Exact Analytical Solutions of the Dunkl-Schrödinger Equation for the Deng-Fan Potential

TL;DR

The paper derives exact analytical solutions for the radial Dunkl-Schrödinger equation with the Deng-Fan potential, where the Dunkl derivative introduces a parity-dependent repulsive core that modifies the centrifugal barrier to $ℓ(ℓ+2μ+1)$. Using the Pekeris approximation and the parametric Nikiforov-Uvarov method, closed-form expressions for the energy spectrum $E_{nℓ}$ and radial wavefunctions in terms of Jacobi polynomials are obtained. The results show a monotonic increase of $E_{nℓ}$ with the Dunkl parameter $μ$, indicating a hard-core behavior and parity-induced symmetry breaking, while the limit $μ\to 0$ recovers standard Deng-Fan spectra. This work demonstrates the Dunkl formalism as a robust framework for modeling parity-dependent exclusion and short-range correlations in molecular quantum systems.

Abstract

We present exact analytical solutions for the radial Dunkl-Schrödinger equation (DSE) confined by the Deng-Fan molecular potential. By employing the Pekeris approximation to resolve the centrifugal singularity and applying the parametric Nikiforov-Uvarov method, we derive closed-form expressions for the energy eigenspectrum and the corresponding radial wavefunctions expressed in terms of Jacobi polynomials. Our investigation reveals that the Dunkl reflection parameter $μ$ fundamentally alters the system's topology by breaking spatial symmetry and introducing a parity-dependent repulsive force. Numerical analysis demonstrates a monotonic increase in energy eigenvalues with increasing $μ$, confirming an effective "hard core" behavior at the origin. The results are shown to be consistent with standard quantum mechanics in the limit $μ\to 0$. This study establishes the Dunkl formalism as a robust tool for modeling quantum systems characterized by parity-dependent exclusion effects and strong short-range correlations.

Figures (3)

• Figure 1: Comparison of the Deng-Fan potential (blue solid line) and the standard Morse potential (red dashed line). Note the correct $1/r^2$ singularity of the Deng-Fan potential at the origin ($r \to 0$), which makes it physically superior for describing Dunkl "hard core" interactions compared to the finite Morse potential.
• Figure 2: Dependence of the energy eigenvalues $E_{n0}$ on the Dunkl parameter $\mu$. The monotonic increase indicates that the Dunkl reflection operator induces a repulsive effect, effectively hardening the potential core.
• Figure 3: Radial probability density of the ground state for varying Dunkl parameters ($\mu=0, 1.5, 3.0$). The peak of the distribution shifts to larger radial distances as $\mu$ increases, visualizing the centrifugal displacement caused by the Dunkl barrier.