Operator equations and Moyal products -- metrics in quasi-hermitian quantum mechanics

Abstract

The Moyal product is used to cast the equation for the metric of a non-hermitian Hamiltonian in the form of a differential equation. For Hamiltonians of the form $p^2+V(ix)$ with $V$ polynomial this is an exact equation. Solving this equation in perturbation theory recovers known results. Explicit criteria for the hermiticity and positive definiteness of the metric are formulated on the functional level.

References added

Cited by in corpus (24)

• Making Sense of Non-Hermitian Hamiltonians
• Pseudo-Hermitian Representation of Quantum Mechanics
• Time evolution of non-Hermitian Hamiltonian systems
• Delta-Function Potential with a Complex Coupling
• Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: from the time-independent to the time dependent quantum mechanical formulation
• Construction of a unique metric in quasi-Hermitian quantum mechanics: non-existence of the charge operator in a 2 x 2 matrix model
• PT-symmetric deformations of integrable models
• Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem
• Non-Hermitian oscillator Hamiltonian and su(1,1): a way towards generalizations
• Conditional observability
• Moyal products -- a new perspective on quasi-hermitian 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
• Differential Realization of Pseudo-Hermiticity: A quantum mechanical analog of Einstein's field equation
• Metric Operators for Quasi-Hermitian Hamiltonians and Symmetries of Equivalent Hermitian Hamiltonians
• Pseudo supersymmetric partners for the generalized Swanson model
• Quasi-hermitian Quantum Mechanics in Phase Space
• Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding
• On the construction of pseudo-hermitian quantum system with a pre-determined metric in the Hilbert space
• Solvable non-Hermitian discrete square well with closed-form physical inner product
• Quasi-Hermitian supersymmetric extensions of a non-Hermitian oscillator Hamiltonian and of its generalizations
• Identification of observables in quantum toboggans
• Green Functions for the Wrong-Sign Quartic
• Exact solution of $D_N$ type quantum Calogero model through a mapping to free harmonic oscillators