))23 b(=)g Fq(;)p Fh(.)36 b(Then)28 b(there)f(is)h(a)f Fr(W)35 b Fq(2)22 b Fr(W)-83 b(W)2776 3075 y(W)2894 3079 y Fh(suc)n(h)27 b(that)418 3237 y Fz(\(1\))1028 b Fq(k)23 b Fr(\037)1669 3249 y Fo(W)1744 3237 y Fz(\()p Fr(Q)p Fz(\))p Fr(\021)s Fz(\()p Fr(H)7 b Fz(\))24 b Fq(k\024)e Fr(":)418 3435 y Fi(Pr)l(o)l(of.)44 b Fz(\(i\))28 b(W)-7 b(e)28 b(use)g(the)g(notation) f Fr(\034)37 b Fz(for)27 b(c)n(haracters)f(in)i(\012\()p Fq(A)p Fz(\))g(and)f(observ)n(e)f(that)804 3596 y Fq(K)868 3562 y Fo(F)947 3596 y Fz(=)c Fq(G)5 b(f)p Fr(')24 b Fq(2)f Fr(C)6 b Fz(\()p Fq(X)12 b Fz(\))24 b Fq(j)f Fr(')p Fq(j)1633 3608 y Fo(F)1712 3596 y Fz(=)g(0)p Fq(g)f Fz(=)g Fq(f)p Fr(')h Fq(2)h(A)f(j)g Fr(\034)9 b Fz(\()p Fr(')p Fz(\))25 b(=)d(0)55 b Fq(8)p Fr(\034)32 b Fq(2)24 b Fr(F)12 b Fq(g)p Fr(:)418 3754 y Fz(So)24 b(if)h Fr(')g Fz(b)r(elongs)e(to)i Fq(K)1139 3724 y Fo(F)1194 3754 y Fz(,)g(then)g(for)f(eac)n(h)g Fr(\016)i(>)c Fz(0)i(there)g(exists)g Fr(W)35 b Fq(2)22 b Fr(W)-82 b(W)2570 3750 y(W)2685 3754 y Fz(suc)n(h)24 b(that)g Fq(j)p Fr(\034)9 b Fz(\()p Fr(')p Fz(\))p Fq(j)25 b(\024)e Fr(\016)418 3853 y Fq(8)p Fr(\034)32 b Fq(2)23 b Fr(i)p Fz(\()p Fr(W)12 b Fz(\).)37 b(Th)n(us,)28 b(if)g Fr(')23 b Fq(2)h(K)1384 3823 y Fo(F)1439 3853 y Fz(,)k(w)n(e)f(ha)n(v)n(e)g Fq(j)c Fr(')p Fz(\()p Fr(x)p Fz(\))h Fq(j\024)f Fr(\016)31 b Fz(for)c(all)g Fr(x)d Fq(2)f Fr(W)12 b Fz(.)518 3953 y(\(ii\))30 b(By)g(the)g(h)n(yp)r(othesis)f(on)h(the)g(supp)r(ort)g(of)g Fr(\021)j Fz(w)n(e)c(ha)n(v)n(e)g Fr(\021)s Fz(\()p Fr(H)7 b Fz(\))27 b Fq(2)g Fj(K)2757 3923 y Fo(F)2843 3953 y Fz(\(see)i(Lemma)h (1\).)418 4053 y(So)23 b(there)f(are)g(a)g(\014nite)i(n)n(um)n(b)r(er)e(of)h (functions)g Fr(')1936 4065 y Fm(1)1974 4053 y Fr(;)14 b(:)g(:)g(:)f(;)h(') 2212 4065 y Fo(N)2299 4053 y Fq(2)23 b(K)2441 4023 y Fo(F)2519 4053 y Fz(and)g Fr( )2730 4065 y Fm(1)2767 4053 y Fr(;)14 b(:)g(:)g(:)g(;)g ( )3006 4065 y Fo(N)3092 4053 y Fq(2)23 b Fr(C)3229 4065 y Fm(0)3267 4053 y Fz(\()p Fr(X)7 b Fz(\))418 4152 y(suc)n(h)27 b(that)1297 4312 y Fq(k)22 b Fr(\021)s Fz(\()p Fr(H)7 b Fz(\))19 b Fq(\000)1678 4209 y Fo(N)1648 4233 y Fe(X)1647 4412 y Fo(k)q Fm(=1)1782 4312 y Fr(')1836 4324 y Fo(k)1877 4312 y Fz(\()p Fr(Q)p Fz(\))p Fr( )2061 4324 y Fo(k)2102 4312 y Fz(\()p Fr(P)12 b Fz(\))23 b Fq(k\024)g Fr("=)p Fz(2)p Fr(:)418 4526 y Fz(W)-7 b(e)28 b(also)f(ha)n(v)n(e)1016 4744 y Fq(k)22 b Fr(\037)1132 4756 y Fo(W)1208 4744 y Fz(\()p Fr(Q)p Fz(\))p Fr(\021)s Fz(\()p Fr(H)7 b Fz(\))24 b Fq(k\024)1706 4640 y Fo(N)1675 4665 y Fe(X)1675 4844 y Fo(k)q Fm(=1)1818 4744 y Fq(k)f Fr(')1937 4756 y Fo(k)2001 4744 y Fq(k)2043 4759 y Fo(L)2089 4743 y Fc(1)2149 4759 y Fm(\()p Fo(W)9 b Fm(\))2276 4744 y Fq(k)23 b Fr( )2395 4756 y Fo(k)2459 4744 y Fq(k)2501 4759 y Fo(L)2547 4743 y Fc(1)2607 4759 y Fm(\()p Fo(X)5 b Fm(\))2745 4744 y Fz(+)1676 5037 y(+)23 b Fq(k)f Fr(\021)s Fz(\()p Fr(H)7 b Fz(\))19 b Fq(\000)2145 4934 y Fo(N)2115 4959 y Fe(X)2114 5137 y Fo(k)q Fm(=1)2249 5037 y Fr(')2303 5049 y Fo(k)2344 5037 y Fz(\()p Fr(Q)p Fz(\))p Fr( )2528 5049 y Fo(k)2569 5037 y Fz(\()p Fr(P)12 b Fz(\))24 b Fq(k)e Fr(:)-2391 b Fz(\(2\))p eop %%Page: 8 8 bop 418 314 a Fx(8)646 b(W.O.)28 b(AMREIN,)g(M.)f(M)1824 297 y(\025)1815 314 y(ANTOIU)h(AND)g(R.)f(PURICE)418 513 y Fz(The)h(\014rst)g (term)g(in)g(the)g(r.h.s.)37 b(of)28 b(\(2\))g(can)g(b)r(e)g(made)g(less)f (than)h Fr("=)p Fz(2)f(b)n(y)h(using)f(the)h(result)g(of)418 629 y(\(i\))g(with)h Fr(\016)d Fz(=)873 562 y Fe(\002)908 629 y Fr(N)h Fq(\001)19 b Fz(sup)1169 641 y Fo(k)q Fm(=1)p Fo(;:::)o(;N)1474 629 y Fq(k)k Fr( )1593 641 y Fo(k)1657 629 y Fq(k)1699 644 y Fo(L)1745 628 y Fc(1)1804 644 y Fm(\()p Fo(X)5 b Fm(\))1919 562 y Fe(\003)1954 579 y Fn(\000)p Fm(1)2061 629 y Fq(\001)19 b Fr("=)p Fz(2,)26 b(so)h(the)h(pro)r(of)f(is)h(\014nished)83 b Fd(\003)418 782 y FB(Corollary.)49 b Fh(Let)30 b Fq(A)p Fh(,)h Fr(F)12 b Fh(,)28 b Fr(W)-82 b(W)1266 778 y(W)1386 782 y Fh(and)29 b Fr(H)37 b Fh(b)r(e)30 b(as)f(in)h(the)g(Theorem.)43 b(Then)30 b(for)f(eac)n(h)g Fr(")e(>)f Fz(0)j Fh(and)418 881 y(eac)n(h)e Fr(\021)f Fq(2)e Fr(C)810 893 y Fm(0)847 881 y Fz(\()p Fp(R)p Fz(\))34 b Fh(with)28 b(supp)q Fr(\021)e Fq(\032)c Fp(R)j Fq(n)18 b Fr(\033)1700 898 y Fg(K)1745 881 y Ff(F)13 b Fz(\(\010)1886 851 y Fo(H)1949 881 y Fz(\))p Fh(,)28 b(there)f(exists)h Fr(W)35 b Fq(2)21 b Fr(W)-82 b(W)2657 877 y(W)2774 881 y Fh(suc)n(h)28 b(that)418 1087 y Fz(\(3\))813 b Fq(k)22 b Fr(\037)1453 1099 y Fo(W)1529 1087 y Fz(\()p Fr(Q)p Fz(\))p Fr(e)1698 1053 y Fn(\000)p Fo(itH)1861 1087 y Fr(\021)s Fz(\()p Fr(H)7 b Fz(\))p Fr(f)32 b Fq(k\024)23 b Fr(")f Fq(k)h Fr(f)32 b Fq(k)418 1289 y Fh(for)27 b(all)g Fr(t)d Fq(2)f Fp(R)34 b Fh(and)27 b(all)g Fr(f)32 b Fq(2)23 b Fr(L)1364 1259 y Fm(2)1401 1289 y Fz(\()p Fr(X)7 b Fz(\))p Fh(.)518 1442 y Fz(\(3\))40 b(is)g(a)g(straigh)n(tforw)n (ard)d(consequence)j(of)g(\(1\).)75 b(Note)40 b(the)h(ob)n(vious)d(fact)j (that)f(one)418 1541 y(ma)n(y)26 b(replace)g(in)i(the)f(corollary)d Fq(f)p Fr(e)1541 1511 y Fn(\000)p Fo(itH)1704 1541 y Fq(g)i Fz(b)n(y)h(an)n(y)f(b)r(ounded)i(family)f(of)g(b)r(ounded)g(op)r(erators)418 1641 y(comm)n(uting)g(with)h Fr(H)7 b Fz(.)518 1741 y FB(Remark.)63 b Fz(The)36 b(corollary)e(giv)n(es)h(the)i(precise)f(meaning,)i(in)f(our)e (framew)n(ork,)i(of)g(the)418 1841 y(notion)31 b(of)g(non-propagation)d (describ)r(ed)j(in)g(the)g(In)n(tro)r(duction.)47 b(T)-7 b(o)30 b(b)r(e)i(more)e(sp)r(eci\014c,)i(let)418 1941 y(us)c(denote)h(b)n(y)f(supp) 1081 1953 y Fo(H)1144 1941 y Fz(\()p Fr(f)9 b Fz(\))28 b Fi(the)j(sp)l(e)l (ctr)l(al)f(supp)l(ort)h(with)g(r)l(esp)l(e)l(ct)f(to)e Fr(H)36 b Fz(of)28 b(the)h(v)n(ector)e Fr(f)32 b Fq(2)25 b(H)q Fz(,)418 2040 y(de\014ned)j(as)f(follo)n(ws)g(in)g(terms)h(of)f(the)h(sp)r(ectral)f (measure)g Fr(E)2341 2052 y Fo(H)2432 2040 y Fz(of)g Fr(H)7 b Fz(:)869 2243 y Fr(\025)33 b(=)-51 b Fq(2)23 b Fz(supp)1190 2255 y Fo(H)1253 2243 y Fz(\()p Fr(f)9 b Fz(\))23 b Fq(,)g(9)p Fr(")g(>)g Fz(0)k(suc)n(h)g(that)h Fr(E)2189 2255 y Fo(H)2252 2243 y Fz(\()p Fr(\025)20 b Fq(\000)e Fr(";)c(\025)k Fz(+)g Fr(")p Fz(\))p Fr(f)32 b Fz(=)22 b(0)p Fr(:)418 2445 y Fz(supp)589 2457 y Fo(H)652 2445 y Fz(\()p Fr(f)9 b Fz(\))29 b(is)f(the)h(smallest)f (closed)f(set)i Fr(J)j Fq(\032)24 b Fp(R)34 b Fz(suc)n(h)28 b(that)h Fr(E)2404 2457 y Fo(H)2467 2445 y Fz(\()p Fr(J)8 b Fz(\))p Fr(f)34 b Fz(=)24 b Fr(f)9 b Fz(.)39 b(Then)28 b(it)h(follo)n(ws)418 2544 y(easily)h(that,)j(under)e(the)g(h)n(yp)r(otheses)g(of)g(the)h (corollary)-7 b(,)29 b(for)i(eac)n(h)f Fr(")f(>)g Fz(0)h(and)h(eac)n(h)g (closed)418 2644 y(subset)d Fr(L)f Fz(of)g Fp(R)e Fq(n)18 b Fr(\033)1037 2660 y Fg(K)1082 2644 y Ff(F)13 b Fz(\(\010)1223 2614 y Fo(H)1286 2644 y Fz(\))28 b(there)f(exists)h(an)f(elemen)n(t)h Fr(W)39 b Fz(of)26 b Fr(W)-82 b(W)2413 2640 y(W)2531 2644 y Fz(suc)n(h)27 b(that)1429 2850 y Fq(k)c Fr(\037)1546 2862 y Fo(W)1621 2850 y Fz(\()p Fr(Q)p Fz(\))p Fr(e)1790 2815 y 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(\014nding)418 3451 y Fr(f)41 b Fz(lo)r(calized)31 b(in)i Fr(W)44 b Fz(is)32 b(less)g(than)g Fr(")1550 3421 y Fm(2)1619 3451 y Fz(at)g(all)g(times.)51 b(If)32 b Fr(K)38 b Fz(is)32 b(a)g(compact)f (subset)i(of)f Fr(X)38 b Fz(and)418 3551 y(if)27 b(the)g(preceding)f(v)n (ector)f Fr(f)36 b Fz(b)r(elongs)26 b(to)g(the)h(absolutely)f(con)n(tin)n (uous)f(subspace)h(of)h Fr(H)7 b Fz(,)26 b(then)418 3650 y(there)33 b(is)h Fr(t)756 3662 y Fm(0)825 3650 y Fq(2)f Fp(R)40 b Fz(suc)n(h)33 b(that)h Fq(k)e Fr(\037)1512 3662 y Fo(K)1576 3650 y Fz(\()p Fr(Q)p Fz(\))p Fr(e)1745 3620 y Fn(\000)p Fo(itH)1908 3650 y Fr(f)41 b Fq(k\024)32 b Fr(")h Fz(for)g(all)g Fr(t)g(>)f(t)2645 3662 y Fm(0)2716 3650 y Fz([2].)54 b(It)33 b(follo)n(ws)g(that)418 3750 y Fq(k)23 b Fr(\037)535 3762 y Fo(K)595 3746 y Ff(c)626 3762 y Fn(\\)p Fo(W)742 3746 y Ff(c)777 3750 y Fz(\()p Fr(Q)p Fz(\))p Fr(e)946 3720 y Fn(\000)p Fo(itH)1109 3750 y Fr(f)32 b Fq(k)1224 