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

We consider a well-posed formulation of the spectral problem for a relativistic analogue of the one-dimensional Schrödinger equation with differential operators replaced with operators of finite purely imaginary argument shifts $\exp ({\pm i\hbar d/dx})$. 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 $\hbar$ and the parameters of the potential and a situation in which the solution for $\hbar \ll 1$ is nevertheless fundamentally different from its Schrödinger analogue is quite possible.