167 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 $K$ one-dimensional
oscillators attached at several different points in the graph. The present
paper is the first one in which the case $K>1$ 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

### Smilansky's model of irreversible quantum graphs, II: the point spectrum

In the model suggested by Smilansky one studies an operator describing the
interaction between a quantum graph and a system of K one-dimensional
oscillators attached at different points of the graph. This paper is a
continuation of our investigation of the case K>1. For the sake of simplicity
we consider K=2, but our argument applies to the general situation. In this
second paper we apply the variational approach to the study of the point
spectrum.Comment: 18 page

### On the negative spectrum of two-dimensional Schr\"odinger operators with radial potentials

For a two-dimensional Schr\"odinger operator $H_{\alpha V}=-\Delta-\alpha V$
with the radial potential $V(x)=F(|x|), F(r)\ge 0$, we study the behavior of
the number $N_-(H_{\alpha V})$ of its negative eigenvalues, as the coupling
parameter $\alpha$ tends to infinity. We obtain the necessary and sufficient
conditions for the semi-classical growth $N_-(H_{\alpha V})=O(\alpha)$ and for
the validity of the Weyl asymptotic law.Comment: 13 page

### Positive-measure self-similar sets without interior

We recall the problem posed by Peres and Solomyak in Problems on self-similar and self-affine sets; an update. Progr. Prob. 46 (2000), 95–106: can one find examples of self-similar sets with positive Lebesgue measure, but with no interior? The method in Properties of measures supported on fat Sierpinski carpets, this issue, leads to families of examples of such sets

### Schr\"odinger operator on homogeneous metric trees: spectrum in gaps

The paper studies the spectral properties of the Schr\"odinger operator
$A_{gV} = A_0 + gV$ on a homogeneous rooted metric tree, with a decaying
real-valued potential $V$ and a coupling constant $g\ge 0$. The spectrum of the
free Laplacian $A_0 = -\Delta$ has a band-gap structure with a single
eigenvalue of infinite multiplicity in the middle of each finite gap. The
perturbation $gV$ gives rise to extra eigenvalues in the gaps. These
eigenvalues are monotone functions of $g$ if the potential $V$ has a fixed
sign. Assuming that the latter condition is satisfied and that $V$ 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 $g\to\infty$. Depending on the sign and decay of $V$,
this asymptotics is either of the Weyl type or is completely determined by the
behaviour of $V$ at infinity.Comment: AMS LaTex file, 47 page

### Spectral estimates for two-dimensional Schroedinger operators with application to quantum layers

A logarithmic type Lieb-Thirring inequality for two-dimensional Schroedinger
operators is established. The result is applied to prove spectral estimates on
trapped modes in quantum layers

### On the "Mandelbrot set" for a pair of linear maps and complex Bernoulli convolutions

We consider the "Mandelbrot set" $M$ for pairs of complex linear maps,
introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and
others. It is defined as the set of parameters $\lambda$ in the unit disk such
that the attractor $A_\lambda$ of the IFS $\{\lambda z-1, \lambda z+1\}$ is
connected. We show that a non-trivial portion of $M$ near the imaginary axis is
contained in the closure of its interior (it is conjectured that all non-real
points of $M$ are in the closure of the set of interior points of $M$). Next we
turn to the attractors $A_\lambda$ themselves and to natural measures
$\nu_\lambda$ supported on them. These measures are the complex analogs of
much-studied infinite Bernoulli convolutions. Extending the results of Erd\"os
and Garsia, we demonstrate how certain classes of complex algebraic integers
give rise to singular and absolutely continuous measures $\nu_\lambda$. Next we
investigate the Hausdorff dimension and measure of $A_\lambda$, for $\lambda$
in the set $M$, for Lebesgue-a.e. $\lambda$. We also obtain partial results on
the absolute continuity of $\nu_\lambda$ for a.e. $\lambda$ of modulus greater
than $\sqrt{1/2}$.Comment: 22 pages, 5 figure

### Singular Continuous Spectrum for the Laplacian on Certain Sparse Trees

We present examples of rooted tree graphs for which the Laplacian has
singular continuous spectral measures. For some of these examples we further
establish fractional Hausdorff dimensions. The singular continuous components,
in these models, have an interesting multiplicity structure. The results are
obtained via a decomposition of the Laplacian into a direct sum of Jacobi
matrices

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