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An-Najah University Journal for Research - A (Natural Sciences)

On the coherent states associated with four-parameter Mittag-Leffler functions

Dušan Popov

Published Dec 1, 2025• 0 citations
Publisher

Abstract

In order to examine the generalized coherent states, defined in the Barut-Girardello manner, whose normalization unction is just the four-parameter generalized Mittag-Leffler function, we used a special pair of non-linear perators which generate these states. Some characteristic properties for pure, as well as mixed (thermal) oherent states are also examined. All calculations are made using the rules of the diagonal ordering of perators (DOOT). Finally, the integral counterpart of the Mittag-Leffler coherent states are examined which is onnected with nu-function. The paper is an example of the application of a mathematical entities (Mittag-Leffler function and nu-function) in quantum mechanics (coherent states formalism)

DOI
10.35552/anujr.a.a2710

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In order to examine the generalized coherent states, defined in the Barut-Girardello manner, whose normalization unction is just the four-parameter generalized Mittag-Leffler function, we used a special pair of non-linear perators which generate these states. Some characteristic properties for pure, as well as mixed (thermal) oherent states are also examined. All calculations are made using the rules of the diagonal ordering of perators (DOOT). Finally, the integral counterpart of the Mittag-Leffler coherent states are examined which is onnected with nu-function. The paper is an example of the application of a mathematical entities (Mittag-Leffler function and nu-function) in quantum mechanics (coherent states formalism)
We defined and used a pair of Hermitian annihilation and creation operators which generate the generalized coherent states, defined in the Barut-Girardello manner, whose normalization function is just the four-parameter generalized Mittag-Leffler function. We examined the characteristic properties for these pure, as well as mixed (thermal) coherent states. All calculations are made using the rules of the technique of diagonal ordering of operators. Finally, the integral counterpart of the Mittag-Leffler coherent states are examined which is connected with nu-function. The paper is an example of the application of a mathematical entity (Mittag-Leffler function) in quantum mechanics (coherent states formalism).
Abstract The purpose of this paper is to show that by using a certain type of discrete-continuous limit, a series of integral entities can be defined (Mittag-Leffler multi-index functions, associated coherent states and their properties), which are counterparts of the corresponding discrete entities. We built and examine the properties of a new aspect of generalized integral multi-index Mittag-Leffler functions and we constructed and examined the properties of coherent states associated with this new function. This approach is motivated through the fact that these functions can be connected with the coherent states of the continuous spectrum, as well as with so-called nu-function.
In this paper we extend the applicability of Fox-Wright functions beyond mathematics, specifically in quantum physics. We focused our attention on a new application, on the connection between the Fox-Wright functions and the generalized coherent states formalism. We constructed the generalized coherent states in the Barut-Girardello manner, in which the Fox-Wright functions play the role of normalization functions, and we demonstrated that the Fox-Wright coherent states satisfy all general conditions imposed on the set of coherent states. In parallel, we examined the properties of both pure and mixed (thermal) Fox-Wright coherent states. All calculations were performed within the diagonal operators ordering technique (DOOT) using the Dirac's bra-ket formalism. Finally, we introduced some (specifically, integral) feedback elements that Fox-Wright coherent states induce in the theory of special functions, including a new integral representation of Fox-Wright functions.
We create a family of boson coherent states using the functions of Mittag-Leffler (ML) E ( z ), ( >0) and their generalizations E , ( z ), ( , >0) instead of exponentials. These states are shown to satisfy the usual requirements of normalizability, continuity in the label and the resolution of unity with a positive weight function. This last quantity is found for arbitrary , >0 as a solution of an associated Stieltjes moment problem. In addition, for = m = 1,2,3 ... and = 1 (corresponding to E m ( z )) we propose and analyse special q -deformations (0< q 1) of the functions E m ( z ) which serve as a tool to define q -deformed coherent states of ML type. We provide the expressions for expectation values of physical quantities for all the above states. We discuss physical properties of these states, noting that they are squeezed. The ML coherent states are sub-Poissonian in nature, whereas the q -deformed ML states can be sub- and super-Poissonian depending on q . All these states are shown to be eigenstates of deformed boson operators whose commutation relations are given.
The technique regarding the integration within a normally ordered product of operators, which refers to the creation and annihilation operators of the harmonic oscillator coherent states, has proved to be very fruitful for different operator identities and applications in quantum optics. In this paper we propose a generalization of this technique by introducing a new operatorial approach—the diagonal ordering operation technique (DOOT)—regarding the calculations connected with the normally ordered product of generalized creation and annihilation operators that generate the generalized hypergeometric coherent states. We have pointed out a number of properties of these coherent states, including the case of mixed (thermal) states. At the same time, by particularizing the obtained results to the one-dimensional harmonic and pseudoharmonic oscillators, we get the well-known results achieved through other methods in the corresponding coherent states representation.