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International Journal of Modern Physics A

Closed-form bound states for two Dunkl–Liénard oscillator systems

Axel Schulze‐Halberg

Published Jan 30, 2024• 12 citations
Publisher

Abstract

In this paper, we construct the general Hamiltonian for position-dependent mass systems within the Dunkl formalism. The associated Schrödinger equation is shown to admit closed-form solutions of bound state type for two particular Dunkl–Liénard systems, given by the Dunkl–Higgs oscillator, and a spiked mass Dunkl oscillator. Differences to the conventional scenario are discussed.

DOI
10.1142/s0217751x24500131

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In this paper, we construct the general Hamiltonian for position-dependent mass systems within the Dunkl formalism. The associated Schrödinger equation is shown to admit closed-form solutions of bound state type for two particular Dunkl–Liénard systems, given by the Dunkl–Higgs oscillator, and a spiked mass Dunkl oscillator. Differences to the conventional scenario are discussed.
Recent studies have shown that the use of Dunkl derivatives instead of ordinary derivatives leads to deriving parity-dependent dynamic solutions. According to this motivation in this manuscript, we formulate the Dunkl-Schrödinger equation within the position-dependent mass formalism and derive an algebraic solution out of it. Our systematic approach lets us observe some new findings in addition to the earlier ones. For example, we find that the solution of the Dunkl-Schröodinger equation with position-dependent mass cannot be considered independent of the choice of parameters. Similarly, through the sl(2) algebra, the energy spectrum and the corresponding wave functions are derived in terms of possible Dunkl, (μμμ), and mass, (ααα), parameters.
In this paper, we investigate the Dunkl–Schrödinger equation in the presence of energy-dependent potentials. A consistent formulation is developed by deriving modified expressions for the probability density, normalization condition, and scalar product, thereby ensuring the conservation of probability. Exact analytical solutions are obtained for both one- and three-dimensional systems. In the one-dimensional case, we study an energy-dependent harmonic oscillator and analyze the resulting energy spectrum for different forms of energy dependence. The analysis is then extended to three dimensions, where central potentials are considered, including the harmonic oscillator and Coulomb interactions. We show that the interplay between the Dunkl deformation and the energy dependence leads to significant modifications of the spectral structure, including nonlinear energy behavior and, in some cases, a finite number of bound states. Concrete physical realizations of such energy-dependent potentials are discussed, including semiconductors with an energy-dependent effective mass, two-dimensional electron gases in strong magnetic fields, and cold atoms in optically engineered traps.
We study persistence of solvability in nonrelativistic quantum systems with positiondependent mass upon introduction of a deformation by Dunkl operators. Conditions are derived for the governing Schrödinger equation of the conventional system to admit the same solutions as in the deformed case, up to a reparametrisation of coupling constants. These conditions require the positiondependent mass or the potential of the system to have a specific form. If this is the case for a particular system, then the Schrödinger equations for its conventional version and for the Dunkl-deformed partner share solutions in the same functional form.
We study the generalized harmonic oscillator that has both the position-dependent mass and the potential depending on the form of mass function in a more general framework. The explicit expressions of the eigenvalue and eigenfunction for such a system are given, they have the same forms as those for the usual harmonic oscillator with constant mass. The coherent state and its properties for the system with PDM are also discussed. We give the corresponding effective potentials for several mass functions, the systems with such potentials are isospectral to the usual harmonic oscillator.