1,264 research outputs found
Universality of the momentum band density of periodic networks
The momentum spectrum of a periodic network (quantum graph) has a band-gap
structure. We investigate the relative density of the bands or, equivalently,
the probability that a randomly chosen momentum belongs to the spectrum of the
periodic network. We show that this probability exhibits universal properties.
More precisely, the probability to be in the spectrum does not depend on the
edge lengths (as long as they are generic) and is also invariant within some
classes of graph topologies
Quantum Graphs via Exercises
Studying the spectral theory of Schroedinger operator on metric graphs (also
known as quantum graphs) is advantageous on its own as well as to demonstrate
key concepts of general spectral theory. There are some excellent references
for this study such as a mathematically oriented book by Berkolaiko and
Kuchment, a review with applications to theoretical physicsby Gnutzmann and
Smilansky, and elementary lecture notes by Berkolaiko. Here, we provide a set
of questions and exercises which can accompany the reading of these references
or an elementary course on quantum graphs. The exercises are taken from courses
on quantum graphs which were taught by the authors
Topological Properties of Neumann Domains
A Laplacian eigenfunction on a two-dimensional manifold dictates some natural
partitions of the manifold; the most apparent one being the well studied nodal
domain partition. An alternative partition is revealed by considering a set of
distinguished gradient flow lines of the eigenfunction - those which are
connected to saddle points. These give rise to Neumann domains. We establish
complementary definitions for Neumann domains and Neumann lines and use basic
Morse homology to prove their fundamental topological properties. We study the
eigenfunction restrictions to these domains. Their zero set, critical points
and spectral properties allow to discuss some aspects of counting the number of
Neumann domains and estimating their geometry
Nodal domains on isospectral quantum graphs: the resolution of isospectrality ?
We present and discuss isospectral quantum graphs which are not isometric.
These graphs are the analogues of the isospectral domains in R2 which were
introduced recently and are all based on Sunada's construction of isospectral
domains. After presenting some of the properties of these graphs, we discuss a
few examples which support the conjecture that by counting the nodal domains of
the corresponding eigenfunctions one can resolve the isospectral ambiguity
Scattering from isospectral quantum graphs
Quantum graphs can be extended to scattering systems when they are connected
by leads to infinity. It is shown that for certain extensions, the scattering
matrices of isospectral graphs are conjugate to each other and their poles
distributions are therefore identical. The scattering matrices are studied
using a recently developed isospectral theory. At the same time, the scattering
approach offers a new insight on the mentioned isospectral construction
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