Bibliographic Details
| Title: |
Formulación espectral de Sturm-Liouville para los polinomios de Gegenbauer y su aplicación al potencial trigonométrico de Pöschl-Teller. |
| Authors: |
Gómez-Hernández, José A.1 jgomez96@uasd.edu.do, Toribio-Milane, Juan1,2, Rosario-Roche, Pablo A.3, Salomón-Martínez, Samuel J.4 |
| Source: |
Latin-American Journal of Physics Education. Mar2026, Vol. 20 Issue 1, p1-16. 16p. |
| Subject Terms: |
Sturm-Liouville equation, Gegenbauer polynomials, Spectral theory, Hilbert space, Eigenvalues, Differential equations, Orthogonal polynomials |
| Abstract (English): |
This paper studies a hypergeometric-type differential equation whose polynomial solutions give rise to the Gegenbauer polynomials. We develop a self-adjoint Sturm-Liouville formulation within a weighted Hilbert-space setting, which provides a natural route to orthogonality, norm formulas, and the associated spectral conditions. We also discuss symmetry and self-adjoint realizations of the differential operator under suitable endpoint conditions. As an application, we treat the trigonometric Pöschl-Teller well: through nondimensionalization, exact singularity extraction, and a change of variables, the model reduces directly to the Gegenbauer framework, yielding an explicit discrete spectrum and a closed family of normalized eigenfunctions. [ABSTRACT FROM AUTHOR] |
| Abstract (Spanish): |
Este trabajo estudia la ecuación diferencial de tipo hipergeométrico que genera, como soluciones polinómicas, a los polinomios de Gegenbauer. Se presenta una formulación en forma autoadjunta de Sturm-Liouville y se establece el problema en un espacio de Hilbert con peso, lo que permite derivar de manera natural la ortogonalidad, las normas y las condiciones espectrales asociadas. Además, se analizan la simetría y las realizaciones autoadjuntas del operador diferencial mediante condiciones de frontera adecuadas. Como aplicación, se aborda el pozo trigonométrico de Pöschl-Teller: mediante escalamiento, extracción de singularidades y un cambio de variable, el problema se reduce directamente al modelo de Gegenbauer, obteniéndose un espectro discreto explícito y una familia cerrada de autofunciones normalizadas. [ABSTRACT FROM AUTHOR] |
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| Database: |
Education Research Complete |
