A new shape invariance form of the trigonometric Scarf potential: Two-parameter cross-additivity shape invariance

Abstract

Abstract Supersymmetric quantum mechanics (SUSYQM) provides an important method for solving the Schrödinger equation rapidly and conveniently. Based on SUSYQM, for the trigonometric Scarf potential, we find that the shape invariance with two parameters shows a new characteristic, i.e., two parameters' cross-additivity . That is different from the parameters' change . The changing of the parameters with cross-additivity brings new characteristic to the wave function and energy spectrum. Based on this cross-additivity characteristic, we discuss in detail the eigenvalues and the eigenfunctions of the Hamiltonian with this potential. And then we get the two-parameter cross-additivity shape invariance again with potential algebra methods and study the energy spectrum. It is shown that the two-parameter cross-additivity shape invariance of the partner potential is completely self-consistent with its potential algebraic form. Our research indicates that the Schrödinger equation with a superpotential with two parameters shows new characteristics.