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 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.