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 Funktsional. Anal. i Prilozhen., 2018, Volume 52, Issue 1, Pages 92–97 (Mi faa3451)

Brief communications

Monodromization and Difference Equations with Meromorphic Periodic Coefficients

A. A. Fedotov

Saint Petersburg State University, St. Petersburg, Russia

Abstract: We consider a system of two first-order difference equations in the complex plane. We assume that the matrix of the system is a 1-periodic meromorphic function having two simple poles per period and bounded as $\operatorname{Im}z\to\pm \infty$. We prove the existence and uniqueness of minimal meromorphic solutions, i.e., solutions having simultaneously a minimal set of poles and minimal possible growth as $\operatorname{Im}z\to\pm \infty$. We consider the monodromy matrix representing the shift-byperiod operator in the space of meromorphic solutions and corresponding to a basis built of two minimal solutions. We check that it has the same functional structure as the matrix of the initial system of equations and, in particular, is a meromorphic periodic function with two simple poles per period. This implies that the initial equation is invariant with respect to the monodromization procedure, that is, a natural renormalization procedure arising when trying to extend the Floquet–Bloch theory to difference equations defined on the real line or complex plane and having periodic coefficients. Our initial system itself arises after one renormalization of a self-adjoint difference Schrödinger equation with 1-periodic meromorphic potential bounded at $\pm i\infty$ and having two poles per period.

Keywords: difference equations in the complex plane, meromorphic periodic coefficients, monodromy matrix, renormalization procedure.

 Funding Agency Grant Number Centre National de la Recherche Scientifique Russian Foundation for Basic Research 17-51-150008 The present work was supported by the Russian foundation of basic research under grant 17-51-150008-CNRS-a.

DOI: https://doi.org/10.4213/faa3451

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English version:
Functional Analysis and Its Applications, 2018, 52:1, 77–81

Bibliographic databases:

UDC: 517.962.22

Citation: A. A. Fedotov, “Monodromization and Difference Equations with Meromorphic Periodic Coefficients”, Funktsional. Anal. i Prilozhen., 52:1 (2018), 92–97; Funct. Anal. Appl., 52:1 (2018), 77–81

Citation in format AMSBIB
\Bibitem{Fed18} \by A.~A.~Fedotov \paper Monodromization and Difference Equations with Meromorphic Periodic Coefficients \jour Funktsional. Anal. i Prilozhen. \yr 2018 \vol 52 \issue 1 \pages 92--97 \mathnet{http://mi.mathnet.ru/faa3451} \crossref{https://doi.org/10.4213/faa3451} \elib{http://elibrary.ru/item.asp?id=32428050} \transl \jour Funct. Anal. Appl. \yr 2018 \vol 52 \issue 1 \pages 77--81 \crossref{https://doi.org/10.1007/s10688-018-0213-8} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000428558200013} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85044767007} 

• http://mi.mathnet.ru/eng/faa3451
• https://doi.org/10.4213/faa3451
• http://mi.mathnet.ru/eng/faa/v52/i1/p92

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. A. Fedotov, “On minimal entire solutions of the one-dimensional difference Schrödinger equation with the potential $v(z)=e^{-2\pi iz}$”, J. Math. Sci. (N. Y.), 238:5 (2019), 750–761
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