发表机构
South China University of Technology; Kyungnam University（华南理工大学; 庆南大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过逆 Stieltjes 变换将 $p$-adic 谱 zeta 函数从离散谱扩展到连续谱，构造谱分布并定义玻色子/费米子 zeta 函数，建立解析性质与特殊值公式，并应用于位置算子。
AI 中文摘要
谱 zeta 函数为量子物理中微分算子的行列式正则化提供了标准工具。在先前的一篇论文（J. Math. Phys. 66: 083505, 2025）中，我们通过局部解析插值函数引入了离散谱的 $p$-adic 谱 zeta 函数。本文利用逆 Stieltjes 变换将框架扩展到连续谱。从有界算子的预解式出发，我们构造了一个广义分布，称为 $p$-adic 谱分布，并定义了玻色子和费米子 zeta 函数及泛函行列式。我们确立了它们的解析性质、特殊值公式和 Stirling 展开，并表明在离散情形下我们之前的框架嵌入到当前框架中。在这种方法中，玻色子和费米子情形表现出有趣的对称性。作为应用，我们考虑了 $\bb{C}_{p}$ 的任意紧子集上 $p$-adic 量子力学中的位置算子。
英文摘要
Spectral zeta functions provide a standard tool for regularizing determinants of differential operators in quantum physics. In a previous paper (J. Math. Phys. 66: 083505, 2025), we introduced a $p$-adic spectral zeta function for discrete spectra via a locally analytic interpolation function. In this paper we extend the framework to continuous spectra using the inverse Stieltjes transform. Starting from the resolvent of a bounded operator, we construct a generalized distribution, called the $p$-adic spectral distribution, and define bosonic and fermionic zeta functions and functional determinants. We establish their analytic properties, special value formulas, and Stirling expansions, and show that our previous framework is embedded into the present one in the discrete case. In this approach, the bosonic and fermionic cases exhibit an interesting symmetry. As an application, we consider the position operator in $p$-adic quantum mechanics on an arbitrary compact subset of $\mathbb{C}_{p}$.
Comments43 pages. Dedicated to the memory of Lev Genrikhovich Shnirelman (1905--1938)
