Systems on Time Scales

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. Copyright q 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. 1. Adiabatic invariant of dynamic systems on time scales Consider the following system with a small parameter ε> 0 on a time scale: vΔt Atεvt, 1.1 where vΔ is the delta derivative, vt is a 2-vector function, and Atε Aτ a11τ εka12τ ε−ka21τ a22τ, τ tε, k is an integer. 1.2 WKB method 1, 2 is a powerful method of the description of behavior of solutions of 1.1 by using asymptotic expansions. It was developed by Carlini 1817, Liouville, Green 1837 and became very useful in the development of quantum mechanics in 1920 1, 3. The discrete WKB approximation was introduced and developed in 4–8. The calculus of times scales was initiated by Aulbach and Hilger 9–11 to unify the discrete and continuous analysis. In this paper, we are developing WKB approximations for the linear dynamic systems on a time scale to unify the discrete and continuousWKB theory. Our formulas forWKB serie