Periodic Jacobi operator with finitely supported perturbation on the half-lattice

Abstract

We consider a periodic Jacobi operator J with finitely supported perturbations on the half-lattice. We describe all eigenvalues and resonances of J and give their properties. We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the Jost functions is one-to-one and onto, we show how the Jost functions can be reconstructed from all eigenvalues, resonances and from the set of zeros of S(λ) − 1, where S(λ) is the scattering matrix.