Construction of the raising operator for Rosen-Morse eigenstates in terms of the Weyl fractional integral

TL;DR

This paper constructs a non-local raising operator for Rosen-Morse eigenstates using Weyl fractional integrals, enabling efficient numerical computation of wave functions and illustrating an application of fractional calculus in quantum systems.

Contribution

It introduces a novel non-local raising operator for Rosen-Morse potentials expressed via fractional calculus, specifically the Weyl fractional integral, which improves computational methods.

Findings

The operator is non-local and involves fractional calculus techniques.

A recurrence relation using Weyl fractional integral is derived.

The method facilitates efficient numerical computation of wave functions.

Abstract

The raising operator relating adjacent bound states for the general, non-symmetric Rosen-Morse potential is constructed explicitly. It is demonstrated that, in constrast to the symmetric (modified P\"oschl-Teller) potential, the operator is non-local and must be expressed applying techniques from fractional calculus. A recurrence relation between adjacent states is derived applying the Weyl fractional integral, which, in contrast to standard recurrence relations, allows the efficient numerical computation of the coefficients of all Jacobi polynomials necessary for the evaluation of the bound state wave functions, providing an application of fractional calculus to exactly solvable quantum systems.