Sur le spectre semi-classique d'un syst\`eme int\'egrable de dimension 1 autour d'une singularit\'e hyperbolique

TL;DR

This paper analyzes the semi-classical spectrum of a one-dimensional Schrödinger operator with a double well potential, focusing on the spectral behavior near a hyperbolic singularity, which models a non-degenerate hyperbolic singularity in integrable systems.

Contribution

It provides a detailed description of the semi-classical spectrum around a hyperbolic singularity for a 1D integrable system with a double well potential.

Findings

Spectrum shape around the local maximum analyzed

Characterization of hyperbolic singularity in integrable systems

Insights into spectral behavior near hyperbolic points

Abstract

In this article we describe the semi-classical spectrum of a Schrodinger operator on $\mathbb{R}$ with a double well potential. We study the shape of spectrum around the local maximum of the potential. In the classification of singularities of integrable systems, the double wells is the model of non-degenerate hyperbolic singularity.