On the invariance of wave operators |
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Authors: | Melvin Mullin |
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Affiliation: | John Jay College of the City University of New York, New York, New York 10019 USA |
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Abstract: | Let H, K be self-adjoint operators on a Hilbert space. Kato's Invariance Principle (T. Kato, “Perturbation Theory for Linear Operators,” Springer-Verlag, Berlin, 1966) states that under certain conditions W(φ(H), φ(K)) = s-limt → + ∞eit(φ)ke?it(φ)H exists and is independent of the monotone function φ whenever W(H, K) exists. The purpose of this paper is to present a new proof of Kato's result based upon a study of the variation of W(φ(H), φ(K)) with respect to φ. It is shown (Theorem 1) that this variation vanishes, and (Theorem 2) that the invariance principle holds provided that K-H belongs to a large subset of the Hilbert-Schmidt class of compact operators. |
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