U.S.-France Cooperative Research: Quantum Mechanics of Ordered and Disordered Media

该奖项将支持数学领域的合作研究 肯塔基州大学的Peter Hislop博士和 Jean-Michel Combes博士，土伦和瓦尔大学，路易斯安那州 Garde，法国。 拟议研究的目的是 研究有序和无序介质量子力学。 带电粒子在磁场中运动的量子行为 原子的周期性排列由允许的能带决定 能源 任何具有这些带能量的粒子 在晶体中传播到无穷远。 从物理上讲，这是一个 理想化的情况，对应于无限的物质 电导率 这些物质的传输特性， 例如电导率，当 材料浸没在电场和磁场中，或者当 晶体的完美有序被杂质破坏了。 这种类型的更改可能会导致系统的状态 在能量和空间中很好地定位。 中的关键成分 这些局域态的形成是量子隧穿 通过潜在的障碍。 一个粒子传播， 能量E将遇到空间区域， 大于E。 传统上，粒子被禁止 穿过这些区域，但从量子力学上讲， 功能将渗透其中。 然而，许多人的净效应 屏障可以是很好地定位粒子。 在这 项目，调查人员将集中在光谱 这些系统的结构，强调的作用 是由量子隧穿作用产生的。 他们将研究 晶体的电子态和散射理论 领域的 他们还将研究无序介质的模型， 该扩散行为被势垒抑制。 该项目将受益于重大和互补的 两名研究人员在量子领域的专业知识 共振和散射理论。

This award will support collaborative research in mathematical physics between Dr. Peter Hislop, University of Kentucky and Dr. Jean-Michel Combes, University of Toulon and Var, La Garde, France. The objective of the proposed research is to investigate quantum mechanics of ordered and disordered media. The quantum behavior of a charged particle moving in a periodic array of atoms is determined by the bands of allowed energy. Any particle with energy in these bands will propagate to infinity in the crystal. Physically, this is an idealized situation which corresponds to matter with infinite conductivity. These transport properties of matter, for example the conductivity, change dramatically when the material is immeresed in electric and magnetic fields or when the perfect order of the crystal is broken by impurities. Changes of this type may cause states of the system to be well-localized in energy and space. A key ingredient in the formation of these localized states is quantum tunneling through potential barriers. A particle propagating with energy E will encounter regions of space where the potential is larger than E. Classically, the particle is forbidden to traverse these regions, but quantum mechnically, the wave function will penetrate them. However, the net effect of many barriers may be to well-localize the particle. In this project, the investigators will concentrate on the spectral structure of these systems with an emphasis on the role played by quantum tunnelling. They will study the resonance states and the scattering theory of crystals in electric fields. They will also study models of disordered media in which diffusive behavior is suppressed by potential barriers. The project will benefit from the significant and complementary expertise of the two investigators in the areas of quantum resonances and scattering theory.