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

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.

17 pages, no figure. Verion in Ann. Phys. Sect. 2 considerably shortened, typos corrected

References in corpus (2)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• A perturbative approach to a class of Fokker-Planck equations

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• Extensions of solvable potentials with finitely many discrete eigenstates
• Exceptional Laguerre polynomials
• Parametric symmetries in exactly solvable real and P T symmetric complex potentials
• Prepotential approach to solvable rational extensions of Harmonic Oscillator and Morse potentials
• Group theoretic approach to rationally extended shape invariant potentials
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• Infinite families of (non)-Hermitian Hamiltonians associated with exceptional $X_m$ Jacobi polynomials
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• DARBOUX partners of pseudoscalar Dirac potentials associated with exceptional orthogonal polynomials
• The Dunkl-Fokker-Planck Equation in $1+1$ Dimensions
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• A class of exactly solvable rationally extended Calogero-Wolfes type 3-body problems
• Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials
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• Quasi-Hermitian Hamiltonians associated with exceptional orthogonal polynomials
• Moment Representations of Exceptional $X_1$ Orthogonal Polynomials
• Relativistic Potentials with Rational Extensions
• Moment Representations of the Exceptional $X_1$-Laguerre Orthogonal Polynomials