The one-dimensional periodic Schrödinger equation

Σχετικά έγγραφα
On homeomorphisms and C 1 maps

Multi-dimensional Central Limit Theorem

Multi-dimensional Central Limit Theorem

12. Radon-Nikodym Theorem

( ) ( t) ( 0) ( ) dw w. = = β. Then the solution of (1.1) is easily found to. wt = t+ t. We generalize this to the following nonlinear differential

Oscillatory integrals

Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών. ΗΥ-570: Στατιστική Επεξεργασία Σήµατος. ιδάσκων : Α. Μουχτάρης. εύτερη Σειρά Ασκήσεων.

Math221: HW# 1 solutions

Areas and Lengths in Polar Coordinates

Areas and Lengths in Polar Coordinates

Math 248 Homework 1. Edward Burkard. Exercise 1. Prove the following Fourier Transforms where a > 0 and c R: f (x) = b. f(x c) = e.

Uniform Convergence of Fourier Series Michael Taylor

SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM

SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit

Ο µετασχηµατισµός Fourier και η κυµατική εξίσωση

Approximation of the Lerch zeta-function

The semiclassical Garding inequality

ΘΕΩΡΗΜΑ (Μέσης Τιμής) Έστω f: [α, β] R συνεχής και παραγωγίσιμη στο (α, β). Τότε υπάρχει ξ (α, β)

Statistical Inference I Locally most powerful tests

Divergence for log concave functions

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3

Iterated trilinear fourier integrals with arbitrary symbols

Κεφάλαιο 1 Πραγματικοί Αριθμοί 1.1 Σύνολα

ΣΥΓΚΛΙΣΗ ΣΥΝΑΡΤΗΣΗΣ: Ορισμός Cauchy

= df. f (n) (x) = dn f dx n

Second Order Partial Differential Equations

Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1

α & β spatial orbitals in

Solutions to Exercise Sheet 5

Πανεπιστήμιο Κρήτης. 19 Οκτωβρίου 2015 Μεταπτυχιακή εργασία στα πλαίσια του προγράμματος "Μαθηματικά και εφαρμογές τους"

forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with

SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions

Λύσεις μερικών ασκήσεων του τέταρτου φυλλαδίου.

Generalized Fibonacci-Like Polynomial and its. Determinantal Identities

Η ϐέλτιστη σταθερά στην ανισότητα Hausdorff-Young

Berry-Esseen Theorem. Po-Ning Chen, Professor. Institute of Communications Engineering. National Chiao Tung University. Hsin Chu, Taiwan 30010, R.O.C.

2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)

Answer sheet: Third Midterm for Math 2339

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)

Homework 8 Model Solution Section

Reminders: linear functions

b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!

Every set of first-order formulas is equivalent to an independent set

8.323 Relativistic Quantum Field Theory I

D Alembert s Solution to the Wave Equation

EE512: Error Control Coding

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS

1. If log x 2 y 2 = a, then dy / dx = x 2 + y 2 1] xy 2] y / x. 3] x / y 4] none of these

ODE, SDE and PDE. Shizan Fang Université de Bourgogne, France. February 29, 2008

Θεωρία Μέτρου και ολοκλήρωσης Ασκήσεις

Εισαγωγή στον Προγραµµατισµό. Ανάλυση (ή Επιστηµονικοί 21Υπολογισµοί)

Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in

Vidyamandir Classes. Solutions to Revision Test Series - 2/ ACEG / IITJEE (Mathematics) = 2 centre = r. a

Nonholomorphic Eisenstein series, the Kronecker limit formula, and the hyperbolic Laplacian

C.S. 430 Assignment 6, Sample Solutions

, P bkc (c[0, 1]) P bkc (L p [0, 1]) (1) 2 P bkc (X) O A (2012) Aumann. R. J., [3]. Feb Vol. 28 No.

