10 research outputs found
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.Comment: 19 pages, 3 figures; substituted a proof for a reference to a
published one, other minor update
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 by short periodic orbits, and construct sequences of
states which have such a limit. Our results are illustrated by numerical
computations.Comment: 4 pages, 3 figures. Final versio
No quantum ergodicity for star graphs
We investigate statistical properties of the eigenfunctions of the
Schrodinger 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.Comment: 26 pages, 5 figure
Three-point correlations for quantum star graphs
We compute the three point correlation function for the eigenvalues of the
Laplacian on quantum star graphs in the limit where the number of edges tends
to infinity. This extends a work by Berkolaiko and Keating, where they get the
2-point correlation function and show that it follows neither Poisson, nor
random matrix statistics. It makes use of the trace formula and combinatorial
analysis.Comment: 10 pages, 2 figure
Intermediate statistics for a system with symplectic symmetry: the Dirac rose graph
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.Comment: 26 pages, references adde
Mathematical Aspects of Vacuum Energy on Quantum Graphs
We use quantum graphs as a model to study various mathematical aspects of the
vacuum energy, such as convergence of periodic path expansions, consistency
among different methods (trace formulae versus method of images) and the
possible connection with the underlying classical dynamics.
We derive an expansion for the vacuum energy in terms of periodic paths on
the graph and prove its convergence and smooth dependence on the bond lengths
of the graph. For an important special case of graphs with equal bond lengths,
we derive a simpler explicit formula.
The main results are derived using the trace formula. We also discuss an
alternative approach using the method of images and prove that the results are
consistent. This may have important consequences for other systems, since the
method of images, unlike the trace formula, includes a sum over special
``bounce paths''. We succeed in showing that in our model bounce paths do not
contribute to the vacuum energy. Finally, we discuss the proposed possible link
between the magnitude of the vacuum energy and the type (chaotic vs.
integrable) of the underlying classical dynamics. Within a random matrix model
we calculate the variance of the vacuum energy over several ensembles and find
evidence that the level repulsion leads to suppression of the vacuum energy.Comment: Fixed several typos, explain the use of random matrices in Section
Quantum graphs where back-scattering is prohibited
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 page
Functionals of the Brownian motion, localization and metric graphs
We review several results related to the problem of a quantum particle in a
random environment.
In an introductory part, we recall how several functionals of the Brownian
motion arise in the study of electronic transport in weakly disordered metals
(weak localization).
Two aspects of the physics of the one-dimensional strong localization are
reviewed : some properties of the scattering by a random potential (time delay
distribution) and a study of the spectrum of a random potential on a bounded
domain (the extreme value statistics of the eigenvalues).
Then we mention several results concerning the diffusion on graphs, and more
generally the spectral properties of the Schr\"odinger operator on graphs. The
interest of spectral determinants as generating functions characterizing the
diffusion on graphs is illustrated.
Finally, we consider a two-dimensional model of a charged particle coupled to
the random magnetic field due to magnetic vortices. We recall the connection
between spectral properties of this model and winding functionals of the planar
Brownian motion.Comment: Review article. 50 pages, 21 eps figures. Version 2: section 5.5 and
conclusion added. Several references adde