[Show abstract][Hide abstract] ABSTRACT: 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.
Preview · Article · Jul 2015 · Annales Henri Poincare
[Show abstract][Hide abstract] ABSTRACT: We study the spectral statistics of the Dirac operator on a rose-shaped
graph---a graph with a single vertex and all bonds connected at both ends to
the vertex. We formulate a secular equation that generically determines the
eigenvalues of the Dirac rose graph, which is seen to generalise the secular
equation for a star graph with Neumann boundary conditions. We derive
approximations to the spectral pair correlation function at large and small
values of spectral spacings, in the limit as the number of bonds approaches
infinity, and compare these predictions with results of numerical calculations.
Our results represent the first example of intermediate statistics from the
symplectic symmetry class.
Preview · Article · May 2012 · Journal of Physics A Mathematical and Theoretical
[Show abstract][Hide abstract] ABSTRACT: We calculate joint moments of the characteristic polynomial of a random
unitary matrix from the circular unitary ensemble and its derivative in the
case that the power in the moments is an odd positive integer. The calculations
are carried out for finite matrix size and in the limit as the size of the
matrices goes to infinity. The latter asymptotic calculation allows us to prove
a long-standing conjecture from random matrix theory.
[Show abstract][Hide abstract] ABSTRACT: 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.
Preview · Article · May 2010 · Journal of Mathematical Physics
[Show abstract][Hide abstract] ABSTRACT: Periodic second-order ordinary differential operators on are known to have the edges of their spectra to occur only at the spectra of periodic and anti-periodic 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 ), 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.
Full-text · Article · Jan 2010 · Journal of Physics A Mathematical and Theoretical
[Show abstract][Hide abstract] ABSTRACT: 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 Seba billiards that localise around a pair of unperturbed eigenfunctions. Comment: 23 pages, 1 figure
[Show abstract][Hide abstract] ABSTRACT: 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.
Full-text · Article · Jan 2008 · Transactions of the American Mathematical Society
[Show abstract][Hide abstract] ABSTRACT: We prove a conditionally convergent trace formula for quantum graphs. By speci- fying an order of summation for the periodic orbit terms we can considerably enlarge the class of test functions for which the trace formula is convergent.
[Show abstract][Hide abstract] ABSTRACT: We describe a new class of scattering matrices for quantum graphs in which back-scattering is prohibited. We discuss some properties of quantum graphs with these scattering matrices and explain the advantages and interest in their study. We also provide two methods to build the vertex scattering matrices needed for their construction. Comment: 15 pages
Preview · Article · Aug 2007 · Journal of Physics A Mathematical and Theoretical
[Show abstract][Hide abstract] ABSTRACT: The article discusses the following frequently arising question on the spectral structure of periodic operators of mathematical physics (e.g., Schroedinger, Maxwell, waveguide operators, etc.). Is it true that one can obtain the correct spectrum by using the values of the quasimomentum running over the boundary of the (reduced) Brillouin zone only, rather than the whole zone? Or, do the edges of the spectrum occur necessarily at the set of ``corner'' high symmetry points? This is known to be true in 1D, while no apparent reasons exist for this to be happening in higher dimensions. In many practical cases, though, this appears to be correct, which sometimes leads to the claims that this is always true. There seems to be no definite answer in the literature, and one encounters different opinions about this problem in the community. In this paper, starting with simple discrete graph operators, we construct a variety of convincing multiply-periodic examples showing that the spectral edges might occur deeply inside the Brillouin zone. On the other hand, it is also shown that in a ``generic'' case, the situation of spectral edges appearing at high symmetry points is stable under small perturbations. This explains to some degree why in many (maybe even most) practical cases the statement still holds. Comment: 25 pages, 10 EPS figures. Typos corrected and a reference added in the new version
Full-text · Article · Feb 2007 · Journal of Physics A Mathematical and Theoretical
[Show abstract][Hide abstract] ABSTRACT: We investigate a class of quantum symmetries of the perturbed cat map which exist only for a subset of possible values of Planck's constant. The effect of these symmetries is to change the spectral statistics along this positive-density subset. The symmetries are shown to be related to some simple classical symmetries of the map.
No preview · Article · Jun 2005 · Journal of Physics A General Physics
[Show abstract][Hide abstract] ABSTRACT: We investigate the semi-classical properties of a two-parameter family of piece-wise linear maps on the torus known as the Casati–Prosen or triangle map. This map is weakly chaotic and has zero Lyapunov exponent. A correspondence between classical and quantum observables is established, leading to an appropriate statement regarding equidistribution of eigenfunctions in the semi-classical limit. We then give a full description of our numerical study of the eigenvalues and eigenvectors of this family of maps. For generic choices of parameters, the spectral and eigenfunction statistics are seen to follow the predictions of the random matrix theory conjecture.
[Show abstract][Hide abstract] ABSTRACT: We prove a Egorov theorem, or quantum-classical correspondence, for the quantised baker's map, valid up to the Ehrenfest time. This yields a logarithmic upper bound for the decay of the quantum variance, and, as a corollary, a quantum ergodic theorem for this map.
Preview · Article · Jan 2005 · Communications in Mathematical Physics
[Show abstract][Hide abstract] ABSTRACT: We investigate statistical properties of the eigenfunctions of the Schrdinger 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 structures are analogous to scars on short unstable periodic orbits.
Full-text · Article · Aug 2004 · Communications in Mathematical Physics
[Show abstract][Hide abstract] ABSTRACT: 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.
Full-text · Article · Oct 2003 · Physical Review Letters
[Show abstract][Hide abstract] ABSTRACT: We compute the value distributions of the eigenfunctions and spectral determinant of the Schrdinger operator on families of star graphs. The values of the spectral determinant are shown to have a Cauchy distribution with respect both to averages over bond lengths in the limit as the wavenumber tends to infinity and to averages over wavenumber when the bond lengths are fixed and not rationally related. This is in contrast to the spectral determinants of random matrices, for which the logarithm is known to satisfy a Gaussian limit distribution. The value distribution of the eigenfunctions also differs from the corresponding random matrix result. We argue that the value distributions of the spectral determinant and of the eigenfunctions should coincide with those of eba-type billiards.
Preview · Article · Jan 2003 · Communications in Mathematical Physics