Form factor for large quantum graphs: evaluating orbits with time reversal

It has been shown that for a certain special type of quantum graphs the random-matrix form factor can be recovered to at least third order in the scaled time τ using periodic-orbit theory. Two types of contributing pairs of orbits were identified: those which require time-reversal symmetry and those which do not. We present a new technique of dealing with contributions from the former type of orbits.

The technique allows us to derive the third-order term of the expansion for general graphs. Although the derivation is rather technical, the advantages of the technique are obvious: it makes the derivation tractable, it identifies explicitly the orbit configurations which give the correct contribution and it is more algorithmic and more system-independent, making possible future applications of the technique to systems other than quantum graphs.

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