The one-dimensional periodic Schrödinger equation
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1 The one-dmensonal perodc Schrödnger equaon Jordan Bell Deparmen of Mahemacs, Unversy of Torono Aprl 23, 26 Translaons and convoluon For y, le τ y f(x f(x y. To say ha f : C s unformly connuous means ha τ h f f b as h, where g b sup g(x. x Le p < and le L (L p ( be he Banach algebra of bounded lnear operaors L p ( L p (, wh he srong operaor opology: a ne T converges o T n he srong operaor opology f and only f for each f L p (, T f T f L p. Lemma. y τ y s connuous L (L p (, usng he srong operaor opology. Proof. For y and f L p (, τ y+h f τ y f L p τ h f f L p. Take ɛ > and le φ C c ( wh f φ L p <. Say supp φ [a, b]. Le K [a, b + ]. For h, f x / K hen x h, x supp φ, and hence τ h φ φ p L φ(x h φ(x p dx p φ(x h φ(x p dx K (b a + 2 τ h φ φ p b (b a + 2 τ φ τ y φ p b.
2 Because φ C c (, φ s unformly connuous on, whence τ h φ φ L p as h, say τ h φ φ L p < ɛ for h h ɛ. Hence τ y+h f τ y f L p τ h f f L p τ h f τ h φ L p + τ h φ φ L p + φ f L p 2 f φ L p + τ h φ L p < 3ɛ. Defne A : by A(x, x 2 x + x 2. If µ, µ 2 are fne Borel measures on, le µ µ 2 be he produc measure on 2, and le µ µ 2 A (µ µ 2 be he pushforward of µ µ 2 by A, called he convoluon of µ and µ 2. If f : [, ] s measurable hen applyng he change of varables formula and hen Tonell s heorem we oban fd(µ µ 2 f Ad(µ µ 2 ( f A(x, x 2 dµ (x dµ 2 (x 2 ( f(x + x 2 dµ (x dµ 2 (x 2. If B s a Borel se n hen applyng he above wh f B, (µ µ 2 (B B d(µ µ 2 ( B (x + x 2 dµ (x dµ 2 (x 2 µ (B x 2 dµ 2 (x 2. 2 Perodc funcons Le T /Z, and le S (T be he collecon of C funcons φ : C sasfyng φ(x + φ(x for all x T. For φ, ψ S (T, for n le and d n (φ, ψ d(φ, ψ sup φ (n (x ψ (n (x x [,] n 2 n d n (φ, ψ + d n (φ, ψ. 2
3 Wh hs merc, S (T s a Fréche space. For n Z, defne e n (x e 2πnx, x. For f L (T, defne f : Z C, for n Z, by φ(n φ(xe n (xdx φ(xe 2πnx dx. Denoe by S (T he dual space of S (T, he collecon of connuous lnear maps S (T C. For L S (T, defne L : Z C by For x, defne δ x : S (T C by δ x belongs o S (T, and L(n Le n. δ x φ φ(x. δ x (n δ x e n e n (x e 2πnx. For f L (T, defne L f S (T by For n Z, L f φ L f (n L f e n f(xφ(xdx, φ S (T. f(xe n (xdx f(n. 3 The Posson summaon formula If f L (, f(x + n dx n+ f(x + n dx f(x dx n f(x dx. Ths mples ha here s a Borel se N f n wh λ(n f such ha for x Nf c, f(x + n <. 3
