Appendix A: the general solution to the evolution operator
We intend in this appendix to give a self-contained proof of Magnus' exponential solution to the equation for the evolution operator. The original proof is to be found in Magnus' paper(1954). Here we will give a short but complete proof introducing in the meanwhile the necessary tools used while working with Magnus' expressions and to get acquainted with algebraic manipulations inherent to problems involving non-commutative operators. Magnus makes use of an original technique to manage the derivative of the exponential of a linear operator (a square matrix in our case). He transforms this derivative into an algebraical expression, and uses it to solve the differential equation.
In what follows w, x, y and z stand for such operators also called Lie elements . The Lie product of two Lie elements x and y is defined by
where w is a new Lie element. This product is usually called commutator . Now an abreviated notation is used for the l-fold Lie product by x of y as
where we remind that the exponential of an operator x is defined as
And we are ready to demonstrate the first two formulas which we will use hereafter.
This formula arises from the straightforward calculation of its left-hand-side. We begin calculating the effect of on the left of the exponential:
Next we multiply on the left by :
so Formula 1 is demonstrated.
The left-hand-side of Formula 2 is obtained by multiplying the left-hand-side of Formula 1 on the left by and on the right by . If we do the same multiplications in the right-hand-side of Formula 1 we obtain
or expanding the right-hand-side
which can be seen to correspond to the expansion of
Next, Magnus demonstrates what can be called
The Magnus' Inversion Lemma: Let and be two power series inxwhich satisfy
Then each of the equations
is a consequence of the other one. Let and , then by hypothesis
Now we can obtain the following equivalent expressions
where we have used notation (A2). Next we can separate indexes and write
If we suppose now that , immediately we obtain that
The inverse implication is completely equivalent.
With the Inversion Lemma and Formula 2, we have all the instruments to solve the equation for the evolution operator
Exponential solution Theorem (Magnus): Let be a known function of t in an associative ring (for our purposes it is a matrix), and let be an unknown function (in our case the evolution operator) satisfying
where is the identity matrix. Then, if certain unspecified conditions of convergence are satisfied, can be written in the form
The vanish for , and , where the (for ) are the Bernoulli numbers.
Let us suppose , then
By using Formula 2 with and , one obtains
which compared with Eq. (A3) gives
We can now apply the Inversion Lemma with the same substitutions in x and y as before, and with and
This expression can be finally expanded using the following power series
where the has the given values.
To integrate the resulting equation for we start at t=0, where to satisfy boundary conditions. Introducing this solution into the equation we obtain a new solution:
We can iterate the procedure to obtain , , ... as
The solution is obtained as the limit of this series:
© European Southern Observatory (ESO) 1999
Online publication: December 22, 1998