AI 中文总结
本文在Bargmann-Fock空间中研究受驱动单模Kerr腔的稳态谱，通过连分式与Heun方程分类获得精确谱条件，并证明强驱动下前导系数，弱驱动展开闭式求和，核心贡献是谱的精确刻画与渐近验证。
AI 中文摘要
我们研究了Bargmann-Fock空间中相干驱动的单模Kerr腔的稳态谱。对于正的Kerr耦合，哈密顿量是自伴的、下有界的，并且具有紧预解式。对于非零驱动，最小解连分式给出了精确的标量谱条件，所有特征值均为单重的。Liouville变换将特征值方程识别为退化的双合流Heun问题，并对其奇点进行了分类。Olver型Volterra构造给出了无穷远处归一化的扇形解，并带有显式误差界。在强驱动区域，证明了前导转折点能量系数，而下一个系数仍取决于所述全局连接猜想；独立的Bogoliubov和复WKB计算提供了一致性检验。从Bargmann递推关系验证了形式弱驱动展开，其上边缘三对角级数以闭式求和。探索性WKB迭代图仅作为视觉辅助，不建立猜想的全局连接。
英文摘要
We study the stationary spectrum of a coherently driven single-mode Kerr cavity in Bargmann--Fock space. For positive Kerr coupling the Hamiltonian is self-adjoint, bounded below, and has compact resolvent. For nonzero drive, a minimal-solution continued fraction yields an exact scalar spectral condition, with all eigenvalues simple. A Liouville transformation identifies the eigenvalue equation with a degenerate double-confluent Heun problem and classifies its singularities. An Olver-type Volterra construction gives normalized sectorial solutions at infinity with explicit error bounds. In the strong-drive regime, the leading turning-point energy coefficient is proved, whereas the next coefficient remains conditional on a stated global-connection conjecture; independent Bogoliubov and complex-WKB calculations provide consistency checks. A formal weak-drive expansion is verified from the Bargmann recurrence, and its upper-edge tridiagonal series is summed in closed form. Exploratory WKB iteration portraits are included only as visual aids and do not establish the conjectural global connection.
