iv : m at h - ph / 0 51 10 78 v 2 2 2 Ja n 20 07 On the Two Spectra Inverse Problem for Semi - Infinite Jacobi Matrices ∗ †

We present results on the unique reconstruction of a semi-infinite Jacobi operator from the spectra of the operator with two different boundary conditions. This is the discrete analogue of the Borg-Marchenko theorem for Schrödinger operators on the half-line. Furthermore, we give necessary and sufficient conditions for two real sequences to be the spectra of a Jacobi operator with different boundary conditions. Mathematics Subject Classification(2000): 47B36, 49N45,81Q10,47A75, 47B37, 47B39. Research partially supported by Universidad Nacional Autónoma de México under Project PAPIITDGAPA IN 105799, and by CONACYT under Project P42553F. Fellow Sistema Nacional de Investigadores.