R\'esonances Semiclassiques Engendr\'ees par des Croisements de Trajectoires Classiques

TL;DR

This paper investigates the asymptotic behavior of resonances in a 2x2 semiclassical Schrödinger system near non-trapping energies, revealing unique resonance widths due to classical trajectory crossings.

Contribution

It demonstrates the existence of specific resonance widths in a coupled system, contrasting with scalar cases, under conditions of crossing classical trajectories forming a periodic orbit.

Findings

Resonances have width proportional to $T^{-1}h\,\log(1/h)$.

Crossing classical trajectories lead to unique resonance phenomena.

Results differ from scalar Schrödinger operator cases.

Abstract

We consider a $2\times2$ system of one-dimensional semiclassical Schr\"odinger operators with small interactions with respect to the semiclassical parameter $h$. We study the asymptotics in the semiclassical limit of the resonances near a non-trapping energy for both corresponding classical Hamiltonians. We show the existence of resonances of width $T^{-1}h\log(1/h)$, contrary to the scalar case, under the condition that two classical trajectories cross and compose a periodic trajectory with period $T$. omposent une trajectoire p\'eriodique de p\'eriode $T$.