Commuting difference operators and the combinatorial Gale transform

2014/03/18 by I. M. Krichever, I. Krichever, Krichever, I.
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Holomorphic and Operator Theory #Mathematical functions and polynomials #Spectral Theory in Mathematical Physics #hep-th #math.AG

paper · pdf · doi:10.48550/arxiv.1403.4629

19 pages, Latex

arxiv created 2014/03/18 · openalex publication_date 2014/03/18 · arxiv updated 2014/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the spectral theory of n-periodic strictly triangular difference operators L=T^-k-1+∑_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)ψ=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↔ \cal L coincides up to a certain involution with the combinatorial Gale transform recently introduced in [21].

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