Numerik und Asymptotik mikroskopischer und makroskopischer Gleichungen für Quantensysteme

出现在量子光学（激光器）、半导体、电磁波和声波传播中的量子力学系统可以通过一系列物理模型来描述，这些模型在数学复杂性上不同，并包含各种时间和长度尺度上的现象。建议通过严格推导简化的（流体动力学和漂移扩散型）量子动力学方程模型来研究这些模型的数学相互作用。了解这些尺度限制为简化模型的有效性范围（和限制）提供了重要信息。第二个主要目标是为这些方程（Wigner-Fokker-Planck，Madelung，Schrödinger型，量子漂移扩散）开发有效的数值方法，以模拟2D中的量子波导和无线电传输问题。这包括边界条件的仔细处理。最终目标是对量子二极管上的各种模型进行数值比较。这将允许识别这样的半导体器件的区域，其中可以使用简化的（因此在数值上更便宜的）模型，以及必须采用全量子动力学模型的区域。

Quantum mechanical systems appearing in quantum optics (lasers), semiconductors, electro-magnetic and acoustic wave propagation can be described by a hierarchy of physical models that differ in mathematical complexity and incorporate phenomena on various time and length scales. It is proposed to investigate the mathematical interplay of these models by rigorously deducing simplified (hydrodynamic- and drift-diffusiontype) models from quantum kinetic equations. Understanding these scaling limits gives important information for the range of validity (and the limitations) of reduced models.The second main aim is to develop efficient numerical methods for these equations (Wigner-Fokker-Planck, Madelung, Schrödinger-type, quantum drift-diffusion) in order to simulate quantum waveguides and radio-transmission problems in 2D. This includes a careful treatment of the boundary conditions.The ultimate goal is a numerical comparison of various models on a quantum diode. This will allow to identify regions of such a semiconductor device where simplified (and hence numerically cheaper) models can be used, and areas where the full quantum kinetic model has to be employed.