36 research outputs found
Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians
In this paper we study in detail some spectral properties of the magnetic
discrete Laplacian. We identify its form-domain, characterize the absence of
essential spectrum and provide the asymptotic eigenvalue distribution.Comment: Few typos. To appear in Journal of Functional Analysi
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
The adjacency matrix and the discrete Laplacian acting on forms
We study the relationship between the adjacency matrix and the discrete
Laplacian acting on 1-forms. We also prove that if the adjacency matrix is
bounded from below it is not necessarily essentially self-adjoint. We discuss
the question of essential self-adjointness and the notion of completeness
Spectral analysis of the Laplacian acting on discrete cusps and funnels
We study perturbations of the discrete Laplacian associated to discrete
analogs of cusps and funnels. We perturb the metric and the potential in a
long-range way. We establish a propagation estimate and a Limiting Absorption
Principle away from the possible embedded eigenvalues. The approach is based on
a positive commutator technique
Erratum: "The problem of deficiency indices for discrete Schr\"odinger operators on locally finite graphs"
In this note we answer negatively to our conjecture concerning the deficiency
indices. More precisely, given any non-negative integer , there is locally
finite graph on which the adjacency matrix has deficiency indices .Comment: Typos and clarifications. The full title is Erratum: "The problem of
deficiency indices for discrete Schr\"odinger operators on locally finite
graphs" [J. Math. Phys. (52), 063512 (2011)]. It will be publish in J. Math.
Phy