Content-Type: multipart/mixed; boundary="-------------0205280817448" This is a multi-part message in MIME format. ---------------0205280817448 Content-Type: text/plain; name="02-244.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-244.comments" e-mail: Marius.Mantoiu@imar.ro Radu.Purice@imar.ro ---------------0205280817448 Content-Type: text/plain; name="02-244.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-244.keywords" C*-algebra, twisted crossed product, magnetic field, Schroedinger operators ---------------0205280817448 Content-Type: application/postscript; name="Mantoiu-Purice-AOMF.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Mantoiu-Purice-AOMF.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: articol.dvi %%Pages: 6 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: 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b(odinger)34 b(Hamiltonian)i Fo(H)44 b Fq(:=)-236 2721 y(\(\005)-142 2691 y Fl(A)-88 2721 y Fq(\))-56 2691 y Fk(2)0 2721 y Fq(+)17 b Fo(V)47 b Fq(in)27 b Fj(H)d Fq(:=)f Fo(L)535 2691 y Fk(2)571 2721 y Fq(\()p Fo(X)7 b Fq(\);)28 b(here)f Fo(A)g Fq(is)h(a)e(v)n(ector)g(p)r(oten)n(tial)i (for)e Fo(B)32 b Fq(and)27 b(\005)2227 2691 y Fl(A)2305 2721 y Fq(:=)22 b Fj(\000)p Fo(i)p Fj(r)17 b(\000)h Fo(A)28 b Fq(is)f(the)g(magnetic)g(momen)n(tum.)-236 2821 y(It)f(is)g(w)n(ell)f (kno)n(wn)g(that)i Fo(H)32 b Fq(can)26 b(b)r(e)g(de\014ned)g(as)f(a)h (self-adjoin)n(t)f(op)r(erator)f(and)i(that)g Fo(C)2516 2791 y Fn(1)2510 2841 y Fk(0)2586 2821 y Fq(\()p Fo(X)7 b Fq(\))26 b(is)g(a)f(core)g(for)g(it)i(\([AHS)q(],)f([LS]\).)-236 2920 y(W)-7 b(e)23 b(shall)f(de\014ne)h(a)f Fo(C)455 2890 y Fn(\003)493 2920 y Fq(-algebra)f(of)h(b)r(ounded)h(op)r(erators) d(in)j Fj(H)h Fq(con)n(taining)d(the)i(resolv)n(en)n(t)e(family)i Fj(f)p Fq(\()p Fo(H)14 b Fj(\000)8 b Fo(z)t Fq(\))3212 2890 y Fn(\000)p Fk(1)3347 2920 y Fj(j)46 b Fo(z)27 b Fj(2)c Fm(C)g Fj(n)8 b Fm(R)e Fj(g)-236 3020 y Fq(and)34 b(th)n(us)g(the)g(en)n(tire)g Fo(C)569 3032 y Fn(1)639 3020 y Fq(-class)f(functional)h(calculus)g(of)g Fo(H)41 b Fq(\(w)n(e)34 b(denote)g(b)n(y)f Fo(C)2411 3032 y Fn(1)2482 3020 y Fq(\()p Fm(R)p Fq(\))41 b(the)34 b(set)g(of)g(con)n(tin)n(uous)f (functions)-236 3120 y(on)g Fm(R)39 b Fq(v)-5 b(anishing)33 b(at)g(in\014nit)n(y\).)54 b(This)33 b Fo(C)1085 3089 y Fn(\003)1124 3120 y Fq(-algebra)e(has)i(a)f(remark)-5 b(able)32 b(structure.)53 b(W)-7 b(e)34 b(consider)e(the)i(natural)e (action)h(b)n(y)-236 3219 y(translations)g Fo(\022)j Fq(of)f Fo(X)41 b Fq(on)34 b Fo(B)t(C)759 3231 y Fl(u)803 3219 y Fq(\()p Fo(X)7 b Fq(\))34 b(\(the)h Fo(C)1224 3189 y Fn(\003)1262 3219 y Fq(-algebra)e(of)h(b)r(ounded,)j(uniformly)d (con)n(tin)n(uous)f(functions)i(on)f Fo(X)7 b Fq(\))34 b(and)g(the)-236 3319 y(imaginary)f(exp)r(onen)n(tial)h(of)h(the)g (\015ux)g(of)f(the)h(magnetic)g(\014eld,)h(that)f(de\014nes)g(a)f(co)r (cycle)g Fo(!)2749 3331 y Fl(B)2841 3319 y Fq(on)g Fo(X)c Fj(\002)23 b Fo(X)41 b Fq(with)35 b(v)-5 b(alues)34 b(in)-236 3418 y(the)g(group)g(of)g(unitary)f(elemen)n(ts)i(in)f Fo(B)t(C)1132 3430 y Fl(u)1176 3418 y Fq(\()p Fo(X)7 b Fq(\).)57 b(F)-7 b(rom)34 b(these)g(one)g(can)g(de\014ne)g(the)h Fo(C)2619 3388 y Fn(\003)2657 3418 y Fq(-algebra)d Fi(C)i Fq(:=)g Fo(B)t(C)3315 3430 y Fl(u)3359 3418 y Fq(\()p Fo(X)7 b Fq(\))23 b Fm(o)3587 3381 y Fl(!)3629 3389 y Fh(B)3587 3443 y Fl(\022)3704 3418 y Fo(X)-236 3518 y Fq(\(the)36 b(t)n(wisted)f(crossed)f(pro)r(duct\))i(that)g(mixes)f(in)g (a)g(subtle)h(non-comm)n(utativ)n(e)e(w)n(a)n(y)g Fo(B)t(C)2720 3530 y Fl(u)2764 3518 y Fq(\()p Fo(X)7 b Fq(\))36 b(and)f Fo(L)3166 3488 y Fk(1)3203 3518 y Fq(\()p Fo(X)7 b Fq(\).)60 b(Then)35 b(an)n(y)-236 3618 y(represen)n(tation)22 b(of)j Fi(C)f Fq(con)n(tains)g(at)g(the)h(same)f(time)g(m)n(ultiplication)h (op)r(erators)d(\(b)n(y)j(functions)g(of)f(class)f Fo(B)t(C)3223 3630 y Fl(u)3267 3618 y Fq(\()p Fo(X)7 b Fq(\)\))25 b(and)f Fo(C)3681 3630 y Fn(1)3752 3618 y Fq(-)-236 3717 y(functions)32 b(of)f(the)h(magnetic)f(momen)n(ta,)h(the)g(comm)n(utation)f(form)n (ulae:)43 b Fo(i)p Fq([)p Fo(Q)2275 3729 y Fl(j)2310 3717 y Fo(;)14 b(Q)2413 3729 y Fl(k)2453 3717 y Fq(])30 b(=)f(0)p Fo(;)43 b(i)p Fq([\005)2822 3687 y Fl(A)2822 3739 y(j)2876 3717 y Fo(;)14 b(Q)2979 3729 y Fl(k)3019 3717 y Fq(])30 b(=)f Fo(\016)3203 3729 y Fl(j)s(k)3275 3717 y Fo(;)43 b(i)p Fq([\005)3455 3687 y Fl(A)3455 3739 y(j)3509 3717 y Fo(;)14 b Fq(\005)3608 3687 y Fl(A)3608 3741 y(k)3662 3717 y Fq(])30 b(=)-236 3817 y Fo(B)-173 3829 y Fl(k)q(j)-73 3817 y Fq(b)r(eing)e(tak)n(en)g(in)n(to)g(accoun)n (t.)38 b(W)-7 b(e)29 b(pro)n(v)n(e)e(in)h(this)h(pap)r(er)f(that)h(an)n (y)e(suc)n(h)h(represen)n(tation)f(also)g(con)n(tains)h(the)g(resolv)n (en)n(t)-236 3917 y(family)f(of)h(the)g(magnetic)f(Sc)n(hr\177)-42 b(odinger)26 b(Hamiltonian)h Fo(H)7 b Fq(.)