Spectral analysis of unbounded Jacobi operators with oscillating entries

Jan Janas, and Marcin Moszyński. "Spectral analysis of unbounded Jacobi operators with oscillating entries." Studia Mathematica 209.2 (2012): 107-133. <http://eudml.org/doc/286625>.

@article{JanJanas2012,
abstract = {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 OP family of Jacobi operators with weight and diagonal sequences wₙ, qₙ, where $wₙ = n^\{α\} + rₙ$, 0 < α < 1 and rₙ, qₙ are given by “essentially oscillating” weighted Stolz D² 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.},
author = {Jan Janas, Marcin Moszyński},
journal = {Studia Mathematica},
keywords = {Jacobi matrix/operator; spectral analysis; absolutely continuous spectrum; point spectrum; discrete spectrum; essential spectrum; subordination theory; asymptotic methods; Levinson theorem; oscillating sequences},
language = {eng},
number = {2},
pages = {107-133},
title = {Spectral analysis of unbounded Jacobi operators with oscillating entries},
url = {http://eudml.org/doc/286625},
volume = {209},
year = {2012},
}

TY - JOUR
AU - Jan Janas
AU - Marcin Moszyński
TI - Spectral analysis of unbounded Jacobi operators with oscillating entries
JO - Studia Mathematica
PY - 2012
VL - 209
IS - 2
SP - 107
EP - 133
AB - 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 OP family of Jacobi operators with weight and diagonal sequences wₙ, qₙ, where $wₙ = n^{α} + rₙ$, 0 < α < 1 and rₙ, qₙ are given by “essentially oscillating” weighted Stolz D² 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.
LA - eng
KW - Jacobi matrix/operator; spectral analysis; absolutely continuous spectrum; point spectrum; discrete spectrum; essential spectrum; subordination theory; asymptotic methods; Levinson theorem; oscillating sequences
UR - http://eudml.org/doc/286625
ER -