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用一系列不同的物理模型来描述，这些模型的数学复杂性不同，并包含了不同时间和长度尺度上的现象。从量子动力学方程出发，严格推导简化模型(流体力学模型和漂移扩散模型)，研究这些模型之间的数学相互作用。了解这些尺度极限对于简化模型的有效性范围(和局限性)提供了重要的信息。第二个主要目的是为这些方程(Wigner-Fokker-Planck，Madelung，Schrödinger-type，量子漂移扩散)发展有效的数值方法，以便在二维中模拟量子波导和无线电传输问题。这包括对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。这将允许识别这种半导体器件中可以使用简化模型(因此在数值上更便宜)的区域，以及必须使用完整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.