-236 4191 y Fr(2)135 b(The)43 b(Twisted)h(Crossed)h(Pro)t(duct)e(Algebra)i(Asso)t (ciated)f(to)h(a)f(Magnetic)-34 4341 y(Field)-236 4522 y Fq(W)-7 b(e)38 b(consider)e(a)h(magnetic)g(\014eld)g(as)g(b)r(eing)h (de\014ned)f(b)n(y)g(a)g(b)r(ounded,)k(uniformly)c(con)n(tin)n(uous,)i (matrix-v)-5 b(alued)36 b(function)-236 4622 y Fo(B)27 b Fq(:)c Fo(X)29 b Fj(!)24 b Fm(M)177 4634 y Fl(n;n)289 4622 y Fq(\()p Fm(R)p Fq(\))34 b(satisfying)27 b(the)h(conditions:)36 b Fo(B)1446 4634 y Fl(j)s(k)1541 4622 y Fq(=)23 b Fj(\000)p Fo(B)1757 4634 y Fl(k)q(j)1828 4622 y Fo(;)37 b(@)1932 4634 y Fl(j)1967 4622 y Fo(B)2030 4634 y Fl(k)q(l)2111 4622 y Fq(+)18 b Fo(@)2238 4634 y Fl(k)2279 4622 y Fo(B)2342 4634 y Fl(lj)2416 4622 y Fq(+)g Fo(@)2543 4634 y Fl(l)2569 4622 y Fo(B)2632 4634 y Fl(j)s(k)2727 4622 y Fq(=)k(0)p Fo(:)-112 4722 y Fq(The)40 b(ph)n(ysical)e(description)h(of)g(a)g (particle)g(mo)n(ving)f(in)h(a)g(magnetic)g(\014eld)h Fo(B)j Fq(is)c(obtained)g(b)n(y)g(replacing)f(the)i(usual)-236 4821 y(momen)n(tum)32 b Fo(p)g Fq(of)h(the)f(particle)g(b)n(y)g(the)g (expression)f Fo(\031)j Fq(:=)d Fo(p)21 b Fj(\000)g Fo(A)p Fq(\()p Fo(x)p Fq(\))34 b(where)d Fo(A)i Fq(is)f(a)g(v)n(ector)f(p)r (oten)n(tial)h(for)g(our)f(magnetic)-236 4921 y(\014eld,)f(i.e.)44 b(a)29 b(v)n(ector)g(function)h Fo(A)d Fq(:)f Fo(X)33 b Fj(!)27 b Fo(X)36 b Fq(satisfying:)41 b Fo(B)1694 4933 y Fl(j)s(k)1792 4921 y Fq(=)26 b Fo(@)1927 4933 y Fl(j)1962 4921 y Fo(A)2024 4933 y Fl(k)2085 4921 y Fj(\000)19 b Fo(@)2213 4933 y Fl(k)2254 4921 y Fo(A)2316 4933 y Fl(j)2352 4921 y Fo(:)30 b Fq(Under)f(our)g(h)n(yp)r(othesis)g(on)h(the)g (function)-236 5020 y Fo(B)g Fq(suc)n(h)c(a)f(v)n(ector)g(p)r(oten)n (tial)h(alw)n(a)n(ys)e(exists)i(but)h(is)f(not)g(unique)g(\(for)g (example)g(an)n(y)f(gradien)n(t)g(of)h(a)f(regular)g(scalar)f(function) -236 5120 y(can)j(b)r(e)h(added)f(to)h Fo(A)p Fq(\).)1751 5517 y(1)p eop %%Page: 2 2 2 1 bop -112 154 a Fq(The)28 b(unitary)g(groups)e(asso)r(ciated)h(to)g (the)i(self-adjoin)n(t)e(op)r(erators)f(\005)2119 124 y Fl(A)2119 176 y(j)2201 154 y Fq(are)h(the)h(magnetic)f(translations)g ([S],)h([Z].)38 b(They)-236 254 y(do)24 b(not)g(comm)n(ute)g(and)g (their)h(comp)r(osition)e(puts)i(in)n(to)f(evidence)g(the)h(follo)n (wing)e(imaginary)g(exp)r(onen)n(tial)h(of)g(the)h(\015ux)f(of)g(the) -236 353 y(magnetic)j(\014eld:)842 498 y Fo(!)894 510 y Fl(B)950 498 y Fq(\()p Fo(x;)14 b(y)s Fq(;)g Fo(q)s Fq(\))24 b(:=)f(exp)1494 381 y Fg(\032)1557 498 y Fj(\000)p Fo(i)1665 385 y Fg(Z)1710 574 y Fl()2232 498 y Fo(B)t Fq(\()p Fo(\030)t Fq(\))p Fo(d\033)s Fq(\()p Fo(\030)t Fq(\))2600 381 y Fg(\033)2679 498 y Fo(;)-236 688 y Fq(where)g Fo(<)g(q)s(;)14 b(q)f Fq(+)e Fo(x;)j(q)f Fq(+)e Fo(x)g Fq(+)g Fo(y)24 b(>)f Fq(is)h(the)g(triangle)e(de\014ned)i(b)n(y)g(the)g(p)r(oin)n(ts:) 35 b Fo(q)s(;)14 b(q)f Fq(+)e Fo(x;)j(q)f Fq(+)e Fo(x)g Fq(+)g Fo(y)s Fq(.)34 b(W)-7 b(e)24 b(consider)e(this)i(function)g(as) -236 788 y(a)g(mapping)h Fo(X)20 b Fj(\002)13 b Fo(X)29 b Fj(3)24 b Fq(\()p Fo(x;)14 b(y)s Fq(\))23 b Fj(7!)g Fo(!)886 800 y Fl(B)943 788 y Fq(\()p Fo(x;)14 b(y)s Fq(;)g Fo(:)p Fq(\))24 b Fj(2)f Fo(C)1356 800 y Fl(u)1400 788 y Fq(\()p Fo(X)7 b Fq(;)14 b Fm(T)1601 758 y Fk(1)1637 788 y Fq(\))26 b(\(where)e Fm(T)2020 758 y Fk(1)2082 788 y Fq(is)h(the)h(m)n(ultiplicativ)n(e)f(group)e(of)i(complex)g(n)n (um)n(b)r(ers)-236 888 y(of)i(mo)r(dulus)h(one\))g(and)f(observ)n(e)f (that:)836 1050 y Fo(!)888 1062 y Fl(B)945 1050 y Fq(\()p Fo(x;)14 b(y)s Fq(\))p Fo(!)1189 1062 y Fl(B)1246 1050 y Fq(\()p Fo(x)19 b Fq(+)f Fo(y)s(;)c(z)t Fq(\))22 b(=)h Fo(\022)r Fq(\()p Fo(x)p Fq(\))p Fj(f)p Fo(!)1939 1062 y Fl(B)1997 1050 y Fq(\()p Fo(y)s(;)14 b(z)t Fq(\))p Fj(g)p Fo(!)2279 1062 y Fl(B)2334 1050 y Fq(\()p Fo(x;)g(y)22 b Fq(+)c Fo(z)t Fq(\))p Fo(;)915 b Fq(\(2.1\))1323 1174 y Fo(!)1375 1186 y Fl(B)1431 1174 y Fq(\()p Fo(x;)14 b Fq(0\))24 b(=)e Fo(!)1784 1186 y Fl(B)1841 1174 y Fq(\(0)p Fo(;)14 b(x)p Fq(\))24 b(=)e(1)p Fo(;)1402 b Fq(\(2.2\))1313 1299 y Fo(!)s Fq(\()p Fo(x;)14 b Fj(\000)p Fo(x)p Fq(\))24 b(=)e Fo(!)s Fq(\()p Fj(\000)p 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(ergence)-236 1860 y(on)e Fe(Aut)p Fq(\()p Fj(A)p Fq(\)\))i(and)d Fo(!)h Fq(:)d Fo(X)f Fj(\002)15 b Fo(X)29 b Fj(!)23 b(U)8 b Fq(\()p Fj(A)p Fq(\))28 b(is)e(a)f(con)n(tin)n(uous)g(mapping)h(in)n (to)g(the)g(group)f(of)h(unitary)g(elemen)n(ts)g(of)g Fj(A)g Fq(satisfying)-236 1959 y(conditions)k(\(2.1,)g(2.2\).)45 b(W)-7 b(e)31 b(sa)n(y)e(that)i Fo(!)i Fq(is)d(a)g Fd(\022)r Ff(-2-co)s(cycle)p Fq(.)45 b(If)31 b(the)g Fo(\022)r Fq(-2-co)r(cycle)e Fo(!)k Fq(also)c(satis\014es)h(\(2.3\))g(w)n(e)g(sa) n(y)f(that)i(w)n(e)-236 2059 y(ha)n(v)n(e)26 b(a)h Ff(Magnetic)32 b(Quan)m(tum)f(Dynamical)g(System)p Fq(,)c(shortened)g(a)g(MQDS.)