The Spinless Relativistic Kink-Like Problem

TL;DR

This paper investigates the bound-state solutions of the spinless Salpeter equation with a kink-like potential, revealing conflicts between common solution procedures and theoretical expectations in a semirelativistic quantum context.

Contribution

It formulates elementary boundary conditions for solutions of the spinless Salpeter equation with kink-like potential and demonstrates conflicts with standard solution methods.

Findings

Standard procedures conflict with theoretical expectations.

Elementary boundary conditions are formulated for the eigenvalue problem.

Results challenge common solution approaches in semirelativistic quantum mechanics.

Abstract

We constrain the possible bound-state solutions of the spinless Salpeter equation (the most obvious semirelativistic generalization of the nonrelativistic Schr\"odinger equation) with an interaction between the bound-state constituents given by the kink-like potential (a central potential of hyperbolic-tangent form) by formulating a bunch of very elementary boundary conditions to be satisfied by all solutions of the eigenvalue problem posed by a bound-state equation of this type, only to learn that all results produced by a procedure very much liked by some quantum-theory practitioners prove to be in severe conflict with our expectations.