关于无限维中的量子控制问题
On the problem of quantum control in infinite dimensions
摘要：
在薛定谔方程的双线性控制框架下，利用有界控制算子，证明了${\cal S}\cap {\cal H}^{2}$中的可达集有稠密补.因此，在这种情况下，在无限维中精确的量子控制是不可能的。另一方面，它是已知的，有一个简单的选择的运营商，当应用到一个任意的状态，产生密集的轨道在希尔伯特空间。本文建立了这两个结果的相容性，特别证明了双线性控制的有界算子可达集的闭包在${\cal S}\cap {\cal H}^{2}$中是稠密的。在无限维中对可控性的要求也与无限维酉群的性质有关。
英文摘要：
In the framework of bilinear control of the Schrödinger equation with bounded control operators, it has been proved that the reachable set has a dense complemement in ${\cal S}\cap {\cal H}^{2}$. Hence, in this setting, exact quantum control in infinite dimensions is not possible. On the other hand it is known that there is a simple choice of operators which, when applied to an arbitrary state, generate dense orbits in Hilbert space. Compatibility of these two results is established in this paper and, in particular, it is proved that the closure of the reachable set of bilinear control by bounded operators is dense in ${\cal S}\cap {\cal H}^{2}$. The requirements for controllability in infinite dimensions are also related to the properties of the infinite dimensional unitary group.
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