Explicit representations of Pollaczek polynomials corresponding to an exactly solvable discretisation of hydrogen radial Schr\"odinger equation

TL;DR

This paper presents an explicit analytical framework for Pollaczek polynomials derived from an exactly solvable discretisation of the hydrogen atom's radial Schrödinger equation, linking spectral and position representations.

Contribution

It introduces explicit solutions and polynomial representations for a discretised hydrogen atom model, connecting spectral properties with Pollaczek polynomials.

Findings

Solutions are entire functions with recursive coefficients.

Established a one-to-one correspondence between spectral and position representations.

Derived explicit expressions for solutions of the discretised equation.

Abstract

We consider an exactly solvable discretisation of the radial Schr\"odinger equation of the hydrogen atom with l=0. We first examine direct solutions of the finite difference equation and remark that the solutions can be analytically continued entire functions. A recursive expression for the coefficients in the solution is obtained. The next step is to identify the related three-term recursion relation for Pollaczek polynomials. One-to-one correspondence between the spectral and position representations facilitates the evaluation of Pollaczek polynomials corresponding to the discrete spectrum. Finally, we obtain two alternative and explicit expressions for the solutions of the original difference equation.