Заседание семинара кафедры высшей математики и математической физики состоится 3 марта (среда) в 18-30.

Докладчик: Григорий Владимирович Розенблюм

Тема: The Birman–Schwinger type operator with singular measure. Eigenvalues analysis, Connes integral and rectifiable sets

Абстракт:
This is an extended version of the authors’ talk on 28.12.2020 at the V.I.Smirnov seminar, containing some new results. We consider the Birman–Schwinger type operator \(\mathbf{T}_{P,\mathfrak{A}}=\mathfrak{A}*P \mathfrak{A}\), where P is a signed measure in \(\mathbb{R}^\mathbf{N}\) and \(\mathfrak{A}\) is a pseudodifferential operator in \(\mathbb{R}^\mathbf{N}\) of order \(-l=-\mathbf{N}\) (in the leading case, \(\mathfrak{A}= (1-\Delta)^{-\mathbf{N}/4}\)). Under rather general conditions we find eigenvalue estimates for this operator, and for measures supported on a Lipschitz surface, find eigenvalue asymptotics. The interesting case is when measure \(P\) contains a singular component. A peculiar feature here is that the order of the eigenvalue estimates and asymptotics does not depend on the dimensional characteristics of the support of the measure, so, contributions of components of different dimensions just add up. Further on the results are carried over to more general measures supported on so-called rectifiable sets. We will discuss relation of our results to spectral theory of fractals, logarithmic potential and, finally, to noncommutative integration of singular measures.

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Meeting ID: 270 950 5573
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