Intermediate statistics for a system with symplectic symmetry: the Dirac rose graph

We study the spectral statistics of the Dirac operator on a rose-shaped graph-a graph with a single vertex and all bonds connected at both ends to the vertex. We formulate a secular equation that generically determines the eigenvalues of the Dirac rose graph, which is seen to generalize the secular...

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Published inJournal of physics. A, Mathematical and theoretical Vol. 45; no. 43; pp. 435101 - 23
Main Authors Harrison, J M, Winn, B
Format Journal Article
LanguageEnglish
Published Bristol IOP Publishing 02.11.2012
IOP
Subjects
Approximation
Dirac operator
Eigenvalues
Exact sciences and technology
Graphs
Infinity
intermediate statistics
Mathematical analysis
Mathematical models
Physics
quantum graph
Spectra
Statistics
Symmetry
Pair correlation function
Neumann problem
Vertex
Eigenvalues
Boundary conditions
Spectral functions
Dirac operators
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Abstract We study the spectral statistics of the Dirac operator on a rose-shaped graph-a graph with a single vertex and all bonds connected at both ends to the vertex. We formulate a secular equation that generically determines the eigenvalues of the Dirac rose graph, which is seen to generalize the secular equation for a star graph with Neumann boundary conditions. We derive approximations to the spectral pair correlation function at large and small values of spectral spacings, in the limit as the number of bonds approaches infinity, and compare these predictions with results of numerical calculations. Our results represent the first example of intermediate statistics from the symplectic symmetry class.
AbstractList We study the spectral statistics of the Dirac operator on a rose-shaped graph-a graph with a single vertex and all bonds connected at both ends to the vertex. We formulate a secular equation that generically determines the eigenvalues of the Dirac rose graph, which is seen to generalize the secular equation for a star graph with Neumann boundary conditions. We derive approximations to the spectral pair correlation function at large and small values of spectral spacings, in the limit as the number of bonds approaches infinity, and compare these predictions with results of numerical calculations. Our results represent the first example of intermediate statistics from the symplectic symmetry class.
Author Harrison, J M
Winn, B
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  organization: Loughborough University Department of Mathematical Sciences, Loughborough LE11 3TU, UK
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Keywords Pair correlation function
Neumann problem
Vertex
Eigenvalues
Boundary conditions
Spectral functions
Dirac operators
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Snippet We study the spectral statistics of the Dirac operator on a rose-shaped graph-a graph with a single vertex and all bonds connected at both ends to the vertex....
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StartPage 435101
SubjectTerms Approximation
Dirac operator
Eigenvalues
Exact sciences and technology
Graphs
Infinity
intermediate statistics
Mathematical analysis
Mathematical models
Physics
quantum graph
Spectra
Statistics
Symmetry
Title Intermediate statistics for a system with symplectic symmetry: the Dirac rose graph
URI https://iopscience.iop.org/article/10.1088/1751-8113/45/43/435101
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