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arXiv:1609.03627 · Modern Physics Letters A

SU(1,1) solution for the Dunkl–Coulomb problem in two dimensions and its coherent states

M. Salazar–Ramírez, D. Ojeda-Guillén, R. D. Mota, V. D. Granados

Published Jun 21, 2018• 42 citations
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Abstract

We study the radial part of the Dunkl–Coulomb problem in two dimensions and show that this problem possesses the SU(1,[Formula: see text]1) symmetry. We introduce two different realizations of the su(1,[Formula: see text]1) Lie algebra and use the theory of irreducible representations to obtain the energy spectrum and eigenfunctions. For the first algebra realization, we apply the Schrödinger factorization to the radial part of the Dunkl–Coulomb problem to construct the algebra generators. In the second realization, we introduce three operators, one of them proportional to the Hamiltonian of the radial Schrödinger equation. Finally, we use the SU(1,[Formula: see text]1) Sturmian basis to construct the radial coherent states in a closed form.

DOI
10.1142/s0217732318501122

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We study the radial part of the Dunkl–Coulomb problem in two dimensions and show that this problem possesses the SU(1,[Formula: see text]1) symmetry. We introduce two different realizations of the su(1,[Formula: see text]1) Lie algebra and use the theory of irreducible representations to obtain the energy spectrum and eigenfunctions. For the first algebra realization, we apply the Schrödinger factorization to the radial part of the Dunkl–Coulomb problem to construct the algebra generators. In the second realization, we introduce three operators, one of them proportional to the Hamiltonian of the radial Schrödinger equation. Finally, we use the SU(1,[Formula: see text]1) Sturmian basis to construct the radial coherent states in a closed form.
We investigate the ddd-dimensional Dunkl--Klein--Gordon equation for a scalar particle within an algebraic framework. By employing Schrödinger factorization, we construct the generators of the su(1,1)\mathfrak{su}(1,1)su(1,1) algebra and establish the associated symmetry of the radial sector. The corresponding energy spectra are derived from unitary irreducible representations, and the Sturmian radial bases are obtained analytically. We analyze the ddd-dimensional Dunkl--Klein--Gordon oscillator and the bound-state sector of the ddd-dimensional Dunkl--Klein--Gordon equation with a Dunkl--Coulomb-like potential. Furthermore, SU(1,1)\mathrm{SU}(1,1)SU(1,1) coherent states are constructed and their time evolution is analyzed, revealing a characteristic radial oscillation behavior. The results show that the Dunkl deformation modifies the spatial and spectral structure of the system and, in the oscillator sector, introduces explicit parity-dependent energy shifts while preserving the underlying radial algebraic dynamics.
We apply the Schrödinger factorization method to the radial second-order equation for the relativistic Kepler-Coulomb problem. From these operators we construct two sets of one-variable radial operators which are realizations for the su(1,1)su(1,1)su(1,1) Lie algebra. We use this algebraic structure to obtain the energy spectrum and the supersymmetric ground state for this system.
We investigate the two-dimensional Dunkl--Pauli equation for a spin--1/21/21/2 particle in the presence of an external magnetic field within an algebraic framework. While previous studies have provided exact solutions of the Dunkl--Pauli system, its underlying dynamical symmetry and algebraic structure remain largely unexplored. By employing Schrödinger factorization, we construct the generators of the su(1,1)\mathfrak{su}(1,1)su(1,1) algebra and establish the associated symmetry of the radial sector. The energy spectrum is derived using irreducible unitary representations, and the corresponding Sturmian radial basis is obtained analytically. Furthermore, SU(1,1)\mathrm{SU}(1,1)SU(1,1) coherent states are constructed and their time evolution is analyzed, revealing a characteristic radial breathing behavior. The results show that the Dunkl deformation introduces parity-dependent modifications in the spatial structure of the system while preserving its underlying algebraic dynamics.
For a particular form of the radial Schrödinger equation it is possible to define SU(1, 1) as a noninvariance group. From the SU(1, 1) algebra and the properties of the unitary irreducible representations, one can easily find the energy eigenvalues. It is observed that the equation is essentially the standard form of the C-type factorization (in the classification of Infeld and Hull). Transformations exist which allow the B, D, and F factorization types to be solved by the same method. General expressions are found for the eigenvalues corresponding to each of the relevant factorization types and a few specific examples are given. The eigenfunctions for the C-type and F-type problems are also discussed.
We study the radial part of the McIntosh-Cisneros-Zwanziger (MICZ)-Kepler problem in an algebraic way by using the 𝔰𝔲(1, 1) Lie algebra. We obtain the energy spectrum and the eigenfunctions of this problem from the 𝔰𝔲(1, 1) theory of unitary representations and the tilting transformation to the stationary Schrödinger equation. We construct the physical Perelomov number coherent states for this problem and compute some expectation values. Also, we obtain the time evolution of these coherent states.