Résonances dans l’approximation de Born-Oppenheimer. I. (Resonances in Born-Oppenheimer approximation. I). (French) Zbl 0737.35046

Martinez, André

J. Differ. Equations 91, No. 2, 204-234 (1991).

Summary: We study the operator \(P=-h^ 2\Delta_ x-\Delta_ y+V(x,y)\) on \(\mathbb{R}^ n_ x\times\mathbb{R}^ p_ y\) when \(h\) tends to zero, in a case where resonances appear. Using the so-called Feshbach method, the study of \(P\) is first reduced to that of a matrix operator on \(L^ 2(\mathbb{R}^ n_ x)\), with principal part \(\hbox{diag}(-h^ 2\Delta+\lambda_ j(x))\) where the \(\lambda_ j\text{'s}\) are the eigenvalues of \(- \lambda_ y+V(x,y)\) on \(L^ 2(\mathbb{R}^ p_ y)\). Under the assumption that \(\lambda_ 2\) admits a non degenerate point well (and additional conditions on \(\lambda_ 1\)), it is then showed that \(P\) has resonances with a real asymptotic expansion in \(h^{1/2}\), close to the eigenvalues of \(-h^ 2\Delta+\lambda_ 2(x)\).
[Part II, see the following review.].

MSC:

35P20	Asymptotic distributions of eigenvalues in context of PDEs
35J10	Schrödinger operator, Schrödinger equation

Keywords:

Feshbach method; resonances; eigenvalues; well; asymptotic expansion

Citations:

Zbl 0737.35047

References:

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