3720 y Fm(2)1261 3750 y Fq(\025)22 b Fz(1)8 b Fq(\000)g Fz(2)p 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(treated)f(analogously\).)43 b(More)30 b(precisely)f Fr(V)418 5013 y Fz(will)34 b(b)r(elong)e(to)h(the)h(C*-algebra)c Fq(A)k Fz(in)n(tro)r(duced)f(b)r(elo)n(w,)h(so)f(that)g Fr(H)39 b Fz(=)32 b Fq(\000)p Fz(\001)22 b(+)g Fr(V)52 b Fz(will)34 b(b)r(e)418 5112 y(a\016liated)27 b(to)h Fj(C)912 5124 y Fn(A)969 5112 y Fz(.)p eop %%Page: 9 9 bop 779 314 a Fx(PR)n(OP)-6 b(A)n(GA)g(TION)29 b(PR)n(OPER)-6 b(TIES)28 b(F)n(OR)g(SCHR)2210 297 y(\177)2200 314 y(ODINGER)h(OPERA)-6 b(TORS)354 b(9)518 513 y Fz(Let)24 b Fr(S)k Fz(b)r(e)d(the)f(unit)g(sphere)f (in)h Fr(X)30 b Fz(=)22 b Fp(R)1750 483 y Fo(n)1802 513 y Fz(.)35 b(F)-7 b(or)24 b Fr(j)k Fz(=)22 b(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(N)33 b Fz(let)24 b(\000)2649 525 y Fo(j)2708 513 y Fz(a)f(p)r(erio)r(dic)h (lattice)g(in)418 618 y Fr(X)33 b Fz(and)26 b(\006)740 630 y Fo(j)801 618 y Fz(a)g(non-empt)n(y)f(op)r(en)i(subset)f(of)g Fr(S)5 b Fz(,)26 b(with)p 2121 552 95 4 v 27 w(\006)2181 630 y Fo(j)2232 618 y Fq(\\)p 2303 552 101 4 v 16 w Fz(\006)2363 630 y Fo(k)2426 618 y Fz(=)d Fq(;)j Fz(if)h Fr(j)h Fq(6)p Fz(=)22 b Fr(k)s Fz(.)37 b(W)-7 b(e)26 b(denote)g(b)n(y)418 718 y Fr(C)477 730 y Fo(j)512 718 y Fz(\()p Fr(X)7 b Fz(\))28 b(the)g(C*-algebra)d Fr(C)1302 730 y Fo(j)1337 718 y Fz(\()p Fr(X)7 b Fz(\))24 b(=)e Fq(f)p Fr(')h Fq(2)h Fr(C)1851 688 y Fo(u)1845 741 y(b)1895 718 y Fz(\()p Fr(X)7 b Fz(\))23 b Fq(j)g Fr(')p Fz(\()p Fr(x)d Fz(+)e Fr(\015)5 b Fz(\))23 b(=)f Fr(')p Fz(\()p Fr(x)p Fz(\))57 b Fq(8)p Fr(x)23 b Fq(2)h Fr(X)r(;)41 b Fq(8)p Fr(\015)27 b Fq(2)d Fz(\000)3331 730 y Fo(j)3365 718 y Fq(g)418 818 y Fz(and)35 b(w)n(e)g(de\014ne)g Fq(A)g Fz(as)g(the)g(set)g(of)g(b)r(ounded,)i(uniformly) e(con)n(tin)n(uous)f(complex)h(functions)418 917 y Fr(')42 b Fz(on)e Fr(X)48 b Fz(suc)n(h)40 b(that)i(for)e(an)n(y)g Fr(j)51 b Fq(2)46 b(f)p Fz(1)p Fr(;)14 b(:)g(:)g(:)e(;)i(N)9 b Fq(g)41 b Fz(there)f(exists)h Fr(')2597 929 y Fo(j)2678 917 y Fq(2)46 b Fr(C)2838 929 y Fo(j)2873 917 y Fz(\()p Fr(X)7 b Fz(\))41 b(suc)n(h)g(that)418 1017 y(lim)533 1029 y Fo(r)r Fn(!1)716 1017 y Fq(j)p Fr(')p Fz(\()p Fr(r)r(!)s Fz(\))10 b Fq(\000)g Fr(')1090 1029 y Fo(j)1126 1017 y Fz(\()p Fr(r)r(!)s Fz(\))p Fq(j)24 b Fz(=)e(0)h(for)g(all)g Fr(!)i Fq(2)f Fz(\006)1933 1029 y Fo(j)1968 1017 y Fz(,)g(uniformly)f(in)g Fr(!)j Fz(on)d(eac)n(h)f (compact)h(subset)418 1116 y(of)30 b(\006)575 1128 y Fo(j)610 1116 y Fz(.)43 b(The)30 b(\(uniquely)g(determined\))h(collection)e Fq(f)p Fr(')2156 1128 y Fo(j)2217 1116 y Fq(j)e Fr(j)32 b Fz(=)26 b(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(N)9 b Fq(g)29 b Fz(corresp)r(onding)f(to)418 1216 y Fr(')f Fq(2)f(A)k Fz(will)g(b)r(e)g(called)f Fi(the)i(asymptotic)i (functions)f(of)e Fr(')p Fz(.)43 b(If)30 b(\006)f(is)h(some)f(subset)g(of)h (\006)3209 1228 y Fo(j)3273 1216 y Fz(and)418 1318 y Fr(R)24 b(>)e Fz(0,)28 b(let)g Fr(W)895 1288 y Fo(R)883 1340 y(j)949 1318 y Fz(\(\006\))g(b)r(e)g(the)g(subset)g Fq(f)p Fr(r)r(!)e Fq(j)d Fr(r)j(>)c(R)q(;)14 b(!)26 b Fq(2)d Fz(\006)p Fq(g)k Fz(of)h Fr(X)7 b Fz(.)518 1418 y(The)22 b(application)g(of)g(the)g(results)g (of)g(Section)g(5)g(leads)g(to)g(the)h(follo)n(wing)e(non-propagation)418 1517 y(prop)r(ert)n(y)27 b(in)n(to)g(the)h(cone)f(subtended)h(b)n(y)f(\006) 1830 1529 y Fo(j)1865 1517 y Fz(:)418 1664 y FB(Prop)s(osition)35 b(1.)47 b Fh(Let)30 b Fr(V)45 b Fq(2)27 b(A)j Fh(b)r(e)g(real)f(and)g(denote) h(its)g(asymptotic)f(functions)h(b)n(y)f Fq(f)p Fr(V)3307 1676 y Fo(j)3342 1664 y Fq(g)p Fh(.)418 1764 y(Set)f Fr(H)i Fz(=)23 b Fq(\000)p Fz(\001)18 b(+)g Fr(V)h Fh(.)37 b(Fix)28 b(a)f(n)n(um)n(b)r(er)g Fr(j)h Fq(2)c(f)p Fz(1)p Fr(;)14 b(:)g(:)g(:)e(;)i(N)9 b Fq(g)27 b Fh(and)h(c)n(ho)r(ose)e Fr(\021)g Fq(2)e Fr(C)2813 1776 y Fm(0)2850 1764 y Fz(\()p Fp(R)p Fz(\))34 b Fh(with)29 b(supp)p Fr(\021)418 1864 y Fh(disjoin)n(t)j(from)f(the)h(sp)r(ectrum)g(of)g(the)g(p)r (erio)r(dic)g(Sc)n(hr\177)-42 b(odinger)29 b(op)r(erator)h Fq(\000)p Fz(\001)21 b(+)g Fr(V)3087 1876 y Fo(j)3122 1864 y Fh(.)50 b(Then,)418 1963 y(giv)n(en)28 b(a)g(compact)g(subset)h Fr(K)34 b Fh(of)29 b Fz(\006)1556 1975 y Fo(j)1620 1963 y Fh(and)f Fr(")d(>)f Fz(0)p Fh(,)29 b(there)f(is)h Fr(R)c Fq(2)h Fz(\(0)p Fr(;)14 b Fq(1)p Fz(\))28 b Fh(suc)n(h)h(that)g(for)f(eac)n(h)418 2063 y Fr(f)k Fq(2)23 b Fr(L)626 2033 y Fm(2)663 2063 y Fz(\()p Fr(X)7 b Fz(\))27 b Fh(w)n(e)h(ha)n(v)n(e)418 2256 y Fz(\(4\))636 b(sup)1167 2326 y Fo(t)p Fn(2)p Fb(R)1309 2256 y Fq(k)22 b Fr(\037)1425 2275 y Fo(W)1496 2255 y Ff(R)1487 2294 y(j)1543 2275 y Fm(\()p Fo(K)t Fm(\))1659 2256 y Fz(\()p Fr(Q)p Fz(\))p Fr(e)1828 2221 y Fn(\000)p Fo(itH)1992 2256 y Fr(\021)s Fz(\()p Fr(H)7 b Fz(\))p Fr(f)32 b Fq(k\024)22 b Fr(")h Fq(k)f Fr(f)32 b Fq(k)22 b Fr(:)518 2498 y Fz(The)27 b(pro)r(of)f(will)h(b)r(e)h(giv)n(en)e (in)h(a)g(series)f(of)g(lemmas.)37 b(The)27 b(v)-5 b(alidit)n(y)27 b(of)g(\(4\))g(is)g(obtained)f(b)n(y)418 2598 y(com)n(bining)g(the)h(last)f (t)n(w)n(o)g(lemmas)g(\(Lemma)h(5)f(and)h(Lemma)f(6\))h(with)g(the)g (Corollary)d(giv)n(en)418 2698 y(in)h(Section)g(5)g(and)f(with)i(the)f 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(than)418 5112 y Fr("=)p Fz(3)j(in)h(the)h(sup)f(norm)g(\(with)g(resp)r(ect)g (to)g Fr(x)p Fz(\))h(since)f(for)g(eac)n(h)f(\014xed)h Fr(x)c Fq(2)h Fr(X)34 b Fz(and)28 b(an)n(y)f Fr(R)e(>)e Fz(0)p eop %%Page: 10 10 bop 418 314 a Fx(10)611 b(W.O.)28 b(AMREIN,)g(M.)f(M)1824 297 y(\025)1815 314 y(ANTOIU)h(AND)g(R.)f(PURICE)418 513 y Fz(one)e(ma)n(y)f (\014nd)h Fr(\015)j Fq(2)c Fz(\000)1110 525 y Fo(j)1169 513 y Fz(suc)n(h)h(that)g Fr(x)13 b Fz(+)g Fr(\015)29 b Fq(2)23 b Fr(W)1909 483 y Fo(R)1897 535 y(j)1963 513 y Fz(\()p Fr(K)6 b Fz(\))26 b(if)f Fr(K)31 b Fz(is)24 b(a)h(compact)f(subset)h(of)g(\006)3185 525 y Fo(j)3245 513 y Fz(with)418 613 y(non-empt)n(y)i(in)n(terior.)518 724 y(No)n(w)33 b(de\014ne)g Fr(')1012 736 y Fo(j)1081 724 y Fz(=)f(lim)1293 736 y Fo(k)q Fn(!1)1481 724 y Fr(')1535 681 y Fm(\()p Fo(k)q Fm(\))1535 747 y Fo(j)1661 724 y Fz(and)i(observ)n(e)d(that) j Fr(')2370 736 y Fo(j)2438 724 y Fq(2)g Fr(C)2586 736 y Fo(j)2621 724 y Fz(\()p Fr(X)7 b Fz(\).)54 b(By)34 b(another)e Fr("=)p Fz(3)418 824 y(t)n(yp)r(e)f(argumen)n(t)e(one)h(then)h(\014nds)f(that)h (these)f(functions)h Fr(')2343 836 y Fo(j)2408 824 y Fz(are)f(asymptotic)f (functions)i(of)418 924 y Fr(')p Fz(.)518 1023 y(Let)h(us)g(sho)n(w)f(that)h Fq(A)g Fz(is)g(in)n(v)-5 b(arian)n(t)31 b(under)h(translations.)48 b(If)32 b Fr(x)f Fq(2)g Fr(X)38 b Fz(and)32 b Fr(')f Fq(2)f(A)i Fz(with)418 1123 y(asymptotic)f(functions)g Fq(f)p Fr(')1306 1135 y Fo(j)1341 1123 y Fq(g)p Fz(,)h(then)g(the)f(collection)g Fq(f)p Fr(a)2236 1135 y Fo(x)2277 1123 y Fz(\()p Fr(')2363 1135 y Fo(j)2399 1123 y Fz(\))p Fq(g)g Fz(are)f(asymptotic)h(functions)418 1223 y(of)d Fr(a)557 1235 y Fo(x)598 1223 y Fz(\()p Fr(')p Fz(\),)h(hence)f Fr(a)1043 1235 y Fo(x)1084 1223 y Fz(\()p Fr(')p Fz(\))c Fq(2)g(A)p Fz(:)468 1427 y Fq(j)p Fr(a)535 1439 y Fo(x)577 1427 y Fz(\()p Fr(')p Fz(\)\()p Fr(r)r(!)s Fz(\))c Fq(\000)e Fr(a)1000 1439 y Fo(x)1042 1427 y Fz(\()p Fr(')1128 1439 y Fo(j)1164 1427 y Fz(\)\()p Fr(r)r(!)s Fz(\))p Fq(j)24 b Fz(=)1489 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(outside)i(the)g(unit)g(ball)g(of)g Fr(X)40 b Fz(and)33 b(satis\014es)g Fr(\022)r Fz(\()p Fr(x)p Fz(\))i(=)e(1)g(on)418 1929 y Fr(W)508 1899 y Fm(1)496 1951 y Fo(j)572 1929 y Fz(and)27 b Fr(\022)r Fz(\()p Fr(x)p Fz(\))e(=)d(0)27 b(on)g Fr(W)1271 1899 y Fm(1)1259 1953 y Fo(k)1335 1929 y Fz(if)h Fr(k)d Fq(6)p Fz(=)e Fr(j)5 b Fz(.)37 b(Then)27 b Fr(')c Fq(2)h(A)p Fz(,)j(with)h(asymptotic)e(functions) i Fr(')3181 1941 y Fo(j)3239 1929 y Fz(=)23 b Fr( )s Fz(,)418 2029 y Fr(')472 2041 y Fo(k)536 2029 y Fz(=)g(0)k(for)g Fr(k)f Fq(6)p Fz(=)d Fr(j)5 b Fz(.)83 b Fd(\003)518 2174 y Fz(W)-7 