Characteristic functions of $p$-adic integral operators

TL;DR

This paper studies $p$-adic integral operators with homogeneous polynomials, showing their eigenvalues relate to zeros of $q$-hypergeometric functions via $q$-Wronskians, linking operator spectra to special functions.

Contribution

It establishes a connection between the spectra of $p$-adic integral operators and $q$-hypergeometric equations through $q$-Wronskians, extending understanding of their eigenstructure.

Findings

Eigenvalues are reciprocals of zeros of $q$-Wronskians.

Characteristic functions are $q$-Wronskians of solutions.

Results apply to operators with homogeneous polynomials.

Abstract

Let $P\in \Bbb Q_p[x,y]$, $s\in \Bbb C$ with sufficiently large real part, and consider the integral operator $ (A_{P,s}f)(y):=\frac{1}{1-p^{-1}}\int_{\Bbb Z_p}|P(x,y)|^sf(x) |dx| $ on $L^2(\Bbb Z_p)$. We show that if $P$ is homogeneous then for each character $\chi$ of $\Bbb Z_p^\times$ the characteristic function $\det(1-uA_{P,s,\chi})$ of the restriction $A_{P,s,\chi}$ of $A_{P,s}$ to the eigenspace $L^2(\Bbb Z_p)_\chi$ is the $q$-Wronskian of a set of solutions of a (possibly confluent) $q$-hypergeometric equation. In particular, the nonzero eigenvalues of $A_{P,s,\chi}$ are the reciprocals of the zeros of such $q$-Wronskian.