Geometric integrators for the Schrödinger equation:<br/><br/>Splitting and Magnus integrators

Monday, January 12, 2009 - 4:20pm - 4:50pm
EE/CS 3-180
Sergio Blanes (Polytechnical University of Valencia)
The time-dependent Schrödinger equation plays an essential role
to understand non-relativistic atomic and molecular processes.
This is a linear partial differential equation with a very
particular structure. A good model for a problem is usually
given by a Hamiltonian operator, which suffices to
describe the evolution of the system (for given initial conditions)
while preserving many qualitative properties (energy, unitarity, etc.).
Unfortunately, in general, analytical solutions for the equations
are unknown, even for most simple models, and numerical methods
are required.

Some techniques frequently used are spectral decomposition
or spatial discretisation. In general, one has to solve a system of
linear ordinary differential equations.
Standard numerical methods do not preserve the qualitative properties
mentioned and usually have a significant error propagation along
the integration. Then, to get accurate and reliable results can
be computationally very expensive. Geometric numerical integration has
been developed during the last years and it intends to build
numerical methods which preserve most qualitative properties
of the exact solution. Some of these methods are developed
for problems with similar structure to the Schrödinger equation leading
in many cases to improved qualitative and quantitative results.
In this talk we review two families of methods: Magnus integrators [1]
(for non-autonomous problems) and splitting methods [2] (for systems
which are separable in solvable parts).

[1] S. Blanes, F. Casas, J.A. Oteo and J. Ros, The Magnus and expansion
and some of its applications. Physics Reports. In Press.

[2] S. Blanes, F. Casas, and A. Murua, Splitting and composition methods in the
numerical integration of differential equations, (arXiv:0812.0377v1).
MSC Code: