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Bemerkungen zur Stetigkeit der Eigenwerte selbstadjungierter Operatoren. (German) Zbl 0532.47016
This is a remark to J. Weidmann’s result [Integral Equations Oper. Theory 3, 138-142 (1980; Zbl 0476.47008)] on the related problem [cf. T. Kato, Perturbation Theory for Linear operators, 2nd Ed. (1976; Zbl 0342.47009), Theorem VIII 3.15 ]. The author notes: Let $$T_ n$$, T be bounded selfadjoint operators in a Hilbert space H with spectral resolutions $$E_ n(\cdot)$$, $$E(\cdot)$$, ad let $$T_ n$$ converge to T strongly. Assume that the negative part $$T^-$$ of T $$(=T^++T^-)$$ is compact, and that there exists a bounded selfadjoint operator $$T_ 0$$ with compact negative part $$T^-_ 0$$ such that $$\lim \inf<(T_ n-T_ 0)f_ n,f_ n>\geq 0$$ for every sequence $$(f_ n)$$ in H with $$f_ n\to^{w}0.$$ Then $$\| E_ n(\lambda)-E(\lambda)\| \to 0$$ for all negative $$\lambda$$ which are not eigenvalues of T. The result is applied to the Dirac operator. Some condition on convergence of $$V_ n$$ to V should be mentioned in Satz 2, e.g., $$V_ n\to V$$ locally in $$L^ 2({\mathbb{R}}^ 3)$$ as $$n\to \infty$$.
Reviewer: T.Ichinose
##### MSC:
 47B25 Linear symmetric and selfadjoint operators (unbounded) 47A10 Spectrum, resolvent 47A55 Perturbation theory of linear operators