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 $(-\Delta +V){\upharpoonright }_{C_0^{\infty }({\mathbb{R}}^n\setminus \Sigma )}$ 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 Schrödinger-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 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.

33 pages

References in corpus (3)

• Inverse Eigenvalue Problems for Perturbed Spherical Schroedinger Operators
• Infinitesimal form boundedness and Trudinger's subordination for the Schrödinger operator
• Dipoles in Graphene Have Infinitely Many Bound States

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