Applications of singular analysis to quantum many-particle models in chemistry

凝聚态物理和化学中的许多有趣问题都可以用量子多粒子系统通过奇异库仑势相互作用来描述。本文的主题是用奇异分析的方法研究这类系统的渐近行为，特别是在电子结构理论中，在粒子的聚合势附近。给出了表示薛定谔方程相应解的渐近行为的量子力学哈密顿量的参数矩阵的显式渐近构造，这可以在具有几何奇性的流形上的一般伪微积分的框架下实现。在这里，库伦势引入了一个自然的奇点等级，可以在微积分中用嵌入的圆锥、边和更高阶角奇点来描述。该项目的一个新的重要特点是不仅考虑薛定谔方程，而且在耦合团簇理论的框架下考虑一些简化的多粒子模型。这为这些模型的渐近行为提供了一个新的视角，并揭示了可能的缺陷，这反过来可能导致电子结构计算模型的改进。

Many interesting problems in condensed matter physics and large parts of chemistry can be described by quantum many-particle systems interacting via singular Coulomb potentials. The subject of this proposl is to study the asymptotic behaviour of such systems, in particular in electronic structure theory, near coalescence ponts of particles using methods from singular analysis. It is intended to provide explicit asymptotic constructions for parametrices of quantum echanical Hamiltonians which encode the asymptotic behaviour of the corresponding solutions of Schrödinger’s equation This can be achieved in the framework of a general pseudo-differential calculus on manifolds with geometric singularities. Here, the Coulomb potentials introduce a natural hierachry of singularities which can be described within the calculus by embedded conical, edge and higher-order corner singularities. A new important feature of the project is to consider not only the Schrödlinger’s equation but also some simplifie many-particle models in the framework of coupled cluster theory. This provides a new perspective concerning the asymptotic behaviour of these models and reveals possible shortcomings which in turn might lead to improved models for electronic structure calculations.