Week of August 11, 2019

Mon Aug 12, 2019
2:00pm - RH 306 - Mathematical Physics
ILYA KACHKOVSKIY - (Michigan State)
A class of Schrodinger operators with convergent perturbation series

Abstract: Rayleigh--Schrodinger perturbation series is one of the main tools of analyzing eigenvalues and eigenvectors of operators in quantum mechanics. The first part of the talk is expository: I will explain a way of representing all terms of the series in terms of graphs with certain structure (similar representations appear in physical literature in various forms). The second part of talk is based on joint work in progress with L. Parnovski and R. Shterenberg. We show that, for a class of lattice Schrodinger operators with unbounded quasiperiodic potentials, one can establish convergence of these series (which is surprising because the eigenvalues are not isolated). The proof is based on the careful analysis of the graphical structure of terms in order to identify cancellations between terms that contain small denominators. The result implies Anderson localization for a class of Maryland-type models on higher-dimensional lattices.