On the solution of linear differential equations in Lie groups

Abstract

The subject matter of this paper is the solution of the linear differential equation y = a(t)y, y(0) = y 0 , where y 0 ∈ G, a(•) : R + → g and g is a Lie algebra of the Lie group G. By building upon an earlier work of Wilhelm Magnus, we represent the solution as an infinite series whose terms are indexed by binary trees. This relationship between the infinite series and binary trees leads both to a convergence proof and to a constructive computational algorithm. This numerical method requires the evaluation of a large number of multivariate integrals, but this can be accomplished in a tractable manner by using quadrature schemes in a novel manner and by exploiting the structure of the Lie algebra.

Numerical integrators for quantum dynamics close to the adiabatic limit

“…To this purpose, we choose better approximations to η(t n + θh) and possibly to W (t n + θh) in (11) instead of just freezing their values at t n as was done in (12). Reinserting η(t n + θh) in (11) once again from an integrated form of (8) leads to…”

On Cayley-Transform Methods for the Discretization of Lie-Group Equations

“…C], m ≥ 1, [12], [13], [20]. Although (1.4) is considerably more complicated than the Liegroup equation (1.3), it has a crucial and most welcome attribute: unlike a Lie group G, the Lie algebra g is a linear space.…”

“…Several discretization methods employ the above approach: Runge-KuttaMunthe-Kaas schemes [20], Fer and Magnus expansions [11], [13], [18]. Moreover, a similar approach can be generalized to homogeneous spaces [21].…”

“…Similarly to the theory of Magnus expansions in [13], we identify terms in a Cayley expansion with rooted trees, which can be constructed recursively. Each such term is an integral over a polytope but all such integrals can be evaluated to high order by using special quadrature formulas similar to the construction in [13]. Truncated Cayley expansions (with exact integrals) need not be time-symmetric, hence the method does not display the usual advantages associated with time symmetry, e.g., even order of approximation.…”

The Second Lie-Group $SO_o(n,1)$ Used to Solve Ordinary Differential Equations

“…One is using the Lie group property and another is employing the Lie algebra structure, of which the geometric integrators developed by Hairer, Lubich and Wanner (2002), Iserles (1984), Iserles and Nørsett (1999), Iserles, Munthe-Kaas, Nørsett and Zanna (2000), Lee and Liu (2009), Munthe-Kaas (1998, 1999, and Deng (2004, 2006) can be referred.…”