Gap-length mapping for periodic Jacobi matrices

Abstract

We construct a real analytic isomorphism between periodic Jacobi operators and the spectral data formed by the gap lengths, the distances between the Dirichlet eigenvalues and the center of the corresponding gap, and some signs. This proves the uniqueness of the solution of the inverse problem and gives a characterization of the solution. Moreover, two-sided a priori estimates of periodic Jacobi operators in terms of gap lengths are obtained.