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

Abstract

We consider a Schrödinger-type differential expression H V=∇ ∗∇+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∈ L loc 1(End E) is a linear self-adjoint bundle map. We define the maximal operator H V,max associated to H V as an operator in L 2( E) given by H V,max u= H V u for all u∈ Dom(H V, max )={u∈L 2(E): Vu∈L loc 1(E), H Vu∈L 2(E)} , where ∇ ∗∇u in H Vu=∇ ∗∇u+Vu is understood in distributional sense. We give a sufficient condition for the self-adjointness of H V,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∈ L 2( M) satisfying the equation ( Δ M + b) u= ν, where Δ M 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.