Inverse resonance scattering for Jacobi operators

The Jacobi operator (Jf) _n = a _n−1 f _n−1 +a _n f _n+1 + b _n f _n on ℤ with real finitely supported sequences (a _n − 1) _n∈ℤ and (b _n ) _n∈ℤ is considered. The inverse problem for two mappings (including their characterization): (a _n , b _n , n ∈ ℤ) → {the zeros of the reflection coefficient} and (a _n , b _n , n ∈ ℤ) → {the eigenvalues and the resonances} is solved. All Jacobi operators with the same eigenvalues and resonances are also described.