On a spectral property of Jacobi matrices

Let $J$ be a Jacobi matrix with elements $b_k$ on the main diagonal and elements $a_k$ on the auxiliary ones. We suppose that $J$ is a compact perturbation of the free Jacobi matrix. In this case the essential spectrum of $J$ coincides with $[-2,2]$, and its discrete spectrum is a union of two sequences $\{x^\pm_j\}, x^+_j>2, x^-_j<-2$, tending to $\pm2$. We denote sequences $\{a_{k+1}-a_k\}$ and $\{a_{k+1}+a_{k-1}-2a_k\}$ by $\partial a$ and $\partial^2 a$, respectively.

The main result of the note is the following theorem.

Theorem.     Let $J$ be a Jacobi matrix described above and $\sigma$ be its spectral measure. Then $a-1,b\in l^4, \partial^2 a,\partial^2 b \in l^2$ if and only if