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.

(Some figures in this article are in colour only in the electronic version)     

A Lower Bound for Nodal Count on Discrete and Metric Graphs

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 a certain genericity condition, we show that the number of nodal domains of the n ^th eigenfunction is bounded below by n  ?  ?, where ? is the number of links that distinguish the graph from a tree.Our results apply to operators on both discrete (combinatorial) and metric (quantum) graphs. They complement already known analogues of a result by Courant who proved the upper bound n for the number of nodal domains.To illustrate that the genericity condition is essential we show that if it is dropped, the nodal count can fall arbitrarily far below the number of the corresponding eigenfunction.In the Appendix we review the proof of the case ?  =  0 on metric trees which has been obtained by other authors.     

Swelling-collapse transition of self-attracting walks

We study the structural properties of self-attracting walks in d dimensions using scaling arguments and Monte Carlo simulations. We find evidence of a transition analogous to the Theta transition of polymers. Above a critical attractive interaction u(c), the walk collapses and the exponents nu and k, characterizing the scaling with time t of the mean square end-to-end distance approximately t(2nu) and the average number of visited sites approximately t(k), are universal and given by nu=1/(d+1) and k=d/(d+1). Below u(c), the walk swells and the exponents are as with no interaction, i.e., nu=1/2 for all d, k=1/2 for d=1 and k=1 for d>/=2. At u(c), the exponents are found to be in a different universality class.     

On infinitely smooth compactly supported almost-wavelets

Translated from Matematicheskie Zametki, Vol. 56, No. 3, pp. 3–12, September, 1994.     

On a Peetre functional

The PeetreK-functional is often used to describe and study the interpolation spaces associated with the real variable method. In the paper a modification of this functional, the PeetreK ₂-functional $$K_2 (t,x) = \mathop {\inf }\limits_{x = x_1 + x_2 } \sqrt {\left\| {x_1 } \right\|_1^2 + t^2 \left\| {x_2 } \right\|_2^2 } ,$$ is treated as a function oft for fixed x, and its properties are studied. Several particular cases are considered and classes of functions expressible asK ₂(t) are investigated.     

Intermediate wave function statistics

We calculate statistical properties of the eigenfunctions of two quantum systems that exhibit intermediate spectral statistics: star graphs and Seba billiards. First, we show that these eigenfunctions are not quantum ergodic, and calculate the corresponding limit distribution. Second, we find that they can be strongly scarred, in the case of star graphs by short (unstable) periodic orbits and, in the case of Seba billiards, by certain families of orbits. We construct sequences of states which have such a limit. Our results are illustrated by numerical computations.