Form factor expansion for large graphs: a diagrammatic approach
Abstract
Description
The form factor of a quantum graph is a function measuring correlations within the spectrum of the graph. It can be expressed as a double sum over the periodic orbits on the graph. We propose a scheme which allows one to evaluate the periodic orbit sum for a special family of graphs and thus to recover the expression for the form factor predicted by the Random Matrix Theory. The scheme, although producing the expected answer, undercounts orbits of a certain structure, raising doubts about an analogous summation recently proposed for quantum billiards.
15 pages, 5 figures, to appear in "Proceedings of Joint Summer Research Conference on Quantum Graphs and Their Applications", AMS Contemporary Mathematics (G.Berkoaliko, R.Carlson, S.Fulling, P.Kuchment, eds)
15 pages, 5 figures, to appear in "Proceedings of Joint Summer Research Conference on Quantum Graphs and Their Applications", AMS Contemporary Mathematics (G.Berkoaliko, R.Carlson, S.Fulling, P.Kuchment, eds)