Spectral properties of the Kramers-Fokker-Planck operator with a long-range potential

TL;DR

This paper investigates the spectral characteristics of the Kramers-Fokker-Planck operator with long-range potentials, establishing properties of eigenvalues, resonances, and decay behaviors of eigenfunctions.

Contribution

It provides new insights into the accumulation points of eigenvalues, the limiting absorption principle, and decay rates of eigenfunctions for this operator.

Findings

Thresholds are the only possible accumulation points of eigenvalues.

The limiting absorption principle holds outside an exceptional set.

Eigenfunctions decay exponentially or polynomially depending on their association.

Abstract

We study real resonances and embedded eigenvalues of the Kramers--Fokker--Planck operator with a long-range potential. We prove that thresholds are only possible accumulation points of eigenvalues and that the limiting absorption principle holds true for energies outside an exceptional set. We also prove that the eigenfunctions associated with discrete eigenvalues decay exponentially and those associated with embedded non-threshold ones decay polynomially.