Resonances in Born-Oppenheimer approximation. II: Length of resonances. (Q1174268)

[Part I, see the preceding review.] The paper concerns the Schrödinger operator in \(L^ 2(\mathbb{R}^ n_ x\times\mathbb{R}^ p_ y)\): \(P=-h^ 2\Delta_ x+Q(x)\), with \(Q(x)=- \Delta_ y+V(x,y)\). The author studies the resonances \(a_ j(h)\) of \(P\), i.e. the eigenvalues of \(P_ z=U_ zPU_ z^{-1}\), where \(U_ zf(x,z)=e^{nz/2}f(xe^ z,y)\). Under suitable assumptions, describing the case of a well of the second electronic level, the following estimates are obtained for \(h\to0\): \[ | Im a_ j(h)|\leq Ce^{- | z|^ 2/ch}. \] This provides the first strict proof of a known result in Atomic and Molecular Physics.