Advances in Difference Equations
Volume 2008 (2008), Article ID 712913, 12 pages
Research Article

WKB Estimates for 2×2 Linear Dynamic Systems on Time Scales

Gro Hovhannisyan

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.


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.