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Ufa Mathematical Journal, 2022, Volume 14, Issue 2, Pages 35–55
DOI: https://doi.org/10.13108/2022-14-2-35
(Mi ufa614)
 

This article is cited in 4 scientific papers (total in 4 papers)

Eta-invariant for parameter-dependent families with periodic coefficients

K. N. Zhuikova, A. Yu. Savinab

a Peoples' Friendship University of Russia, Moscow
b Leibniz Universität Hannover, Welfengarten 1, D - 30167 Hannover, Germany
References:
Abstract: On a closed smooth manifold, we consider operator families being linear combinations of parameter-dependent pseudodifferential operators with periodic coefficients. Such families arise in studying nonlocal elliptic problems on manifolds with isolated singularities and/or with cylindrical ends. The aim of the work is to construct the $\eta$-invariant for invertible families and to study its properties. We follow Melrose's approach who treated the $\eta$-invariant as a generalization of the winding number being equal to the integral the trace of the logarithmic derivative of the family. At the same time, the Melrose $\eta$-invariant is equal to the regularized integral of the regularized trace of the logarithmic derivative of the family. In our situation, for the trace regularization, we employ the operator of difference differentiating instead of the usual differentation used by Melrose. The main technical result is the fact that the operator of difference differentiation is an isomorphism between the spaces of functions with conormal asymptotics at infinity and this allows us to determine the regularized trace. Since the obtained regularized trace can increase at infinity, we also introduce a regularization for the integral. Our integral regularization involves an averaging operation. Then we establish the main properties of the $\eta$-invariant. Namely, the $\eta$-invariant in the sense of this work satisfies the logarithmic property and is a generalization of Melrose's $\eta$-invariant, that is, it coincides with it for usual parameter-dependent pseudodifferential operators. Finally, we provide a formula for the variation of the $\eta$-invariant under a variation of the family.
Keywords: elliptic operator, parameter-dependent operator, $\eta$-invariant, difference differentiation.
Received: 21.04.2021
Russian version:
Ufimskii Matematicheskii Zhurnal, 2022, Volume 14, Issue 2, Pages 37–57
Bibliographic databases:
Document Type: Article
UDC: 515.168.5
MSC: Primary 58J28; Secondary 58J40
Language: English
Original paper language: Russian
Citation: K. N. Zhuikov, A. Yu. Savin, “Eta-invariant for parameter-dependent families with periodic coefficients”, Ufimsk. Mat. Zh., 14:2 (2022), 37–57; Ufa Math. J., 14:2 (2022), 35–55
Citation in format AMSBIB
\Bibitem{ZhuSav22}
\by K.~N.~Zhuikov, A.~Yu.~Savin
\paper Eta-invariant for parameter-dependent families with periodic coefficients
\jour Ufimsk. Mat. Zh.
\yr 2022
\vol 14
\issue 2
\pages 37--57
\mathnet{http://mi.mathnet.ru/ufa614}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4448017}
\transl
\jour Ufa Math. J.
\yr 2022
\vol 14
\issue 2
\pages 35--55
\crossref{https://doi.org/10.13108/2022-14-2-35}
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  • https://doi.org/10.13108/2022-14-2-35
  • https://www.mathnet.ru/eng/ufa/v14/i2/p37
  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Уфимский математический журнал
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    English version PDF:15
    References:17
     
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