Spectral analysis of unbounded Jacobi operators with oscillating entries
Tom 209 / 2012
Streszczenie
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.
