90 research outputs found
Wegner bounds for a two-particle tight binding model
We consider a quantum two-particle system on a d-dimensional lattice with
interaction and in presence of an IID external potential. We establish
Wegner-typer estimates for such a model. The main tool used is Stollmann's
lemma
Multi-Particle Anderson Localisation: Induction on the Number of Particles
This paper is a follow-up of our recent papers \cite{CS08} and \cite{CS09}
covering the two-particle Anderson model. Here we establish the phenomenon of
Anderson localisation for a quantum -particle system on a lattice
with short-range interaction and in presence of an IID external potential with
sufficiently regular marginal cumulative distribution function (CDF). Our main
method is an adaptation of the multi-scale analysis (MSA; cf. \cite{FS},
\cite{FMSS}, \cite{DK}) to multi-particle systems, in combination with an
induction on the number of particles, as was proposed in our earlier manuscript
\cite{CS07}. Similar results have been recently obtained in an independent work
by Aizenman and Warzel \cite{AW08}: they proposed an extension of the
Fractional-Moment Method (FMM) developed earlier for single-particle models in
\cite{AM93} and \cite{ASFH01} (see also references therein) which is also
combined with an induction on the number of particles.
An important role in our proof is played by a variant of Stollmann's
eigenvalue concentration bound (cf. \cite{St00}). This result, as was proved
earlier in \cite{C08}, admits a straightforward extension covering the case of
multi-particle systems with correlated external random potentials: a subject of
our future work. We also stress that the scheme of our proof is \textit{not}
specific to lattice systems, since our main method, the MSA, admits a
continuous version. A proof of multi-particle Anderson localization in
continuous interacting systems with various types of external random potentials
will be published in a separate papers
The Density of States and the Spectral Shift Density of Random Schroedinger Operators
In this article we continue our analysis of Schroedinger operators with a
random potential using scattering theory. In particular the theory of Krein's
spectral shift function leads to an alternative construction of the density of
states in arbitrary dimensions. For arbitrary dimension we show existence of
the spectral shift density, which is defined as the bulk limit of the spectral
shift function per unit interaction volume. This density equals the difference
of the density of states for the free and the interaction theory. This extends
the results previously obtained by the authors in one dimension. Also we
consider the case where the interaction is concentrated near a hyperplane.Comment: 1 figur
Leaky quantum graphs: approximations by point interaction Hamiltonians
We prove an approximation result showing how operators of the type in , where is a graph,
can be modeled in the strong resolvent sense by point-interaction Hamiltonians
with an appropriate arrangement of the potentials. The result is
illustrated on finding the spectral properties in cases when is a ring
or a star. Furthermore, we use this method to indicate that scattering on an
infinite curve which is locally close to a loop shape or has multiple
bends may exhibit resonances due to quantum tunneling or repeated reflections.Comment: LaTeX 2e, 31 pages with 18 postscript figure
Essential self-adjointness, generalized eigenforms, and spectra for the -Neumann problem on -manifolds
Let be a strongly pseudoconvex complex manifold which is also the total
space of a principal -bundle with a Lie group and compact orbit space
. Here we investigate the -Neumann Laplacian on . We
show that it is essentially self-adjoint on its restriction to compactly
supported smooth forms. Moreover we relate its spectrum to the existence of
generalized eigenforms: an energy belongs to if there is a
subexponentially bounded generalized eigenform for this energy. Vice versa,
there is an expansion in terms of these well-behaved eigenforms so that,
spectrally, almost every energy comes with such a generalized eigenform.Comment: 25 page
Nonrelativistic hydrogen type stability problems on nonparabolic 3-manifolds
We extend classical Euclidean stability theorems corresponding to the
nonrelativistic Hamiltonians of ions with one electron to the setting of non
parabolic Riemannian 3-manifolds.Comment: 20 pages; to appear in Annales Henri Poincar
Localization for a matrix-valued Anderson model
We study localization properties for a class of one-dimensional,
matrix-valued, continuous, random Schr\"odinger operators, acting on
L^2(\R)\otimes \C^N, for arbitrary . We prove that, under suitable
assumptions on the F\"urstenberg group of these operators, valid on an interval
, they exhibit localization properties on , both in the
spectral and dynamical sense. After looking at the regularity properties of the
Lyapunov exponents and of the integrated density of states, we prove a Wegner
estimate and apply a multiscale analysis scheme to prove localization for these
operators. We also study an example in this class of operators, for which we
can prove the required assumptions on the F\"urstenberg group. This group being
the one generated by the transfer matrices, we can use, to prove these
assumptions, an algebraic result on generating dense Lie subgroups in
semisimple real connected Lie groups, due to Breuillard and Gelander. The
algebraic methods used here allow us to handle with singular distributions of
the random parameters
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