Nodal Statistics on Quantum Graphs

Abstract

It has been suggested that the distribution of the suitably normalized number of zeros of Laplacian eigenfunctions contains information about the geometry of the underlying domain. We study this distribution (more precisely, the distribution of the "nodal surplus") for Laplacian eigenfunctions of a metric graph. The existence of the distribution is established, along with its symmetry. One consequence of the symmetry is that the graph's first Betti number can be recovered as twice the average nodal surplus of its eigenfunctions. Furthermore, for graphs with disjoint cycles it is proven that the distribution has a universal form -it is binomial over the allowed range of values of the surplus. To prove the latter result, we introduce the notion of a local nodal surplus and study its symmetry and dependence properties, establishing that the local nodal surpluses of disjoint cycles behave like independent Bernoulli variables.

Oscillation Properties of Singular Quantum Trees

“…Another complication arises from the cycles; the papers by R. Band, G. Berkolaiko a.o. [19][20][21][22][23] discuss deep dependence between the nodal counts, Betty numbers, and geometric structure of the underlying metric graphs.…”

On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs

“…For example, it has been shown in [5] that Courant's bound applies to quantum graphs as well. Yet, for graphs there are generically infinitely many Courant sharp eigenfunctions [7,6]. For tree graphs it has been proven that all generic eigenfunctions are Courant sharp, i.e., ν n = n [8,9] .…”

“…Some statistical properties of the nodal count are also known [6], but to date there is no general explicit formula or a full statistical description of the nodal count.…”

Understanding Quantum Graphs

“…Quantum graphs -ordinary differential equations on metric graphs -have attracted a lot of attention in recent years [1][2][3]. These models were used by physicists due to their simplicity and rich spectral properties, in particular as an explicit model for transition from regular to chaotic behaviour (see, e.g., [4][5][6][7][8]). One has even performed experiments using microwave networks to simulate quantum graphs [9,10].…”