Dunkl-Schrödinger Models with Energy-Dependent Couplings: Exact Solutions and Confinement-Like Spectra

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.