Description of the algebro-geometric Sturm–Liouville coefficients

It was discovered recently that there is a class of Sturm–Liouville operators whose coefficients are related to algebro-geometric data via a construction analogous to that carried out in the 1970s for the Schrödinger operator by Dubrovin, Matveev, and Novikov. In this paper, we study the “algebro-geometric” Sturm–Liouville coefficients in detail. Emphasis is placed on their recurrence properties. We make systematic use of the theory of abelian group extensions.