Self-adjointness of Schrodinger-type operators with singular potentials on manifolds of bounded geometry
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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 ∈ L2(E)}. The proof follows the scheme of T. Kato, but it requires the use of 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 (∆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.
CitationMilatovic, O. (2003). Self-adjointness of Schrodinger-type operators with singular potentials on manifolds of bounded geometry. Electronic Journal of Differential Equations, 2003(64), pp. 1-8.
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