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b(a)i(p)q(erio)q(dic)e(electric)e(p)q(oten)o(tial)j Fx(V)33 b Fz(=)1090 200 y(\026)1083 213 y Fx(V)f Fz(is)21 b(absolutely)f(con)o(tin)o(uous)h(in)228 271 y(the)h(case)g Fx(d)i Fz(=)g(3)e(\(see)g(also)h([9])e(for)i(generalisation)f(to)g(all) g Fx(d)i Ft(\025)g Fz(2\).)39 b(Later)228 329 y(the)22 b(Thomas')g(idea)g(w)o(as)h(used)f(b)o(y)g(L.)h(Danilo)o(v)f([2)o(])g (to)h(pro)o(v)o(e)f(the)g(absolute)228 387 y(con)o(tin)o(uit)o(y)12 b(of)i(the)g(Dirac)g(op)q(erator)i(with)e(a)g(p)q(erio)q(dic)g (electric)e(p)q(oten)o(tial.)20 b(The)228 445 y(k)o(ey)c(p)q(oin)o(t)i (of)g(the)g(Thomas')f(approac)o(h)i(is)f(to)g(extend)f(the)g(op)q (erator)i Ft(A)1598 452 y Fs(k)1640 445 y Fz(in)o(to)228 503 y(the)e(complex)f(plane)i(in)f(one)h(of)g(the)g(comp)q(onen)o(ts)f (of)h(the)g(quasi-momen)o(tum)o(,)228 561 y(for)e(example)e(in)i Fx(k)575 568 y Fw(1)595 561 y Fz(,)g(and)h(establish)f(an)h(estimate)d (of)j(the)f(form)779 651 y Ft(kA)844 630 y Fq(\000)p Fw(1)844 665 y Fs(k)891 651 y Ft(k)e(\024)g Fx(C)t Fz(\(Im)6 b Fx(k)1132 658 y Fw(1)1152 651 y Fz(\))-943 b(\(1.2\))228 743 y(with)17 b Fx(C)t Fz(\()p Fx(t)p Fz(\))c Ft(!)i Fz(0)i(as)h Fx(t)d Ft(!)f(1)p Fz(.)23 b(Then)17 b(due)g(to)g(the)g (analyticit)o(y)e(of)i Ft(A)1504 750 y Fs(k)1544 743 y Fz(in)g Fx(k)1627 750 y Fw(1)1664 743 y Fz(for)228 801 y(the)d(Hamiltonian)e(\(1.1\))i(this)g(estimate)f(guaran)o(tees)h (the)g(absence)g(of)h(constan)o(t)228 859 y(eigen)o(v)m(alues)g Fx(E)516 866 y Fu(l)529 859 y Fz(\()p FC(k)p Fz(\),)h(whic)o(h)f(in)h (its)g(turn)g(pro)o(vides)g(absolute)h(con)o(tin)o(uit)o(y)l(.)278 917 y(The)g(Thomas')g(approac)o(h)i(w)o(as)f(applied)f(b)o(y)g(R.)g (Hemp)q(el)e(and)k(I.)d(Herbst)i(in)228 975 y([5])e(\(see)f(also)i ([6]\))f(to)h(the)f(Sc)o(hr\177)-24 b(odinger)15 b(op)q(erator)639 1065 y Fx(H)j Fz(=)c(\()p FC(D)d Ft(\000)g FC(a)p Fz(\))918 1045 y Fw(2)949 1065 y Fz(+)g Fx(V)s(;)40 b FC(D)14 b Fz(=)g Ft(\000)p Fx(i)p Ft(r)p Fx(;)-1076 b Fz(\(1.3\))228 1155 y(with)18 b(a)i(p)q(erio)q(dic)e(v)o(ector)g(p)q(oten)o(tial)g FC(a)h Fz(and)h Fx(V)29 b Fz(=)18 b(0.)29 b(It)19 b(w)o(as)g(found)g (that)g(the)228 1213 y(sp)q(ectrum)13 b(of)j Fx(H)j Fz(is)14 b(absolutely)h(con)o(tin)o(uous)g(if)f(the)h(p)q(oten)o(tial)f FC(a)i Fz(is)e(su\016cien)o(tly)228 1271 y(small.)29 b(Note)19 b(that)h(if)f(the)h Fr(magnetic)e(\014eld)h FC(B)g Fz(=)g(curl)8 b FC(a)20 b Fz(is)f(p)q(erio)q(dic,)h(then)f FC(a)228 1329 y Fz(ma)o(y)d(not)j(b)q(e)g(p)q(erio)q(dic,)f(and)g(then) h(the)f(sp)q(ectrum)f(ma)o(y)f(ha)o(v)o(e)i(eigen)o(v)m(alues)f(of)228 1387 y(in\014nite)e(m)o(ultiplic)o(it)o(y)e(\(see)i([3]\).)278 1445 y(Although)20 b(R.)f(Hemp)q(el)f(and)j(I.)e(Herbst)g(conjectured)h (that)g(absolute)h(con-)228 1504 y(tin)o(uit)o(y)15 b(m)o(ust)h(b)q(e)i (true)f(for)g(all)g FC(a)p Fz('s,)g(the)g(straigh)o(tforw)o(ard)h (application)f(of)g(the)228 1562 y(Thomas')k(idea)g(could)h(not)g (justify)f(this)h(conclusion,)g(for)g(the)g(p)q(erturbation)228 1620 y(due)17 b(to)g(the)g(magnetic)e(\014eld)i(is)f(of)i(the)e (\014rst)i(order.)23 b(The)17 b(latter)g(obstacle)g(w)o(as)228 1678 y(o)o(v)o(ercome)e(in)j(the)f(recen)o(t)g(pap)q(er)h([1])g(b)o(y)f (M.)g(Birman)f(and)i(T.)g(Suslina)g(in)f(the)228 1736 y(case)22 b Fx(d)h Fz(=)g(2,)g(who)g(pro)o(v)o(ed)e(absolute)h(con)o (tin)o(uit)o(y)e(of)i Fx(H)k Fz(for)c(all)f(con)o(tin)o(uous)228 1794 y FC(a)p Fz('s.)37 b(The)22 b(sp)q(ecial)f(role)g(in)h(their)f (approac)o(h)h(w)o(as)g(pla)o(y)o(ed)f(b)o(y)g(the)g(2)p Fx(D)q Fz(-P)o(auli)228 1852 y(op)q(erator,)e(whic)o(h)e(is)g(kno)o(wn) h(to)g(admit)f(a)h(factorisation)g(in)o(to)f(t)o(w)o(o)h(\014rst)g (order)228 1910 y(op)q(erators.)34 b(This)20 b(fact)g(allo)o(ws)g(one)g (to)h(treat)f(the)f(p)q(erturbation)i(due)f(to)g(the)228 1969 y(v)o(ector-p)q(oten)o(tial)e FC(a)h Fz(as)h(a)g(m)o(ultiplic)o (ativ)n(e)c(one.)31 b(This)19 b(w)o(as)h(the)f(main)e(reason)228 2027 y(wh)o(y)12 b(the)h(Thomas')g(idea)f(pro)o(v)o(ed)h(to)g(b)q(e)g (e\016cien)o(t.)18 b(Although)13 b(b)q(eing)g(v)o(ery)f(clear)228 2085 y(and)18 b(elegan)o(t,)f(the)g(pro)q(of)i(from)e([1])g(uses)h(hea) o(vily)e(the)h(t)o(w)o(o-dimensionalit)o(y)e(of)228 2143 y(the)21 b(problem)g(and)h(apparen)o(tly)g(cannot)h(b)q(e)f (generalised)f(to)h(higher)g(dimen-)228 2201 y(sions.)278 2259 y(In)11 b(the)g(presen)o(t)g(pap)q(er)i(w)o(e)e(pro)o(v)o(e)g (that)h(the)f(magnetic)f(p)q(erio)q(dic)i(Sc)o(hr\177)-24 b(odinger)228 2317 y(op)q(erator)23 b Fx(H)j Fz(is)c(absolutely)f(con)o (tin)o(uous)h(in)f(all)h(dimensions)e(\(see)i(Theorem)228 2375 y(2.1)f(for)h(the)f(precise)f(statemen)o(t\).)34 b(The)22 b(pro)q(of)g(follo)o(ws)f(the)g(Thomas')g(idea.)228 2433 y(Ho)o(w)o(ev)o(er,)12 b(as)j(sho)o(ws)g(a)g(coun)o(terexample)c (constructed)j(b)o(y)g(T.)g(Suslina)g([11])g(for)228 2492 y Fx(d)h Fz(=)g(3,)i(extending)f Fx(k)625 2499 y Fw(1)662 2492 y Fz(in)o(to)h(the)f(complex)f(plane)i(cannot)g(pro)o (vide)f(the)h(b)q(ound)228 2550 y(\(1.2\))f(for)f(all)g(v)o(ector)g(p)q (oten)o(tials)g FC(a)p Fz(.)21 b(Instead)16 b(w)o(e)f(in)o(v)o (estigate)f(an)i(extension)f(of)228 2608 y(the)j(op)q(erator)h Ft(A)552 2615 y Fs(k)594 2608 y Fz(to)g(the)f(complex)e(plain)i(with)g (resp)q(ect)g(to)h(the)f(comp)q(onen)o(t)p eop %%Page: 3 3 3 2 bop 604 127 a Fo(PERIODIC)16 b(SCHR)926 117 y(\177)921 127 y(ODINGER)i(OPERA)m(TOR)356 b(3)228 213 y Fz(of)15 b FC(k)f Fz(along)h(a)g(sp)q(eci\014c)g(direction)e Fn(\013)h Ft(2)g Fv(R)1001 195 y Fu(d)1018 213 y Fz(.)21 b(The)14 b(c)o(hoice)g(of)h Fn(\013)f Fz(dep)q(ends)h(on)g(the)228 271 y(v)o(ector-p)q(oten)o(tial)h FC(a)p Fz(.)24 b(A)17 b(detailed)f(plan)h(of)g(the)g(pro)q(of)h(is)f(giv)o(en)f(in)h(Sect.3)g (and)228 329 y(w)o(e)12 b(do)h(not)g(commen)o(t)d(on)j(it)f(here.)20 b(W)l(e)12 b(only)h(men)o(tion)e(that)i(our)g(approac)o(h)g(still)228 387 y(con)o(tains)j(some)f(traces)h(of)g(the)g(t)o(w)o(o-dimensional)f (sc)o(heme)e(from)i([1].)21 b(Namely)l(,)228 445 y(one)14 b(of)h(the)f(most)f(imp)q(ortan)o(t)g(observ)m(ations)i(is)f(that)h (the)f(p)q(erturbation)h(due)f(to)228 503 y(the)h(magnetic)e(p)q(oten)o (tial)i(can)h(b)q(e)f(represen)o(ted)f(as)i(a)f(m)o(ultiplicativ)n(e)d (one,)j(just)228 561 y(as)j(in)g([1].)26 b(Ho)o(w)o(ev)o(er,)16 b(instead)i(of)h(m)o(ultipli)o(cation)c(b)o(y)j(a)g(function)g(one)g (has)h(to)228 619 y(deal)d(with)g(a)h(m)o(ultipli)o(cation)c(b)o(y)j(a) h(pseudo-di\013eren)o(tial)f(op)q(erator.)278 678 y(Note)c(that)h(the)f (Thomas')g(metho)q(d)g(is)g(not)h(the)g(only)f(one)h(whic)o(h)f(can)g (b)q(e)h(used)228 736 y(to)k(tac)o(kle)f(the)i(absolute)f(con)o(tin)o (uit)o(y)f(problem.)22 b(An)17 b(alternativ)o(e)f(approac)o(h)i(is)228 794 y(describ)q(ed)i(in)h([7].)36 b(V)l(ery)20 b(roughly)l(,)i(its)f (idea)g(is)g(to)h(sho)o(w)g(that)f(an)h(eigefunc-)228 852 y(tion)g(has)g(to)g(deca)o(y)f(\\extremely)e(rapidly"and)j(then,)h (relying)d(on)i(a)h(sort)f(of)228 910 y(unique)16 b(con)o(tin)o(uation) g(theorem)e(at)j(in\014nit)o(y)l(,)e(conclude)h(that)h(suc)o(h)f(a)h (deca)o(y)e(is)228 968 y(imp)q(ossible)j(unless)i(the)g(eigenfunction)g (is)g(iden)o(tically)d(zero.)33 b(In)20 b(particular,)228 1026 y(this)f(approac)o(h)g(guaran)o(tees)h(absolute)f(con)o(tin)o(uit) o(y)e(for)j(div)o(ergence)d(t)o(yp)q(e)h(sec-)228 1084 y(ond)i(order)g(p)q(erio)q(dic)g(elliptic)d(op)q(erators.)34 b(On)20 b(the)f(other)h(hand)h(it)e(do)q(es)i(not)228 1142 y(allo)o(w)16 b(one)g(to)h(handle)f(the)g(magnetic)f (Hamiltonians.)278 1201 y(The)g(pap)q(er)h(is)f(organised)h(as)g(follo) o(ws.)21 b(The)15 b(main)f(result)h(is)g(stated)g(in)g(Sect.)228 1259 y(2.)20 b(Here)10 b(w)o(e)g(also)i(reduce)e(the)h(problem)e(to)i (the)g(estimate)e(of)j(the)e(form)g(\(1.2\))i(\(see)228 1317 y(Theorem)18 b(2.2\),)j(repro)q(ducing)g(the)e(Thomas')h(argumen)o (t.)31 b(Sect.)h(3)20 b(con)o(tains)228 1375 y(an)c(elab)q(orate)h (plan)f(of)g(the)g(pro)q(of)h(of)g(Theorem)d(2.2)j(with)f(precise)f (statemen)o(ts)228 1433 y(of)k(in)o(termediate)c(results.)28 b(Their)18 b(pro)q(ofs)i(are)e(p)q(ostp)q(oned)j(un)o(til)c(Sect.)28 b(5)18 b(-)h(7.)228 1491 y(Sect.)29 b(4)19 b(con)o(tains)g(some)f (results)h(from)f(the)h(calculus)f(of)i(pseudo-di\013eren)o(tial)228 1549 y(op)q(erators)d(on)g(the)f(torus.)278 1607 y(W)l(e)11 b(p)q(oin)o(t)g(out)h(in)e(conclusion)h(that)h(the)f(successful)g (solution)g(of)h(the)f(problem)228 1665 y(for)17 b Fx(d)g Fz(=)e(2)j(in)f(pap)q(er)h([1])f(has)h(pla)o(y)o(ed)f(a)h(crucial)e (role)h(in)g(author's)h(decision)f(to)228 1724 y(attac)o(k)f(the)g (absolute)g(con)o(tin)o(uit)o(y)f(problem)g(for)h(higher)g(dimensions.) 228 1805 y FC(Notation.)21 b Fz(By)16 b(b)q(old)h(lo)o(w)o(ercase)f (letters)g(w)o(e)g(denote)g(v)o(ectors)g(in)g Fv(R)1526 1787 y Fu(d)1560 1805 y Fz(and)h Fv(Z)1691 1787 y Fu(d)1709 1805 y Fz(,)228 1863 y(e.g.)38 b Fn(\013)24 b Ft(2)g Fv(R)498 1845 y Fu(d)515 1863 y Fx(;)30 b FC(m)22 b Ft(2)i Fv(Z)723 1845 y Fu(d)740 1863 y Fz(.)39 b(Bold)21 b(upp)q(ercase)i (letters)e(are)h(used)g(for)g Fx(d)16 b Ft(\002)f Fx(d)p Fz(-)228 1922 y(matrices,)k(e.g.)34 b FC(G)p Fz(.)g(The)21 b(only)f(exception)g(is)g(made)g(for)h(letter)e FC(D)i Fz(whic)o(h)e(is)228 1980 y(the)c(standard)h(notation)g(for)f Ft(\000)p Fx(i)p Ft(r)p Fz(,)f(i.e.)19 b FC(D)14 b Fz(=)g(\()p Fx(D)1157 1987 y Fw(1)1177 1980 y Fx(;)8 b(D)1239 1987 y Fw(2)1259 1980 y Fx(;)g(:)g(:)g(:)f(;)h(D)1408 1987 y Fu(d)1429 1980 y Fz(\))p Fx(;)23 b(D)1525 1987 y Fu(j)1557 1980 y Fz(=)14 b Ft(\000)p Fx(i@)1691 1987 y Fu(j)1709 1980 y Fz(.)228 2039 y(The)h(notations)h FC(ab)g Fz(and)g FC(aGb)f Fz(stand)h(for)g(the)f(scalar)g(pro)q(duct)h(in)f Fv(R)1528 2021 y Fu(d)1561 2039 y Fz(and)g(the)228 2097 y(quadratic)h(form)f(of)i(the)f(matrix)e FC(G)i Fz(resp)q(ectiv)o(ely)l (.)278 2155 y(When)i(w)o(e)g(use)h(the)f(notation)h FC(D)909 2137 y Fs(s)946 2155 y Fz(w)o(e)f(alw)o(a)o(ys)g(assume)g(that)h FC(s)g Fz(is)f(a)h(v)o(ector)228 2213 y(from)h Fv(Z)384 2195 y Fu(d)423 2213 y Fz(with)h(non-negativ)o(e)g(comp)q(onen)o(ts)f Fx(s)1127 2220 y Fw(1)1147 2213 y Fx(;)8 b(s)1192 2220 y Fw(2)1211 2213 y Fx(;)g(:)g(:)g(:)g(;)g(s)1344 2220 y Fu(d)1385 2213 y Fz(and)22 b(then)f FC(D)1644 2195 y Fs(s)1684 2213 y Fz(=)228 2271 y Fx(D)269 2250 y Fu(s)285 2255 y Fm(1)268 2284 y Fw(1)305 2271 y Fx(D)346 2250 y Fu(s)362 2255 y Fm(2)345 2284 y Fw(2)391 2271 y Ft(\001)8 b(\001)g(\001)h Fx(D)499 2248 y Fu(s)515 2254 y Fl(d)498 2285 y Fu(d)536 2271 y Fz(.)789 2409 y(2.)27 b Fy(Main)18 b(resul)m(t)228 2496 y Fz(2.1.)28 b FC(Main)18 b(result.)23 b Fz(W)l(e)16 b(study)h(in)f Fx(L)965 2478 y Fw(2)985 2496 y Fz(\()p Fv(R)1043 2478 y Fu(d)1060 2496 y Fz(\))p Fx(;)24 b(d)14 b Ft(\025)g Fz(3)p Fx(;)i Fz(the)g(op)q(erator)551 2588 y Fx(H)i Fz(=)c(\()p FC(D)d Ft(\000)f FC(a)p Fz(\))p FC(G)p Fz(\()p FC(D)i Ft(\000)e FC(a)p Fz(\))i(+)f Fx(V)s(;)24 b FC(D)14 b Fz(=)g Ft(\000)p Fx(i)p Ft(r)p Fx(;)-1165 b Fz(\(2.1\))p eop %%Page: 4 4 4 3 bop 228 118 a Fo(4)587 b(A.V.)15 b(SOBOLEV)228 213 y Fz(where)23 b FC(G)j Fz(=)h Ft(f)p Fx(g)559 220 y Fu(j)r(l)588 213 y Ft(g)p Fx(;)k(j;)8 b(l)27 b Fz(=)g(1)p Fx(;)8 b Fz(2)p Fx(;)g(:)g(:)g(:)g(;)g(d)24 b Fz(is)f(a)h(symmetri)o(c)d(p)q (ositiv)o(e-de\014nite)228 271 y(matrix)10 b(with)i(real-v)m(alued)f (en)o(tries,)g FC(a)j Fz(=)g(\()p Fx(a)1036 278 y Fw(1)1055 271 y Fx(;)8 b(a)1103 278 y Fw(2)1123 271 y Fx(;)19 b(:)8 b(:)g(:)g(;)g(a)1270 278 y Fu(d)1289 271 y Fz(\))k(is)g(a)g(v)o (ector-function)228 329 y(\(magnetic)20 b(v)o(ector-p)q(oten)o(tial\))h (with)g(real-v)m(alued)h(comp)q(onen)o(ts)f(and)h Fx(V)33 b Fz(is)22 b(a)228 387 y(real-v)m(alued)12 b(function)g(\(electric)e(p) q(oten)o(tial\).)19 b(W)l(e)12 b(supp)q(ose)i(that)e FC(a)p Fx(;)c(V)24 b Fz(are)12 b(b)q(oth)228 445 y(p)q(erio)q(dic.)19 b(Ob)o(viously)10 b(the)h(op)q(erator)h Fx(H)j Fz(can)c(b)q(e)g (reduced)g(b)o(y)f(a)i(suitable)e(c)o(hange)228 503 y(of)18 b(v)m(ariables)f(to)h(\(1.3\).)25 b(Ho)o(w)o(ev)o(er)16 b(it)h(will)f(b)q(e)i(more)e(con)o(v)o(enien)o(t)f(to)j(deal)f(with)228 561 y(\(2.1\))k(for)g(without)g(loss)h(of)f(generalit)o(y)f(w)o(e)g(ma) o(y)g(supp)q(ose)i(that)f(the)g(v)o(ector-)228 619 y(p)q(oten)o(tial)f FC(a)h Fz(and)g(the)f(electric)e(p)q(oten)o(tial)j Fx(V)31 b Fz(are)21 b(p)q(erio)q(dic)f(with)h(resp)q(ect)f(to)228 678 y(the)c(cubic)f(lattice)g(\000)g(=)e(\(2)p Fx(\031)r Fv(Z)-11 b Fz(\))812 659 y Fu(d)829 678 y Fz(.)22 b(Denote)554 763 y(\012)14 b(=)g([0)p Fx(;)8 b Fz(2)p Fx(\031)r Fz(\))788 742 y Fu(d)807 763 y Fx(;)41 b Fv(T)6 b Fz(=)14 b Fv(R)1000 742 y Fu(d)1017 