Page 394 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 394
ANALYSIS ON GRAPHS (MS-48)
Neumann domains on metric graphs
Ram Band, ramband@technion.ac.il
Technion - Israel Institute of Technology, Israel
Coauthor: Lior Alon
The Neumann points of an eigenfunction f on a quantum (metric) graph are the interior zeros of
f . The Neumann domains of f are the sub-graphs bounded by the Neumann points. Neumann
points and Neumann domains are the counterparts of the well-studied nodal points and nodal
domains.
We present the following three main properties of Neumann domains: their count, wave-
length capacity and spectral position. We study their bounds and probability distributions and
use those to investigate inverse spectral problems.
The relevant probability distributions are rigorously defined in terms of selected random
variables for quantum graphs. To this end, we provide conditions for considering spectral func-
tions of quantum graphs as random variables with respect to the natural density on N.
The talk is based on joint work with Lior Alon.
Universality of nodal count statistics for large quantum graphs
Gregory Berkolaiko, gberkolaiko@tamu.edu
Texas A&M University, United States
Coauthors: Lior Alon, Ram Beand
An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros
due to the graph’s notrivial topology. This number, called the nodal surplus, is an integer be-
tween 0 and the first Betti number of the graph. We study the value distribution of the nodal
surplus within the countably infinite spectrum of the graph. We conjecture that this distribution
converges to Gaussian in any sequence of graphs of growing Betti number. We prove this con-
jecture for several special graph sequences and test it numerically for some other well-known
types of graphs. An accurate computation of the distribution is made possible by a formula
expressing the distribution as an integral over a high-dimensional torus with uniform measure.
The density of states for periodic Jacobi matrices on trees
Jonathan Breuer, jbreuer@math.huji.ac.il
The Hebrew University of Jerusalem, Israel
Periodic Jacobi matrices on trees are a natural generalization of one dimensional periodic
Schroedinger operators. While the one dimensional theory is very well developed, very lit-
tle is known about the general tree case. In this talk we will review some of the few known
results with a focus on the question of convergence of the appropriately defined finite volume
approximations. If time permits, we will also discuss some open problems. Based on joint
works with Nir Avni, Gil Kalai and Barry Simon.
392
Neumann domains on metric graphs
Ram Band, ramband@technion.ac.il
Technion - Israel Institute of Technology, Israel
Coauthor: Lior Alon
The Neumann points of an eigenfunction f on a quantum (metric) graph are the interior zeros of
f . The Neumann domains of f are the sub-graphs bounded by the Neumann points. Neumann
points and Neumann domains are the counterparts of the well-studied nodal points and nodal
domains.
We present the following three main properties of Neumann domains: their count, wave-
length capacity and spectral position. We study their bounds and probability distributions and
use those to investigate inverse spectral problems.
The relevant probability distributions are rigorously defined in terms of selected random
variables for quantum graphs. To this end, we provide conditions for considering spectral func-
tions of quantum graphs as random variables with respect to the natural density on N.
The talk is based on joint work with Lior Alon.
Universality of nodal count statistics for large quantum graphs
Gregory Berkolaiko, gberkolaiko@tamu.edu
Texas A&M University, United States
Coauthors: Lior Alon, Ram Beand
An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros
due to the graph’s notrivial topology. This number, called the nodal surplus, is an integer be-
tween 0 and the first Betti number of the graph. We study the value distribution of the nodal
surplus within the countably infinite spectrum of the graph. We conjecture that this distribution
converges to Gaussian in any sequence of graphs of growing Betti number. We prove this con-
jecture for several special graph sequences and test it numerically for some other well-known
types of graphs. An accurate computation of the distribution is made possible by a formula
expressing the distribution as an integral over a high-dimensional torus with uniform measure.
The density of states for periodic Jacobi matrices on trees
Jonathan Breuer, jbreuer@math.huji.ac.il
The Hebrew University of Jerusalem, Israel
Periodic Jacobi matrices on trees are a natural generalization of one dimensional periodic
Schroedinger operators. While the one dimensional theory is very well developed, very lit-
tle is known about the general tree case. In this talk we will review some of the few known
results with a focus on the question of convergence of the appropriately defined finite volume
approximations. If time permits, we will also discuss some open problems. Based on joint
works with Nir Avni, Gil Kalai and Barry Simon.
392
