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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
47B25 Linear symmetric and selfadjoint operators (unbounded)
47A10 Spectrum, resolvent
47A55 Perturbation theory of linear operators