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 $H=D^*D+V$, where $D$ is a first order elliptic differential operator acting on sections of a Hermitian vector bundle over a Riemannian manifold $M$, and $V$ is a Hermitian bundle endomorphism. In the case when $M$ is geodesically complete, we establish the essential self-adjointness of positive integer powers of $H$. In the case when $M$ is not necessarily geodesically complete, we give a sufficient condition for the essential self-adjointness of $H$, expressed in terms of the behavior of $V$ relative to the Cauchy boundary of $M$

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 mm-sectorial Schrödinger-type operators with singular potentials on manifolds of bounded geometry

summary:We consider a Schrödinger-type differential expression $H_V=\nabla^*\nabla+V$, where $\nabla$ is a $C^{\infty}$-bounded Hermitian connection on a Hermitian vector bundle $E$ of bounded geometry over a manifold of bounded geometry $(M,g)$ with metric $g$ and positive $C^{\infty}$-bounded measure $d\mu$, and $V$ is a locally integrable section of the bundle of endomorphisms of $E$. We give a sufficient condition for $m$-sectoriality of a realization of $H_V$ in $L^2(E)$. In the proof we use generalized Kato's inequality as well as a result on the positivity of $u\in L^2(M)$ satisfying the equation $(\Delta _M+b)u=\nu$, where $\Delta _M$ is the scalar Laplacian on $M$, $b>0$ is a constant and $\nu\geq 0$ is a positive distribution on $M$

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. Survey