Non-Hermitian superintegrable systems

Abstract

A non-Hermitian generalisation of the Marsden--Weinstein reduction method is introduced to construct families of quantum $\mathcal{P}\mathcal{T}$-symmetric superintegrable models over an $n$-dimensional sphere $S^n$. The mechanism is illustrated with one- and two-dimensional examples, related to $u(2)$ and $u(3)$ Lie algebras respectively, providing new quantum models with real spectra and spontaneous $\mathcal{P}\mathcal{T}$-symmetric breaking. In certain limits, the models reduce to known non-Hermitian systems and complex extensions of previously studied real superintegrable systems.

arXiv comment18 pages, no figures

References in corpus (36)

• Making Sense of Non-Hermitian Hamiltonians
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Classical and Quantum Superintegrability with Applications
• Completeness of superintegrability in two-dimensional constant curvature spaces
• Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems
• Pseudospectra in non-Hermitian quantum mechanics
• Exact solution for Morse oscillator in PT-symmetric quantum mechanics
• Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics II:Painleve transcendent potentials
• Complex Calogero model with real energies
• Aharonov-Bohm effect on AdS_2 and nonlinear supersymmetry of reflectionless Poschl-Teller system
• Spectral singularities in PT-symmetric periodic finite-gap systems
• PT-symmetric Deformations of the Korteweg-de Vries Equation
• PT-Symmetric Extension of the Korteweg-de Vries Equation
• Integrable nonlocal Hirota equations
• PT-symetrically regularized Eckart,Poeschl-Teller and Hulthen potentials
• On an exactly solvable $B_N$ type Calogero model with nonhermitian PT invariant interaction
• PT-symmetric deformations of integrable models
• Position-dependent mass, finite-gap systems, and supersymmetry
• Non-Hermitian Hamiltonians of Lie algebraic type
• Regularized degenerate multi-solitons
• Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
• A note on the integrability of non-Hermitian extensions of Calogero-Moser-Sutherland models
• The Conformal Group SU(2,2) and Integrable Systems on a Lorentzian Hyperboloid
• On realizations of polynomial algebras with three generators via deformed oscillator algebras
• Superintegrable quantum u(3)--systems and higher rank factorizations
• Pseudo-orthogonal groups and integrable dynamical systems in two dimensions
• The Racah algebra: An overview and recent results
• Symmetry algebra for the generic superintegrable system on the sphere
• Polynomial algebras from $su(3)$ and the generic model on the two sphere
• Non-Hermitian multi-particle systems from complex root spaces
• The general Racah algebra as the symmetry algebra of generic systems on pseudo--spheres
• The theory of contractions of 2D 2nd order quantum superintegrable systems and its relation to the Askey scheme for hypergeometric orthogonal polynomials
• Intertwining Symmetry Algebras of Quantum Superintegrable Systems
• (Quasi)-exact-solvability on the sphere $S^n$
• Integrable systems on the sphere, ellipsoid and hyperboloid