# 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).

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:

35Q41

Keywords: