2015/09/30 by Fritz Gesztesy, Marius Mitrea, Irina Nenciu +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Combinatorics #Decoupling (probability) #Gravitational singularity #Mathematical analysis #Mathematical physics #Mathematics #Measure (data warehouse) #Numerical methods in inverse problems #Operator (biology) #Physics #Quantum mechanics #Sigma #Singularity #Spectral Theory in Mathematical Physics #Type (biology) #math-ph #math.AP #math.MP #msc:35J10 #msc:35P05 #msc:47B25 #msc:81Q10
paper · pdf · doi:10.1016/j.aim.2016.08.008
published as Adv. Math. 301, 1022-1061 (2016) · 33 pages
openalex publication_date 2016/08/19 · arxiv created 2016/08/20 · arxiv updated 2016/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We investigate closed, symmetric L2(ℝn)-realizations H of Schrödinger-type operators (- Δ+V)\upharpoonrightC0∞(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= \bigcupj ∈ 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)\upharpoonrightC0∞(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\upharpoonrightH2(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.