b(e)28 b(next)g(sho)n(w)e(that)i(there)g(is)g(a)f(canonical)f(iden)n (ti\014cation)i(of)f Fp(T)2614 2186 y Fo(j)2677 2174 y Fz(with)h(a)f(closed)g (subset)418 2274 y Fq(T)463 2286 y Fo(j)531 2274 y Fz(of)32 b Fq(Z)37 b Fz(=)31 b Fq(X)i(n)22 b Fr(i)p Fz(\()p Fr(X)7 b Fz(\),)32 b(where)g Fq(X)43 b Fz(=)31 b(\012\()p Fq(A)p Fz(\))i(as)e(b)r (efore.)51 b(This)32 b(is)g(a)g(direct)g(consequence)f(of)418 2373 y(the)g(fact)h(that)f(there)g(is)f(a)h(surjectiv)n(e)f Fq(\003)p Fz(-homomorphism)f(\010)2384 2385 y Fo(j)2448 2373 y Fz(:)g Fr(C)6 b Fz(\()p Fq(Z)h Fz(\))29 b Fq(!)g Fr(C)6 b Fz(\()p Fp(T)2990 2385 y Fo(j)3024 2373 y Fz(\),)33 b(deduced)418 2473 y(from)23 b(\011)675 2485 y Fo(j)732 2473 y Fz(:)h Fq(A)f(!)g Fr(C)1033 2485 y Fo(j)1068 2473 y Fz(\()p Fr(X)7 b Fz(\),)24 b(whic)n(h)f(is)g(a)g(surjectiv)n(e)f Fq(\003)p Fz(-homomorphism)f(with)j(k)n (ernel)e(including)418 2572 y Fr(C)477 2584 y Fm(0)515 2572 y Fz(\()p Fr(X)7 b Fz(\),)24 b(and)e(from)h(the)g(natural)f(isomorphisms)g Fq(A)p Fr(=C)2155 2584 y Fm(0)2192 2572 y Fz(\()p Fr(X)7 b Fz(\))2355 2550 y Fq(\030)2355 2577 y Fz(=)2443 2572 y Fr(C)f Fz(\()p Fq(Z)h Fz(\))23 b(and)g Fr(C)2878 2584 y Fo(j)2913 2572 y Fz(\()p Fr(X)7 b Fz(\))3076 2550 y Fq(\030)3076 2577 y Fz(=)3164 2572 y Fr(C)f Fz(\()p Fp(T)3317 2584 y Fo(j)3352 2572 y Fz(\).)418 2672 y(Belo)n(w)27 b(w)n(e)g(shall)g(mak)n(e)g(the)h (construction)f(as)g(explicit)g(as)g(p)r(ossible.)518 2772 y(F)-7 b(or)36 b(this)g(w)n(e)g(in)n(tro)r(duce)g(a)g(mapping)g Fr(i)1812 2784 y Fo(j)1885 2772 y Fz(:)h Fp(T)2001 2784 y Fo(j)2073 2772 y Fq(!)h(X)49 b Fz(that)37 b(asso)r(ciates)d(to)j Fr(\020)44 b Fq(2)38 b Fp(T)3221 2784 y Fo(j)3292 2772 y Fz(the)418 2890 y(c)n(haracter)18 b Fr(i)802 2902 y Fo(j)837 2890 y Fz(\()p Fr(\020)6 b Fz(\))24 b Fq(\021)f Fr(\034)1100 2847 y Fm(\()p Fo(j)s Fm(\))1091 2916 y Fo(\020)1210 2890 y Fq(2)h(X)32 b Fz(giv)n(en)20 b(as)g Fr(\034)1730 2847 y Fm(\()p Fo(j)s Fm(\))1721 2916 y Fo(\020)1817 2890 y Fz(\()p Fr(')p Fz(\))k(=)f Fr(')2101 2902 y Fo(j)2136 2890 y Fz(\()p Fr(z)t Fz(\),)f(where)e Fr(')2575 2902 y Fo(j)2630 2890 y Fz(is)g(the)h Fr(j)5 b Fz(-th)20 b(asymptotic)418 3022 y(function)i(of)e Fr(')i Fz(and)e Fr(z)k Fz(is)d(an)n(y)f(represen)n (tativ)n(e)f(of)i(the)g(class)f Fr(\020)6 b Fz(.)35 b(In)21 b(other)g(terms)f Fr(\034)2960 2979 y Fm(\()p Fo(j)s Fm(\))2951 3047 y Fo(\020)3071 3022 y Fz(=)i Fr(\034)3194 3034 y Fo(z)3238 3022 y Fq(\016)5 b Fz(\011)3350 3034 y Fo(j)3384 3022 y Fz(,)418 3129 y(where)27 b Fr(\034)694 3141 y Fo(z)760 3129 y Fz(is)h(in)n(terpreted)f (as)g(a)g(c)n(haracter)f(of)h Fr(C)6 b Fz(\()p Fp(T)2052 3141 y Fo(j)2087 3129 y Fz(\).)37 b(W)-7 b(e)28 b(set)g Fq(T)2497 3141 y Fo(j)2555 3129 y Fz(=)23 b Fr(i)2672 3141 y Fo(j)2706 3129 y Fz(\()p Fp(T)2794 3141 y Fo(j)2829 3129 y Fz(\).)418 3274 y FB(Lemma)32 b(4.)47 b Fh(\(a\))27 b Fq(T)1057 3286 y Fo(j)1120 3274 y Fh(is)h(con)n(tained)f(in)h Fq(X)i(n)18 b Fr(i)p Fz(\()p Fr(X)7 b Fz(\))p Fh(.)518 3393 y(\(b\))28 b(The)g(corresp)r (ondence)e Fr(\020)j Fq(7!)23 b Fr(\034)1623 3349 y Fm(\()p Fo(j)s Fm(\))1614 3418 y Fo(\020)1738 3393 y Fh(is)28 b(injectiv)n(e)g(and)f (con)n(tin)n(uous.)518 3500 y(\(c\))h(The)g(set)f Fq(T)992 3512 y Fo(j)1055 3500 y Fh(is)h(closed.)518 3599 y(\(d\))g(The)g(set)f Fq(T)1001 3611 y Fo(j)1064 3599 y Fh(is)h(in)n(v)-5 b(arian)n(t)26 b(under)i(all)f(translations.)418 3763 y Fi(Pr)l(o)l(of.)44 b Fz(\(a\))25 b(If)h Fr(\020)j Fq(2)24 b Fp(T)1091 3775 y Fo(j)1125 3763 y Fz(,)i(then)f Fr(\034)1405 3720 y Fm(\()p Fo(j)s Fm(\))1396 3788 y Fo(\020)1518 3763 y Fz(do)r(es)f(not)h(b)r(elong)g(to)g Fr(i)p Fz(\()p Fr(X)7 b Fz(\):)35 b(if)25 b Fr(x)f Fq(2)f Fr(X)7 b Fz(,)25 b(c)n(ho)r(ose)f(a)g(function)418 3895 y Fr(')f Fq(2)h Fr(C)633 3907 y Fm(0)670 3895 y Fz(\()p Fr(X)7 b Fz(\))20 b(suc)n(h)f(that)h Fr(')p Fz(\()p Fr(x)p Fz(\))k Fq(6)p Fz(=)f(0:)32 b(then)20 b Fr(')1790 3907 y Fo(j)1848 3895 y Fz(=)j(0,)d(so)f(that)h Fr(\034)2332 3852 y Fm(\()p Fo(j)s Fm(\))2323 3920 y Fo(\020)2419 3895 y Fz(\()p Fr(')p Fz(\))k(=)f(0)g Fq(6)p Fz(=)f Fr(')p Fz(\()p Fr(x)p Fz(\))j(=)d([)p Fr(i)p Fz(\()p Fr(x)p Fz(\)]\()p Fr(')p Fz(\).)418 4026 y(Th)n(us)27 b Fr(\034)673 3983 y Fm(\()p Fo(j)s Fm(\))664 4051 y Fo(\020)784 4026 y Fq(6)p Fz(=)c Fr(i)p Fz(\()p Fr(x)p Fz(\))28 b(for)f(eac)n(h)g Fr(x)c Fq(2)h Fr(X)7 b Fz(.)518 4133 y(\(b\))32 b(Assume)g(that)f Fr(z)1194 4145 y Fm(1)1252 4133 y Fq(\000)21 b Fr(z)1377 4145 y Fm(2)1452 4133 y Fr(=)-51 b Fq(2)30 b Fz(\000)1580 4145 y Fo(j)1615 4133 y Fz(.)49 b(Cho)r(ose)30 b Fr(')2033 4145 y Fo(j)2098 4133 y Fq(2)g Fr(C)2242 4145 y Fo(j)2277 4133 y Fz(\()p Fr(X)7 b Fz(\))32 b(suc)n(h)f(that)h Fr(')2878 4145 y Fo(j)2913 4133 y Fz(\()p Fr(z)2984 4145 y Fm(1)3021 4133 y Fz(\))e Fq(6)p Fz(=)f Fr(')3231 4145 y Fo(j)3266 4133 y Fz(\()p Fr(z)3337 4145 y Fm(2)3375 4133 y Fz(\))418 4252 y(and)g(let)h Fr(')c Fq(2)g(A)k Fz(b)r(e)f(suc)n(h)g(that)h Fr(')1499 4264 y Fo(j)1560 4252 y Fz(=)25 b(\011)1715 4264 y Fo(j)1750 4252 y Fz(\()p Fr(')p Fz(\))30 b(\(Lemma)f(3.\(b\)\).)43 b(Then)29 b Fr(\034)2764 4209 y Fm(\()p Fo(j)s Fm(\))2755 4277 y Fo(\020)2785 4285 y Fk(1)2852 4252 y Fz(\()p Fr(')p Fz(\))d(=)g Fr(')3141 4264 y Fo(j)3176 4252 y Fz(\()p Fr(z)3247 4264 y Fm(1)3284 4252 y Fz(\))g Fq(6)p Fz(=)418 4383 y Fr(')472 4395 y Fo(j)507 4383 y Fz(\()p Fr(z)578 4395 y Fm(2)616 4383 y Fz(\))d(=)h Fr(\034)805 4340 y Fm(\()p Fo(j)s Fm(\))796 4408 y Fo(\020)826 4416 y Fk(2)892 4383 y Fz(\()p Fr(')p Fz(\),)29 b(so)e Fr(\034)1209 4340 y Fm(\()p Fo(j)s Fm(\))1200 4408 y Fo(\020)1230 4416 y Fk(1)1320 4383 y Fq(6)p Fz(=)d Fr(\034)1454 4340 y Fm(\()p Fo(j)s Fm(\))1445 4408 y Fo(\020)1475 4416 y Fk(2)1541 4383 y Fz(.)38 b(Th)n(us)28 b(the)g(mapping)f Fr(i)2326 4395 y Fo(j)2389 4383 y Fz(is)h(injectiv)n(e.)38 b(Its)28 b(con)n(tin)n(uit)n(y)f(is)418 4485 y(easy)g(to)g(establish.)518 4585 y(\(c\))32 b Fq(T)696 4597 y Fo(j)764 4585 y Fz(is)f(the)i(con)n(tin)n (uous)e(image)g(of)h(the)g(compact)g(space)f Fp(T)2528 4597 y Fo(j)2562 4585 y Fz(,)j(hence)e(it)g(is)g(a)f(compact)418 4685 y(subset)d(of)f Fq(Z)7 b Fz(.)518 4784 y(\(d\))24 b(Let)g Fr(z)j Fq(2)c Fr(X)30 b Fz(b)r(e)25 b(a)e(represen)n(tativ)n(e)f(of)i Fr(\020)29 b Fq(2)24 b Fp(T)2036 4796 y Fo(j)2070 4784 y Fz(.)36 b(F)-7 b(or)23 b Fr(x)g Fq(2)h Fr(X)7 b Fz(,)24 b(denote)g(the)g(class)f(of)h Fr(z)14 b Fz(+)d Fr(x)418 4884 y Fz(in)28 b Fp(T)571 4896 y Fo(j)633 4884 y Fz(b)n(y)f Fr(\020)e Fz(+)18 b Fr(\030)t Fz(.)37 b(Then)483 4999 y Fe(h)522 5092 y Fr(\032)565 5104 y Fo(x)607 5092 y Fz(\()p Fr(\034)684 5048 y Fm(\()p Fo(j)s Fm(\))675 5117 y Fo(\020)772 5092 y Fz(\))804 4999 y Fe(i)857 5092 y Fz(\()p Fr(')p Fz(\))24 b(=)f Fr(\034)1132 5048 y Fm(\()p Fo(j)s Fm(\))1123 5117 y Fo(\020)1219 5092 y Fz([)p Fr(a)1286 5104 y Fo(x)1328 5092 y Fz(\()p Fr(')p Fz(\)])h(=)e([)p Fr(a)1647 5104 y Fo(x)1689 5092 y Fz(\()p Fr(')p Fz(\)])1830 5104 y Fo(j)1866 5092 y Fz(\()p Fr(z)t Fz(\))h(=)g([)p Fr(a)2151 5104 y Fo(x)2192 5092 y Fz(\()p Fr(')2278 5104 y Fo(j)2314 5092 y Fz(\)]\()p Fr(z)t Fz(\))g(=)g Fr(')2641 5104 y Fo(j)2676 5092 y Fz(\()p Fr(z)f Fz(+)c Fr(x)p Fz(\))24 b(=)f Fr(\034)3088 5048 y Fm(\()p Fo(j)s Fm(\))3079 5117 y Fo(\020)t Fm(+)p Fo(\030)3200 5092 y Fz(\()p Fr(')p Fz(\))p Fr(:)p eop %%Page: 11 11 bop 779 314 a Fx(PR)n(OP)-6 b(A)n(GA)g(TION)29 b(PR)n(OPER)-6 b(TIES)28 b(F)n(OR)g(SCHR)2210 297 y(\177)2200 314 y(ODINGER)h(OPERA)-6 b(TORS)318 b(11)418 517 y Fz(Hence)35 b Fr(\032)715 529 y Fo(x)756 517 y Fz(\()p Fr(\034)833 474 y Fm(\()p Fo(j)s Fm(\))824 542 y Fo(\020)921 517 y Fz(\))f(=)g Fr(\034)1131 474 y Fm(\()p Fo(j)s Fm(\))1122 542 y Fo(\020)t Fm(+)p Fo(\030)1244 517 y Fz(,)i(the)e(c)n(haracter)f(asso)r(ciated)f(to)j(the)f(class)f(of)i Fr(z)26 b Fz(+)c Fr(x)35 b Fz(in)f Fp(T)3177 529 y Fo(j)3212 517 y Fz(.)57 b(W)-7 b(e)418 619 y(conclude)27 b(that)h Fq(T)984 631 y Fo(j)1047 619 y Fz(is)g(a)f(single)g(orbit)g(under)h(the)g(represen)n (tation)d