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

Abstract

We consider a family of Schrödinger-type differential expressions L ( κ ) = D 2 + V + κ V ( 1 ) , where κ ∈ C , and D is the Dirac operator associated with a Clifford bundle ( E , ∇ 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 End E. We also consider the family I ( κ ) = ( ∇ F ) * ∇ F + V + κ V ( 1 ) , where κ ∈ C , and ∇ F is 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 self-adjoint locally integrable sections of End F. We give sufficient conditions for L ( κ ) and I ( κ ) 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 Kato's inequality for Bochner Laplacian operator and Weitzenböck formula.