Decoupling Of Deficiency Indices And Applications To Schrodinger-Type Operators With Possibly Strongly Singular Potentisld.
journal contributionposted on 13.12.2017 by GESZTESY F, MITREA M, NENCIU I, TESCHL G
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We investigate closed, symmetric L2 (Rn)-realizations H of Schr¨odinger-type operators (−∆ + V ) C∞0 (Rn\Σ) whose potential coefficient V has a countable number of well-separated singularities on compact sets Σj , j ∈ J, of n-dimensional Lebesgue measure zero, with J ⊆ N an index set and Σ = S j∈J Σj . We show that the defect, def(H), of H can be computed in terms of the individual defects, def(Hj ), of closed, symmetric L2 (Rn)-realizations of (−∆ + Vj ) C∞0 (Rn\Σj ) with potential coefficient Vj localized around the singularity Σj , j ∈ J, where V = P j∈J Vj . In particular, we prove def(H) = X j∈J def(Hj ), including the possibility that one, and hence both sides equal ∞. We first develop an abstract approach to the question of decoupling of deficiency indices and then apply it to the concrete case of Schr¨odinger-type operators in L2 (Rn). Moreover, we also show how operator (and form) bounds for V relative to H0 = −∆ H2(Rn) can be estimated in terms of the operator (and form) bounds of Vj , j ∈ J, relative to H0. Again, we first prove an abstract result and then show its applicability to Schr¨odinger-type operators in L2 (Rn). Extensions to second-order (locally uniformly) elliptic differential operators on Rn with a possibly strongly singular potential coefficient are treated as well.