Finite local propagation rate of a hyperbolic equation in the problem of selfadjointness of powers of a second order elliptic differential operator

Let $S$ be a formally selfadjoint second order elliptic expression and $H$ the minimal nonclosed operator in $L_2(\mathbf R^m)$, $m\geqslant1$, generated by it. The property of finite local propagation rate of the hyperbolic equation $\frac{\partial^2u}{\partial t^2}+S[u]=0$ is applied to obtain new criteria for the essential selfadjointness of $H$ and its powers. In these criteria restrictions are imposed on the coefficients of $S$ along a sequence of nonintersecting solid layers diverging to infinity.
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