Commuting difference operators and the combinatorial Gale transform

Abstract

We study 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 $L$ 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 [21].

19 pages, Latex

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