763 y Fx(=)p Fz(\000)p Fx(;)1164 750 y Fz(~)1159 763 y(\012)g(=)f([0)p Fx(;)8 b Fz(1\))1362 742 y Fu(d)1383 763 y Fx(:)228 844 y Fz(Besides,)15 b(it)g(will)h(b)q (e)g(con)o(v)o(enien)o(t)e(to)j(assume)e(the)h(follo)o(wing)g (conditions:)862 925 y FC(DGa)e Fz(=)g(0)p Fx(;)-852 b Fz(\(2.2\))817 964 y Fk(Z)844 1076 y Fw(\012)880 1031 y FC(a)p Fz(\()p FC(x)p Fz(\))p Fx(d)p FC(x)14 b Fz(=)f(0)p Fx(:)-905 b Fz(\(2.3\))228 1141 y(The)20 b(equalit)o(y)e(\(2.2\))i(is)g (understo)q(o)q(d)h(in)e(the)h(distributional)f(sense.)32 b(The)19 b(re-)228 1199 y(quiremen)o(ts)d(\(2.2\),)j(\(2.3\))h(can)f(b) q(e)g(alw)o(a)o(ys)f(satis\014ed)i(b)o(y)e(c)o(ho)q(osing)i(an)f (appro-)228 1257 y(priate)d(gauge.)22 b(Under)16 b(the)g(condition)g (\(2.2\))g(the)g(op)q(erator)i Fx(H)i Fz(has)d(the)f(form)500 1338 y Fx(H)i Fz(=)c FC(DGD)d Fz(+)g Fx(X)16 b Fz(+)11 b Fx(V)s(;)40 b(X)18 b Fz(=)c Ft(\000)p Fz(2)p FC(aGD)e Fz(+)f FC(aGa)p Fx(:)228 1420 y Fz(Observ)o(e)i(that)i(the)f(p)q (erturbation)h Fx(X)c Fz(+)c Fx(V)26 b Fz(will)13 b(b)q(e)i (in\014nitesimally)c(small)h(with)228 1478 y(resp)q(ect)18 b(to)h(the)f(op)q(erator)h FC(DGD)g Fz(if)f FC(a)f Ft(2)h Fx(L)1067 1460 y Fq(1)1104 1478 y Fz(\()p Fv(R)1162 1460 y Fu(d)1180 1478 y Fz(\),)g(and)h Fx(V)29 b Ft(2)17 b Fx(L)1468 1454 y Fu(p)1468 1492 y Fw(lo)q(c)1515 1478 y Fz(\()p Fv(R)1573 1460 y Fu(d)1590 1478 y Fz(\))h(with)228 1536 y Fx(p)c Fz(=)g(2\()p Fx(d)g Fz(=)g(2)p Fx(;)8 b Fz(3\))14 b(and)g Fx(p)g(>)g(d=)p Fz(2\()p Fx(d)h Ft(\025)f Fz(4\).)20 b(This)14 b(guaran)o(tees)g(self-adjoin)o(tness)f(of)228 1594 y Fx(H)20 b Fz(on)15 b(the)h(domain)e Fx(D)q Fz(\()p Fx(H)t Fz(\))h(=)f Fx(D)q Fz(\()p FC(DGD)p Fz(\))g(=)g Fx(H)1119 1576 y Fw(2)1139 1594 y Fz(\()p Fv(R)1197 1576 y Fu(d)1214 1594 y Fz(\).)21 b(The)16 b(ab)q(o)o(v)o(e)f(conditions)228 1652 y(are)d(satis\014ed)h(in)f(the)g(follo)o(wing)g(Theorem)f(whic)o (h)g(constitutes)i(the)f(main)f(result)228 1710 y(of)16 b(the)g(pap)q(er:)228 1799 y FC(Theorem)g(2.1.)24 b Fj(L)n(et)e Fx(d)i Ft(\025)g Fz(3)p Fj(.)39 b(L)n(et)22 b FC(a)i Ft(2)g Fx(C)1092 1781 y Fw(2)p Fu(d)p Fw(+3)1175 1799 y Fz(\()p Fv(T)-5 b Fz(\))20 b Fj(b)n(e)j(a)g(ve)n(ctor-p)n(otential,) 228 1857 y(satisfying)g Fz(\(2.2\))p Fj(,)d Fz(\(2.3\))p Fj(,)f(and)g(let)h Fx(V)27 b Ft(2)17 b Fx(L)1023 1834 y Fu(p)1023 1871 y Fi(lo)n(c)1067 1857 y Fz(\()p Fv(R)1125 1839 y Fu(d)1142 1857 y Fz(\))p Fx(;)27 b(p)16 b(>)h(d)12 b Ft(\000)g Fz(1)19 b Fj(b)n(e)g(a)g(function,)228 1916 y(b)n(oth)d(p)n(erio)n(dic)f(with)h(r)n(esp)n(e)n(ct)f(to)h(the)h (lattic)n(e)g Fz(\000)d(=)g(\(2)p Fx(\031)r Fv(Z)-11 b Fz(\))1276 1898 y Fu(d)1293 1916 y Fj(.)22 b(Then)16 b(the)h(sp)n(e)n(ctrum)228 1974 y(of)g(the)h(op)n(er)n(ator)e Fx(H)22 b Fj(is)17 b(absolutely)i(c)n(ontinuous.)228 2063 y Fz(2.2.)28 b FC(Flo)r(quet)11 b(decomp)r(osition.)21 b Fz(The)11 b(pro)q(of)h(of)f(Theorem)e(2.1)i(will)f(b)q(e)h(based)228 2121 y(on)18 b(the)f(Flo)q(quet)g(decomp)q(osition)g(for)h(the)f(op)q (erator)i Fx(H)j Fz(in)17 b(a)h(direct)e(in)o(tegral.)228 2179 y(Let)c(us)h(in)o(tro)q(duce)f(necessary)g(notation.)21 b(W)l(e)12 b(iden)o(tify)e(the)j(space)f Fx(L)1501 2161 y Fw(2)1521 2179 y Fz(\()p Fv(R)1579 2161 y Fu(d)1596 2179 y Fz(\))g(with)228 2238 y(the)k(direct)f(in)o(tegral)795 2342 y Fh(H)f Fz(=)897 2274 y Fk(Z)928 2379 y Fw(~)924 2387 y(\012)960 2342 y Fx(L)993 2322 y Fw(2)1013 2342 y Fz(\(\012\))p Fx(d)p FC(k)p Fx(:)228 2449 y Fz(The)i(iden)o (ti\014cation)f(is)h(implem)o(en)o(te)o(d)e(b)o(y)h(the)h(Gelfand)h (transform)500 2538 y(\()p Fx(U)5 b(f)g Fz(\)\()p FC(x)p Fx(;)j FC(k)p Fz(\))14 b(=)f Fx(e)813 2518 y Fq(\000)p Fu(i)p Fs(kx)919 2491 y Fk(X)906 2601 y Fs(m)p Fq(2)p Fg(Z)988 2591 y Fl(d)1012 2538 y Fx(e)1035 2518 y Fq(\000)p Fu(i)p Fw(2)p Fu(\031)q Fs(km)1170 2538 y Fx(f)5 b Fz(\()p FC(x)11 b Fz(+)g(2)p Fx(\031)r FC(m)p Fz(\))p Fx(;)p eop %%Page: 5 5 5 4 bop 604 127 a Fo(PERIODIC)16 b(SCHR)926 117 y(\177)921 127 y(ODINGER)i(OPERA)m(TOR)356 b(5)228 213 y Fz(whic)o(h)17 b(is)h(initially)e(de\014ned)i(on)g(functions)g(from)f(the)h(Sc)o(h)o (w)o(arz)f(class)h(and)h(ex-)228 271 y(tends)i(b)o(y)g(con)o(tin)o(uit) o(y)e(to)i(a)h(unitary)e(mapping)h(from)f Fx(L)1316 253 y Fw(2)1336 271 y Fz(\()p Fv(R)1394 253 y Fu(d)1411 271 y Fz(\))h(on)o(to)g Fh(H)p Fz(.)36 b(F)l(or)228 329 y(ev)o(ery)15 b FC(k)e Ft(2)h Fv(R)486 311 y Fu(d)519 329 y Fz(w)o(e)i(de\014ne)g(on) h(the)f(p)q(erio)q(dic)g(functions)g(the)g(op)q(erators)585 382 y Fk(8)585 427 y(>)585 442 y(<)585 532 y(>)585 547 y(:)630 431 y Fx(H)t Fz(\()p FC(k)p Fz(\))d(=)81 b Fx(H)914 438 y Fw(0)934 431 y Fz(\()p FC(k)p Fz(\))11 b(+)g Fx(X)t Fz(\()p FC(k)p Fz(\))g(+)g Fx(V)s(;)630 501 y(H)670 508 y Fw(0)690 501 y Fz(\()p FC(k)p Fz(\))i(=)65 b(\()p FC(D)11 b Fz(+)g FC(k)p Fz(\))p FC(G)p Fz(\()p FC(D)g Fz(+)g FC(k)p Fz(\))p Fx(;)630 571 y(X)t Fz(\()p FC(k)p Fz(\))i(=)81 b Ft(\000)p Fz(2)p FC(aG)p Fz(\()p FC(D)12 b Fz(+)f FC(k)p Fz(\))f(+)h FC(aGa)p Fx(:)228 499 y Fz(\(2.4\))228 676 y(F)l(or)23 b(eac)o(h)f FC(k)j Ft(2)h Fv(R)591 658 y Fu(d)631 676 y Fz(the)d(p)q(erturbation)g Fx(H)t Fz(\()p FC(k)p Fz(\))16 b Ft(\000)f Fx(H)1236 683 y Fw(0)1256 676 y Fz(\()p FC(0)p Fz(\))25 b(=)g Fx(H)t Fz(\()p FC(k)p Fz(\))16 b Ft(\000)f FC(DGD)228 735 y Fz(is)i(in\014nitesimally)e Fx(H)636 742 y Fw(0)656 735 y Fz(\()p FC(0)p Fz(\)-b)q(ounded,)k(so)f (that)g(the)g(op)q(erators)h Fx(H)t Fz(\()p FC(k)p Fz(\))f(are)g(self-) 228 793 y(adjoin)o(t)23 b(on)g(the)g(domain)f Fx(D)q Fz(\()p Fx(H)t Fz(\()p FC(k)p Fz(\)\))k(=)g Fx(D)q Fz(\()p Fx(H)1126 800 y Fw(0)1147 793 y Fz(\()p FC(0)p Fz(\)\))f(=)g Fx(H)1364 775 y Fw(2)1384 793 y Fz(\()p Fv(T)-5 b Fz(\).)39 b(Moreo)o(v)o(er,)228 851 y(since)16 b Fx(H)388 858 y Fw(0)408 851 y Fz(\()p FC(0)p Fz(\))i(has)g(a)f(compact)f(resolv)o(en)o (t,)g(so)i(do)q(es)g Fx(H)t Fz(\()p FC(k)p Fz(\).)24 b(The)17 b(family)e Fx(H)t Fz(\()p FC(k)p Fz(\))228 909 y(represen)o(ts)g(the)h(decomp)q(osition)g(of)g Fx(H)21 b Fz(in)16 b(the)g(direct)f(in)o(tegral.)20 b(Namely)l(,)736 1030 y Fx(U)5 b(H)t(U)856 1010 y Fq(\000)p Fw(1)918 1030 y Fz(=)970 962 y Fk(Z)1002 1067 y Fw(~)998 1075 y(\012)1033 1030 y Fx(H)t Fz(\()p FC(k)p Fz(\))p Fx(d)p FC(k)p Fx(:)-986 b Fz(\(2.5\))278 1160 y(Let)20 b(us)h(single)e(out)i(a)g(v)o(ector)e