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Exact spectrum and wave functions of the hyperbolic Scarf potential in terms of finite Romanovski polynomials

D.E. Alvarez–Castillo and M. Kirchbach

Instituto de Física, Universidad Autónoma de San Luis Potosí, Av. Manuel Nava 6, San Luis Potosí, S.L.P 78290, México.

Recibido el 1 de septiembre de 2006
Aceptado el 24 de febrero de 2007

Abstract

The Schrödinger equation with the hyperbolic Scarf potential reported so far in the literature is somewhat artificially manipulated into the form of the Jacobi equation with an imaginary argument and parameters that are complex conjugate to each other. Instead we here solve the former equation anew and make the case that it reduces straight forward to a particular form of the generalized real hypergeometric equation whose solutions are referred to in the mathematics literature as the finite Romanovski polynomials, in reference to the observation that for any parameter set only a finite number of such polynomials appear to be orthogonal. This is a qualitatively new integral property that does not copy any of the features of the Jacobi polynomials. In this manner, the finite number of bound states within the hyperbolic Scarf potential is brought into correspondence with a finite system of a new class of orthogonal polynomials. This work adds a new example to the circle of the problems on the Schrödinger equation. The techniques used by us extend the teachings on the Sturm–Liouville theory of ordinary differential equations beyond their standard presentation in the textbooks on mathematical methods in physics.

Keywords: Schrödinger equation; Scarf potentials; Romanovski polynomials.

Resumen

La solución a la ecuación de Schrödinger con el potencial de Scarf hiperbólico reportada hasta ahora en la literatura física está manipulada artificialmente para obtenerla en la forma de los polinomios de Jacobi con argumentos imaginarios y parámetros que son complejos conjugados entre ellos. En lugar de eso, nosotros resolvimos la nueva ecuación obtenida y desarrollamos el caso en el que realmente se reduce a una forma particular de la ecuación hipergeométrica generalizada real, cuyas soluciones se refieren en la literatura matemática como los polinomios finitos de Romanovski. La notación de finito se refiere a que, para cualquier parámetro fijo, solo un número finito de dichos polinomios son ortogonales. Esta es una nueva propiedad cualitativa de la integral que no surge como copia de ninguna de las características de los polinomios de Jacobi. De esta manera, el número finito de estados en el potencial de Scarf hiperbólico es consistente en correspondencia a un sistema finito de polinomios ortogonales de una nueva clase.

Descriptores: Ecuación de Schrödinger; potenciales de Scarf; polinomios de Romanovski.

PACS: 02.30.Gp; 03.65.Ge; 12.60.Jv

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Acknowledgments

We are indebted to Dr. Jose–Luis López Bonilla for bringing the important Refs. 20 and 39 to our attention. We furthermore wish to thank Drs. Hans–Jürgen Weber and Alvaro Pérez Raposo for many insightful discussions on the orthogonality issue. We benefited from the lectures on supersymmetric quantum mechanics at the 35th Latin American School (ELAF), "Super–symmetries in Physics and Applications", held in Mexico in the Summer of 2004.

Work supported by Consejo Nacional de Ciencia y Technología (CONACyT) México under grant number C01–39280.

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