Geometric methods for nonlinear many-body quantum systems
Geometric methods for nonlinear many-body quantum systems
复制标题
非线性多体量子系统的几何方法
DOI：
10.1016/j.jfa.2010.11.017
复制
发表时间：
2010
影响因子：
1.7
通讯作者：
Mathieu Lewin
中科院分区：
文献类型：
--
作者：
Mathieu Lewin
几何技术在70年代对多体薛定谔算子谱的研究中发挥了重要作用。在本文中，我们提供了一个形式主义，也可以研究非线性系统。我们首先定义了多体状态的弱拓扑，它适当地描述了系统在缺乏紧致性的情况下的物理行为，即当一些粒子在无穷远处丢失时。我们提供了几个重要的性质，这个拓扑结构，并用它们写一个简单的证明著名的HVZ定理在排斥的情况下。在第二步中，我们回顾Derezienski和Gérard提出的Fock空间中的几何局部化方法，并将此工具与我们的弱拓扑联系起来。然后我们提供几个应用程序。我们从研究所谓的有限秩近似开始，这种近似要求多体波函数可以用100多个单体函数展开。因此，我们强调几何性质的Hartree-Fock状态和证明非线性版本的HVZ定理，在Friesecke的作品的精神。在最后一节中，我们研究了包含非线性项的摄动不变多体系统，它有效地描述了与第二系统的相互作用。作为一个例子，我们证明了在Pekar-Tomasevich近似下，对于一定的耦合常数，多重极化子的存在性。
Geometric techniques have played an important role in the seventies, for the study of the spectrum of many-body Schrödinger operators. In this paper we provide a formalism which also allows to study nonlinear systems. We start by defining a weak topology on many-body states, which appropriately describes the physical behavior of the system in the case of lack of compactness, that is when some particles are lost at infinity. We provide several important properties of this topology and use them to write a simple proof of the famous HVZ theorem in the repulsive case. In the second step we recall the method of geometric localization in Fock space as proposed by Dereziński and Gérard, and we relate this tool to our weak topology. We then provide several applications. We start by studying the so-called finite-rank approximation which consists in imposing that the many-body wavefunction can be expanded using finitely many one-body functions. We thereby emphasize geometric properties of Hartree–Fock states and prove nonlinear versions of the HVZ theorem, in the spirit of works of Friesecke. In the last section we study translation-invariant many-body systems comprising a nonlinear term, which effectively describes the interactions with a second system. As an example, we prove the existence of the multi-polaron in the Pekar–Tomasevich approximation, for certain values of the coupling constant.