Implications of Spectral Interlacing for Quantum Graphs
Quantum graphs are ideally suited to studying the spectral statistics of chaotic systems. Depending on the boundary conditions at the vertices, there are Neumann and Dirichlet graphs. The latter ones correspond to totally disassembled graphs with a spectrum being the superposition of the spectra of...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-01-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/25/1/109 |
| _version_ | 1797442956410486784 |
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| author | Junjie Lu Tobias Hofmann Ulrich Kuhl Hans-Jürgen Stöckmann |
| author_facet | Junjie Lu Tobias Hofmann Ulrich Kuhl Hans-Jürgen Stöckmann |
| author_sort | Junjie Lu |
| collection | DOAJ |
| description | Quantum graphs are ideally suited to studying the spectral statistics of chaotic systems. Depending on the boundary conditions at the vertices, there are Neumann and Dirichlet graphs. The latter ones correspond to totally disassembled graphs with a spectrum being the superposition of the spectra of the individual bonds. According to the interlacing theorem, Neumann and Dirichlet eigenvalues on average alternate as a function of the wave number, with the consequence that the Neumann spectral statistics deviate from random matrix predictions. There is, e.g., a strict upper bound for the spacing of neighboring Neumann eigenvalues given by the number of bonds (in units of the mean level spacing). Here, we present analytic expressions for level spacing distribution and number variance for ensemble averaged spectra of Dirichlet graphs in dependence of the bond number, and compare them with numerical results. For a number of small Neumann graphs, numerical results for the same quantities are shown, and their deviations from random matrix predictions are discussed. |
| first_indexed | 2024-03-09T12:49:09Z |
| format | Article |
| id | doaj.art-8e2dfd68c1f94fb789b98591550fc005 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-09T12:49:09Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-8e2dfd68c1f94fb789b98591550fc0052023-11-30T22:08:47ZengMDPI AGEntropy1099-43002023-01-0125110910.3390/e25010109Implications of Spectral Interlacing for Quantum GraphsJunjie Lu0Tobias Hofmann1Ulrich Kuhl2Hans-Jürgen Stöckmann3Institut de Physique de Nice, CNRS, Université Côte d’Azur, 06108 Nice, FranceFachbereich Physik, Philipps-Universität Marburg, 35032 Marburg, GermanyInstitut de Physique de Nice, CNRS, Université Côte d’Azur, 06108 Nice, FranceFachbereich Physik, Philipps-Universität Marburg, 35032 Marburg, GermanyQuantum graphs are ideally suited to studying the spectral statistics of chaotic systems. Depending on the boundary conditions at the vertices, there are Neumann and Dirichlet graphs. The latter ones correspond to totally disassembled graphs with a spectrum being the superposition of the spectra of the individual bonds. According to the interlacing theorem, Neumann and Dirichlet eigenvalues on average alternate as a function of the wave number, with the consequence that the Neumann spectral statistics deviate from random matrix predictions. There is, e.g., a strict upper bound for the spacing of neighboring Neumann eigenvalues given by the number of bonds (in units of the mean level spacing). Here, we present analytic expressions for level spacing distribution and number variance for ensemble averaged spectra of Dirichlet graphs in dependence of the bond number, and compare them with numerical results. For a number of small Neumann graphs, numerical results for the same quantities are shown, and their deviations from random matrix predictions are discussed.https://www.mdpi.com/1099-4300/25/1/109quantum graphsinterlacing theoremrandom matrix theory |
| spellingShingle | Junjie Lu Tobias Hofmann Ulrich Kuhl Hans-Jürgen Stöckmann Implications of Spectral Interlacing for Quantum Graphs Entropy quantum graphs interlacing theorem random matrix theory |
| title | Implications of Spectral Interlacing for Quantum Graphs |
| title_full | Implications of Spectral Interlacing for Quantum Graphs |
| title_fullStr | Implications of Spectral Interlacing for Quantum Graphs |
| title_full_unstemmed | Implications of Spectral Interlacing for Quantum Graphs |
| title_short | Implications of Spectral Interlacing for Quantum Graphs |
| title_sort | implications of spectral interlacing for quantum graphs |
| topic | quantum graphs interlacing theorem random matrix theory |
| url | https://www.mdpi.com/1099-4300/25/1/109 |
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