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 (2)

• An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential
• An explicitly solvable model of the spontaneous PT-symmetry breaking

Cited by in corpus (7)

• 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
• Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem
• Isospectral Hamiltonians from Moyal products
• 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
• Metric Operators for Quasi-Hermitian Hamiltonians and Symmetries of Equivalent Hermitian Hamiltonians