Near boundary behavior of elliptic operator potential garanteeing its essential self-adjointness

For the class of elliptic operations in space $L_2(G)$ ($G$ is an arbitrary open set in $R^N$), containing the Schrödinger operator with electromagnetic potential, conditions are obtained on near boundary behavior of the coefficients under which the operator was essential self-adjoint on $C_0^\infty(G)$. The closeness of the sufficient conditions derived to the necessary ones is discussed by examples.