发表机构
East Carolina University; The Ohio State University（东卡罗来纳大学; 俄亥俄州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对具有 $N$ 个广义点势的 Sturm--Liouville 算子，通过围道积分构造其谱 zeta 函数，并示例展示解析延拓及正则化行列式的计算。
AI 中文摘要
本文分析了与具有 $N$ 个广义点势的正则和奇异 Sturm--Liouville 算子相关的谱 zeta 函数。点相互作用被刻画为定义在 $N+1$ 个相邻区间链上的 Sturm--Liouville 算子的一类特定自伴延拓。每个相互作用由 $SL(2,\R)$ 匹配条件描述，这些条件关联相邻区间上的广义边界值。我们通过涉及适当特征函数的围道积分，构造了具有 $N$ 个广义点势的 Sturm--Liouville 算子的谱 zeta 函数。通过具体例子说明了将谱 zeta 函数解析延拓到原点邻域的过程，并在这些例子中显式计算了 $\zeta$-正则化的函数行列式。
英文摘要
This work analyzes the spectral zeta function associated with regular and singular Sturm--Liouville operators endowed with $N$ generalized point potentials. The point interactions are characterized as a particular class of self-adjoint extensions of Sturm--Liouville operators defined on a chain of $N+1$ adjacent intervals. Each interaction is described by $SL(2,\R)$ matching conditions relating generalized boundary values across neighboring intervals. We construct the spectral zeta function associated with Sturm--Liouville operators endowed with $N$ generalized point potentials in terms of a contour integral involving an appropriate characteristic function. The process of analytic continuation of the spectral zeta function to a neighborhood of the origin is illustrated by means of specific examples where the $ζ$-regularized functional determinant is also explicitly computed.
Comments32 Pages