Fr(\032)j Fz(of)g Fr(X)34 b Fz(in)27 b Fq(X)12 b Fz(.)84 b Fd(\003)518 760 y Fz(W)-7 b(e)22 b(need)f(to)g(\014nd)h(subsets)f (of)h Fr(i)1496 730 y Fn(\000)p Fm(1)1584 760 y Fz([)p Fq(W)13 b(\\)6 b Fr(i)p Fz(\()p Fr(X)h Fz(\)])22 b(easy)e(to)h(express)f(in)i(terms)f (of)h(the)f(geometry)418 860 y(of)28 b Fr(X)7 b Fz(,)27 b(for)g(arbitrary)e (neigh)n(b)r(ourho)r(o)r(ds)i Fq(W)34 b Fz(of)28 b Fq(T)1976 872 y Fo(j)2011 860 y Fz(.)418 1002 y FB(Lemma)38 b(5.)47 b Fh(Let)32 b Fq(W)40 b Fh(b)r(e)33 b(a)g(neigh)n(b)r(ourho)r(o)r(d)e(of)i Fq(T)2070 1014 y Fo(j)2138 1002 y Fh(in)g Fq(X)45 b Fh(and)33 b(set)g Fr(W)43 b Fz(=)32 b Fr(i)2893 971 y Fn(\000)p Fm(1)2981 1002 y Fz([)p Fq(W)d(\\)22 b Fr(i)p Fz(\()p Fr(X)7 b Fz(\)])p Fh(.)418 1103 y(Giv)n(en)28 b(an)n(y)e(compact)h(subset)h Fr(K)33 b Fh(of)28 b Fz(\006)1661 1115 y Fo(j)1723 1103 y Fh(there)g(is)f Fr(R)d Fq(2)f Fz(\(0)p Fr(;)14 b Fq(1)p Fz(\))28 b Fh(suc)n(h)f(that)h Fr(W)2895 1073 y Fo(R)2883 1125 y(j)2950 1103 y Fz(\()p Fr(K)6 b Fz(\))23 b Fq(\032)f Fr(W)12 b Fh(.)418 1245 y Fi(Pr)l(o)l(of.)44 b Fz(By)33 b(Lemma)f(2)h(applied)f(to)h Fr(F)44 b Fz(=)31 b Fq(T)1831 1257 y Fo(j)1866 1245 y Fz(,)j(there)f(are)f Fr(")f(>)g Fz(0)i(and)f(a)h(\014nite)g(family)g Fq(F)39 b Fz(=)418 1345 y Fq(f)p Fr(')514 1357 y Fm(1)551 1345 y Fr(;)14 b(:)g(:)g(:)g(;)g(')790 1357 y Fo(m)853 1345 y Fq(g)31 b Fz(of)h(elemen)n(ts)g(of)f Fq(A)h Fz(suc)n(h)g(that)g Fq([)1996 1357 y Fo(\034)7 b Fn(2T)2117 1365 y Ff(j)2151 1345 y Fq(V)2202 1357 y Fn(F)f Fo(;")2310 1345 y Fz(\()p Fr(\034)j Fz(\))32 b Fq(\032)d(W)7 b Fz(,)33 b(in)f(other)f(terms)g(suc)n(h)418 1468 y(that)d Fq(\\)653 1480 y Fo(i)681 1468 y Fq(f)p Fr(\034)768 1438 y Fn(0)815 1468 y Fq(2)c(X)36 b(j)24 b(j)p Fr(\034)1104 1438 y Fn(0)1128 1468 y Fz(\()p Fr(')1214 1480 y Fo(i)1242 1468 y Fz(\))19 b Fq(\000)f Fr(\034)1421 1425 y Fm(\()p Fo(j)s Fm(\))1412 1493 y Fo(\020)1509 1468 y Fz(\()p Fr(')1595 1480 y Fo(i)1623 1468 y Fz(\))p Fq(j)24 b Fr(<)f(")p Fq(g)g(\032)g(W)35 b Fz(for)27 b(eac)n(h)g Fr(\020)j Fq(2)24 b Fp(T)2614 1480 y Fo(j)2648 1468 y Fz(.)38 b(Up)r(on)28 b(restricting)f(to)418 1575 y(c)n(haracters)22 b Fr(\034)855 1545 y Fn(0)904 1575 y Fz(b)r(elonging)i(to)g Fr(i)p Fz(\()p Fr(X)7 b Fz(\))24 b(and)h(denoting)f(the)h Fr(j)5 b Fz(-th)25 b(asymptotic)f(function)h(of)g Fr(')3267 1587 y Fo(i)3319 1575 y Fz(b)n(y)418 1675 y Fr(')472 1687 y Fo(i;j)551 1675 y Fz(,)g(w)n(e)f(get)g (immediately)g(that)g Fq(\\)1553 1687 y Fo(i)1581 1675 y Fq(f)p Fr(x)f Fq(2)h Fr(X)29 b Fq(j)23 b(j)p Fr(')1993 1687 y Fo(i)2021 1675 y Fz(\()p Fr(x)p Fz(\))12 b Fq(\000)g Fr(')2275 1687 y Fo(i;j)2354 1675 y Fz(\()p Fr(x)p Fz(\))p Fq(j)24 b Fr(<)f(")p Fq(g)f(\032)h(W)7 b Fz(.)35 b(No)n(w)24 b(for)g(eac)n(h)418 1784 y Fr(i)32 b Fz(there)g(is)g Fr(R)847 1796 y Fo(i)906 1784 y Fq(2)f Fz(\(0)p Fr(;)14 b Fq(1)p Fz(\))32 b(suc)n(h)g(that)h Fq(j)p Fr(')1704 1796 y Fo(i)1732 1784 y Fz(\()p Fr(x)p Fz(\))22 b Fq(\000)f Fr(')2005 1796 y Fo(i;j)2084 1784 y Fz(\()p Fr(x)p Fz(\))p Fq(j)32 b Fr(<)e(")i Fz(for)g(all)g Fr(x)f Fq(2)g Fr(W)2922 1747 y Fo(R)2972 1755 y Ff(i)2910 1807 y Fo(j)3003 1784 y Fz(\()p Fr(K)6 b Fz(\).)51 b(Then)418 1889 y(clearly)27 b(the)h(assertion)e(of)h(the) h(Lemma)f(holds)h(for)f Fr(R)d Fz(=)e(max)p Fq(f)p Fr(R)2486 1901 y Fm(1)2522 1889 y Fr(;)14 b(:)g(:)g(:)g(;)g(R)2770 1901 y Fo(m)2833 1889 y Fq(g)p Fz(.)83 b Fd(\003)518 2030 y Fz(W)-7 b(e)28 b(\014nally)f(sp)r(ecify)h(the)g Fj(K)1385 2000 y Fn(T)1423 2008 y Ff(j)1458 2030 y Fz(-essen)n(tial)f(sp)r(ectrum)g(of)h Fr(H)7 b Fz(.)418 2172 y FB(Lemma)32 b(6.)47 b Fh(The)28 b(set)g Fr(\033)1227 2199 y Fg(K)1272 2172 y Fc(T)1306 2185 y Ff(j)1343 2172 y Fz(\(\010)1435 2142 y Fo(H)1498 2172 y Fz(\))g Fh(coincides)g(with)g (the)g(\(band\))h(sp)r(ectrum)f(of)g(the)g(p)r(erio)r(dic)418 2271 y(Sc)n(hr\177)-42 b(odinger)26 b(op)r(erator)g Fr(H)1277 2283 y Fo(j)1335 2271 y Fz(=)c Fq(\000)p Fz(\001)d(+)f Fr(V)1706 2283 y Fo(j)1741 2271 y Fh(.)418 2415 y Fi(Pr)l(o)l(of.)44 b Fz(Let)33 b(us)f(denote)h(b)n(y)f(\005)1399 2427 y Fo(j)1465 2415 y Fz(:)g Fj(C)1571 2427 y Fn(A)1659 2415 y Fq(!)g Fj(C)1825 2427 y Fn(A)1882 2415 y Fr(=)p Fj(K)1979 2385 y Fn(T)2017 2393 y Ff(j)2084 2415 y Fz(the)h(canonical)f Fq(\003)p Fz(-homomorphism.)49 b(By)418 2517 y(de\014nition,)24 b Fr(\033)853 2544 y Fg(K)898 2517 y Fc(T)932 2530 y Ff(j)969 2517 y Fz(\(\010)1061 2487 y Fo(H)1124 2517 y Fz(\))e(is)g(the)h(sp)r(ectrum)f(of)g(the)g(observ)-5 b(able)21 b(\005)2438 2529 y Fo(j)2480 2517 y Fq(\016)7 b Fz(\010)2589 2487 y Fo(H)2674 2517 y Fz(a\016liated)22 b(to)g Fj(C)3157 2529 y Fn(A)3214 2517 y Fr(=)p Fj(K)3311 2487 y Fn(T)3349 2495 y Ff(j)3384 2517 y Fz(.)418 2623 y(It)40 b(is)f(enough)f(to)h(sho)n(w)g(that) g Fj(C)1485 2635 y Fn(A)1543 2623 y Fr(=)p Fj(K)1640 2592 y Fn(T)1678 2600 y Ff(j)1752 2623 y Fz(is)g(isomorphic)f(to)h Fj(C)2443 2638 y Fo(C)2491 2646 y Ff(j)2521 2638 y Fm(\()p Fo(X)5 b Fm(\))2676 2623 y Fz(and)39 b(that)g(the)h(image)418 2732 y(of)h(\005)588 2744 y Fo(j)650 2732 y Fq(\016)27 b Fz(\010)779 2702 y Fo(H)882 2732 y Fz(under)41 b(this)f(isomorphism)g(is)g(\010)1960 2702 y Fo(H)2014 2710 y Ff(j)2050 2732 y Fz(;)47 b(this)41 b(will)g(conclude)f(the)h(pro)r(of,)i(since)418 2832 y(isomorphisms)26 b(of)i(C*-algebras)d(lea)n(v)n(e)h(the)i(sp)r(ectra)f(of)g(observ)-5 b(ables)26 b(in)n(v)-5 b(arian)n(t.)518 2931 y(F)e(or)27 b(an)n(y)g Fr(M)k Fq(2)24 b Fp(N)t Fz(,)33 b Fr(')1179 2943 y Fm(1)1217 2931 y Fr(;)14 b(:)g(:)g(:)f(;)h(')1455 2943 y Fo(M)1552 2931 y Fq(2)24 b(A)k Fz(and)f Fr( )1940 2943 y Fm(1)1978 2931 y Fr(;)14 b(:)g(:)g(:)f(;)h( )2216 2943 y Fo(M)2313 2931 y Fq(2)23 b Fr(C)2450 2943 y Fm(0)2488 2931 y Fz(\()p Fr(X)7 b Fz(\))27 b(w)n(e)h(set)1158 3176 y(\002)1223 3188 y Fo(j)1272 3034 y Fe(")1345 3072 y Fo(M)1320 3097 y Fe(X)1326 3274 y Fo(i)p Fm(=1)1454 3176 y Fr(')1508 3188 y Fo(i)1536 3176 y Fz(\()p Fr(Q)p Fz(\))p Fr( )1720 3188 y Fo(i)1748 3176 y Fz(\()p Fr(P)12 b Fz(\))1877 3034 y Fe(#)1949 3176 y Fz(=)2061 3072 y Fo(M)2036 3097 y Fe(X)2042 3274 y Fo(i)p Fm(=1)2170 3176 y Fr(')2224 3188 y Fo(i;j)2302 3176 y Fz(\()p Fr(Q)p Fz(\))p Fr( )2486 3188 y Fo(i)2514 3176 y Fz(\()p Fr(P)g Fz(\))p Fr(;)418 3417 y Fz(where)35 b Fr(')720 3429 y Fo(i;j)834 3417 y Fq(2)i Fr(C)985 3429 y Fo(j)1020 3417 y Fz(\()p Fr(X)7 b Fz(\))35 b(is)g(the)h Fr(j)5 b Fz(-th)35 b(asymptotic)g(function)h(of)f Fr(')2541 3429 y Fo(i)2569 3417 y Fz(.)60 b(By)35 b(the)h(discussion)f(in)418 3516 y(Section)d(4,)h(\002)876 3528 y Fo(j)943 3516 y Fz(extends)f(to)g(a)g(surjectiv)n(e)g Fq(\003)p Fz(-homomorphism)e Fj(C)2518 3528 y Fn(A)2606 3516 y Fq(!)g Fj(C)2770 3531 y Fo(C)2818 3539 y Ff(j)2849 3531 y Fm(\()p Fo(X)5 b Fm(\))2996 3516 y Fz(with)33 b(k)n(ernel)418 3626 y Fj(K)473 3596 y Fn(T)511 3604 y Ff(j)574 3626 y Fz(=)27 b([)p Fq(G)5 b Fr(C)808 3596 y Fn(T)846 3604 y Ff(j)881 3626 y Fz(\()p Fq(X)12 b Fz(\)])21 b Fp(o)f Fr(X)7 b Fz(.)44 b(A)31 b(simple)f(argumen)n(t)f(in)i(terms)f(of)g(the)g(Neumann)h(series)e(sho)n(ws) 418 3725 y(that)f(\002)663 3737 y Fo(j)698 3725 y Fz([\()p Fr(H)d Fq(\000)18 b Fr(z)t Fz(\))1005 3695 y Fn(\000)p Fm(1)1094 3725 y Fz(])23 b(=)g(\()p Fr(H)1329 3737 y Fo(j)1382 3725 y Fq(\000)18 b Fr(z)t Fz(\))1540 3695 y Fn(\000)p Fm(1)1629 3725 y Fz(,)28 b(so)f(that)h(\002)2027 3737 y Fo(j)2061 3725 y Fz(\(\010)2153 3695 y Fo(H)2216 3725 y Fz(\))c(=)e(\010)2419 3695 y Fo(H)2473 3703 y Ff(j)2509 3725 y Fz(.)83 b Fd(\003)518 3869 y FB(Remark.)48 b Fz(Since)32 b Fr(K)6 b Fz(\()p Fq(H)q Fz(\))30 b(=)f Fj(C)1532 3884 y Fo(C)1580 3892 y Fk(0)1612 3884 y Fm(\()p Fo(X)5 b Fm(\))1756 3869 y Fz(=)30 b Fj(K)1906 3839 y Fn(Z)1993 3869 y Fq(\032)f Fj(K)2142 3839 y Fn(T)2180 3847 y Ff(j)2215 3869 y Fz(,)k(w)n(e)e(ha)n(v)n(e) f Fq([)2647 3839 y Fo(N)2647 3891 y(j)s Fm(=1)2767 3869 y Fr(\033)s Fz(\()p Fr(H)2918 3881 y Fo(j)2954 3869 y Fz(\))g Fq(\032)f Fr(\033)3157 3881 y Fm(ess)3244 3869 y Fz(\()p Fr(H)7 b