Fn(\013)h Fz(=)h(\()p Fx(\013)1028 1167 y Fw(1)1048 1160 y Fx(;)8 b(\013)1101 1167 y Fw(2)1120 1160 y Fx(;)g(:)g(:)g(:)g(;)g (\013)1261 1167 y Fu(d)1281 1160 y Fz(\))20 b Ft(2)h Fv(R)1413 1142 y Fu(d)1450 1160 y Fz(and)g(set)f Ft(L)h Fz(=)228 1218 y Ft(f)p FC(k)13 b Ft(2)h Fv(R)382 1200 y Fu(d)413 1218 y Fz(:)g Fn(\013)o FC(k)g Fz(=)g(0)p Ft(g)p Fz(.)21 b(Consider)c(the)f(family)451 1308 y(~)438 1320 y Fx(H)t Fz(\()p Fx(k)r Fz(\))e(=)g Fx(H)657 1280 y Fk(\000)680 1320 y Fn(\013)p Fz(\()p Fx(k)f Ft(\000)e Fn(\026)860 1332 y Fq(?)890 1320 y FC(G)p Fn(\013)p Fz(\))g(+)g Fn(\026)1086 1332 y Fq(?)1115 1280 y Fk(\001)1138 1320 y Fx(;)24 b(k)16 b Ft(2)e Fv(R)p Fx(;)21 b Fn(\026)1374 1332 y Fq(?)1417 1320 y Ft(2)14 b(L)p Fx(:)-1284 b Fz(\(2.6\))228 1418 y(The)16 b(cen)o(tral)g(role)g(will)f(b)q(e)i(pla)o(y)o(ed)e(b)o (y)h(the)g(op)q(erator)i(obtained)f(b)o(y)f(extending)228 1476 y(this)h(family)e(to)i(large)h(complex)d(v)m(alues)i(of)g Fx(k)r Fz(.)24 b(Namely)l(,)15 b(set)i Fx(k)g Fz(=)f Fx(\014)e Fz(+)d Fx(i\032)17 b Fz(with)228 1534 y Fx(\014)f Ft(2)e Fv(R)f Fz(and)k Fx(\032)d(>)g Fz(0)i(and)h(de\014ne)775 1634 y Ft(H)p Fz(\()p Fx(\032)p Fz(\))e(=)959 1621 y(~)947 1634 y Fx(H)t Fz(\()p Fx(\014)e Fz(+)e Fx(i\032)p Fz(\))p Fx(:)228 1734 y Fz(W)l(e)16 b(shall)g(b)q(e)h(in)o(terested)e(in)h(b)q (eha)o(viour)g(of)h Ft(H)p Fz(\()p Fx(\032)p Fz(\))g(in)f Fx(\032)f Ft(!)f(1)i Fz(uniformly)e(with)228 1792 y(resp)q(ect)i(to)g Fn(\026)490 1804 y Fq(?)535 1792 y Fz(in)g(a)h(compact)e(set.)21 b(Let)476 1890 y Fn(\026)14 b Fz(=)g Fn(\026)612 1902 y Fq(k)643 1890 y Fz(+)d Fn(\026)728 1902 y Fq(?)757 1890 y Fx(;)57 b Fn(\026)863 1902 y Fq(k)896 1890 y Fz(=)14 b Fn(\013)p Fz(\()p Fx(\014)f Ft(\000)e Fn(\026)1132 1902 y Fq(?)1161 1890 y FC(G)p Fn(\013)p Fz(\))p Fx(;)24 b Fn(\026)1335 1902 y Fq(?)1379 1890 y Ft(2)14 b(L)p Fx(:)-1246 b Fz(\(2.7\))228 1991 y(W)l(e)19 b(denote)h(b)o(y)f Ft(H)589 1998 y Fs(a)611 1991 y Fz(\()p Fx(\032)p Fz(\))h(the)g(op)q (erator)h Ft(H)p Fz(\()p Fx(\032)p Fz(\))f(with)g Fx(V)31 b Fz(=)20 b(0)g(and)h(b)o(y)e Ft(H)1595 1998 y Fw(0)1615 1991 y Fz(\()p Fx(\032)p Fz(\))h({)228 2049 y(with)15 b FC(a)f Fz(=)g(0)p Fx(;)8 b(V)25 b Fz(=)14 b(0.)21 b(V)l(ector)15 b Fn(\026)g 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Fz(\))p Ft(j)k(\025)i Fx(!)r(\032;)24 b Ft(8)p FC(m)11 b Ft(2)j Fh(N)p Fz(\()p Fx(\032;)8 b(!)r Fz(\))p Fx(:)-1075 b Fz(\(3.2\))278 1709 y(Let)15 b Fx(P)7 b Fz(\()p FC(m)p Fz(;)h Fx(\032;)g(!)r Fz(\))p Fx(;)g(Q)p Fz(\()p FC(m)p Fz(;)g Fx(\032;)g(!)r Fz(\))k(b)q(e)k(the)f(c)o (haracteristic)e(functions)j(of)f(the)g(sets)228 1767 y Fx(M)r(;)8 b(N)23 b Fz(resp)q(ectiv)o(ely)l(.)h(In)o(tro)q(duce)17 b(also)i(smo)q(oth)f(cut-o\013)g(functions)g(supp)q(orted)228 1826 y(near)e(the)g(shell)f Ft(j)p FC(F)p Fz(\()p FC(m)9 b Fz(+)i Fn(\026)p Fz(\))p Ft(j)g(\000)f Fx(\032)p Ft(j)k(\024)f Fx(!)r Fz(.)22 b(Let)16 b Fx(\020)i Ft(2)c Fx(C)1218 1807 y Fq(1)1214 1838 y Fw(0)1255 1826 y Fz(\()p Fv(R)p Fz(\))f(b)q(e)j(a)g(non-negativ)o(e)228 1884 y(function)c(suc)o(h)g (that)g Fx(\020)t Fz(\()p Fx(t)p Fz(\))i(=)g(1)e(for)h Ft(j)p Fx(t)p Ft(j)g(\024)h Fz(1)p Fx(;)32 b(\020)t Fz(\()p Fx(t)p Fz(\))14 b(=)f(0)g(for)g Ft(j)p Fx(t)p Ft(j)g(\025)g Fz(2)g(and)g Fx(\020)t Fz(\()p Fx(t)p Fz(\))g Ft(\024)h Fz(1)228 1942 y(for)i(all)g Fx(t)p Fz(.)21 b(Then)16 b(for)h(an)o(y)f Fx(!)g(>)e Fz(0)i(and)h Fx(j)g Fz(=)d(1)p Fx(;)8 b Fz(2)17 b(de\014ne)434 2028 y Fx(\020)455 2035 y Fu(j)474 2028 y Fz(\()p FC(m)p Fz(;)8 b Fx(\032;)g(!)r Fz(\))k(=)i Fx(\020)750 1988 y Fk(\000)773 2028 y Fz(\()p Ft(j)p FC(F)p Fz(\()p FC(m)c Fz(+)h Fn(\026)p Fz(\))p Ft(j)g(\000)g Fx(\032)p Fz(\)\()p Fx(j)s(!)r Fz(\))1233 2007 y Fq(\000)p Fw(1)1280 1988 y Fk(\001)1303 2028 y Fx(;)24 b FC(m)13 b Ft(2)h Fv(Z)1485 2007 y Fu(d)1502 2028 y Fx(:)-1288 b Fz(\(3.3\))228 2119 y(No)o(w)16 b(de\014ne)g(the)g (subspaces)h(of)f Fx(L)875 2101 y Fw(2)895 2119 y Fz(\(\012\))h(asso)q (ciated)g(with)f(the)g(sets)g Fx(M)r(;)8 b(N)d Fz(:)361 2206 y Fh(M)14 b Fz(=)i Fh(M)p Fz(\()p Fx(\032;)8 b(!)r Fz(\))14 b(=)g Ft(f)p Fx(u)g Ft(2)g Fx(L)863 2185 y Fw(2)883 2206 y Fz(\(\012\))g(:)i(^)-27 b Fx(u)p Fz(\()p FC(m)p Fz(\))12 b(=)i(0)p Fx(;)25 b Ft(8)p FC(m)17 b Fx(=)-30 b Ft(2)14 b Fx(M)5 b Fz(\()p Fx(\032;)j(!)r Fz(\))p Ft(g)p Fx(;)372 2286 y Fh(N)14 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Fz(\))k(on)i(the)e(subspace)h Fh(N)p Fz(\()p Fx(\032;)c(!)r Fz(\))k(whic)o(h)f(almost)228 2608 y(imme)o(diately)g(leads)j(to)h(a)f(b)q(ound)i(of)f(the)f(form)f (\(2.10\))i(for)f(the)h(op)q(erator)g Ft(H)1648 2615 y Fs(a)1670 2608 y Fx(Q)p Fz(.)p eop %%Page: 9 9 9 8 bop 604 127 a Fo(PERIODIC)16 b(SCHR)926 117 y(\177)921 127 y(ODINGER)i(OPERA)m(TOR)356 b(9)228 213 y Fz(On)16 b(the)g(con)o(trary)l(,)f(estimating)g Ft(H)878 220 y Fs(a)899 213 y Fx(P)24 b Fz(is)16 b(far)g(from)f(b)q(eing)h(easy)g(for) g(the)g(op)q(era-)228 271 y(tors)i Ft(H)369 278 y Fw(0)389 271 y Fz(\()p Fx(\032)p Fz(\))g(and)h Ft(X)7 b Fz(\()p Fx(\032)p Fz(\))18 b(are)g(of)g(the)g(same)f(order)h(of)g(magnitude)f (on)h Fh(M)p Fz(\()p Fx(\032;)8 b(!)r Fz(\).)228 329 y(T)l(o)20 b(o)o(v)o(ercome)d(this)i(di\016cult)o(y)f(w)o(e)h(pro)o(v)o (e)g(\014rst)g(of)h(all)f(that)h(on)g(the)g(subspace)228 387 y Fh(M)p Fz(\()p Fx(\032;)8 b(!)r Fz(\))16 b(the)g(op)q(erator)i Ft(H)736 394 y Fs(a)758 387 y Fz(\()p Fx(\032)p 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Fz(in)f(the)g(form)g(\(2.7\),)g(for)h(an)o(y)f(v)o(ec-)228 1477 y(tor)f Fn(\013)h Fz(satisfying)g(\(2.9\),)f(w)o(e)f(ha)o(v)o(e) 228 1585 y(\(3.7\))58 b(min)378 1620 y Fs(m)p Fq(2)p Fg(Z)460 1610 y Fl(d)484 1585 y Ft(j)p Fz(\()p FC(m)9 b Fz(+)i Fn(\026)q Fz(\))p FC(F)-5 b(e)p Ft(j)13 b Fz(=)22 b(min)813 1620 y Fs(m)p Fq(2)p Fg(Z)896 1610 y Fl(d)919 1585 y Ft(j)p Fz(\()p FC(m)10 b Fz(+)h Fn(\026)1094 1596 y Fq(k)1125 1585 y Fz(+)g Fn(\026)1209 1596 y Fq(?)1239 1585 y Fz(\))p FC(G)p Fn(\013)o Ft(j)1055 1687 y(\025)j(j)p Fx(\014)s Ft(j)p Fx(;)23 b Fz(if)32 b Ft(j)p Fx(\014)s Ft(j)13 b(\024)g(j)p FC(G)p Fn(\013)p Ft(j)p Fz(\(2)p Ft(j)p FC(n)p Ft(j)p Fz(\))1620 1667 y Fq(\000)p Fw(1)1667 1687 y Fx(:)228 1783 y Fz(Since)i Ft(j)p FC(G)p Fn(\013)p Ft(j)f Fz(=)f Ft(j)p FC(F)-5 b(e)p Ft(j)14 b(\025)g Fx(f)705 1790 y