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

Abstract

We investigate closed, symmetric $L^2(\mathbb{R}^n)$-realizations $H$ of Schrödinger-type operators $(- Δ+V)\upharpoonright_{C_0^{\infty}(\mathbb{R}^n \setminus Σ)}$ whose potential coefficient $V$ has a countable number of well-separated singularities on compact sets $Σ_j$, $j \in J$, of $n$-dimensional Lebesgue measure zero, with $J \subseteq \mathbb{N}$ an index set and $Σ= \bigcup_{j \in J} Σ_j$. We show that the defect, $\mathrm{def}(H)$, of $H$ can be computed in terms of the individual defects, $\mathrm{def}(H_j)$, of closed, symmetric $L^2(\mathbb{R}^n)$-realizations of $(- Δ+ V_j)\upharpoonright_{C_0^{\infty}(\mathbb{R}^n \setminus Σ_j)}$ with potential coefficient $V_j$ localized around the singularity $Σ_j$, $j \in J$, where $V = \sum_{j \in J} V_j$. In particular, we prove \[ \mathrm{def}(H) = \sum_{j \in J} \mathrm{def}(H_j), \] including the possibility that one, and hence both sides equal $\infty$. 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 $L^2(\mathbb{R}^n)$. Moreover, we also show how operator (and form) bounds for $V$ relative to $H_0= - Δ\upharpoonright_{H^2(\mathbb{R}^n)}$ can be estimated in terms of the operator (and form) bounds of $V_j$, $j \in J$, relative to $H_0$. Again, we first prove an abstract result and then show its applicability to Schrödinger-type operators in $L^2(\mathbb{R}^n)$. Extensions to second-order (locally uniformly) elliptic differential operators on $\mathbb{R}^n$ with a possibly strongly singular potential coefficient are treated as well.