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TMF, 1971, Volume 8, Number 1, Pages 49–54 (Mi tmf4357)  

Invariance principle for generalized wave operators

V. B. Matveev


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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English version:
Theoretical and Mathematical Physics, 1971, 8:1, 663–667

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Received: 26.10.1970

Citation: V. B. Matveev, “Invariance principle for generalized wave operators”, TMF, 8:1 (1971), 49–54; Theoret. and Math. Phys., 8:1 (1971), 663–667

Citation in format AMSBIB
\Bibitem{Mat71}
\by V.~B.~Matveev
\paper Invariance principle for generalized wave operators
\jour TMF
\yr 1971
\vol 8
\issue 1
\pages 49--54
\mathnet{http://mi.mathnet.ru/tmf4357}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=473877}
\zmath{https://zbmath.org/?q=an:0218.47035}
\transl
\jour Theoret. and Math. Phys.
\yr 1971
\vol 8
\issue 1
\pages 663--667
\crossref{https://doi.org/10.1007/BF01038674}


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  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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