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

Abstract

We propose a new, exactly solvable Schrödinger 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 Schrödinger 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. Schrödinger 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.

11 pages. arXiv admin note: substantial text overlap with arXiv:2102.02775