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

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.