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Commuting difference operators and the combinatorial Gale transform

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Abstract

We develop the spectral theory of n-periodic strictly triangular difference operators L = T -k-1 + ∑ k j=1 a j i T−j and the spectral theory of the “superperiodic” operators for which all solutions of the equation (L + 1)ψ = 0 are (anti)periodic. We show that, for a superperiodic operator L of order k+1, there exists a unique superperiodic operator L of order n-k-1 which commutes with L and show that the duality L ↔ L coincides, up to a certain involution, with the combinatorial Gale transform recently introduced in [21].

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Author information

Authors and Affiliations

  1. Department of Mathematics, Columbia University, New York, USA

    I. M. Krichever

  2. Laboratory of Representation Theory and Mathematical Physics, National Research University Higher School of Economics, New York, USA

    I. M. Krichever

  3. Institute for Information Transmission Problems, Russian Academy of Sciences, St. Petersburg, Russia

    I. M. Krichever

  4. Landau Institute for Theoretical Physics, Russian Academy of Sciences, St. Petersburg, Russia

    I. M. Krichever

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  1. I. M. Krichever
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Correspondence to I. M. Krichever.

Additional information

The research was carried out at the RAS Institute for Information Transmission Problems at the expense of a grant from the Russian Science Foundation (project no. 14-50-00150).

Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 49, No. 3, pp. 22–40, 2015

Original Russian Text Copyright © by I. M. Krichever

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Krichever, I.M. Commuting difference operators and the combinatorial Gale transform. Funct Anal Its Appl 49, 175–188 (2015). https://doi.org/10.1007/s10688-015-0102-3

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  • DOI: https://doi.org/10.1007/s10688-015-0102-3

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