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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

47A55 Perturbation theory of linear operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
81Q15 Perturbation theories for operators and differential equations in quantum theory
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