Mathematical Analysis in Condensed Matter and Atomic Physics

这个项目是为了研究那些由物理应用驱动的问题，在这些问题上，传统的数学方法失败了，需要开发新的和原创的技术，这将超出这些具体案例的背景。Lieb-Thirring不等式在许多数学和物理问题中起着重要作用。一个目标是通过利用共形对称性来提高它们的常数。另一个目标是将这些不等式扩展到应用中出现的复值势，并推导出类似的边界，在这种情况下，粒子的背景密度而不是真空，就像在实际固体中发生的那样。在量子场论中，Frohlich极化子既可以作为离子晶体中电子的模型，也可以作为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得到扩展。有一个关于有效极化子质量的具有挑战性的猜想需要解决。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最近提供了分数阶拉普拉斯方程的第一个鲁棒唯一性证明。它的目的是将这种技术扩展到关于玻色子星方程的猜想。我们的目标是从BCS理论中严格推导出金兹堡-朗道超导理论，该理论在最小正则性条件下有效地达到半经典。进一步的问题，在标准的半经典微积分是不适用的，是追求对物理直觉的眼睛。我们将尝试找到两个泛函不等式的尖锐形式，即由Bloch相干自旋态定义的熵界和关于薛定谔方程解衰减的Strichartz不等式。更广泛的影响：来自物理学的问题常常促进数学的进步，而新的数学工具使物理学能够增强我们对自然界中发生的复杂现象的定性和定量理解。关于这些领域界面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。

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.