Finite Gap Jacobi Matrices: A Review

Perhaps the most common theme in Fritz Gesztesy’s broad opus is the study of problems with periodic or almost periodic finite gap differential and difference equations, especially those connected to integrable systems. The present paper reviews recent progress in the understanding of finite gap Jacobi matrices and their perturbations. We’d like to acknowledge our debt to Fritz as a collaborator and friend. We hope Fritz enjoys this birthday bouquet! We consider Jacobi matrices, J , on l2({1, 2, . . . , }) indexed by {an, bn}n=1, an > 0, bn ∈ R, where (u0 ≡ 0) (Ju)n = anun+1 + bnun + an−1un−1 (1.1) or its two-sided analog on l(Z) where an, bn, un are indexed by n ∈ Z and J is still given by (1.1) (we refer to “Jacobi matrix” for the one-sided objects and “two-sided Jacobi matrix” for the Z analog). Here the a’s and b’s parametrize the operator J and {un} ∈ l. We recall that associated to each bounded Jacobi matrix, J , there is a unique probability measure, μ, of compact support in R characterized by either of the equivalent (a) J is unitarily equivalent to multiplication by x on L(R, dμ) by a unitary with (Uδ1)(x) ≡ 1. (b) {an, bn}n=1 are the recursion parameters for the orthogonal polynomials for μ. We’ll call μ the spectral measure for J . By a finite gap Jacobi matrix, we mean one whose essential spectrum is a finite union σess(J) = e ≡ [α1, β1] ∪ · · · ∪ [αl+1, βl+1] (1.2) where α1 < β1 < · · · < αl+1 < βl+1 (1.3)