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Dévelopements asymptotiques et effet tunnel dans l’approximation de Born-Oppenheimer. (Asymptotic expansions and tunnel effect in Born- Oppenheimer approximation). (French) Zbl 0689.35063
Summary: We study the discrete spectrum of \[ P=-h^ 2\Delta_ x-\Delta_ y+V(x,y) \] on \({\mathbb{R}}^ n_ x\times {\mathbb{R}}^ p_ y\) when h tends to zero, in the case where V is a smooth potential, and the first eigenvalue \(\lambda_ 1(x)\) of \(Q(x)=-\Delta_ y+V(x,y)\) on \({\mathbb{R}}^ p_ y\) admits one (or several) non degenerate point-wells. We show the existence of asymptotic expansions in \(h^{1/2}\) for the eigenvalues of P, as well as WKB-type expansions for the associated normalized eigenfunctions. Then, we apply this to the study of tunnel effect between two wells of \(\lambda_ 1\).

MSC:
35P05 General topics in linear spectral theory for PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35J10 Schrödinger operator, Schrödinger equation
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