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 L2(ℝn)-realizations H of Schrödinger-type
operators (- Δ +V)|C0∞(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 = ∪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
(- Δ + Vj)|C0∞(ℝn ∖ Σj) with potential
coefficient Vj localized around the singularity Σj, j ∈ J, where
V = ∑j ∈ J Vj. In particular, we prove
def(H) = ∑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 Schrödinger-type operators in
L2(ℝn).
Moreover, we also show how operator (and form) bounds for V relative to H0= - Δ|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 Schrödinger-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.
MSC2010: Primary 35J10, 35P05; Secondary 47B25, 81Q10.
Keywords: Strongly singular potentials, deficiency indices, self-adjointness.
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