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

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.

💡 Research Summary

The paper investigates the role of the recently discovered exceptional Xℓ Laguerre (and Jacobi) polynomials in concrete physical models. Unlike classical orthogonal polynomials, which begin with a constant term, the exceptional families start at degree ℓ ≥ 1 yet still form a complete set with respect to a positive‑definite weight. After a concise review of the mathematical construction of these polynomials via Darboux‑Crum transformations and supersymmetric (SUSY) shape‑invariance, the authors embed them into three types of exactly solvable systems.

First, they consider a one‑dimensional radial oscillator whose conventional pre‑potential is W₀(x;g)=−½ ω x²+g ln x. By deforming W₀ with a rational factor ξℓ(η;g) (two families L1 and L2), they obtain new pre‑potentials Wℓ(x;g) that generate deformed Hamiltonians H₊ℓ(g) and their SUSY partners H₋ℓ(g). The eigenfunctions of H₊ℓ are proportional to e^{Wℓ} multiplied by exceptional Laguerre polynomials Pℓ,n(η;g), which are degree ℓ + n polynomials beginning at ℓ. The spectra remain iso‑spectral to the original oscillator (Eₙ=4n ω), while the wavefunctions acquire the new polynomial structure.

Second, the authors embed these constructions into the (2+1)-dimensional Dirac equation with a cylindrically symmetric magnetic field Aφ(r). By writing the Dirac operator in polar coordinates, the two spinor components f₊(r) and f₋(r) satisfy SUSY‑partner Schrödinger equations with superpotential W′=g r−Aφ(r) (g=m+½). Choosing Aφ(r) so that the SUSY Hamiltonian A₋A₊ coincides with H₊ℓ yields a “deformed Landau” system: the magnetic field is modified by terms involving ξℓ and its derivative, yet the energy levels stay at E²−M²=4n ω, identical to the ordinary relativistic Landau problem. Both components of the Dirac spinor are then expressed through the exceptional Laguerre polynomials.

Alternatively, if Aφ(r) is chosen such that A₋A₊ matches H₋ℓ, the lower spinor component f₋ alone contains the exceptional polynomial, while the upper component remains a conventional Laguerre function. In this case the spectrum acquires ℓ‑dependence (E²−M²=4ω