Chris Marx
Oberlin College
Title
Ergodic Schrödinger Operators on the Bethe Lattice and a Modified Thouless Formula
Abstract
Random Schrödinger operators on the Bethe lattice received considerable attention in the literature as an interesting `intermediary’ between one-dimensional and multidimensional results. While the tree structure of the Bethe lattice gives rise to a recursive structure similar to one-dimensional Schrödinger operators, the geometry of the Bethe lattice implies exponential growth of the surface-volume ratio, which is in stark contrast to Schrödinger operators on $Z^d$. This feature presents difficulties when trying to approximate infinite volume quantities, e.g. spectral averages for functions of the infinite volume Hamiltonian, by finite volume restrictions.
In this talk, we will explain the basic set-up for general ergodic Schrödinger operators on the Bethe lattice and present results on the limiting behavior of finite volume restrictions of functions of the Hamiltonian. We will use these results to present a generalization of Thouless’ formula to ergodic Schrödinger operators on the Bethe lattice. The Thouless formula connects the Lyapunov exponent (defined as the exponential decay rate of the Green function) to the density of states.The talk is based on joint work with Peter D. Hislop.