### Abstract

Language | English |
---|---|

Pages | 8373-8392 |

Number of pages | 19 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 36 |

Issue number | 31 |

DOIs | |

Publication status | Published - 2003 |

### Fingerprint

### Keywords

- quantum graphs
- form factor
- statistics
- time-reversal symmetry
- orbits

### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*36*(31), 8373-8392. https://doi.org/10.1088/0305-4470/36/31/303

}

*Journal of Physics A: Mathematical and Theoretical*, vol. 36, no. 31, pp. 8373-8392. https://doi.org/10.1088/0305-4470/36/31/303

**Form factor for a family of quantum graphs: an expansion to third order.** / Berkolaiko, G.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Form factor for a family of quantum graphs: an expansion to third order

AU - Berkolaiko, G.

PY - 2003

Y1 - 2003

N2 - 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 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 also applies at higher orders in the expansion. This expectation is in full agreement with the fact that in this case the linear-τ contribution, the diagonal approximation, already reproduces the random matrix form factor for τ < 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 +2τ3, 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.

AB - 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 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 also applies at higher orders in the expansion. This expectation is in full agreement with the fact that in this case the linear-τ contribution, the diagonal approximation, already reproduces the random matrix form factor for τ < 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 +2τ3, 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.

KW - quantum graphs

KW - form factor

KW - statistics

KW - time-reversal symmetry

KW - orbits

UR - http://dx.doi.org/10.1088/0305-4470/36/31/303

U2 - 10.1088/0305-4470/36/31/303

DO - 10.1088/0305-4470/36/31/303

M3 - Article

VL - 36

SP - 8373

EP - 8392

JO - Journal of Physics A: Mathematical and Theoretical

T2 - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 0305-4470

IS - 31

ER -