Fq(\000)734 1783 y Fz(,)i(this)g(implies)e(that)372 1877 y Ft(k)p Fx(T)426 1884 y Fw(0)445 1877 y Fz(\()p Fx(\032)p Fz(\))p Fx(u)p Ft(k)g(\025)f Fx(f)651 1884 y Fq(\000)681 1877 y Fz(\(2)p Ft(j)p FC(n)p Ft(j)p Fz(\))802 1856 y Fq(\000)p Fw(1)849 1877 y Ft(k)p Fx(u)p Ft(k)p Fx(;)24 b(\014)16 b Fz(=)e Ft(j)p FC(G)p Fn(\013)p Ft(j)p Fz(\(2)p Ft(j)p FC(n)p Ft(j)p Fz(\))1292 1856 y Fq(\000)p Fw(1)1339 1877 y Fx(;)40 b Ft(8)p Fx(u)12 b Ft(2)i Fx(C)1548 1856 y Fq(1)1585 1877 y Fz(\()p Fv(T)-5 b Fz(\))p Fx(;)-1445 b Fz(\(3.8\))228 1972 y(uniformly)14 b(in)h Fx(\032)f Ft(\025)g Fz(0.)21 b(T)l(o)c(reduce)e Fx(T)912 1979 y Fs(a)933 1972 y Fz(\()p Fx(\032)p Fz(\))h(to)g Fx(T)1100 1979 y Fw(0)1120 1972 y Fz(\()p Fx(\032)p Fz(\))g(w)o(e)f(shall)h (\014nd)g(a)g(similarit)o(y)228 2031 y(transformation)24 b(\(\\gauge)i(transformation"\))e Fx(W)31 b Fz(suc)o(h)24 b(that)h(the)f(op)q(erator)228 2089 y Fx(W)7 b(T)310 2096 y Fs(a)331 2089 y Fz(\()p Fx(\032)p Fz(\))p Fx(W)447 2071 y Fq(\000)p Fw(1)509 2089 y Fz(w)o(ould)14 b(coincide)f(with)h Fx(T)972 2096 y Fw(0)992 2089 y Fz(\()p Fx(\032)p Fz(\))g(up)h(to)f (some)g(insigni\014can)o(t)f(errors.)228 2147 y(Belo)o(w)i(w)o(e)h (explain)f(ho)o(w)i(to)f(\014nd)h(suc)o(h)f(a)h(transformation.)228 2263 y(3.2.)28 b FC(Gauge)e(transformation.)c Fz(W)l(e)g(shall)h (\014nd)f(a)h(b)q(oundedly)g(in)o(v)o(ertible)228 2321 y(op)q(erator)h Fx(W)30 b Fz(suc)o(h)23 b(that)h Fx(W)7 b(T)819 2328 y Fs(a)840 2321 y Fz(\()p Fx(\032)p Fz(\))p Fx(W)956 2303 y Fq(\000)p Fw(1)1026 2321 y Fz(is)23 b(close)g(to)g Fx(T)1301 2328 y Fw(0)1321 2321 y Fz(\()p Fx(\032)p Fz(\))g(in)g(some)f (sense.)228 2379 y(W)l(e)h(b)q(egin)h(with)g(constructing)f(suc)o(h)h (a)g(transformation)g(for)g(the)f(op)q(erator)228 2437 y(obtained)e(from)f(\(3.4\))h(b)o(y)f(replacing)g Fn(\027)s Fz(\()p FC(D)p Fz(\))h(with)g(a)g(\014xed)f(unit)h(v)o(ector)f Fn(\015)25 b Ft(2)228 2495 y Fv(S)258 2477 y Fu(d)p Fq(\000)p Fw(1)321 2495 y Fz(:)500 2588 y Fx(T)536 2568 y Fw(\()p Ff(\015)q Fw(\))529 2601 y Fs(a)602 2588 y Fz(=)13 b Fx(T)689 2568 y Fw(\()p Ff(\015)q Fw(\))682 2601 y Fs(a)741 2588 y Fz(\()p Fx(\032)p Fz(\))h(=)g(\()p FC(D)d Fz(+)g Fn(\026)g Ft(\000)g FC(a)p Fz(\))p FC(F)p Fz(\()p Fx(i)p FC(e)g Fz(+)g Fn(\015)s Fz(\))g Ft(\000)g Fx(\032;)p eop %%Page: 10 10 10 9 bop 228 118 a Fo(10)568 b(A.V.)15 b(SOBOLEV)228 221 y Fz(A)i(reduction)h(to)h Fx(T)599 196 y Fw(\()p Fu(\015)r Fw(\))592 234 y(0)666 221 y Fz(w)o(ould)f(b)q(e)g(secured)g (if)f(w)o(e)h(\014nd)g(a)h(scalar)f(p)q(oten)o(tial)g(for)228 288 y Fx(T)264 263 y Fw(\()p Ff(\015)q Fw(\))257 294 y Fs(a)316 288 y Fz(,)g(that)g(is)g(a)g(p)q(erio)q(dic)g(complex-v)m (alued)d(smo)q(oth)j(function)g Fx( )g Fz(=)f Fx( )1586 295 y Ff(\015)1628 288 y Fz(suc)o(h)228 346 y(that)659 425 y(\()p Fx(i)p FC(e)10 b Fz(+)h Fn(\015)s Fz(\))p FC(FD)p Fx( )16 b Fz(=)e Ft(\000)p Fz(\()p Fx(i)p FC(e)c Fz(+)h Fn(\015)s Fz(\))p FC(F)-5 b(a)p Fx(:)-1063 b Fz(\(3.9\))228 503 y(Indeed,)15 b(if)h(suc)o(h)g(a)g(function)g(exists)g(then)800 588 y Fx(e)823 567 y Fu( )849 588 y Fx(T)885 567 y Fw(\()p Ff(\015)q Fw(\))878 600 y Fs(a)951 588 y Fz(=)d Fx(T)1038 562 y Fw(\()p Fu(\015)r Fw(\))1031 600 y(0)1087 588 y Fx(e)1110 567 y Fu( )1136 588 y Fx(:)-922 b Fz(\(3.10\))228 666 y(Ho)o(w)o(ev)o(er,)14 b(to)i(\014nd)h Fx( )h Fz(w)o(e)e(need)f(to) i(mak)o(e)d(t)o(w)o(o)i(assumptions:)278 735 y(1.)k(The)12 b(v)o(ector-p)q(oten)o(tial)g FC(a)g Fz(is)g(a)h(trigonometric)d(p)q (olynomial,)h(i.e.)19 b(for)12 b(some)336 793 y Fx(L)k Ft(2)g Fv(N)j Fz(and)f Fx(K)625 800 y Fu(L)667 793 y Fz(=)e Ft(f)p FC(m)f Ft(2)h Fv(Z)894 775 y Fu(d)927 793 y Fz(:)g(0)g Fx(<)g Ft(j)p FC(m)p Ft(j)e(\024)i Fx(L)p Ft(g)h Fz(it)g(can)h(b)q(e)f(represen)o(ted)336 851 y(as)g(follo)o(ws:) 629 929 y FC(a)p Fz(\()p FC(x)p Fz(\))d(=)f(\(2)p Fx(\031)r Fz(\))881 909 y Fq(\000)913 895 y Fl(d)p 914 901 17 2 v 914 922 a Fm(2)984 882 y Fk(X)945 990 y Fw(0)p Fu(<)p Fq(j)p Fs(m)p Fq(j\024)p Fu(L)1105 928 y Fz(^)1103 929 y FC(a)q Fz(\()p FC(m)p Fz(\))p Fx(e)1240 909 y Fu(i)p Fs(mx)1307 929 y Fx(:)-1093 b Fz(\(3.11\))278 1062 y(2.)20 b(The)d(v)o(ector)e FC(e)h Fz(is)g(c)o(hosen)g(in)g(suc)o(h)g(a)g(w)o (a)o(y)g(that)695 1141 y FC(eFm)c Ft(6)p Fz(=)i(0)33 b(for)16 b(all)g FC(m)c Ft(2)i Fx(K)1215 1148 y Fu(L)1242 1141 y Fx(:)-1028 b Fz(\(3.12\))228 1219 y(In)16 b(Lemma)e(3.7)i(w)o(e) g(construct)h(suc)o(h)f(an)g FC(e)g Fz(explicitly)l(.)228 1304 y FC(Lemm)o(a)g(3.2.)24 b Fj(L)n(et)17 b FC(a)i Fj(b)n(e)f(a)f(trigonometric)i(p)n(olynomial)f(of)f(the)i(form)i Fz(\(3.11\))p Fj(.)228 1362 y(Supp)n(ose)i(that)h FC(e)h Ft(2)g Fv(S)665 1344 y Fu(d)p Fq(\000)p Fw(1)751 1362 y Fj(ob)n(eys)i Fz(\(3.12\))q Fj(.)40 b(Then)24 b(for)f(any)h Fn(\015)k Ft(2)e Fv(S)1532 1344 y Fu(d)p Fq(\000)p Fw(1)1618 1362 y Fj(ther)n(e)228 1420 y(exists)16 b(a)g(function)h Fx( )f Fz(=)d Fx( )721 1427 y Ff(\015)760 1420 y Ft(2)h Fx(C)846 1402 y Fq(1)883 1420 y Fz(\()p Fv(T)-5 b Fz(\))13 b Fj(satisfying)j(the)g(e)n(quation)21 b Fz(\(3.9\))p Fj(.)h(If)15 b(one)228 1478 y(r)n(e)n(quir)n(es)763 1547 y Fz(^)753 1560 y Fx( )q Fz(\()p FC(m)p Fz(\))e(=)h(0)p Fx(;)25 b FC(m)19 b Fx(=)-30 b Ft(2)14 b Fx(K)1149 1567 y Fu(L)1175 1560 y Fx(;)-961 b Fz(\(3.13\))228 1638 y Fj(then)18 b Fx( )h Fj(is)e(de\014ne)n(d)i(uniquely)g(and)e (satis\014es)i(the)e(upp)n(er)g(b)n(ounds)757 1717 y Fz(sup)767 1756 y Fs(x)p Fu(;)p Ff(\015)839 1717 y Ft(j)p FC(D)896 1696 y Fs(s)914 1717 y Fx( )946 1724 y Ff(\015)970 1717 y Fz(\()p FC(x)p Fz(\))p Ft(j)d(\024)f Fx(C)1153 1724 y Fs(s)1171 1717 y Fx(;)-957 b Fz(\(3.14\))228 1827 y Fj(and)633 1905 y Fz(sup)659 1945 y Fs(x)715 1905 y Ft(j)p FC(D)772 1885 y Fs(s)789 1865 y Fk(\000)812 1905 y Fx( )844 1912 y Ff(\015)880 1905 y Ft(\000)11 b Fx( )962 1912 y Ff(\021)986 1865 y Fk(\001)1009 1905 y Ft(j)j(\024)f Fx(C)1124 1912 y Fs(s)1142 1905 y Ft(j)p Fn(\015)i Ft(\000)c Fn(\021)q Ft(j)p Fx(;)-1081 b Fz(\(3.15\))228 2009 y Fj(with)18 b(c)n(onstants)g Fx(C)583 2016 y Fs(s)618 2009 y Fj(indep)n(endent)i(of)d Fn(\015)s Fx(;)8 b Fn(\021)r Fj(,)17 b(that)h(dep)n(end)g(only)g(on)f FC(a)p Fx(;)8 b FC(e)p Fj(.)278 2093 y Fz(See)15 b(subsection)i(5.1)f(for)h(pro)q (of.)278 2151 y(Note)e(that)i(this)f(Lemma)e(do)q(es)j(not)g(require)e (that)i Fn(\013)f