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

Yu. B. Orochko

Izvestiya. Mathematics , Tome 61 (1997) no. 5, pp. 969-994

Résumé

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 $\mathcal L$.