Analysis of quantum many-particle operators with disorder

无序量子多粒子算子分析

• 批准号: 0701181
• 负责人: Simone Warzel
• 金额: $ 14.06万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701181&HistoricalAwards=false
• 关键词: Analysis; quantum; many; particle; operators

The goal of this project is to address fundamental issues in the analysis of random quantum many-particle operators. So far, most of the mathematical research in the field of random operators has been focused on the spectral theory of random single-particle operators. A key theorem here addresses localization for large disorder: any given interval of the almost-sure spectrum belongs to the pure point spectrum provided the randomness is large enough; the corresponding eigenfunctions are exponentially localized. The PI's research program aims to investigate the fate of this strong localization regime in the interacting case. The influence of many-particle correlations will be examined for both fermionic and bosonic systems.Random quantum many-particle operators arise in models of semiconductors, dirty superconductors, or Bose-Einstein condensates in disordered media. In such systems, the interplay between quantum and statistical fluctuations due to the disorder gives rise to interesting physical phenomena like the suppression of electronic transport or of Bose-Einstein condensation. The mathematical study of these phenomena involves combining techniques from different areas of mathematics, in particular, functional analysis and probability.

该项目的目标是解决随机量子多粒子算子分析中的基本问题。 迄今为止，随机算子领域的数学研究主要集中在随机单粒子算子的谱理论上。 这里的一个关键定理解决了大无序的局部化问题：如果随机性足够大，几乎确定谱的任何给定区间都属于纯点谱;相应的本征函数是指数局部化的。PI的研究计划旨在调查这种强本地化制度在相互作用情况下的命运。多粒子关联对费米子和玻色子系统的影响将被研究。随机量子多粒子算符出现在半导体、脏超导体或无序介质中的玻色-爱因斯坦凝聚体的模型中。 在这样的系统中，由于无序而引起的量子和统计涨落之间的相互作用引起了有趣的物理现象，如电子输运或玻色-爱因斯坦凝聚的抑制。 对这些现象的数学研究涉及到结合不同数学领域的技术，特别是泛函分析和概率。