Daniel Lenz
A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states.
Sat, 03/20/2010 - 16:01 — SomNambulAuthors: Daniel Lenz, Fabian Schwarzenberger, Ivan Veselić
Subjects: Mathematical Physics
AbstractIn this paper we consider bounded operators on infinite graphs, in particular
Cayley graphs of amenable groups. The operators satisfy an equivariance
condition which is formulated in terms of a colouring of the vertex set of the
underlying graph. In this setting it is natural to expect that the integrated
density of states (IDS), or spectral distribution function, exists. We show
that it can be defined as the uniform limit of approximants associated to
finite matrices. The proof is based on a Banach space valued ergodic theorem
which even allows explicit convergence estimates.Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals.
Sat, 03/20/2010 - 16:01 — SomNambulAuthors: Daniel Lenz, Peter Stollmann, Steffen Klassert
Subjects: Mathematical Physics
AbstractWe study measures on the real line and present various versions of what it
means for such a measure to take only finitely many values. We then study
perturbations of the Laplacian by such measures. Using Kotani-Remling theory,
we show that the resulting operators have empty absolutely continuous spectrum
if the measures are not periodic. When combined with Gordon type arguments this
allows us to prove purely singular continuous spectrum for some continuum
models of quasicrystals.On the spectral theory of trees with finite forward cone type.
Thu, 01/21/2010 - 16:01 — SomNambulAuthors: Daniel Lenz, Simone Warzel, Matthias Keller
Subjects: Spectral Theory
AbstractWe study basic spectral features of graph Laplacians associated to a class of
rooted trees which contains all regular trees. Trees in this class can be
generated by substitution processes. Their spectra are shown to be purely
absolutely continuous and to consist of finitely many bands. The main result
gives stability of absolutely continuous spectrum under sufficiently small
radially label symmetric perturbations for non regular trees in this class.Note on powers in three interval exchange transformations.
Tue, 09/08/2009 - 12:16 — SomNambulAuthors: Daniel Lenz, Zuzana Masakova, Edita Pelantova
Subjects: Combinatorics
AbstractWe study repetitions in infinite words coding exchange of three intervals
with permutation (3,2,1), called 3iet words. The language of such words is
determined by two parameters $\varepsilon,\ell$. We show that finiteness of the
index of 3iet words is equivalent to boundedness of the coefficients of the
continued fraction of $\varepsilon$. In this case we also give an upper and
lower estimate on the index of the corresponding 3iet word.Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms.
Tue, 09/08/2009 - 12:16 — SomNambulAuthors: Daniel Lenz, Peter Stollmann, Ivan Veselic
Subjects: Spectral Theory
AbstractWe present an introduction to the framework of strongly local Dirichlet forms
and discuss connections between the existence of certain generalized
eigenfunctions and spectral properties within this framework. The range of
applications is illustrated by a list of examples.
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