Microlocal scattering matrix for coupled Schrödinger operators and applications to semiclassical resonances

Abstract

We consider a one-dimensional Schrödinger operator with matrix-valued potential. For such an operator, the WKB construction breaks down, not only at the turning points, but also at the crossing points where the potential has multiple eigenvalues. We study the connection problem at the crossing points and compute the semiclassical asymptotics of what we call microlocal scattering matrix. We apply this result to the semiclassical resonance width arising from the molecule predissociation in quantum chemistry.