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Oscillation of elliptic operators and the structure of their spectrum. (English. Russian original) Zbl 0592.35043
Oscillation and non oscillation criteria of Kneser types in an unbounded domain $\Omega \subset {\bbfR}\sp n$ are obtained for the equation $$ (- 1)\sp m\sum\sb{\vert \alpha \vert =\vert \beta \vert =m}D\sp{\alpha}a\sb{\alpha \beta}(x)\quad D\sp{\beta} u+a(x)u=0, $$ where $a\sb{\alpha \beta}(x)=a\sb{\beta \alpha}(x)$ and a(x) are measurable and locally bounded and $\sum\sb{\vert \alpha \vert =\vert \beta \vert =m}a\sb{\alpha \beta}(x) \xi\sp{\alpha +\beta}\ge c \vert \xi \vert\sp{2m}$, $x\in \Omega$, $c=const$. A criterion for discreteness of the spectrum for the associated Dirichlet operator in $L\sb 2(\Omega)$ is also obtained.
Reviewer: A.Bove

35J40Higher order elliptic equations, boundary value problems
35D05Existence of generalized solutions of PDE (MSC2000)
35P99Spectral theory and eigenvalue problems for PD operators