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

该奖项将支持肯塔基大学的Peter Hislop博士和法国土伦大学的Jean-Michel Combes博士和法国拉加德的Var博士在数学物理方面的合作研究。这项研究的目的是研究有序和无序介质的量子力学。带电粒子在原子的周期性阵列中运动的量子行为由允许的能带决定。在这些带中有能量的任何粒子都将在晶体中传播到无限远。在物理上，这是一种理想化的情况，对应于具有无限导电性的物质。当物质浸入电场和磁场中，或者当晶体的完美秩序被杂质破坏时，物质的这些输运性质，例如导电性，会发生巨大的变化。这种类型的变化可能导致系统的状态在能量和空间中得到很好的局部化。形成这些局域态的一个关键因素是通过势垒的量子隧穿。以能量E传播的粒子将遇到势能大于E的空间区域。经典地说，粒子被禁止穿过这些区域，但从量子力学的角度来看，波函数会穿透这些区域。然而，许多障碍的净效果可能是将粒子很好地局部化。在这个项目中，研究人员将专注于这些系统的光谱结构，重点是量子隧道所起的作用。他们将研究晶体在电场中的共振态和散射理论。他们还将研究无序介质的模型，在无序介质中，传播行为被潜在的障碍抑制。该项目将受益于两位研究人员在量子共振和散射理论领域的重要和互补的专业知识。

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.