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Decoupling of deficiency indices and applications to Schrödinger-type operators with possibly strongly singular potentials. (English) Zbl 1357.35099
The study of deficiency indices of symmetric operators in Hilbert spaces is one of the central problems in spectral theory of unbounded linear operators. One of the main objects in the paper under review are closed symmetric realizations \(H\) in \(L^2(\mathbb{R}^n)\) of Schrödinger-type operators \[ (-\Delta + V)\upharpoonright_{C_0^\infty(\mathbb{R}^n\setminus\Sigma)} \] whose potential \(V\) 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 \(\Sigma=\bigcup_{j\in J}\Sigma_j\).
The central result in the paper under review is an abstract approach to the question of decoupling of deficiency indices. Then the authors apply it to the concrete case of Schrödinger type operators in \(L^2(\mathbb{R}^n)\) and to second-order elliptic differential operators on \(\mathbb{R}^n\). For example, it is proved that under certain assumptions on \(V\) and \(\Sigma\) the defect \(\text{Def}(H)\) of \(H\) can be computed in terms of the individual defects \(\text{Def}(H_j)\) of \(H_j\), where \(H_j\) is a closed symmetric realization of \((-\Delta + V)\upharpoonright_{C_0^\infty(\mathbb{R}^n\setminus\Sigma_j)}\). More specifically, the authors derive the results that the defect of \(H\) can be computed as the sum of over the individual defects of \(H_j\), \[ \text{Def}(H) = \sum_{j\in J} \text{Def}(H_j). \] The possibility that one and hence both sides equal \(\infty\) is included. Here \[ \text{Def}(A):= \frac{n_+(A) + n_-(A)}{2}, \] and \(n_\pm(A)\) are the standard deficiency indices of \(A\). Notice that \(\text{Def}(A) = n_\pm(A)\) if \(n_+(A)=n_-(A)\) (the latter holds, in particular, when \(A\) is lower semibounded or commutes with a conjugation).
Finally, let us mention that applications to Schrödinger operators with multipolar inverse-square potentials allow to considerably improve the results obtained in [V. Felli et al., J. Funct. Anal. 250, No. 2, 265–316 (2007; Zbl 1222.35074)].

MSC:
35J10 Schrödinger operator, Schrödinger equation
35P05 General topics in linear spectral theory for PDEs
47B25 Linear symmetric and selfadjoint operators (unbounded)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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