The Multidimensional Weyl Theorem and Covering Families

Abstract

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 $q{\text{(}}x) \geqslant const$ 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 $\mathbb{R}^\user1{n}$.