Asymptotics of the extreme eigenvalues for some Toeplitz-type operators

Hamiltonians in Magnetic Fields

18 October 15:30 - 16:30

Alexander Sobolev - University College London

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).
Rafael D. Benguria
Pontificia Universidad Católica de Chile
Arne Jensen
Aalborg University
Georgi Raikov
Pontificia Universidad Católica de Chile
Grigori Rozenblioum
Chalmers/University of Gothenburg
Jan Philip Solovej
University of Copenhagen