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

• URL: http://arxiv.org/abs/2103.08066v1
• Date: Sun, 14 Mar 2021 23:35:20 GMT
• Title: Solvable Schrodinger Equations of Shape Invariant Potentials with Superpotential $W(x,A,B)=A\tanh 3px-B\coth px$
• Authors: Jamal Benbourenane
• Abstract summary: We propose a new, exactly solvable Schr"odinger equation. We derive entirely the exact solutions of this family of Schr"odinger equations. This result has potential applications in nuclear physics and chemistry.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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.

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