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Astron. Astrophys. 342, 201-211 (1999)

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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

[EQUATION]

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

[EQUATION]

with [FORMULA]. With this notation, it is easy to show by a straightforward calculation that

[EQUATION]

where we remind that the exponential of an operator x is defined as

[EQUATION]

Following Magnus, we extend the previous notation to a polynom [FORMULA] in an evident form:

[EQUATION]

where

[EQUATION]

And we are ready to demonstrate the first two formulas which we will use hereafter.

Formula 1

[EQUATION]

This formula arises from the straightforward calculation of its left-hand-side. We begin calculating the effect of [FORMULA] on the left of the exponential:

[EQUATION]

Next we multiply on the left by [FORMULA]:

[EQUATION]

Now,

[EQUATION]

so Formula 1 is demonstrated.

Formula 2

[EQUATION]

The left-hand-side of Formula 2 is obtained by multiplying the left-hand-side of Formula 1 on the left by [FORMULA] and on the right by [FORMULA]. If we do the same multiplications in the right-hand-side of Formula 1 we obtain

[EQUATION]

or expanding the right-hand-side

[EQUATION]

where

[EQUATION]

which can be seen to correspond to the expansion of

[EQUATION]

Next, Magnus demonstrates what can be called

The Magnus' Inversion Lemma: Let [FORMULA] and [FORMULA] be two power series inxwhich satisfy

[EQUATION]

Then each of the equations

[EQUATION]

is a consequence of the other one. Let [FORMULA] and [FORMULA], then by hypothesis

[EQUATION]

Now we can obtain the following equivalent expressions

[EQUATION]

where we have used notation (A2). Next we can separate indexes and write

[EQUATION]

If we suppose now that [FORMULA], immediately we obtain that

[EQUATION]

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 [FORMULA] be a known function of t in an associative ring (for our purposes it is a matrix), and let [FORMULA] be an unknown function (in our case the evolution operator) satisfying

[EQUATION]

where [FORMULA] is the identity matrix. Then, if certain unspecified conditions of convergence are satisfied, [FORMULA] can be written in the form

[EQUATION]

where

[EQUATION]

The [FORMULA] vanish for [FORMULA], and [FORMULA], where the [FORMULA] (for [FORMULA]) are the Bernoulli numbers.

Integration of this equation by iteration leads to an infinite series for [FORMULA] the first terms of which (up to terms involving 4 [FORMULA]'s) are

[EQUATION]

Let us suppose [FORMULA], then

[EQUATION]

By using Formula 2 with [FORMULA] and [FORMULA], one obtains

[EQUATION]

which compared with Eq. (A3) gives

[EQUATION]

We can now apply the Inversion Lemma with the same substitutions in x and y as before, and with [FORMULA] and

[EQUATION]

to obtain

[EQUATION]

and

[EQUATION]

This expression can be finally expanded using the following power series

[EQUATION]

where the [FORMULA] has the given values.

To integrate the resulting equation for [FORMULA] we start at t=0, where [FORMULA] to satisfy boundary conditions. Introducing this solution into the equation we obtain a new solution:

[EQUATION]

We can iterate the procedure to obtain [FORMULA], [FORMULA], ... as

[EQUATION]

The solution is obtained as the limit of this series:

[EQUATION]

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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