Analysis of Multi-Particle Systems

9706981 Ruskai这项研究涉及多粒子系统的数学分析中的问题。主要集中在原子和分子哈密顿算符的束缚态上，新的重点是磁场中的系统。特别是，一维模型将被用来研究强磁场中原子的最大负电离。我们还将利用非对易概率空间上的单调黎曼度量来研究一般量子系统的相对熵的收缩，并考虑一些相关的算子不等式。由于真实的原子和分子存在于普通的3维空间中，核电荷的整数值已经确定，因此具有分数或渐近无限核电荷的一维或二维系统的模型可能看起来有点人造和深奥。然而，真正的原子并不是孤立存在的，而是存在于从微型计算机芯片到恒星等各种大小不等的材料中。电子在这种复杂材料中的行为可以通过使用分数核电荷或假设它们被限制在一维或两维来精确建模。例如，半导体科学家最近制造了被称为“量子点”的微观材料，它们的行为就像具有分数核电荷的二维原子。在另一个极端，人们发现，在极强磁场中的原子，比如那些在中子星表面发现的原子，其行为就像电子被限制在一维空间中一样。因此，建议研究的系统具有广泛的适用性。这一建议的另一部分涉及将一类重要的熵不等推广到量子系统。熵及其相关泛函在经济学、统计学、人口生物学、信息论、物理学等多个领域有着重要的应用。鉴于量子计算的最新进展，本文提出的关于量子力学熵的工作有望对信息论和物理学产生影响。

9706981 Ruskai This research is concerned with problems in the mathematical analysis of multi-particle systems. The main focus is on bound states of atomic and molecular Hamiltonians with a new emphasis on systems in magnetic fields. In particular, one-dimensional models will be used to study the maximum negative ionization of atoms in strong magnetic fields. The contraction of relative entropy for general quantum systems will also be studied using monotone Riemannian metrics on non-commutative probability spaces, and some related operator inequalities will be considered. Because real atoms and molecules exist in ordinary 3-dimensional space with well established integer values for the nuclear charges, models of systems in one or two dimensions with fractional, or asymptotically infinite, nuclear charge may seem somewhat artificial and esoteric. However, real atoms do not exist in isolation, but within various kinds of materials ranging in size from microscopic computer chips to stars. The behavior of electrons within such complex materials may be accurately modelled by using a fractional nuclear charge or by supposing that they are confined to one or two dimensions. For example, semiconductor scientists have recently manufactured microscopic materials called "quantum dots" which behave like two-dimensional atoms with fractional nuclear charge. At the other extreme, one finds that atoms in extremely strong magnetic fields, such as those found on the surface of a neutron star, behave as if the electrons were confined to one dimension. Thus, the systems proposed for study have a wide range of applicability. The other part of this proposal is concerned with the generalization of an important class of entropy inequalities to quantum systems. Entropy and the related functionals have significant applications in such diverse fields as economics, statistics, population biology, information theory, and physics. In view of recent advances in quantum co mputing, the work on quantum mechanical entropy proposed here is expected to have an impact on information theory as well as physics.