Fz(\).)418 3969 y(The)27 b(b)r(eha)n(viour)f(of)i(the)f(b)r(ounded,)h(uniformly)f(con)n (tin)n(uous)f(function)i Fr(V)46 b Fz(outside)27 b(the)h(cones)418 4068 y Fq(fC)504 4080 y Fo(j)538 4068 y Fq(g)580 4080 y Fo(j)s Fm(=1)p Fo(;:::;N)882 4068 y Fz(is)e(submitted)h(to)f(no)g(constrain)n(t)f (and)g(the)i(asymptotic)e(functions)i Fr(V)3089 4080 y Fo(j)3150 4068 y Fz(are)e(not)418 4170 y(related.)64 b(So)36 b(it)h(is)g(p)r(ossible)f (to)h(ha)n(v)n(e)e(a)i(large)e(set)i Fr(\033)2173 4182 y Fm(ess)2259 4170 y Fz(\()p Fr(H)7 b Fz(\))25 b Fq(n)f([)2545 4140 y Fo(N)2545 4192 y(j)s Fm(=1)2665 4170 y Fr(\033)s Fz(\()p Fr(H)2816 4182 y Fo(j)2851 4170 y Fz(\))38 b(on)e(whic)n(h)h(the)418 4281 y(result)31 b(of)f(the)i(Prop)r(osition)d(\(non-propagation)f(in)n(to)j(the)g (asymptotic)g(part)f(of)h Fq([)3090 4251 y Fo(N)3090 4303 y(j)s Fm(=1)3209 4281 y Fq(C)3253 4293 y Fo(j)3288 4281 y Fz(\))g(is)418 4381 y(relev)-5 b(an)n(t)24 b(and)h(non-trivial.)35 b(Of)25 b(course)f(the)i(simplest)f(situation)g(is)g(that)g(where)f Fr(N)32 b Fz(=)23 b(1,)i(and)418 4481 y(the)h(general)e(case)g(\()p Fr(N)32 b(>)23 b Fz(1\))i(can)g(b)r(e)g(reduced)g(to)g(it)h(since)f(Prop)r (osition)e(1)i(in)n(v)n(olv)n(es)f(only)g(one)418 4580 y(v)-5 b(alue)28 b(of)f Fr(j)5 b Fz(.)1264 4764 y FA(7.)45 b(Other)31 b(Examples,)f(Comments)518 4913 y Fz(Some)k(other)f(examples)g(will)i(b)r(e)f (discussed)g(brie\015y)f(in)i(the)f(presen)n(t)f(section.)56 b(Most)34 b(of)418 5013 y(the)27 b(pro)r(ofs)f(consist)h(in)g(suitable)g (adaptations)e(of)i(argumen)n(ts)f(already)f(used)i(ab)r(o)n(v)n(e)f(and)g(w) n(e)418 5112 y(shall)h(just)h(sk)n(etc)n(h)f(them.)p eop %%Page: 12 12 bop 418 314 a Fx(12)611 b(W.O.)28 b(AMREIN,)g(M.)f(M)1824 297 y(\025)1815 314 y(ANTOIU)h(AND)g(R.)f(PURICE)418 513 y FB(Example)j(1.)41 b(P)m(oten)m(tials)31 b(that)f(are)g(asymptotically)i(p)s(erio)s(dic)f(in)f (a)f(half-space.)518 613 y Fz(This)24 b(is)f(the)h(example)g(that)g(is)g(the) g(most)f(close)g(to)h(that)g(treated)g(in)g([3].)35 b(It)24 b(is)g(also)f(related)418 712 y(to)28 b(Section)f(6.)518 812 y(Let)20 b(us)g(write)h Fr(X)29 b Fz(=)23 b Fp(R)t Fq(\002)t Fr(X)1353 782 y Fn(0)1380 812 y Fz(.)35 b(F)-7 b(or)20 b(a)f(p)r(erio)r(dic)i (lattice)f(\000)2251 824 y Fm(+)2326 812 y Fz(of)h Fr(X)7 b Fz(,)21 b(w)n(e)f(denote)g(b)n(y)g Fr(C)3076 824 y Fm(+)3131 812 y Fz(\()p Fr(X)7 b Fz(\))21 b(the)418 911 y(C*-algebra)e(of)i(all)g (complex)g(con)n(tin)n(uous)f(functions)i(on)f Fr(X)27 b Fz(that)22 b(are)e(\000)2672 923 y Fm(+)2727 911 y Fz(-p)r(erio)r(dic.)35 b(W)-7 b(e)21 b(shall)418 1011 y(consider)28 b(the)i(unital)g(C*-algebra)c Fq(A)1617 1023 y Fm(+)1702 1011 y Fz(of)j(all)g(b)r(ounded,)h(uniformly)f (con)n(tin)n(uous)g(functions)418 1111 y Fr(')c Fz(:)g Fr(X)31 b Fq(!)25 b Fp(C)50 b Fz(suc)n(h)28 b(that)h(there)f(exist)h(a)f (\(necessarily)f(unique\))i(elemen)n(t)g Fr(')2804 1123 y Fm(+)2884 1111 y Fq(2)c Fr(C)3023 1123 y Fm(+)3079 1111 y Fz(\()p Fr(X)7 b Fz(\))28 b(suc)n(h)418 1210 y(that)e Fq(j)e Fr(')p Fz(\()p Fr(x)776 1222 y Fm(1)814 1210 y Fr(;)14 b(x)898 1180 y Fn(0)922 1210 y Fz(\))h Fq(\000)g Fr(')1103 1222 y Fm(+)1159 1210 y Fz(\()p Fr(x)1238 1222 y Fm(1)1276 1210 y Fr(;)f(x)1360 1180 y Fn(0)1383 1210 y Fz(\))24 b Fq(j!)f Fz(0)i(when)i Fr(x)1898 1222 y Fm(1)1958 1210 y Fq(!)c Fz(+)p Fq(1)p Fz(,)k(uniformly)f(in)g Fr(x)2781 1180 y Fn(0)2828 1210 y Fq(2)d Fr(X)2982 1180 y Fn(0)3005 1210 y Fz(.)36 b(As)26 b(b)r(efore)418 1310 y(w)n(e)d(call)g Fr(')738 1322 y Fm(+)816 1310 y Fi(the)j(asymptotic)h(function)c Fz(of)g Fr(')p Fz(.)36 b(It)23 b(is)g(easy)f(to)h(see)g(that)h Fq(A)2698 1322 y Fm(+)2776 1310 y Fz(is)f(in)n(v)-5 b(arian)n(t)22 b(under)418 1410 y(translations.)518 1509 y(Since)35 b(w)n(e)f(imp)r(osed)h (no)g(conditions)g(on)f(the)i(b)r(eha)n(viour)d(of)i Fr(')h Fz(outside)e(a)h(remote)f(half-)418 1609 y(space,)45 b(w)n(e)d(cannot)g (determine)g(the)h(c)n(haracter)d(space)h Fq(X)55 b Fz(of)42 b Fq(A)2567 1621 y Fm(+)2665 1609 y Fz(precisely)-7 b(.)80 b(But)42 b(it)h(is)418 1708 y(straigh)n(tforw)n(ard)18 b(to)j(sho)n(w)f(that) h(the)g(torus)f Fp(T)1861 1720 y Fm(+)1939 1708 y Fz(=)i Fr(X=)p Fz(\000)2189 1720 y Fm(+)2264 1708 y Fz(is)f(a)f(closed)g(in)n(v)-5 b(arian)n(t)20 b(subset)h(of)g(its)418 1808 y(fron)n(tier)28 b(and)h(that)g(an)n(y)g(neigh)n(b)r(ourho)r(o)r(d)f(of)g Fp(T)1938 1820 y Fm(+)2022 1808 y Fz(in)h Fq(X)42 b Fz(con)n(tains)28 b Fq(f)p Fz(\()p Fr(x)2669 1820 y Fm(1)2706 1808 y Fr(;)14 b(x)2790 1778 y Fn(0)2814 1808 y Fz(\))26 b Fq(2)f Fr(X)32 b Fq(j)26 b Fr(x)3149 1820 y Fm(1)3212 1808 y Fr(>)f(R)q Fq(g)418 1908 y Fz(for)i(some)g Fr(R)d(>)f Fz(0)k(large)f(enough)h(\(use)h(Lemma)f (2\).)518 2007 y(It)k(is)g(also)e(easy)h(to)h(sho)n(w)f(that)h(the)g(quotien) n(t)g(C*-algebra)d Fj(C)2501 2019 y Fn(A)2554 2027 y Fk(+)2605 2007 y Fr(=)p Fj(K)2702 1977 y Fb(T)2747 1985 y Fk(+)2823 2007 y Fz(is)i(isomorphic)g(to)418 2107 y Fj(C)469 2122 y Fo(C)517 2130 y Fk(+)563 2122 y Fm(\()p Fo(X)5 b Fm(\))712 2107 y Fz(in)34 b(suc)n(h)f(a)g(w)n(a)n(y)f(that)i Fr(')p Fz(\()p Fr(Q)p Fz(\))p Fr( )s Fz(\()p Fr(P)12 b Fz(\))34 b(corresp)r(onds)d(to)j Fr(')2470 2119 y Fm(+)2525 2107 y Fz(\()p Fr(Q)p Fz(\))p Fr( )s Fz(\()p Fr(P)12 b Fz(\);)37 b(here)c Fr(')g Fq(2)g(A)3328 2119 y Fm(+)3384 2107 y Fz(,)418 2208 y Fr(')472 2220 y Fm(+)551 2208 y Fq(2)23 b Fr(C)688 2220 y Fm(+)743 2208 y Fz(\()p Fr(X)7 b Fz(\))28 b(is)f(its)h(asymptotic)f(function)i(and)e Fr( )f Fq(2)d Fr(C)2241 2220 y Fm(0)2279 2208 y Fz(\()p Fr(X)7 b Fz(\).)518 2307 y(By)27 b(applying)g(the)h(results)f(of)h(Section)f(5)g(and)h(the)g(discussion)f(ab)r (o)n(v)n(e)f(one)h(gets)418 2453 y FB(Prop)s(osition)47 b(2.)g Fh(Let)40 b Fr(V)63 b Fq(2)45 b(A)1509 2465 y Fm(+)1605 2453 y Fh(b)r(e)40 b(real)g(and)g(let)g Fr(V)2259 2465 y Fm(+)2359 2453 y Fq(2)45 b Fr(C)2518 2465 y Fm(+)2573 2453 y Fz(\()p Fr(X)7 b Fz(\))40 b Fh(b)r(e)h(its)f(asymptotic)418 2552 y(function.)61 b(Let)36 b Fr(H)43 b Fz(=)36 b Fq(\000)p Fz(\001)23 b(+)h Fr(V)54 b Fh(and)35 b Fr(H)1755 2564 y Fm(+)1847 2552 y Fz(=)h Fq(\000)p Fz(\001)23 b(+)g Fr(V)2241 2564 y Fm(+)2332 2552 y Fh(b)r(e)36 b(the)g(asso)r(ciated)e(self-adjoin)n(t)418 2652 y(op)r(erators)28 b(in)i Fq(H)e Fz(=)e Fr(L)1133 2622 y Fm(2)1170 2652 y Fz(\()p Fr(X)7 b Fz(\))p Fh(.)44 b(Let)30 b Fr(\021)g Fq(2)d Fr(C)1740 2664 y Fm(0)1777 2652 y Fz(\()p Fp(R)q Fz(\))36 b Fh(with)30 b(supp)p Fr(\021)24 b Fq(\\)c Fr(\033)s Fz(\()p Fr(H)2585 2664 y Fm(+)2641 2652 y Fz(\))27 b(=)f Fq(;)p Fh(.)43 b(Then)30 b(for)g(eac)n(h)418 2752 y Fr(")23 b(>)g Fz(0)k Fh(there)g(exist)h Fr(R)23 b(>)g Fz(0)k Fh(suc)n(h)g(that)1269 2933 y Fq(k)22 b Fr(\037)p Fz(\()p Fr(Q)1483 2945 y Fm(1)1544 2933 y Fq(\025)g Fr(R)q Fz(\))p Fr(e)1766 2899 y Fn(\000)p Fo(itH)1929 2933 y Fr(\021)s Fz(\()p Fr(H)7 b Fz(\))p Fr(f)32 b Fq(k\024)23 b Fr(")f Fq(k)h Fr(f)32 b Fq(k)418 3114 y Fh(for)27 b(all)g Fr(f)32 b Fq(2)24 b(H)k Fh(and)g(all)f Fr(t)c Fq(2)g Fp(R)p Fh(.)518 3259 y Fz(Of)j(course,)g(one)g(can)g(also)g(in)n(tro)r(duce)g(the)h (C*-algebra)d Fq(A)2374 3271 y Fn(\000)2430 3259 y Fz(,)j(consisting)e(in)i (all)g(b)r(ounded,)418 3359 y(uniformly)k(con)n(tin)n(uous)f(functions)h (that)g(b)r(ecome)g(\000)2116 3371 y Fn(\000)2172 3359 y Fz(-p)r(erio)r(dic)g (\(for)f(some)h(other)f(p)r(erio)r(dic)418 3459 y(lattice)f(\000)725 3471 y Fn(\000)781 3459 y Fz(\))g(at)f Fr(x)991 3471 y Fm(1)1054 3459 y Fz(=)c Fq(\0001)p Fz(,)29 b(uniformly)g(in)g(the)g(orthogonal)d(v)-5 b(ariable)28 b Fr(x)2741 3428 y Fn(0)2764 3459 y Fz(.)41 b(The)28 b(elemen)n(ts)h(of)418 3558 y Fq(A)24 b Fz(=)f Fq(A)662 3570 y Fn(\000)736 3558 y Fq(\\)c(A)876 3570 y Fm(+)960 3558 y Fz(are)26 b(b)r(ounded,)j(uniformly)e(con)n(tin)n(uous)g(functions)h(whic)n(h)g(ha)n(v) n(e)e(\(di\013eren)n(t\))418 3658 y(p)r(erio)r(dic)k(limits)g(at)g Fr(x)1123 3670 y Fm(1)1188 3658 y Fz(=)c Fq(\0061)p Fz(;)31 b(their)f(b)r(eha)n(viour)f(in)h(a)f(v)n(ertical)g(strip)h(is)g(submitted)g (to)g(no)418 3757 y(constrain)n(t.)418 3903 y FB(Example)i(2.)42 b(P)m(oten)m(tials)34 b(with)e(asymptotic)h(v)-5 b(anishing)33 b(oscillation.)518 4002 y Fz(W)-7 b(e)28 b(sa)n(y)e(that)i(a)f(function)h Fr(')23 b Fq(2)h Fr(C)1593 4014 y Fo(b)1627 4002 y Fz(\()p Fr(X)7 b Fz(\))27 b(has)g Fi(asymptotic)k(vanishing)g(oscil)t(lation)p Fz(,)f(and)d(w)n(e)418 4102 y(write)22 b Fr(')i Fq(2)f Fr(V)c(O)r Fz(\()p Fr(X)7 b Fz(\),)24 b(if)f(the)f(function)h Fr(x)h Fq(7!)f Fz(sup)1930 4122 y Fn(j)p Fo(y)r Fn(j\024)p Fm(1)2117 4102 y Fq(j)g Fr(')p Fz(\()p Fr(x)8 b Fz(+)g Fr(y)s Fz(\))g Fq(\000)g Fr(')p Fz(\()p Fr(x)p Fz(\))24 b Fq(j)45 b Fz(is)22 b(of)g(class)f Fr(C)3206 4114 y Fm(0)3244 4102 y Fz(\()p Fr(X)7 b Fz(\).)418 4216 y(W)-7 b(e)22 b(remark)f(that)h Fr(V)d(O)r Fz(\()p Fr(X)7 b Fz(\))22 b(con)n(tains)f(the)i(C*-algebra)c Fr(C)2239 4186 y Fm(rad)2339 4216 y Fz(\()p Fr(X)7 b Fz(\))22 b(of)g(all)g(complex)f(con)n (tin)n(uous)418 4315 y(functions)26 b(on)f Fr(X)32 b Fz(that)26 b(ha)n(v)n(e)e(radial)h(limits)h(at)f(in\014nit)n(y)-7 b(,)27 b(uniformly)e(in)h(all)f(directions.)36 b(Sums)418 4415 y(of)22 b(the)g(form)f Fr(')888 4427 y Fm(0)932 4415 y Fz(+)7 b Fr(')1058 4427 y Fm(1)1117 4415 y Fz(with)22 b Fr(')1354 4427 y Fm(0)1415 4415 y Fq(2)h Fr(C)1552 4427 y Fm(0)1590 4415 y Fz(\()p Fr(X)7 b Fz(\))21 b(and)h Fr(')1961 4427 y Fm(1)2020 4415 y Fz(con)n(tin)n(uous)f (and)g(homogeneous)f(of)i(degree)418 4515 y(zero)k(outside)g(a)h(ball)g(are)e (the)j(most)e(general)g(elemen)n(ts)g(of)h Fr(C)2373 4484 y Fm(rad)2473 4515 y Fz(\()p Fr(X)7 b Fz(\).)37 b(Note)27 b(that)g Fr(V)19 b(O)r Fz(\()p Fr(X)7 b Fz(\))27 b(is)418 4614 y(considerably)h (larger)g(than)i Fr(C)1397 4584 y Fm(rad)1497 4614 y Fz(\()p Fr(X)7 b Fz(\).)43 b(A)30 b Fr(C)1860 4584 y Fm(1)1898 4614 y Fz(-function)g Fr(')g Fz(with)g Fr(@)2572 4626 y Fo(j)2607 4614 y Fr(')d Fq(2)g Fr(C)2829 4626 y Fm(0)2866 4614 y Fz(\()p Fr(X)7 b Fz(\))30 b(for)f(all)g Fr(j)35 b Fz(is)418 4714 y(in)30 b Fr(V)19 b(O)r Fz(\()p Fr(X)7 b Fz(\).)43 b(This)29 b(includes)h Fr(')p Fz(\()p Fr(x)p Fz(\))d(=)f Fr(\036)14 b Fz([)q(\(1)k(+)g Fq(j)p Fr(x)p Fq(j)p Fz(\))2037 4684 y Fo(p)2076 4714 y Fz(])30 b(for)f Fr(p)d(<)f Fz(1)k(and)h Fr(\036)p Fz(,)g Fr(\036)2802 4684 y Fn(0)2856 4714 y Fz(con)n(tin)n(uous)e(and)418 4813 y(b)r(ounded.)41 b(W)-7 b(e)29 b(p)r(oin)n(t)g(out)g(that)g(Prop)r(osition)f (3)g(will)h(b)r(e)g(particularly)f(simple)h(to)f(in)n(terpret)418 4913 y(for)f(p)r(oten)n(tials)g(in)h Fr(C)1090 4883 y Fm(rad)1190 4913 y Fz(\()p Fr(X)7 b Fz(\).)518 5013 y(Let)19 b Fq(X)31 b Fz(b)r(e)19 b(the)h(c)n(haracter)c(space)j(of)f Fr(V)h(O)r Fz(\()p Fr(X)7 b Fz(\).)35 b(By)18 b(iden)n(tifying)i Fr(X)25 b Fz(with)19 b(its)g(homeomorphic)418 5112 y(image)27 b(in)i Fq(X)12 b Fz(,)29 b(w)n(e)e(can)h(express)f(this)h(c)n(haracter)e(space)i(as) f(the)i(disjoin)n(t)f(union)g Fq(X)36 b Fz(=)24 b Fr(X)h Fq(t)19 b(Z)7 b Fz(.)p eop %%Page: 13 13 bop 779 314 a Fx(PR)n(OP)-6 b(A)n(GA)g(TION)29 b(PR)n(OPER)-6 b(TIES)28 b(F)n(OR)g(SCHR)2210 297 y(\177)2200 314 y(ODINGER)h(OPERA)-6 b(TORS)318 b(13)418 513 y Fz(The)40 b(nice)g(feature)f(is)h(that)g Fr(V)19 b(O)r Fz(\()p Fr(X)7 b Fz(\))40 b(is)g(the)g(largest)e(unital,)44 b(translation)38 b(in)n(v)-5 b(arian)n(t)39 b(C*-)418 613 y(subalgebra)34 b(of)h Fr(C)1009 582 y Fo(u)1003 636 y(b)1053 613 y Fz(\()p Fr(X)7 b Fz(\))35 b(suc)n(h)g(that)h(all)f(the)h(elemen)n(ts)f(of)g Fq(Z)42 b Fz(are)35 b(\014xed)g(p)r(oin)n(ts)g(under)h(the)418 712 y(extension)g(of)h(the)g(action)f(of)h(the)g(group)f Fr(X)7 b Fz(.)63 b(This)37 b(w)n(as)f(used)g(in)h([9])g(to)f(sho)n(w)g(that,)k(for) 418 812 y Fr(V)57 b Fq(2)38 b Fr(V)19 b(O)r Fz(\()p Fr(X)7 b Fz(\),)40 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4558 y Fh(.)67 b(Let)418 4658 y Fr(G)27 b Fq(\032)g Fp(R)36 b Fh(b)r(e)30 b(a)g(closed)f(set)h(suc)n(h)g (that)g Fr(G)1695 4628 y Fo(o)1753 4658 y Fq(\\)20 b Fr(V)f Fz(\()p Fr(X)7 b Fz(\))2035 4670 y Fm(asy)2160 4658 y Fq(6)p Fz(=)27 b Fq(;)p Fh(,)j(and)g(let)g Fr(")d(>)f Fz(0)p Fh(.)44 b(Then)30 b(for)g(eac)n(h)418 4758 y Fr(\021)c Fq(2)e Fr(C)623 4770 y Fm(0)660 4758 y Fz(\()p Fp(R)q Fz(\))33 b Fh(with)28 b(supp)p Fr(\021)e Fq(\032)d Fz(\()p Fq(\0001)p Fr(;)14 b Fz(min)p Fq(f)p Fr(G)k Fq(\\)h Fr(V)g Fz(\()p Fr(X)7 b Fz(\))2088 4770 y Fm(asy)2186 4758 y Fq(g)p Fz(\))27 b Fh(there)g(is)g(a)g(compact)f(subset)h Fr(K)33 b Fh(of)418 4858 y(X)28 b(suc)n(h)f(that)1208 4957 y Fq(k)c Fr(\037)1325 4973 y Fo(V)1378 4956 y Fc(\000)p Fk(1)1456 4973 y Fm(\()p Fo(G)p Fm(\))p Fn(n)p Fo(K)1657 4957 y Fz(\()p Fr(Q)p Fz(\))p Fr(e)1826 4923 y Fn(\000)p Fo(itH)1990 4957 y Fr(\021)s Fz(\()p Fr(H)7 b Fz(\))p Fr(f)32 b Fq(k\024)22 b Fr(")h Fq(k)f Fr(f)32 b Fq(k)418 5112 y Fh(for)27 b(all)g Fr(f)32 b Fq(2)24 b(H)k Fh(and)g(all)f Fr(t)c Fq(2)g Fp(R)p Fh(.)p eop %%Page: 14 14 bop 418 314 a Fx(14)611 b(W.O.)28 b(AMREIN,)g(M.)f(M)1824 297 y(\025)1815 314 y(ANTOIU)h(AND)g(R.)f(PURICE)518 513 y Fz(T)-7 b(o)40 b(illustrate)g(this)h(result,)i(let)e(us)f(tak)n(e)g Fr(G)45 b Fz(=)f([)p Fr(\025;)14 b Fq(1)p Fz(\))41 b(with)g(min)14 b([)q Fr(V)k Fz(\()p Fr(X)7 b Fz(\))2974 525 y Fm(asy)3073 513 y Fz(])44 b Fr(<)g(\025)h(<)418 613 y Fz(max)13 b([)q Fr(V)18 b Fz(\()p Fr(X)7 b Fz(\))816 625 y Fm(asy)915 613 y Fz(].)50 b(Then,)34 b(roughly)-7 b(,)32 b(scattering)f(states)g(at)h(energies)f (situated)h(b)r(elo)n(w)g Fr(\025)h Fz(will)418 712 y(not)40 b(propagate)e(in)n(to)i(the)g(asymptotic)g(part)f(of)h(the)g(set)g Fq(f)p Fr(x)k Fq(2)g Fr(X)50 b Fq(j)44 b Fr(V)19 b Fz(\()p Fr(x)p Fz(\))44 b Fq(\025)g Fr(\025)p Fq(g)p Fz(.)74 b(F)-7 b(or)418 812 y(a)34 b(one-dimensional)f(system)h(with)h(a)f(slo)n(wly)g (oscillating)f(p)r(oten)n(tial,)j(this)f(corresp)r(onds)d(to)418 911 y(tunneling)26 b(through)f(an)h(in\014nite)h(sequence)e(of)h(more)f(and)h (more)f(widely)g(separated)g(barriers)418 1011 y(of)d(increasing)f(length.)36 b(The)22 b(e\013ectiv)n(e)g(parts)g(of)h(these)f(barriers,)f(for)h(states)g (with)h(energy)e(less)418 1111 y(than)f Fr(\025)p Fz(,)i(o)r(ccup)n(y)d(the)h (in)n(terv)-5 b(als)18 b Fq(f)p Fr(x)23 b Fq(2)h Fp(R)r Fq(n)r Fr(K)35 b Fq(j)23 b Fr(V)c Fz(\()p Fr(x)p Fz(\))24 b Fq(\025)e Fr(\025)p Fq(g)e Fz(for)f(some)g(compact)g(set)h Fr(K)28 b Fq(\032)23 b Fp(R)i Fz(\(as)418 1210 y(an)20 b(example)g(one)f(ma)n(y)h (consider)f(a)h(p)r(oten)n(tial)g(that)g(is)g(asymptotically)f(of)h(the)h (form)f(cos)o(\()p Fq(j)p Fr(x)p Fq(j)3329 1180 y Fo(\014)3375 1210 y Fz(\))418 1310 y(with)30 b(0)25 b Fr(<)g(\014)k(<)c Fz(1;)30 b(the)f(essen)n(tial)f(sp)r(ectrum)h(of)g(the)h(asso)r(ciated)d (Sturm-Liouville)i(op)r(erator)418 1410 y(will)e(often)g(b)r(e)g(con)n(tin)n (uous,)f(cf.)37 b(Theorem)26 b(4)g(in)h([5],)g(in)g(particular)e(v)n(ectors)g (with)i(a)g(sp)r(ectral)418 1509 y(supp)r(ort)i(in)h(the)g(in)n(terv)-5 b(al)29 b(\()p Fq(\000)p Fz(1)p Fr(;)14 b(\025)p Fz(\))29 b(will)h(propagate) d(a)n(w)n(a)n(y)g(from)i(eac)n(h)g(compact)g(set)g Fr(K)35 b Fz(and)418 1609 y(th)n(us)23 b(undergo)e(tunneling)i(of)f(the)h(indicated)g (t)n(yp)r(e\).)35 b(F)-7 b(or)22 b(m)n(ulti-dimensional)g(systems)g(there)418 1708 y(are)38 b(v)-5 b(arious)37 b(p)r(ossibilities:)58 b(for)38 b(a)g(spherically)f(symmetric)h(slo)n(wly)g(oscillating)f(p)r(oten)n(tial)418 1808 y(there)30 b(will)g(b)r(e)g(an)f(in\014nite)i(sequence)e(of)h (spherically)e(symmetric)i(barriers)d(arranged)h(\(as)h(a)418 1908 y(function)k(of)g(the)g(radial)e(v)-5 b(ariable)32 b Fr(r)r Fz(\))h(in)g(analogy)e(with)i(the)g(one-dimensional)e(case;)j(for)e(a)418 2007 y(p)r(oten)n(tial)27 b(ha)n(ving)f(radial)h(limits)g(\()p Fr(V)42 b Fq(2)24 b Fr(C)1767 1977 y Fm(rad)1867 2007 y Fz(\()p Fr(X)7 b Fz(\)\))27 b(there)g(will)h(b)r(e)g(no)f(propagation)e(in)n(to)i (the)418 2107 y(asymptotic)d(part)f(of)h(the)g(cone)g(subtended)g(b)n(y)g Fq(f)p Fr(!)h Fq(2)e Fr(S)28 b Fq(j)23 b Fz(lim)2376 2119 y Fo(r)r Fn(!1)2559 2107 y Fr(V)c Fz(\()p Fr(r)r(!)s