Inverse problems for periodic generalized Jacobi matrices

TL;DR

This paper explores inverse problems related to semi-infinite periodic generalized Jacobi matrices, introducing a generalized Abel criterion and linking solvability to the existence of a specific monodromy matrix.

Contribution

It generalizes the Abel criterion for inverse problems of periodic generalized Jacobi matrices and connects solvability to a normalized J-unitary matrix polynomial.

Findings

Generalization of the Abel criterion for these matrices

Equivalence between Pell-Abel equation solvability and monodromy matrix existence

New conditions for inverse problem solvability

Abstract

Some inverse problems for semi-infinite periodic generalized Jacobi matrices are considered. In particular, a generalization of the Abel criterion is presented. The approach is based on the fact that the solvability of the Pell-Abel equation is equivalent to the existence of a certainly normalized $J$-unitary $2\times 2$-matrix polynomial (the monodromy matrix).