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Nov 15, 2019 at 16:59 comment added asv Many thanks. This answers my question. Let me comment that your argument shows that the required integral does converge provided $\hat\phi_0 =0$. This fact is sufficient for the application to scattering theory I am interested in -proof of existence of Moller operators. This is due to the fact that functions satisfying this condition are still dense in $L^2$.
Nov 15, 2019 at 16:07 vote accept asv
Nov 15, 2019 at 15:55 history answered Willie Wong CC BY-SA 4.0