Spectral properties of some Jacobi matrices with double weights

We study the spectral properties of Jacobi matrices with the weights satisfying λ2n−1 = λ2n = na or λ2n = λ2n+1 = na,a >0.We show that for a = 1 these are cases of spectral phase transitions in a. We use a new method of estimating transfer matrix products to describe the absolutely continuous part of these operators. For a = 1 the existence of a spectral gap is proved. We also show how the results for double weights can be used for the spectral analysis of the Jacobi matrices related to some birth and death processes, previously studied by Janas and Naboko.  2003 Elsevier Science (USA). All rights reserved