Alexander Sobolev: Asymptotics of the extreme eigenvalues for some Toeplitz-type operators

We study the compact integral operator of the form $B_\alpha u(x) = \alpha^d \int_\Lambda \rho(\alpha(x-y)) u(y) dy, $ acting in $L^2(\Lambda)$, where $\Lambda\subset\mathbb R^d, d\ge 1,$ is a bounded domain, $\rho$ is a real-valued $L^1$-function, and $\alpha\ge 1$ is a parameter. The objective is to describe the asymptotics of the top eigenvalue $\mu_1(B_\alpha)$ of $B_\alpha$ as $\alpha\to\infty$. If $B_\alpha$ is self-adjoint then one can write asymptotic formulas for all eigenvalues $\mu_j(B_\alpha)$, $\alpha\to\infty$ for fixed $j\ge 1$. This work is motivated by the random walk problem. Joint with A. Nazarov (St Petersburg).