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This article is cited in 11 scientific papers (total in 11 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.
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
Linking options:
https://www.mathnet.ru/eng/faa3204https://doi.org/10.4213/faa3204 https://www.mathnet.ru/eng/faa/v49/i3/p22
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