On holomorphic families of Schrodinger-type operators with singular potentials on manifolds of bounded geometry

We consider a family of Schrodinger-type differential expressions L (kappa)=D^2+V+(kappa)V^(1), where (kappa)(element of)C, and D is the Dirac operator associated with a Clifford bundle (E(nabla)^E) of bounded geometry over a manifold of bounded geometry (M,g) with metric g, and V and V^(1) are self-adjoint locally integrable sections of EndE. We also consider the family I(kappa)=((nabla)^f)^(pi)(nabla) ^f+V+(kappa)V^1, where (kappa)(element of)C, and (nabla)^fis a Hermitian connection on a Hermitian vector bundle F of bonded geometry over a manifold of bounded geometry (M,g), and V and V^(1) are selfadjoint locally integrable sections of EndF. We give sufficient conditions for L(kappa) and I(kappa) to have a realization in L^2(E) and L^2(F), respectively, as self-adjoint holomorphic families of type (B). In the proofs we use Katoʹs inequality for Bochner Laplacian operator and Weitzenbock formula.