Marcin Moszyński

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.

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

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.

New discrete Levinson type asymptotics of solutions of linear systems

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.

Topological chaos: When topology meets medicine

In the present note, the chaotic behaviour of a class of infinite system of linear ODEs (with variable coefficients) describing the population of neoplastic cells divided into subpopulations characterized by different levels of resistance to drugs is discussed. The result of [1] is extended to a wider class of sequences defining the parameters of the system.

A discrete Levinson theorem for systems with singular limit and estimates of generalized eigenvectors of some Jacobi operators

We prove a discrete Levinson type theorem on asymptotic properties of solutions of discrete d-dimensional systems with being bounded variation sequence and with the perturbation being ‘appropriately small’, for the case when has zero eigenvalue. We use this result to obtain estimates for some generalized eigenvectors of Jacobi operators. In particular we get estimates of the eigenvectors decay for Jacobi matrices with ‘shaky weights’.

A quantum version of the Nekhoroshev theorem

Using a simple method, it is proved that the quantum propagator for some polynomially perturbed harmonic oscillators is close to the propagator for the unperturbed oscillators for a small coupling constant and arbitrarily large times.