Wavelets and Basis Set Optimization for Molecular and Other Few-Body Quantum Calculations

0070879Littlejohn量子力学是描述原子、分子和原子核行为的基本理论。 这项工作涉及相对简单的，或“少体”系统。在某些情况下，可以使用量子理论和计算机计算来预测分子反应的结果（应用的一个例子），这些反应在实验室中很难或不可能测试，最终对生物学，医学和环境科学等领域产生重要影响。 量子理论的应用包括找到某些波函数，这些波函数在实践中用更多的基本波（基波函数）表示。 寻找最优基的问题是量子理论中的一个老问题，但近年来出现了许多与这个问题相关的新思想，这些新思想来自各个领域，包括纯数学和应用数学、物理学、化学和工程学。 这项工作的目的是整合这些新的发展，寻找和利用基本的，统一的原则，并将其应用到基组选择在少体量子物理。新的想法包括以下内容。 第一个是小波，这是应用数学中的一个相对较新的发展，它已经对信号和数据的处理、传输和存储产生了重要影响。 第二种方法涉及相空间或半经典方法，在数学文献中称为“微局部分析”。 这些方法的特点是同时考虑位置和动量，这是量子力学中一个不寻常且相对陌生的观点，海森堡不确定性原理限制了这些量的同时知识。 第三个是关于在少体量子问题中抽象的高维空间的数学，即所谓的“几何”方法。 近年来，这种方法一直是数学中非常活跃的领域，并且在物理学的某些领域中是众所周知的。 然而，它们在少体量子力学中并没有得到太多的利用，尽管它们对于理解少体问题的“内部”空间的性质至关重要。 最后，这项工作将采取语料库的方法，目前使用的原子，分子和核物理的基组选择，比较它们，整合它们，寻找推广和改进，并应用它们。

0070879Littlejohn Quantum mechanics is the fundamental theory which describes the behavior of atoms, molecules and nuclei. This work concerns relatively simple, or ``few-body,'' systems. In some cases it is possible to use quantum theory and computer calculations to predict the outcome of molecular reactions (one example of an application) which are difficult or impossible to test in the laboratory, with ultimately an important impact on biology, medicine and environmental sciences, among other fields. The application of quantum theory involves finding certain wave functions, which in practice are expressed in terms of more elementary waves (the basis wave functions). The problem of finding an optimal basis is an old one in quantum theory, but in recent years there has arisen a host of new ideas relevant to this question from a variety of fields, including pure and applied mathematics, physics, chemistry and engineering. The object of this work is to integrate these new developments, to search for and exploit fundamental, unifying principles, and to apply them to basis set selection in few-body quantum physics. The new ideas include the following. The first is wavelets, arelatively recent development in applied mathematics, which has already had an important impact on the processing, transmission and storage of signals and data. A second concerns phase space or semiclassical methods, which are called ``microlocal analysis'' in the mathematics literature. These methods are characterized by a consideration of position and momentum together, an unusual and relatively unfamiliar point of view in quantum mechanics where the Heisenberg uncertainly principle limits the simultaneous knowledge of these quantities. A third concerns the mathematics of abstract, higher dimensional spaces in few-body quantum problems, the so-called ``geometrical'' methods. Such methods have been a very active area of mathematics in recent years and are well known in certain areas of physics. They have not, however, been exploited much in few-body quantum mechanics, although they are vital for understanding the nature of the ``internal'' spaces of few-body problems. Finally, this work will take the corpus of methods which are in current use in atomic, molecular and nuclear physics for basis set selection, compare them, integrate them, search for generalizations and improvements, and apply them.