Spectral analysis of unbounded Jacobi operators with oscillating entries

Jan Janas¹ ; Marcin Moszyński²

¹Instytut Matematyczny Polska Akademia Nauk Św. Tomasza 30 31-027 Kraków, Poland
²Wydział Matematyki, Informatyki i Mechaniki Uniwersytet Warszawski Banacha 2 02-097 Warszawa, Poland

Studia Mathematica, Tome 209 (2012) no. 2, pp. 107-133

Résumé

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 family of Jacobi operators with weight and diagonal sequences $\{w_n\}$, $\{q_n\}$, where $w_n=n^{\alpha}+r_n$, $0\alpha1$ 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.

Détail

DOI : 10.4064/sm209-2-2

Keywords: describe spectra jacobi operators irregular entries divide mathbb three spectral regions using subordinacy method asymptotic methods based particular discrete versions levinsons theorem prove absolute continuity first region pure pointness second third region information given above methods call uncertainty region illustration introduce analyse family jacobi operators weight diagonal sequences where alpha alpha given essentially oscillating weighted stolz sequences mixed periodic sequences particular limit point set typically infinite family get extra information subsets uncertainty region contained essential spectrum subsets pure point region contained discrete spectrum