发表机构
Université Constantine 1-Frères Mentouri（康斯坦丁一世大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出广义Dunkl算子与能量依赖相互作用的双参数扩展，精确求解谐振子并分析其热力学性质，揭示参数对谱与热特性的调控作用。
AI 中文摘要
本研究探讨了一类由广义Dunkl-薛定谔方程描述的、具有能量依赖相互作用的精确可解量子系统。我们引入了Dunkl微分算子的双参数扩展，该扩展同时包含传统的反射贡献和额外的导数-反射耦合。由此产生的形式体系导出了修正的海森堡代数，并从中推导出适用于能量依赖势的连续性方程、守恒概率流和广义归一化条件。作为应用，我们详细研究了一个能量依赖的谐振子。我们以关联拉盖尔多项式形式获得了本征函数和能谱的精确解析表达式。结果表明，维格纳参数、形变参数和能量依赖参数的联合效应会产生宇称分裂、非线性能谱和不均匀的能级间距，同时保持精确可解性。随后，我们通过正则配分函数并使用欧拉-麦克劳林求和公式检验了该模型的热力学性质。我们推导并数值分析了配分函数、内能、熵和比热的解析表达式。结果表明，广义Dunkl形变和能量依赖相互作用能够有效控制系统的光谱和热特性，从而扩展了具有反射对称性的精确可解量子模型类别。
英文摘要
This work investigates a class of exactly solvable quantum systems described by a generalized Dunkl--Schrödinger equation with energy-dependent interactions. A two-parameter extension of the Dunkl differential operator is introduced, incorporating both the conventional reflection contribution and an additional derivative-reflection coupling. The resulting formalism leads to a modified Heisenberg algebra, from which the continuity equation, conserved probability current, and generalized normalization condition are derived for energy-dependent potentials. As an application, an energy-dependent harmonic oscillator is studied in detail. Exact analytical expressions for the eigenfunctions and energy spectrum are obtained in terms of associated Laguerre polynomials. The combined effects of the Wigner parameter, the deformation parameter, and the energy-dependence parameter are shown to produce parity splitting, nonlinear energy spectra, and nonuniform level spacing while preserving exact solvability. The thermodynamic properties of the model are then examined through the canonical partition function using the Euler--Maclaurin summation formula. Analytical expressions for the partition function, internal energy, entropy, and specific heat are derived and analyzed numerically. The results demonstrate that the generalized Dunkl deformation and energy-dependent interaction provide effective control over both the spectral and thermal characteristics of the system, extending the class of exactly solvable quantum models with reflection symmetry.
Comments16 pages, 12 figures
