Laguerre-Sobolev orthogonal Polynomials and Boundary Value Problems on a semi-infinite domain

TL;DR

This paper introduces Laguerre--Sobolev orthogonal polynomials tailored for boundary value problems on semi-infinite domains, enabling efficient spectral methods for Schrödinger equations with singular potentials.

Contribution

It develops explicit formulas, recurrence relations, and a spectral method based on these polynomials, improving numerical solutions for singular Schrödinger problems.

Findings

Spectral accuracy confirmed by numerical experiments

Method avoids solving linear systems, enabling recursive implementation

Effective for Schrödinger equations with inverse-distance potentials

Abstract

We study a family of Laguerre--Sobolev orthogonal polynomials associated with a Sobolev inner product arising from second--order boundary value problems on the semi--infinite interval $(0,+\infty)$. These polynomials generate an orthogonal basis of test functions vanishing at the endpoints and are especially well suited for the spectral approximation of Schr\"odinger--type problems with singular potentials. Explicit connection formulas with classical Laguerre polynomials are obtained, together with recurrence relations and asymptotic properties of the corresponding coefficients. A generating function involving Bessel functions is also derived. As an application, we develop a fully diagonalized Laguerre--Sobolev spectral method for Dirichlet problems with singular potentials. The method avoids the solution of linear systems and can be implemented recursively. Numerical experiments for a Schr\"odinger--type equation with inverse--distance potential confirm spectral accuracy and exponential convergence.