• Corpus ID: 222178214

# Quantum graphs -- Generic eigenfunctions and their nodal count and Neumann count statistics

@article{Alon2020QuantumG,
title={Quantum graphs -- Generic eigenfunctions and their nodal count and Neumann count statistics},
author={Lior Alon},
journal={arXiv: Mathematical Physics},
year={2020}
}
• Lior Alon
• Published 6 October 2020
• Mathematics
• arXiv: Mathematical Physics
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\textquoteright s…
2 Citations
Generic Laplace eigenfunctions on metric graphs
. It is known, that up to certain pathalogies, a compact metric graph with standard vertex conditions has a Baire generic set of choices of edge lengths such that all Laplace eigenvalues are simple
Universality of nodal count distribution in large metric graphs
• Mathematics
• 2021
. An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph’s non-trivial topology. This number, called the nodal surplus, is an integer between 0

## References

SHOWING 1-10 OF 138 REFERENCES
The Number of Nodal Domains on Quantum Graphs as a Stability Index of Graph Partitions
• Mathematics
• 2012
The Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of a
Nodal Statistics on Quantum Graphs
• Mathematics
• 2017
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
A Lower Bound for Nodal Count on Discrete and Metric Graphs
The number of nodal domains (maximal connected regions on which a function has constant sign) of the eigenfunctions of Schrödinger operators on graphs is shown to be bounded below by n, where ℓ is the number of links that distinguish the graph from a tree.
RELATIONSHIP BETWEEN SCATTERING MATRIX AND SPECTRUM OF QUANTUM GRAPHS
• Mathematics
• 2008
AbstractWe investigate the equivalence between spectral characteristics of the Laplace op-erator on a metric graph, and the associated unitary scattering operator. We provethat the statistics of
Dynamics of Nodal Points and the Nodal Count on a Family of Quantum Graphs
• Mathematics
• 2010
We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schrödinger-type differential operator). Using tools such as scattering approach and
Neumann Domains on Quantum Graphs
• Mathematics
• 2019
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 subgraphs bounded by the Neumann points. Neumann points and
Quantum Ergodicity for Graphs Related to Interval Maps
• Mathematics
• 2007
We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakónski et al (J. Phys. A, 34, 9303-9317
Periodic Orbit Theory and Spectral Statistics for Quantum Graphs
• Physics, Mathematics
• 1998
Abstract We quantize graphs (networks) which consist of a finite number of bonds and vertices. We show that the spectral statistics of fully connected graphs is well reproduced by random matrix
Surgery principles for the spectral analysis of quantum graphs
• Mathematics
Transactions of the American Mathematical Society
• 2019
We present a systematic collection of spectral surgery principles for the Laplacian on a metric graph with any of the usual vertex conditions (natural, Dirichlet or $\delta$-type), which show how
Quantum Graphs which Optimize the Spectral Gap
• Mathematics
• 2016
A finite discrete graph is turned into a quantum (metric) graph once a finite length is assigned to each edge and the one-dimensional Laplacian is taken to be the operator. We study the dependence of