Ad Hoc Physical Hilbert Spaces in Quantum Mechanics

0451498 - ÚJF 2016 RIV US eng J - Journal Article
Fernandez, F. M. - Garcia, J. - Semorádová, Iveta - Znojil, Miloslav
Ad Hoc Physical Hilbert Spaces in Quantum Mechanics.
International Journal of Theoretical Physics. Roč. 54, č. 12 (2015), s. 4187-4203. ISSN 0020-7748. E-ISSN 1572-9575
Institutional support: RVO:61389005
Keywords : quantum mechanics * physical Hilbert spaces * ad hoc inner product * singular potentials regularized * low lying energies
Subject RIV: BE - Theoretical Physics
Impact factor: 1.041, year: 2015

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.
Permanent Link: http://hdl.handle.net/11104/0252644