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Bohr Sommerfeld phases for avoided crossings Yves Colin de Verdiere

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19 pages
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


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Bohr-Sommerfeld phases for avoided crossingsYvesColindeVerdie`re∗†‡June 8, 2006
Introduction0.1 Adiabatic limit in quantum mechanicsThe problem of adiabatic limit in quantum mechanics with avoided eigenvaluescrossings will be the basic example in our paper.Let us describe it in a precise way. We start with a smooth family ofN×NHermitian matricesA(t) withatband consider the following linear systemof differential equations:hddXt=A(t)X(1)iSolving this equation gives an unitary mapShdefined bySh(X(a)) =X(b). Wecall it the scattering matrix of the problem. Adiabatic Theorems describe theasymptotic behaviour ofShwhenh0.1. The clasical adiabatic theorem [3, 17] concerns the case where the eigenval-ues ofA(t) satisfy, for allt:λ1(t)< λ2(t)<∙ ∙ ∙< λN(t)(2)In this case, if we start withXj(a) an eigenvector ofA(a) with eigen-valueλj(a), we haveXj(b) = exp(iRabλj(s)ds)Yj(b) +O(h) whereYj(b)is obtained by parallel transportingXj(a) along [a b] in the eigenbundleLj(t) = ker(A(t)λj(t)) w.r. to the geometric (or Berry) connection de-fined bydYr∂∂tY(t) = projLj(t)dtInstitut Fourier, Unit´ ixte de recherche CNRS-UJF 5582, BP 74, 38402-Saint Martine mdH`eresCedex(France);yves.colin-de-verdiere@ujf-grenoble.frKey-words: adiabatic limit, semi-classical analysis, Landau-Zener formula, avoided cross-ings, pseudo-differential operatorsMSC 2000: 34E05, 35S30, 81Q20, 81Q70
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