Transition of the semiclassical resonance widths across a tangential crossing energy-level

Abstract

We consider a 1D $2\times 2$ matrix-valued operator (1.1) with two semiclassical Schrödinger operators on the diagonal entries and small interactions on the off-diagonal ones. When the two potentials cross at a turning point with contact order n, the corresponding two classical trajectories at the crossing level intersect at one point in the phase space with contact order 2n. Below this level, they have no intersection, which suggests exponentially small widths of resonances (see e.g., [1], [2]), while above this level, on the contrary, they intersect at two points, which implies a polynomial order of the widths as proved in [3]. We prove that the transition of the resonance widths near the crossing level is described in terms of a generalized Airy function. This result generalizes [4] to the tangential crossing and [3] to the crossing at a turning point. Our approach is based on the computation of the microlocal transfer matrix at the crossing point between the incoming and outgoing microlocal solutions.