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Unformatted text preview: Supplementary notes: Wei-Norman Theorem Consider the linear operator differential equation of the first order dU ( t ) dt = H ( t ) U ( t ) ; U (0) = 1 (1) where H and U are both time-dependent linear operators in a Banach space or a finite-dimensional space. According to the Wei-Norman theorem 1 , if the operator H can be expressed as H ( t ) = N X n =1 a n ( t ) L n , (2) where a n ’s are scalar functions of time and L n are the generators of an N-dimensional solvable Lie algebra or the real split 3-dimensional simple Lie algebra, then the oper- ator U can be expressed as U ( t ) = N Y n =1 exp[ g n ( t ) L n ] . (3) Here the g n ’s are time-dependent scalar functions to be determined. To find the g n ’s, we simply substitute Eq.(2) and Eq.(3) into Eq.(1), and compare the two sides term by term to obtain a set of coupled nonlinear differential equations dg n ( t ) dt = N X m =1 η nm a m ( t ) , g n (0) = 0 (4) where η nm are nonlinear functions of g n ’s. Thus, we have reduced the linear operator’s....
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- Spring '11