Coordinate representation for non-Hermitian position and momentum operators

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Abstract

In this paper, we undertake an analysis of the eigenstates of two non-self-adjoint operators q^q^ and p^p^ similar, in a suitable sense, to the self-adjoint position and momentum operators q^0 q^0 and p^0 p^0 usually adopted in ordinary quantum mechanics. In particular, we discuss conditions for these eigenstates to be biorthogonal distributions, and we discuss a few of their properties. We illustrate our results with two examples, one in which the similarity map between the self-adjoint and the non-self-adjoint is bounded, with bounded inverse, and the other in which this is not true. We also briefly propose an alternative strategy to deal with q^ q^ and p^ p^, based on the so-called quasi *-algebras.
Lingua originaleEnglish
pagine (da-a)1-13
Numero di pagine13
RivistaPROCEEDINGS OF THE ROYAL SOCIETY OF LONDON. SERIES A
Volume473
Stato di pubblicazionePublished - 2017

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)

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