A class of exactly solvable models for the Schrodinger equation

TL;DR

This paper introduces a new class of exactly solvable confining potentials for the one-dimensional Schrödinger equation, enabling solutions via Bessel or Whittaker equations with explicit wavefunctions and eigenvalues.

Contribution

It provides a novel class of potentials that simplify solving the Schrödinger equation by reducing it to well-known differential equations with explicit solutions.

Findings

Closed-form wavefunctions for symmetric and antisymmetric states

Transcendental equations for eigenvalues derived

Includes potentials with single and double wells

Abstract

We present a class of confining potentials which allow one to reduce the one-dimensional Schroodinger equation to a named equation of mathematical physics, namely either Bessel's or Whittaker's differential equation. In all cases, we provide closed form expressions for both the symmetric and antisymmetric wavefunction solutions, each along with an associated transcendental equation for allowed eigenvalues. The class of potentials considered contains an example of both cusp-like single wells and a double-well.