Bi-Confluent Heun Potentials for a Stationary Relativistic Wave Equation for a Spinless Particle

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 Schrödinger 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 Schrödinger 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 Schrödinger case.

Exact solutions of the 2D Schrödinger equation with the inverse square root potential

“…As shown in figure 1, this is a more longer range potential than the Coulomb potential due to the factor r −1/2 . In addition, the relativistic equations including the Klein-Gordon and Dirac cases with this potential have also been worked out [14,15]. For example, Li and Dai considered this potential in 3D Schrödinger * Authors to whom any correspondence should be addressed.…”