Solvable Schrodinger Equations of Shape Invariant Potentials with Superpotential $W(x,A,B)=A\tanh 3px-B\coth px$

TL;DR

This paper introduces a new exactly solvable Schrödinger equation with a specific shape-invariant potential, providing explicit solutions and eigenvalues, which advances the understanding of solvable quantum models with potential applications in physics and chemistry.

Contribution

The paper presents a novel shape-invariant potential and derives its exact solutions using supersymmetric methods, expanding the class of solvable Schrödinger equations.

Findings

Exact eigenvalues are derived explicitly.

Eigenfunctions are expressed in closed form.

Potential applications in nuclear physics and chemistry.

Abstract

We propose a new, exactly solvable Schr\"{o}dinger equation. The potential partner is given by \[{ V=}-Bp\operatorname{csch}[px]^{2}-9p(B+p)\operatorname*{sech}[3px]^{2}+(B\coth[px]-3(B+p)\tanh[3px])^{2}.\] obtained using supersymmetric method with shape invariance property having a superpotential $W(x,A,B)=A\tanh 3px-B\coth px.$ We derive entirely the exact solutions of this family of Schr\"{o}dinger equations with the eigenvalue given by $E_{n}^{\left( -\right) }=(A-B)^{2}-(A-B-4np)^{2}% $ and the corresponding eigenfunctions are determined exactly and in closed form. Schr\"{o}dinger equations, and Sturm-Liouville equations in general, are challenging to solve in closed form, and only a few of them are known. Therefore, in a strict mathematical sense, discovering new solvable equations is essential in understanding the eluded solutions' underpinnings. This result has potential applications in nuclear physics and chemistry, and other fields of science.