Analysis of Multi-Particle Systems

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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.