Coordinate representation for non-Hermitian position and momentum operators

In this paper, we undertake an analysis of the eigenstates of two non-self-adjoint operators q^ and p^ similar, in a suitable sense, to the self-adjoint position and momentum operators q^0 and 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^ and p^, based on the so-called quasi *-algebras.

[1]  F. Gieres.  Mathematical surprises and Dirac's formalism in quantum mechanics , 1999, quant-ph/9907069.

[2]  Fabio Bagarello,et al.  Deformed Canonical (anti‐)commutation relations and non‐self‐adjoint hamiltonians , 2015 .

[3]  N. Bebiano,et al.  Mathematical Aspects of Quantum Systems with a Pseudo-Hermitian Hamiltonian , 2016 .

[4]  M. Znojil,et al.  Problem of the coexistence of several non-Hermitian observables in PT -symmetric quantum mechanics , 2016, 1610.09396.

[5]  Fabio Bagarello,et al.  Non-selfadjoint operators in quantum physics : mathematical aspects , 2015 .

[6]  Fabio Bagarello,et al.  kq-Representation for pseudo-bosons, and completeness of bi-coherent states , 2017, 1701.05180.

[7]  A. F. Adams,et al.  The Survey , 2021, Dyslexia in Higher Education.

[8]  F. Bagarello.  Some results on the dynamics and transition probabilities for non self-adjoint hamiltonians , 2015, 1502.07175.

[9]  C. Bender,et al.  Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry , 1997, physics/9712001.

[10]  F. Bagarello,et al.  Model pseudofermionic systems: Connections with exceptional points , 2014, 1402.6201.

[11]  Anatolii A. Logunov,et al.  Introduction to axiomatic quantum field theory , 1975 .

[12]  M. Sakamoto,et al.  Spectral intertwining relations in exactly solvable quantum-mechanical systems , 2017, 1701.04307.

[13]  D. Krejčiřík,et al.  On the metric operator for the imaginary cubic oscillator , 2012, 1208.1866.

[14]  J. Zak.  Dynamics of Electrons in Solids in External Fields , 1968 .

[15]  Carl M. Bender,et al.  Interactions of Hermitian and non-Hermitian Hamiltonians , 2007, 0709.3605.

[16]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[17]  ${\mathcal{PT}}$-Symmetric Square Well-Perturbations and the Existence of Metric Operator , 2011 .

[18]  J. Antoine,et al.  Partial *- Algebras and Their Operator Realizations , 2002 .

[19]  A. Kuijlaars,et al.  Spectral Asymptotics of the Non‐Self‐Adjoint Harmonic Oscillator , 2004 .

[20]  R. Parthasarathy.  A Modern Introduction to Quantum Field Theory , 2005 .

[21]  F. Bagarello,et al.  𝒟\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {D}$\end{document}-Deformed Harmonic Oscillators , 2014, International Journal of Theoretical Physics.

[22]  Factorization Method and the Position-dependent Mass Problem , 2013 .

[23]  Jun-Qing Li,et al.  Investigation of PT-symmetric Hamiltonian Systems from an Alternative Point of View , 2012, 1204.6544.

[24]  E. Davies,et al.  Pseudo–spectra, the harmonic oscillator and complex resonances , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[25]  B. Bagchi,et al.  Minimal length in quantum mechanics and non-Hermitian Hamiltonian systems , 2009, 0907.5354.

[26]  N. Bebiano,et al.  Implications of losing Hermiticity in quantum mechanics , 2017 .

[27]  Jean-Pierre Antoine,et al.  Partial Inner Product Spaces: Theory and Applications , 2009 .

[28]  Jean-Pierre Antoine,et al.  Partial Inner Product Spaces , 2009 .

[29]  F. Bagarello.  Transition probabilities for non self-adjoint Hamiltonians in infinite dimensional Hilbert spaces , 2015, 1508.02572.

[30]  Fabio Bagarello,et al.  Algebras of unbounded operators and physical applications: a survey , 2007, 0903.5446.

[31]  C. Bender.  Introduction to 𝒫𝒯-symmetric quantum theory , 2005, quant-ph/0501052.

[32]  Naila Amir,et al.  Algebraic solutions of shape-invariant position-dependent effective mass systems , 2016, 1606.08125.

[33]  Fabio Bagarello,et al.  Intertwining operators for non-self-adjoint Hamiltonians and bicoherent states , 2016, 1609.07579.