Spectral analysis of unbounded Jacobi operators with oscillating entries
We describe the spectra of Jacobi operators J with some irregular entries. We divide R into three “spectral regions” for J and using the subordinacy method and asymptotic methods based on some particular discrete versions of Levinson’s theorem we prove the absolute continuity in the first region and the pure pointness in the second. In the third region no information is given by the above methods, and we call it the “uncertainty region”. As an illustration, we introduce and analyse the O&P family of Jacobi operators with weight and diagonal sequences {wn}, {qn}, where wn = n+rn, 0 < α < 1 and {rn}, {qn} are given by “essentially oscillating” weighted Stolz D sequences, mixed with some periodic sequences. In particular, the limit point set of {rn} is typically infinite then. For this family we also get extra information that some subsets of the uncertainty region are contained in the essential spectrum, and that some subsets of the pure point region are contained in the discrete spectrum. 0. Introduction. In this work we are concerned with spectral properties of a new family of unbounded self-adjoint Jacobi operators acting in `2(N). Most papers dealing with unbounded Jacobi operators are restricted to “regular” sequences of weights and diagonals. However, recently, several works concerning the “irregular” case have appeared: see e.g. [10], [7], [11], [5], [4], [18]. The irregular sequences in those works were mostly given as periodic perturbations (modulations) of regular ones, but also more irregular behaviour of the entries has been studied lately in [2], [3], [12] and [15]. The main goal of studying various kinds of deformations of regular unbounded entries is to illustrate and understand the somewhat delicate influence of such deformations on spectral properties of the operator. This general idea of deformations allowed several examples with new spectral properties to be constructed. In particular, in [2] concrete classes of weights defining Jacobi operators having a few gaps in the essential spectrum were found. 2010 Mathematics Subject Classification: 47B36, 47B39, 47B25, 39A22, 47A10, 81Q10.