About one of Weil's theorems for many-dimensional case

It is given the new theorem, which extends the known Weil's theorem about Shturm–Liuvill's operator self-adjointness in $L_2(-\infty;+\infty)$ to elliptic second-order operators in $L_2(G)$ ($G\subseteq R^n$). Many-dimensional Weil's theorem is followed from more general theorem, for statement which special construction of covering collection is built. Given results contain the known analogs of many-dimensional Weil's theorem and, as distinguished from them, the results refer to the domain $G$, which may be proper subset of $R^n$.