Harmonic Analysis with Applications to Mathematical Physics

Pi：加州TechnologyDMS-0300081----------------------------------------------------Abstract：---------------------------------------------This学院的Wilhelm Schlag的提案处理了数学物理和调和分析之间的接口的几个问题。作者打算继续研究既有确定性势又有随机势的薛定谔方程。关于具有准周期势的线上的离散薛定谔算子仍然存在一些问题，而由非独立但比准周期更强的混合动力学给出的势的情况提出了许多严峻的挑战，在这方面需要做更多的工作。对于依赖时间的方程，仍然存在关于色散估计的问题，包括依赖于时间的势和与时间无关的势。对于后者，尚不清楚通常的色散估计是否适用于衰减速度快于平方反比的势，至少对于二维或更大的维度是如此。作者最近在一维上证明了这一点，但在二维中，强多项式衰减假设下的色散估计是未知的。然而，他认为，在假设势能足够快的情况下，二维情况下的色散估计是一个可以解决的问题。线性估计的主要兴趣在于非线性应用。其中一个例子是由Rodnianski、Soffe和作者最近建立的弱相互作用多孤子解的渐近稳定性的证明。它严重依赖于电荷转移模型的色散估计。在这一领域，无论是在一般的非线性薛定谔方程(临界散焦三维方程的整体解)方面，还是在关于非线性束束态(孤子)的动力学问题上，仍有许多工作要做。此外，作者打算研究调和分析中的问题或将其应用于数学物理之外的问题。科学和工程的成功很大程度上取决于它在计算机建模和数值研究方面对数学工具的有效使用。数学家在开发这些方法并将其提供给科学家和工程师方面发挥了重要作用。这项建议旨在解决大部分源于数学物理的数学问题。前述的非线性薛定谔方程在光学等领域有着广泛的应用。这种非线性方程的束缚态(孤子)表示一个粒子或光束在运动中不解体。一个重要的问题是了解这样一个物体的稳定性或不稳定性。显然，孤子在光学介质中的任何商业应用都需要孤子的稳定性。事实证明，对这些问题的理论理解非常困难，往往需要对数学问题有新的见解。这项建议旨在解决这些问题。

PI: Wilhelm Schlag, California Institute of TechnologyDMS-0300081----------------------------------------------------Abstract:---------------------------------------------This proposal deals with several problems on the interface between mathematical physics and harmonic analysis. The author intends to pursue his work on Schroedinger equations with both deterministic and random potentials. Some questions remain on discrete Schroedinger operators on the line with quasi-periodic potentials, whereas the case of potentials given by non-independent but more strongly mixing dynamics than quasi-periodic presents many serious challenges and more needs to be done in this area. For time-dependent equations questions remain concerning dispersive estimates, both for time-dependent and time-independent potentials. For the latter, it is unknown whether or not the usual dispersive estimate holds for potentials that decay faster than an inverse square power, at least for dimensions two and larger. The author recently established this in one dimension, but in two dimensions dispersive estimatesunder the assumption of strong polynomial decay are unknown. He believes, however, that dispersive estimates in the two-dimensional case under the assumption of sufficiently fast decaying potentials is an accessible problem. The main interest in linear estimates lies with nonlinear applications. One example is given by the proof of asymptotic stability of weakly interacting multi-soliton solutions, which was recently established by Rodnianski, Soffer, and the author. It relied heavily on dispersive estimates for charge transfer models. Much remains to be done in this area, both in terms of nonlinear Schroedinger equations in general (global solutions for the critical defocusing three-dimensional equation), as well as questions concerning the dynamics of nonlinear bound states (solitons). In addition, the author intends to work on problems in harmonic analysis or applications thereof to problems outside of mathematical physics.Much of the success of science and engineering lies with its effective use of mathematical tools, both in terms of modeling and numerical studies on computers. Mathematicians play an important role in developing those methods and making them available to scientist and engineers. This proposal aims at addressing mathematical problems that for the most part originate in mathematical physics. The aforementioned nonlinear Schroedinger equations arise in variousapplications, e.g., optics. A bound state (soliton) for such a nonlinear equation represents a particle or beam that travels without disintegrating. An important issue is to understand the stability or instability of such an object. I.e., do they persist under small perturbations or not?Clearly, any commercial application of a soliton in optical media will require stability of the soliton. It turns out that the theoretical understanding of these issues is very difficult, often requiring new insights into mathematical problems. This proposal aims at addressing such problems.