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Perturbation of a self-adjoint operator by a subordinate symmetric operator. (Russian. English summary) Zbl 0587.47017
The author states the following variant of the Rellich-Kato theorem: let A, B be operators in some Hilbert space $${\mathcal H}$$, $$A=A^*$$, $$B=B^*$$, $${\mathcal D}(B)\supset {\mathcal D}(A)$$, $$\exists \gamma >-1:$$ (Bu,u)$$\geq \gamma (Au,u)$$, $$\forall u\in {\mathcal D}(A)$$, then the operator $$A+B$$ is self- adjoint on the domain $${\mathcal D}(A).$$
The above theorem is invalid. It may be seen from the following example:
$${\mathcal H}=L^ 2({\mathbb{R}}^{\ell})$$, $$\ell \geq 5$$, $$A=-\Delta$$, $$B=\frac{\gamma (\ell -2)^ 2}{4(x)^ 2}$$, $$\gamma\in (-1,0)$$. The non- self-adjointness of the operator $$A+B$$ is known from the Weyl theorem.
Reviewer: M.A.Perelmuter

##### MSC:
 47A55 Perturbation theory of linear operators 47B25 Linear symmetric and selfadjoint operators (unbounded) 81Q15 Perturbation theories for operators and differential equations in quantum theory
##### Keywords:
Rellich-Kato theorem; Weyl theorem
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