Trace formulas and inverse spectral theory for Jacobi operators (Q1265482)

The author considers Jacobi operators on \(l^2 (\mathbb{Z})\) associated with the difference expression \[ (\tau f)(n) = a(n) f(n+1) + a(n-1) f(n-1) + b(n) f(n), \] where \(a, b \in l(\mathbb{Z})\), \(a(n) \in \mathbb{R} \setminus \{ 0 \}\), \(b(n) \in \mathbb{R}\), \(n \in \mathbb{Z}\), \(l(\mathbb{Z})\) is the set of complex-valued sequences and \(l^p (\mathbb{Z})\) the set of sequences \(u \in l(\mathbb{Z})\) such that \(| u| ^p\) is summable over \(\mathbb{Z}.\) Using asymptotic expansions and Herglotz properties of Green and Weyl \(m\)-functions infinite series of trace formulas for Jacobi operators are derived. A simple recursive method of reconstructing the sequences \(a^2, b\) when the Weyl matrix is known for one fixed \(n \in \mathbb{Z}\) and new uniqueness results are presented. These results are applied to the inverse spectral problem of a class of reflectionless Jacobi operators and to scattering theory with periodic backgrounds.