### Time dependent (magnetic) Schrödinger equations (Norbert J. Mauser)

[P5, Mauser]

deals with the modeling, (asymptotic) analysis, numerical simulation, and application of time dependent nonlinear (semi-relativistic) Schrödinger (NLS) equations, such as the Gross-Pitaevskii equation (GPE) for Bose Einstein Condensates (BEC) and the Pauli equation, coupled to the magneto-static Maxwell equations for electrons.
The analysis focusses on asymptotic limits of solutions of equations in a model hierarchy, where some appropriately defined small parameter vanishes, e.g. confinement limits to "low dimensional models" or "semiclassical limits" for vanishing Planck's constant.

Numerical methods for NLS are developed or adapted to the time dependent problems considered, extending the time-split-spectral methods already used by the group, including (time) adaptive methods, and implementation on parallel computers. The problem of finite domain of computation is dealt with by Absorbing Boundary Conditions (ABC).

A long term goal is to go beyond simple models approximating the many particle system with a single deterministic NLS, in particular the inclusion of stochastic terms into NLS in order to describe "dissipation"/thermalization of (systems of coupled) BECs.

[P5, Mauser] is a focus point for SFB activities in quantum mechanics, including numerics for Schrödinger equations, tobe integrated in the SFB software platform in collaboration with [P10, Schöberl]. Derivation of low dimensional structures will use the expertise of [P4, Markowich] and [P9, Schmeiser]; for numerics for NLS the expertise of [P6, Melenk] on Transparent BC will be valuable. Numerics for micromagnetism will be studied in collaboration with [P8, Praetorius].

### Members of this project part

Norbert J. Mauser (Project Leader)