Exceptional orthogonal polynomials, QHJ formalism and SWKB quantization condition

Abstract

We study the quantum Hamilton-Jacobi (QHJ) equation of the recently obtained exactly solvable models, related to the newly discovered exceptional polynomials and show that the QHJ formalism reproduces the exact eigenvalues and the eigenfunctions. The fact that the eigenfunctions have zeros and poles in complex locations leads to an unconventional singularity structure of the quantum momentum function p(x), the logarithmic derivative of the wave function, which forms the crux of the QHJ approach to quantization. A comparison of the singularity structure for these systems with the known exactly solvable and quasi-exactly solvable models reveals interesting differences. We find that the singularity structure of the momentum function for these new potentials lies between the above two distinct models, sharing similarities with both of them. This prompted us to examine the exactness of the supersymmetric WKB (SWKB) quantization condition. The interesting singularity structure of p(x) and of the superpotential for these models has important consequences for the SWKB rule and in our proof of its exactness for these quantal systems.

Supersymmetric Quantum Mechanics and Solvable Models

“…In each case, the new potential is isospectral with a potential that arises from one of the "conventional" superpotentials listed in Table 1. Authors of [52][53][54] have added an infinite number of potentials that belong in this class, and extended shape invariant potentials continue to be objects of intense research [55,56].…”

Inter-relations between additive shape invariant superpotentials

“…These superpotentials were then generalized in Ref. [26-29, 32, 33, 37], and some of their properties have been further studied [34,35]. Since these superpotentials could not be generated from the two PDEs, they must contain explicit -dependence.…”

Quantum Hamilton–Jacobi route to exceptional Laguerre polynomials and the corresponding rational potentials

“…In the QHJF the singularity structure analysis of the QMF in the complex plane, allows us to calculate the eigenvalues and eigenfunctions for a given potential V − (x) [40], [41]. From the same analysis we can get the entire information required to construct all the superpotentials associated with V − (x).…”