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

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 straightforward 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.