Form factor for a family of quantum graphs: an expansion to third order
Berkolaiko, G. (2003) Form factor for a family of quantum graphs: an expansion to third order. Journal of Physics A: Mathematical and Theoretical, 36 (31). pp. 83738392. ISSN 03054470 (http://dx.doi.org/10.1088/03054470/36/31/303)
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For certain types of quantum graphs we show that the random matrix form factor can be recovered to at least third order in the scaled time τ from periodicorbit theory. We consider the contributions from pairs of periodic orbits represented by diagrams with up to two selfintersections connected by up to four arcs and explain why all other diagrams are expected to give higherorder corrections only. For a large family of graphs with ergodic classical dynamics the diagrams that exist in the absence of timereversal symmetry sum to zero. The mechanism for this cancellation is rather general which suggests that it also applies at higher orders in the expansion. This expectation is in full agreement with the fact that in this case the linearτ contribution, the diagonal approximation, already reproduces the random matrix form factor for τ < 1. For systems with timereversal symmetry there are more diagrams which contribute at third order. We sum these contributions for quantum graphs with uniformly hyperbolic dynamics, obtaining +2τ3, in agreement with random matrix theory. As in the previous calculation of the leadingorder correction to the diagonal approximation we find that the thirdorder contribution can be attributed to exceptional orbits representing the intersection of diagram classes.


Item type: Article ID code: 2119 Dates: DateEvent2003PublishedKeywords: quantum graphs, form factor, statistics, timereversal symmetry, orbits, Mathematics, Statistics Subjects: Science > Mathematics
Social Sciences > StatisticsDepartment: Faculty of Science > Mathematics and Statistics > Mathematics Depositing user: Strathprints Administrator Date deposited: 17 Nov 2006 Last modified: 04 Jan 2024 13:47 URI: https://strathprints.strath.ac.uk/id/eprint/2119