arXiv subjects

R. D. Mota

Publications and source records attributed to R. D. Mota.

At least 19 recordsLinked to original sources

arXiv2026

Quantum Dynamics and Collapse-and-Revival Phenomena in the Dunkl Anharmonic Oscillator

We study the Dunkl anharmonic oscillator (Kerr medium) Hamiltonian from an algebraic approach of the $SU(1,1)$ group. In order to obtain the exact energy spectrum of this problem, we write its Hamiltonian in terms of the Dunkl creation and annihilation operators, which close the $su(1,1)$ Lie algebra. This allows us to exactly solve this Hamiltonian and obtain its parity-dependent energy spectrum. Then, we investigate the quantum dynamics of the system, particularly the collapse and revival phenomena, by using an initial state given by a superposition of even and odd Dunkl coherent states. We compute the field quadrature and the survival probability, showing that the Dunkl parameter $μ$ modulates the fractional revivals and produces perfect state reconstructions at half-periods for specific deformation values. We analyze the quadrature variance to show that the Dunkl deformation generates interference-induced squeezed states around $t \approx π$. The standard Kerr medium dynamics are exactly recovered in the limit $μ\rightarrow 0$.

quant-ph↗

arXiv2026

A Generalization of the Parametric Amplifier with Dunkl Derivative: Spectral and Statistical Properties

We study the parametric amplifier Hamiltonian within the framework of the Dunkl formalism. We introduce the Dunkl creation and annihilation operators and show that their quadratic combinations generate an $su(1,1)$ Lie algebra. The spectral problem is solved exactly using two algebraic methods: the $su(1,1)$ tilting transformation and the generalized Bogoliubov transformation. The exact energy spectrum and the corresponding eigenfunctions are obtained in terms of the Dunkl number coherent states. Furthermore, we compute the Mandel $Q$ parameter and the second-order correlation function $g^{(2)}(0)$ to analyze the statistical properties of the Dunkl squeezed states. We show that, for the squeezed vacuum, the Mandel parameter remains independent of the Dunkl deformation, whereas the correlation function exhibits an explicit dependence on the Dunkl parameter $μ$, which modifies the photon bunching effects. Finally, we show that our results reduce to the standard parametric amplifier case in the limit of vanishing Dunkl deformation parameter.

quant-ph↗

arXiv2025

Exact Solutions of the Schrödinger-Dunkl Equation for a Free Particle in a Finite and Infinite Cylindrical Well

In this paper, we study the Schrödinger equation with Dunkl derivative for a free particle confined in a cylindrical potential well. We consider both the finite and infinite height cases. The Dunkl formalism introduces reflection operators that modify the structure of the Hamiltonian and affect the parity of the solutions. By working in cylindrical coordinates, we obtain exact analytical expressions for the radial and axial wavefunctions in terms of Bessel functions. The energy spectrum and the solutions are classified according to the eigenvalues of the reflection operators in the three coordinates. We analyze in detail the conditions under which the wavefunctions acquire definite parity and discuss the resulting constraints on the Dunkl parameters.

quant-ph↗

arXiv2025

Algebraic approach for the one-dimensional Dirac-Dunkl oscillator

We extend the $(1+1)$-dimensional Dirac-Moshinsky oscillator by changing the standard derivative by the Dunkl derivative. We demonstrate in a general way that for the Dirac-Dunkl oscillator be parity invariant, one of the spinor component must be even, and the other spinor component must be odd, and vice versa. We decouple the differential equations for each of the spinor component and introduce an appropriate $su(1,1)$ algebraic realization for the cases when one of these functions is even and the other function is odd. The eigenfunctions and the energy spectrum are obtained by using the $su(1,1)$ irreducible representation theory. Finally, by setting the Dunkl parameter to vanish, we show that our results reduce to those of the standard Dirac-Moshinsky oscillator.

quant-ph↗

arXiv2025

Landau levels for the $(2+1)$ Dunkl-Klein-Gordon oscillator

In this paper we study the $(2+1)$-dimensional Klein-Gordon oscillator coupled to an external magnetic field, in which we change the standard partial derivatives for the Dunkl derivatives. We find the energy spectrum (Landau levels) in an algebraic way, by introducing three operators that close the $su(1,1)$ Lie algebra and from the theory of unitary representations. Also we find the energy spectrum and the eigenfunctions analytically, and we show that both solutions are consistent. Finally, we demonstrate that when the magnetic field vanishes or when the parameters of the Dunkl derivatives are set zero, our results are adequately reduced to those reported in the literature.

