Ognjen Milatovic
On $m$-sectorial Schr\"odinger-type operators with singular potentials on manifolds of bounded geometry

Comment.Math.Univ.Carolinae 45,1 (2004) 91-100.

Abstract:We consider a Schr\"odinger-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 $M$.

Keywords: Schr\"odinger operator, $m$-sectorial, manifold, bounded geometry, singular potential
AMS Subject Classification: Primary 35P05, 58J50; Secondary 47B25, 81Q10