# Polyanalytic relativistic second Bargmann transforms

## Abstract

We construct coherent states through special superpositions of eigenstates of the relativistic isotonic oscillator. In each superposition, the coefficients are chosen to be L{sup 2}-eigenfunctions of a σ-weight Maass Laplacian on the Poincaré disk, which are associated with the eigenvalue 4m(σ−1−m), m∈Z{sub +}∩[0,(σ−1)/2]. For each nonzero m, the associated coherent states transform constitutes the m-true-polyanalytic extension of a relativistic version of the second Bargmann transform, whose integral kernel is expressed in terms of a special Appel-Kampé de Fériet’s hypergeometric function. The obtained results could be used to extend the known semi-classical analysis of quantum dynamics of the relativistic isotonic oscillator.

- Authors:

- Faculty of Sciences and Technics (M’Ghila), P.O. Box 523, Béni Mellal (Morocco)

- Publication Date:

- OSTI Identifier:
- 22403148

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 56; Journal Issue: 5; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANNIHILATION OPERATORS; EIGENFUNCTIONS; EIGENSTATES; EIGENVALUES; HYPERGEOMETRIC FUNCTIONS; KERNELS; LAPLACIAN; OSCILLATORS

### Citation Formats

```
Mouayn, Zouhaïr.
```*Polyanalytic relativistic second Bargmann transforms*. United States: N. p., 2015.
Web. doi:10.1063/1.4919544.

```
Mouayn, Zouhaïr.
```*Polyanalytic relativistic second Bargmann transforms*. United States. doi:10.1063/1.4919544.

```
Mouayn, Zouhaïr. Fri .
"Polyanalytic relativistic second Bargmann transforms". United States. doi:10.1063/1.4919544.
```

```
@article{osti_22403148,
```

title = {Polyanalytic relativistic second Bargmann transforms},

author = {Mouayn, Zouhaïr},

abstractNote = {We construct coherent states through special superpositions of eigenstates of the relativistic isotonic oscillator. In each superposition, the coefficients are chosen to be L{sup 2}-eigenfunctions of a σ-weight Maass Laplacian on the Poincaré disk, which are associated with the eigenvalue 4m(σ−1−m), m∈Z{sub +}∩[0,(σ−1)/2]. For each nonzero m, the associated coherent states transform constitutes the m-true-polyanalytic extension of a relativistic version of the second Bargmann transform, whose integral kernel is expressed in terms of a special Appel-Kampé de Fériet’s hypergeometric function. The obtained results could be used to extend the known semi-classical analysis of quantum dynamics of the relativistic isotonic oscillator.},

doi = {10.1063/1.4919544},

journal = {Journal of Mathematical Physics},

issn = {0022-2488},

number = 5,

volume = 56,

place = {United States},

year = {2015},

month = {5}

}