-236 2159 y Ff(De\014nition)d(2.2.)33 b Fq(A)22 b Ff(co)m(v)-5 b(arian)m(t)26 b(represen)m(tation)c Fq(of)f(a)g(TQDS)h Fj(f)p Fo(X)r(;)14 b Fj(A)p Fo(;)g(\022)r(;)g(!)s Fj(g)20 b Fq(is)i(a)f(triple)g Fj(fH)q Fo(;)14 b(U;)g(\032)p Fj(g)20 b Fq(where)h Fj(H)i Fq(is)e(a)g(Hilb)r(ert)-236 2258 y(space,)27 b Fo(U)33 b Fq(:)24 b Fo(X)30 b Fj(!)24 b(U)8 b Fq(\()p Fj(H)q 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Ff(The)k(Twisted)f(Con)m(v)m(olution)h(Algebra.)41 b Fq(Let)32 b(us)f(\014x)h(a)g(Haar)f(measure)f Fo(dx)j Fq(on)f Fo(X)38 b Fq(and)32 b(consider)f(the)h(complex)f(linear)-29 2819 y(space)22 b Fo(L)245 2789 y Fk(1)282 2819 y Fq(\()p Fo(X)7 b Fq(;)14 b Fj(A)p Fq(\))22 b(of)g(Bo)r(c)n(hner)f(in)n (tegrable)g(v)n(ector)g(functions)i(on)f Fo(X)29 b Fq(with)22 b(v)-5 b(alues)22 b(in)h Fj(A)p Fq(,)g(with)g(the)g Fo(L)3150 2789 y Fk(1)3187 2819 y Fq(-norm)e Fj(k)p Fo(f)9 b Fj(k)3560 2831 y Fk(1)p Fl(;)p Fn(A)3692 2819 y Fq(:=)-29 2852 y Fg(R)11 2948 y Fl(X)87 2918 y Fo(dx)p Fj(k)p Fo(f)g Fq(\()p Fo(x)p Fq(\))p Fj(k)422 2930 y Fn(A)480 2918 y Fo(:)28 b Fq(W)-7 b(e)28 b(de\014ne)g(the)g(comp)r(osition)f(giv)n (en)g(b)n(y)g(the)h(follo)n(wing)e('t)n(wisted)i(con)n(v)n(olution':) 373 3139 y(\()p Fo(f)g(?)516 3105 y Fl(!)516 3160 y(\022)582 3139 y Fo(g)s Fq(\)\()p Fo(x)p Fq(\))c(:=)902 3026 y Fg(Z)948 3215 y Fl(X)1025 3139 y Fo(dy)1126 3022 y Fg(\032)1188 3139 y Fo(\022)1244 3022 y Fg(\022)1315 3083 y Fo(y)d Fj(\000)d Fo(x)p 1315 3120 193 4 v 1390 3196 a Fq(2)1517 3022 y Fg(\023)1592 3139 y Fq([)p Fo(f)9 b Fq(\()p Fo(y)s Fq(\)])1796 3022 y Fg(\033)1872 3047 y(n)1928 3139 y Fo(\022)1983 3047 y Fg(\020)2042 3083 y Fo(y)p 2042 3120 44 4 v 2043 3196 a Fq(2)2096 3047 y Fg(\021)2159 3139 y Fq([)p Fo(g)s Fq(\()p Fo(x)19 b Fj(\000)f Fo(y)s Fq(\)])2505 3047 y Fg(o)c(n)2630 3139 y Fo(\022)2685 3047 y Fg(\020)2734 3139 y Fj(\000)2809 3083 y Fo(x)p 2809 3120 48 4 v 2812 3196 a Fq(2)2866 3047 y Fg(\021)2930 3139 y Fo(!)s Fq(\()p Fo(y)s(;)g(x)k Fj(\000)g Fo(y)s Fq(\))3322 3047 y Fg(o)-29 3357 y Fq(and)29 b(an)f(in)n(v)n(olution)g(de\014ned)h(b)n(y)g Fo(f)1095 3326 y Fl(?)1132 3357 y Fq(\()p Fo(x)p Fq(\))d(:=)f Fo(f)9 b Fq(\()p Fj(\000)p Fo(x)p Fq(\))1608 3326 y Fn(\003)1646 3357 y Fo(:)29 b Fq(W)-7 b(e)29 b(shall)f(denote)h(the)g(structure)f (th)n(us)h(de\014ned)g(b)n(y)f Fo(L)3451 3326 y Fk(1)3451 3380 y Fl(\022)3488 3357 y Fq(\()p Fo(X)7 b Fq(;)14 b Fj(A)p Fq(\))3731 3326 y Fl(!)-29 3456 y Fq(and)24 b(call)g(it)h Fp(the)i(twiste)l(d)g(c)l(onvolution)g(algebr)l(a)f Fq(asso)r(ciated)d (to)h(the)g(TQDS;)h(it)g(is)f(not)g(di\016cult)h(to)f(v)n(erify)g(that) g(it)h(forms)-29 3556 y(a)i(Banac)n(h)335 3526 y Fn(\003)373 3556 y Fq(-algebra.)35 b(Let)27 b(us)g(observ)n(e)e(that)i(w)n(e)f(use) h(an)g(isomorphic)e(form)i(of)g(the)g(usual)f(t)n(wisted)h(crossed)f (pro)r(duct,)-29 3655 y(that)i(in)g(the)g(absence)f(of)g(the)h (magnetic)g(\014eld)g(leads)f(to)g(the)h(W)-7 b(eyl)28 b(form)f(of)h(the)g(sym)n(b)r(olic)f(calculus.)-29 3784 y(Giv)n(en)d(a)f(Banac)n(h)566 3754 y Fn(\003)604 3784 y Fq(-algebra)f Fj(B)s Fq(,)i(an)n(y)f Fo(C)1242 3754 y Fn(\003)1280 3784 y Fq(-seminorm)g(on)g(it)i(is)e(b)r(ounded)i(b)n(y) e(the)h(giv)n(en)f(norm,)h(so)g(that)g(the)g(suprem)n(um)-29 3884 y(of)f(these)g Fo(C)334 3854 y Fn(\003)372 3884 y Fq(-seminorms)f(exists)g(and)h(satis\014es)f(the)h(same)g(b)r(ound.) 35 b(W)-7 b(e)23 b(call)g(the)g Fo(C)2606 3854 y Fn(\003)2644 3884 y Fq(-algebra)e(obtained)i(b)n(y)f(separation)-29 3983 y(and)d(completion)g(its)g Fp(enveloping)k Fo(C)1111 3953 y Fn(\003)1150 3983 y Fp(-algebr)l(a)d Fq([D],)h(denoted)e(b)n(y)g Fd(C)2088 3947 y Fn(\003)2126 3983 y Fq([)p Fj(B)s Fq(];)i(let)f Fi(j)j Fq(:)g Fj(B)i(!)e Fd(C)2738 3947 y Fn(\003)2776 3983 y Fq([)p Fj(B)s Fq(])18 b(b)r(e)i(the)f(natural)f(morphism)-29 4083 y(th)n(us)32 b(obtained.)47 b(Then)32 b Fd(C)836 4047 y Fn(\003)888 4016 y Fg(\002)923 4083 y Fo(L)980 4053 y Fk(1)980 4107 y Fl(\022)1017 4083 y Fq(\()p Fo(X)7 b Fq(;)14 b Fj(A)p Fq(\))1260 4053 y Fl(!)1308 4016 y Fg(\003)1372 4083 y Fj(\021)29 b(A)21 b Fm(o)1618 4053 y Fl(!)1618 4107 y(\022)1687 4083 y Fo(X)37 b Fq(is)31 b(called)g Ff(the)36 b(t)m(wisted)g(crossed)g(pro)s(duct)g(of)g Fj(A)g Ff(b)m(y)h Fo(X)7 b Fq(.)