Separation of variables in the Kramers equation

TL;DR

This paper investigates the separation of variables in the Kramers equation with quadratic potentials, providing a complete solution and explicit separated solutions when symmetry conditions are met.

Contribution

It offers a complete method for separating variables in the Kramers equation with quadratic potentials and constructs explicit solutions.

Findings

Complete solution for separation of variables with quadratic potentials

Explicit separated solutions of the Kramers equation

Conditions under which separation of variables is possible

Abstract

We consider the problem of separation of variables in the Kramers equation admitting a non-trivial symmetry group. Provided the external potential $V(x)$ is at most quadratic, a complete solution of the problem of separation of variables is obtained. Furthermore, we construct solutions of the Kramers equation with separated variables in explicit form.