Semi-analytic version of shooting method and the bound states in non-analytic potential $V(x)= -g^2\exp (-|x|)$

TL;DR

This paper introduces a semi-analytic approach to find bound states in a class of potentials that are neither exactly solvable nor purely numerical, exemplified by the exponential-well potential.

Contribution

The paper proposes a semi-analytic method for bound state calculation in non-analytic potentials, bridging the gap between exact solutions and numerical techniques.

Findings

Demonstrates the method on the exponential-well potential

Shows the potential class is neither exactly solvable nor purely numerical

Provides insights into bound state properties in intermediate cases

Abstract

People construct quantum bound states either non-numerically (in not too many exceptional cases) or, in general, numerically (mainly using majorization or discretization techniques). We point out that there exists a class of interactions "in between". Being, in conventional terminology, neither exactly solvable nor purely numerical. The idea is illustrated by the elementary exponential-well toy-model potential $V(x)= -g^2\exp (-|x|)$.