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The Kato-Rosenblum theorem says that if $H_0, H$ are self-adjoint operators on a Hilbert space such that the difference $H-H_0$ belongs to the trace class, then the strong limit of $\exp(itH)\exp(-itH_0)P_0$ exists as $t\to\pm\infty$, where $P_0$ is the orthogonal projection onto the absolutely continuous subspace of $H_0$. What happens if we remove the projection $P_0$? It is more or less clear that the direct analog of the result cannot be true, but is it possible to construct limits in some weaker sense?

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