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

G. Berkolaiko

    Research output: Contribution to journalArticle

    23 Citations (Scopus)

    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 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.
    LanguageEnglish
    Pages8373-8392
    Number of pages19
    JournalJournal of Physics A: Mathematical and Theoretical
    Volume36
    Issue number31
    DOIs
    Publication statusPublished - 2003

    Fingerprint

    Quantum Graphs
    Form Factors
    form factors
    Orbits
    Diagram
    diagrams
    expansion
    Time Reversal
    orbits
    Random Matrices
    intersections
    Periodic Orbits
    Higher Order
    Symmetry
    Self-intersection
    Random Matrix Theory
    matrix theory
    symmetry
    Approximation
    Cancellation

    Keywords

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

    Cite this

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

    In: Journal of Physics A: Mathematical and Theoretical, Vol. 36, No. 31, 2003, p. 8373-8392.

    Research output: Contribution to journalArticle

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    AU - Berkolaiko, G.

    PY - 2003

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    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.

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