Eigenvalue Splitting of Polynomial Order for a System of Schrödinger Operators with Energy-Level Crossing

Abstract

We consider a 1D 2 × 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. We compute the transfer matrix at this point between the incoming and outgoing microlocal solutions and apply it to the semiclassical distribution of resonances at the energy crossing level. It is described in terms of a generalized Airy function. This result generalizes [5] to the tangential crossing and [3] to the crossing at a turning point.

Resonance free domain for a system of Schrödinger operators with energy-level crossings

“…In this section, we recall the microlocal connection formulae at crossing and turning points from [FMW3,Sec. 5] and [AsFu,Sec. 4].…”

“…These microlocal connection formulae were established in [AsFu,FMW3] using normal form in the spirit of [CdvPa,HeSj2,Sj1] (see also [BFRZ1,BFRZ2,Cdv]).…”