-29 4183 y(In)39 b(this)g(case)f(the)h (application)e Fi(j)i Fq(is)g(injectiv)n(e)g(so)f(that)g Fo(L)1897 4153 y Fk(1)1897 4206 y Fl(\022)1934 4183 y Fq(\()p Fo(X)7 b Fq(;)14 b Fj(A)p Fq(\))2177 4153 y Fl(!)2264 4183 y Fq(is)39 b(isomorphic)e(to)h(a)h(dense)3220 4153 y Fn(\003)3258 4183 y Fq(-subalgebra)e(of)-29 4282 y Fj(A)19 b Fm(o)121 4252 y Fl(!)121 4306 y(\022)187 4282 y Fo(X)7 b Fq(.)36 b(Let)28 b(us)g(observ)n(e)d(that)j(in)g(the)g (literature)f(there)g(are)g(equiv)-5 b(alen)n(t)27 b(de\014nitions)h (of)g(this)f(structure)g(but)i(w)n(e)e(do)-29 4382 y(not)g(w)n(an)n(t)f (to)h(insist)g(up)r(on)g(this)g(p)r(oin)n(t.)37 b(It)27 b(is)g(kno)n(wn)f(that)h(the)h(non-degenerate)d(represen)n(tations)g (of)h Fj(A)18 b Fm(o)3406 4352 y Fl(!)3406 4405 y(\022)3470 4382 y Fo(X)33 b Fq(are)26 b(in)-29 4482 y(a)e(one-to-one)e(corresp)r (ondence)g(with)i(the)h(co)n(v)-5 b(arian)n(t)22 b(represen)n(tations)f (of)j(the)g(TQDS)g Fj(f)p Fo(X)r(;)14 b Fj(A)p Fo(;)g(\022)r(;)g(!)s Fj(g)p Fq(.)35 b(F)-7 b(or)23 b(a)h(co)n(v)-5 b(arian)n(t)-29 4581 y(represen)n(tation)26 b Fj(fH)q Fo(;)14 b(U;)g(\032)p Fj(g)25 b Fq(w)n(e)h(denote)h(b)n(y)g Fo(\032)17 b Fm(o)g Fo(U)36 b Fq(the)27 b(asso)r(ciated)f(represen)n(tation)f(of)i(the)g(t) n(wisted)g(crossed)f(pro)r(duct)-29 4681 y(and)i(w)n(e)f(ha)n(v)n(e:) 775 4802 y(\()p Fo(\032)18 b Fm(o)g Fo(U)9 b Fq(\)\()p Fo(f)g Fq(\))23 b(:=)1297 4689 y Fg(Z)1343 4878 y Fl(X)1420 4802 y Fo(dx)h(\032)1591 4710 y Fg(\020)1640 4802 y Fo(\022)r Fq(\()1723 4746 y Fo(x)p 1723 4783 V 1726 4859 a Fq(2)1781 4802 y(\))p Fo(f)9 b Fq(\()p Fo(x)p Fq(\))1974 4710 y Fg(\021)2039 4802 y Fo(U)g Fq(\()p Fo(x)p Fq(\))p Fo(;)180 b Fj(8)p Fo(f)31 b Fj(2)23 b Fo(L)2673 4768 y Fk(1)2710 4802 y Fq(\()p Fo(X)7 b Fq(;)14 b Fj(A)p Fq(\))p Fo(:)-29 5010 y Ff(Prop)s(osition)31 b(2.3.)36 b Fp(F)-6 b(or)30 b(any)g(TQDS:)45 5168 y(1.)42 b(The)37 b(r)l(epr)l(esentation)g Fo(U)43 b Fq(:)35 b Fo(X)42 b Fj(!)35 b(U)8 b Fq(\()p Fj(H)q Fq(\))37 b Fp(induc)l(es)f(a)h(line)l(ar)g(c)l(ontr)l(action) 2572 5147 y Fq(~)2558 5168 y Fo(U)44 b Fq(:)35 b Fo(L)2774 5138 y Fk(1)2810 5168 y Fq(\()p Fo(X)7 b Fq(\))35 b Fj(!)g(B)s Fq(\()p Fj(H)q Fq(\))p Fp(,)j(with)f Fo(L)3603 5138 y Fk(1)3640 5168 y Fq(\()p Fo(X)7 b Fq(\))154 5268 y Fp(c)l(onsider)l(e)l (d)31 b(as)f(a)g(c)l(omplex)g(Banach)i(sp)l(ac)l(e)1578 5247 y Fq(~)1563 5268 y Fo(U)9 b Fq(\()p Fo(\036)p Fq(\))24 b(:=)1877 5201 y Fg(R)1916 5297 y Fl(X)1993 5268 y Fo(dx)30 b(\036)p Fq(\()p Fo(x)p Fq(\))p Fo(U)9 b Fq(\()p Fo(x)p Fq(\))p Fp(;)1751 5517 y Fq(2)p eop %%Page: 3 3 3 2 bop 45 154 a Fp(2.)42 b(The)36 b(image)g Fq(\()p Fo(\032)22 b Fm(o)g Fo(U)9 b Fq(\))p Fj(fA)22 b Fm(o)1053 124 y Fl(!)1053 177 y(\022)1123 154 y Fo(X)7 b Fj(g)34 b Fp(is)h(e)l(qual)g(to)g(the)g(norm)g(closur)l(e)g(of)h(the)f(line)l (ar)h(sp)l(ac)l(e)f(gener)l(ate)l(d)g(by)h(the)f(set)154 262 y Fj(f)p Fo(\032)p Fq(\()p Fo(a)p Fq(\))361 241 y(~)347 262 y Fo(U)9 b Fq(\()p Fo(\036)p Fq(\))57 b Fj(j)f(8)p Fo(a)26 b Fj(2)j(A)p Fo(;)42 b Fj(8)p Fo(\036)27 b Fj(2)i Fo(L)1258 232 y Fk(1)1295 262 y Fq(\()p Fo(X)7 b Fq(\))p Fj(g)p Fp(.)46 b(The)33 b(statement)f(r)l(emains)g(true)g(for)h(the)g (line)l(ar)g(sp)l(ac)l(e)g(gener)l(ate)l(d)g(by)154 362 y(the)d(set)f(of)i(pr)l(o)l(ducts)f(taken)f(in)h(the)g(r)l(everse)l(d)h (or)l(der.)-29 524 y Fq(The)d(\014rst)f(p)r(oin)n(t)h(is)f(ob)n(vious;) f(the)i(second)f(one)g(follo)n(ws)g(b)n(y)g(rephrasing)e(the)j(pro)r (of)f(in)h([GI1)o(])g(for)f(the)h(un)n(t)n(wisted)f(case.)-236 696 y Ff(The)32 b(Sc)m(hr\177)-48 b(odinger)32 b(Represen)m(tation.)40 b Fq(The)26 b(argumen)n(ts)f(b)r(elo)n(w)h(ma)n(y)f(b)r(e)h (generalized)f(to)h(an)n(y)f(TQDS)h(due)g(to)g(the)g(triv-)-29 795 y(ialit)n(y)31 b(of)g(the)g Fo(\022)r Fq(-2-cohomology)d(group)i (discussed)g(in)h([GI2],)h(but)f(for)g(space)f(reasons)f(w)n(e)h(shall) h(concen)n(trate)f(on)g(the)-29 895 y(ph)n(ysical)d(case)g(of)g(in)n (terest.)-29 1025 y(T)-7 b(o)27 b Fp(a)i(quantum)f(p)l(article)i(in)f (a)g(magnetic)g(\014eld)f Fq(one)e(can)g(asso)r(ciate)f(in)i(a)f (natural)g(w)n(a)n(y)g(a)g(MQDS)h(and,)g(once)f(a)g(v)n(ector)-29 1125 y(p)r(oten)n(tial)21 b Fo(A)f Fq(is)h(c)n(hosen,)g(a)f(co)n(v)-5 b(arian)n(t)18 b(represen)n(tation)h(of)h(it.)35 b(In)21 b(fact)f(one)g(tak)n(es:)33 b Fo(X)c Fq(:=)23 b Fm(R)2808 1095 y Fl(n)2880 1125 y Fq(the)d(group)g(of)g(translations;)-29 1225 y Fj(A)25 b Fq(:=)f Fo(B)t(C)300 1237 y Fl(u)344 1225 y Fq(\()p Fm(R)430 1195 y Fl(n)481 1225 y Fq(\))29 b(the)f(algebra)f(of)h(observ)-5 b(ables)27 b(asso)r(ciated)g(to)h(the) h(p)r(osition)f(op)r(erator;)f(\()p Fo(\022)r Fq(\()p Fo(x)p Fq(\))p Fo(a)p Fq(\)\()p Fo(y)s Fq(\))f(:=)e Fo(a)p Fq(\()p Fo(y)e Fq(+)c Fo(x)p Fq(\))29 b(the)-29 1324 y(standard)f(represen)n(tation)f(of)h(translations)f(on)i(this)g (algebra;)e Fo(!)g Fq(:=)d