We investigate closed, symmetric L2(ℝn)
operators (- Δ +V)|C0∞(ℝn ∖ Σ)
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
set and Σ = ∪j ∈ J Σj
. We show that the defect, def(H)
can be computed in terms of the
individual defects, def(Hj)
, of closed, symmetric L2(ℝn)
(- Δ + Vj)|C0∞(ℝn ∖ Σj)
localized around the singularity Σj
, j ∈ J
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
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.