Scattering and Perturbation Theory for Discrete-Time Dynamics

Perturbation Theory and Discrete Hamiltonian Dynamics

In this paper we discuss a weak version of KAM theory for symplectic maps which arise from the discretization of the minimal action principle. These maps have certain invariant sets, the Mather sets, which are the generalization of KAM tori in the non-differentiable case. These sets support invariant measures, the Mather measures, which are action minimizing measures. We generalize viscosity so...

Scattering theory and discrete-time quantum walks

We study quantum walks on general graphs from the point of view of scattering theory. For a general finite graph we choose two vertices and attach one half line to each, and consider walks that proceed from one half line, through the graph, to the other. The probability of starting on one line and reaching the other after n steps can be expressed in terms of the transmission amplitude for the g...

Discrete wavelets and perturbation theory

We show with the help of examples that discrete wavelets can be a useful tool in perturbation theory of finite-dimensional quantum Hamilton systems. PACS numbers: 02.30.Nw, 02.70.−c In perturbation theory the Hamilton operator Ĥ is given by Ĥ = Ĥ 0 + Ĥ 1 where Ĥ 0 and Ĥ 1 are self-adjoint operators in a Hilbert space [1]. It is assumed that the perturbation Ĥ 1 is relatively ‘small’ in comparis...

Dynamics of a discrete-time plant-herbivore model

Time-Dependent Perturbation Theory

Our starting point is the set of eigenstates n of the unperturbed Hamiltonian 0 n H n E n = , notice we are not labeling with a zero, no 0 n E , because with a time-dependent Hamiltonian, energy will not be conserved, so it is pointless to look for energy corrections. What happens instead, provided the perturbation is not too large, is that the system makes transitions between the eigenstates n...