Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: from the time-independent to the time dependent quantum mechanical formulation

Abstract

We provide a reviewlike introduction into the quantum mechanical formalism related to non-Hermitian Hamiltonian systems with real eigenvalues. Starting with the time-independent framework we explain how to determine an appropriate domain of a non-Hermitian Hamiltonian and pay particular attention to the role played by PT-symmetry and pseudo-Hermiticity. We discuss the time-evolution of such systems having in particular the question in mind of how to couple consistently an electric field to pseudo-Hermitian Hamiltonians. We illustrate the general formalism with three explicit examples: i) the generalized Swanson Hamiltonians, which constitute non-Hermitian extensions of anharmonic oscillators, ii) the spiked harmonic oscillator, which exhibits explicit supersymmetry and iii) the -x^4-potential, which serves as a toy model for the quantum field theoretical phi^4-theory.

14 pages, 3 figures, to appear in Laser Physics, minor typos corrected

References in corpus (9)

• An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential
• Time evolution of non-Hermitian Hamiltonian systems
• Equivalence of a Complex $\cP\cT$-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly
• Perturbation theory of PT-symmetric Hamiltonians
• Operator equations and Moyal products -- metrics in quasi-hermitian quantum mechanics
• Moyal products -- a new perspective on quasi-hermitian quantum mechanics
• Isospectral Hamiltonians from Moyal products
• Differential Realization of Pseudo-Hermiticity: A quantum mechanical analog of Einstein's field equation
• Detecting Broken PT-Symmetry

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