Mathematical Analysis in Condensed Matter and Atomic Physics

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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.