邓克利量子力学中对称伍兹 - 萨克逊势的束缚态、散射态和共振态
Bound, Scattering, and Resonance States of the Symmetric Woods--Saxon Potential in Dunkl Quantum Mechanic
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中文总结 AI 辅助
该研究在邓克利量子力学框架下，对一维对称伍兹 - 萨克逊势的多种态进行统一解析描述。通过佩克里斯型近似转化问题，推导出相关解，揭示了邓克利形变对谱性质的影响，此框架为研究有限范围势及反射形变量子系统提供了通用方法。
中文摘要 AI 辅助
我们在邓克利量子力学框架内，首次对一维对称伍兹 - 萨克逊势的束缚态、散射态和伽莫夫（亚稳态）共振态进行了统一的解析描述，建立了一个能一致描述所有物理相关谱域的单一解析框架。邓克利形变引入了与反射扇区相关的平方反比相互作用，使得相应薛定谔方程无法直接进行解析处理。通过采用佩克里斯型近似，将问题转化为超几何微分方程，从而在统一形式下推导出解析波函数、束缚态量子化条件、散射振幅和伽莫夫共振解。结果表明，邓克利形变通过产生与反射扇区相关的束缚态谱分裂以及反射和透射概率的定量变化，从根本上改变了系统的谱性质。窄透射峰源于长寿命的伽莫夫共振态，其复能量、衰变宽度和寿命通过解析延拓到复能平面来确定。在邓克利形变消失的极限情况下，能完全恢复传统的伍兹 - 萨克逊结果，证实了所提形式体系的一致性。所提出的解析框架为研究邓克利量子力学中的有限范围势提供了一种通用方法，并为超越伍兹 - 萨克逊相互作用的反射形变量子系统的系统研究开辟了道路。
英文摘要
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.
