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

doi:10.1016/j.aop.2010.12.006

Annals of Physics

Volume 326, Issue 4, April 2011, Pages 797-807

https://doi.org/10.1016/j.aop.2010.12.006 Get rights and content

Abstract

An interesting discovery in the last two years in the field of mathematical physics has been the exceptional X_ℓ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have lowest degree ℓ = 1, 2, … , and yet they form complete set with respect to some positive-definite measure. While the mathematical properties of these new X_ℓ 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 highlights

► Physical examples involving exceptional orthogonal polynomials. ► Exceptional polynomials as deformations of classical orthogonal polynomials. ► Exceptional polynomials from Darboux–Crum transformation.

Introduction

The past two years have witnessed some interesting developments in the area of exactly solvable models in quantum mechanics: the number of exactly solvable shape-invariant models has been greatly increased owing to the discovery of new types of orthogonal polynomials, called the exceptional X_ℓ polynomials. Two families of such polynomials, namely, the Laguerre- and Jacobi-type X₁ polynomials, corresponding to ℓ = 1, were first proposed by Gómez-Ullate et al. [1], within the Sturm–Liouville theory, as solutions of second-order eigenvalue equations with rational coefficients. Unlike the classical orthogonal polynomials, these new polynomials have the remarkable properties that they still form complete set with respect to some positive-definite measure, although they start with a linear polynomials instead of a constant. The results in [1] were then reformulated by Quesne in the framework of quantum mechanics and shape-invariant potentials, first in [2] by the point canonical transformation method, and then in [3] by supersymmetric (SUSY) method [4] (or the Darboux–Crum transformation [5]). Soon after these works, such kind of exceptional polynomials were generalized by Odake and Sasaki to all integral ℓ = 1, 2, … [6] (the case of ℓ = 2 was also discussed in [3]). By construction these new polynomials satisfy the Schrödinger equation and yet they start at degree ℓ > 0 instead of the degree zero constant term. Thus they are not constrained by Bochner’s theorem [7], which states that the orthogonal polynomials (starting with degree 0) satisfying a second order differential equations can only be the classical orthogonal polynomials, i.e., the Hermite, Laguerre, Jacobi and Bessel polynomials.

Later, equivalent but much simpler looking forms of the Laguerre- and Jacobi-type X_ℓ polynomials than those originally presented in [6] were given in [8]. These nice forms were derived based on an analysis of the second order differential equations for the X_ℓ polynomials within the framework of the Fuchsian differential equations in the entire complex x-plane. They allow us to study in-depth some important properties of the X_ℓ polynomials, such as the actions of the forward and backward shift operators on the X_ℓ polynomials, Gram–Schmidt orthonormalization for the algebraic construction of the X_ℓ polynomials, Rodrigues formulas, and the generating functions of these new polynomials.

Recently, in [9] the X₁ Laguerre polynomials in [1] are generalized to the X_ℓ Laguerre polynomials with higher ℓ based on the Darboux–Crum transformation. Then in [10] such transformation was successfully employed to generate the X_ℓ Jabobi as well as the X_ℓ Laguerre polynomials as given in [8].

While the mathematical properties of these new X_ℓ polynomials deserve further analysis, it is also of interest to see if they play any role in physical systems. It is the purpose of the present work to indicate some physical models that involve these new polynomials. We shall be mainly concerned with the X_ℓ Laguerre polynomials for clarity of presentation. Models that are linked with the X_ℓ Jacobi polynomials will be briefly mentioned at the end.

The plan of the paper is as follows. First we review the deformed radial oscillators associated with the exceptional X_ℓ Laguerre polynomials in Section 2. In Section 3 the Dirac equation minimally coupled with an external magnetic field is considered. We present the forms of the vector potentials such that the eigenfunctions of the Dirac equation are related to the X_ℓ Laguerre polynomials. Dirac equations with non-minimal couplings are then mentioned in Section 4, and the Fokker–Planck equations are considered in Section 5. Section 6 concludes the paper.

Section snippets

Exceptional X_ℓ Laguerre polynomials

Consider a generic one-dimensional quantum mechanical system described by a Hamiltonian H = −d²/dx² + V₀(x). Suppose the ground state is given by the wave function ϕ₀(x) with zero energy: Hϕ₀ = 0. By the well known oscillation theorem ϕ₀ is nodeless, and thus can be written as $ϕ_{0} \equiv e^{W_{0} (x)}$ , where W₀(x) is a regular function of x. This implies that the function W₀(x) completely determines the potential $V_{0} : V_{0} = W_{0}^{' 2} + W_{0}^{″}$ (the prime here denotes derivatives with respect to x). Thus W₀(x) is sometimes called a

Dirac equation with magnetic field in cylindrical coordinates

Having reviewed the deformed radial oscillators associated with the exceptional X_ℓ Laguerre polynomials, we will like to see if there are physical systems in which these new polynomials could play a role. It turns out that there are indeed such systems. In what follows we shall indicate some of them.

First let us consider the Dirac equation in 2+1 dimensions coupling minimally with a cylindrically symmetric magnetic field. The discussion can be extended to 3+1 dimensions where the magnetic field

Dirac equations with non-minimal coupling

The example discussed in the previous section illustrates how physical Dirac systems whose eigenfunctions are related to the exceptional orthogonal polynomials can be constructed. The construction relies mainly on the fact that the two radial components of the wave function form a SUSY pair as in (26), (27). Thus, as long as a Dirac equation can be reduced to such a form, this construction applies and one can obtain new Dirac systems that involve the new polynomials. In this section we indicate

Fokker–Planck equations

Finally, we discuss briefly how the exceptional orthogonal polynomials can appear in the Fokker–Planck (FP) equations. In one dimension, the FP equation of the probability density $P (x, t)$ is [22] $\begin{matrix} \frac{\partial}{\partial t} P (x, t) = LP (x, t), \\ L \equiv - \frac{\partial}{\partial x} D^{(1)} (x) + \frac{\partial^{2}}{\partial x^{2}} D^{(2)} (x) . \end{matrix}$ The functions D⁽¹⁾(x) and D⁽²⁾(x) in the FP operator $L$ are, respectively, the drift and the diffusion coefficient (we consider only time-independent case). The drift coefficient represents the external force acting on the particle, while the diffusion

Summary

The discovery of the exceptional X_ℓ Laguerre and Jacobi polynomials has opened up new avenues in the area of mathematical physics and in classical analysis. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have lowest degree ℓ = 1, 2, …, and yet they form complete set with respect to some positive-definite measure. Some properties of these new polynomials have been studied, and many more have yet to be investigated.

In this paper we have

Acknowledgments

This work is supported in part by the National Science Council (NSC) of the Republic of China under Grant NSC 96–2112-M-032–007-MY3 and NSC-99–2112-M-032–002-MY3. I thank S. Odake and R. Sasaki for many helpful discussions on the exceptional polynomials.

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

Abstract

Research highlights

Introduction

Section snippets

Exceptional Xℓ Laguerre polynomials

Dirac equation with magnetic field in cylindrical coordinates

Dirac equations with non-minimal coupling

Fokker–Planck equations

Summary

Acknowledgments

J. Approx. Theory

J. Math. Anal. Appl.

J. Phys.

C. R. Acad. Paris

Quart. J. Math. Oxford Ser.

Phys. Rev. Lett.

Phys. Rep.

SIGMA

Phys. Rep.

Phys. Lett.

Phys. Lett.

J. Math. Phys.

Math. Zeit.

J. Phys.

J. Phys.

JETP Lett.

Exceptional X_ℓ Laguerre polynomials