Fz(satis\014es)g(\(2.9\))q(.)278 2218 y(In)10 b(con)o(trast)i(to)f Fx(T)607 2193 y Fw(\()p Ff(\015)q Fw(\))600 2224 y Fs(a)670 2218 y Fz(the)f(required)g(gauge)i (transformation)f(for)g(the)g(op)q(erator)228 2276 y(\(3.4\))23 b(will)f(b)q(e)h(p)q(erformed)f(b)o(y)g(a)i(pseudo-di\013eren)o(tial)e (op)q(erator)i(\(PDO\))g(on)228 2334 y(the)19 b(torus)h Fv(T)12 b Fz(\(see)19 b(\(4.1\))g(for)h(de\014nition\).)30 b(Assuming)19 b(as)h(ab)q(o)o(v)o(e)f(that)h FC(a)g Fz(is)f(a)228 2392 y(trigonometric)9 b(p)q(olynomial)h(\(3.11\))i(and)g FC(e)f Fz(satis\014es)g(\(3.12\))q(,)h(let)e Fx( )1458 2399 y Ff(\015)1497 2392 y Fz(=)k Fx( )1581 2399 y Ff(\015)1606 2392 y Fz(\()p FC(x)p Fz(;)8 b FC(a)p Fz(\))228 2452 y(b)q(e)19 b(the)f(function)h(constructed)g(in)f(Lemma)f(3.2)i(with)g Fn(\015)j Fz(=)c Fn(\027)s Fz(\()p FC(m)p Fz(\))p Fx(;)26 b FC(m)17 b Ft(2)h Fv(Z)1691 2434 y Fu(d)1709 2452 y Fz(.)228 2510 y(Denote)571 2588 y Fx( )r Fz(\()p FC(x)p Fx(;)8 b FC(m)p Fz(;)g FC(a)p Fz(;)g Fx(\032;)g(!)r Fz(\))k(=)i Fx( )989 2595 y Ff(\015)1013 2588 y Fz(\()p FC(x)p Fz(;)8 b FC(a)p Fz(\))p Fx(\020)1151 2595 y Fw(2)1171 2588 y Fz(\()p FC(m)p Fz(;)g Fx(\032;)g(!)r Fz(\))p Fx(;)-1144 b Fz(\(3.16\))p eop %%Page: 11 11 11 10 bop 604 127 a Fo(PERIODIC)16 b(SCHR)926 117 y(\177)921 127 y(ODINGER)i(OPERA)m(TOR)337 b(11)228 213 y Fz(with)22 b Fx(\032;)8 b(!)26 b(>)e Fz(0.)39 b(T)l(o)23 b(a)o(v)o(oid)f(cum)o(b)q (ersome)d(notation,)24 b(when)f(this)f(do)q(es)h(not)228 271 y(cause)14 b(an)o(y)g(confusion,)h(w)o(e)f(omit)f(some)g(or)i(all)e (of)i(the)f(parameters)f(on)i(whic)o(h)f Fx( )228 329 y Fz(dep)q(ends)j(and)g(write)e(simply)g Fx( )r Fz(\()p FC(x)p Fx(;)8 b FC(m)p Fz(\))14 b(or)i(just)h Fx( )r Fz(.)k(The)16 b(role)g(of)h(the)f(m)o(ultipli-)228 387 y(cation)f(b)o(y)f Fx(e)462 369 y Fu( )503 387 y Fz(in)h(\(3.10\))g (will)f(b)q(e)h(no)o(w)h(pla)o(y)o(ed)e(b)o(y)g(the)h(PDO)g(with)g(the) g(sym)o(b)q(ol)228 445 y Fx(b)e Fz(=)h(exp)8 b Fx( )r Fz(:)228 539 y(\(3.17\))42 b(\()p Fx(W)7 b(u)p Fz(\)\()p FC(x)p Fz(\))13 b(=)h(\()p Fx(W)7 b Fz(\()p FC(a)p Fz(;)h Fx(\032;)g(!)r Fz(\))p Fx(u)p Fz(\)\()p FC(x)p Fz(\))455 630 y(=)14 b(\(2)p Fx(\031)r Fz(\))599 609 y Fq(\000)631 595 y Fl(d)p 631 601 17 2 v 631 622 a Fm(2)662 582 y Fk(X)681 687 y Fs(m)742 630 y Fz(exp)817 589 y Fk(\000)840 630 y Fx( )r Fz(\()p FC(x)p Fx(;)8 b FC(m)p Fz(;)g FC(a)p Fz(;)g Fx(\032;)g(!)r Fz(\))1162 589 y Fk(\001)1185 630 y Fz(^)-27 b Fx(u)p Fz(\()p FC(m)p Fz(\))p Fx(e)1319 609 y Fu(i)p Fs(mx)1387 630 y Fx(;)24 b Ft(8)p Fx(u)12 b Ft(2)i Fx(L)1574 609 y Fw(2)1594 630 y Fz(\(\012\))p Fx(:)228 752 y Fz(The)i(prop)q(erties)g(of)h Fx(W)23 b Fz(are)16 b(summarised)e(in)228 840 y FC(Lemm)o(a)i(3.3.)24 b Fj(L)n(et)g Fz(\(3.11\))q Fj(,)c Fz(\(3.12\))g Fj(b)n(e)g(ful\014l)r (le)n(d)i(and)e(let)h(the)f(op)n(er)n(ator)e Fx(W)25 b Fz(=)228 898 y Fx(W)7 b Fz(\()p FC(a)p Fz(;)h Fx(\032;)g(!)r Fz(\))p Fx(;)g(\032)17 b(>)g Fz(0)p Fx(;)8 b(!)20 b(>)d Fz(0)j Fj(b)n(e)f(as)g(de\014ne)n(d)h(ab)n(ove.)29 b(Then)20 b Ft(k)p Fx(W)7 b Ft(k)16 b(\024)h Fx(C)t Fj(.)28 b(Mor)n(e-)228 956 y(over,)278 1025 y Fz(1.)20 b Fj(The)e(op)n(er)n(ator)e Fx(W)7 b Fz(\()p FC(a)p Fz(;)h Fx(\032;)g(!)r Fz(\))17 b Fj(maps)g Fx(C)1030 1007 y Fq(1)1067 1025 y Fz(\()p Fv(T)-5 b Fz(\))15 b Fj(into)i(itself;)278 1083 y Fz(2.)j Fj(F)l(or)d(any)h Fx(v)d Ft(2)f Fx(C)646 1065 y Fw(2)p Fu(d)p Fw(+3)728 1083 y Fz(\()p Fv(T)l Fz(\))228 1177 y(\(3.18\))42 b Ft(k)p Fz([)p Fx(W)7 b Fz(\()p FC(a)p Fz(;)h Fx(\032;)g(!)r Fz(\))p Fx(;)g(v)r Fz(]\()p FC(D)h Fz(+)i Fn(\026)p Fz(\))888 1157 y Fs(s)907 1177 y Ft(k)i(\024)h Fx(C)t Fz(\()p Fx(\032)d Fz(+)g Fx(!)i Fz(+)e(1\))1276 1157 y Fu(s)1295 1177 y Fz(\(1)g(+)g Fx(!)r Fz(\))1449 1157 y Fq(\000)p Fw(1)1496 1177 y Fx(;)1152 1253 y(\032)j Ft(\025)g Fz(6)p Fx(!)i(>)e Fz(0)p Fx(;)26 b(s)13 b Fz(=)h Ft(j)p FC(s)p Ft(j)g(\024)f Fz(1;)278 1335 y(3.)20 b Fj(F)l(or)j(a)g(su\016ciently)h(lar)n(ge)f Fx(!)885 1342 y Fw(0)929 1335 y Fz(=)h Fx(!)1021 1342 y Fw(0)1041 1335 y Fz(\()p FC(a)p 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2492 y Fz(F)l(or)16 b Fx(q)f Fz(=)f Fx(p)e Fz(+)f Fx(d)g Fz(+)g Fx(t)16 b Fz(and)h Fx(t)c(>)h Fv({)r Fz(,)i(this)g(pro)q(duces)h(the)f (b)q(ound)h(\(4.5\))q(.)278 2550 y(As)23 b(far)g(as)h(the)f (inequalities)e(\(4.6\))i(and)h(\(4.7\))g(are)f(concerned,)h(w)o(e)e (shall)228 2608 y(pro)o(v)o(e)17 b(only)h(\(4.6\))q(.)26 b(Applying)17 b FC(D)867 2590 y Fs(s)903 2608 y Fz(to)h(\(4.8\))g (directly)l(,)e(w)o(e)i(conclude)f(that)h(the)p eop %%Page: 17 17 17 16 bop 604 127 a Fo(PERIODIC)16 b(SCHR)926 117 y(\177)921 127 y(ODINGER)i(OPERA)m(TOR)337 b(17)228 213 y Fz(function)15 b FC(D)461 195 y Fs(s)487 213 y Fz(Op\()p Fx(b)592 220 y Fw(1)612 213 y Fz(\))8 b(Op\()p Fx(b)744 220 y Fw(2)763 213 y Fz(\))p Fx(u)p Fz(\()p FC(x)p Fz(\))15 b(is)g(a)h(linear)e(com)o (bination)g(of)i(terms)d(ha)o(ving)228 271 y(the)j(form)278 337 y Fk(X)302 441 y Fs(n)358 384 y FC(n)389 363 y Fs(s)405 368 y Fm(2)424 384 y FC(D)467 363 y Fs(s)483 368 y Fm(1)503 384 y Fx(b)524 391 y Fw(1)543 384 y Fz(\()p FC(y)q Fx(;)8 b 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Fz(In)16 b(view)f(of)20 b(\(4.9\))d(and)g(\(4.4\))453 961 y Ft(j)-8 b(j)g(j)g(j)p Fz(~)-25 b Fx(g)r Ft(j)-8 b(j)g(j)g(j)539 976 y Fu(p)573 961 y Ft(\024)14 b Fx(C)t Ft(j)-8 b(j)g(j)g(j)m Fx(b)715 968 y Fw(1)735 961 y Ft(j)g(j)g(j)g(j) 765 976 y Fu(p)p Fw(+)p Fu(s)845 961 y Ft(j)g(j)g(j)g(j)o Fx(b)897 968 y Fw(2)916 961 y Ft(j)g(j)g(j)g(j)947 976 y Fu(q)987 961 y Fz(sup)974 1002 y Fs(m)p Fq(2)p Fw(\003)1057 1007 y Fm(2)1089 914 y Fk(X)1081 1020 y Fs(n)p Fq(2)p Fw(\003)1152 1025 y Fm(1)1178 961 y Ft(j)p FC(n)p Ft(j)1237 941 y Fu(s)1255 961 y Fz(\(1)11 b(+)g Ft(j)p FC(n)g Ft(\000)g FC(m)p Ft(j)p Fz(\))1545 941 y Fu(p)p Fq(\000)p Fu(q)1608 961 y Fx(:)228 1097 y Fz(If)20 b(\003)315 1104 y Fw(1)357 1097 y Ft(\032)i Fz(\003,)g(then)f Ft(j)p FC(n)p Ft(j)663 1079 y Fu(s)703 1097 y Ft(\024)h Fx(R)801 1079 y Fu(s)841 1097 y Fz(and)f(th)o(us)h(\(4.6\))f(follo)o(ws.)36 b(If)20 b(\003)1455 1104 y Fw(2)1497 1097 y Ft(\032)i Fz(\003,)f(then)228 1155 y Ft(j)p FC(n)p Ft(j)287 1137 y Fu(s)319 1155 y Ft(\024)13 b Fx(C)t Fz(\()p Ft(j)p FC(n)e Ft(\000)g FC(m)p Ft(j)597 1137 y Fu(s)625 1155 y Fz(+)g Fx(R)711 1137 