Semiclassical Calculation of the C Operator in PT-Symmetric Quantum Mechanics

Abstract

To determine the Hilbert space and inner product for a quantum theory defined by a non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian $H$, it is necessary to construct a new time-independent observable operator called $C$. It has recently been shown that for the {\it cubic} $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian $H=p^2+x^2+i\epsilon x^3$ one can obtain $\mathcal{C}$ as a perturbation expansion in powers of $\epsilon$. This paper considers the more difficult case of noncubic Hamiltonians of the form $H=p^2+x^2(ix{)}^{\delta }$ ($\delta \ge 0$). For these Hamiltonians it is shown how to calculate $\mathcal{C}$ by using nonperturbative semiclassical methods.

arXiv comment11 pages, 1 figure

References in corpus (15)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Complex Extension of Quantum Mechanics
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• PT-Symmetric Quantum Mechanics
• Pseudo-Hermiticity versus PT-Symmetry II: A complete characterizatio n of non-Hermitian Hamiltonians with a real spectrum
• Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Must a Hamiltonian be Hermitian?
• Quantum Entanglement of Moving Bodies
• Supersymmetry and the spontaneous breakdown of PT symmetry
• Solution of Schwinger-Dyson Equations for $\mathcal{P}\mathcal{T}$-Symmetric Quantum Field Theory
• Space of State Vectors in PT Symmetrical Quantum Mechanics
• Non-Hermitian Hamiltonians with real and complex eigenvalues in a Lie-algebraic framework
• Calculation of the Hidden Symmetry Operator in PT-Symmetric Quantum Mechanics
• Quantum Complex Henon-Heiles Potentials

Cited by in corpus (23)

• Making Sense of Non-Hermitian Hamiltonians
• Introduction to PT-Symmetric Quantum Theory
• Dual PT-Symmetric Quantum Field Theories
• An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential
• PT-Symmetric Cubic Anharmonic Oscillator as a Physical Model
• Ghost Busting: PT-Symmetric Interpretation of the Lee Model
• Construction of a unique metric in quasi-Hermitian quantum mechanics: non-existence of the charge operator in a 2 x 2 matrix model
• The C Operator in PT-Symmetric Quantum Field Theory Transforms as a Lorentz Scalar
• A note on the integrability of non-Hermitian extensions of Calogero-Moser-Sutherland models
• Nonunique C operator in PT Quantum Mechanics
• Application of Pseudo-Hermitian Quantum Mechanics to a Complex Scattering Potential with Point Interactions
• $J$-self-adjoint operators with $\mathcal{C}$-symmetries: extension theory approach
• PT-Symmetric Quantum Theory Defined in a Krein Space
• PT symmetric fermionic field theories with axions: Renormalization and dynamical mass generation
• Analysis of the interaction of an electron with radial electric fields in the presence of a disclination
• Unbounded C-symmetries and their nonuniqueness
• Entangled Quantum State Discrimination using Pseudo-Hermitian System
• A semiclassical treatment of the interaction of non-uniform electric fields with the electric quadrupole moment of a neutral particle
• Small-$\epsilon$ behavior of the Non-Hermitian PT-Symmetric Hamiltonian $H=p^2+x^2(ix{)}^{\epsilon }$
• Time-dependent C-operators as Lewis-Riesenfeld invariants in non-Hermitian theories
• $\mathcal{P}\mathcal{T}$ symmetric models with nonlinear pseudo supersymmetry
• A non-trivial PT-symmetric continuum Hamiltonian and its Eigenstates and Eigenvalues
• A selection rule for transitions in PT-symmetric quantum theory