Tunnel effect and symmetries for Kramers Fokker-Planck type operators

TL;DR

This paper analyzes the spectral properties of Kramers-Fokker-Planck operators in the semiclassical limit, revealing the behavior of small eigenvalues related to the Morse function's local minima.

Contribution

It provides a detailed semiclassical asymptotic analysis of the first eigenvalues of Kramers-Fokker-Planck operators with Morse potential functions, including their reality and exponential smallness.

Findings

First $n_0$ eigenvalues are real and exponentially small.

Complete semiclassical asymptotics for these eigenvalues are established.

Eigenvalues correspond to local minima of the Morse function.

Abstract

We study operators of Kramers-Fokker-Planck type in the semiclassical limit, assuming that the exponent of the associated Maxwellian is a Morse function with a finite number $n_0$ of local minima. Under suitable additional assumptions, we show that the first $n_0$ eigenvalues are real and exponentially small, and establish the complete semiclassical asymptotics for these eigenvalues.