Title: A gap labelling theorem for Schrödinger operators on graphs

Abstract: Given a Schrödinger operator on Rn or Zn, we are often interested in its integrated density of states, which roughly measures the number of states per unit volume in the system below a given energy. Special attention is often given to the values the integrated density of states attains at spectral gaps, known as gap labels, which are of physical importance. For instance, in the integer quantum Hall effect, the gap labels correspond to the quantized values of the Hall conductance. However, predicting these gap labels often requires the use of complicated machinery, such as K-theory, making the proofs quite challenging.

In this talk, we present a more accessible approach to developing gap labelling theorems, based on a method developed by Johnson and Moser. This is done using an object known as the Schwartzman group, and involves computing the rotation of the Prüfer angle for the associated generalized eigenfunctions. Mainly, we present a gap labelling theorem for Schrödinger operators on discrete and metric graphs, providing a simple way to predict the possible gap labels of graphs with the geometric structure of one-dimensional aperiodic tilings. Time permitting, we will also discuss the implications of this to the 'Dry Ten Martini Problem' for Schrödinger operators on graphs with a Sturmian structure.

Based on joint work with Ram Band.