85 research outputs found
Smilansky's model of irreversible quantum graphs, I: the absolutely continuous spectrum
In the model suggested by Smilansky one studies an operator describing the
interaction between a quantum graph and a system of one-dimensional
oscillators attached at several different points in the graph. The present
paper is the first one in which the case is investigated. For the sake of
simplicity we consider K=2, but our argument is of a general character. In this
first of two papers on the problem, we describe the absolutely continuous
spectrum. Our approach is based upon scattering theory
Schr\"odinger operator on homogeneous metric trees: spectrum in gaps
The paper studies the spectral properties of the Schr\"odinger operator
on a homogeneous rooted metric tree, with a decaying
real-valued potential and a coupling constant . The spectrum of the
free Laplacian has a band-gap structure with a single
eigenvalue of infinite multiplicity in the middle of each finite gap. The
perturbation gives rise to extra eigenvalues in the gaps. These
eigenvalues are monotone functions of if the potential has a fixed
sign. Assuming that the latter condition is satisfied and that is
symmetric, i.e. depends on the distance to the root of the tree, we carry out a
detailed asymptotic analysis of the counting function of the discrete
eigenvalues in the limit . Depending on the sign and decay of ,
this asymptotics is either of the Weyl type or is completely determined by the
behaviour of at infinity.Comment: AMS LaTex file, 47 page
Irreversible quantum graphs
Irreversibility is introduced to quantum graphs by coupling the graphs to a
bath of harmonic oscillators. The interaction which is linear in the harmonic
oscillator amplitudes is localized at the vertices. It is shown that for
sufficiently strong coupling, the spectrum of the system admits a new continuum
mode which exists even if the graph is compact, and a {\it single} harmonic
oscillator is coupled to it. This mechanism is shown to imply that the quantum
dynamics is irreversible. Moreover, it demonstrates the surprising result that
irreversibility can be introduced by a "bath" which consists of a {\it single}
harmonic oscillator
Limit theorems for self-similar tilings
We study deviation of ergodic averages for dynamical systems given by
self-similar tilings on the plane and in higher dimensions. The main object of
our paper is a special family of finitely-additive measures for our systems. An
asymptotic formula is given for ergodic integrals in terms of these
finitely-additive measures, and, as a corollary, limit theorems are obtained
for dynamical systems given by self-similar tilings.Comment: 36 pages; some corrections and improved exposition, especially in
Section 4; references adde
An estimate for the Morse index of a Stokes wave
Stokes waves are steady periodic water waves on the free surface of an
infinitely deep irrotational two dimensional flow under gravity without surface
tension. They can be described in terms of solutions of the Euler-Lagrange
equation of a certain functional. This allows one to define the Morse index of
a Stokes wave. It is well known that if the Morse indices of the elements of a
set of non-singular Stokes waves are bounded, then none of them is close to a
singular one. The paper presents a quantitative variant of this result.Comment: This version contains an additional reference and some minor change
Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies
Delone sets of finite local complexity in Euclidean space are investigated.
We show that such a set has patch counting and topological entropy 0 if it has
uniform cluster frequencies and is pure point diffractive. We also note that
the patch counting entropy is 0 whenever the repetitivity function satisfies a
certain growth restriction.Comment: 16 pages; revised and slightly expanded versio
Aperiodic order and pure point diffraction
We give a leisurely introduction into mathematical diffraction theory with a
focus on pure point diffraction. In particular, we discuss various
characterisations of pure point diffraction and common models arising from cut
and project schemes. We finish with a list of open problems.Comment: 14 page
Golden gaskets: variations on the Sierpi\'nski sieve
We consider the iterated function systems (IFSs) that consist of three
general similitudes in the plane with centres at three non-collinear points,
and with a common contraction factor \la\in(0,1).
As is well known, for \la=1/2 the invariant set, \S_\la, is a fractal
called the Sierpi\'nski sieve, and for \la<1/2 it is also a fractal. Our goal
is to study \S_\la for this IFS for 1/2<\la<2/3, i.e., when there are
"overlaps" in \S_\la as well as "holes". In this introductory paper we show
that despite the overlaps (i.e., the Open Set Condition breaking down
completely), the attractor can still be a totally self-similar fractal,
although this happens only for a very special family of algebraic \la's
(so-called "multinacci numbers"). We evaluate \dim_H(\S_\la) for these
special values by showing that \S_\la is essentially the attractor for an
infinite IFS which does satisfy the Open Set Condition. We also show that the
set of points in the attractor with a unique ``address'' is self-similar, and
compute its dimension.
For ``non-multinacci'' values of \la we show that if \la is close to 2/3,
then \S_\la has a nonempty interior and that if \la<1/\sqrt{3} then \S_\la$
has zero Lebesgue measure. Finally we discuss higher-dimensional analogues of
the model in question.Comment: 27 pages, 10 figure
Harmonic analysis of iterated function systems with overlap
In this paper we extend previous work on IFSs without overlap. Our method
involves systems of operators generalizing the more familiar Cuntz relations
from operator algebra theory, and from subband filter operators in signal
processing.Comment: 37 page
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