Bohr Sommerfeld phases for avoided crossings Yves Colin de Verdiere

Yves Colin De Verdière

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Bohr-Sommerfeld phases for avoided crossings Yves Colin de Verdiere ?†‡ June 8, 2006 Introduction 0.1 Adiabatic limit in quantum mechanics The problem of adiabatic limit in quantum mechanics with avoided eigenvalues crossings will be the basic example in our paper. Let us describe it in a precise way. We start with a smooth family of N ?N Hermitian matrices A(t) with a ≤ t ≤ b and consider the following linear system of differential equations: h i dX dt = A(t)X (1) Solving this equation gives an unitary map Sh defined by Sh(X(a)) = X(b). We call it the scattering matrix of the problem. Adiabatic Theorems describe the asymptotic behaviour of Sh when h ? 0. 1. The clasical adiabatic theorem [3, 17] concerns the case where the eigenval- ues of A(t) satisfy, for all t: ?1(t) < ?2(t) < · · · < ?N(t) . (2) In this case, if we start with Xj(a) an eigenvector of A(a) with eigen- value ?j(a), we have Xj(b) = exp(i ∫ b a ?j(s)ds)Yj(b) + O(h) where Yj(b) is obtained by parallel transporting Xj(a) along [a, b] in the eigenbundle Lj(t) = ker(A(t) ? ?j(t))

• following smoothly


• landau- zener normal


• self-adjoint semi- classical


• bohr-sommerfeld phases


• expansion along

Informations

• Publié par: pefav
• Nombre de lectures: 5
• Langue: English

Extrait

Bohr-Sommerfeld phases for avoided crossings YvesColindeVerdie`re ∗†‡ June 8, 2006

Introduction 0.1 Adiabatic limit in quantum mechanics The problem of adiabatic limit in quantum mechanics with avoided eigenvalues crossings will be the basic example in our paper. Let us describe it in a precise way. We start with a smooth family ofN×N Hermitian matricesA(t) witha≤t≤band consider the following linear system of differential equations: hd d X t =A(t)X(1) i Solving this equation gives an unitary mapS h defined byS h (X(a)) =X(b). We call it the scattering matrix of the problem. Adiabatic Theorems describe the asymptotic behaviour ofS h whenh→0. 1. The clasical adiabatic theorem [3, 17] concerns the case where the eigenval- ues ofA(t) satisfy, for allt: λ 1 (t)< λ 2 (t)<∙ ∙ ∙< λ N (t)(2) In this case, if we start withX j (a) an eigenvector ofA(a) with eigen- valueλ j (a), we haveX j (b) = exp(i R a b λ j (s)ds)Y j (b) +O(h) whereY j (b) is obtained by parallel transportingX j (a) along [a b] in the eigenbundle L j (t) = ker(A(t)−λ j (t)) w.r. to the geometric (or Berry) connection de- fined by dY r ∂∂t Y(t) = proj L j (t) dt ∗ Institut Fourier, Unit´ ixte de recherche CNRS-UJF 5582, BP 74, 38402-Saint Martine m d’H`eresCedex(France);[e-mail] † Key-words: adiabatic limit, semi-classical analysis, Landau-Zener formula, avoided cross- ings, pseudo-differential operators ‡ MSC 2000: 34E05, 35S30, 81Q20, 81Q70

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