Bi-confluent Heun potentials for a stationary relativistic wave equation for a spinless particle

TL;DR

This paper explores bi-confluent Heun potentials in a relativistic wave equation for spinless particles, revealing how the uncertainty principle influences the energy spectrum and quantum states differently than in Schrödinger equations.

Contribution

It introduces bi-confluent Heun potentials for a relativistic wave equation and analyzes how the uncertainty principle restricts the energy spectrum and quantum states.

Findings

The potentials are applicable to the relativistic wave equation similarly to Schrödinger's.

The uncertainty principle imposes a lower bound on the quantum number of bound states.

Physically feasible states have a higher minimum quantum number compared to the Schrödinger case.

Abstract

The variety of bi-confluent Heun potentials for a stationary relativistic wave equation for a spinless particle is presented. The physical potentials and energy spectrum of this wave equation are related to those for a corresponding Schr\"odinger equation in the sense that all the potentials derived for the latter equation are also applicable for the wave equation under consideration. We show that in contrast to the Schr\"odinger equation the characteristic spatial length of the potential imposes a restriction on the energy spectrum that directly reflects the uncertainty principle. Studying the inverse-square-root bi-confluent Heun potential, it is shown that the uncertainty principle limits, from below, the principal quantum number for the bound states, i.e., physically feasible states have an infimum cut so that the ground state adopts a higher quantum number as compared to the Schr\"odinger case.