39 research outputs found
Self-adjointness of perturbed bi-Laplacians on infinite graphs
We give a sufficient condition for the essential self-adjointness of a
perturbation of the square of the magnetic Laplacian on an infinite weighted
graph. The main result is applicable to graphs whose degree function is not
necessarily bounded. The result allows perturbations that are not necessarily
bounded from below by a constant.Comment: We edited the introduction and updated the bibliographical
information for some references. We moved the examples towards the front of
the articl
Self-adjoint extensions of differential operators on Riemannian manifolds
We study , where is a first order elliptic differential
operator acting on sections of a Hermitian vector bundle over a Riemannian
manifold , and is a Hermitian bundle endomorphism. In the case when
is geodesically complete, we establish the essential self-adjointness of
positive integer powers of .
In the case when is not necessarily geodesically complete, we give a
sufficient condition for the essential self-adjointness of , expressed in
terms of the behavior of relative to the Cauchy boundary of
Maximal accretive extensions of Schr\"odinger operators on vector bundles over infinite graphs
Given a Hermitian vector bundle over an infinite weighted graph, we define
the Laplacian associated to a unitary connection on this bundle and study the
essential self-adjointness of a perturbation of this Laplacian by an
operator-valued potential. Additionally, we give a sufficient condition for the
resulting Schr\"odinger operator to serve as the generator of a strongly
continuous contraction semigroup in the corresponding l^{p}-space.Comment: We have made significant revisions of the previous version. In
particular, this version has a new title: "Maximal Accretive Extensions of
Schr\"odinger Operators on Vector Bundles over Infinite Graphs." The final
version will appear in Integral Equations and Operator Theory and will be
availableat Springer via http://dx.doi.org/10.1007/s00020-014-2196-
On -sectorial Schrödinger-type operators with singular potentials on manifolds of bounded geometry
summary:We consider a Schrödinger-type differential expression , where is a -bounded Hermitian connection on a Hermitian vector bundle of bounded geometry over a manifold of bounded geometry with metric and positive -bounded measure , and is a locally integrable section of the bundle of endomorphisms of . We give a sufficient condition for -sectoriality of a realization of in . In the proof we use generalized Kato's inequality as well as a result on the positivity of satisfying the equation , where is the scalar Laplacian on , is a constant and is a positive distribution on
Essential self-adjointness of Schroedinger type operators on manifolds
We obtain several essential self-adjointness conditions for a Schroedinger
type operator D*D+V acting in sections of a vector bundle over a manifold M.
Here V is a locally square-integrable bundle map. Our conditions are expressed
in terms of completeness of certain metrics on M; these metrics are naturally
associated to the operator. We do not assume a priori that M is endowed with a
complete Riemannian metric. This allows us to treat e.g. operators acting on
bounded domains in the euclidean space.
For the case when the principal symbol of the operator is scalar, we
establish more precise results. The proofs are based on an extension of the
Kato inequality which modifies and improves a result of Hess, Schrader and
Uhlenbrock.Comment: 52 pages, Minor corrections are made; To appear in Russian Math.
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