Nodal Statistics On Quantum Graphs
arXiv:1709.10413 · doi:10.1007/s00220-018-3111-2
Abstract
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 distribution (more precisely, the distribution of the "nodal surplus") for Laplacian eigenfunctions of a metric graph. The existence of the distribution is established, along with its symmetry. One consequence of the symmetry is that the graph's first Betti number can be recovered as twice the average nodal surplus of its eigenfunctions. Furthermore, for graphs with disjoint cycles it is proven that the distribution has a universal form --- it is binomial over the allowed range of values of the surplus. To prove the latter result, we introduce the notion of a local nodal surplus and study its symmetry and dependence properties, establishing that the local nodal surpluses of disjoint cycles behave like independent Bernoulli variables.
arXiv comment39 pages, 16 color figures; to appear in Communications in Mathematical Physics
References in corpus (28)
- Periodic Orbit Theory and Spectral Statistics for Quantum Graphs
- Kirchhoff's Rule for Quantum Wires
- Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
- Chaotic Scattering on Graphs
- Nodal domains statistics - a criterion for quantum chaos
- Percolation model for nodal domains of chaotic wave functions
- On the level spacing distribution in quantum graphs
- The Generalized Star Product and the Factorization of Scattering Matrices on Graphs
- Quantum wires with magnetic fluxes
- Nodal domains of Maass forms I
- On the Spectral Gap of a Quantum Graph
- Nodal domains on quantum graphs
- A lower bound for nodal count on discrete and metric graphs
- Resolving isospectral "drums" by counting nodal domains
- Index theorems for quantum graphs
- Can one count the shape of a drum?
- Quantization of Sinai's Billiard - A Scattering Approach
- Quantum graphs which optimize the spectral gap
- Nodal domains on isospectral quantum graphs: the resolution of isospectrality ?
- Universality of the momentum band density of periodic networks
- Non-Weyl asymptotics for quantum graphs with general coupling conditions
- Relationship between scattering matrix and spectrum of quantum graphs
- Dynamics of nodal points and the nodal count on a family of quantum graphs
- Counting nodal domains on surfaces of revolution
- Periodic quantum graphs from the Bethe-Sommerfeld perspective
- Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions
- On Bogomolny-Schmit conjecture
- Exterior-Interior Duality for Discrete Graphs
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- Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs
- Concrete method for recovering the Euler characteristic of quantum graphs
- On the nodal structure of nonlinear stationary waves on star graphs
- Time evolution and the Schrödinger equation on time dependent quantum graphs
- On fully supported eigenfunctions of quantum graphs
- Dirac particles on periodic quantum graphs