Self-adjointness of Schrodinger-type operators with singular potentials on manifolds of bounded geometry
Date
2003-06-11
Authors
Milatovic, Ognjen
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
We consider the 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 = V1 + V2, where 0 ≤ V1 ∈ L1 loc (End E) and 0 ≥ V2 ∈ L1 loc (End E) are linear self-adjoint bundle endomorphisms. We give a sufficient condition for self-adjointness of the operator S in L2(E) defined by Su = HVu for all u ∈ Dom(s) = {u ∈ W1,2(E): ∫⟨V1u, u⟩dμ < +∞ and HVu for all u ∈ L2(E)}. The proof follows the scheme of T. Kato, but it requires the use of more general vision of Kato's inequality for Bochner Laplacian operator as well as a result on the positivity of u ∈ L2(M) satisfying the equation (∆M + b)u = v, where ∆M is the scalar Laplacian on M, b > 0 is a constant and v ≥ 0 is a positive distribution on M.
Description
Keywords
Schrodinger operator, Self-adjointness, Manifold, Bounded geometry, Singular potential
Citation
Milatovic, O. (2003). Self-adjointness of Schrodinger-type operators with singular potentials on manifolds of bounded geometry. <i>Electronic Journal of Differential Equations, 2003</i>(64), pp. 1-8.
Rights
Attribution 4.0 International