Advances in Difference Equations
Volume 2008 (2008), Article ID 712913, 12 pages
doi:10.1155/2008/712913

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

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.