Advances in Difference Equations
Volume 2008 (2008), Article ID 712913, 12 pages
doi:10.1155/2008/712913
Research Article
WKB Estimates for 2×2 Linear Dynamic Systems on Time Scales
Kent State University, Stark Campus, 6000 Frank Avenue NW, Canton, OH 44720-7599, USA
Received 3 May 2008; Accepted 26 August 2008
Academic Editor: Ondřej Došlý
Copyright © 2008 Gro Hovhannisyan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We establish WKB estimates for 2×2 linear dynamic systems
with a small parameter ε on a time scale unifying continuous and discrete
WKB method. We introduce an adiabatic invariant for 2×2 dynamic system
on a time scale, which is a generalization of adiabatic invariant of Lorentz's
pendulum. As an application we prove that the change of adiabatic invariant
is vanishing as ε approaches zero. This result was known before only for a
continuous time scale. We show that it is true for the discrete scale only for
the appropriate choice of graininess depending on a parameter ε. The proof is
based on the truncation of WKB series and WKB estimates.