No Quantum Ergodicity for Star Graphs

Abstract

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 ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average. We further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds. We describe how to construct such subsequences explicitly. These constructions are analogous to scars on short unstable periodic orbits.Recently, authors have begun to investigate the wave-functions of quantum graphs. Kaplan [K2] studied eigenfunction statistics for ring-graphs using a combination of numerical techniques and analytical calculations of the short-time semiclassical behaviour of a wave-packet close to a 1-bond periodic orbit. The inverse participation ratio (a measure of localisation in a given state) was found to be well-described by this contribution, and shows deviation from the ergodically expected behaviour. Similar deviations were noticed for lattice-graphs. Remarkably, Schanz and Kottos [SK] observed that it would be impossible for the shortest orbits that are responsible for this enhanced localisation to support strong scarring. They wrote down an explicit criterion which must be satisfied by the energy of any strongly scarred state, and deduced asymptotics of the probability distribution of scarring strengths. In [KMW] a study was made of the eigenfunctions of a family of graphs known as star graphs (the name being derived from the connectivity of graphs in the family). The value distribution for the amplitude of eigenfunctions on a single bond of the graph, subject to an appropriate normalisation, was rigorously calculated

Relationship between scattering matrix and spectrum of quantum graphs

“…The weighting on the left-hand side of (3.4) shows that there is an exact equivalence of moments only in the (additional) limit ΔL → 0, as was taken in [4,6]. However, provided ΔL is bounded away from zero and infinity, the centered moments would approach the zero limit either simultaneously or not at all.…”

Localized eigenfunctions in Šeba billiards

“…26 and mirrors a related result proved for quantum graphs with a star-shaped connectivity. 27 This so-called quantum star graph can be considered as a singular perturbation of a disconnected set of onedimensional bonds, each supporting a wave function. In Ref.…”

“…In Ref. 27 the existence of subsequences of eigenfunctions that become localized on a pair of bonds was proved. This is exactly analogous to the localization onto a pair of unperturbed billiard eigenfunctions in Theorem 4.4.…”

“…In both Ref. 27 and Theorem 4.4 the main idea of the proof is to show localization in an eigenfunction with eigenvalue lying between two closely spaced eigenvalues of the unperturbed problem.…”

Arbeitsgemeinschaft: Quantum Ergodicity

“…For star graphs (graphs which consists of αv vertices which are connected only to one central vertex) it turns out that AQE cannot hold. With a specific observable which is the indicator function of the first v bonds of the graph and with a direct calculation it can be proved that these family of quantum graphs is not asymptotic quantum ergodic [Ber04]. On the other hand in [Ber07] asymptotic quantum ergodicity is proved for a specific construction of graphs related to interval maps.…”