A remark on essential self-adjointness of nonsemi-bounded elliptic operators in $L_2(G)$

Conditions are obtained for the general-type symmetric elliptic second-order operator $L$ in space $L_2(G)$ ($D_L=C_0^{\infty}(G)$, $G$ is an arbitrary open set in $R^n$)under which self-adjointness $\overline{L}$ follows from the essential self-adjointness of some semi-bounded elliptic operator.