Spectral analysis of unbounded Jacobi operators with oscillating entries

• Autorzy: Jan Janas , Marcin Moszyński
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 47B39: Difference operators
• 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis
• 47B25: Symmetric and selfadjoint operators (unbounded)
• 39A22: Growth, boundedness, comparison of solutions
• 47B36: Jacobi (tridiagonal) operators (matrices) and generalizations
• 47A10: Spectrum, resolvent

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2012
• Tom: 209
• Numer: 2
• Strony: 107-133

Daty

wydano

2012

Twórcy

autor

Jan Janas

• Instytut Matematyczny, Polska Akademia Nauk, Św. Tomasza 30, 31-027 Kraków, Poland

autor

Marcin Moszyński

• Wydział Matematyki, Informatyki i Mechaniki, Uniwersytet Warszawski, Banacha 2, 02-097 Warszawa, Poland

Identyfikatory

DOI

10.4064/sm209-2-2