Controllability of schrödinger equations

在过去的几十年里，物理、化学和工程领域的发展，如设备的不断小型化，量子计算范式的出现，或影响分子反应和操纵超冷原子凝聚物（玻色-爱因斯坦凝聚物）的愿望，引发了人们对量子尺度控制系统的兴趣日益浓厚；也就是说，受量子效应严重影响的系统。本研究项目旨在通过研究量子理论基本方程Schrödinger方程（及其非线性变体）的控制特性，为新兴的“量子工程”领域的理论做出贡献。基本问题是给定的控制目标（如激发态或叠加态的制备，玻色-爱因斯坦凝聚物的转移，或分子反应产率的增加）是否可以通过对系统应用适当设计的控制场（通常是激光场）来实现。虽然偏微分方程的数学控制理论已经发展得很好，许多物理和工程中出现的方程已经得到了非常详细的研究，但Schrödinger方程的理论还不完整。这个研究项目旨在缩小这一差距。这项研究的结果可能会对半导体技术、纳米科学或量子计算等领域产生潜在影响。

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.