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Superintegrability and Deformed Oscillator Realizations of Quantum TTW Hamiltonians on Constant-Curvature Manifolds and with Reflections in a Plane

Ian Marquette, Anthony Parr

TL;DR

This work develops an expansion-based framework to construct higher-order symmetry operators for two-dimensional quantum Hamiltonians on constant-curvature manifolds, avoiding explicit eigenfunctions. By decomposing integrals into angular ladder operators $J^u$ and radial operators $K^u$, the authors derive explicit ladder structures for TTW and PVZ models (including reflections), obtain their full symmetry algebras, and realize these algebras as deformed oscillators to recover finite-dimensional spectra algebraically. The analysis on curved spaces yields higher-order polynomial algebras with central elements and Casimirs, and reveals how curvature and parity (q even/odd) change the algebraic structure, including Racah-type reductions in the flat limit. The results provide a comprehensive algebraic mechanism to determine spectra and degeneracies for these superintegrable systems and suggest broad applicability to other deformations and non-elementary angular components.

Abstract

We extend the method for constructing symmetry operators of higher order for two-dimensional quantum Hamiltonians by Kalnins, Kress and Miller (2010). This expansion method expresses the integral in a finite power series in terms of lower degree integrals so as to exhibit it as a first-order differential operators. One advantage of this approach is that it does not require the a priori knowledge of the explicit eigenfunctions of the Hamiltonian nor the action of their raising and lowering operators as in their recurrence approach (2011). We obtain insight into the two-dimensional Hamiltonians of radial oscillator type with general second-order differential operators for the angular variable. We then re-examine the Hamiltonian of Tremblay, Turbiner and Winternitz (2009) as well as a deformation discovered by Post, Vinet and Zhedanov (2011) which possesses reflection operators. We will extend the analysis to spaces of constant curvature. We present explicit formulas for the integrals and the symmetry algebra, the Casimir invariant and oscillator realisations with finite-dimensional irreps which fill a gap in the literature.

Superintegrability and Deformed Oscillator Realizations of Quantum TTW Hamiltonians on Constant-Curvature Manifolds and with Reflections in a Plane

TL;DR

This work develops an expansion-based framework to construct higher-order symmetry operators for two-dimensional quantum Hamiltonians on constant-curvature manifolds, avoiding explicit eigenfunctions. By decomposing integrals into angular ladder operators and radial operators , the authors derive explicit ladder structures for TTW and PVZ models (including reflections), obtain their full symmetry algebras, and realize these algebras as deformed oscillators to recover finite-dimensional spectra algebraically. The analysis on curved spaces yields higher-order polynomial algebras with central elements and Casimirs, and reveals how curvature and parity (q even/odd) change the algebraic structure, including Racah-type reductions in the flat limit. The results provide a comprehensive algebraic mechanism to determine spectra and degeneracies for these superintegrable systems and suggest broad applicability to other deformations and non-elementary angular components.

Abstract

We extend the method for constructing symmetry operators of higher order for two-dimensional quantum Hamiltonians by Kalnins, Kress and Miller (2010). This expansion method expresses the integral in a finite power series in terms of lower degree integrals so as to exhibit it as a first-order differential operators. One advantage of this approach is that it does not require the a priori knowledge of the explicit eigenfunctions of the Hamiltonian nor the action of their raising and lowering operators as in their recurrence approach (2011). We obtain insight into the two-dimensional Hamiltonians of radial oscillator type with general second-order differential operators for the angular variable. We then re-examine the Hamiltonian of Tremblay, Turbiner and Winternitz (2009) as well as a deformation discovered by Post, Vinet and Zhedanov (2011) which possesses reflection operators. We will extend the analysis to spaces of constant curvature. We present explicit formulas for the integrals and the symmetry algebra, the Casimir invariant and oscillator realisations with finite-dimensional irreps which fill a gap in the literature.
Paper Structure (13 sections, 9 theorems, 142 equations)

This paper contains 13 sections, 9 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Models, Eigenfunctions and Spectra
  3. General Form of the Symmetry
  4. Ladder Operators
  5. PVZ Angular Operators
  6. TTW Angular Operators
  7. Radial Operators
  8. Symmetry Algebra
  9. PVZ q even
  10. PVZ q odd
  11. TTW on constant-curvature space
  12. Spectrum
  13. Conclusion

Key Result

Lemma 4.1

For all odd integers $u$, $J^u(\theta)(-ku)=J^u_{0,0}(-ku)$.

Theorems & Definitions (16)

  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Lemma 4.3
  • proof
  • Theorem 4.4
  • proof
  • Corollary 4.5
  • Proposition 4.6
  • ...and 6 more