107 research outputs found
Erratum to: From Uncertainty Principles to Wegner Estimates
Erratum to "From Uncertainty Principles to Wegner Estimates"
Lower bounds for Dirichlet Laplacians and uncertainty principles
We prove lower bounds for the Dirichlet Laplacian on possibly unbounded
domains in terms of natural geometric conditions. This is used to derive
uncertainty principles for low energy functions of general elliptic second
order divergence form operators with not necessarily continuous main part.Comment: 25 pages, v2: final versio
An ergodic theorem for Delone dynamical systems and existence of the integrated density of states
We study strictly ergodic Delone dynamical systems and prove an ergodic
theorem for Banach space valued functions on the associated set of pattern
classes. As an application, we prove existence of the integrated density of
states in the sense of uniform convergence in distribution for the associated
random operators.Comment: 19 page
Delone dynamical systems and associated random operators
We carry out a careful study of basic topological and ergodic features of
Delone dynamical systems. We then investigate the associated topological
groupoids and in particular their representations on certain direct integrals
with non constant fibres. Via non-commutative-integration theory these
representations give rise to von Neumann algebras of random operators. Features
of these algebras and operators are discussed. Restricting our attention to a
certain subalgebra of tight binding operators, we then discuss a Shubin trace
formula.Comment: 19 pages; revised versio
On the decomposition principle and a Persson type theorem for general regular Dirichlet forms
We present a decomposition principle for general regular Dirichlet forms
satisfying a spatial local compactness condition. We use the decomposition
principle to derive a Persson type theorem for the corresponding Dirichlet
forms. In particular our setting covers Laplace-Beltrami operators on
Riemannian manifolds, and Dirichlet forms associated to -stable
processes in Euclidean space.Comment: 19 pages, corrected reference
Generic sets in spaces of measures and generic singular continuous spectrum for Delone Hamiltonians
We show that geometric disorder leads to purely singular continuous spectrum
generically.
The main input is a result of Simon known as the ``Wonderland theorem''.
Here, we provide an alternative approach and actually a slight strengthening by
showing that various sets of measures defined by regularity properties are
generic in the set of all measures on a locally compact metric space.
As a byproduct we obtain that a generic measure on euclidean space is
singular continuous
The Kato class on compact manifolds with integral bounds on the negative part of Ricci curvature
We show that under Ricci curvature integral assumptions the dimension of the
first cohomology group can be estimated in terms of the Kato constant of the
negative part of the Ricci curvature. Moreover, this provides quantitative
statements about the cohomology group, contrary to results by Elworthy and
Rosenberg.Comment: 13 page
Lifshitz asymptotics for percolation Hamiltonians
We study a discrete Laplace operator on percolation subgraphs of an
infinite graph. The ball volume is assumed to grow at most polynomially. We are
interested in the behavior of the integrated density of states near the lower
spectral edge. If the graph is a Cayley graph we prove that it exhibits
Lifshitz tails. If we merely assume that the graph has an exhausting sequence
with positive -dimensional density, we obtain an upper bound on the
integrated density of states of Lifshitz type
The Allegretto-Piepenbrink Theorem for strongly local forms
The existence of positive weak solutions is related to spectral information
on the corresponding partial differential operator
Bounds on the first Betti number - an approach via Schatten norm estimates on semigroup differences
We derive new estimates for the first Betti number of compact Riemannian
manifolds. Our approach relies on the Birman-Schwinger principle and Schatten
norm estimates for semigroup differences. In contrast to previous works we do
not require any a priori ultracontractivity estimates and we provide bounds
which explicitly depend on suitable integral norms of the Ricci tensor
- …