Invariance principle for generalized wave operators

V. B. Matveev

Teoretičeskaâ i matematičeskaâ fizika, Tome 8 (1971) no. 1, pp. 49-54

Résumé

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.