Fo(!)2250 1336 y Fl(B)2336 1324 y Fq(the)29 b Fo(\022)r Fq(-2-co)r(cycle)e(de\014ned)i(b)n(y)f (the)h(magnetic)-29 1424 y(\014eld;)24 b Fj(H)g Fq(:=)f Fo(L)432 1394 y Fk(2)468 1424 y Fq(\()p Fm(R)555 1394 y Fl(n)606 1424 y Fq(\),)g(\()p Fo(\032)p Fq(\()p Fo(a)p Fq(\))p Fo(u)p Fq(\)\()p Fo(x)p Fq(\))h Fj(\021)f Fq(\()p Fo(a)p Fq(\()p Fo(Q)p Fq(\))p Fo(u)p Fq(\)\()p Fo(x)p Fq(\))h(:=)f Fo(a)p Fq(\()p Fo(x)p Fq(\))p Fo(u)p Fq(\()p Fo(x)p Fq(\),)h Fo(U)2120 1436 y Fl(A)2174 1424 y Fq(\()p Fo(x)p Fq(\))g(=)f(\003)2455 1436 y Fl(A)2508 1424 y Fq(\()p Fo(x)p Fq(\))p Fo(T)12 b Fq(\()p Fo(x)p Fq(\))23 b(\(the)f(magnetic)e(translation\),)-29 1524 y(where:)34 b(\()p Fo(T)12 b Fq(\()p Fo(x)p Fq(\))p Fo(u)p Fq(\)\()p Fo(y)s Fq(\))24 b(:=)f Fo(u)p Fq(\()p Fo(y)12 b Fq(+)d Fo(x)p Fq(\),)24 b(\003)1159 1536 y Fl(A)1213 1524 y Fq(\()p Fo(x)p Fq(\))g(:=)f(exp)p Fj(f\000)p Fo(i)p Fq(\000)1774 1536 y Fl(A)1826 1524 y Fq([)p Fo(Q;)14 b(Q)9 b Fq(+)g Fo(x)p Fq(])p Fj(g)23 b(2)g(B)s Fq(\()p Fj(H)q Fq(\))g(with)h(\000)2767 1536 y Fl(A)2821 1524 y Fq([)p Fo(x;)14 b(y)s Fq(])23 b(:=)3129 1457 y Fg(R)3168 1553 y Fk([)p Fl(x;y)r Fk(])3317 1524 y Fo(A)p Fq(\()p Fo(\030)t Fq(\))9 b Fj(\001)g Fo(d\030)29 b Fq(\(the)-29 1638 y(circulation)j(of)h Fo(A)g Fq(along)f(the)h(line)g (segmen)n(t)f([)p Fo(x;)14 b(y)s Fq(]\).)54 b(W)-7 b(e)33 b(call)f(this)h(co)n(v)-5 b(arian)n(t)31 b(represen)n(tation)g Ff(the)38 b(Sc)m(hr\177)-48 b(odinger)-29 1738 y(represen)m(tation)32 b(with)g(v)m(ector)h(p)s(oten)m(tial)e Fo(A)p Fq(.)37 b(W)-7 b(e)28 b(denote)f Fi(R)2114 1750 y Fl(A)2191 1738 y Fj(\021)c Fo(\032)18 b Fm(o)g Fo(U)2480 1750 y Fl(A)2562 1738 y Fq(and)27 b(w)n(e)g(ha)n(v)n(e)1079 1948 y Fi(R)1148 1960 y Fl(A)1201 1948 y Fq(\()p Fo(f)9 b Fq(\))p Fo(u)23 b Fq(:=)1497 1835 y Fg(Z)1543 2023 y Fl(X)1620 1948 y Fo(dxf)1774 1855 y Fg(\020)1824 1948 y Fo(x;)14 b(Q)k Fq(+)2085 1891 y Fo(x)p 2085 1928 48 4 v 2088 2004 a Fq(2)2142 1855 y Fg(\021)2206 1948 y Fq(\003)2264 1960 y Fl(A)2318 1948 y Fq(\()p Fo(x)p Fq(\))p Fo(T)12 b Fq(\()p Fo(x)p Fq(\))p Fo(u:)937 b Fq(\(2.4\))-29 2161 y(It)26 b(is)g(easy)e(to)i(v)n(erify)e(that)i(this)g(represen)n(tation)e(is)h (injectiv)n(e)h(and)f(the)h(follo)n(wing)f('gauge)f(co)n(v)-5 b(ariance')23 b(relation)i(holds:)890 2333 y Fo(\032)p Fq(\()p Fo(e)1004 2299 y Fl(i\025)1071 2333 y Fq(\))p 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bop -236 154 a Fq(con)n(tin)n(ue)27 b(to)i(denote)f(b)n(y)g Fo(?)625 117 y Fl(!)667 125 y Fh(B)625 179 y Fl(\022)747 154 y Fq(the)h(asso)r(ciativ)n(e)d(op)r(eration)i(on)f(this)i(larger)d (algebra.)37 b(By)28 b(the)h(same)e(form)n(ula)h(\(3.5\))f(w)n(e)h(ma)n (y)-236 254 y(extend)i(the)h Fo(?)224 217 y Fl(!)266 225 y Fh(B)224 279 y Fl(\022)318 254 y Fq(-op)r(eration)e(to)h(the)g (pair)g(\()p Fi(h)1218 266 y Fl(z)1257 254 y Fo(;)14 b(f)9 b Fq(\))29 b(with)i Fo(f)36 b Fj(2)27 b Fo(L)1813 223 y Fk(1)1813 277 y Fl(\022)1850 254 y Fq(\()p Fo(X)7 b Fq(;)14 b Fj(A)p Fq(\))2093 223 y Fl(!)2135 231 y Fh(B)2189 254 y Fq(,)30 b(under)g(some)g(regularit)n(y)e(assumptions)h(on)h(the) -236 353 y(magnetic)d(\014eld)h Fo(B)t Fq(.)-236 453 y Ff(Hyp)s(othesis)22 b(3.4.)33 b Fe(The)20 b(magnetic)h(\014eld)f(is)h (giv)n(en)f(b)n(y)g(a)g(b)r(ounded)h(matrix-v)-5 b(alued)19 b Fo(C)2496 423 y Fk(2)2534 453 y Fe(-function)i(with)g(b)r(ounded)g (deriv)-5 b(ativ)n(es)-236 552 y(up)28 b(to)f(second)g(order.)-112 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Fq(\()p Fm(R)2689 821 y Fl(n)2740 851 y Fq(\))p Fp(,)31 b(satisfying)f(the)g(homo)l(geneity) -236 951 y(c)l(ondition)c Fo(f)163 963 y Fl(\033)208 951 y Fq(\()p Fo(p)p Fq(\))d(=)g Fo(\033)475 921 y Fl(a)515 951 y Fo(f)556 963 y Fk(1)593 951 y Fq(\()p Fo(p=\033)759 921 y Fl(b)792 951 y Fq(\))p Fp(.)38 b(Then)26 b Fj(F)8 b Fo(f)1208 963 y Fl(\033)1275 951 y Fj(2)24 b Fo(L)1411 921 y Fk(1)1447 951 y Fq(\()p Fm(R)1534 921 y Fl(n)1585 951 y Fq(\))h Fp(for)i(any)e Fo(\033)k Fp(and)d(we)g(have)g(the)g (estimate)f Fj(kF)8 b Fo(f)3074 963 y Fl(\033)3118 951 y Fj(k)3160 966 y Fl(L)3206 949 y Fc(1)3265 951 y Fq(=)23 b Fo(\033)3403 921 y Fl(a)3443 951 y Fj(kF)8 b Fo(f)3594 963 y Fk(1)3630 951 y Fj(k)3672 966 y Fl(L)3718 949 y Fc(1)3754 951 y Fp(.)-112 1051 y Fq(This)25 b(lemma)h(follo)n(ws)e (easily)g(from)h(Prop)r(osition)e(1.3.6)h(of)i([ABG])f(and)g(the)h (fact)f(that)g(for)g Fo(n)e(<)g Fq(2)p Fo(m)h Fq(an)n(y)h(sym)n(b)r(ol) f(of)i(class)-236 1150 y Fo(S)-180 1120 y Fn(\000)p Fl(m)-38 1150 y Fq(is)h(in)g Fo(L)198 1120 y Fk(2)235 1150 y Fq(,)g(so)g(that)g (its)h(F)-7 b(ourier)26 b(transform)g(is)h(also)f(in)h Fo(L)1752 1120 y Fk(2)1812 1150 y Fj(\032)c Fo(L)1957 1120 y Fk(1)1957 1174 y(lo)r(c)2043 1150 y Fq(.)37 b(A)28 b(straigh)n(tforw)n(ard)c(computation)j(leads)f(from)h(the)-236 1250 y(homogeneit)n(y)f(of)i Fo(f)384 1262 y Fl(\033)456 1250 y Fq(to)f(that)h(of)g(the)g Fo(L)1032 1220 y Fk(1)1068 1250 y Fq(-norm)f(of)h(its)f(F)-7 b(ourier)27 b(transform.)-112 1349 y(Using)h(the)g(Lemma)f(ab)r(o)n(v)n(e)f(for)h Fj(f)p Fo(x)1012 1319 y Fl(\013)1060 1349 y Fo(@)1109 1319 y Fl(\014)1153 1349 y Fo(G)1218 1361 y Fl(i\033)1287 1349 y Fj(g)1329 1361 y Fl(\033)1400 1349 y Fq(\(with)i Fj(j)p Fo(\013)p Fj(j)23 b(\024)g Fq(1)p Fo(;)41 b Fj(j)p Fo(\014)t Fj(j)24 b(\024)e Fq(1\))28 b(w)n(e)f(get:)986 1532 y Fj(k)p Fo(x)1075 1498 y Fl(\013)1122 1532 y Fo(@)1171 1498 y Fl(\014)1215 1532 y Fo(G)1280 1544 y Fl(i\033)1349 1532 y Fj(k)1391 1547 y Fl(L)1437 1530 y Fc(1)1495 1532 y Fj(\024)c Fo(\033)1633 1498 y Fn(\000)p Fk(1)p Fn(\000)p Fk(\()p Fn(j)p Fl(\013)p Fn(\000j)p Fl(\014)s Fn(j)p Fk(\))p Fl(=)p Fk(2)2088 1532 y Fj(k)p Fo(x)2177 1498 y Fl(\013)2225 1532 y Fo(@)2274 1498 y Fl(\014)2318 1532 y Fo(G)2383 1544 y Fl(i)2411 1532 y Fj(k)2453 1547 y Fl(L)2499 1530 y Fc(1)2535 1532 y Fo(:)1051 b Fq(\(3.7\))-112 1715 y(F)-7 b(or)33 b(space)g(reasons)e(w)n(e)i(shall)g(con)n(tin)n(ue) g(the)g(argumen)n(ts)f(in)i(a)f(minimal)g(v)n(ersion,)h(suitable)f(for) g(our)f(explicit)i(form)f(of)-236 1814 y Fi(h)h Fq(and)g(lea)n(v)n(e)e (the)j(general)d(problem)i(of)f(extending)h(the)g Fo(?)1670 1777 y Fl(!)1712 1785 y Fh(B)1670 1839 y Fl(\022)1765 1814 y Fq(-op)r(eration)f(for)g(a)g(forthcoming)h(pap)r(er.)55 b(W)-7 b(e)34 b(shall)g(consider)-236 1914 y(a)29 b(molli\014er)h (family)g Fj(f)p Fi(h)502 1884 y Fl(\017)502 1935 y(z)540 1914 y Fj(g)582 1926 y Fl(\017>)p Fk(0)728 1914 y Fq(for)f Fi(h)f Fj(2)f(E)1061 1884 y Fn(0)1054 1935 y Fk(2)1091 1914 y Fq(\()p Fo(X)7 b Fq(\))30 b(\(the)g(space)g(of)g(second)f(order) f(compactly)i(supp)r(orted)f(distributions)h(on)g Fo(X)7 b Fq(\).)-236 2014 y(Then)28 b(for)f(an)n(y)f Fo(g)g Fj(2)d Fo(L)465 1984 y Fk(1)502 2014 y Fq(\()p Fo(X)7 b Fq(\))28 b(w)n(e)f(de\014ne)h(for)f Fo(')c Fj(2)h(S)6 b Fq(\()p Fo(X)h Fq(\):)1004 2196 y Fo(<)23 b Fi(h)1135 2208 y Fl(z)1192 2196 y Fo(?)1234 2159 y Fl(!)1276 2167 y Fh(B)1234 2221 y Fl(\022)1347 2196 y Fo(g)s(;)14 b(')23 b(>)p Fq(:=)28 b(lim)1679 2247 y Fl(\017)p Fn(!)p Fk(0)1829 2196 y Fo(<<)22 b Fi(h)2024 2165 y Fl(\017)2024 2216 y(z)2063 2196 y Fo(;)14 b Fq(\011)2165 2165 y Fl(B)2165 2216 y(g)2222 2196 y Fq(\()p Fo(')p Fq(\))24 b Fo(>>;)1004 2332 y(<)f(g)e(?)1195 2295 y Fl(!)1237 2303 y Fh(B)1195 2357 y Fl(\022)1308 2332 y Fi(h)1351 2344 y Fl(z)1390 2332 y Fo(;)14 b(')23 b(>)p Fq(:=)28 b(lim)1679 2383 y Fl(\017)p Fn(!)p Fk(0)1829 2332 y Fo(<<)22 b Fi(h)2024 2301 y Fl(\017)2024 2352 y(z)2063 2332 y Fo(;)2111 2311 y Fq(~)2100 2332 y(\011)2165 2301 y Fl(B)2165 2352 y(g)2222 2332 y Fq(\()p Fo(')p Fq(\))i Fo(>>)e(:)-236 2540 y Fq(F)-7 b(or)27 b Fo(g)e Fq(=)e Fo(G)131 2552 y Fl(z)197 2540 y Fq(w)n(e)k(ha)n(v)n(e)1014 2634 y(lim)1008 2686 y Fl(\017)p Fn(!)p Fk(0)1158 2634 y Fo(<<)22 b Fi(h)1353 2604 y Fl(\017)1353 2655 y(z)1392 2634 y Fo(;)14 b Fq(\011)1494 2604 y Fl(B)1494 2655 y(g)1551 2634 y Fq(\()p Fo(')p Fq(\))23 b Fo(>>)p Fq(=)f Fo(')p Fq(\(0\)+)h Fo(<)g(\026)2295 2646 y Fl(z)2333 2634 y Fo(;)14 b(')23 b(>;)1014 2770 y Fq(lim)1008 2822 y Fl(\017)p Fn(!)p Fk(0)1158 2770 y Fo(<<)f Fi(h)1353 2740 y Fl(\017)1353 2791 y(z)1392 2770 y Fo(;)1440 2750 y Fq(~)1429 2770 y(\011)1494 2740 y Fl(B)1494 2791 y(g)1551 2770 y Fq(\()p Fo(')p Fq(\))h Fo(>>)p Fq(=)f Fo(')p Fq(\(0\)+)h Fo(<)g(\027)2286 2782 y Fl(z)2324 2770 y Fo(;)14 b(')23 b(>;)-236 2946 y Fq(where)409 3075 y Fo(\026)459 3087 y Fl(z)498 3075 y Fq(\()p Fo(x)p Fq(\))h(:=)e Fj(\000)14 b Fq(\()p Fj(r)p Fo(G)988 3087 y Fl(z)1027 3075 y Fq(\))g(\()p Fo(x)p Fq(\))19 b Fj(\001)1245 2982 y Fg(h\020)1334 3075 y Fj(r)1352 3126 y Fk(1)1421 3075 y Fj(\000)f(r)1522 3126 y Fk(2)1574 2982 y Fg(\021)1637 3075 y Fo(!)1689 3087 y Fl(B)1746 2982 y Fg(i)1799 3075 y Fq(\(0)p Fo(;)c(x)p Fq(\))19 b(+)f Fo(G)2156 3087 y Fl(z)2194 3075 y Fq(\()p Fo(x)p Fq(\))2319 2957 y Fg(\024)2364 2982 y(\020)2414 3075 y Fj(r)2432 3126 y Fk(1)2501 3075 y Fj(\000)g(r)2602 3126 y Fk(2)2653 2982 y Fg(\021)2703 3000 y Fk(2)2754 3075 y Fo(!)2806 3087 y Fl(B)2863 2957 y Fg(\025)2920 3075 y Fq(\(0)p Fo(;)c(x)p Fq(\))p Fo(;)425 3373 y(\027)466 3385 y Fl(z)505 3373 y Fq(\()p Fo(x)p Fq(\))24 b(:=)e Fj(\000)14 b Fq(\()p Fj(r)p Fo(G)995 3385 y Fl(z)1034 3373 y Fq(\))g(\()p Fo(x)p Fq(\))19 b Fj(\001)1252 3281 y Fg(h\020)1341 3373 y Fj(r)1359 3425 y Fk(2)1428 3373 y Fj(\000)f(r)1529 3425 y Fk(1)1581 3281 y Fg(\021)1644 3373 y Fo(!)1696 3385 y Fl(B)1753 3281 y Fg(i)1806 3373 