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International Journal of Theoretical Physics

, Volume 54, Issue 12, pp 4187–4203 | Cite as

Ad Hoc Physical Hilbert Spaces in Quantum Mechanics

  • Francisco M. Fernández
  • Javier Garcia
  • Iveta Semorádová
  • Miloslav Znojil
Article
  • 144 Downloads

Abstract

The overall principles of what is now widely known as PT-symmetric quantum mechanics are listed, explained and illustrated via a few examples. In particular, models based on an elementary local interaction V(x) are discussed as motivated by the naturally emergent possibility of an efficient regularization of an otherwise unacceptable presence of a strongly singular repulsive core in the origin. The emphasis is put on the constructive aspects of the models. Besides the overall outline of the formalism we show how the low-lying energies of bound states may be found in closed form in certain dynamical regimes. Finally, once these energies are found real we explain that in spite of a manifest non-Hermiticity of the Hamiltonian the time-evolution of the system becomes unitary in a properly amended physical Hilbert space.

Keywords

Quantum mechanics Physical Hilbert spaces Ad hoc inner product Singular potentials regularized Low lying energies 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Francisco M. Fernández
    • 1
  • Javier Garcia
    • 1
  • Iveta Semorádová
    • 2
  • Miloslav Znojil
    • 2
  1. 1.INIFTA (UNLP, CCT La Plata-CONICET), División Química TeóricaLa PlataArgentina
  2. 2.Nuclear Physics Institute ASCRŘežCzech Republic

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