Spectral theory for a class of periodically perturbed unbounded Jacobi matrices: elementary methods

We use elementary methods to give a full characterization of the spectral properties of unbounded Jacobi matrices with zero diagonal and off-diagonal entries of the type λn=nα+cn, where 12<α⩽1 and cn is a real periodic sequence. The spectral properties depend strongly on the parity of the minimal period of cn. The methods used are asymptotic diagonalization techniques, including the finite difference version of Levinsonʹs theorem, subordinacy theory, and the variational principle.