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Let $A$ be self-adjoint (possibly unbounded) operator on Hilbert space $\mathcal{H}$. Under what conditions $w-\lim_{t\rightarrow\infty} e^{i A t}=P_0$, where $w-\lim$ - the limit in weak operator topology and $P_0$ - spectral projection on eigenvalue $0$. If pure-point spectrum of $A$ consists of only one point $0$ and there is no singular-continuous spectrum then this assertion follows from Riemann-Lebesgue theorem. Is the absence of singular-continuous spectrum really necessary?

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