## Spectral analysis of unbounded Jacobi operators with oscillating entries

### Volume 209 / 2012

#### Abstract

We describe the spectra of Jacobi operators $J$ with some
irregular entries. We divide $\mathbb R$ 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 $\{w_n\}$, $\{q_n\}$, where $w_n=n^{\alpha}+r_n$,
$0<\alpha<1$ and $\{r_n\}$, $\{q_n\}$ are given by “essentially
oscillating” weighted Stolz $D^2$ sequences, mixed
with some periodic sequences. In particular, the limit point set of
$\{r_n\}$ 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.