2 Composition. Invertible Mappings

REMARK ON THE OPTIMAL REGULARITY FOR EQUATIONS OF WAVE MAPS TYPE

On Pseudo-Differential Operator Associated with Bessel Operator

SPECIAL FUNCTIONS and POLYNOMIALS

(s n (f)) g = s n (f g) = f (s n (g)). s n (f) g = (f D n ) g = f (D n g) = f (g D n ) = f s n (g). K n (x)g δ (x) dx. K n (x) dx.

F19MC2 Solutions 9 Complex Analysis

Σύγκλιση σειρών Fourier σε χώρους L p

ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΑΝΟΙΚΤΑ ΑΚΑΔΗΜΑΙΚΑ ΜΑΘΗΜΑΤΑ

8.1 The Nature of Heteroskedasticity 8.2 Using the Least Squares Estimator 8.3 The Generalized Least Squares Estimator 8.

Lecture 21: Properties and robustness of LSE

4.6 Autoregressive Moving Average Model ARMA(1,1)

6.1. Dirac Equation. Hamiltonian. Dirac Eq.

k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +

New symmetries of Black-Scholes equation

Partial Differential Equations in Biology The boundary element method. March 26, 2013

Finite Field Problems: Solutions

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics

Other Test Constructions: Likelihood Ratio & Bayes Tests

Inverse trigonometric functions & General Solution of Trigonometric Equations

Lecture 13 - Root Space Decomposition II


Mock Exam 7. 1 Hong Kong Educational Publishing Company. Section A 1. Reference: HKDSE Math M Q2 (a) (1 + kx) n 1M + 1A = (1) =

Sequent Calculi for the Modal µ-calculus over S5. Luca Alberucci, University of Berne. Logic Colloquium Berne, July 4th 2008

On the Galois Group of Linear Difference-Differential Equations

Bessel functions. ν + 1 ; 1 = 0 for k = 0, 1, 2,..., n 1. Γ( n + k + 1) = ( 1) n J n (z). Γ(n + k + 1) k!

1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r(t) = 3cost, 4t, 3sint

Spherical Coordinates

Jordan Journal of Mathematics and Statistics (JJMS) 4(2), 2011, pp

PROPERTIES OF SCALES OF KATO CLASSES, BESSEL POTENTIALS, MORREY SPACES, AND A WEAK HARNACK INEQUALITY FOR NON-NEGATIVE SOLUTIONS OF ELLIPTIC EQUATIONS

THE SECOND ISOMORPHISM THEOREM ON ORDERED SET UNDER ANTIORDERS. Daniel A. Romano

Parametrized Surfaces

Homomorphism in Intuitionistic Fuzzy Automata

Πρόταση. f(x) ομοιόμορφα συνεχής στο I. δ (ɛ) > 0 : x, ξ I, x ξ < δ (ɛ, ξ) f(x) f(ξ) < ɛ. ɛ > 0, δ > 0 : ΜΗ ομοιόμορφα συνεχής.

STAT200C: Hypothesis Testing

Π Ο Λ Ι Τ Ι Κ Α Κ Α Ι Σ Τ Ρ Α Τ Ι Ω Τ Ι Κ Α Γ Ε Γ Ο Ν Ο Τ Α

One and two particle density matrices for single determinant HF wavefunctions. (1) = φ 2. )β(1) ( ) ) + β(1)β * β. (1)ρ RHF

Solution Series 9. i=1 x i and i=1 x i.

Exercise 1.1. Verify that if we apply GS to the coordinate basis Gauss form ds 2 = E(u, v)du 2 + 2F (u, v)dudv + G(u, v)dv 2

SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES. Reading: QM course packet Ch 5 up to 5.6

Lecture 15 - Root System Axiomatics

A NONSTANDARD VERSION OF THE FOKKER-PLANCK EQUATION

DiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation

de Rham Theorem May 10, 2016

Transcript:

The one-dmensonal perodc Schrödnger equaon Jordan Bell jordan.bell@gmal.com 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.

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

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

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

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 ( + 2 2 ( 2π 4π2 x 2 4π 2 4π 2 xγ(, x. Γ(, x Γ(, x 5

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

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

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

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