p-adic spectral zeta functions via the inverse Stieltjes transform
Abstract
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 ℂp.
Related
- Tropicalization of Fock's inverse spectral transform
- Annular Frobenius Classification of p-adic Stieltjes--Schwarzian Equations
- Bernstein and Stieltjes properties of some functions involving logarithmic functions
- On p-Adic Valuation Trees Arising from Laplace Transform of Fibonacci Polynomials
- Stieltjes constants for Hurwitz-type L-series attached to periodic arithmetic functions
BibTeX
@misc{hu2026padic,
title = {{$p$-adic spectral zeta functions via the inverse Stieltjes transform}},
author = {Su Hu and Min-Soo Kim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.33660},
eprint = {2609.33660},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33660},
}
RIS
TY - UNPB
TI - p-adic spectral zeta functions via the inverse Stieltjes transform
AU - Su Hu
AU - Min-Soo Kim
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2609.33660
UR - https://arxiv.org/abs/2609.33660
AB - 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 ℂ_p.
ER -