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Decoupling Of Deficiency Indices And Applications To Schrodinger-Type Operators With Possibly Strongly Singular Potentisld.

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journal contribution
posted on 13.12.2017 by GESZTESY F, MITREA M, NENCIU I, TESCHL G
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.

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Publisher Statement

This is the author’s version of a work that was accepted for publication in Advances in Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Advances in Mathematics. 2016. 301: 1022-1061. doi: 10.1016/j.aim.2016.08.008.

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Elsevier Inc.

Language

en_US

issn

0001-8708

Issue date

01/10/2016

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