Abstract:
In a Hilbert space $\mathfrak H$ a study is made of limits of the form $W_{\pm}(H,H_0|\Lambda)=\displaystyle\operatornamewithlimits{s-lim}_{t\to\pm\infty}\exp\{it H\}\Lambda(t)$
it being assumed that $\varphi(H)W_{\pm}=W_{\pm}\varphi(H_0)$ for any function $\varphi$ that the operators $H$ and $H_0$ are selfadjoint, and that $\Lambda(t)$ is bounded. The invariance principle states that the limit
$\displaystyle\operatornamewithlimits{s-lim}_{t\to\pm\infty}\exp\{if(H,t)\}Q(\varphi,t)$, where $Q$ is a certain operator constructed explicitly from $\Lambda$ and $f$, is independent of the choice of $f$ and is identical with $W_{\pm}(H,H_0|\Lambda)$. In some cases the invariance
principle can be justified by invoking a theorem proved in the paper. Applications
of this theorem to the Schrödinger equation are considered.
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