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arXiv:2009.09921 · Journal of Mathematical Physics

Coherent states of systems with pure continuous energy spectra

Zouhaı̈r Mouayn, Hashim A. Yamani

Published Jun 1, 2021• 2 citations
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Abstract

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.

DOI
10.1063/5.0030759

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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.
We explicitly construct a Hamiltonian whose exact eigenfunctions are the generalized Laguerre functions. Moreover, we present the related raising and lowering operators. We investigate the corresponding coherent states by adopting the Gazeau–Klauder approach, where resolution of unity and overlapping properties are examined. Coherent states are found to be similar to those found for a particle trapped in a Pöschl–Teller potential of the trigonometric type. Some comparisons with Barut–Girardello and Klauder–Perelomov methods are noted.
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
A new approach is developed for solving spectral problems for operators with continuous spectrum, which consists in the integral transform of the problem by using coherent Schwartz distributions. The constructed family of coherent distributions is a complete analogue of the family of ordinary coherent states. More precisely, it satisfies all Gazeau–Klauder axioms satisfied by the usual coherent states. But in contrast to the coherent states belonging to the point spectrum of the annihilation operator (or operators), the coherent distributions belong to the continuous spectrum of some Hermitian operators. Therefore, the coherent distributions work better than the coherent states as the kernel of the integral representation of generalized eigenfunctions of operators with continuous spectrum. In this work, this approach is demonstrated with an example of solving a basic problem of quantum mechanics, i.e., the problem of the continuous part of the spectrum of the Hamiltonian of the hydrogen atom.
We consider an example of a quantum system with a continuous and unbounded spectrum—a particle in the parabolic potential barrier. Conditionally, we call this system the inverse oscillator. Following the method used in our previous publication (Bagrov et al 2012 J. Phys. A: Math. Theor. 45 125306), we construct different families of generalized coherent states (GCS). We discuss properties of the constructed GCS and their relation to what is already known in the literature about semiclassical states for the inverse oscillator. Then in the same manner, we construct families of GCS for the normal oscillator. Using special variables, we succeeded in constructing GCS for the normal oscillator that admit different limiting cases: free particle GCS, the usual Schrödinger, Glauber CS and GCS of the inverse oscillator.
We are continuing here the study of generalized coherent states of Barut-Girardello type for the oscillator-like systems connected with the given set of orthogonal polynomials. In this work we construct the family of coherent states associated with discrete qqq-Hermite polynomials of the II-type and prove the over-completeness of this family of states by constructing the measure for unity decomposition for this family of coherent states.