Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials

Abstract

We introduce an extended Kepler-Coulomb quantum model in spherical coordinates. The Schrödinger 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.

20 pages

References in corpus (13)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Solvable rational extensions of the isotonic oscillator
• Exceptional orthogonal polynomials and the Darboux transformation
• Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics I:Rational function potentials
• Superintegrability of the Caged Anisotropic Oscillator
• Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials
• Superintegrable systems with position dependent mass
• The generalized quantum isotonic oscillator
• Quadratic algebra structure and spectrum of a new superintegrable system in N-dimension
• DARBOUX partners of pseudoscalar Dirac potentials associated with exceptional orthogonal polynomials
• A new family of $N$ dimensional superintegrable double singular oscillators and quadratic algebra $Q(3)\oplus so(n)\oplus so(N-n)$
• Recurrence approach and higher rank cubic algebras for the $N$-dimensional superintegrable systems

Cited by in corpus (4)

• An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension
• Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators
• Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces
• General solution of the exceptional Hermite differential equation and its minimal surface representation