y Fq(\()p Fo(x;)c Fq(0\))19 b(+)f Fo(G)2163 3385 y Fl(z)2201 3373 y Fq(\()p Fo(x)p Fq(\))2326 3256 y Fg(\024)2371 3281 y(\020)2421 3373 y Fj(r)2439 3425 y Fk(2)2508 3373 y Fj(\000)g(r)2609 3425 y Fk(1)2660 3281 y Fg(\021)2710 3298 y Fk(2)2761 3373 y Fo(!)2813 3385 y Fl(B)2870 3256 y Fg(\025)2928 3373 y Fq(\()p Fo(x;)c Fq(0\))-236 3573 y(as)23 b(maps)g(in)h Fj(B)s Fq(\()p Fj(S)6 b Fq(\()p Fo(X)h Fq(\);)14 b Fj(A)p Fq(\))24 b(\(here)g Fj(r)841 3625 y Fl(j)915 3573 y Fq(signi\014es)f(the)h(gradien)n(t)f(with)h (resp)r(ect)g(to)f(the)h(j-th)h(argumen)n(t\).)35 b(F)-7 b(or)23 b(a)g(magnetic)g(\014eld)h Fo(B)-236 3711 y Fq(satisfying)i (the)h(Hyp)r(othesis)f(3.4,)h(all)f(the)h(factors)f(con)n(taining)f (deriv)-5 b(ativ)n(es)26 b(of)h(the)g Fo(\022)r Fq(-2-co)r(cycle)e Fo(!)2910 3723 y Fl(B)2993 3711 y Fq(in)i(the)g(ab)r(o)n(v)n(e)f(form)n (ulae)-236 3810 y(are)d(con)n(tin)n(uous)h(functions)h(from)f Fo(X)19 b Fj(\002)13 b Fo(X)30 b Fq(in)n(to)25 b Fo(B)t(C)1417 3822 y Fl(u)1461 3810 y Fq(\()p Fo(X)7 b Fq(\),)25 b(b)r(ounded)g(b)n (y)f(quadratic)g(p)r(olynomials)g(in)h(the)g(v)-5 b(ariable)23 b Fo(x)p Fq(.)37 b(Using)-236 3910 y(these)27 b(facts)h(and)f(the)h (estimate)g(\(3.7\))f(w)n(e)g(obtain:)-236 4009 y Ff(Lemma)32 b(3.6.)44 b Fp(We)32 b(have)i Fi(h)689 4021 y Fl(z)747 4009 y Fo(?)789 3972 y Fl(!)831 3980 y Fh(B)789 4034 y Fl(\022)904 4009 y Fo(G)969 4021 y Fl(z)1035 4009 y Fq(=)27 b Fo(\016)c Fq(+)d Fo(\026)1322 4021 y Fl(z)1360 4009 y Fp(,)33 b Fo(G)1483 4021 y Fl(z)1542 4009 y Fo(?)1584 3972 y Fl(!)1626 3980 y Fh(B)1584 4034 y Fl(\022)1699 4009 y Fi(h)1742 4021 y Fl(z)1808 4009 y Fq(=)27 b Fo(\016)c Fq(+)d Fo(\027)2086 4021 y Fl(z)2124 4009 y Fp(.)46 b(F)-6 b(or)32 b(any)h Fo(z)d Fj(2)e Fm(C)41 b Fj(n)20 b Fm(R)38 b Fp(the)32 b(two)g(maps)h Fo(\026)3466 4021 y Fl(z)3537 4009 y Fp(and)f Fo(\027)3741 4021 y Fl(z)-236 4109 y Fp(b)l(elong)e(to)g Fo(L)171 4079 y Fk(1)171 4133 y Fl(\022)208 4109 y Fq(\()p Fo(X)7 b Fq(;)14 b Fj(A)p Fq(\))451 4079 y Fl(!)493 4087 y Fh(B)576 4109 y Fp(and)30 b(we)g(have)h(the)f (estimates:)1058 4292 y Fj(k)p Fo(\026)1150 4304 y Fn(\006)p Fl(i\033)1269 4292 y Fj(k)1311 4307 y Fl(L)1357 4290 y Fc(1)1416 4292 y Fj(\024)23 b Fo(\033)1554 4257 y Fn(\000)p Fk(1)1643 4292 y Fo(C)q(;)100 b Fj(k)p Fo(\027)1909 4304 y Fn(\006)p Fl(i\033)2028 4292 y Fj(k)2070 4307 y Fl(L)2116 4290 y Fc(1)2175 4292 y Fj(\024)22 b Fo(\033)2312 4257 y Fn(\000)p Fk(1)2402 4292 y Fo(C)q(:)-112 4474 y Fq(Let)28 b(us)g(de\014ne:)426 4736 y Fi(r)458 4748 y Fn(\006)p Fl(i\033)602 4736 y Fq(:=)22 b Fo(G)777 4748 y Fn(\006)p Fl(i\033)916 4736 y Fo(?)958 4699 y Fl(!)1000 4707 y Fh(B)958 4761 y Fl(\022)1071 4566 y Fg(8)1071 4640 y(<)1071 4790 y(:)1172 4632 y Fn(1)1145 4657 y Fg(X)1148 4834 y Fl(j)s Fk(=0)1279 4736 y Fq(\()p Fo(\026)1361 4748 y Fn(\006)p Fl(i\033)1481 4736 y Fq(\))1513 4694 y Fl(?j)1582 4566 y Fg(9)1582 4640 y(=)1582 4790 y(;)1670 4736 y Fq(;)1868 4732 y(~)1873 4736 y Fi(r)1905 4748 y Fn(\006)p Fl(i\033)2048 4736 y Fq(:=)2159 4566 y Fg(8)2159 4640 y(<)2159 4790 y(:)2260 4632 y Fn(1)2233 4657 y Fg(X)2235 4834 y Fl(j)s Fk(=0)2367 4736 y Fq(\()p Fo(\027)2440 4748 y Fn(\006)p Fl(i\033)2560 4736 y Fq(\))2592 4694 y Fl(?j)2661 4566 y Fg(9)2661 4640 y(=)2661 4790 y(;)2754 4736 y Fo(?)2796 4699 y Fl(!)2838 4707 y Fh(B)2796 4761 y Fl(\022)2909 4736 y Fo(G)2974 4748 y Fn(\006)p Fl(i\033)3094 4736 y Fo(:)492 b Fq(\(3.8\))-236 5024 y(Here)39 b Fo(f)22 4994 y Fl(?j)130 5024 y Fq(means)g(the)g Fo(?)592 4988 y Fl(!)634 4996 y Fh(B)592 5049 y Fl(\022)687 5024 y Fq(-pro)r(duct)g(of)g Fo(j)44 b Fq(factors)39 b Fo(f)9 b Fq(.)71 b(W)-7 b(e)40 b(de\014ne)g Fi(h)2099 5036 y Fn(\006)p Fl(i\033)2245 5024 y Fo(?)2287 4988 y Fl(!)2329 4996 y Fh(B)2287 5049 y Fl(\022)2408 5024 y Fi(r)2440 5036 y Fn(\006)p Fl(i\033)2600 5024 y Fq(and)2769 5020 y(~)2773 5024 y Fi(r)2805 5036 y Fn(\006)p Fl(i\033)2952 5024 y Fo(?)2994 4988 y Fl(!)3036 4996 y Fh(B)2994 5049 y Fl(\022)3115 5024 y Fi(h)3158 5036 y Fn(\006)p Fl(i\033)3317 5024 y Fq(b)n(y)f(the)h(same)-236 5124 y(regularization)19 b(metho)r(d)j(as)f(ab)r(o)n(v)n(e.)34 b(Using)21 b(the)h(asso)r (ciativit)n(y)e(prop)r(ert)n(y)g(of)i(the)g Fo(?)2377 5087 y Fl(!)2419 5095 y Fh(B)2377 5149 y Fl(\022)2472 5124 y Fq(-pro)r(duct)f(and)g(some)g(ob)n(vious)f(algebraic)-236 5224 y(manipulations,)28 b(w)n(e)g(get:)38 b Fi(h)666 5236 y Fn(\006)p Fl(i\033)805 5224 y Fo(?)847 5187 y Fl(!)889 5195 y Fh(B)847 5249 y Fl(\022)961 5224 y Fi(r)993 5236 y Fn(\006)p Fl(i\033)1138 5224 y Fq(=)23 b Fo(\016)s Fq(,)1314 5220 y(~)1318 5224 y Fi(r)1350 5236 y Fn(\006)p Fl(i\033)1490 5224 y Fo(?)1532 5187 y Fl(!)1574 5195 y Fh(B)1532 5249 