12,347 research outputs found
Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation
We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We
focus on phenomena related to unboundedness of the Laplacians. This includes
(failure of) essential selfadjointness, absence of essential spectrum and
stochastic incompleteness
A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains
In this note we investigate the asymptotic behaviour of the -numbers of
the resolvent difference of two generalized self-adjoint, maximal dissipative
or maximal accumulative Robin Laplacians on a bounded domain with
smooth boundary . For this we apply the recently introduced
abstract notion of quasi boundary triples and Weyl functions from extension
theory of symmetric operators together with Krein type resolvent formulae and
well-known eigenvalue asymptotics of the Laplace-Beltrami operator on
. It will be shown that the resolvent difference of two
generalized Robin Laplacians belongs to the Schatten-von Neumann class of any
order for which . Moreover, we also give a simple
sufficient condition for the resolvent difference of two generalized Robin
Laplacians to belong to a Schatten-von Neumann class of arbitrary small order.
Our results extend and complement classical theorems due to M.Sh.Birman on
Schatten-von Neumann properties of the resolvent differences of Dirichlet,
Neumann and self-adjoint Robin Laplacians
Variable exponent Hardy spaces associated with discrete Laplacians on graphs
In this paper we develop the theory of variable exponent Hardy spaces
associated with discrete Laplacians on infinite graphs. Our Hardy spaces are
defined by square integrals, atomic and molecular decompositions. Also we study
boundedness properties of Littlewood-Paley functions, Riesz transforms, and
spectral multipliers for discrete Laplacians on variable exponent Hardy spaces
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