受Braak启发的狄拉克束缚态问题谱行列式方法
A Braak-Inspired Spectral Determinant Approach to a Dirac Bound-State Problem
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中文总结 AI 辅助
本文针对吸引型软芯库仑势中的一维狄拉克粒子，构建宇称分辨谱连接结构，通过引入t=√(x²+β²)处理分支点，结合准精确Bethe-ansatz扇区，阐明局域准精确可解性与全局谱行列式的区别，受量子拉比模型启发但未等同Braak的G函数。
中文摘要 AI 辅助
我们针对处于吸引型软芯库仑势中的一维狄拉克粒子，构建了宇称分辨的谱连接结构。该势的平方根依赖关系会在复平面上产生代数分支点，我们通过引入\nt = √(x²+β²)来均匀化这些分支点，同时保留由Π=σ_z P_x生成的全直线Z₂对称性。在每个宇称扇区中，所得问题被表述为在物理原点选取的解与无穷远处衰减解之间的全局连接问题，这定义了宇称分辨的谱连接函数，其零点决定了束缚态能量。我们还直接从同一狄拉克方程构建了准精确Bethe-ansatz扇区，并将其多项式解作为谱条件的独立基准。该比较阐明了局域准精确可解性与全局谱行列式之间的区别。该结构受量子拉比模型中离散对称性和全局解析性的作用启发，但未将当前谱函数与Braak的G函数等同。
英文摘要
We develop a parity-resolved spectral-connection construction for a one-dimensional Dirac particle in an attractive soft-core Coulomb potential. The square-root dependence of the potential produces algebraic branch points in the coordinate plane. We uniformize these branch points by introducing \(t=\sqrt{x^2+β^2}\), while retaining the full-line \(Z_2\) symmetry generated by \(Π=σ_z P_x\). In each parity sector the resulting problem is formulated as a global connection problem between a solution selected at the physical origin and the decaying solution at infinity. This defines parity-resolved spectral connection functions whose zeros determine bound-state energies. We further construct a quasi-exact Bethe-ansatz sector directly from the same Dirac equations and use its polynomial solutions as independent benchmarks of the spectral condition. The comparison clarifies the distinction between local quasi-exact solvability and the global spectral determinant. The construction is inspired by the role of discrete symmetry and global analyticity in the quantum Rabi model, without identifying the present spectral function with Braak's \(G\)-function.
