Form factor expansion for large graphs: a diagrammatic approach

Abstract

Abstract. 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.

Bipartite separability and nonlocal quantum operations on graphs

“…The operator acts on functions defined on each edge of the graph when the edges are equipped with compact real intervals. [9,10]. (b).…”

Relationship between scattering matrix and spectrum of quantum graphs

“…We refer the reader to the edited volumes [3,11] for a potpourri of new results. Recent reviews, dedicated to quantum graphs, include [25,17].…”

“…One may consider a self-adjoint realisation of the Laplace operator, or a scattering matrix approach. Mathematically, the scattering approach appears to be more tractable, and has formed the basis of most investigations [6,7,8,9,10]. Moreover, the spectra arising from these quantizations are subtly different.…”

Degeneracy in the length spectrum for metric graphs

“…A particular problem is to understand the distribution of differences {λ n − λ m } ∞ n,m=0 and, via a trace formula, this is related to the differences {l(γ ) − l(γ )} for pairs of closed cycles in G [2][3][4][5]12,29]. Indeed, a important role is played by summations over pairs of closed cycles (γ , γ ) with l(γ ) = l(γ ).…”