Inverse resonance scattering for Jacobi operators

We consider the Jacobi operator $(Jf{)}_n=a_{n-1}{f}_{n-1}+a_n{f}_{n+1}+b_n{f}_n$ on $\mathbb{Z}$ with a real compactly supported sequences $(a_n-1{)}_{n\in \mathbb{Z}}$ and $(b_n{)}_{n\in \mathbb{Z}}$. We give the solution of two inverse problems (including characterization): $(a,b)\to \{$zeros of the reflection coefficient$\}$ and $(a,b)\to \{$bound states and resonances$\}$. We describe the set of “iso-resonance operators $J$”, i.e., all operators $J$ with the same resonances and bound states.