How to solve Fokker-Planck equation treating mixed eigenvalue spectrum?

TL;DR

This paper extends the analytical solution methods for the Fokker-Planck equation by applying quantum mechanics techniques to cases with mixed eigenvalue spectra, including bounded and free states, validated through specific potential examples.

Contribution

It demonstrates how to solve the Fokker-Planck equation with mixed spectra using quantum analogies, expanding beyond previous discrete-spectrum limitations.

Findings

Successfully applied method to constant potential case

Extended approach to Pöschl-Teller potential

Results align with expected Fokker-Planck dynamics

Abstract

An analogy of the Fokker-Planck equation (FPE) with the Schr\"odinger equation allows us to use quantum mechanics technique to find the analytical solution of the FPE in a number of cases. However, previous studies have been limited to the Schr\"odinger potential with a discrete eigenvalue spectrum. Here, we will show how this approach can be also applied to a mixed eigenvalue spectrum with bounded and free states. We solve the FPE with boundaries located at x=\pm L/2 and take the limit L\rightarrow\infty, considering the examples with constant Schr\"{o}dinger potential and with P\"{o}schl-Teller potential. An oversimplified approach was proposed earlier by M.T. Araujo and E. Drigo Filho. A detailed investigation of the two examples shows that the correct solution, obtained in this paper, is consistent with the expected Fokker-Planck dynamics.