Resonance free domain for a system of Schr\"odinger operators with energy-level crossings

TL;DR

This paper investigates the absence of resonances near non-trapping energies in a 2x2 semiclassical Schr"odinger system with potential crossings, providing lower bounds on resonance widths related to bicharacteristic cycles.

Contribution

It introduces conditions for resonance-free domains in a coupled Schr"odinger system with potential crossings, extending semiclassical resonance theory.

Findings

Resonance widths are bounded below by Mh log(1/h).

Resonance-free domains are characterized near non-trapping energies.

The coefficient M relates to directed cycles of bicharacteristics.

Abstract

We consider a $2\times 2$ system of 1D semiclassical differential operators with two Schr\"odinger operators in the diagonal part and small interactions of order $h^\nu$ in the off-diagonal part, where $h$ is a semiclassical parameter and $\nu$ is a constant larger than $1/2$. We study the absence of resonance near a non-trapping energy for both Schr\"odinger operators in the presence of crossings of their potentials. The width of resonances is estimated from below by $Mh\log(1/h)$ and the coefficient $M$ is given in terms of the directed cycles of the generalized bicharacteristics induced by two Hamiltonians.