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Vector coherent states and intertwining operators

F. Bagarello

Published Jan 1, 2009• 17 citations
OpenAlex

Abstract

In this paper we discuss a general strategy to construct vector coherent states of the Gazeau-Klauder type and we use them to built up examples of isospectral hamiltonians. For that we use a general strategy recently proposed by the author and which extends well known facts on intertwining operators. We also discuss the possibility In a recent paper, [1], we have proposed a new procedure which gives rise, given few ingredients, to a hamiltonian h2 which has the same spectrum of a given hamiltonian h1 and whose respective eigenstates are related by a given intertwining operator. These results extend what was discussed in the previous literature on this subject, [2], and have

Related papers

Vector coherent states and intertwining operators

F Bagarello

In this paper, we discuss a general strategy to construct vector coherent states of the Gazeau–Klauder type and we use them to build up examples of isospectral Hamiltonians. For that we use a general strategy recently proposed by the author, which extends well-known facts on intertwining operators. We also discuss the possibility of constructing non-isospectral Hamiltonians with related eigenstates.

arXiv:0904.0201•Journal of Physics A Mathematical and Theoretical•Jan 21, 2009•18 citations

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S. Twareque Ali, Fabio Bagarello

In the spirit of some earlier work on the construction of vector coherent states (VCS) over matrix domains, we compute here such states associated to some physical Hamiltonians. In particular, we construct vector coherent states of the Gazeau–Klauder type. As a related problem, we also suggest a way to handle degeneracies in the Hamiltonian for building coherent states. Specific physical Hamiltonians studied include a single photon mode interacting with a pair of fermions, a Hamiltonian involving a single boson and a single fermion, a charged particle in a three-dimensional harmonic force field and the case of a two-dimensional electron placed in a constant magnetic field, orthogonal to the plane which contains the electron. In this last example, which is related to the fractional quantum Hall effect, an interesting modular structure emerges for two underlying von Neumann algebras, related to opposite directions of the magnetic field. This leads to the existence of coherent states built out of Kubo-Martin-Schwinger (KMS) states for the system.

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Coherent states of systems with pure continuous energy spectra

Zouhaı̈r Mouayn, Hashim A. Yamani

While dealing with a Hamiltonian with a continuous spectrum, we use a tridiagonal method involving orthogonal polynomials to construct a set of coherent states obeying a Glauber-type condition. We perform a Bayesian decomposition of the weight function of the orthogonality measure to show that the obtained coherent states can be recast in the Gazeau–Klauder approach. The Hamiltonian of the ℓ-wave free particle is treated as an example to illustrate the method.

arXiv:2009.09921•Journal of Mathematical Physics•Jun 1, 2021•2 citations
In this paper we discuss a general strategy to construct vector coherent states of the Gazeau-Klauder type and we use them to built up examples of isospectral hamiltonians. For that we use a general strategy recently proposed by the author and which extends well known facts on intertwining operators. We also discuss the possibility In a recent paper, [1], we have proposed a new procedure which gives rise, given few ingredients, to a hamiltonian h2 which has the same spectrum of a given hamiltonian h1 and whose respective eigenstates are related by a given intertwining operator. These results extend what was discussed in the previous literature on this subject, [2], and have
In the spirit of some earlier work on the construction of vector coherent states (VCS) over matrix domains, we compute here such states associated to some physical Hamiltonians. In particular, we construct vector coherent states of the Gazeau–Klauder type. As a related problem, we also suggest a way to handle degeneracies in the Hamiltonian for building coherent states. Specific physical Hamiltonians studied include a single photon mode interacting with a pair of fermions, a Hamiltonian involving a single boson and a single fermion, a charged particle in a three-dimensional harmonic force field and the case of a two-dimensional electron placed in a constant magnetic field, orthogonal to the plane which contains the electron. In this last example, which is related to the fractional quantum Hall effect, an interesting modular structure emerges for two underlying von Neumann algebras, related to opposite directions of the magnetic field. This leads to the existence of coherent states built out of Kubo-Martin-Schwinger (KMS) states for the system.
As a substantial generalization of the technique for constructing canonical and the related nonlinear and q -deformed coherent states, we present here a method for constructing vector coherent states (VCS) in the same spirit. These VCS may have a finite or an infinite number of components. The resulting formalism, which involves an assumption on the existence of a resolution of the identity, is broad enough to include all the definitions of coherent states existing in the current literature, subject to this restriction. As examples, we first apply the technique to construct VCS using the Plancherel isometry for groups and VCS associated with Clifford algebras, in particular quaternions. As physical examples, we discuss VCS for a quantum optical model and finally apply the general technique to build VCS over certain matrix domains.