Spectral asymptotics of pseudodifferential operators with discontinuous symbols

TL;DR

This paper investigates the spectral behavior of certain pseudodifferential operators with discontinuous symbols, establishing sharper decay rates for singular values in polygonal domains and deriving asymptotic formulas in sector domains.

Contribution

It introduces a dual symbol approach to analyze spectral asymptotics, providing new bounds for smooth symbols and confirming the sharpness of decay rates for specific geometries.

Findings

Singular values decay as O(k^{-1} log k) for polygonal domains.

Asymptotic formulas are derived for sector domains.

New bounds for singular values of pseudodifferential operators with smooth symbols.

Abstract

We study discrete spectrum of self-adjoint Weyl pseudodifferential operators with discontinuous symbols of the form $1_\Omega \phi$ where $1_\Omega$ is the indicator of a domain in $\Omega\subset\mathbb R^2$, and $\phi\in C^\infty_0(\mathbb R^2)$ is a real-valued function. It was known that in general, the singular values $s_k$ of such an operator satisfy the bound $s_k = O(k^{-3/4})$, $k = 1, 2, \dots$. We show that if $\Omega$ is a polygon, the singular values decrease as $O(k^{-1}\log k)$. In the case where $\Omega$ is a sector, we obtain an asymptotic formula which confirms the sharpness of the above bound. Our main technical tool is the reduction to another symbol that we call \textit{dual}, which is automatically smooth. To analyse the dual symbol we find new bounds for singular values of pseudodifferential operators with smooth symbols in $L^2(\mathbb R^d)$ for arbitrary dimension $d\ge 1$.