On the Spectral Asymptotics of Operators on Manifolds with Ends

TL;DR

This paper investigates the asymptotic behavior of the eigenvalue counting function for certain differential operators on noncompact manifolds with ends, improving remainder estimates in Weyl's law based on operator order ratios and manifold dimension.

Contribution

It introduces new techniques using pseudodifferential and Fourier Integral Operators to refine the spectral asymptotics for operators on manifolds with ends, considering their double order structure.

Findings

Improved remainder estimates in Weyl's law for operators with different orders.

Dependence of spectral asymptotics on the ratio of operator orders and manifold dimension.

Application of global pseudodifferential calculus on noncompact manifolds with ends.

Abstract

We deal with the asymptotic behaviour for $\lambda\to+\infty$ of the counting function $N_P(\lambda)$ of certain positive selfadjoint operators $P$ with double order $(m,\mu)$, $m,\mu>0$, $m\not=\mu$, defined on a manifold with ends $M$. The structure of this class of noncompact manifolds allows to make use of calculi of pseudodifferential operators and Fourier Integral Operators associated with weighted symbols globally defined on $\mathbb{R}^n$. By means of these tools, we improve known results concerning the remainder terms of the Weyl Formulae for $N_P(\lambda)$ and show how their behaviour depends on the ratio $\frac{m}{\mu}$ and the dimension of $M$.