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تسجيل مستخدم جديدInfinitesimal spectral flow and scattering matrix
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تأليف
Nurulla Azamov
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In this note the notion of infinitesimal scattering matrix is introduced. It is shown that under certain assumption, the scattering operator of a pair of trace compatible operators is equal to the chronological exponential of the infinitesimal scattering matrix and that the trace of the infinitesimal scattering matrix is equal to the absolutely continuous part of the infinitesimal spectral flow. As a corollary, a variant of the Birman-Krein formula is derived. An interpretation of Pushnitskis $mu$-invariant is given.
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It has been shown recently that spectral flow admits a natural integer-valued extension to essential spectrum. This extension admits four different interpretations; two of them are singular spectral shift function and total resonance index. In this w
ork we study resonance index outside essential spectrum. Among results of this paper are the following. 1. Total resonance index satisfies Robbin-Salamon axioms for spectral flow. 2. Direct proof of equality total resonance index = intersection number. 3. Direct proof of equality total resonance index = total Fredholm index. 4. (a) Criteria for a perturbation~$V$ to be tangent to the~resonance set at a point~$H,$ where the resonance set is the infinite-dimensional variety of self-adjoint perturbations of the initial self-adjoint operator~$H_0$ which have~$lambda$ as an eigenvalue. (b) Criteria for the order of tangency of a perturbation~$V$ to the resonance set. 5. Investigation of the root space of the compact operator $(H_0+sV-lambda)^{-1}V$ corresponding to an eigenvalue $(s-r_lambda)^{-1},$ where $H_0+r_lambda V$ is a point of the resonance set. This analysis gives a finer information about behaviour of discrete spectrum compared to spectral flow. Finally, many results of this paper are non-trivial even in finite dimensions, in which case they can be and were tested in numerical experiments.
The spectral flow is a classical notion of functional analysis and differential geometry which was given different interpretations as Fredholm index, Witten index, and Maslov index. The classical theory treats spectral flow outside the essential spec
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We show that for a Jacobi operator with coefficients whose (j+1)th moments are summable the jth derivative of the scattering matrix is in the Wiener algebra of functions with summable Fourier coefficients. We use this result to improve the known disp
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This paper is a continuation of the study of spectral flow inside essential spectrum initiated in cite{AzSFIES}. Given a point $lambda$ outside the essential spectrum of a self-adjoint operator $H_0,$ the resonance set, $mathcal R(lambda),$ is an ana
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