y Fl(\022)1645 5224 y Fi(h)1688 5236 y Fn(\006)p Fl(i\033)1833 5224 y Fq(=)h Fo(\016)s Fq(,)2009 5220 y(~)2013 5224 y Fi(r)2045 5236 y Fn(\006)p Fl(i\033)2190 5224 y Fq(=)2274 5220 y(~)2279 5224 y Fi(r)2311 5236 y Fn(\006)p Fl(i\033)2450 5224 y Fo(?)2492 5187 y Fl(!)2534 5195 y Fh(B)2492 5249 y Fl(\022)2606 5224 y Fi(h)2649 5236 y Fn(\006)p Fl(i\033)2788 5224 y Fo(?)2830 5187 y Fl(!)2872 5195 y Fh(B)2830 5249 y Fl(\022)2944 5224 y Fi(r)2976 5236 y Fn(\006)p Fl(i\033)3120 5224 y Fq(=)g Fi(r)3241 5236 y Fn(\006)p Fl(i\033)3362 5224 y Fq(,)k Fi(r)3445 5236 y Fl(i\033)3538 5224 y Fq(=)c Fi(r)3659 5194 y Fl(?)3659 5245 y Fn(\000)p Fl(i\033)1751 5517 y Fq(4)p eop %%Page: 5 5 5 4 bop -236 154 a Fq(and)23 b Fi(r)-47 166 y Fl(i\033)32 154 y Fj(\000)11 b Fi(r)140 166 y Fn(\000)p Fl(i\033)283 154 y Fq(=)23 b(2)p Fo(i\033)s Fi(r)524 166 y Fl(i\033)592 154 y Fi(r)624 166 y Fn(\000)p Fl(i\033)744 154 y Fq(.)36 b(By)23 b(the)h(Neumann)g(series,)g(Lemma)f(3.6)g(and)h(the)g(last)f(t) n(w)n(o)g(form)n(ulae)g(ab)r(o)n(v)n(e,)g(w)n(e)g(can)g(de\014ne)-236 254 y(b)n(y)k(analytic)g(extension)g(a)g(resolv)n(en)n(t)f(function)i Fm(C)40 b Fj(n)18 b Fm(R)29 b Fj(3)23 b Fo(z)k Fj(!)c Fi(r)1809 266 y Fl(z)1870 254 y Fj(2)h Fi(C)j Fq(\(extending)h Fi(r)2472 266 y Fn(\006)p Fl(i\033)2592 254 y Fq(\).)38 b(This)27 b(giv)n(es)g(the)h(follo)n(wing:)-236 353 y Ff(Theorem)e(3.7.)36 b Fp(L)l(et)26 b Fo(B)31 b Fp(b)l(e)26 b(a)h(magnetic)g(\014eld)g(satisfying)h(our)f(Hyp)l(othesis)g(3.4.)39 b(Ther)l(e)28 b(exists)e(an)h(observable)h Fq(\010)3404 365 y Fb(h)3471 353 y Fp(a\016liate)l(d)-236 453 y(to)d Fo(B)t(C)-15 465 y Fl(u)28 453 y Fq(\()p Fo(X)7 b Fq(\))h Fm(o)241 416 y Fl(!)283 424 y Fh(B)241 478 y Fl(\022)343 453 y Fo(X)f Fp(,)26 b(such)f(that)g(for)g(the)g(extende)l(d)g(op)l(er) l(ation)h(de\014ne)l(d)f(ab)l(ove)h(we)f(have)h Fi(h)2613 465 y Fl(z)2659 453 y Fo(?)2701 416 y Fl(!)2743 424 y Fh(B)2701 478 y Fl(\022)2803 453 y Fq(\010)2863 465 y Fb(h)2904 453 y Fq(\()p Fo(r)2973 465 y Fl(z)3012 453 y Fq(\))d(=)g(\010)3215 465 y Fb(h)3255 453 y Fq(\()p Fo(r)3324 465 y Fl(z)3363 453 y Fq(\))8 b Fo(?)3445 416 y Fl(!)3487 424 y Fh(B)3445 478 y Fl(\022)3547 453 y Fi(h)3590 465 y Fl(z)3652 453 y Fq(=)22 b Fo(\016)-236 552 y Fp(with)30 b Fo(r)-19 564 y Fl(z)20 552 y Fq(\()p Fo(t)p Fq(\))23 b(:=)g(\()p Fo(t)c Fj(\000)f Fo(z)t Fq(\))487 522 y Fn(\000)p Fk(1)576 552 y Fp(.)-112 652 y Fq(It)27 b(is)g(easy)f(to)g(see)g(that)h(for)f(a)h(magnetic)f(\014eld)h (satisfying)f(Hyp)r(othesis)h(3.4)e(one)i(can)f(alw)n(a)n(ys)f(c)n(ho)r 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1120 y Fk(2)2687 1150 y Fq(\()p Fo(X)7 b Fq(\))p Fp(.)39 b(Its)29 b(closur)l(e)h Fo(H)3369 1120 y Fl(A)3452 1150 y Fp(satis\014es:)970 1399 y Fo(H)1046 1365 y Fl(A)1122 1399 y Fq(=)1250 1296 y Fl(n)1210 1321 y Fg(X)1213 1497 y Fl(j)s Fk(=1)1330 1399 y Fq(\(\005)1424 1365 y Fl(A)1424 1420 y(j)1479 1399 y Fq(\))1511 1365 y Fk(2)1548 1399 y Fo(;)184 b Fi(R)1824 1411 y Fl(A)1878 1399 y Fq(\()p Fi(r)1942 1411 y Fl(z)1980 1399 y Fq(\))24 b(=)e(\()p Fo(H)2231 1365 y Fl(A)2304 1399 y Fj(\000)c Fo(z)t Fq(\))2462 1365 y Fn(\000)p Fk(1)2551 1399 y Fo(:)-236 1666 y Ff(Conclusion.)39 b Fq(The)28 b(magnetic)f(Sc)n(hr\177)-42 b(odinger)26 b(Hamiltonian)h(de\014nes)h(an)f(observ)-5 b(able)27 b(a\016liated)g(to)g(the)h Fo(C)3189 1636 y Fn(\003)3228 1666 y Fq(-algebra)-29 1766 y Fi(R)40 1778 y Fl(A)108 1766 y Fj(f)p Fo(B)t(C)276 1778 y Fl(u)319 1766 y Fq(\()p Fo(X)7 b Fq(\))19 b Fm(o)543 1729 y Fl(!)585 1737 y Fh(B)543 1791 y Fl(\022)656 1766 y Fo(X)7 b Fj(g)o Fq(.)-236 1948 y(P)12 b(r)g(o)g(o)g(f)50 b(o)12 b(f)51 b(P)12 b(r)g(o)g(p)g(o)g(s)g(i) g(t)g(i)g(o)g(n)34 b(3.9.)64 b(Let)37 b Fo(u)i Fj(2)g Fo(C)1362 1918 y Fn(1)1356 1969 y Fk(0)1432 1948 y Fq(\()p Fo(X)7 b Fq(\))37 b(and)g Fo(v)42 b Fj(2)d Fo(L)2013 1918 y Fk(2)2049 1948 y Fq(\()p Fo(X)7 b Fq(\);)42 b(w)n(e)36 b(ha)n(v)n(e)g Fo(u)i Fj(2)h(D)r Fq(\(\005)2926 1918 y Fl(A)2926 1970 y(j)2981 1948 y Fq(\),)h Fo(u)e Fj(2)h(\\)3311 1918 y Fl(n)3311 1970 y(j)s Fk(=1)3430 1948 y Fj(D)r Fq(\(\(\005)3622 1918 y Fl(A)3622 1970 y(j)3678 1948 y Fq(\))3710 1918 y Fk(2)3747 1948 y Fq(\))-236 2059 y([LS])c(and)h Fj(\000)p Fq(\001)247 2071 y Fl(x)325 2059 y Fo(<)f(v)s(;)14 b(U)571 2029 y Fl(A)625 2059 y Fq(\()p Fo(x)p Fq(\))p Fo(u)37 b(>)p Fq(=)987 1997 y Fg(P)1074 2018 y Fl(n)1074 2084 y(j)s Fk(=1)1229 2059 y Fo(<)f(v)s(;)14 b Fq(\(\005)1504 2029 y Fl(A)1504 2081 y(j)1559 2059 y Fq(\))1591 2029 y Fk(2)1629 2059 y Fo(u)35 b(>)p Fq(.)61 b(Using)35 b(the)h(molli\014er)f(family)g(for)g Fi(h)h Fq(and)g(the)f(Dominated)-236 2174 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