Spectral properties of Jacobi matrices by asymptotic analysis Original Research Article

We consider two classes of Jacobi matrix operators in l2 with zero diagonals and with weights of the form nα+cn for 0<α⩽1 or of the form nα+cnnα−1 for α>1, where {cn} 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.