math-ph↗

arXiv2025

Exact solutions of the Schrödinger Equation with Dunkl Derivative for the Free-Particle Spherical Waves, the Pseudo-Harmonic Oscillator and the Mie-type Potential

We solve exactly the Schrödinger equation for the free-particle, the pseudo-harmonic oscillator and the Mie-type potential in three dimensions with the Dunkl derivative. The equations for the radial and angular parts are obtained by using spherical coordinates and separation of variables. The wave functions and the energy spectrum for these potentials are derived in an analytical way and it is shown that our results are adequately reduced to those previously reported when we remove the Dunkl derivative parameters.

quant-ph↗

arXiv2025

Effect of the two-parameter generalized Dunkl derivative on the two-dimensional Schrödinger equation

We introduce a generalization of the Dunkl-derivative with two parameters to study the Schrödinger equation in Cartesian and polar coordinates in two dimensions. The eigenfunctions and the energy spectrum for the harmonic oscillator and the Coulomb problem are derived in an analytical way and it is shown that our results are properly reduced to those previously reported for the Dunkl derivative with a single parameter.

quant-ph↗

arXiv2025

$SU(1,1)\times SU(2)$ approach and the Mandel parameter to the Hamiltonian of two oscillators with weak coupling

We study the Hamiltonian of two isotropic oscillators with weak coupling from an algebraic approach. We write the Hamiltonian of this problem in terms of the boson generators of the $SU(1,1)$ and $SU(2)$ groups. This allows us to apply two tilting transformations based on both group similarity transformations to obtain its energy spectrum and eigenfunctions. Then, we obtain the Mandel $Q$-parameter and the second-order correlation function $g^2(0)$ of the photon numbers $n_a$ and $n_b$. It is important to note that in our procedure we consider the case of weak coupling.

quant-ph↗

arXiv2024

Exact solution of the isotropic and anisotropic Hamiltonian of two coupled harmonic oscillators

We study the isotropic and anisotropic Hamiltonian of two coupled harmonic oscillators from an algebraic approach of the $SU(1,1)$ and $SU(2)$ groups. In order to obtain the energy spectrum and eigenfunctions of this problem, we write its Hamiltonian in terms of the boson generators of the $SU(1,1)$ and $SU(2)$ groups. We use the one boson and two boson realizations of the $su(1,1)$ Lie algebra, and the one boson realization of the $su(2)$ Lie algebra to apply three tilting transformations to diagonalize the original Hamiltonian. These transformations let us to obtain the exact solutions of the isotropic and the anisotropic cases, from which the particular expected results are obtained for the cases where the coupling is neglected.

quant-ph↗

arXiv2024

Berry phase and the Mandel parameter of the non-degenerate parametric amplifier

We study the non-degenerate parametric amplifier problem from an algebraic approach of the $SU(1,1)$ group. We write the Hamiltonian of this problem in terms of the boson generators of the $SU(1,1)$ group and the difference operator. We apply the tilting transformation to our results to exactly solve this Hamiltonian and obtain its energy spectrum and eigenfunctions. Then, by assuming that our Hamiltonian is an explicit function of time we calculate its Berry phase. Finally we obtain the Mandel $Q-$parameter of the photon numbers $n_a$ and $n_b$.

quant-ph↗

arXiv2024

The Dunkl-Fokker-Planck Equation in $1+1$ Dimensions

By replacing the spatial derivative with the Dunkl derivative, we generalize the Fokker-Planck equation in (1+1) dimensions. We obtain the Dunkl-Fokker-Planck eigenvalues equation and solve it for the harmonic oscillator plus a centrifugal-type potential. Furthermore, when the drift function is odd, we reduce our results to those of the recently developed Wigner-Dunkl supersymmetry.

math-ph↗

arXiv2024

The Generalized Fokker-Planck Equation in terms of Dunkl-type Derivatives

In this work we introduce two different generalizations of the Fokker-Planck equation in (1+1) dimensions by replacing the spatial derivatives in terms of generalized Dunkl-type derivatives involving reflection operators. As applications of these results, we solve exactly the generalized Fokker-Planck equations for the harmonic oscillator and the centrifugal-type potentials.

math-ph↗

arXiv2021

Algebraic solution and coherent states for the Dirac oscillator interacting with the Aharonov-Casher system in the cosmic string background

