Quasiexact Solution of a Relativistic Finite-Difference Analogue of the Schrödinger Equation for a Rectangular Potential Well

Abstract

We consider a well-posed formulation of the spectral problem for a relativistic analogue of the one-dimensional Schrodinger equation with differential operators replaced with operators of finite purely imaginary argument shifts exp(±iℏd/dx). We find effective solution methods that permit determining the spectrum and investigating the properties of wave functions in a wide parameter range for this problem in the case of potentials of the type of a rectangular well. We show that the properties of solutions of these equations depend essentially on the relation between ℏ and the parameters of the potential and a situation in which the solution for ℏ≪1 is nevertheless fundamentally different from its Schrodinger analogue is quite possible.