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 $1_\Omega \phi$ 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 $d\ge 1$.