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

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Quantum theory
Algebra
Momentum
momentum
operators
Non-self-adjoint Operator
Operator
Quantum Mechanics
eigenvectors
Alternatives
quantum mechanics
algebra
Strategy
Similarity

All Science Journal Classification (ASJC) codes

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

Cita questo

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title = "Coordinate representation for non-Hermitian position and momentum operators",
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.",
author = "Fabio Bagarello and Francesco Gargano and Salvatore Triolo and Salvatore Spagnolo and Bagarello",
year = "2017",
language = "English",
volume = "473",
pages = "1--13",
journal = "Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences",
issn = "0080-4630",
publisher = "The Royal Society",

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TY - JOUR

T1 - Coordinate representation for non-Hermitian position and momentum operators

AU - Bagarello, Fabio

AU - Gargano, Francesco

AU - Triolo, Salvatore

AU - Spagnolo, Salvatore

AU - Bagarello, null

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10447/241108

UR - http://rspa.royalsocietypublishing.org/content/473/2205/20170434

M3 - Article

VL - 473

SP - 1

EP - 13

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0080-4630

ER -