y Fu(s)730 1155 y Fz(\))16 b(whic)o(h)f(again)i(leads)g(to)f(\(4.6\))q (.)p 1687 1155 2 33 v 1689 1124 30 2 v 1689 1155 V 1718 1155 2 33 v 278 1246 a(This)g(Lemma)e(has)j(the)f(follo)o(wing)g (simple)e(but)i(useful)g(consequences:)228 1337 y FC(Corollary)j(4.3.)k Fj(L)n(et)e Fx(b)e Ft(2)i Fx(C)810 1319 y Fu(q)828 1337 y Fz(\()p Fv(T)-5 b Fz(\))18 b Fj(b)n(e)j(a)g(symb)n(ol,)g(ob)n(eying) 26 b Fz(\(4.2\))p Fj(,)c Fz(\(4.3\))f Fj(and)228 1395 y(let)d Fx(q)e(>)d Fz(2)p Fx(d)f Fz(+)f(1)h(+)f Fv({)r Fj(.)22 b(Supp)n(ose)c(that)600 1480 y Ft(j)p Fx(b)p Fz(\()p FC(x)p Fx(;)8 b FC(m)p Fz(\))p Ft(j)j(\025)i Fx(c)h(>)g Fz(0)p Fx(;)8 b Ft(8)p FC(x)k Ft(2)i Fz(\012)p Fx(;)26 b FC(m)12 b Ft(2)i Fv(Z)1320 1459 y Fu(d)1337 1480 y Fx(:)228 1565 y Fj(Then)i(ther)n(e)g(exists)h(a)f(numb)n(er)h Fx(\016)841 1572 y Fw(0)874 1565 y Fx(>)c Fz(0)p Fj(,)k(which)g(dep)n (ends)f(only)h(on)f Fx(d;)8 b Fv({)r Fx(;)g(q)r Fj(,)16 b(such)228 1623 y(that)h(under)h(the)g(c)n(ondition)584 1708 y Ft(h)p Fx(b)p Ft(i)643 1715 y Fu(d)p Fw(+1)p Fu(;)p Fg({)743 1708 y Ft(j)-8 b(j)g(j)g(j)o Fx(b)795 1687 y Fq(\000)p Fw(1)842 1708 y Ft(j)g(j)g(j)g(j)872 1722 y Fu(q)902 1708 y Fz(+)11 b Ft(h)p Fx(b)991 1687 y Fq(\000)p Fw(1)1038 1708 y Ft(i)1057 1715 y Fu(d)p Fw(+1)p Fu(;)p Fg({)1158 1708 y Ft(j)-8 b(j)g(j)g(j)n Fx(b)p Ft(j)g(j)g(j)g(j)1240 1722 y Fu(q)1272 1708 y Ft(\024)14 b Fx(\016)1347 1715 y Fw(0)228 1799 y Fj(the)g(op)n(er)n(ator)e Fx(T)20 b Fz(=)14 b(Op\()p Fx(b)p Fz(\))f Fj(is)h(b)n(ounde)n(d)r(ly)g (invertible,)i Fx(T)1261 1781 y Fq(\000)p Fw(1)1321 1799 y Fj(is)e(de\014ne)n(d)g(on)g Fx(L)1629 1781 y Fw(2)1649 1799 y Fz(\(\012\))228 1857 y Fj(and)552 1942 y Ft(k)p Fx(T)7 b Ft(k)13 b(\024)g Fx(C)t Ft(j)-8 b(j)g(j)g(j)n Fx(b)p Ft(j)g(j)g(j)g(j)823 1956 y Fu(d)p Fw(+1)889 1942 y Fx(;)42 b Ft(k)p Fx(T)1006 1921 y Fq(\000)p Fw(1)1053 1942 y Ft(k)13 b(\024)h Fx(C)t Ft(j)-8 b(j)g(j)g(j)n 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Fq(\000)p Fw(1)1397 2588 y Fx(:)p eop %%Page: 18 18 18 17 bop 228 118 a Fo(18)568 b(A.V.)15 b(SOBOLEV)228 213 y Fz(Consequen)o(tly)433 311 y Ft(k)p Fx(T)494 290 y Fq(\000)p Fw(1)540 311 y Ft(k)f(\024)f Fz(\(1)f Ft(\000)f Fx(C)t(\016)797 318 y Fw(0)815 311 y Fz(\))834 290 y Fq(\000)p Fw(1)881 311 y Ft(k)p Fx(T)942 290 y Fq(y)959 311 y Ft(k)p Fx(;)41 b Ft(k)p Fz(\()p Fx(T)1119 290 y Fq(y)1136 311 y Fz(\))1155 290 y Fq(\000)p Fw(1)1202 311 y Ft(k)14 b(\024)f Fz(\(1)f Ft(\000)e Fx(C)t(\016)1458 318 y Fw(0)1477 311 y Fz(\))1496 290 y Fq(\000)p Fw(1)1543 311 y Ft(k)p Fx(T)d Ft(k)p Fx(:)228 412 y Fz(No)o(w)16 b(the)g(estimates)e(\(4.10\))j(follo)o(w)f(from)f(Lemma)g(4.1.)p 1687 412 2 33 v 1689 381 30 2 v 1689 412 V 1718 412 2 33 v 228 511 a FC(Corollary)k(4.4.)k Fj(L)n(et)g Fx(b)700 518 y Fw(1)719 511 y Fx(;)8 b(b)762 518 y Fw(2)805 511 y Ft(2)24 b Fx(C)901 493 y Fu(q)920 511 y Fz(\()p Fv(T)-5 b Fz(\))p Fx(;)27 b(q)f Fz(=)d(2)p Fx(d)16 b Fz(+)f(1)h(+)f Fx(t;)30 b(t)23 b(>)h Fz(0)p 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y(sym)o(b)q(ol)d(\(3.1\).)21 b(W)l(rite)269 821 y(\()p FC(m)10 b Fz(+)h Fn(\026)p Fz(\))p FC(G)p Fz(\()p FC(m)f Fz(+)h Fn(\026)p Fz(\))g Ft(\000)g Fx(\032)759 801 y Fw(2)449 901 y Fz(=)i Ft(j)p FC(F)p Fz(\()p FC(m)d Fz(+)h Fn(\026)p Fz(\))p Ft(j)743 881 y Fw(2)774 901 y Ft(\000)g Fx(\032)849 881 y Fw(2)882 901 y Fz(=)j(\()p Ft(j)p FC(F)p Fz(\()p FC(m)c Fz(+)h Fn(\026)p Fz(\))p Ft(j)g(\000)g Fx(\032)p Fz(\)\()p Ft(j)p FC(F)p Fz(\()p FC(m)f Fz(+)h Fn(\026)p Fz(\))p Ft(j)g Fz(+)g Fx(\032)p Fz(\))p Fx(:)228 982 y Fz(Therefore,)k(taking)i(in)o(to)e(accoun)o(t)i(the)f(estimate) 555 1062 y Ft(jj)p FC(F)p Fz(\()p FC(m)9 b Fz(+)i Fn(\026)p Fz(\))p Ft(j)g(\000)g Fx(\032)p Ft(j)j(\024)g Fz(2)p Fx(!)r(;)24 b FC(m)13 b Ft(2)h Fx(M)5 b Fz(\()p Fx(\032;)j Fz(2)p Fx(!)r Fz(\))p Fx(;)228 1142 y Fz(w)o(e)16 b(ha)o(v)o(e)474 1222 y Ft(j)p FC(F)p Fz(\()p FC(m)10 b Fz(+)h Fn(\026)p Fz(\))p Ft(j)717 1202 y Fw(2)748 1222 y Ft(\000)g Fx(\032)823 1202 y Fw(2)853 1222 y Ft(\000)g Fz(2)p Fx(\032)p Fz(\()p Ft(j)p 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Fz(+)d(1\))776 1538 y Fq(\000)p Fw(1)824 1518 y Fk(\000)846 1558 y Ft(j)-8 b(j)g(j)g(j)o FC(a)p Ft(j)g(j)g(j)g(j)934 1534 y Fw(2)934 1573 y(2)p Fu(d)p Fw(+3)1029 1558 y Fz(+)11 b Ft(j)-8 b(j)g(j)g(j)n FC(a)p Ft(j)g(j)g(j)g(j)1166 1573 y Fw(2)p Fu(d)p Fw(+3)1260 1558 y Fz(+)11 b Ft(j)-8 b(j)g(j)g(j)n FC(a)1366 1538 y Fq(0)1378 1558 y Ft(j)g(j)g(j)g(j)1408 1573 y Fw(2)p Fu(d)p Fw(+3)1491 1518 y Fk(\001)228 1640 y Fz(b)o(y)13 b(virtue)g(of)k(\(4.14\))q(.)j(Along)14 b(with)g(\(3.24\))g(this)g(estimate)d(leads)j(to)g(\(3.25\))q(.)p 1687 1640 V 1689 1608 30 2 v 1689 1640 V 1718 1640 2 33 v 465 1738 a(6.)27 b Fy(Lo)o(wer)18 b(bounds:)23 b(pr)o(oof)17 b(of)i(Theorem)e(3.6)228 1826 y Fz(6.1.)28 b FC(Lo)n(w)n(er)f(b)r (ounds)g(on)g(the)f(subspaces)h Fh(M)g FC(and)h Fh(N)p FC(.)c Fz(The)f(pro)q(of)i(of)228 1884 y(Theorem)15 b(3.6)h(relies)f (on)i(the)f(follo)o(wing)g(t)o(w)o(o)g(lemmas:)228 1970 y FC(Lemm)o(a)g(6.1.)24 b Fj(L)n(et)e Fn(\013)g Fj(b)n(e)g(as)g(in)k Fz(\(2.9\))p Fj(,)d FC(a)g Ft(2)g Fx(L)1163 1952 y Fq(1)1200 1970 y Fz(\(\012\))f Fj(and)h FC(a)1422 1952 y Fq(00)1465 1970 y Fj(ob)n(ey)k Fz(\(3.26\))p Fj(.)228 2028 y(L)n(et)20 b Fn(\026)h Fj(b)n(e)g(as)g(in)k Fz(\(2.7\))c Fj(with)g Fx(\014)h Fz(=)e Ft(j)p FC(G)p Fn(\013)p Ft(jj)p Fz(2)p FC(n)p Ft(j)1098 2010 y Fq(\000)p Fw(1)1145 2028 y Fj(.)32 b(Then)22 b(ther)n(e)e(exist)i(p)n(ositive)228 2086 y(numb)n(ers)17 b Fx(!)451 2093 y Fw(0)485 2086 y Fz(=)d Fx(!)567 2093 y Fw(0)587 2086 y Fz(\()p FC(a)p Fz(\))p Fx(;)26 b(\032)717 2093 y Fw(0)750 2086 y Fz(=)14 b Fx(\032)827 2093 y Fw(0)847 2086 y Fz(\()p FC(a)p Fx(;)8 b(!)r Fz(\))18 b Fj(such)g(that)433 2178 y Ft(k)478 2166 y Fz(~)458 2178 y Ft(H)500 2185 y Fs(a)522 2178 y Fz(\()p Fx(\032;)8 b(!)r Fz(\))p Fx(P)f Fz(\()p Fx(\032;)h(!)r Fz(\))p Fx(u)p Ft(k)14 b(\025)f Fx(f)937 2185 y Fq(\000)1001 2145 y Fx(\032)p 972 2167 84 2 v 972 2213 a Fz(2)p Ft(j)p FC(n)p Ft(j)1060 2178 y(k)p Fx(P)7 b Fz(\()p Fx(\032;)h(!)r Fz(\))p Fx(u)p Ft(k)p Fx(;)25 b Ft(8)p Fx(u)12 b Ft(2)i Fx(C)1487 2158 y Fq(1)1524 2178 y Fz(\()p Fv(T)-5 b Fz(\))p Fx(;)-1384 b Fz(\(6.1\))228 2287 y Fj(for)17 b(al)r(l)h Fx(!)e Ft(\025)e Fx(!)504 2294 y