No abstract

Using periodic-orbit theory beyond the diagonal approximation we investigate the form factor, K(tau), of a generic quantum graph with mixing classical dynamics and time-reversal symmetry. We calculate the contribution from pairs of self-intersecting orbits that differ from each other only in the orientation of a single loop. In the limit of large graphs, these pairs produce a contribution -2tau(2) to the form factor which agrees with random-matrix theory.

spectral statistics, graph theory, combinatoricsThe eigenvalues of the Schrodinger operator on a graph G are related via an exact trace formula to periodic orbits on G. This connection is used to calculate two-point spectral statistics for a particular family of graphs, called star graphs, in the limit as the number of edges tends to infinity. Combinatorial techniques are used to evaluate both the diagonal (same orbit) and offdiagonal (different orbit) contributions to the sum over pairs of orbits involved. In this way, a general formula is derived for terms in the (short-time) expansion of the form factor K (>) in powers of >, and the first few are computed explicitly. The result demonstrates that K ( >) is neither Poissonian nor random-matrix, but intermediate between the two. Off-diagonal pairs of orbits are shown to make a significant contribution to all but the first few coefficients.

We present a systematic collection of spectral surgery principles for the Laplacian on a compact metric graph with any of the usual vertex conditions (natural, Dirichlet or δ-type), which show how various types of changes of a local or localised nature to a graph impact on the spectrum of the Laplacian. Many of these principles are entirely new; these include "transplantation" of volume within a graph based on the behaviour of its eigenfunctions, as well as "unfolding" of local cycles and pendants. In other cases we establish sharp generalisations, extensions and refinements of known eigenvalue inequalities resulting from graph modification, such as vertex gluing, adjustment of vertex conditions and introducing new pendant subgraphs.To illustrate our techniques we derive a new eigenvalue estimate which uses the size of the doubly connected part of a compact metric graph to estimate the lowest nontrivial eigenvalue of the Laplacian with natural vertex conditions. This quantitative isoperimetric-type inequality interpolates between two known estimates -one assuming the entire graph is doubly connected and the other making no connectivity assumption (and producing a weaker bound) -and includes them as special cases. Contents2010 Mathematics Subject Classification. 34B45 (05C50 35P15 81Q35).

For chaotic cavities with scattering leads attached, transport properties can be approximated in terms of the classical trajectories which enter and exit the system. With a semiclassical treatment involving fine correlations between such trajectories we develop a diagrammatic technique to calculate the moments of various transport quantities. Namely, we find the moments of the transmission and reflection eigenvalues for systems with and without time reversal symmetry. We also derive related quantities involving an energy dependence: the moments of the Wigner delay times and the density of states of chaotic Andreev billiards, where we find that the gap in the density persists when subleading corrections are included. Finally, we show how to adapt our techniques to non-linear statistics by calculating the correlation between transport moments.In each setting, the answer for the n-th moment is obtained for arbitrary n (in the form of a moment generating function) and for up to the three leading orders in terms of the inverse channel number. Our results suggest patterns which should hold for further corrections and by matching with the low order moments available from random matrix theory we derive likely higher order generating functions.