We develop the spectral theory of n-periodic strictly triangular difference operators L = T-k-1 + ∑j=1 k ai jT−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 .
- commuting difference operators
- frieze patterns
- Gale transform
- moduli spaces of n-gons
- spectral theory of linear difference operators