Spectral asymptotics of pseudodifferential operators with discontinuous symbols

Abstract

We study discrete spectrum of self-adjoint Weyl pseudodifferential operators with discontinuous symbols of the form $1_{\Omega }\varphi$ where $1_{\Omega }$ is the indicator of a domain in $\Omega \subset {\mathbb{R}}^2$, and $\varphi \in C_0^{\infty }({\mathbb{R}}^2)$ 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 $d\ge 1$.

27 pages