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Keywords:
Schrödinger operator; $m$-sectorial; manifold; bounded geometry; singular potential
Schrödinger operator; $m$-sectorial; manifold; bounded geometry; singular potential
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 $M$.
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