Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials

TL;DR

This paper introduces an extended Kepler-Coulomb quantum model with wave functions expressed via exceptional orthogonal polynomials, demonstrating superintegrability through algebraic methods and spectrum degeneracy analysis.

Contribution

It develops a superintegrable quantum model using exceptional orthogonal polynomials and constructs algebraic integrals of motion to analyze its spectrum.

Findings

Wave functions involve Laguerre, Legendre, and exceptional Jacobi polynomials.

Higher-order polynomial algebra of integrals confirms superintegrability.

Degeneracy of energy spectrum derived from deformed oscillator algebra structure.

Abstract

We introduce an extended Kepler-Coulomb quantum model in spherical coordinates. The Schr\"{o}dinger equation of this Hamiltonian is solved in these coordinates and it is shown that the wave functions of the system can be expressed in terms of Laguerre, Legendre and exceptional Jacobi polynomials (of hypergeometric type). We construct ladder and shift operators based on the corresponding wave functions and obtain their recurrence formulas. These recurrence relations are used to construct higher-order, algebraically independent integrals of motion to prove superintegrability of the Hamiltonian. The integrals form a higher rank polynomial algebra. By constructing the structure functions of the associated deformed oscillator algebras we derive the degeneracy of energy spectrum of the superintegrable system.