PT symmetric models in more dimensions and solvable square-well versions of their angular Schroedinger equations

Abstract

For any central potential V in D dimensions, the angular Schroedinger equation remains the same and defines the so called hyperspherical harmonics. For non-central models, the situation is more complicated. We contemplate two examples in the plane: (1) the partial differential Calogero's three-body model (without centre of mass and with an impenetrable core in the two-body interaction), and (2) the Smorodinsky-Winternitz' superintegrable harmonic oscillator (with one or two impenetrable barriers). These examples are solvable due to the presence of the barriers. We contemplate a small complex shift of the angle. This creates a problem: the barriers become "translucent" and the angular potentials cease to be solvable, having the sextuple-well form for Calogero model and the quadruple or double well form otherwise. We mimic the effect of these potentials on the spectrum by the multiple, purely imaginary square wells and tabulate and discuss the result in the first nontrivial double-well case.

21 pages, 5 figures (see version 1), amendment (a single comment added on p. 7)

References in corpus (34)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• Localization transitions in non-Hermitian quantum mechanics
• PT-Symmetric Quantum Mechanics
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Schrödinger operators with complex potential but real spectrum
• Quasi-exactly solvable quartic potential
• Complex periodic potentials with real band spectra
• Non-Hermitean Localization and De-Localization
• PT symmetric square well
• Generalized Continuity Equation and Modified Normalization in PT-Symmetric Quantum Mechanics
• Path Integral Discussion for Smorodinsky-Winternitz Potentials: I.\ Two- and Three Dimensional Euclidean Space
• Exact Solvability of Superintegrable Systems
• Exact Solvability of the Calogero and Sutherland Models
• Non-Hermitian Hamiltonians with real and complex eigenvalues in a Lie-algebraic framework
• Some properties of eigenvalues and eigenfunctions of the cubic oscillator with imaginary coupling constant
• Nonlinear Supersymmetry, Quantum Anomaly and Quasi-Exactly Solvable Systems
• Complex Calogero model with real energies
• Quantum Complex Henon-Heiles Potentials
• Variational Ansatz for PT-Symmetric Quantum Mechanics
• Conditions for complex spectra in a class of PT symmetric potentials
• Spontaneous breakdown of PT symmetry in the solvable square-well model
• Complex Optical Potentials and Pseudo-Hermitian Hamiltonians
• The energy spectrum of complex periodic potentials of the Kronig-Penney type
• On the uncertainty relations and squeezed states for the quantum mechanics on a circle
• On an exactly solvable $B_N$ type Calogero model with nonhermitian PT invariant interaction
• Complexified PSUSY and SSUSY interpretations of some PT-symmetric Hamiltonians possessing two series of real energy eigenvalues
• Quasi-exactly solvable quartic potentials with centrifugal and Coulombic terms
• Non-Hermitian matrix description of the PT symmetric anharmonic oscillators
• Hidden Algebra of Three-Body Integrable Systems
• Pseudohermitian Hamiltonians, time-reversal invariance and Kramers degeneracy
• Poeschl-Teller paradoxes
• Exactly solvable three-body Calogero-type model with translucent two-body barriers
• Comment on "On the uncertainty relations and squeezed states for the quantum mechanics on a circle"

Cited by in corpus (8)

• Solvable PT-symmetric model with a tunable interspersion of non-merging levels
• Eigenstate entanglement entropy in $PT$ invariant non-Hermitian system
• Experiments in PT-symmetric quantum mechanics
• Deconfinement to Confinement as PT phase transition
• Matching method and exact solvability of discrete PT-symmetric square wells
• Exactly solvable models with PT-symmetry and with an asymmetric coupling of channels
• Periodic square-well potential and spontaneous breakdown of PT-symmetry
• PT-symmetric regularizations in supersymmetric quantum mechanics