Fz(\))24 b Fq(\025)f Fr(\025)p Fq(g)p Fz(;)i(for)f(certain)418 2207 y(slo)n(wly)39 b(oscillating)g(non-spherically)f(symmetric)h(p)r(oten)n (tials)h(there)f(ma)n(y)g(b)r(e)i(an)e(in\014nite)418 2306 y(collection)27 b(of)h(inaccessible)e(regions)g(of)i(increasing)e(size)h(to)n (w)n(ards)f(in\014nit)n(y)-7 b(.)518 2415 y(The)25 b(ab)r(o)n(v)n(e)f(t)n(yp) r(e)i(of)f(b)r(eha)n(viour)f(is)h(sp)r(eci\014c)h(to)f(the)h(class)e Fr(V)19 b(O)r Fz(\()p Fr(X)7 b Fz(\))26 b(and)f(is)h(related)e(to)i(the)418 2515 y(fact)i(that)h(the)g(action)e(of)i(translations)d(on)i(the)h(fron)n (tier)e Fq(Z)35 b Fz(is)28 b(trivial.)39 b(If)28 b Fr(V)48 b Fz(w)n(ould)27 b(tend)i(at)418 2614 y(in\014nit)n(y)d(to)g(a)f(p)r(erio)r (dic)h(function)g(for)g(instance,)g(the)g(connection)f(b)r(et)n(w)n(een)h (the)g(lo)r(calization)418 2714 y(in)35 b(energy)f(and)g(the)h(domains)f(of)h (non-propagation)d(has)i(a)g(di\013eren)n(t)h(nature,)h(as)e(seen)h(in)418 2813 y(Section)28 b(6.)418 2990 y FB(Example)k(3.)42 b(P)m(oten)m(tials)34 b(with)e(cartesian)i(anisotrop)m(y.)518 3099 y Fz(W)-7 b(e)33 b(shall)g(w)n(ork)e(here)h(with)i Fr(X)k Fz(=)32 b Fp(R)1712 3069 y Fm(2)1787 3099 y Fq(\021)f Fp(R)1937 3111 y Fm(1)2003 3099 y Fq(\002)21 b Fp(R)2143 3111 y Fm(2)2186 3099 y Fz(.)53 b(The)33 b(generalization)e(to)i(arbitrary)418 3199 y(dimension)28 b(is)f(straigh)n(tforw)n(ard.)518 3307 y(F)-7 b(or)39 b Fr(j)48 b Fz(=)43 b(1)p Fr(;)14 b Fz(2)39 b(let)h(us)f(denote)h(b)n(y)g Fq(A)1753 3319 y Fo(j)1828 3307 y Fz(the)g(C*-algebra)d(of)j(all)f(con)n(tin) n(uous)g(functions)418 3411 y Fr(')23 b Fz(:)h Fp(R)596 3423 y Fo(j)660 3411 y Fq(!)f Fp(C)47 b Fz(suc)n(h)25 b(that)i(the)f(limits)g Fr(c)1622 3375 y Fn(\006)1622 3434 y Fo(j)1701 3411 y Fz(=)d(lim)1904 3423 y Fo(x)1942 3431 y Ff(j)1973 3423 y Fn(!\0061)2175 3411 y Fr(')p Fz(\()p Fr(x)2308 3423 y Fo(j)2344 3411 y Fz(\))k(exist.)36 b(Then)26 b Fq(A)d Fz(=)g Fq(A)3089 3423 y Fm(1)3142 3411 y Fq(\012)15 b(A)3288 3423 y Fm(2)3351 3411 y Fz(is)418 3516 y(an)27 b(unital)g(C*-subalgebra)d(of)i Fr(C)1474 3486 y Fo(u)1468 3539 y(b)1518 3516 y Fz(\()p Fr(X)7 b Fz(\))27 b(that)g(is)f(in)n(v)-5 b(arian)n(t)26 b(under)g(translations)g(b)n(y)g(elemen)n(ts)418 3615 y(of)i Fr(X)7 b Fz(.)518 3724 y(Let)676 3703 y Fe(f)670 3724 y Fp(R)724 3736 y Fo(j)793 3724 y Fz(=)28 b Fp(R)940 3736 y Fo(j)1002 3724 y Fq([)21 b(f\0001)1268 3736 y Fo(j)1302 3724 y Fr(;)14 b Fz(+)p Fq(1)1487 3736 y Fo(j)1521 3724 y Fq(g)31 b Fz(the)g(t)n(w)n(o-p)r(oin)n(t)f(compacti\014cation)g(of)h Fp(R)2895 3736 y Fo(j)2935 3724 y Fz(.)47 b(Then)3231 3703 y Fe(f)3225 3724 y Fp(R)3279 3736 y Fo(j)3351 3724 y Fz(is)418 3840 y(the)30 b(sp)r(ectrum)h(of)e Fq(A)1091 3852 y Fo(j)1157 3840 y Fz(and)h Fq(X)39 b Fz(=)1517 3818 y Fe(f)1510 3840 y Fp(R)1564 3852 y Fm(1)1628 3840 y Fq(\002)1719 3818 y Fe(f)1712 3840 y Fp(R)1766 3852 y Fm(2)1839 3840 y Fz(is)30 b(the)h(sp)r(ectrum)f(of)g Fq(A)p Fz(.)44 b(The)30 b(closed)f(in)n(v)-5 b(arian)n(t)418 3940 y(subsets)28 b(of)g(the)h(fron)n(tier)f Fq(Z)j Fz(=)24 b Fq(X)31 b(n)18 b Fr(X)35 b Fz(are)27 b(as)h(follo)n(ws:)37 b(the)29 b(four)f(corners)f Fq(f)p Fz(\()p Fq(\0061)3052 3952 y Fm(1)3088 3940 y Fr(;)14 b Fq(\0061)3273 3952 y Fm(2)3310 3940 y Fz(\))p Fq(g)p Fz(,)418 4052 y(the)21 b(four)g(edges)943 4031 y Fe(f)936 4052 y Fp(R)990 4064 y Fm(1)1038 4052 y Fq(\002)5 b(f\0061)1298 4064 y Fm(2)1334 4052 y Fq(g)20 b Fz(and)h Fq(f\0061)1741 4064 y Fm(1)1777 4052 y Fq(g)5 b(\002)1901 4031 y Fe(f)1894 4052 y Fp(R)1948 4064 y Fm(2)2011 4052 y Fz(and)21 b(all)g(their)f(unions.)35 b(W)-7 b(e)21 b(shall)f(illustrate)418 4165 y(our)27 b(Theorem)g(for)g Fr(F)35 b Fz(=)22 b Fq(f)p Fz(\(+)p Fq(1)1441 4177 y Fm(1)1478 4165 y Fr(;)14 b Fz(+)p Fq(1)1663 4177 y Fm(2)1700 4165 y Fz(\))p Fq(g)27 b Fz(and)h(for)f Fr(F)35 b Fz(=)2272 4144 y Fe(f)2265 4165 y Fp(R)2319 4177 y Fm(1)2381 4165 y Fq(\002)18 b(f)p Fz(+)p Fq(1)2654 4177 y Fm(2)2690 4165 y Fq(g)p Fz(.)518 4274 y(So)h(let)g(us)h (consider)e(the)i(self-adjoin)n(t)f(op)r(erator)e Fr(H)30 b Fz(=)23 b Fq(\000)p Fz(\001)r(+)r Fr(V)37 b Fz(in)20 b Fq(H)k Fz(=)f Fr(L)2833 4244 y Fm(2)2869 4274 y Fz(\()p Fr(X)7 b Fz(\),)21 b(for)e Fr(V)42 b Fq(2)24 b(A)418 4373 y Fz(a)i(real)f(function.)37 b(It)27 b(is)f(easy)f(to)h(sho)n(w)g(that)g Fr(V)45 b Fz(has)26 b(the)h(follo)n(wing)e(prop)r(ert)n(y)g(\(whic)n(h)h(in)h(fact)418 4474 y(c)n(haracterizes)g(the)k(elemen)n(ts)e(of)h Fq(A)p Fz(\):)42 b(for)29 b Fr(j;)14 b(k)30 b Fq(2)d(f)p Fz(1)p Fr(;)14 b Fz(2)p Fq(g)28 b Fz(and)i Fr(k)f Fq(6)p Fz(=)e Fr(j)5 b Fz(,)30 b(the)g(limits)h Fr(V)3113 4438 y Fn(\006)3094 4497 y Fo(j)3169 4474 y Fz(\()p Fr(x)3248 4486 y Fo(j)3283 4474 y Fz(\))c(=)418 4579 y(lim)533 4591 y Fo(x)571 4600 y Ff(k)607 4591 y Fn(!\0061)791 4600 y Ff(k)846 4579 y Fr(V)19 b Fz(\()p Fr(x)p Fz(\))27 b(exist)g(uniformly)f(in)h Fr(x)1767 4591 y Fo(j)1826 4579 y Fq(2)c Fp(R)1958 4591 y Fo(j)2026 4579 y Fz(and)j(de\014ne)h(elemen)n(ts)f(of)h Fq(A)2923 4591 y Fo(j)2958 4579 y Fz(.)37 b(The)26 b(v)-5 b(alues)418 4679 y(tak)n(en)23 b(b)n(y)f(the)i(con)n(tin)n(uous)e(extension)g(of)h Fr(V)42 b Fz(to)23 b Fq(X)35 b Fz(on)23 b(the)g(four)g(edges)f(coincide)h (resp)r(ectiv)n(ely)418 4778 y(with)j Fr(V)672 4743 y Fn(\006)653 4801 y Fo(j)728 4778 y Fz(,)g Fr(j)i Fz(=)23 b(1)p Fr(;)14 b Fz(2.)35 b(Its)25 b(v)-5 b(alues)26 b(at)f(the)h(four)f(corners)f(will)h(b) r(e)h(denoted)g(b)n(y)f Fr(c)2894 4748 y Fm(++)3000 4778 y Fz(,)h Fr(c)3085 4748 y Fm(+)p Fn(\000)3192 4778 y Fz(,)g Fr(c)3277 4748 y Fn(\000)p Fm(+)3384 4778 y Fz(,)418 4892 y Fr(c)454 4862 y Fn(\000\000)562 4892 y Fz(.)37 b(Note)28 b(that)f(for)h(example)f Fr(c)1491 4862 y Fm(++)1620 4892 y Fz(=)22 b(lim)1823 4904 y Fo(x)1861 4912 y Fk(1)1893 4904 y Fn(!)p Fm(+)p Fn(1)2076 4912 y Fk(1)2126 4892 y Fr(V)2193 4857 y Fm(+)2174 4915 y(1)2248 4892 y Fz(\()p Fr(x)2327 4904 y Fm(1)2365 4892 y Fz(\))i(=)e(lim)2624 4904 y Fo(x)2662 4912 y Fk(2)2694 4904 y Fn(!)p Fm(+)p Fn(1)2877 4912 y Fk(2)2928 4892 y Fr(V)2994 4857 y Fm(+)2976 4915 y(2)3050 4892 y Fz(\()p Fr(x)3129 4904 y Fm(2)3167 4892 y Fz(\).)518 5004 y(Let)f(us)g(consider)f(the)i(op)r(erator)d Fr(H)1618 4968 y Fm(+)1611 5026 y(1)1696 5004 y Fz(=)k Fq(\000)p Fz(\001)6 b(+)g Fr(V)2060 4968 y Fm(+)2043 5026 y(1)2138 5004 y Fz(=)2226 4936 y Fe(\000)2264 5004 y Fq(\000)p Fz(\001)2398 5016 y Fm(1)2453 5004 y Fz(+)18 b Fr(V)2603 4968 y Fm(+)2584 5026 y(1)2658 4936 y Fe(\001)2702 5004 y Fq(\012)6 b Fz(1)2815 5016 y Fm(2)2856 5004 y Fz(+)g(1)2969 5016 y Fm(1)3011 5004 y Fq(\012)g Fz(\()p Fq(\000)p Fz(\001)3248 5016 y Fm(2)3284 5004 y Fz(\))22 b(in)418 5112 y(the)28 b(represen)n(tation)e Fr(L)1161 5082 y Fm(2)1198 5112 y Fz(\()p Fp(R)1284 5124 y Fm(1)1328 5112 y Fz(\))18 b Fq(\012)h Fr(L)1519 5082 y Fm(2)1555 5112 y Fz(\()p Fp(R)1642 5124 y Fm(2)1685 5112 y Fz(\).)38 b(Its)28 b(sp)r(ectrum)g(equals)f([)p Fr(a)2583 5077 y Fm(+)2583 5135 y(1)2638 5112 y Fr(;)14 b Fq(1)p Fz(\),)28 b(where)f Fr(a)3125 5077 y Fm(+)3125 5135 y(1)3208 5112 y Fz(is)h(the)p eop %%Page: 15 15 bop 779 314 a Fx(PR)n(OP)-6 b(A)n(GA)g(TION)29 b(PR)n(OPER)-6 b(TIES)28 b(F)n(OR)g(SCHR)2210 297 y(\177)2200 314 y(ODINGER)h(OPERA)-6 b(TORS)318 b(15)418 513 y Fz(in\014m)n(um)23 b(of)g(the)g(sp)r(ectrum)g(of)g (the)g(op)r(erator)e Fr(H)1958 483 y Fm(1)p Fo(;)p Fm(+)2089 513 y Fz(=)h Fq(\000)p Fz(\001)2310 525 y Fm(1)2356 513 y Fz(+)9 b Fr(V)2496 477 y Fm(+)2478 535 y(1)2574 513 y Fz(acting)22 b(in)h Fq(H)2980 525 y Fm(1)3041 513 y Fz(=)f Fr(L)3185 483 y Fm(2)3222 513 y Fz(\()p Fp(R)3308 525 y Fm(1)3352 513 y Fz(\).)418 613 y(Three)d(other)f(op)r(erators)f(of)i(this)h(kind)f(are)f(a)n(v)-5 b(ailable)18 b(and,)j(with)f(ob)n(vious)d(notations,)j(w)n(e)f(ha)n(v)n(e)418 712 y Fr(\033)465 724 y Fm(ess)552 712 y Fz(\()p Fr(H)7 b Fz(\))23 b(=)803 645 y Fe(\002)837 712 y Fz(min)q Fq(f)p Fr(a)1062 677 y Fm(+)1062 734 y(1)1116 712 y Fr(;)14 b(a)1197 677 y Fn(\000)1197 734 y