Extending Romanovski polynomials in quantum mechanics

TL;DR

This paper explores extensions of Romanovski polynomials related to quantum potentials, establishing new orthogonality relations and analyzing potential regularity, thereby broadening the mathematical tools for quantum mechanics models.

Contribution

It introduces generalized Romanovski polynomials with new orthogonality properties and examines their role in extended quantum potentials, expanding the mathematical framework in quantum mechanics.

Findings

Generalized polynomials satisfy finite orthogonality for Scarf II potentials.

Infinite relations among polynomials with degree-dependent parameters for Rosen-Morse I.

Potential regularity confirmed via disconjugacy properties of differential equations.

Abstract

Some extensions of the (third-class) Romanovski polynomials (also called Romanovski/pseudo-Jacobi polynomials), which appear in bound-state wavefunctions of rationally-extended Scarf II and Rosen-Morse I potentials, are considered. For the former potentials, the generalized polynomials satisfy a finite orthogonality relation, while for the latter an infinite set of relations among polynomials with degree-dependent parameters is obtained. Both types of relations are counterparts of those known for conventional polynomials. In the absence of any direct information on the zeros of the Romanovski polynomials present in denominators, the regularity of the constructed potentials is checked by taking advantage of the disconjugacy properties of second-order differential equations of Schr\"odinger type. It is also shown that on going from Scarf I to Scarf II or from Rosen-Morse II to Rosen-Morse I potentials, the variety of rational extensions is narrowed down from types I, II, and III to type III only.