PT -Symmetric Potentials from the Confluent Heun Equation.

Géza Lévai

doi:10.3390/e23010068

Abstract

We derive exactly solvable potentials from the formal solutions of the confluent Heun equation and determine conditions under which the potentials possess symmetry. We point out that for the implementation of symmetry, the symmetrical canonical form of the Heun equation is more suitable than its non-symmetrical canonical form. The potentials identified in this construction depend on twelve parameters, of which three contribute to scaling and shifting the energy and the coordinate. Five parameters control the function that detemines the variable transformation taking the Heun equation into the one-dimensional Schrödinger equation, while four parameters play the role of the coupling coefficients of four independently tunable potential terms. The potentials obtained this way contain Natanzon-class potentials as special cases. Comparison with the results of an earlier study based on potentials obtained from the non-symmetrical canonical form of the confluent Heun equation is also presented. While the explicit general solutions of the confluent Heun equation are not available, the results are instructive in identifying which potentials can be obtained from this equation and under which conditions they exhibit symmetry, either unbroken or broken.

Highlights

The efforts of extending and perfecting the mathematical formulation of quantum mechanics are as old as quantum mechanics itself
In the present study we discussed quantum mechanical potentials originating from the confluent Heun equation and discussed conditions under which they can satisfy the P T
This work was motivated by our earlier systematic study in which we discussed the construction of the P T -symmetric version of general Natanzon-class potentials

Summary

Introduction

The efforts of extending and perfecting the mathematical formulation of quantum mechanics are as old as quantum mechanics itself. (See the much earlier reference [48], where potentials derivable from the Heun, confluent, biconfluent and double confluent Heun equation are discussed in some detail.) The general structure of the potentials has been outlined, together with the solutions written formally in terms of confluent Heun functions Since these functions are not known in closed form in general, the complete solutions (with energy eigenvalues) could not be presented. We revisit this problem with the intention of implementing P T symmetry to potentials that can be derived from the confluent Heun differential equation.

Exactly Solvable Potentials from Special Functions of Mathematical Physics

Application to the Confluent Heun Equation

Implementing P T Symmetry to the Potentials

Parity Considerations

P T -Parity Considerations

Summary