标题： 准正则Sturm--Liouville算子的谱$ζ$-函数 显示英文标题

标题： The spectral $ζ$-function for quasi-regular Sturm--Liouville operators

摘要： 在本工作中，我们分析与下有界准正则Sturm--Liouville算子的自伴扩张$T_{A,B}$相关的谱$\zeta$-函数。 通过利用格林函数形式，我们找到特征函数，该函数隐式地提供了与给定自伴扩张$T_{A,B}$相关的特征值。 然后将特征函数用于构建$T_{A,B}$的谱$\zeta$-函数的围道积分表示。 通过假设特征函数渐近展开的一般形式，我们描述了$\zeta$-函数到复平面上更大区域的解析延拓。 We also present a method for computing the value of the spectral $\zeta$-function of $T_{A,B}$ at all positive integers. We provide two examples to illustrate the methods developed in the paper: the generalized Bessel and Legendre operators. We show that in the case of the generalized Bessel operator, the spectral $\zeta$-function develops a branch point at the origin, while in the case of the Legendre operator it presents, more remarkably, branch points at every nonpositive integer value of $s$. 显示英文摘要

摘要： In this work we analyze the spectral $\zeta$-function associated with the self-adjoint extensions, $T_{A,B}$, of quasi-regular Sturm--Liouville operators that are bounded from below. By utilizing the Green's function formalism, we find the characteristic function which implicitly provides the eigenvalues associated with a given self-adjoint extension $T_{A,B}$. The characteristic function is then employed to construct a contour integral representation for the spectral $\zeta$-function of $T_{A,B}$. By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the $\zeta$-function to a larger region of the complex plane. We also present a method for computing the value of the spectral $\zeta$-function of $T_{A,B}$ at all positive integers. We provide two examples to illustrate the methods developed in the paper: the generalized Bessel and Legendre operators. We show that in the case of the generalized Bessel operator, the spectral $\zeta$-function develops a branch point at the origin, while in the case of the Legendre operator it presents, more remarkably, branch points at every nonpositive integer value of $s$.