Decoupling of Deficiency Indices and Applications to Schr\"odinger-Type Operators with Possibly Strongly Singular Potentials

TL;DR

This paper establishes a method to compute the deficiency indices of Schrödinger operators with multiple singularities by decoupling them into contributions from individual singularities, and extends the approach to more general elliptic operators.

Contribution

It introduces an abstract decoupling framework for deficiency indices and applies it to Schrödinger operators with strongly singular potentials, including operator bounds and extensions to elliptic operators.

Findings

Deficiency index of the combined operator equals the sum of individual indices.

Operator bounds for the potential can be estimated from localized potentials.

Framework applies to second-order elliptic differential operators with singular potentials.

Abstract

We investigate closed, symmetric $L^2(\mathbb{R}^n)$-realizations $H$ of Schr\"odinger-type operators $(- \Delta +V)\upharpoonright_{C_0^{\infty}(\mathbb{R}^n \setminus \Sigma)}$ whose potential coefficient $V$ has a countable number of well-separated singularities on compact sets $\Sigma_j$, $j \in J$, of $n$-dimensional Lebesgue measure zero, with $J \subseteq \mathbb{N}$ an index set and $\Sigma = \bigcup_{j \in J} \Sigma_j$. 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 $(- \Delta + V_j)\upharpoonright_{C_0^{\infty}(\mathbb{R}^n \setminus \Sigma_j)}$ with potential coefficient $V_j$ localized around the singularity $\Sigma_j$, $j \in J$, 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 Schr\"odinger-type operators in $L^2(\mathbb{R}^n)$. Moreover, we also show how operator (and form) bounds for $V$ relative to $H_0= - \Delta\upharpoonright_{H^2(\mathbb{R}^n)}$ 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 Schr\"odinger-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.