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 Izv. RAN. Ser. Mat., 1997, Volume 61, Issue 5, Pages 71–98 (Mi izv152)

Some properties of the deficiency indices of symmetric singular elliptic second-order operators in $L^2(\mathbb R^m)$

Yu. B. Orochko

Abstract: We consider the minimal operator $H$ in $L^2(\mathbb R^m)$, $m\geqslant 2$, generated by a real formally self-adjoint singular elliptic second-order differential expression (DE) $\mathcal L$. The example of the differential operator $H=H_0$ corresponding to the DE $\mathcal L=\mathcal L_0=-\operatorname{div}a(|x|)\operatorname{grad}$, where $a(r)$, $r\in[0,+\infty)$, is a non-negative scalar function, is studied to determine the dependence of the deficiency indices of $H$ on the degree of smoothness of the leading coefficients in $\mathcal L$. The other result of this paper is a test for the self-adjontness of an operator $H$ without any conditions on the behaviour of the potential of $\mathcal L$ as $|x|\to+\infty$. These results have no direct analogues in the case of an elliptic DE $\mathcal L$.

DOI: https://doi.org/10.4213/im152

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English version:
Izvestiya: Mathematics, 1997, 61:5, 969–994

Bibliographic databases:

MSC: 47B25, 35J70

Citation: Yu. B. Orochko, “Some properties of the deficiency indices of symmetric singular elliptic second-order operators in $L^2(\mathbb R^m)$”, Izv. RAN. Ser. Mat., 61:5 (1997), 71–98; Izv. Math., 61:5 (1997), 969–994

Citation in format AMSBIB
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