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

Cleonice F. Bracciali, Miguel A. Piñar

TL;DR

This work addresses solving Schrödinger-type boundary value problems on the half-line $(0,+\infty)$ with a singular potential $V(x)=1/x$ by developing Laguerre-Sobolev polynomials associated with a Sobolev inner product. It builds a diagonalized spectral method by expanding the solution in the basis $\{S_n(x) x e^{-x/2}\}$, using explicit connection formulas to classical Laguerre polynomials, and a generating function involving $J_1$ to derive recurrence for the coefficients. The paper derives asymptotics for the connection coefficients, provides explicit formulas for $a_n$ and a practical recurrence, and demonstrates how to implement the diagonalized solver without linear systems. Numerical experiments with Schrödinger-type equations confirm spectral accuracy and exponential convergence for decaying solutions, while highlighting limitations for non-decaying cases, validating the approach for singular half-line problems.

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 Schrödinger--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 Schrödinger--type equation with inverse--distance potential confirm spectral accuracy and exponential convergence.

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

TL;DR

This work addresses solving Schrödinger-type boundary value problems on the half-line with a singular potential by developing Laguerre-Sobolev polynomials associated with a Sobolev inner product. It builds a diagonalized spectral method by expanding the solution in the basis , using explicit connection formulas to classical Laguerre polynomials, and a generating function involving to derive recurrence for the coefficients. The paper derives asymptotics for the connection coefficients, provides explicit formulas for and a practical recurrence, and demonstrates how to implement the diagonalized solver without linear systems. Numerical experiments with Schrödinger-type equations confirm spectral accuracy and exponential convergence for decaying solutions, while highlighting limitations for non-decaying cases, validating the approach for singular half-line problems.

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 . These polynomials generate an orthogonal basis of test functions vanishing at the endpoints and are especially well suited for the spectral approximation of Schrödinger--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 Schrödinger--type equation with inverse--distance potential confirm spectral accuracy and exponential convergence.
Paper Structure (5 sections, 7 theorems, 80 equations, 3 figures)

This paper contains 5 sections, 7 theorems, 80 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Laguerre orthogonal polynomials
  3. Laguerre--Sobolev orthogonal polynomials
  4. The diagonalized spectral method for BVP
  5. Numerical experiment

Key Result

Theorem 2.1

Let $\alpha,\beta>-1$, $j\in\mathbb{Z}$ and $z\in\mathbb{C}\setminus [0,\infty)$, and fix an integer $d\geq 1$, then the ratio of arbitrary Laguerre polynomials has the following asymptotic expansion as $n\to\infty$: where the first coefficients are The error term holds uniformly for $z$ in compact sets of $\mathbb{C}\setminus[0,\infty)$.

Figures (3)

  • Figure 5.1: The solution $u(x)$ and the Fourier-Sobolev partial sums $\mathcal{S}_n(u,x)$ for $n = 6, 9, 12, 15$.
  • Figure 5.2: A logarithmic plot of errors for $n = 0, 1, \ldots, 20$.
  • Figure 5.3: A logarithmic plot of errors for $n = 0, 1, \ldots, 20$ for a solution with rational decay.

Theorems & Definitions (12)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 2 more