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