One-dimensional Schr\"odinger equation with non-analytic potential $V(x)= -g^2\exp (-|x|)$ and its exact Bessel-function solvability

TL;DR

This paper introduces a new exactly solvable quantum model with a non-analytic potential, demonstrating solutions in terms of Bessel functions and expanding the class of potentials considered exactly solvable.

Contribution

It proposes that many non-traditional potentials with piecewise special function solutions should be considered exactly solvable, exemplified by the exponential potential model.

Findings

Bound states described by Bessel functions

Scattering solutions matched at the origin

Potential expands the class of exactly solvable models

Abstract

Exact solvability (ES) of one-dimensional quantum potentials $V(x)$ is a vague concept. We propose that beyond its most conventional range the ES status should be attributed also to many less common interaction models for which the wave functions remain piecewise proportional to special functions. The claim is supported by constructive analysis of a toy model $V(x)= -g^2\exp (-|x|)$. The detailed description of the related bound-state and scattering solutions of Schr\"{o}dinger equation is provided in terms of Bessel functions which are properly matched in the origin.