Résonances dans l'approximation de Born-Oppenheimer I

Abstract

We study the operator P = − h 2 Δ x − Δ y + V( x, y) on R x n × R x p 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 R x n × R y p , with principal part diag(− h 2 Δ + λ j ( x)) where the λ j ' s are the eigenvalues of − Δ y + V( x, y) on L 2( R y p . Under the assumption that λ 2 admits a non degenerate point well (and additional conditions on λ 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 Δ + λ 2( x) (see Theorem 1.1).