The eigenvalues of the Schr ödinger 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… (More)

We study the number of nodal domains (maximal connected regions on which a function has constant sign) of the eigenfunctions of Schrödinger operators on graphs. Under certain genericity condition, we… (More)

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… (More)

The Wigner time delay is a measure of the time spent by a particle inside the scattering region of an open system. For chaotic systems, the statistics of the individual delay times (whose average is… (More)

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… (More)

We consider stochastic difference equation xn+1 = xn ( 1 − hf(xn) + √ hg(xn)ξn+1 ) , n = 0, 1, . . . , where functions f and g are nonlinear and bounded, random variables ξi are independent and h > 0… (More)

We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakónski et al (J. Phys. A, 34, 9303-9317… (More)

We study the territory covered by N Lévy flights by calculating the mean number of distinct sites, ^SN(n)&, visited aftern time steps on ad-dimensional,d>2, lattice. The Le ́vy flights are initially… (More)

We derive an exact expression for the two-point correlation function for quantum star graphs in the limit as the number of bonds tends to infinity. This turns out to be identical to the corresponding… (More)

We investigate statistical properties of the eigenfunctions of the Schrödinger operator on families of star graphs with incommensurate bond lengths. We show that these eigenfunctions are not quantum… (More)