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

Abstract

Supersymmetric quantum mechanics is one of the most important branches, providing an important method for the rapid and convenient solution of the Schrödinger equation. For the trigonometric Scarf potential, we find that the shape invariance with two parameters shows a new characteristic, i.e., two parameters’ cross-additivity ( A1→B0+α/2, B1→A0+α/2). That is different with the parameters’ change ( A1→A0+α/2, B1→B0+α/2). 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.

Analysis of quantum systems subject to inverse trigonometry Scarf plus Coulomb potential in N-dimensions

“…Since they describe the various phenomena such as scattering, and vibrational properties of molecules, bound state solutions of these potentials are collectively essential in literature. Analytical investigations of the Schrödinger and Dirac equations under the Scarf-type potentials have been carried out within the framework of different methods in the literature [24][25][26][27][28][29][30][31][32][33]. More recently, Onate and his collaborators have obtained the non-relativistic energy spectrum and the corresponding wavefunction for its inverse potential by using the Nikiforov-Uvarov method in hyperspherical coordinates [11].…”