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Funktsional. Anal. i Prilozhen., 2015, Volume 49, Issue 3, Pages 22–40 (Mi faa3204)  

This article is cited in 3 scientific papers (total in 3 papers)

Commuting Difference Operators and the Combinatorial Gale Transform

I. M. Kricheverabcd

a Columbia University
b L.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences
c Department of Mathematics, National Research University "Higher School of Economics"
d Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow

Abstract: We develop the spectral theory of $n$-periodic strictly triangular difference operators $L=T^{-k-1}+\sum_{j=1}^k a_i^j T^{-j}$ and the spectral theory of the “superperiodic” operators for which all solutions of the equation $(L+1)\psi=0$ are (anti)periodic. We show that, for a superperiodic operator $L$ of order $k+1$, there exists a unique superperiodic operator $\mathcal{L}$ of order $n-k-1$ which commutes with $L$ and show that the duality $L\leftrightarrow \mathcal{L}$ coincides, up to a certain involution, with the combinatorial Gale transform recently introduced in [S. Morier-Genoud, V. Ovsienko, R. E. Schwartz, S. Tabachnikov, Linear difference equations, frieze patterns and combinatorial Gale transform, Forum Math. Sigma, 2 (2014), e22].

Keywords: spectral theory of linear difference operators, commuting difference operators, frieze patterns, moduli spaces of $n$-gons, Gale transform

Funding Agency Grant Number
Russian Science Foundation 14-50-00150


DOI: https://doi.org/10.4213/faa3204

Full text: PDF file (261 kB)
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English version:
Functional Analysis and Its Applications, 2015, 49:3, 175–188

Bibliographic databases:

Document Type: Article
UDC: 512.77+517.984
Received: 18.05.2015

Citation: I. M. Krichever, “Commuting Difference Operators and the Combinatorial Gale Transform”, Funktsional. Anal. i Prilozhen., 49:3 (2015), 22–40; Funct. Anal. Appl., 49:3 (2015), 175–188

Citation in format AMSBIB
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\pages 175--188
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Mauleshova G.S., Mironov A.E., “One-point commuting difference operators of rank 1”, Dokl. Math., 93:1 (2016), 62–64  crossref  mathscinet  zmath  isi  elib  scopus
    2. A. V. Il'ina, I. M. Krichever, “Triangular Reductions of the $2D$ Toda Hierarchy”, Funct. Anal. Appl., 51:1 (2017), 48–65  mathnet  crossref  crossref  mathscinet  isi  elib
    3. V. Ovsienko, “Partitions of unity in $\mathrm{SL}(2,\mathbb{Z})$, negative continued fractions, and dissections of polygons”, Res. Math. Sci., 5 (2018), 21, 25 pp.  crossref  mathscinet  zmath  isi  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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