PT-Symmetric Cubic Anharmonic Oscillator as a Physical Model

Abstract

We perform a perturbative calculation of the physical observables, in particular pseudo-Hermitian position and momentum operators, the equivalent Hermitian Hamiltonian operator, and the classical Hamiltonian for the PT-symmetric cubic anharmonic oscillator, $H=p^1/(2m)+{\mu }^2{x}^2/2+i\epsilon x^3$. Ignoring terms of order ${\epsilon }^4$ and higher, we show that this system describes an ordinary quartic anharmonic oscillator with a position-dependent mass and real and positive coupling constants. This observation elucidates the classical origin of the reality and positivity of the energy spectrum. We also discuss the quantum-classical correspondence for this PT-symmetric system, compute the associated conserved probability density, and comment on the issue of factor-ordering in the pseudo-Hermitian canonical quantization of the underlying classical system.

16 pages, 3 figures, Erratum to published version added

References in corpus (2)

• Physical Aspects of Pseudo-Hermitian and $PT$-Symmetric Quantum Mechanics
• PT-Symmetric Quantum Mechanics: A Precise and Consistent Formulation

Cited by in corpus (93)

• Perturbation theory in a framework of iteration methods
• Solutions of PT symmetric tight-binding chain and its equivalent Hermitian counterpart
• First-order intertwining operators and position-dependent mass Schrodinger equations in d dimensions
• Time evolution of non-Hermitian Hamiltonian systems
• Metric Operator in Pseudo-Hermitian Quantum Mechanics and the Imaginary Cubic Potential
• Closed formula for the metric in the Hilbert space of a PT-symmetric model
• Conceptual Problems in Scattering from Localized non-Hermitian Potentials
• Multi-Instantons and Exact Results III: Unified Description of the Resonances of Even and Odd Anharmonic Oscillators
• Physics counterpart of the PT non-hermitian tight-binding chain
• Delta-Function Potential with a Complex Coupling
• Biorthogonal Quantum Systems
• Topological phases in Kitaev chain with imbalanced pairing
• PT-Symmetric Versus Hermitian Formulations of Quantum Mechanics
• Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation
• A spin chain model with non-Hermitian interaction: The Ising quantum spin chain in an imaginary field
• Comments on the Dynamics of the Pais-Uhlenbeck Oscillator
• Pseudo-Hermiticity and some consequences of a generalized quantum condition
• Interactions of Hermitian and non-Hermitian Hamiltonians
• Perfect State Transfer in PT-symmetric Non-Hermitian Networks
• Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: from the time-independent to the time dependent quantum mechanical formulation
• Energy Observable for a Quantum System with a Dynamical Hilbert Space and a Global Geometric Extension of Quantum Theory
• Construction of a unique metric in quasi-Hermitian quantum mechanics: non-existence of the charge operator in a 2 x 2 matrix model
• Supersymmetry vs ghosts
• Non-Hermitian anisotropic XY model with intrinsic rotation-time reversal symmetry
• A generalized quantum nonlinear oscillator
• Application of Pseudo-Hermitian Quantum Mechanics to a PT-Symmetric Hamiltonian with a Continuum of Scattering States
• The PT-symmetric brachistochrone problem, Lorentz boosts and non-unitary operator equivalence classes
• Cryptogauge symmetry and cryptoghosts for crypto-Hermitian Hamiltonians
• $\mathcal{P}\mathcal{T}$-symmetric, non-Hermitian quantum many-body physics -- a methodological perspective
• Permanent Underdetermination from Approximate Empirical Equivalence in Field Theory: Massless and Massive Scalar Gravity, Neutrino, Electromagnetic, Yang-Mills and Gravitational Theories
• The disappearing $Q$ operator
• Hermitian scattering behavior for the non-Hermitian scattering center
• A physical interpretation for the non-Hermitian Hamiltonian
• Coupled-channel version of PT-symmetric square well
• Isospectral Hamiltonians from Moyal products
• Quantum Mechanics of Proca Fields
• A note on the integrability of non-Hermitian extensions of Calogero-Moser-Sutherland models
• Metrics and isospectral partners for the most generic cubic PT-symmetric non-Hermitian Hamiltonian
