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Vladimir Kapustin: WAVE OPERATORS ON THE SINGULAR SPECTRUM AND FUNCTIONAL MODELS |
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| Vladimir Kapustin St. Petersburg Wave operators of a (discrete time) system describe asymptotic behavior of the system. They are defined as limits of the operator sequences of the form $U_2^n X U_1^{-n}$ (as $n\ightarrow\infty$), where $U_1, U_2$ are unitary operators and $X$ is a bounded operator; the natural assumption is that the commutator $K=U_2 X-XU_1$ is small. The classical scattering theory provides the wave operators in the case where the spectral measures of $U_1, U_2$ are absolutely continuous (with $K$ from the trace class), while the existence of a wave operator in the case of singular measures is still a challenging problem. In the talk we will discuss some reductions of the general problem to some special cases connected with function model theory. In particular, we give a solution (namely, the type of convergence of $U_2^n X U_1^{-n}$) for the case $\mathrm{rank}K=1$, and we discuss the case $\mathrm{rank}K=2$. |
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| Location : Fejér terem | ||||
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