Absolute continuity for unbounded Jacobi matrices with constant row sums (Q1599123)

In the present paper the authors study two classes of Jacobi matrices. The first one consists of the matrices with zero diagonal \((b_n=0)\) and the off diagonal sequences \(a_n\) which satisfy \[ a_{2n-1}=a_{2n}=n+\varepsilon, \qquad \varepsilon\geq 0, \quad n\geq 1. \] The authors show that the spectral measures of such matrices are absolutely continuous. The second class includes generic Jacobi matrices with \[ a_n=(n+\varepsilon_n)^2, \qquad a_{n-1}+b_n+a_n=0. \] The corresponding spectral measures are proved to have an absolutely continuous part as long as \(\varepsilon_n\) is of bounded variation and \(\varepsilon_n \geq 0\).