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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
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1594 y Fx(H)20 b Fz(on)15 b(the)h(domain)e Fx(D)q Fz(\()p
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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
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Ft(2)g Fx(C)1092 1781 y Fw(2)p Fu(d)p Fw(+3)1175 1799
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228 1857 y(satisfying)g Fz(\(2.2\))p Fj(,)d Fz(\(2.3\))p
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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
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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
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2063 y Fz(2.2.)28 b FC(Flo)r(quet)11 b(decomp)r(osition.)21
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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
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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
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eop
%%Page: 5 5
5 4 bop 604 127 a Fo(PERIODIC)16 b(SCHR)926 117 y(\177)921
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y Fz(=)970 962 y Fk(Z)1002 1067 y Fw(~)998 1075 y(\012)1033
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Ft(\000)g Fx(\032)1579 2322 y Fw(2)1599 2341 y Fx(;)469
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6 5 bop 228 118 a Fo(6)587 b(A.V.)15 b(SOBOLEV)228 213
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900 y Fr(\014xed)i Fz(v)m(alue)g Fn(\026)507 912 y Fq(?)536
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1126 y(from)15 b(the)h(pap)q(er)h([1].)278 1188 y(Since)j
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1238 y(~)228 1250 y Fx(H)t Fz(\()p Fx(k)r Fz(\))c(in)g(\(2.6\))h(form)e
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1295 y Fq(\003)1545 1313 y Fz(=)1612 1300 y(~)1599 1313
y Fx(H)5 b Fz(\()1664 1300 y(\026)1663 1313 y Fx(k)r
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825 1362 y(~)813 1375 y Fx(H)t Fz(\()p Fx(k)r Fz(\))8
b Ft(\000)g Fx(z)r Fz(\))1021 1357 y Fq(\000)p Fw(1)1081
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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
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1155 y Ft(j)p FC(n)p Ft(j)287 1137 y Fu(s)319 1155 y
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Fj(L)n(et)e Fx(b)e Ft(2)i Fx(C)810 1319 y Fu(q)828 1337
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26 b Fz(\(4.2\))p Fj(,)c Fz(\(4.3\))f Fj(and)228 1395
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Fj(.)22 b(Supp)n(ose)c(that)600 1480 y Ft(j)p Fx(b)p
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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
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Fq(\000)p Fw(1)842 1708 y Ft(j)g(j)g(j)g(j)872 1722 y
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1722 y Fu(q)1272 1708 y Ft(\024)14 b Fx(\016)1347 1715
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1799 y Fz(\(\012\))228 1857 y Fj(and)552 1942 y Ft(k)p
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%%Page: 18 18
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%%Page: 22 22
22 21 bop 228 118 a Fo(22)568 b(A.V.)15 b(SOBOLEV)228
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eop
%%Page: 23 23
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1203 y Fk(\000)1281 1244 y Fx(i)p FC(e)q Fz(+)q Fn(\027)s
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1326 y Fw(1)1184 1319 y Fx(;)d FC(a)p Fz(])j(+)g([)p
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1407 y Fw(1)1074 1400 y Fx(;)d FC(a)1123 1379 y Fq(0)1134
1400 y Fz(])p FC(F)1183 1359 y Fk(\000)1206 1400 y Fx(i)p
FC(e)i Fz(+)h Fn(\027)s Fz(\()p FC(D)p Fz(\))1421 1359
y Fk(\001)1444 1400 y Fx(\020)1465 1407 y Fw(2)1485 1400
y Fz(\()p FC(D)p Fz(;)d Fx(\032;)g(!)r Fz(\))p Fx(:)228
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y Fz(b)o(y)13 b(virtue)g(of)k(\(4.14\))q(.)j(Along)14
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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
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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
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%%Page: 24 24
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960 y Fw(+)1465 953 y Fx(A\032)g Fz(+)g Fx(\032)p Fz(\))p
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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
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%%Page: 25 25
25 24 bop 604 127 a Fo(PERIODIC)16 b(SCHR)926 117 y(\177)921
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%%Page: 26 26
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%%Page: 27 27
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Fx(C)t(\032)1258 1165 y Fq(\000)1302 1152 y Fm(2)p Fl(t)p
1290 1158 V 1290 1178 a Fm(2)p Fe(\000)p Fl(t)1346 1165
y Fw(+)p Fu(d)p Fq(\000)p Fw(1)1439 1187 y Fx(:)228 1326
y Fz(T)l(o)g(handle)f(\006)486 1333 y Fw(1)520 1326 y
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