  We consider the Jacobi operator $(Jf)_n= a_{n-1}f_{n-1}+a_nf_{n+1}+b_nf_n$ on
$\Z$ with a real compactly supported sequences $(a_n-1)_{n\in\Z}$ and
$(b_n)_{n\in\Z}$. 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.
