03/25/2025
By Joris Roos
The Department of Mathematics and Statistics invites you to attend a colloquium lecture by Ilya Kachkovskiy, Michigan State University, on Wednesday, April 9.
Title: Quasiperiodic operators with monotone potentials: Robust localization and Cantor spectra
Time: 11 a.m. to Noon
Room: Southwick Hall, Room 350W
Everyone is welcome!
Abstract
The general subject of the talk is spectral theory of discrete (tight-binding) Schrodinger operators on $d$-dimensional lattices. For operators with periodic potentials, it is known that the spectra of such operators are purely absolutely continuous. For random i.i.d. potentials, such as the Anderson model, it is expected and can be proved in many cases that the spectra are almost surely purely point with exponentially decaying eigenfunctions (Anderson localization). Quasiperiodic operators can be placed somewhere in between: depending on the potential sampling function and the Diophantine properties of the frequency and the phase, one can have a large variety of spectral types.
We will consider a class of quasiperiodic operators $$ (H(x)\psi)_n=\epsilon(\Delta\psi)_n+f(x+n\cdot\omega)\psi_n,\quad n\in \mathbb Z^d, $$ where $\Delta$ is the discrete Laplacian, $\omega$ is a vector with rationally independent components, and $f$ is a $1$-periodic function on $\mathbb R$, monotone on $(0,1)$ with a positive lower bound on the derivative.
The above condition of monotonicity, as it turns out, implies several unique phenomena, including a robust version of uniform Anderson localization and a new connection between spectral gaps, vanishing of eigenfunctions, and rank one perturbations. In particular, we show that a large class of such operators in 1D has Cantor spectrum.
The talk is based on joint works with S. Jitomirskaya, L. Parnovski and R. Shterenberg.
See the Mathematics and Statistics Colloquium site for more information.