Semi-classical resonances associated with a periodic orbit of hyperbolic type

TL;DR

This paper investigates semi-classical resonances linked to hyperbolic periodic orbits in quantum systems, extending previous frameworks to include mixed eigenvalues and semi-excited states with specific imaginary parts.

Contribution

It generalizes existing methods to handle hyperbolic and elliptic eigenvalues, analyzing semi-excited resonances with novel imaginary part scales.

Findings

Extended the framework to include hyperbolic and elliptic eigenvalues.

Identified semi-excited resonances with imaginary parts of order -h log h or h^s.

Applied results to Schrödinger operators with Stark effect and geodesic flows.

Abstract

We consider in this Note resonances for a $h$-Pseudo-Differential Operator $H(x,hD_x;h)$ on $L^2(M)$ induced by a periodic orbit of hyperbolic type, as arises for Schr\"odinger operator with AC Stark effect when $M={\bf R}^n$, or the geodesic flow on an axially symmetric manifold $M$, extending Poincar\'e example of Lagrangian systems with 2 degrees of freedom. We generalize the framework of [G\'eSj], in the sense that we allow for hyperbolic and elliptic eigenvalues of Poincar\'e map, and look for so-called semi-excited resonances with imaginary part of magnitude $-h\log h$, or $h^s$, with $0<s<1$.