Onm-accretive Schrödinger-type operators with singular potentials on manifolds of bounded geometry

Ognjen Milatovic

doi:10.1155/s0161171203209212

Abstract

We consider a Schrödinger-type differential expression∇∗ ∇+V, where∇is aC∞-bounded Hermitian connection on a Hermitian vector bundleEof bounded geometry over a manifold of bounded geometry(M,g)with positiveC∞-bounded measuredμ, andVis a locally integrable linear bundle endomorphism. We define a realization of∇∗ ∇+VinL2(E)and give a sufficient condition for itsm-accretiveness. The proof essentially follows the scheme of T. Kato, but it requires the use of a more general version of Kato's inequality for Bochner Laplacian operator as well as a result on the positivity of solution to a certain differential equation onM.

Highlights

We will assume that M has bounded geometry
We will assume that E is a bundle of bounded geometry (i.e., it is supplied by an additional structure: trivializations of E on every canonical coordinate neighborhood U such that the corresponding matrix transition functions hU,U on all intersections U U of such neighborhoods are C∞-bounded, that is, all derivatives ∂yαhU,U (y), where α is a multiindex, with respect to canonical coordinates, are bounded with bounds Cα which do not depend on the chosen pair U, U )
Be a Hermitian connection on E which is C∞-bounded as a linear differential operator, that is, in any canonical coordinate system U (with the chosen trivializations of E|U and (T ∗M ⊗ E)|U ), ∇ is written in the form aα (y )∂yα

Summary

OGNJEN MILATOVIC

We consider a Schrödinger-type differential expression ∇∗∇ + 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 positive C∞-bounded measure dμ, and V is a locally integrable linear bundle endomorphism. We define a realization of ∇∗∇ + V in L2(E) and give a sufficient condition for its maccretiveness. The proof essentially follows the scheme of T. Kato, but it requires the use of a more general version of Kato’s inequality for Bochner Laplacian operator as well as a result on the positivity of solution to a certain differential equation on M. 2000 Mathematics Subject Classification: 35P05, 58J50, 47B25, 81Q10

Introduction and the main result

We will consider a Schrödinger type differential expression of the form