Spectral properties of Jacobi matrices by asymptotic analysis

We consider two classes of Jacobi matrix operators in l² with zero diagonals and with weights of the form n^α+c_n for 0<α1 or of the form n^α+c_nn^α−1 for α>1, where {c_n} is periodic. We study spectral properties of these operators (especially for even periods), and we find asymptotics of some of their generalized eigensolutions. This analysis is based on some discrete versions of the Levinson theorem, which are also proved in the paper and may be of independent interest.