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In this paper, 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. [ABSTRACT FROM AUTHOR]

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. • We generalize the Fokker–Planck equation through Dunkl-type derivatives. • We solve exactly the Dunkl–Fokker–Planck equation for the centrifugal-type potential. • We solve exactly the Dunkl–Fokker–Planck equation for the harmonic oscillator. • We compute the energy spectrum and eigenfunctions of these potentials. [ABSTRACT FROM AUTHOR]

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 d d x . Also, by changing the derivative d d x by the Dunkl derivative in the creation and annihilation-type operators of the standard inverted harmonic oscillator H SIO = − 1 2 d 2 d x 2 − 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. [ABSTRACT FROM AUTHOR]

We introduce a generalization of the Jaynes-Cummings model and study some of its properties. We obtain the energy spectrum and eigenfunctions of this model by using the tilting transformation and the squeezed number states of the onedimensional harmonic oscillator. As physical applications, we connect this new model to two important and novelty problems: the relativistic parametric amplifier and the quantum simulation of a single trapped ion. [ABSTRACT FROM AUTHOR]

We construct the Perelomov number coherent states for an arbitrary su(1, 1) group operation and study some of their properties. We introduce three operators which act on Perelomov number coherent states and close the su(1, 1) Lie algebra. By using the tilting transformation we apply our results to obtain the energy spectrum and eigenfunctions of the non-degenerate parametric amplifier. We show that these eigenfunctions are the Perelomov number coherent states of the two-dimensional harmonic oscillator. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]