Controllability of schrödinger equations

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In the last few decades, developments in physics, chemistry, and engineering, such as the ever-increasing miniaturization of devices, the emergence of the quantum-computation paradigm, or the desire to influence molecular reactions and to manipulate condensates of ultra-cold atoms (Bose-Einstein condensates), have triggered a growing interest in controlling systems at the quantum scale; i.e., systems that are critically influenced by quantum effects. This research project is aimed at contributing to the theory of the emerging field of "quantum engineering'' by investigating the control properties of the fundamental equation of quantum theory, the Schrödinger equation (and its nonlinear variants). The basic question is whether a given control objective (such as the preparation of an excited or superposition state, the transfer of a Bose-Einstein condensate, or the increase in the yield of a molecular reaction) can be realized by applying suitably designed control fields (usually laser fields) to the system. Although the mathematical control theory for partial differential equations is well developed and many equations arising in physics and engineering have been studied in great detail, the theory for Schrödinger's equation is less complete. This research project is aimed at closing this gap. The results of this research may potentially have implications for areas such as semiconductor technology, nanoscience, or quantum computing.