Ph.D Thesis


Ph.D StudentAlon Lior
SubjectQuantum Graghs - Generic Eigenfunctions and their
Nodal Count and Neumann Count Statistics
DepartmentDepartment of Mathematics
Supervisor ASSOCIATE PROF. Ram Band


Abstract

In this thesis, we study Laplacian eigenfunctions on metric graphs, also known as quantum graphs. We restrict the discussion to standard quantum graphs. These are finite connected metric graphs with functions that satisfy Neumann vertex conditions.

The first goal of this thesis is the study of the nodal count problem. That is the number of points on which the n-th eigenfunction vanishes. We provide a probabilistic setting using which we are able to define the nodal count statistics. We show that the nodal count statistics admits a topological symmetry by which the first Betti number of the graph can be obtained. This result generalizes a result by which the nodal count is 0,1,2,3... if and only if the graph is a tree. We revise a conjecture that predicts a universal Gaussian behavior of the nodal count statistics for large graphs and prove it for a certain family of graphs which we call trees of cycles.

The second goal is to formulate and study a new closely related counting problem which we call the Neumann count, in which one counts the number of local extrema of the n-th eigenfunction. This counting problem is motivated by the Neumann partitions of planar domains, a novel concept in spectral geometry. We provide uniform bounds on the Neumann count and investigate the Neumann statistics using our probabilistic setting. We show that the Neumann count statistics admits a symmetry by which the number of leaves of the graph can be obtained. In particular, we show that the Neumann count provides a complementary geometrical information to that obtained from the nodal count. We show that for a certain family of tree graphs the Neumann count statistics can be calculated explicitly and it approaches a Gaussian distribution for large enough graphs, similarly to the nodal count conjecture.

The third goal is a genericity result, which justifies the generality of the Neumann count discussion. To this day it was known that generically, eigenfunctions do not vanish on vertices. We generalize this result to derivatives at vertices as well. That is, generically, the derivatives of an eigenfunction on interior vertices do not vanish.