“Localized” self-adjointness of Schrödinger type operators on Riemannian manifolds

We prove self-adjointness of the Schrödinger type operator HV =∇∗∇ +V, where ∇ is a Hermitian connection on a Hermitian vector bundle E over a complete Riemannian manifold M with positive smooth measure dμ which is fixed independently of the metric, and V ∈ L1 loc(EndE) is a Hermitian bundle endomorphism. Self-adjointness of HV is deduced from the self-adjointness of the corresponding “localized” operator. This is an extension of a result by Cycon. The proof uses the scheme of Cycon, but requires a refined integration by parts technique as well as the use of a family of cut-off functions which are constructed by a non-trivial smoothing procedure due to Karcher.  2003 Elsevier Inc. All rights reserved