Mostafa Sabri

Recent results of quantum ergodicity on graphs

Given a Schrödinger operator on an infinite graph, we may understand the fact that it exhibits a delocalized behavior from several points of view. At the spectral level, we expect absolutely continuous spectrum. At the spatial level, the wave functions should not concentrate on small regions. Finally, on a dynamical level, the waves should diffuse on the graph as time goes on, for example at a linear rate.

In this talk, I will discuss a result showing that in some cases, spectral delocalization implies a form of spatial delocalization. More precisely, consider a Schrödinger operator on an infinite tree whose spectrum (in a certain region) is absolutely continuous. Consider a sequence of finite graphs "converging" to this tree. Then the wave functions become somehow equidistributed on the graph, as the graph gets large. This is called a quantum ergodicity theorem. We can actually consider random models. This theorem applies in particular to the case of the Anderson model on a uniform tree.I will also discuss more recent results of quantum ergodicity on quantum graphs.

Most of the results are based on joint work with Nalini Anantharaman.