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

Faria, C. F. D. M. & Fring, A. (2007). Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: From the time-independent to the time-dependent quantum mechanical formulation. Laser Physics, 17(4), pp. 424-437. doi: 10.1134/S1054660X07040196

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Abstract

We provide a reviewlike introduction to 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 PJ 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 super-symmetry, and (iii) the −x 4-potential, which serves as a toy model for the quantum field theoretical ϕ4-theory.

Item Type: Article
Uncontrolled Keywords: PT-SYMMETRY, PSEUDO-HERMITICITY, MOYAL PRODUCTS, POTENTIALS, SPECTRUM, STABILIZATION, OSCILLATORS, IONIZATION, OPERATORS, EQUATIONS
Subjects: Q Science > QC Physics
Divisions: School of Engineering & Mathematical Sciences > Department of Mathematical Science
URI: http://openaccess.city.ac.uk/id/eprint/696

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