Laser Physics

, Volume 17, Issue 4, pp 424–437 | Cite as

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

  • C. F. M. Faria
  • A. Fring
Strong Field Phenomena

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 −x4-potential, which serves as a toy model for the quantum field theoretical ϕ4-theory.

PACS numbers

03.65.Ca 03.65.Ta 03.65.-w 32.80.Rm 02.30.Mv 

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Copyright information

© MAIK “Nauka/Interperiodica” 2007

Authors and Affiliations

  • C. F. M. Faria
    • 1
  • A. Fring
    • 1
  1. 1.Centre for Mathematical ScienceCity UniversityLondonUK

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