开放量子系统的散射理论
Scattering Theory for Open Quantum Systems
摘要：
与环境相互作用的量子系统通常由最大耗散算子或所谓的伪哈密顿算子来建模。本文考虑了这种开放系统的散射理论。首先假设在Hilbert空间$\sH$中用单个极大耗散算子$A_D$描述开放量子系统。在这种情况下，$A_D$的最小自伴伸缩$\widetilde K$可以被看作是包含开放系统$\{A_D,\sH\}$的封闭系统的哈密顿量，但是由于$\widetilde K$不一定从下面是半有界的，所以这个模型很难从物理的角度来解释。本文的第二部分用依赖于能量$μ$的最大耗散算符族$\{A(μ)\}$模拟了一个开放量子系统，并证明了开放系统可以嵌入到哈密顿量是半有界的封闭系统中.令人惊讶的是，原来，相应的散射矩阵可以完全恢复从散射矩阵的单伪哈密顿在第一部分的文件。一般结果被应用到一类Sturm-Liouville算子中产生的耗散和量子传输薛定谔-泊松系统。
英文摘要：
Quantum systems which interact with their environment are often modeled by maximal dissipative operators or so-called Pseudo-Hamiltonians. In this paper the scattering theory for such open systems is considered. First it is assumed that a single maximal dissipative operator $A_D$ in a Hilbert space $\sH$ is used to describe an open quantum system. In this case the minimal self-adjoint dilation $\widetilde K$ of $A_D$ can be regarded as the Hamiltonian of a closed system which contains the open system $\{A_D,\sH\}$, but since $\widetilde K$ is necessarily not semibounded from below, this model is difficult to interpret from a physical point of view. In the second part of the paper an open quantum system is modeled with a family $\{A(μ)\}$ of maximal dissipative operators depending on energy $μ$, and it is shown that the open system can be embedded into a closed system where the Hamiltonian is semibounded. Surprisingly it turns out that the corresponding scattering matrix can be completely recovered from scattering matrices of single Pseudo-Hamiltonians as in the first part of the paper. The general results are applied to a class of Sturm-Liouville operators arising in dissipative and quantum transmitting Schrödinger-Poisson systems.
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