2,239 research outputs found
Essential spectrum and Weyl asymptotics for discrete Laplacians
In this paper, we investigate spectral properties of discrete Laplacians. Our
study is based on the Hardy inequality and the use of super-harmonic functions.
We recover and improve lower bounds for the bottom of the spectrum and of the
essential spectrum. In some situation, we obtain Weyl asymptotics for the
eigenvalues. We also provide a probabilistic representation of super-harmonic
functions. Using coupling arguments, we set comparison results for the bottom
of the spectrum, the bottom of the essential spectrum and the stochastic
completeness of different discrete Laplacians. The class of weakly spherically
symmetric graphs is also studied in full detail
Spectral Theory of Infinite Quantum Graphs
We investigate quantum graphs with infinitely many vertices and edges without
the common restriction on the geometry of the underlying metric graph that
there is a positive lower bound on the lengths of its edges. Our central result
is a close connection between spectral properties of a quantum graph and the
corresponding properties of a certain weighted discrete Laplacian on the
underlying discrete graph. Using this connection together with spectral theory
of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new
results on spectral properties of quantum graphs. Namely, we prove several
self-adjointness results including a Gaffney type theorem. We investigate the
problem of lower semiboundedness, prove several spectral estimates (bounds for
the bottom of spectra and essential spectra of quantum graphs, CLR-type
estimates) and study spectral types.Comment: Dedicated to the memory of M. Z. Solomyak (16.05.1931 - 31.07.2016
Regular sequences and random walks in affine buildings
We define and characterise regular sequences in affine buildings, thereby
giving the "-adic analogue" of the fundamental work of Kaimanovich. As
applications we prove limit theorems for random walks on affine buildings and
their automorphism groups
- …