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Báo cáo hóa học: " Research Article WKB Estimates for 2 × 2 Linear Dynamic Systems on Time Scales"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article WKB Estimates for 2 × 2 Linear Dynamic Systems on Time Scales | Hindawi Publishing Corporation Advances in Difference Equations Volume 2008 Article ID 712913 12 pages doi 10.1155 2008 712913 Research Article WKB Estimates for 2 X 2 Linear Dynamic Systems on Time Scales Gro Hovhannisyan Kent State University Stark Campus 6000 Frank Avenue NW Canton OH 44720-7599 USA Correspondence should be addressed to Gro Hovhannisyan ghovhann@kent.edu Received 3 May 2008 Accepted 26 August 2008 Recommended by Ondrej Dosly We establish WKB estimates for 2 X 2 linear dynamic systems with a small parameter e on a time scale unifying continuous and discrete WKB method. We introduce an adiabatic invariant for 2 X 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 e 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 e. The proof is based on the truncation of WKB series and WKB estimates. 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. 1. Adiabatic invariant of dynamic systems on time scales Consider the following system with a small parameter e 0 on a time scale vA t A t v C 1.1 where vA is the delta derivative v t is a 2-vector function and A tA - A t _ ai1 T e ai2 .T T _ te k is an intep-er C12f Ate AiT e-fcữ21 T a22 r te k 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 .