The cross-additivity-two parameters shape invariance of superpotential Bcscαx-Acotαx based on SUSYQM

Abstract

Supersymmetric quantum mechanics is an effective method to solve the exact solution of the Schrödinger equation. This paper studies the solution of the Schrödinger equation with the partner potentials generated by the superpotential (Bcscαx-Acotαx) with two parameters (A and B). Firstly, the shape invariance of the partner potentials generated by the superpotential is obtained. The parametric additivity of shape invariance satisfies a special additivity characteristic: the two-parameter cross-additivity(A→B+α2,B→A+α2), which is completely different from the general additivity characteristic(A→A+α2,B→B+α2). Secondly, we discuss the case that belongs to two-parameter cross additive shape invariance in detail, and find that this two-parameter cross-additivity resulted in partial states missing. The existing energy spectrum and eigenfunctions of the Schrödinger equation with this new parametric transformation are worked out. Thirdly, we discuss the Shape invariance of the partner potentials generated again by the two parameters with cross additive characteristics through the potential algebra method. Lastly, the conclusions and discussions are made.

Keywords

• Supersymmetric quantum mechanics
• Two-parameter cross additive shape invariance
• Potential algebra
• Superpotential