Let $A,B$ be self-adjoint operators on $H$,then we can define the strong limit $$ W=s-\lim_{t\to+\infty}e^{iBt}e^{-iAt} $$ If the limit exists, then W is called the wave operator, which is fundamental in the scattering theory. My question here is that if instead, we consider the operator norm above,then what can be said about it ? I was told that the the limit exists if and only if $A=B$, but I don't know how to prove this statement.
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6If $A$ and $B$ commute, you get $e^{iBt}e^{-iAt}=e^{i(B-A)t}$. If $B-A\neq 0$, take a nonzero eigenvalue and a nonzero eigenvector to see that the strong limit does not exist, therefore the operator norm limit does not exist. When $A$ and $B$ do not commute, it is less easy... – Julien Mar 03 '13 at 13:14
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@1015: I'm having the same problem. ^^ – C-star-W-star Jan 26 '15 at 22:04