Commuting difference operators and the combinatorial Gale transform

TL;DR

This paper explores the spectral theory of specific periodic difference operators and reveals a duality that aligns with the combinatorial Gale transform, connecting spectral properties with combinatorial structures.

Contribution

It establishes a unique dual operator for superperiodic difference operators and links this duality to the combinatorial Gale transform.

Findings

Existence of a unique superperiodic operator commuting with a given operator.

Duality between operators coincides with the combinatorial Gale transform.

Spectral theory results for periodic difference operators.

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 ${\cal L}$ of order $(n-k-1)$ which commutes with $L$ and show that the duality $L\leftrightarrow {\cal L}$ coincides up to a certain involution with the combinatorial Gale transform recently introduced in [21].