We consider the mininial operator H in, in Z 2, generated by a real formally self-adjoint singular elliptic second-order differential expression (DE) C. The example of the differential operator H = Ha corresponding to the DE grad, where a(r), r 6 .[0, +oo)s is a uon-nfigative scalar function, is studied to determine the dependence of the deficiency indices of H on the degree of smoothness of the (leading coefficients in. The other result of this paper is a test for the seli-adjontness of an operator H without any conditions on the behaviour of the potential of C as |x| -+ +oo. These results have no direct analogues in the case of an elliptic DE.
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