• Real Description of Classical Hamiltonian Dynamics Generated by a Complex Potential
• A Positive-Definite Scalar Product for Free Proca Particle
• On the 'toy model" in the Reggeon field theory
• Krein-Space Formulation of PT-Symmetry, CPT-Inner Products, and Pseudo-Hermiticity
• Application of Pseudo-Hermitian Quantum Mechanics to a Complex Scattering Potential with Point Interactions
• Nonunique C operator in PT Quantum Mechanics
• Spectrum generating algebras for position-dependent mass oscillator Schrodinger equations
• Scaling behavior and phase diagram of a PT-symmetric non-Hermitian Bose-Hubbard system
• Differential Realization of Pseudo-Hermiticity: A quantum mechanical analog of Einstein's field equation
• Simple harmonic oscillation in non-Hermitian SSH chain at exceptional point
• An introduction to PT-symmetric quantum mechanics -- time-dependent systems
• Pseudo-Hermitian versus Hermitian position-dependent-mass Hamiltonians in a perturbative framework
• Effect of Position-dependent Mass on Dynamical Breaking of Type B and Type X_2 N-fold Supersymmetry
• General Aspects of PT-Symmetric and P-Self-Adjoint Quantum Theory in a Krein Space
• Exceptional points in quantum and classical dynamics
• Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics
• From Pseudo-bosons to Pseudo-Hermiticity via multiple generalized Bogoliubov transformations
• Path-Integral Formulation of Pseudo-Hermitian Quantum Mechanics and the Role of the Metric Operator
• Non-perturbative tests for the Asymptotic Freedom in the $\mathcal{P}\mathcal{T}$-symmetric $(-{\varphi }^4{)}_{3+1}$ theory
• QT-Symmetry and Weak Pseudo-Hermiticity
• Non-Hermitian quantum mechanics: the case of bound state scattering theory
• Repeatability of measurements: Equivalence of hermitian and non-hermitian observables
• Exact Isospectral Pairs of PT-Symmetric Hamiltonians
• Is Weak Pseudo-Hermiticity Weaker than Pseudo-Hermiticity?
• Cryptoreality of nonanticommutative Hamiltonians
• Resonant generation of $p$-wave Cooper pair in non-Hermitian Kitaev chain at exceptional point
• On the construction of pseudo-hermitian quantum system with a pre-determined metric in the Hilbert space
• Gauging non-Hermitian Hamiltonians
• Non-Hermitian systems of Euclidean Lie algebraic type with real eigenvalue spectra
• PT-symmetric quartic anharmonic oscillator and position-dependent mass in a perturbative approach
• Anomalous real spectra of non-Hermitian quantum graphs in strong-coupling regime
• Anderson localization induced by complex potential
• On Existence of a Biorthonormal Basis Composed of Eigenvectors of Non-Hermitian Operators
• A squeeze-like operator approach to position-dependent mass in quantum mechanics
• Perturbative approach for strong and weakly coupled time-dependent non-Hermitian quantum systems
• Origin of branch points in the spectrum of PT-symmetric periodic potentials
• Equivalent Hamiltonian for Lee Model
• Sturm-Schroedinger equations: formula for metric
• Entropy and entanglement in a bipartite quasi-Hermitian system and its Hermitian counterparts
• Non-Hermitian description of the dynamics of inter-chain pair tunnelling
• On Supergroups with Odd Clifford Parameters and Supersymmetry with Modified Leibniz Rule
• An Algebraic PT-Symmetric Quantum Theory with a Maximal Mass
• Topological phase in Kitaev chain with spatially separated pairing processes
• PT Symmetry and Hermithean Hamiltonian in the Local Supercritical Pomeron Model
• Physics of crypto-Hermitian and/or cryptosupersymmetric field theories
• Infinite families of position-dependent mass Schrödinger equations with known ground and first excited states
• New Ansatz for Metric Operator Calculation in Pseudo-Hermitian Field Theory
• Pseudo-Hermiticity, PT-symmetry, and the Metric Operator
• Classical-quantum correspondence for two-level pseudo-Hermitian systems
• Ring-Frustrated Non-Hermitian $XY$ Model
• Deformations of Yang-Mills theory
• Hidden Gauge Symmetry in Holomorphic Models
• Parallel dynamics between non-Hermitian and Hermitian systems
• Linked $\mathcal{P}\mathcal{T}$-symmetry to Supersymmetry in a class of non-Hermitian Hamiltonians
• Consistent treatment of quantum systems with a time-dependent Hilbert space