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Spectral statistics for the Dirac operator on graphs

Jens Bolte and Jonathan Harrison

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We determine conditions for the quantization of graphs using the Dirac operator for both two- and four-component spinors. According to the Bohigas–Giannoni–Schmit conjecture for such systems with time-reversal symmetry the energy level statistics are expected, in the semiclassical limit, to correspond to those of random matrices from the Gaussian symplectic ensemble. This is confirmed by numerical investigation. The scattering matrix used to formulate the quantization condition is found to be independent of the type of spinor. We derive an exact trace formula for the spectrum and use this to investigate the form factor in the diagonal approximation.


PACS

03.65.Sq Semiclassical theories and applications

03.65.Fd Algebraic methods

MSC

47A10 Spectrum, resolvent

05C50 Graphs and matrices

81R25 Spinor and twistor methods (See also 32L25)

81U20 S-matrix theory, etc.

Subjects

Quantum information and quantum mechanics

Dates

Issue 11 (21 March 2003)

Received 23 October 2002, in final form 29 January 2003

Published 6 March 2003



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