Morse potential, symmetric Morse potential and bracketed bound-state energies

TL;DR

This paper introduces an advanced method for solving Schrödinger equations with piece-wise analytic potentials, utilizing special functions like Whittaker functions, and highlights a novel approach to bound-state energy localization.

Contribution

It proposes a new non-perturbative approach to solvability for Schrödinger equations with piece-wise analytic potentials, including a unique treatment of non-analytic points.

Findings

Bound states can be localized on both sides of the potential.

The method employs Whittaker functions for solving differential equations.

The approach allows for symbolic manipulation and improved energy localization.

Abstract

An upgraded concept of solvability of Schr\"{o}dinger-type equations is proposed. In a broader methodical context of non-perturbative quantum theory the innovation involves potentials which are piece-wise analytic yielding differential equations solvable in terms of special functions. In our illustrative example, Whittaker functions are employed and a single point of non-analyticity is admitted in the origin. In a symbolic-manipulation-based practical implementation of the method a serendipitous advantage of the construction of bound states is found in the both-sided nature of the numerical localization of their energies.