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Essential self-adjointness and self-adjointness for even order elliptic operators. (English) Zbl 0535.35018
The subject of this paper are symmetric elliptic operators on $$L^ 2({\mathbb{R}}^ n)$$, of the form $$T=\sum_{0<| \alpha |,| \beta | \leq m}(-1)^{| \alpha |}D^{\alpha}a_{\alpha \beta}(x)D^{\beta}+q(x).$$ For the first it is proved that T is essentially self adjoint on $$C_ 0\!^{\infty}({\mathbb{R}}^ n)$$ if the $$a_{\alpha \beta}$$ are sufficiently smooth and bounded and $$q(x)\geq - cons\tan t\quad | x|^{2m/(2m-1)}.$$ Then it is proved that such an operator is self-adjoint on $$H^{2m}({\mathbb{R}}^ n)\cap D(q)$$ if q is positive and $$| D^{\alpha}q| \leq \quad cons\tan t\quad q^{1+| \alpha | /2m},$$ for $$1<| \alpha | \leq m$$. This extends earlier results which had more restrictions on the operator.
Reviewer: G.Gudmundsdottir

##### MSC:
 35J30 Higher-order elliptic equations 47B25 Linear symmetric and selfadjoint operators (unbounded) 35P05 General topics in linear spectral theory for PDEs 35P25 Scattering theory for PDEs
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##### References:
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