WKB Estimates for 2×2 Linear Dynamic Systems on Time Scales. (3rd September 2008)

• Record Type: Journal Article
• Authors: Hovhannisyan, Gro
• Other Names: Došlý Ondřej Academic Editor.
• Is Part Of: Advances in difference equations. Volume 2008(2008)
• Publication Date: 2008-09-03
• Subjects: Difference equations -- Periodicals; 515.625
• DOI: 10.1155/2008/712913
• Languages: English
• ISSNs: 1687-1847

Abstract

Abstract : We establish WKB estimates for2 × 2 linear dynamic systems with a small parameterε on a time scale unifying continuous and discrete WKB method. We introduce an adiabatic invariant for2 × 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 asε 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 parameterε . The proof is based on the truncation of WKB series and WKB estimates.