Deformed Fokker-Planck Equations

TL;DR

This paper introduces deformed Fokker-Planck equations linked to discrete quantum mechanics, providing exactly solvable models with eigenfunctions related to deformed orthogonal polynomials, bridging continuous and discrete quantum systems.

Contribution

It proposes a new class of deformed Fokker-Planck equations associated with discrete quantum mechanics, expanding solvable models and eigenfunction structures.

Findings

Derived exactly solvable deformed FP equations

Connected eigenfunctions to deformations of classical orthogonal polynomials

Established a link between discrete QM and Fokker-Planck dynamics

Abstract

Based on the well-known relation between Fokker-Planck equations and Schroedinger equations of quantum mechanics (QM), we propose new deformed Fokker-Planck (FP) equations associated with the Schroedinger equations of "discrete" QM. The latter is a natural discretization of QM and its Schroedinger equations are difference instead of differential equations. Exactly solvable FP equations are obtained corresponding to exactly solvable "discrete" QM, whose eigenfunctions include various deformations of the classical orthogonal polynomials.