December 31, 2013 – Roman Schubert, University of Bristol

Title: Entropy of eigenfunctions on Quantum Graphs

We are interested in the distribution of eigenfunctions on quantum graphs, in particular how they depend on the
topology of the graph. As a measure for the distribution we consider the entropy, if an eigenfunction has a large entropy it implies that it cannot be concentrated on a small set of edges. We will focus on two classes of graphs, star graphs and regular graphs.
For star graphs we show that the average of the entropies of eigenfunctions is small, indicating eigenfunctions which localise on few bonds.
In contrast for regular graphs with large girth we show that the entropy of eigenfunctions is large. The strongest estimates we obtain for expanders where we choose the length of the bonds randomly, then we can show that with large probability the entropy is at least half as large as the maximal possible value. This is analogous to the results by Anantharaman and Nonnenmacher on the entropy of quantum limits on Anosov manifolds, and we in particular borrow one of their tools, the entropic uncertainty principle by Maassen and Uffink. This is joint work with Lionel Kameni.Entropy of eigenfunctions on Quantum Graphs