Mathematical Analysis in Condensed Matter and Atomic Physics

凝聚态与原子物理中的数学分析

• 批准号: 1068285
• 负责人: Rupert Frank
• 金额: $ 18.32万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1068285&HistoricalAwards=false
• 关键词: Mathematical; Analysis; Condensed; Matter; Atomic

This project is for research on problems, motivated by physical applications, where traditional mathematical methods fail and where novel and original techniques need to be developed, which will be relevant beyond the context of these concrete cases. Lieb-Thirring inequalities play an important role in numerous problems from mathematics and physics. One goal is to improve their constants by exploiting a conformal symmetry. Another goal is to extend these inequalities to complex valued potentials arising in applications and to derive analogous bounds in the case in which instead of the vacuum there is a background density of particles, as occurs in real solids. The Frohlich polaron serves both as a model for an electron in an ionic crystal and as a simple model for a dressed particle in quantum field theory. The PI's recent results about the binding of several polarons will be extended. There is a challenging conjecture about the effective polaron mass to be addressed. Uniqueness of ground states is important for the understanding of finite time blow up of dispersive, non-linear equations. The PI recently provided the first robust uniqueness proof for fractional Laplace equations. It is intended to extend this technique towards a conjecture about the boson star equation. A goal is to rigorously derive Ginzburg-Landau theory of superconductivity from BCS theory, which effectively amounts to semi-classics under minimal regularity conditions. Further problems, where the standard semi-classical calculus is not applicable, are to be pursued with an eye towards physical intuition. An attempt will be made to find the sharp form of two functional inequalities, namely a bound on the entropy defined via Bloch coherent spin states and the Strichartz inequality about the decay of solutions of the Schrodinger equation. Broader Impact: Problems from physics have often fostered progress in mathematics, while new tools of mathematics allow physics to enhance our qualitative and quantitative understanding of complex phenomena occuring in nature. The proposed project on the interface of these fields will strenghten the interdisciplinary bonds between the communities of mathematicians and physicists and promote the relevance of modern methods of mathematical analysis to problems of atomic and condensed matter physics.

这个项目是研究问题，物理应用的动机，传统的数学方法失败，需要开发新的和原始的技术，这将是相关的超出这些具体情况的背景。Lieb-Thirring不等式在许多数学和物理问题中起着重要的作用。一个目标是通过利用共形对称来改进它们的常数。另一个目标是将这些不等式扩展到复值的应用程序中产生的潜力，并得出类似的边界的情况下，而不是真空中有一个背景密度的粒子，发生在真实的固体。Frohlich极化子既是离子晶体中电子的模型，也是量子场论中修饰粒子的简单模型。PI最近关于几个极化子结合的结果将被推广。关于有效极化子质量有一个具有挑战性的猜想。基态的唯一性对于理解色散非线性方程的有限时间爆破是很重要的。PI最近为分数拉普拉斯方程提供了第一个鲁棒唯一性证明。它的目的是扩展这种技术对玻色子星星方程的猜想。一个目标是严格地从BCS理论推导出超导的金兹伯格-朗道理论，该理论在最小正则性条件下有效地达到了半经典。进一步的问题，在标准的半经典微积分是不适用的，是追求与眼睛对物理的直觉。试图找到两个函数不等式的精确形式，即通过布洛赫相干自旋态定义的熵的界和关于薛定谔方程解的衰减的Hohartz不等式。更广泛的影响：物理学的问题往往促进了数学的进步，而新的数学工具使物理学能够提高我们对自然界中发生的复杂现象的定性和定量理解。关于这些领域接口的拟议项目将加强数学家和物理学家社区之间的跨学科联系，并促进现代数学分析方法与原子和凝聚态物理问题的相关性。