A conditionally integrable Schr\"odinger potential of a bi-confluent Heun class

TL;DR

This paper introduces a new exactly solvable Schr"odinger potential related to the bi-confluent Heun class, providing explicit solutions, energy spectrum, and accurate bound-state energy approximations.

Contribution

It presents a novel conditionally integrable potential within the bi-confluent Heun class, with explicit solutions and energy spectrum derivation.

Findings

Exact solutions expressed as Hermite functions

Derived energy spectrum equation

Accurate approximation for bound-state energies

Abstract

We present a bi-confluent Heun potential for the Schr\"odinger equation involving inverse fractional powers and a repulsive centrifugal-barrier term the strength of which is fixed to a constant. This is an infinite potential well defined on the positive half-axis. Each of the fundamental solutions for this conditionally integrable potential is written as an irreducible linear combination of two Hermite functions of a shifted and scaled argument. We present the general solution of the problem, derive the exact energy spectrum equation and construct a highly accurate approximation for the bound-state energy levels.