Spectral convergence of self-adjoint operators and generalized wave operators

Jingbo Xia

doi:10.1016/0022-1236(88)90083-3

Spectral convergence of self-adjoint operators and generalized wave operators

Jingbo Xia

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

It is well known that if the difference of self-adjoint operators A and B is of trace class, then the generalized wave operators W_±(B, A) = s-lim_{t → ± ∞} exp(itB) exp(-itA) P_ac(A) exist. We give a sufficient condition for s-lim_{n → ∞} W_±(B_n, A_n) = W_±(B, A) when sequences of operators are considered. The condition involves the convergence of the spectral density of A_n. Examples of operators with this property are given. This investigation is also related to the absolute continuity of self-adjoint Operators.

Original language	English
Pages (from-to)	176-197
Number of pages	22
Journal	Journal of Functional Analysis
Volume	77
Issue number	1
DOIs	https://doi.org/10.1016/0022-1236(88)90083-3
State	Published - Mar 1988

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abstract = "It is well known that if the difference of self-adjoint operators A and B is of trace class, then the generalized wave operators W±(B, A) = s-limt → ± ∞ exp(itB) exp(-itA) Pac(A) exist. We give a sufficient condition for s-limn → ∞ W±(Bn, An) = W±(B, A) when sequences of operators are considered. The condition involves the convergence of the spectral density of An. Examples of operators with this property are given. This investigation is also related to the absolute continuity of self-adjoint Operators.",

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AB - It is well known that if the difference of self-adjoint operators A and B is of trace class, then the generalized wave operators W±(B, A) = s-limt → ± ∞ exp(itB) exp(-itA) Pac(A) exist. We give a sufficient condition for s-limn → ∞ W±(Bn, An) = W±(B, A) when sequences of operators are considered. The condition involves the convergence of the spectral density of An. Examples of operators with this property are given. This investigation is also related to the absolute continuity of self-adjoint Operators.

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Spectral convergence of self-adjoint operators and generalized wave operators

Abstract

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