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periodic elliptic operators, band gap structure, spectral gaps
---------------0207160616603
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e
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e
n
1234 5234 m
scm
n
1234 5234 m
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e
n
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n
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1234 5316 m
1234 5316 l
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e
n
1234 5316 m
scm
n
1234 5316 m
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Fm(M)2918 2101 y Fl(0)2985 2086 y Fo(=)28 b Fm(Q)22 b
Fn(\\)h Fm(M)10 b Fo(.)240 2387 y(factor)36 b(in)g(the)h(metric)f
(tensor)i(is)e(discussed)i(in)e(Section)h(5.)56 b(W)-8
b(e)37 b(also)f(refer)h(to)f(construc-)240 2503 y(tions)d(of)f(a)h
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240 2852 y(crystals.)967 3143 y(1.)48 b Fp(Preliminaries)38
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(enough)g(to)h(consider)240 3550 y(\000)40 b(=)h Fk(Z)527
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Fm(Q)g Fo(=)f(\(0)p Fm(;)17 b Fo(1])3594 3514 y Fj(m)240
3666 y Fo(if)39 b(\000)h(=)g Fk(Z)623 3630 y Fj(m)687
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(spaces)i(are)f Fn(H)f Fo(=)e Fm(L)1893 4146 y Fl(2)1933
4131 y Fo(\()p Fk(R)2037 4095 y Fj(m)2110 4131 y Fo(\))i(and)i
Fn(H)2462 4095 y Fl(1)2501 4131 y Fo(\()p Fk(R)2605 4095
y Fj(m)2678 4131 y Fo(\),)f(the)h(Sob)s(olev)e(space)i(of)240
4247 y Fm(L)306 4262 y Fl(2)346 4247 y Fo(-functions)32
b(that)g(ha)m(v)m(e)i(w)m(eak)g(deriv)-5 b(ativ)m(es)33
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y(All)39 b(our)i(op)s(erators)h(are)f(semi-b)s(ounded,)i(self-adjoin)m
(t)d(op)s(erators)h Fm(T)55 b Fo(acting)40 b(in)h(some)240
4480 y Fm(L)306 4495 y Fl(2)346 4480 y Fo(-space;)47
b(they)c(are)f(obtained)f(from)g(some)h(closed,)i(semi-b)s(ounded)e
(quadratic)f(form)g Fg(t)240 4596 y Fo(via)d(the)i(usual)f(represen)m
(tation)h(theorem)f(\([K,)g(RS-I]\))g(and)g(satisfy)g
Fm(D)s Fo(\()p Fm(T)14 b Fo(\))39 b Fn(\032)g Fm(D)s
Fo(\()p Fg(t)p Fo(\))g(and)240 4712 y(the)33 b(relation)1018
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Fm(D)s Fo(\()p Fm(T)14 b Fo(\))p Fm(;)113 b(v)32 b Fn(2)c
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Fn(\032)g Fm(D)s Fo(\()p Fg(t)3583 5046 y Fl(1)3622 5031
y Fo(\))240 5147 y(and)g Fg(t)458 5162 y Fl(2)498 5147
y Fo([)p Fm(u;)17 b(u)p Fo(])26 b Fn(\025)j Fg(t)873
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b(for)g(all)e Fm(u)h Fn(2)h Fm(D)s Fo(\()p Fg(t)1785
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%%Page: 3 3
3 2 bop 120 66 a Fh(SPECTRAL)33 b(GAPS)h(F)n(OR)e(PERIODIC)h(ELLIPTIC)h
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b(VIEW)-38 b(3)220 266 y Fo(F)-8 b(or)33 b(a)g(sequence)k(of)d
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Fm(T)2593 281 y Fj(n)2674 266 y Fo(in)f Fn(H)i Fo(w)m(e)g(sa)m(y)g
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Fo(i\013)972 540 y(\()p Fm(T)1067 555 y Fj(n)1136 540
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540 y Fm(f)38 b Fn(!)28 b Fo(\()p Fm(T)1726 555 y Ff(1)1823
540 y Fo(+)22 b Fm(I)8 b Fo(\))2010 499 y Ff(\000)p Fl(1)2104
540 y Fm(f)d(;)212 b(f)38 b Fn(2)28 b(H)q Fo(;)651 b(\(1)p
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y(Most)e(of)e(our)i(results)f(hinge)g(up)s(on)h(the)g(follo)m(wing)c
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1048 y(forms)32 b(\([RS-I],)g([S]\):)120 1223 y(1.1)p
Fx(.)45 b(Theorem.)h Fe(L)-5 b(et)41 b Fo(\()p Fm(T)1105
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1339 y(op)-5 b(er)g(ators.)63 b(Then)40 b(ther)-5 b(e)41
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Fn(Q)1808 1762 y Ff(1)1914 1747 y Fo(=)30 b Fn(f)p Fm(u)g
Fn(2)g(\\)2318 1711 y Ff(1)2318 1772 y Fj(n)p Fl(=1)2456
1747 y Fn(Q)2537 1762 y Fj(n)2584 1747 y Fo(;)17 b(sup)g
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(smaller)f(Hilb)s(ert)g(space)i(than)g(the)g Fm(T)3321
1878 y Fj(n)3368 1863 y Fo(.)56 b(In)120 1979 y(this)31
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2640 y Fl(1)3105 2676 y Fo(\()p Fm(M)3237 2691 y Fl(0)3276
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2818 y(k)3407 2792 y Fo(\()p Fm(\025)p Fo(\))120 2908
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Fn(H)2949 2872 y Fl(1)2948 2933 y(0)2989 2908 y Fo(\()p
Fm(Q)p Fo(\),)i(the)f Fn(H)3468 2872 y Fl(1)3507 2908
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3049 y Fj(c)708 3025 y Fo(\()p Fm(Q)p Fo(\),)h(as)g(transition)e(op)s
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3199 y(5.1)p Fx(.)27 b(Theorem.)35 b Fo(\([P-1,2]\))27
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y Fo(\))h Fe(c)-5 b(onver)g(ges)25 b(to)i(the)g(sp)-5
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Fk(R)1864 2744 y Fl(2)1945 2780 y Fo(or)g(in)g Fk(R)2252
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3852 y Fw([A])207 b(C.)25 b(Ann)n(\023)-39 b(e,)26 b
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3852 y([KK])139 b(P)-7 b(.)20 b(Kuc)n(hmen)n(t)g(and)h(L.)g(Kun)n(y)n
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b(anal-)435 4848 y(ysis.)f(R)l(evise)l(d)f(and)g(enlar)l(ge)l(d)h(e)l
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12 11 bop 240 66 a Fh(12)963 b(RAINER)23 b(HEMPEL)k(AND)d(OLAF)h(POST)
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Fv(Institut)j(f)764 944 y(\177)761 951 y(ur)i(Anal)-6
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b(WTH)45 b(Aa)n(chen,)j(Templer-)240 1419 y(graben)32
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%%Trailer
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%%EOF

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