Fw(0)524 2287 y Fx(;)25 b(\032)14 b Ft(\025)g Fx(\032)680 2294 y Fw(0)700 2287 y Fj(,)j(uniformly)g(in)h Fn(\026)1049 2299 y Fq(?)1093 2287 y Fz(:)13 b Ft(j)p Fn(\026)1169 2299 y Fq(?)1199 2287 y Ft(j)h(\024)f Fx(C)t Fj(.)228 2373 y(Pr)n(o)n(of.)19 b Fz(F)l(or)d Fx(P)7 b Fz(\()p Fx(\032;)h(!)r Fz(\))14 b(=)g Fx(P)7 b Fz(\()p Fx(\032;)h(!)r Fz(\))p Fx(\020)861 2380 y Fw(1)882 2373 y Fz(\()p FC(D)p Fz(;)g Fx(\032;)g(!)r Fz(\),)15 b(w)o(e)h(ha)o(v)o(e)f (for)i Fx(u)d Fz(=)f Fx(P)7 b Fz(\()p Fx(\032;)h(!)r Fz(\))p Fx(u)257 2514 y Ft(k)296 2501 y Fz(~)282 2514 y Ft(H)324 2521 y Fs(a)345 2514 y Fz(\()p Fx(\032;)g(!)r Fz(\))p Fx(u)p Ft(k)14 b(\025)g Fz(2)p Fx(\032)p Ft(k)p Fx(T)685 2521 y Fw(0)704 2514 y Fz(\()p Fx(\032)p Fz(\))p Fx(u)p Ft(k)d(\000)g(k)906 2474 y Fk(\000)943 2501 y Fz(~)929 2514 y Ft(H)971 2521 y Fs(a)993 2514 y Fz(\()p 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Fn(\026)p Fz(\))1475 1109 y Fk(\003)1496 1150 y Fx(u)p Ft(k)752 1230 y(\024)16 b(kH)874 1237 y Fw(0)894 1230 y Fz(\()p Fx(\032)p Fz(\))p Fx(u)p Ft(k)11 b Fz(+)g Fx(\032)1095 1209 y Fw(2)1114 1230 y Ft(k)p Fx(u)p Ft(k)p Fx(;)228 1314 y Fz(w)o(e)16 b(obtain)g(from)g(here)f(that)540 1398 y Ft(kX)7 b Fz(\()p Fx(\032)p Fz(\))p Fx(u)p Ft(k)13 b(\024)h Fx(\032)815 1378 y Fq(\000)p Fw(1)862 1398 y Ft(kH)929 1405 y Fw(0)949 1398 y Fz(\()p Fx(\032)p Fz(\))p Fx(u)p Ft(k)d Fz(+)g Fx(C)t Fz(\()p Fx(\032)f Fz(+)h(1\))p Ft(k)p Fx(u)p Ft(k)p Fx(;)-1174 b Fz(\(6.3\))228 1483 y(with)16 b(some)f(constat)i Fx(C)j Fz(dep)q(ending)c(on)h FC(a)f Fz(and)h Fx(f)1151 1490 y Fw(+)1181 1483 y Fz(.)228 1573 y Fj(Pr)n(o)n(of)f(of)h(L)n(emma)f(6.2.)21 b Fz(Since)c(the)g (second)h(op)q(erator)h(in)f(the)g(r.h.s.)25 b(of)d(\(3.20\))228 1631 y(is)13 b(b)q(ounded)i(b)o(y)e Fx(C)t(\032;)8 b(C)16 b Fz(=)e Fx(C)t Fz(\()p FC(a)p Fz(\),)f(uniformly)f(in)h Fx(!)r Fz(,)h(it)f(su\016ces)g(to)h(establish)f(the)228 1689 y(b)q(ound)k(of)g(the)f(form)f(\(6.2\))h(for)h(the)f(op)q(erator)h Ft(H)1146 1696 y Fs(a)1168 1689 y Fz(\()p Fx(\032)p Fz(\))g(only)l(.)k (Precisely)l(,)589 1798 y Ft(kH)656 1805 y Fs(a)678 1798 y Fz(\()p Fx(\032)p Fz(\))p Fx(Qu)p Ft(k)13 b(\025)904 1764 y Fz(3)p Fx(!)r(\032)p 904 1786 82 2 v 932 1832 a Fz(4)990 1798 y Ft(k)p Fx(Qu)p Ft(k)e(\000)g Fx(C)t(\032)p Ft(k)p Fx(Qu)p Ft(k)p Fx(:)-1135 b Fz(\(6.4\))228 1899 y(It)16 b(follo)o(ws)g(from)f(\(2.8\))h(and)h(\(6.3\))g(that)f(for)h Fx(\032)d Ft(\025)g Fz(4)i(one)h(has)594 1984 y Ft(kH)661 1991 y Fs(a)683 1984 y Fz(\()p Fx(\032)p Fz(\))p Fx(u)p Ft(k)c(\025)h Fz(3)p Fx(=)p Fz(4)p Ft(kH)1004 1991 y Fw(0)1025 1984 y Fz(\()p Fx(\032)p Fz(\))p Fx(u)p Ft(k)d(\000)f Fx(C)t(\032)p Ft(k)p Fx(u)p Ft(k)p Fx(:)228 2071 y Fz(No)o(w)17 b(\(6.4\))f(follo)o(ws)g(from)f(\(3.2\))q(.)p 1687 2071 2 33 v 1689 2039 30 2 v 1689 2071 V 1718 2071 2 33 v 228 2169 a(6.2.)28 b FC(Pro)r(of)e(of)g(Theorem)d(3.6.)h Fz(Let)f Fx(P)31 b Fz(=)25 b Fx(P)7 b Fz(\()p 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(Septem)o(b)q(er)d(1997.)p eop %%Page: 28 28 28 27 bop 228 118 a Fo(28)568 b(A.V.)15 b(SOBOLEV)830 213 y Fy(References)249 291 y FB(1.)20 b(M.)12 b(Birman,)f(T.)h (Suslina,)f Fp(Two-dimensional)j(p)n(erio)n(dic)f(magnetic)h (Hamiltonian)g(is)f(ab-)302 341 y(solutely)h(c)n(ontinuous)p FB(,)h(Ro)o(y)o(al)d(Institute)i(of)g(T)m(ec)o(hnology)m(,)e(preprin)o (t,)i(Septem)o(b)q(er)g(1996.)249 391 y(2.)20 b(L.)d(Danilo)o(v,)f Fp(On)i(the)h(sp)n(e)n(ctrum)f(of)g(the)g(Dir)n(ac)h(op)n(er)n(ator)e (with)h(p)n(erio)n(dic)g(p)n(otential)g(in)302 441 y Fd(R)335 426 y Fc(n)354 441 y FB(,)c(Theor.)f(and)h(Math.)g(Ph)o(ys.)f Fb(85)h FB(\(1990\),)f(1039{1048.)249 491 y(3.)20 b(B.A.)11 b(Dubro)o(vin,)g(S.P)m(.)g(No)o(vik)o(o)o(v,)f Fp(Gr)n(ound)k(states)e (in)i(a)f(p)n(erio)n(dic)f(\014eld.)h(Magnetic)h(Blo)n(ch)302 541 y(functions)h(and)h(ve)n(ctor)e(bund)r(les)i FB(,)d(So)o(viet)h (Math.)f(Dokl.)f Fb(22)p FB(,)h(1)h(\(1980\),)f(240{244.)249 590 y(4.)20 b(G.)14 b(Eskin,)i(J.)g(Ralston,)e Fp(Inverse)j(sc)n (attering)f(pr)n(oblem)g(for)g(the)h(Schr\177)-21 b(odinger)17 b(e)n(quation)302 640 y(with)i(magnetic)i(p)n(otential)f(at)h(a)g (\014xe)n(d)g(ener)n(gy)p FB(,)g(Comm.)16 b(Math.)k(Ph)o(ys.)g Fb(173)f FB(\(1995\),)302 690 y(199{224.)249 740 y(5.)h(R.)10 b(Hemp)q(el,)h(I.)g(Herbst,)i Fp(Str)n(ong)g(magnetic)g(\014elds,)g (Dirichlet)f(b)n(oundaries,)i(and)f(sp)n(e)n(ctr)n(al)302 790 y(gaps)p FB(,)g(Comm.)e(Math.)i(Ph)o(ys.)h Fb(164)p FB(\(1995\),)e(237{259.)249 839 y(6.)p 302 839 125 2 v 151 w(,)g Fp(Bands)i(and)g(gaps)g(for)f(p)n(erio)n(dic)f(magnetic)i (Hamiltonians)p FB(,)e(in)g(Op)q(erator)h(The-)302 889 y(ory:)18 b(Adv)n(ances)c(and)g(Applications,)f(v)o(ol.)f(78,)h (Birkh\177)-21 b(auser,)14 b(1995,)f(175{184.)249 939 y(7.)20 b(P)m(.)i(Kuc)o(hmen)o(t,)k Fp(Flo)n(quet)d(the)n(ory)h(for)f (p)n(artial)g(di\013er)n(ential)h(e)n(quations)p FB(,)i(Birkh\177)-21 b(auser,)302 989 y(Basel,)13 b(1993.)249 1039 y(8.)20 b(T.)13 b(Kato,)g Fp(Perturb)n(ation)i(the)n(ory)f(for)h(line)n(ar)f (op)n(er)n(ators)p FB(,)f(Springer,)h(New)g(Y)m(ork,)f(1966.)249 1088 y(9.)20 b(M.)c(Reed,)h(B.)f(Simon,)f Fp(Metho)n(ds)j(of)f(mo)n (dern)h(mathematic)n(al)f(physics,)h(IV)p FB(,)e(Academic)302 1138 y(Press,)f(New)f(Y)m(ork,)f(1975.)228 1188 y(10.)20 b(Z.)9 b(Sun,)i Fp(A)o(n)g(inverse)g(b)n(oundary)h(vaue)g(pr)n(oblem)e (for)h(Schr\177)-21 b(odinger)11 b(op)n(er)n(ators)g(with)g(ve)n(ctor) 302 1238 y(p)n(otentials)p FB(,)i(T)m(rans.)g(of)g(AMS)i Fb(338)e FB(\(1993\),)g(953{969.)228 1288 y(11.)20 b(T.)13 b(Suslina,)g Fp(Private)h(c)n(ommunic)n(ation)p FB(.)228 1338 y(12.)20 b(L.E.)10 b(Thomas,)f Fp(Time)j(dep)n(endent)h(appr)n(o)n (ach)f(to)g(sc)n(attering)g(fr)n(om)f(impurities)g(in)h(a)h(crys-)302 1387 y(tal)p FB(,)g(Comm.)d(Math.)j(Ph)o(ys.)h Fb(33)g FB(\(1973\),)e(335{343.)228 1437 y(13.)20 b(A.)13 b(Zygm)o(und,)f Fp(T)m(rigonometric)i(series,)g(II)p FB(,)f(Cam)o(bridge)f(Univ)o (ersit)o(y)i(Press,)h(1959.)278 1531 y FA(Centre)e(f)o(or)h(Ma)m(thema) m(tical)g(Anal)m(ysis)g(and)h(Its)d(Applica)m(tions,)i(University)228 1580 y(of)i(Sussex,)h(F)-5 b(almer,)17 b(Brighton,)f(BN1)f(9QH,)h(UK) 278 1630 y Fp(E-mail)e(addr)n(ess)s FB(:)k Fa(A.V.Sobolev@sussex.a)o (c.uk)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ENDBODY