Publications (16)27.4 Total impact
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ABSTRACT: We give an estimate of the quantum variance for $d$regular graphs quantised with boundary scattering matrices that prohibit backscattering. 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.  [Show abstract] [Hide abstract]
ABSTRACT: We study the spectral statistics of the Dirac operator on a roseshaped grapha 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.  [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 longstanding 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.  [Show abstract] [Hide abstract]
ABSTRACT: Periodic secondorder ordinary differential operators on are known to have the edges of their spectra to occur only at the spectra of periodic and antiperiodic boundary value problems. The multidimensional 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 antiperiodic 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 antiperiodic spectra.  [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.  [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 backscattering 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  [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 multiplyperiodic 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  [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 positivedensity subset. The symmetries are shown to be related to some simple classical symmetries of the map.  [Show abstract] [Hide abstract]
ABSTRACT: We investigate the semiclassical properties of a twoparameter family of piecewise 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 semiclassical 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 quantumclassical 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.  [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.  [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.  [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 ebatype billiards.
Publication Stats
185  Citations  
27.40  Total Impact Points  
Top Journals
Institutions

20082012

Loughborough University
 • Department of Mathematical Sciences
 • School of Mathematics
Loughborough, England, United Kingdom


20052008

Texas A&M University
 Department of Mathematics
College Station, Texas, United States 
University of Bologna
 Department of Mathematics MAT
Bolonia, EmiliaRomagna, Italy


2007

Weizmann Institute of Science
 Department of Physics of Complex Systems
Israel


20032004

University of Bristol
 School of Mathematics
Bristol, England, United Kingdom
