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Form factor for a family of quantum graphs: An expansion to third order | | Gregory Berkolaiko
; Holger Schanz
; Robert S. Whitney
; | | Date: |
8 May 2002 | | Subject: | Chaotic Dynamics; Mesoscopic Systems and Quantum Hall Effect | nlin.CD cond-mat.mes-hall | | Abstract: | 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 $ au$ from periodic-orbit theory. We consider the contributions from pairs of periodic orbits represented by diagrams with up to two self-intersections connected by up to four arcs and explain why all other diagrams are expected to give higher-order corrections only. For a large family of graphs with ergodic classical dynamics the diagrams that exist in the absence of time-reversal symmetry sum to zero. The mechanism for this cancellation is rather general which suggests that it may also apply at higher-orders in the expansion. This expectation is in full agreement with the fact that in this case the linear-$ au$ contribution, the diagonal approximation, already reproduces the random-matrix form factor for $ au<1$. For systems with time-reversal symmetry there are more diagrams which contribute at third order. We sum these contributions for quantum graphs with uniformly hyperbolic dynamics, obtaining $+2 au^{3}$, in agreement with random-matrix theory. As in the previous calculation of the leading-order correction to the diagonal approximation we find that the third order contribution can be attributed to exceptional orbits representing the intersection of diagram classes. | | Source: | arXiv, nlin.CD/0205014 | | Services: | Forum | Review | PDF | Favorites | |
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