09-21 B. Messirdi, A. Senoussaoui

Resonances for a General Hamiltonian in the Born-Oppenheimer Approximation (221K, LaTeX 2e) Feb 8, 09

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Abstract. We study the discrete spectrum of a general class of Born-Oppenheimer Hamiltonians of the type: \begin{equation*} H=-h^{2}\Delta _{x}+P\left( x,y,D_{y}\right) \text{ on }L^{2}\left( \mathbb{R% }_{x}^{n}\times \mathbb{R}_{y}^{p}\right) ,n,p\in \mathbb{N}^{\ast } \end{equation*}% {\small when }$h${\small \ tends to }$0^{+}${\small , here }$P\left( x,y,D_{y}\right) ${\small \ is a pseudodifferential operator on }$% L^{2}\left( \mathbb{R}_{y}^{p}\right) .$ {\small In the case where the first eigenvalue }$\lambda _{1}\left( x\right) ${\small \ of }$P\left( x,y,D_{y}\right) ${\small \ on }$L^{2}\left( \mathbb{R}_{y}^{p}\right) $% {\small \ admits one non degenerate point-well, we obtain WKB-type expansions for all order in }${\small h}^{{\small 1/2}}${\small \ of eigenvalues (in the interval }$[0,C_{0}h],${\small \ }$C_{0}>0)${\small \ and associated normalized eigenfunctions of }$H,${\small \ and this for all orders in }$h^{1/2}$.

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