The Multidimensional Weyl Theorem and Covering Families

The well-known theorem of Weyl about the essential self-adjointness of the Sturm–Liouville operator $Lu=-(p(x)u')'+q(x)u$ in $L_2(\mathbb R^1)$ with $D_L=C_0^\infty(\mathbb R^1)$, $p(x)>0$, and $q(x)\ge\operatorname{const}$ is generalized to second-order elliptic operators in $L_2(G)$ ($G\subseteq\mathbb R^n$). The multidimensional Weyl theorem is derived from a more general theorem; to state and prove the latter, a special covering family is constructed. The results obtained imply the known multidimensional analogs of the Weyl theorem and, unlike these analogs, apply to open proper subsets $G$ in $\mathbb R^n$ .