Levinson Theorem for 2 x 2 System and Applications to the Asymptotic Stability and Schrodinger Equation
by
Gro Hovhannisyan
Kent State University, Stark

We prove new asymptotical stability and instability theorems for non autonomous 2 x 2 system of first-order linear differential equations by using a new version of the classical Levinson asymptotic theorem for 2 x 2 systems. The proof of this version is based on the construction of approximate fundamental solution of the original system in the special form with unknown phase functions and error estimates formulated in the terms of generalized characteristic functional. In the case of system with constant coefficients generalized characteristic functional turns to the usual characteristic polynomial and by choosing phase functions as eigenvalues of the corresponding matrix the error could be eliminated. As another application we derive a transition probability formula for the two level atom in the external electromagnetic field described by Schrodinger system.

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Date received: August 15, 2007