Differential Realization of Pseudo-Hermiticity: A quantum mechanical analog of Einstein's field equation

Abstract

For a given pseudo-Hermitian Hamiltonian of the standard form: H=p^2/2m+v(x), we reduce the problem of finding the most general (pseudo-)metric operator ηsatisfying H^\dagger=ηH η^{-1} to the solution of a differential equation. If the configuration space is the real line, this is a Klein-Gordon equation with a nonconstant mass term. We obtain a general series solution of this equation that involves a pair of arbitrary functions. These characterize the arbitrariness in the choice of η. We apply our general results to calculate ηfor the PT-symmetric square well, an imaginary scattering potential, and a class of imaginary delta-function potentials. For the first two systems, our method reproduces the known results in a straightforward and extremely efficient manner. For all these systems we obtain the most general ηup to second order terms in the coupling constants.

14 pages, slightly expanded (published) version

References in corpus (22)

• Complex Extension of Quantum Mechanics
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• 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
• Physical realization of $\mathcal{P}\mathcal{T}$-symmetric potential scattering in a planar slab waveguide
• Physical Aspects of Pseudo-Hermitian and $PT$-Symmetric Quantum Mechanics
• Pseudo-Hermiticity and Generalized PT- and CPT-Symmetries
• Pseudo-Supersymmetric Quantum Mechanics and Isospectral Pseudo-Hermi tian Hamiltonians
• PT symmetric square well
• On Pseudo-Hermitian Hamiltonians and Their Hermitian Counterparts
• On the dimensionality of spacetime
• An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential
• PT-Symmetric Cubic Anharmonic Oscillator as a Physical Model
• PT-symmetric square well and the associated SUSY hierarchies
• Calculation of the Hidden Symmetry Operator for a $\cP\cT$-Symmetric Square Well
• Application of Pseudo-Hermitian Quantum Mechanics to a PT-Symmetric Hamiltonian with a Continuum of Scattering States
• Is Pseudo-Hermitian Quantum Mechanics an Indefinite-Metric Quantum Theory?
• Time-Dependent Hilbert Spaces, Geometric Phases, and General Covariance in Quantum Mechanics
• Solvability and PT-symmetry in a double-well model with point interactions
• Operator equations and Moyal products -- metrics in quasi-hermitian quantum mechanics
• Pseudo-Hermitian versus Hermitian position-dependent-mass Hamiltonians in a perturbative framework
• An explicitly solvable model of the spontaneous PT-symmetry breaking

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• Pseudo-Hermitian Representation of Quantum Mechanics
• Quantum Brachistochrone Problem and the Geometry of the State Space in Pseudo-Hermitian Quantum Mechanics
• Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation
• Self-dual Spectral Singularities and Coherent Perfect Absorbing Lasers without PT-symmetry
• Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: from the time-independent to the time dependent quantum mechanical formulation
• Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem
• Isospectral Hamiltonians from Moyal products
• Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric Aspect of Quantum Mechanics
• Metrics and isospectral partners for the most generic cubic PT-symmetric non-Hermitian Hamiltonian
• Krein-Space Formulation of PT-Symmetry, CPT-Inner Products, and Pseudo-Hermiticity
• Complete Set of Inner Products for a Discrete PT-symmetric Square-well Hamiltonian
• Application of Pseudo-Hermitian Quantum Mechanics to a Complex Scattering Potential with Point Interactions
• Non-Hermitian Hamiltonians with a Real Spectrum and Their Physical Applications
• Metric Operators for Quasi-Hermitian Hamiltonians and Symmetries of Equivalent Hermitian Hamiltonians
• Is Weak Pseudo-Hermiticity Weaker than Pseudo-Hermiticity?
• Solvable non-Hermitian discrete square well with closed-form physical inner product
• Consistent treatment of quantum systems with a time-dependent Hilbert space