Quasi-exactly solvable Fokker-Planck equations

TL;DR

This paper explores exact and quasi-exact solutions of the one-dimensional Fokker-Planck equation by linking it to Schrödinger equations, using prepotentials and Bethe ansatz, and extends classification with new examples.

Contribution

It provides a unified framework for solvability of Fokker-Planck equations and introduces a new $sl(2)$-based example outside Turbiner's classification.

Findings

Lists quasi-exactly solvable Fokker-Planck equations related to $sl(2)$ systems.

Provides a new $sl(2)$-based example not in Turbiner's scheme.

Establishes a connection between Fokker-Planck and Schrödinger equations.

Abstract

We consider exact and quasi-exact solvability of the one-dimensional Fokker-Planck equation based on the connection between the Fokker-Planck equation and the Schr\"odinger equation. A unified consideration of these two types of solvability is given from the viewpoint of prepotential together with Bethe ansatz equations. Quasi-exactly solvable Fokker-Planck equations related to the $sl(2)$-based systems in Turbiner's classification are listed. We also present one $sl(2)$-based example which is not listed in Turbiner's scheme.