Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials

TL;DR

This paper explores the role of exceptional Laguerre and Jacobi polynomials in physical models, demonstrating their appearance in solutions to Dirac and Fokker-Planck equations, thus expanding the set of exactly solvable systems.

Contribution

It introduces physical systems where exceptional orthogonal polynomials serve as eigenfunctions, extending the class of exactly solvable models in mathematical physics.

Findings

Exceptional polynomials appear in Dirac equation solutions.

They also feature in Fokker-Planck equations.

The work broadens the scope of solvable physical models.

Abstract

An interesting discovery in the last two years in the field of mathematical physics has been the exceptional $X_\ell$ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have lowest degree $\ell=1,2,...$, and yet they form complete set with respect to some positive-definite measure. While the mathematical properties of these new $X_\ell$ polynomials deserve further analysis, it is also of interest to see if they play any role in physical systems. In this paper we indicate some physical models in which these new polynomials appear as the main part of the eigenfunctions. The systems we consider include the Dirac equations coupled minimally and non-minimally with some external fields, and the Fokker-Planck equations. The systems presented here have enlarged the number of exactly solvable physical systems known so far.