Laguerre-Sobolev orthogonal Polynomials and Boundary Value Problems on a semi-infinite domain
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| Format: | Preprint |
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2026
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| author | Bracciali, Cleonice F. Piñar, Miguel A. |
| author_facet | Bracciali, Cleonice F. Piñar, Miguel A. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_06685 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Laguerre-Sobolev orthogonal Polynomials and Boundary Value Problems on a semi-infinite domain Bracciali, Cleonice F. Piñar, Miguel A. Numerical Analysis Classical Analysis and ODEs 33C47, 65L60, 65T50 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. |
| title | Laguerre-Sobolev orthogonal Polynomials and Boundary Value Problems on a semi-infinite domain |
| topic | Numerical Analysis Classical Analysis and ODEs 33C47, 65L60, 65T50 |
| url | https://arxiv.org/abs/2602.06685 |