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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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