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A Weyl-Titchmarsh type formula for a discrete Schrödinger operator with Wigner-von Neumann potential

100%
Janas J., Simonov S.
Studia Mathematica
|
2010
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tom 201
|
nr 2
167-189
EN
We consider a discrete Schrödinger operator 𝒥 with Wigner-von Neumann potential not belonging to l². We find the asymptotics of orthonormal polynomials associated to 𝒥. We prove a Weyl-Titchmarsh type formula, which relates the spectral density of 𝒥 to a coefficient in the asymptotics of the orthonormal polynomials.
2
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Unbounded Jacobi Matrices with Empty Absolutely Continuous Spectrum

100%
Cojuhari P., Janas J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2008
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tom 56
|
nr 1
39-51
EN
Sufficient conditions for the absence of absolutely continuous spectrum for unbounded Jacobi operators are given. A class of unbounded Jacobi operators with purely singular continuous spectrum is constructed as well.
3
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Selfadjoint operator matrices with finite rows

100%
Janas J., Stochel J.
Annales Polonici Mathematici
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1997
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tom 66
|
nr 1
155-172
EN
A generalization of the Carleman criterion for selfadjointness of Jacobi matrices to the case of symmetric matrices with finite rows is established. In particular, a new proof of the Carleman criterion is found. An extension of Jørgensen's criterion for selfadjointness of symmetric operators with "almost invariant" subspaces is obtained. Some applications to hyponormal weighted shifts are given.
4
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Spectral analysis of unbounded Jacobi operators with oscillating entries

100%
Janas J., Moszyński M.
Studia Mathematica
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2012
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tom 209
|
nr 2
107-133
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.
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