Self-adjointness of Schr"odinger-type operators with singular potentials on manifolds of bounded geometry

We consider the Schrodinger type differential expression $$ H_V= abla^* abla+V, $$ where $ abla$ 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 $dmu$, and $V=V_1+V_2$, where $0leq V_1in L_{ m loc}^1(mathop{ m End} E)$ and $0geq V_2in L_{ m loc}^1(mathop{ m End} E)$ are linear self-adjoint bundle endomorphisms. We give a sufficient condition for self-adjointness of the operator $S$ in $L^2(E)$ defined by $Su=H_Vu$ for all $uinmathop{ m Dom}(S)={uin W^{1,2}(E)colon intlangle V_1u,u angle,dmu<+infty hbox{ and }H_Vuin L^2(E)}$. The proof follows the scheme of Kato, but it requires the use of more general version of Kato's inequality for Bochner Laplacian operator as well as a result on the positivity of $uin L^2(M)$ satisfying the equation $(Delta_M+b)u= u$, where $Delta_M$ is the scalar Laplacian on $M$, $b>0$ is a constant and $ ugeq 0$ is a positive distribution on $M$.