Exceptional orthogonal polynomials, QHJ formalism and SWKB quantization condition

TL;DR

This paper explores the quantum Hamilton-Jacobi formalism applied to new exactly solvable models based on exceptional polynomials, revealing unique singularity structures and confirming the exactness of the SWKB quantization rule.

Contribution

It demonstrates that the QHJ formalism accurately reproduces eigenvalues and eigenfunctions for models involving exceptional polynomials and establishes the exactness of the SWKB quantization condition for these systems.

Findings

QHJ formalism reproduces eigenvalues and eigenfunctions accurately.

Unique singularity structures differ from known solvable models.

SWKB quantization condition is exact for these new models.

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 singularities of the momentum function for these new potentials lie 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.