Jan Janas

Spectral properties of Jacobi matrices by asymptotic analysis

We consider two classes of Jacobi matrix operators in l2 with zero diagonals and with weights of the form nα + cn for 0 1, where {cn} is periodic. We study spectral properties of these operators (especially for even periods), and we find asymptotics of some of their generalized eigensolutions. This analysis is based on some discrete versions of the Levinson theorem, which are also proved in the paper and may be of independent interest.

Infinite Jacobi Matrices with Unbounded Entries: Asymptotics of Eigenvalues and the Transformation Operator Approach

In this paper the exactasymptotics of eigenvalues

Alternative approaches to the absolute continuity of Jacobi matrices with monotonic weights

\lambda_n (J), \ n \to \infty,

New discrete Levinson type asymptotics of solutions of linear systems

of a class of unbounded self-adjoint Jacobi matrices J with discrete spectrum are given. Their calculation is based on a successive diagonalization approach---a new version of the classical transformation operator method. The approximations of the transformation operator are constructed step by step using a successive diagonalization procedure, which results in higher order approximations of the

The asymptotic analysis of generalized eigenvectors of some Jacobi operators. Jordan box case

\lambda_n (J).

Spectral properties of selfadjoint Jacobi matrices coming from birth and death processes

We present two approaches to the spectral studies for infinite Jacobi matrices with monotonic or “near-to-monotonic” weights. The first one is based on the subordination theory due to Khan and Pearson [17] combined with the detailed analysis of the transfer matrices for the solutions of the formal eigenequation. The second one uses an extension of the commutator approach developed by Putnam in [19]. Applying these methods we prove the absolute continuity for several classes of weights and diagonals. For some other cases we prove the emptiness of the point spectrum. The results are illustrated with examples and compared with the results of Dombrowski [7]-[13], Clark [2] and of Máté and Nevai [18]. We show that some of our results are stronger.

Spectral Theory and Analysis : Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2008, Bedlewo, Poland

We prove new discrete versions of Levinson type theorems describing asymptotic behavior of solutions of systems of linear difference equations. We show that for several cases of equations with coefficients possessing some “essential” oscillations the asymptotics should be also essentially corrected, comparing with the classical Levinsons cases studied, e.g. in [2,5,9]. The results obtained here allow to study the asymptotics for some systems with coefficients which are not necessary convergent. As an illustration, an application to spectral studies of some Jacobi matrices is presented, by using the asymptotics of generalized eigenvectors.

Asymptotics of Generalized Eigenvectors for Unbounded Jacobi Matrices with Power-like Weights, Pauli Matrices Commutation Relations and Cesaro Averaging

This paper presents two methods of finding asymptotic formulae for a basis of solutions of the second order difference equations in the Jordan box case. An application to spectral analysis of Jacobi operators is also sketched.

Asymptotic Behavior of Generalized Eigenvectors of Jacobi Matrices in the Critical (“Double Root”) Case

Let J be a Jacobi matrix defined in 12 as ReW, where W is a unilateral weighted shift with nonzero weights Ak such that limk Ak = 1. Define the seqences: Ek : = 1, k, k=k := 26k + Ek If Ek = 0(k-a), = k-Y), 2 3/2 and y > 3/4, then J has an absolutely continuous spectrum covering (-2,2). Moreover, the asymptotics of the solution Ju = Au, A E R is also given.