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Slowly oscillating perturbations of periodic Jacobi operators in l²(ℕ)

100%

Moszyński M.

Studia Mathematica

2009

tom 192

nr 3

259-279

We prove that the absolutely continuous part of the periodic Jacobi operator does not change (modulo unitary equivalence) under additive perturbations by compact Jacobi operators with weights and diagonals defined in terms of the Stolz classes of slowly oscillating sequences. This result substantially generalizes many previous results, e.g., the one which can be obtained directly by the abstract trace class perturbation theorem of Kato-Rosenblum. It also generalizes several results concerning perturbations of the discrete (free or periodic) Schrödinger operator. The paper concerns "one-sided" Jacobi operators (i.e. in l²(ℕ)) and is based on the method of subordinacy. We provide some spectral results for the unperturbed, periodic case, and also an appendix containing some subordination theory tools.

Spectral analysis of unbounded Jacobi operators with oscillating entries

63%

Janas J., Moszyński M.

Studia Mathematica

2012

tom 209

nr 2

107-133

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 $wₙ = n^{α} + rₙ$, 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.

Ograniczanie wyników

2 Studia Mathematica

2 Moszyński M.

1 Janas J.

1 2012

1 2009

Wyniki wyszukiwania

Slowly oscillating perturbations of periodic Jacobi operators in l²(ℕ)

Spectral analysis of unbounded Jacobi operators with oscillating entries