We introduce an $SU(1,1)$ algebraic approach to study the $(2+1)$-Dirac oscillator in the presence of the Aharonov-Casher effect coupled to an external electromagnetic field in the Minkowski spacetime and the cosmic string spacetime. This approach is based on a quantum mechanics factorization method that allows us to obtain the $su(1,1)$ algebra generators, the energy spectrum and the eigenfunctions. We obtain the coherent states and their temporal evolution for each spinor component of this problem. Finally, for these problems, we calculate some matrix elements and the Schrödinger uncertainty relationship for two general $SU(1,1)$ operators.

quant-ph↗

arXiv2021

Exact Solutions of the 2D Dunkl--Klein--Gordon Equation: The Coulomb Potential and the Klein--Gordon Oscillator

We introduce the Dunkl--Klein--Gordon (DKG) equation in 2D by changing the standard partial derivatives by the Dunkl derivatives in the standard Klein--Gordon (KG) equation. We show that the generalization with Dunkl derivative of the $z$-component of the angular momentum is what allows the separation of variables of the DKG equation. Then, we compute the energy spectrum and eigenfunctions of the DKG equations for the 2D Coulomb potential and the Klein--Gordon oscillator analytically and from an ${\rm su}(1,1)$ algebraic point of view. Finally, we show that if the parameters of the Dunkl derivative vanish, the obtained results suitably reduce to those reported in the literature for these 2D problems.

math-ph↗

arXiv2018

Exact Solution of the Relativistic Dunkl Oscillator in $(2+1)$ Dimensions

In this paper we study the $(2+1)$-dimensional Dirac-Dunkl oscillator coupled to an external magnetic field. Our Hamiltonian is obtained from the standard Dirac oscillator coupled to an external magnetic field by changing the partial derivatives by the Dunkl derivatives. We solve the Dunkl-Klein-Gordon-type equations in polar coordinates in a closed form. The angular part eigenfunctions are given in terms of the Jacobi-Dunkl polynomials and the radial functions in terms of the Laguerre functions. Also, we compute the energy spectrum of this problem and show that, in the non-relativistic limit, it properly reduces to the Hamiltonian of the two dimensional harmonic oscillator.

math-ph↗

arXiv2018

An algebraic approach to a charged particle in an uniform magnetic field

We study the problem of a charged particle in a uniform magnetic field with two different gauges, known as Landau and symmetric gauges. By using a similarity transformation in terms of the displacement operator we show that, for the Landau gauge, the eigenfunctions for this problem are the harmonic oscillator number coherent states. In the symmetric gauge, we calculate the $SU(1,1)$ Perelomov number coherent states for this problem in cylindrical coordinates in a closed form. Finally, we show that these Perelomov number coherent states are related to the harmonic oscillator number coherent states by the contraction of the $SU(1,1)$ group to the Heisenberg-Weyl group.

math-ph↗

arXiv2018

Non-Hermitian inverted Harmonic Oscillator-Type Hamiltonians Generated from Supersymmetry with Reflections

By modifying and generalizing known supersymmetric models we are able to find four different sets of one-dimensional Hamiltonians for the inverted harmonic oscillator. The first set of Hamiltonians is derived by extending the supersymmetric quantum mechanics with reflections to non-Hermitian supercharges. The second set is obtained by generalizing the supersymmetric quantum mechanics valid for non-Hermitian supercharges with the Dunkl derivative instead of $\frac{d}{dx}$. Also, by changing the derivative $\frac{d}{dx}$ by the Dunkl derivative in the creation and annihilation-type operators of the standard inverted Harmonic oscillator $H_{SIO}=-\frac{1}{2}\frac{d^2}{dx^2}-\frac{1}{2}x^2$, we generate the third set of Hamiltonians. The fourth set of Hamiltonians emerges by allowing a parameter of the supersymmetric two-body Calogero-type model to take imaginary values. The eigensolutions of definite parity for each set of Hamiltonians are given.

hep-th↗

arXiv2016

$SU(1,1)$ and $SU(2)$ Perelomov number coherent states: algebraic approach for general systems

We study some properties of the $SU(1,1)$ Perelomov number coherent states. The Schrödinger's uncertainty relationship is evaluated for a position and momentum-like operators (constructed from the Lie algebra generators) in these number coherent states. It is shown that this relationship is minimized for the standard coherent states. We obtain the time evolution of the number coherent states by supposing that the Hamiltonian is proportional to the third generator $K_0$ of the $su(1,1)$ Lie algebra. Analogous results for the $SU(2)$ Perelomov number coherent states are found. As examples, we compute the Perelomov coherent states for the pseudoharmonic oscillator and the two-dimensional isotropic harmonic oscillator.

math-ph↗