Fm(1)1253 712 y Fr(;)g(a)1334 677 y Fm(+)1334 734 y(2)1389 712 y Fr(;)g(a)1470 677 y Fn(\000)1470 734 y Fm(2)1526 712 y Fq(g)p Fr(;)g Fq(1)1688 645 y Fe(\001)1725 712 y Fz(.)37 b(This)28 b(follo)n(ws)f(quite)g(easily)g(from)g(our)g(argumen)n(ts)418 812 y(and)f(w)n(as)g(pro)n(v)n(ed)f(in)i(a)f(greater)f(generalit)n(y)g(in)h Fq(x)p Fz(3)g(of)h([9].)36 b(Remark)26 b(also)f(that)i(the)g(sp)r(ectrum)418 912 y(of)c Fr(H)584 882 y Fn(\006\006)715 912 y Fz(=)g Fq(\000)p Fz(\001)10 b(+)g Fr(c)1058 882 y Fn(\006\006)1188 912 y Fz(is)23 b([)p Fr(c)1326 882 y Fn(\006\006)1434 912 y Fr(;)14 b Fq(1)p Fz(\))23 b(and)g(that)h(inequalities)f(suc)n(h)f(as)h Fr(a)2698 877 y Fm(+)2698 935 y(1)2776 912 y Fq(\024)g Fz(min)p Fq(f)p Fr(c)3080 882 y Fn(\000)p Fm(+)3186 912 y Fr(;)14 b(c)3259 882 y Fm(++)3365 912 y Fq(g)418 1012 y Fz(are)27 b(true.)518 1112 y(A)j(neigh)n(b)r(ourho)r(o)r(d)f(base)g(of)h(the)g(p)r(oin)n(t)g Fq(f)p Fz(\(+)p Fq(1)2050 1124 y Fm(1)2087 1112 y Fr(;)14 b Fz(+)p Fq(1)2272 1124 y Fm(2)2308 1112 y Fz(\))p Fq(g)30 b Fz(is)g(comp)r(osed)f(of)h(all)f(the)i(rect-)418 1211 y(angles)i Fq(f)p Fz(\()p Fr(y)789 1223 y Fm(1)826 1211 y Fr(;)14 b Fz(+)p Fq(1)1011 1223 y Fm(1)1047 1211 y Fz(])23 b Fq(\002)f Fz(\()p Fr(y)1253 1223 y Fm(2)1290 1211 y Fr(;)14 b Fz(+)p Fq(1)1475 1223 y Fm(2)1512 1211 y Fz(])34 b Fq(j)f Fr(y)1666 1223 y Fm(1)1736 1211 y Fq(2)h Fp(R)1879 1223 y Fm(1)1922 1211 y Fr(;)14 b(y)2000 1223 y Fm(2)2071 1211 y Fq(2)33 b Fp(R)2213 1223 y Fm(2)2257 1211 y Fq(g)g Fz(and)h(a)f(neigh)n(b)r(ourho)r(o)r(d)g(base)g(of)425 1303 y Fe(f)418 1324 y Fp(R)472 1336 y Fm(1)534 1324 y Fq(\002)18 b(f)p Fz(+)p Fq(1)807 1336 y Fm(2)843 1324 y Fq(g)27 b Fz(consists)g(of)h (the)g(rectangles)e Fq(f)1893 1303 y Fe(f)1886 1324 y Fp(R)1940 1336 y Fm(1)2001 1324 y Fq(\002)18 b Fz(\()p Fr(y)2157 1336 y Fm(2)2194 1324 y Fr(;)c Fz(+)p Fq(1)2379 1336 y Fm(2)2416 1324 y Fz(])23 b Fq(j)g Fr(y)2549 1336 y Fm(2)2609 1324 y Fq(2)h Fp(R)2742 1336 y Fm(2)2785 1324 y Fq(g)p Fz(.)36 b(W)-7 b(e)28 b(get:)418 1473 y FB(Prop)s(osition)k(4.)47 b Fh(Let)28 b Fr(")23 b(>)f Fz(0)p Fh(.)518 1573 y(\(a\))33 b(F)-7 b(or)33 b(an)n(y)f Fr(\021)k Fq(2)d Fr(C)1198 1585 y Fm(0)1236 1573 y Fz(\()p Fp(R)p Fz(\))40 b Fh(with)34 b(supp)p Fr(\021)i Fq(\032)c Fz(\()p Fq(\0001)p Fr(;)14 b(c)2187 1543 y Fm(++)2293 1573 y Fz(\))33 b Fh(there)g(exist)h Fr(y)2820 1585 y Fm(1)2889 1573 y Fq(2)f Fp(R)3031 1585 y Fm(1)3074 1573 y Fr(;)14 b(y)3152 1585 y Fm(2)3222 1573 y Fq(2)33 b Fp(R)3364 1585 y Fm(2)418 1673 y Fh(suc)n(h)27 b(that)h(for)f(all)h Fr(f)j Fq(2)24 b(H)1035 1872 y(k)e Fr(\037)p Fz(\()p Fr(Q)1249 1884 y Fm(1)1309 1872 y Fr(>)h(y)1438 1884 y Fm(1)1475 1872 y Fz(\))p Fr(\037)p Fz(\()p Fr(Q)1657 1884 y Fm(2)1717 1872 y Fr(>)g(y)1846 1884 y Fm(2)1883 1872 y Fz(\))p Fr(e)1954 1838 y Fn(\000)p Fo(itH)2117 1872 y Fr(\021)s Fz(\()p Fr(H)7 b Fz(\))p Fr(f)33 b Fq(k\024)22 b Fr(")h Fq(k)f Fr(f)32 b Fq(k)22 b Fr(:)518 2072 y Fh(\(b\))32 b(F)-7 b(or)31 b(an)n(y)f Fr(\021)i Fq(2)e Fr(C)1190 2084 y Fm(0)1228 2072 y Fz(\()p Fp(R)p Fz(\))38 b Fh(with)31 b(supp)q Fr(\021)h Fq(\032)d Fz(\()p Fq(\0001)p Fr(;)14 b(a)2176 2037 y Fm(+)2176 2094 y(1)2231 2072 y Fz(\))31 b Fh(there)g(exists)g Fr(y)2784 2084 y Fm(2)2851 2072 y Fq(2)e Fp(R)2989 2084 y Fm(2)3064 2072 y Fh(suc)n(h)i(that)418 2172 y(for)c(all)g Fr(f)32 b Fq(2)24 b(H)1239 2271 y(k)e Fr(\037)p Fz(\()p Fr(Q)1453 2283 y Fm(2)1513 2271 y Fr(>)h(y)1642 2283 y Fm(2)1679 2271 y Fz(\))p Fr(e)1750 2237 y Fn(\000)p Fo(itH)1913 2271 y Fr(\021)s Fz(\()p Fr(H)7 b Fz(\))p Fr(f)33 b Fq(k\024)22 b Fr(")g Fq(k)h Fr(f)32 b Fq(k)22 b Fr(:)518 2430 y Fz(So,)h(at)g(energies)e (b)r(elo)n(w)h Fr(a)1332 2395 y Fm(+)1332 2452 y(1)1410 2430 y Fz(there)g(is)g(no)h(propagation)d(to)n(w)n(ards)h Fr(x)2614 2442 y Fm(2)2675 2430 y Fz(=)h(+)p Fq(1)2910 2442 y Fm(2)2947 2430 y Fz(.)35 b(A)n(t)23 b(energies)418 2532 y(comprised)k(b)r(et)n(w)n(een) g Fr(a)1175 2496 y Fm(+)1175 2554 y(1)1258 2532 y Fz(and)g Fr(c)1455 2502 y Fm(++)1589 2532 y Fz(this)g(b)r(ecomes)h(p)r(ossible,)f(but) h(then)g(the)g(observ)-5 b(able)26 b Fr(Q)3370 2544 y Fm(2)418 2632 y Fz(cannot)d(div)n(erge)g(through)f(p)r(ositiv)n(e)i(v)-5 b(alues.)35 b(Results)24 b(of)f(this)h(t)n(yp)r(e)g(are)f(b)n(y)g(no)g(means) h(trivial)418 2731 y(in)i(the)g(sense)f(that,)h(in)g(certain)f(situations,)g (propagation)f(a)n(w)n(a)n(y)f(from)i(an)n(y)g(compact)g(subset)418 2831 y(of)30 b Fr(X)35 b Fz(do)r(es)30 b(o)r(ccur)e(at)i(energies)e(as)h(ab)r (o)n(v)n(e)f(\(under)i(suitable)f(assumptions)g(on)g Fr(V)48 b Fz(there)30 b(will)418 2930 y(b)r(e)23 b(in)n(terv)-5 b(als)21 b(of)i(purely)f(absolutely)f(con)n(tin)n(uous)h(sp)r(ectrum)g(of)g Fr(H)29 b Fz(in)23 b(the)g(considered)e(energy)418 3030 y(range,)26 b(and)i(asso)r(ciated)e(states)h(m)n(ust)h(propagate)e(to)h(in\014nit)n(y)h (b)n(y)f(the)h(results)f(of)h([2]\).)518 3130 y FB(Ac)m(kno)m(wledgmen)m(t:) 36 b Fz(W)-7 b(e)24 b(are)f(grateful)g(to)h(G)r(\177)-44 b(un)n(ter)24 b(Stolz)f(for)h(corresp)r(ondence)e(and)i(for)418 3229 y(p)r(oin)n(ting)k (out)f(to)h(us)f(the)h(pap)r(er)f([5].)1670 3429 y FA(References)461 3561 y Fx(1.)d(W.)29 b(O.)f(Amrein,)g(A.)h(Boutet)h(de)g(Mon)n(v)n(el)f(and)g (V.)g(Georgescu,)h Fw(C)2393 3570 y Fu(0)2428 3561 y Fl(-Gr)l(oups,)j (Commutator)f(Metho)l(ds)540 3644 y(and)26 b(Sp)l(e)l(ctr)l(al)h(The)l(ory)f (of)g(N-Bo)l(dy)g(Hamiltonians)p Fx(,)f(Birkh\177)-35 b(auser-V)-6 b(erlag,)22 b(Basel-Boston-Berlin,)h(1996.)461 3728 y(2.)h(W.)k(O.)g(Amrein)e (and)j(V.)f(Georgescu,)h Fl(On)h(the)g(Char)l(acterization)h(of)f(Bound)i (States)d(and)i(Sc)l(attering)540 3811 y(States)25 b(in)g(Quantum)i(Me)l (chanics)p Fx(,)c(Helv.)g(Ph)n(ys.)h(Acta)g Fa(46)f Fx(\(1973\),)i(635-658.) 461 3894 y(3.)f(E.)19 b(B.)h(Da)n(vies)g(and)g(B.)g(Simon,)e Fl(Sc)l(attering)k(The)l(ory)h(for)g(Systems)e(with)i(Di\013er)l(ent)e(Sp)l (atial)j(Asymptotics)540 3977 y(on)i(the)f(L)l(eft)g(and)h(R)n(ight)p Fx(,)d(Comm)n(un.)f(Math.)i(Ph)n(ys.)f Fa(63)g Fx(\(1978\),)i(277-301.)461 4060 y(4.)f(V.)h(Georgescu)i(and)f(A.)f(Iftimo)n(vici,)f Fl(C*-A)n(lgebr)l (as)k(of)g(Ener)l(gy)g(Observables:)36 b(I.)27 b(Gener)l(al)i(The)l(ory)f (and)540 4143 y(Bumps)e(A)n(lgebr)l(as)p Fx(,)e(preprin)n(t)g(00-520)g(at)h Fw(http)19 b Fx(:)h Fw(==w)r(w)r(w)r(:ma:utexas:edu=mp)p Fx(-)p Fw(ar)r(c=)p Fx(.)461 4226 y(5.)k(A.)32 b(Y)-6 b(a.)33 b(Gordon,)h Fl(Pur)l(e)h(Point)g(Sp)l(e)l(ctrum)g(under)g(1-Par)l(ameter)h(Perturb)l (ations)f(and)g(Instability)g(of)540 4309 y(A)n(nderson)26 b(L)l(o)l(c)l(alizaton)p Fx(,)g(Comm)n(un.)21 b(Math.)j(Ph)n(ys.)f Fa(164)g Fx(\(1994\),)i(489-505.)461 4392 y(6.)f(A.)f(Guic)n(hardet,)h Fl(Sp)l(e)l(cial)j(T)-5 b(opics)26 b(in)f(T)-5 b(op)l(olo)l(gic)l(al)29 b(A)n(lgebr)l(as)p Fx(,)23 b(Gordon)i(and)f(Breac)n(h,)g(New)g(Y)-6 b(ork,)23 b(1968.)461 4475 y(7.)h(R.)i(V.)h(Kadison)g(and)h(J.)f(R.)f (Ringrose,)h Fl(F)-5 b(undamentals)31 b(of)e(the)g(The)l(ory)g(of)g(Op)l(er)l (ator)h(A)n(lgebr)l(as)p Fx(,)e(Aca-)540 4558 y(demic)23 b(Press,)f(New)i(Y) -6 b(ork,)23 b(1983.)461 4641 y(8.)h(M.)15 b(M\025)-35 b(an)n(toiu,)17 b Fl(On)i(a)g(Class)h(of)f(A)n(nisotr)l(opic)i(Schr\177)-36 b(odinger)19 b(Op)l(er)l(ators)p Fx(,)f(preprin)n(t)e(01-201)i(at)f Fl(http://www.)540 4724 y(ma.utexas.e)l(du/mp)p Fv(\000)p Fl(ar)l(c/)p Fx(.)461 4807 y(9.)24 b(M.)19 b(M\025)-35 b(an)n(toiu,)20 b Fl(C*-A)n(lgebr)l(as,)k(Dynamic)l(al)f(Systems)f(at)g(In\014nity)g(and)h(the) g(Essential)f(Sp)l(e)l(ctrum)i(of)e(Gen-)540 4890 y(er)l(alize)l(d)29 b(Schr\177)-36 b(odinger)29 b(Op)l(er)l(ators)p Fx(,)e(preprin)n(t)f(01-298)i (at)e Fl(http://www.ma.utexas.e)l(du/mp)p Fv(\000)p Fl(ar)l(c/)32 b Fx(and)540 4973 y(to)24 b(app)r(ear)g(in)g(J.)f(Reine)h(Angew.)g(Math..)426 5056 y(10.)g(G.)f(J.)g(Murph)n(y,)h Fl(C*-A)n(lgebr)l(as)i(and)g(Op)l(er)l (ator)h(The)l(ory)p Fx(,)d(Academic)f(Press,)g(Boston,)h(1990.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0112050226516--