Morse momentum wavefunctions and rational functions

Abstract

We revisit the bound states of the Morse potential in the momentum representation. After the ground-state factor is extracted, the remaining factors are finite rational functions of the momentum variable. These functions are eigenfunctions of a second-order difference operator and are identified with the symmetric specialization of a finite family of biorthogonal rational functions introduced by Koepf and Masjed-Jamei. They also satisfy a generalized eigenvalue problem in the degree variable, thereby placing the Morse momentum wavefunctions within the framework of rational bispectrality and $R_{II}$-type systems. Finally, after extraction of their poles, the same wavefunctions are expressed in terms of Meixner--Pollaczek polynomials with degree-dependent parameters. This gives a simple description of their zeros. The Morse potential thus provides a concrete quantum-mechanical realization of finite biorthogonal rational functions.