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Zapiski Nauchnykh Seminarov LOMI, 1977, Volume 73, Pages 102–117
(Mi znsl1947)
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This article is cited in 2 scientific papers (total in 2 papers)
Local conditions for the existence of the spectral shift function
L. S. Koplienko
Abstract:
Let $U_0$, $U_1$ be unitary operators in a Hilbert space. If the operator $U_1-U_0$ is nuclear, then (as M. G. Krein established) there exists a function $\eta$ on the unit circle $\mathbf T$, $\eta=\eta(U_1,U_0)$, $\eta\in L_1(\mathbf T)$ satisfying the equality
\begin{gather}
tr(\varphi(U_1)-\varphi(U_0))=\int_{\mathbf T}\eta(\zeta)\varphi'(\zeta)d\zeta
\end{gather}
for all functions $\varphi$ with derivative $\varphi'$ from the Wiener class. M. Sh. Rirman and M. G. Krein proved that the function $\varphi'$ is connected with the scattering matrix $S$ for the pair $U_0$, $U_1$ by
\begin{gather}
\det S(\zeta)=\exp(-2\pi i\eta(\zeta)),
\tag{2}
\end{gather}
In this paper (1) and (2) are proved under more general (local) conditions on the pair $U_0$, $U_1$. Under these conditions we investigate some properties of the function n and describe the class of functions $\eta$, which are admissible in (1). Applications to differential operators are given.
Citation:
L. S. Koplienko, “Local conditions for the existence of the spectral shift function”, Investigations on linear operators and function theory. Part VIII, Zap. Nauchn. Sem. LOMI, 73, "Nauka", Leningrad. Otdel., Leningrad, 1977, 102–117; J. Soviet Math., 34:6 (1986), 2080–2090
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https://www.mathnet.ru/eng/znsl1947 https://www.mathnet.ru/eng/znsl/v73/p102
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