On the construction of pseudo-hermitian quantum system with a pre-determined metric in the Hilbert space

Abstract

A class of pseudo-hermitian quantum system with an explicit form of the positive-definite metric in the Hilbert space is presented. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators those are hermitian with respect to a pre-determined positive definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-hermitian quantum systems admitting entirely real spectra and unitary time evolution. The examples considered include simple harmonic oscillators with complex angular frequencies, Stark(Zeeman) effect with complex electric(magnetic) field, non-hermitian general quadratic form of N boson(fermion) operators, symmetric and asymmetric XXZ spin-chain in complex magnetic field, non-hermitian Haldane-Shastry spin-chain and Lipkin-Meshkov-Glick model.

29 pages, revtex, minor changes, version to appear in Journal of Physics A(v3)

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• Non-Hermitian multi-particle systems from complex root spaces
• Constructing Exactly Solvable Pseudo-hermitian Many-particle Quantum Systems by Isospectral Deformation
• On regular and chaotic dynamics of a non-$\mathcal{P}\mathcal{T}$-symmetric Hamiltonian system of a coupled Duffing oscillator with balanced loss and gain
• Deconstructing non-Dirac-hermitian supersymmetric quantum systems
• Supersymmetric Many-particle Quantum Systems with Inverse-square Interactions
• PT-symmetrically deformed shock waves
• Deconstructing non-dissipative non-Dirac-hermitian relativistic quantum systems
• Edge states and persistent current in a PT-symmetric extended Su-Schrieffer-Heeger model with generic boundary conditions