Variational supersymmetric approach and Gram–Schmidt process for evaluating Fokker–Planck probabilities

In this work, an alternative method for solving eigenvalue equations is investigated, with a specific application to the Schrödinger-type Fokker–Planck equation. This method is based on combined eigenfunctions through the Gram–Schmidt orthogonalization process, coupled with the well-formalized factorization technique in supersymmetric quantum mechanics. Eigenvalues are obtained via the variational method, using numerical computation. The aim is to obtain solutions for two polynomial potentials of the form V1(x)=x66−x44 and V2(x)=x44−x35−x22, in order to obtain the probability distributions at different times t and initial conditions represented by x0. The results for the symmetric potential V1(x) are compared with values found in the literature. For the asymmetric potential V2(x), the solution is compared only with numerical results, also demonstrating a low margin of error. In both cases, the proposed technique generates probability distributions that respect the typical behavior of the Fokker–Planck equation, with percentage errors below 0.5% compared to reference methods.