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Zbl 0592.35043
Toraev, A.
Oscillation of elliptic operators and the structure of their spectrum.
(English. Russian original)
[J] Sov. Math., Dokl. 30, 663-666 (1984); translation from Dokl. Akad. Nauk SSSR, 279, 306-309 (1984). ISSN 0197-6788

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.
[A.Bove]
MSC 2000:
*35J40 Higher order elliptic equations, boundary value problems
35D05 Existence of generalized solutions of PDE
35P99 Spectral theory and eigenvalue problems for PD operators

Keywords: Oscillation and non oscillation criteria; Kneser types; discreteness of the spectrum; Dirichlet operator

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