Weyl asymptotics of Bisingular Operators and Dirichlet Divisor Problem

TL;DR

This paper investigates the spectral asymptotics of a class of pseudodifferential operators on product manifolds, connecting Weyl law expansions with the Dirichlet Divisor Problem in number theory.

Contribution

It extends Weyl asymptotics to bisingular operators and links these results to classical problems in analytic number theory, providing explicit asymptotic terms.

Findings

First term of Weyl asymptotics determined for the class of operators.

Second term identified in a special case, refining spectral estimates.

Connections established between spectral asymptotics and the Dirichlet Divisor Problem.

Abstract

We consider a class of pseudodifferential operators defined on the product of two closed manifolds, with crossed vector valued symbols. We study the asymptotic expansion of Weyl counting function of positive selfadjoint operators in this class. Exploiting a general Theorem of J. Aramaki, we determine the first term of the asymptotic expansion of Weyl counting function and, in a special case, we find the second term. We finish with some examples, emphasizing connections with problems of analytic number theory, in particular with Dirichlet Divisor Problem.