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Functional Analysis and Its Applications

, Volume 49, Issue 3, pp 175–188 | Cite as

Commuting difference operators and the combinatorial Gale transform

  • I. M. Krichever
Article

Abstract

We develop the spectral theory of n-periodic strictly triangular difference operators L = T -k-1 + ∑ j=1 k a i j Tj 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 LL coincides, up to a certain involution, with the combinatorial Gale transform recently introduced in [21].

Keywords

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

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Laboratory of Representation Theory and Mathematical PhysicsNational Research University Higher School of EconomicsNew YorkUSA
  3. 3.Institute for Information Transmission ProblemsRussian Academy of SciencesSt. PetersburgRussia
  4. 4.Landau Institute for Theoretical PhysicsRussian Academy of SciencesSt. PetersburgRussia

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