• 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… 
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
. 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

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