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Oscillation of elliptic operators and the structure of their spectrum. (English. Russian original) Zbl 0592.35043

Sov. Math., Dokl. 30, 663-666 (1984); translation from Dokl. Akad. Nauk SSSR, 279, 306-309 (1984).
Oscillation and non oscillation criteria of Kneser types in an unbounded domain \(\Omega \subset {\mathbb{R}}^ n\) are obtained for the equation \[ (- 1)^ m\sum_{| \alpha | =| \beta | =m}D^{\alpha}a_{\alpha \beta}(x)\quad D^{\beta} u+a(x)u=0, \] where \(a_{\alpha \beta}(x)=a_{\beta \alpha}(x)\) and a(x) are measurable and locally bounded and \(\sum_{| \alpha | =| \beta | =m}a_{\alpha \beta}(x) \xi^{\alpha +\beta}\geq c | \xi |^{2m}\), \(x\in \Omega\), \(c=const\). A criterion for discreteness of the spectrum for the associated Dirichlet operator in \(L_ 2(\Omega)\) is also obtained.
Reviewer: A.Bove

MSC:

35J40 Boundary value problems for higher-order elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35P99 Spectral theory and eigenvalue problems for partial differential equations
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