Functional Analysis and Its Applications

, Volume 49, Issue 3, pp 175–188

# 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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