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Implications of Spectral Interlacing for Quantum Graphs
Junjie Lu1, Tobias Hofmann2, Ulrich Kuhl1,2
1Institut de Physique de Nice, CNRS, Université Côte d'Azur, 06108 Nice, France.
Entropy (Basel, Switzerland)
|January 21, 2023
Summary
Quantum graphs reveal spectral statistics of chaotic systems. Dirichlet graphs offer analytic expressions for level spacing and number variance, while Neumann graphs show deviations from random matrix predictions.
Area of Science:
- Quantum physics
- Spectral statistics
- Chaos theory
Background:
- Quantum graphs are models for chaotic systems.
- Neumann and Dirichlet boundary conditions yield different spectral properties.
- Neumann eigenvalues deviate from random matrix predictions due to interlacing theorems.
Purpose of the Study:
- Derive analytic expressions for Dirichlet graphs.
- Compare analytic and numerical results for Dirichlet graphs.
- Analyze spectral statistics of small Neumann graphs.
Main Methods:
- Ensemble averaging of spectra.
- Derivation of analytic expressions for level spacing and number variance.
- Numerical simulations for Dirichlet and Neumann graphs.
Main Results:
- Analytic expressions for Dirichlet graphs' spectral statistics presented.
- Numerical results confirm analytic predictions for Dirichlet graphs.
- Deviations from random matrix predictions observed for Neumann graphs.
Conclusions:
- Dirichlet graphs provide a framework for understanding spectral statistics.
- Neumann graphs exhibit distinct statistical behaviors compared to random matrix theory.
- The bond number influences spectral statistics in quantum graphs.
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