Article

**Adv. Math. 301, 1022-1061 (2016)**[DOI: 10.1016/j.aim.2016.08.008]

## Decoupling of Deficiency Indices and Applications to Schrödinger-Type Operators with Possibly Strongly Singular Potentials

### Fritz Gesztesy, Marius Mitrea, Irina Nenciu, and Gerald Teschl

We investigate closed, symmetric
including the possibility that one, and hence both sides equal

*L*-realizations^{2}(ℝ^{n})*H*of Schrödinger-type operators*(- Δ +V)|*whose potential coefficient_{C0∞}(ℝ^{n}∖ Σ)*V*has a countable number of well-separated singularities on compact sets*Σ*,_{j}*j ∈ J*, of*n*-dimensional Lebesgue measure zero, with*J ⊆ ℕ*an index set and*Σ = ∪*. We show that the defect,_{j ∈ J}Σ_{j}*def(H)*, of*H*can be computed in terms of the individual defects,*def(H*, of closed, symmetric_{j})*L*-realizations of^{2}(ℝ^{n})*(- Δ + V*with potential coefficient_{j})|_{C0∞}(ℝ^{n}∖ Σ_{j})*V*localized around the singularity_{j}*Σ*,_{j}*j ∈ J*, where*V = ∑*. In particular, we prove_{j ∈ J}V_{j}*def(H) = ∑*

_{j ∈ J}def(H_{j}),*∞*. We first develop an abstract approach to the question of decoupling of deficiency indices and then apply it to the concrete case of Schrödinger-type operators in*L*.^{2}(ℝ^{n})
Moreover, we also show how operator (and form) bounds for *V* relative to
*H _{0}= - Δ|_{H2(ℝn)}* can be estimated in terms of the operator (and form) bounds
of

*V*,

_{j}*j ∈ J*, relative to

*H*. Again, we first prove an abstract result and then show its applicability to Schrödinger-type operators in

_{0}*L*.

^{2}(ℝ^{n})
Extensions to second-order (locally uniformly) elliptic differential operators on *ℝ ^{n}* with
a possibly strongly singular potential coefficient are treated as well.

** MSC2010:** Primary 35J10, 35P05; Secondary 47B25, 81Q10.

**Keywords:** *Strongly singular potentials, deficiency indices, self-adjointness.*

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