Magnus expansion

In mathematics and physics, the Magnus expansion, named after Wilhelm Magnus (1907–1990), provides an exponential representation of the solution of a first order linear homogeneous differential equation for a linear operator. In particular it furnishes the fundamental matrix of a system of linear ordinary differential equations of order \scriptstyle n with varying coefficients. The exponent is built up as an infinite series whose terms involve multiple integrals and nested commutators.

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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

Y^{\prime}(t)=A(t)Y(t),\qquad\qquad Y(t_0)=Y_{0}

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

When n = 1, the solution reads

Y(t)= \exp \left( \int_{t_0}^{t}A(s)ds \right) \, Y_{0}.

This is still valid for n > 1 if the matrix A(t) satisfies A(t_1)A(t_2)=A(t_2)A(t_1) for any pair of values of t, t1 and t2. In particular, this is the case if the matrix A 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 \Omega(t,t_0),

Y(t)=\exp \left( \Omega (t,t_0)\right) \, Y_0

which is subsequently constructed as a series expansion,

\Omega(t)=\sum_{k=1}^{\infty}\Omega_{k}(t),

where, for the sake of simplicity, it is customary to write down \Omega (t) for \Omega (t,t_0) 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

\Omega_1(t) =\int_0^t A(t_1)~\text{d}t_1,
\Omega_2(t) =\frac{1}{2}\int_0^t \text{d}t_1 \int_0^{t_1} \text{d}t_2\ \left[ A(t_1),A(t_2)\right]
\Omega_3(t) =\frac{1}{6} \int_0^t \text{d}t_1 \int_0^{t_{1}}\text{d} t_2 \int_0^{t_{2}} \text{d}t_3 \ (\left[ A(t_1),\left[ A(t_2),A(t_3)\right] \right] %2B\left[ A(t_3),\left[ A(t_2),A(t_{1})\right] \right] )
\Omega_4(t) =\frac{1}{12} \int_0^t \text{d}t_1 \int_0^{t_{1}}\text{d} t_2 \int_0^{t_{2}} \text{d}t_3 \int_0^{t_{3}} \text{d}t_4 \ (\left[\left[\left[A_1,A_2\right], A_3\right],A_4\right]%2B \left[A_1,\left[\left[A_2,A_3\right],A_4\right]\right]%2B \left[A_1,\left[A_2,\left[A_3,A_4\right]\right]\right]%2B \left[A_2,\left[A_3,\left[A_4,A_1\right]\right]\right] )

where \scriptstyle \left[ A,B\right] \equiv AB-BA is the matrix commutator of A and B.

These equations may be interpreted as follows: \Omega_1(t) 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).

Convergence of the expansion

From a mathematical point of view, the convergence problem is the following: given a certain matrix A(t), when can the exponent \Omega(t) be obtained as the sum of the Magnus series? A sufficient condition for this series to converge for \scriptstyle t \in [0,T) is

\int_0^T \|A(s)\| ds < \pi

where \scriptstyle \| \cdot \| denotes a matrix norm. This result is generic, in the sense that one may consider specific matrices A(t) for which the series diverges for any t > T.

Magnus generator

It is possible to design a recursive procedure to generate all the terms in the Magnus expansion. Specifically, with the matrices \textstyle S_n^{(k)} defined recursively through

S_{n}^{(j)} =\sum_{m=1}^{n-j}\left[ \Omega_{m},S_{n-m}^{(j-1)}\right],\qquad\qquad 2\leq j\leq n-1
S_{n}^{(1)} =\left[ \Omega _{n-1},A \right] ,\qquad S_{n}^{(n-1)}= \mathrm{ad} _{\Omega _{1}}^{n-1} (A)

one has

\Omega _{1} =\int_{0}^{t}A (\tau )d\tau
\Omega _{n} =\sum_{j=1}^{n-1}\frac{B_{j}}{j!}\int_{0}^{t}S_{n}^{(j)}(\tau)d\tau ,\qquad\qquad n\geq 2.

Here \mathrm{ad}^k is a shorthand for an iterated commutator,

\mathrm{ad}_{\Omega}^0 A = A, \qquad \mathrm{ad}_{\Omega}^{k%2B1} A = [ \Omega, \mathrm{ad}_{\Omega}^k A ],

and B_j are the Bernoulli numbers.

When this recursion is worked out explicitly, it is possible to express \Omega_n as a linear combination of n-fold integrals of n-1 nested commutators containing n matrices A,

\Omega_n(t) = \sum_{j=1}^{n-1} \frac{B_j}{j!} \, \sum_{ k_1 %2B \cdots %2B k_j = n-1 \atop k_1 \ge 1, \ldots, k_j \ge 1} \, \int_0^t \, \mathrm{ad}_{\Omega_{k_1}(\tau )} \, \mathrm{ad}_{\Omega_{k_2}(\tau )} \cdots \, \mathrm{ad}_{\Omega_{k_j}(\tau )} A(\tau ) \, d\tau \qquad n \ge 2,

an expression that becomes increasingly intricate with n.

Applications

Since the 1960s, the Magnus expansion has been successfully applied as a perturbative tool in numerous areas of physics and chemistry, from atomic and molecular physics to nuclear magnetic resonance and quantum electrodynamics. It has been also used since 1998 as a tool to construct practical algorithms for the numerical integration of matrix linear differential equations. As they inherit from the Magnus expansion the preservation of qualitative traits of the problem, the corresponding schemes are prototypical examples of geometric numerical integrators.

See also

References