Magnus Expansion - Magnus Approach and Its Interpretation

Magnus Approach and Its Interpretation

Given the n × n coefficient matrix A(t) we want to solve the initial value problem associated with the linear ordinary differential equation

for the unknown n-dimensional vector function Y(t).

When n = 1, the solution reads

This is still valid for n > 1 if the matrix A(t) satisfies for any pair of values of t, t₁ and t₂. In particular, this is the case if the matrix is constant. In the general case, however, the expression above is no longer the solution of the problem.

The approach proposed by Magnus to solve the matrix initial value problem is to express the solution by means of the exponential of a certain n × n matrix function ,

which is subsequently constructed as a series expansion,

where, for simplicity, it is customary to write down for and to take t₀ = 0. The equation above constitutes the Magnus expansion or Magnus series for the solution of matrix linear initial value problem.

The first four terms of this series read

where is the matrix commutator of A and B.

These equations may be interpreted as follows: coincides exactly with the exponent in the scalar (n = 1) case, but this equation cannot give the whole solution. If one insists in having an exponential representation the exponent has to be corrected. The rest of the Magnus series provides that correction.

In applications one can rarely sum exactly the Magnus series and has to truncate it to get approximate solutions. The main advantage of the Magnus proposal is that, very often, the truncated series still shares with the exact solution important qualitative properties, at variance with other conventional perturbation theories. For instance, in classical mechanics the symplectic character of the time evolution is preserved at every order of approximation. Similarly the unitary character of the time evolution operator in quantum mechanics is also preserved (in contrast to the Dyson series).