4 We defne P f(x f(x + n for x N c f and P f(x for x N f. Thus makes sense o defne P : L ( L ( by P f(x f(x + n, n oher words, P f τ n f. Then Tha s, P f(xe 2πmx dx ( f(x + n e 2πmx dx n+ f(x + ne 2πmx dx f(xe 2πmx dx n f(xe 2πmx dx f(m. P f(m f(m. Supposng ha P f(x P f(ne 2πnx, P f(x and supposng P f(x f(x + n, f(x + n he Posson summaon formula. For N, le For n Z, L N (n N N j L N N δ j/n e n N f(ne 2πnx N j N j f(ne 2πnx, δ j/n. e n (j/n N N j If n NZ hen L N (n and oherwse L N (n. Tha s, L N N N j δ j/n k Z L N (ke k k Z e Nk. e 2πnj/N. 4
5 4 The hea kernel For x and > defne H (x Usng e 4π2 ξ 2 e 2πξx dξ. ( 2π 2 aw2 + Jw dw a ( J 2 for 2 a 4π2 we ge a 8π 2 and J 2πx, and we calculae ( 2π H (x 8π 2 6π 2 4π2 x 2 ( 4π x2. 4 By he Fourer nverson heorem, For f L (, τ y f(ξ Ĥ (ξ e 4π2 ξ 2. 2a, f(x ye 2πξx dx e 2πξy f(ξ e n (y f(ξ. 5 The Schrödnger equaon on Le whch sasfes and x Γ(, x 2πx Γ(, x 2 e πx /, Γ(, x, 2 xγ(, x 4π2 x 2 Γ(, x 2 Γ(, x + πx 2 2 Γ(, x. 2 Γ(, x 2π Γ(, x Ths sasfes Γ(, x 2πx2 ( ( 2π 4π2 x 2 4π 2 4π 2 xγ(, x. Γ(, x Γ(, x 5
6 For f : C, le ψ(f(, x f Γ(, (x Ths sasfes ψ(f(, x f(yγ(, x ydy. f(y Γ(, x ydy f(y 4π 2 xψ(f(, x. We also calculae ψ(f(, x f(y Γ(, x ydy f(y e π(x y f(y ( πx2 Γ(, x f(y Le Usng f(y 4π 2 xγ(, x ydy 2 / dy + 2πxy ( π (y2 2xy f(xe 2πxy dx. ( 2π 2 aw2 + Jw dw a πy2 dy. ( J 2 2a, dy 6
7 we ge, wh a 2π and J 2πu, Γ(, x ψ( f( /, x/ Γ(, x f(yγ (, x y dy Γ(, x f ( x y Γ (, y dy ( 2 e πx / f(ue 2πu( x y du e πy2 dy ( e πx2 / f(ue 2πux/ e 2πuy+πy2 dy du ( e πx2 / 2π f(ue 2πux/ 2π 4π (2πu2 du e πx2 / f(ue 2πux/ ( πu2 du f(u ( πx2 + 2πux πu2 du f(ue π(x u2 du f(uγ(, x udu In oher words, ψ(f(, x. ψ(f(, x Γ(, x ψ( f( /, x/ 2 e πx / f(ξ ( π ( x 2 ξ dξ f(ξ ( πx2 + πx2 + 2πxξ + πξ 2 dξ f(ξe 2πxξ+πξ2 dξ. 7
8 6 The Schrödnger equaon on T Gven and x, le γ(y Γ(, x y. We calculae γ(ξ γ(ye 2πξy dy 2 e π(x y / e 2πξy dy ( πx2 + 2πxy πy2 Usng wh a 2π 2πξy dy. ( 2π 2 aw2 + Jw dw ( a J 2 2a and J 2πx 2πξ, for whch J 2 4π2 x 2 2 ( πx2 γ(ξ ( πx2 ( 2πxξ + πξ 2. The Posson summaon formula ells us.e. Defne γ(n ( 4π J 2 8π2 xξ ( πx 2 2πxξ + πξ 2 γ(n, Γ(, x n e 2πnx+πn2 Θ(, x e π(n2 +2xn e 2πnx+πn2. + 4π 2 ξ 2, e πn2 e 2πxn Γ(, x n. For φ S, namely a Schwarz funcon, defne Θ φ(x φ(xe πn2 e 2πxn dx, whch sasfes Θ φ(x φ( ne πn2 φ(ne πn2. If f s -perodc, for n Z le f(n f(ye 2πny dy. 8
9 Defne whch sasfes ψ(f(, x Θ f(x ψ(f(, x Θ(, x yf(ydy, e πn2 e 2π(x yn f(ydy e πn2 e 2πxn f(ye 2πny dy e πn2 e 2πxn f(n. We remnd ourselves Θ(, x Θ (x e πn2 e 2πxn and Say 2M N. Then for k Z, Θ (n e πn2. Θ (k + N (π 2MN (k + N2 (π 2MN k2. 9
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