Bound, Scattering, and Resonance States of the Symmetric Woods--Saxon Potential in Dunkl Quantum Mechanic

Mebarek Heddar; Can Ertuğay; Bekir Can Lütfüoğlu

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arXiv · 2607.16944

Bound, Scattering, and Resonance States of the Symmetric Woods--Saxon Potential in Dunkl Quantum Mechanic

Mebarek Heddar·Can Ertuğay·Bekir Can Lütfüoğlu

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Abstract

We present the first unified analytical description of the bound, scattering, and Gamow (metastable) resonance states of the one-dimensional symmetric Woods--Saxon potential within the framework of Dunkl quantum mechanics, establishing a single analytical framework that consistently describes all physically relevant spectral regimes. The Dunkl deformation introduces a reflection-sector-dependent inverse-square interaction that precludes a direct analytical treatment of the corresponding Schrödinger equation. By employing a Pekeris-type approximation, the problem is transformed into a hypergeometric differential equation, enabling the derivation of analytical wave functions, bound-state quantization conditions, scattering amplitudes, and Gamow resonance solutions within a unified formalism. The results reveal that the Dunkl deformation fundamentally modifies the system's spectral properties by generating a reflection-sector-dependent splitting of the bound-state spectrum, along with quantitative changes in the reflection and transmission probabilities. The narrow transmission peaks are shown to originate from long-lived Gamow resonance states, whose complex energies, decay widths, and lifetimes are determined through analytic continuation into the complex-energy plane. In the limit of vanishing Dunkl deformation, the conventional Woods--Saxon results are fully recovered, confirming the consistency of the proposed formalism. The proposed analytical framework provides a versatile approach for investigating finite-range potentials in Dunkl quantum mechanics and opens the way to systematic studies of reflection-deformed quantum systems beyond the Woods--Saxon interaction.

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Mebarek Heddar, Can Ertuğay, Bekir Can Lütfüoğlu. 2026-07-18. Bound, Scattering, and Resonance States of the Symmetric Woods--Saxon Potential in Dunkl Quantum Mechanic. https://arxiv.org/abs/2607.16944

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