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 in | Journal of physics. A, Mathematical and theoretical Vol. 45; no. 43; pp. 435101 - 23 |
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Main Authors | , |
Format | Journal Article |
Language | English |
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Bristol
IOP Publishing
02.11.2012
IOP |
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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. |
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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 |
Author_xml | – sequence: 1 givenname: J M surname: Harrison fullname: Harrison, J M email: jon_harrison@baylor.edu organization: Baylor University Department of Mathematics, One Bear Place, Waco, TX 76798, USA – sequence: 2 givenname: B surname: Winn fullname: Winn, B organization: Loughborough University Department of Mathematical Sciences, Loughborough LE11 3TU, UK |
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Cites_doi | 10.1088/0951-7715/15/4/302 10.1088/0305-4470/32/4/006 10.1023/A:1004108022199 10.1098/rspa.1985.0078 10.1103/PhysRevE.63.068201 10.1103/PhysRevE.47.R3822 10.1088/0951-7715/15/5/311 10.1088/0305-4470/36/27/101 10.1007/s100510170357 10.1007/3-540-13392-5_1 10.1023/A:1026495012522 10.1088/1367-2630/12/12/123021 10.1007/BF02798790 10.1007/s002200100516 10.1209/0295-5075/5/5/001 10.1103/PhysRevE.59.R1315 10.1088/0305-4470/33/50/305 10.1007/BF01057882 10.1088/0305-4470/29/14/015 10.1088/0305-4470/37/28/L01 10.1103/PhysRevE.51.198 10.1088/0305-4470/32/45/302 10.1088/0305-4470/34/3/301 10.1088/0305-4470/36/11/307 10.1088/0959-7174/14/1/014 10.1006/aphy.1999.5904 10.1016/0375-9601(89)90420-9 10.1103/PhysRevA.44.3457 10.1088/0305-4470/31/37/003 10.1103/PhysRevLett.64.1855 10.1088/0305-4470/36/40/L02 10.1103/PhysRevLett.52.1 10.1103/PhysRevE.63.036206 10.1103/PhysRevE.52.3341 10.1103/PhysRevE.65.056214 10.1103/PhysRevB.47.11487 10.1103/PhysRevLett.85.2486 |
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Keywords | Pair correlation function Neumann problem Vertex Eigenvalues Boundary conditions Spectral functions Dirac operators |
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References | 22 23 Giraud O (17) 2004; 37 29 Bolte J (31) 2003; 36 Rahav S (26) 2003; 36 Berkolaiko G (39) 2000 Berkolaiko G (27) 1999; 32 Bolte J (30) 2003; 36 10 Rahav S (25) 2002; 15 11 12 34 13 35 Tudorovskiy T (16) 2010; 12 Bogomolny E (24) 2002; 15 Kostrykin V (33) 1999; 32 37 38 18 19 Gradshteyn I S (36) 2007 Parab H D (15) 1996; 29 1 2 Harmer M (32) 2000; 33 4 Grémaud B (14) 1998; 31 5 6 Scharf R (3) 1988; 5 7 8 9 Berkolaiko G (28) 2001; 34 20 21 |
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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 |
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