Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1802.03369 · Journal of Physics A Mathematical and Theoretical

Generalized Heisenberg algebra and (non linear) pseudo-bosons

F Bagarello, Evaldo M. F. Curado, J P Gazeau

Published Feb 7, 2018• 16 citations
arXivHTMLPDF

Abstract

Abstract We propose a deformed version of the generalized Heisenberg algebra by using techniques borrowed from the theory of pseudo-bosons. In particular, this analysis is relevant when non self-adjoint Hamiltonians are needed to describe a given physical system. We also discuss relations with nonlinear pseudo-bosons. Several examples are discussed.

DOI
10.1088/1751-8121/aaad6d

Related papers

Pseudo-bosons and bi-coherent states out of ℒ2(ℝ)

Fabio Bagarello

Abstract In this paper we continue our analysis on deformed canonical commutation relations and on their related pseudo-bosons and bi-coherent states. In particular, we show how to extend the original approach outside the Hilbert space ℒ 2 ( ℝ ) , leaving untouched the possibility of defining eigenstates of certain number-like operators, manifestly non self-adjoint, but opening to the possibility that these states are not square-integrable. We also extend this possibility to bi-coherent states, and we discuss in many details an example based on a couple of superpotentials first introduced in [30]. The results deduced here belong to the same distributional approach to pseudo-bosons first proposed in [29].

Journal of Physics Conference Series•Oct 1, 2021•8 citations

Pseudo-Hermitian position and momentum operators, Hermitian Hamiltonian, and deformed oscillators

Alexandre M. Gavrilik, І. І. Качурик

The recently introduced by us, two- and three-parameter (p, q)- and (p, q, [Formula: see text])-deformed extensions of the Heisenberg algebra were explored under the condition of their direct link with the respective (nonstandard) deformed quantum oscillator algebras. In this paper, we explore certain Hermitian Hamiltonians build in terms of non-Hermitian position and momentum operators obeying definite [Formula: see text](N)-pseudo-hermiticity properties. A generalized nonlinear (with the coefficients depending on the particle number operator N) one-mode Bogoliubov transformation is developed as main tool for the corresponding study. Its application enables to obtain the spectrum of “almost free” (but essentially nonlinear) Hamiltonian.

arXiv:1808.04714•Modern Physics Letters A•Dec 14, 2018•1 citation

New results for Heisenberg dynamics for non self-adjoint Hamiltonians

Fabio Bagarello

In a previous paper we began our analysis on the role of non self-adjoint Hamiltonians in connection with the Heisenberg dynamics for quantum systems. Here, motivated by the growing interest on this topic and on some recent results on dynamical systems, we continue this analysis focusing on what we believe is an unexplored (or, at least, not so explored! aspect of Heisenberg dynamics, related to the need for using vectors which are {\em brute-force normalized}. Our main interest is on conserved quantities, and on conditions which guarantee that some observables of the system, or their mean values, do not evolve in time.

math-phCross-listedarXiv:2603.06419•Mar 9, 2026•0 citations

New results for Heisenberg dynamics for nonself-adjoint Hamiltonians

Bagarello F.

Abstract In a previous paper, we began our analysis on the role of non-self-adjoint Hamiltonians in connection with the Heisenberg dynamics for quantum systems. Here, motivated by the growing interest on this topic and on some recent results on dynamical systems, we continue this analysis focusing on what we believe is an unexplored (or, at least, not so explored! aspect of Heisenberg dynamics, related to the need for using vectors which are brute-force normalized . Our main interest is on conserved quantities and on conditions which guarantee that some observables of the system, or their mean values, do not evolve in time.

Zeitschrift für angewandte Mathematik und Physik•Mar 23, 2026•0 citations

Spectral analysis of a selected non self-adjoint Hamiltonian in an infinite dimensional Hilbert space

Natália Bebiano, João da Providência, João P. da Providência

The so-called equation of motion method is useful to obtain the explicit form of the eigenvectors and eigenvalues of certain non self-adjoint bosonic Hamiltonians with real eigenvalues. These operators can be diagonalized when they are expressed in terms of pseudo-bosons, which do not behave as ordinary bosons under the adjoint transformation, but obey the Weil-Heisenberg commutation relations.

arXiv:1509.00653•Sep 2, 2015•0 citations

Damping and pseudo-fermions

Fabio Bagarello

After a short abstract introduction on the time evolution driven by non-self-adjoint Hamiltonians, we show how the recently introduced concept of pseudo-fermion can be used in the description of damping in finite dimensional quantum systems, and we compare the results deduced adopting the Schrödinger and the Heisenberg representations.

arXiv:1212.3663•Journal of Mathematical Physics•Feb 1, 2013•9 citations
Abstract We propose a deformed version of the generalized Heisenberg algebra by using techniques borrowed from the theory of pseudo-bosons. In particular, this analysis is relevant when non self-adjoint Hamiltonians are needed to describe a given physical system. We also discuss relations with nonlinear pseudo-bosons. Several examples are discussed.
Abstract In this paper we continue our analysis on deformed canonical commutation relations and on their related pseudo-bosons and bi-coherent states. In particular, we show how to extend the original approach outside the Hilbert space ℒ 2 ( ℝ ) , leaving untouched the possibility of defining eigenstates of certain number-like operators, manifestly non self-adjoint, but opening to the possibility that these states are not square-integrable. We also extend this possibility to bi-coherent states, and we discuss in many details an example based on a couple of superpotentials first introduced in [30]. The results deduced here belong to the same distributional approach to pseudo-bosons first proposed in [29].
The recently introduced by us, two- and three-parameter (p, q)- and (p, q, [Formula: see text])-deformed extensions of the Heisenberg algebra were explored under the condition of their direct link with the respective (nonstandard) deformed quantum oscillator algebras. In this paper, we explore certain Hermitian Hamiltonians build in terms of non-Hermitian position and momentum operators obeying definite [Formula: see text](N)-pseudo-hermiticity properties. A generalized nonlinear (with the coefficients depending on the particle number operator N) one-mode Bogoliubov transformation is developed as main tool for the corresponding study. Its application enables to obtain the spectrum of “almost free” (but essentially nonlinear) Hamiltonian.
In a previous paper we began our analysis on the role of non self-adjoint Hamiltonians in connection with the Heisenberg dynamics for quantum systems. Here, motivated by the growing interest on this topic and on some recent results on dynamical systems, we continue this analysis focusing on what we believe is an unexplored (or, at least, not so explored! aspect of Heisenberg dynamics, related to the need for using vectors which are {\em brute-force normalized}. Our main interest is on conserved quantities, and on conditions which guarantee that some observables of the system, or their mean values, do not evolve in time.
Abstract In a previous paper, we began our analysis on the role of non-self-adjoint Hamiltonians in connection with the Heisenberg dynamics for quantum systems. Here, motivated by the growing interest on this topic and on some recent results on dynamical systems, we continue this analysis focusing on what we believe is an unexplored (or, at least, not so explored! aspect of Heisenberg dynamics, related to the need for using vectors which are brute-force normalized . Our main interest is on conserved quantities and on conditions which guarantee that some observables of the system, or their mean values, do not evolve in time.