Weyl asymptotics for tensor products of operators and Dirichlet divisors

TL;DR

This paper investigates the eigenvalue counting function for tensor product operators, exploring their asymptotic behavior and connections to number theory, especially Dirichlet divisor functions, on manifolds and within Shubin classes.

Contribution

It provides new asymptotic formulas for eigenvalues of tensor product operators and links these results to classical problems in analytic number theory.

Findings

Derived Weyl asymptotics for tensor product operators

Established connections between eigenvalue counts and Dirichlet divisor functions

Extended results to perturbations within Shubin classes

Abstract

We study the counting function of the eigenvalues for tensor products of operators, and their perturbations, in the context of Shubin classes and closed manifolds. We emphasize connections with problems of analytic number theory, concerning in particular generalized Dirichlet divisor functions.