Self-adjointness of Schrödinger-type operators with locally integrable potentials on manifolds of bounded geometry

We consider a Schrödinger-type differential expression HV = ∇∗∇ + V, where ∇ is a C∞- 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∞-bounded measure dμ, and V ∈ L1 loc(EndE) is a linear self-adjoint bundle map. We define the maximal operator HV,max associated to HV as an operator in L2(E) given by HV,maxu = HV u for all u ∈ Dom(HV,max) = {u ∈ L2(E): Vu ∈ L1 loc(E), HV u ∈ L2(E)}, where ∇∗∇u in HV u = ∇∗∇u + Vu is understood in distributional sense. We give a sufficient condition for the self-adjointness of HV,max. The proof adopts Kato’s technique to our setting, but it requires a more general version of Kato’s inequality for Bochner Laplacian operator as well as a result on the positivity of u ∈ L2(M) satisfying the equation (ΔM + b)u = ν, where ΔM is the scalar Laplacian on M, b >0 is a constant and ν 0 is a positive distribution on M. For local estimates, we use a family of cut-off functions constructed with the help of regularized distance on manifolds of bounded geometry.  2004 Elsevier Inc. All rights reserved