Relationship between scattering matrix and spectrum of quantum graphs

We investigate the equivalence between spectral characteristics of the Laplace operator on a metric graph and the associated unitary scattering operator. We prove that the statistics of level spacings and moments of observations in the eigenbases coincide in the limit that all bond lengths approach a positive constant value. © 2010 American Mathematical Society

Maximal scarring for eigenfunctions of quantum graphs

We prove the existence of scarred eigenstates for star graphs with scattering matrices at the central vertex which are either a Fourier transform matrix, or a matrix that prohibits back-scattering. We prove the existence of scars that are half-delocalised on a single bond. Moreover we show that the scarred states we construct are maximal in the sense that it is impossible to have quantum eigenfunctions with a significantly lower entropy than our examples. These scarred eigenstates are on graphs that exhibit generic spectral statistics of random matrix type in the large graph limit, and, in contrast to other constructions, correspond to non-degenerate eigenvalues; they exist for almost all choices of length

Quantum ergodicity for quantum graphs without back-scattering

We give an estimate of the quantum variance for d-regular graphs quantised with boundary scattering matrices that prohibit back-scattering. For families of graphs that are expanders, with few short cycles, our estimate leads to quantum ergodicity for these families of graphs. Our proof is based on a uniform control of an associated random walk on the bonds of the graph. We show that recent constructions of Ramanujan graphs, and asymptotically almost surely, random d-regular graphs, satisfy the necessary conditions to conclude that quantum ergodicity holds

Localized eigenfunctions in Seba billiards

We describe some new families of quasimodes for the Laplacian perturbed by the addition of a potential formally described by a Dirac delta function. As an application, we find, under some additional hypotheses on the spectrum, subsequences of eigenfunctions of Šeba billiards that localize around a pair of unperturbed eigenfunctions

On the location of spectral edges in Z-periodic media

Periodic second-order ordinary differential operators on R are known to have the edges of their spectra to occur only at the spectra of periodic and antiperiodic boundary value problems. The multi-dimensional analog of this property is false, as was shown in a 2007 paper by some of the authors of this paper. However, one sometimes encounters the claims that in the case of a single periodicity (i.e., with respect to the lattice Z), the 1D property still holds, and spectral edges occur at the periodic and anti-periodic spectra only. In this work, we show that even in the simplest case of quantum graphs this is not true. It is shown that this is true if the graph consists of a 1D chain of finite graphs connected by single edges, while if the connections are formed by at least two edges, the spectral edges can already occur away from the periodic and anti-periodic spectra

Moments of the eigenvalue densities and of the secular coefficients of β-ensembles

© 2017 IOP Publishing Ltd & London Mathematical Society.We compute explicit formulae for the moments of the densities of the eigenvalues of the classical β-ensembles for finite matrix dimension as well as the expectation values of the coefficients of the characteristic polynomials. In particular, the moments are linear combinations of averages of Jack polynomials, whose coefficients are related to specific examples of Jack characters

Quantum ergodicity for large equilateral quantum graphs

Consider a sequence of finite regular graphs converging, in the sense of Benjamini-Schramm, to the infinite regular tree. We study the induced quantum graphs with equilateral edge lengths, Kirchhoff conditions (possibly with a non-zero coupling constant α) and a symmetric potential U on the edges. We show that in the spectral regions where the infinite quantum tree has absolutely continuous spectrum, the eigenfunctions of the converging quantum graphs satisfy a quantum ergodicity theorem. In case α = 0 and U = 0, the limit measure is the uniform measure on the edges. In general, it has an explicit C 1 density. We finally prove a stronger quantum ergodicity theorem involving integral operators, the purpose of which is to study eigenfunction correlations

Empirical spectral measures of quantum graphs in the Benjamini-Schramm limit

We introduce the notion of Benjamini-Schramm convergence for quantum graphs. This notion of convergence, intended to play the role of the already existing notion for discrete graphs, means that the restriction of the quantum graph to a randomly chosen ball has a limiting distribution. We prove that any sequence of quantum graphs with uniformly bounded data has a convergent subsequence in this sense. We then consider the empirical spectral measure of a convergent sequence (with general boundary conditions and edge potentials) and show that it converges to the expected spectral measure of the limiting random rooted quantum graph. These results are similar to the discrete case, but the proofs are significantly different

Quantum ergodicity for expanding quantum graphs in the regime of spectral delocalization

We consider a sequence of finite quantum graphs with few loops, so that they converge, in the sense of Benjamini-Schramm, to a random infinite quantum tree. We assume these quantum trees are spectrally delocalized in some interval I, in the sense that their spectrum in I is purely absolutely continuous and their Green’s functions are well controlled near the real axis. We furthermore suppose that the underlying sequence of discrete graphs is expanding. We deduce a quantum ergodicity result, showing that the eigenfunctions with eigenvalues lying in I are spatially delocalized

Absolutely continuous spectrum for quantum trees

We study the spectra of quantum trees of finite cone type. These are quantum graphs whose geometry has a certain homogeneity, and which carry a finite set of edge lengths, coupling constants and potentials on the edges. We show the spectrum consists of bands of purely absolutely continuous spectrum, along with a discrete set of eigenvalues. Afterwards, we study random perturbations of such trees, at the level of edge length and coupling, and prove the stability of pure AC spectrum, along with resolvent estimates