Sharp Weyl Estimates for Tensor Products of Pseudodifferential Operators

TL;DR

This paper investigates the asymptotic distribution of eigenvalues for tensor products of pseudodifferential operators, establishing sharp remainder estimates in Weyl laws through explicit example analysis.

Contribution

It provides the first sharp remainder estimates in Weyl formulas for tensor products of pseudodifferential operators on closed manifolds and $ ext{R}^n$.

Findings

Sharp Weyl remainder estimates are obtained.

Explicit examples demonstrate the sharpness of the results.

Results apply to both closed manifolds and $ ext{R}^n$ cases.

Abstract

We study the asymptotic behavior of the counting function of tensor products of operators, in the cases where the factors are either pseudodifferential operators on closed manifolds, or pseudodifferential operators of Shubin type on $\mathbb{R}^n$, respectively. We obtain, in particular, the sharpness of the remainder term in the corresponding Weyl formulae, which we prove by means of the analysis of some explicit examples.