Harmonic Analysis with Applications to Mathematical Physics

主要研究者：Wilhelm Schlag，加州理工学院DMS-0300081摘要：本文讨论了数学物理与调和分析之间的接口问题.作者打算继续研究具有确定势和随机势的薛定谔方程。一些问题仍然存在于离散薛定谔运营商的线与准周期的潜力，而非独立的情况下，但更强的混合动力学比准周期的潜力提出了许多严峻的挑战，需要做更多的工作在这方面。对于含时方程的问题仍然是关于色散估计，无论是对含时和不含时的潜力。对于后者，我们不知道通常的色散估计是否适用于衰减速度超过平方幂倒数的势，至少对于2维和更大的维。作者最近在一维中建立了这一点，但在二维中，在强多项式衰减的假设下的色散估计是未知的。然而，他认为，在二维情况下的色散估计假设足够快的衰减潜力是一个容易解决的问题。线性估计的主要兴趣在于非线性应用。一个例子是由Rodnianski，Soffer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分散估计。在这一领域还有很多工作要做，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决方案），以及有关的非线性束缚态（孤子）的动力学问题。此外，作者还打算研究调和分析中的问题或其在数学物理以外的问题中的应用。科学和工程的成功很大程度上取决于它在计算机上的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使其可供科学家和工程师使用方面发挥着重要作用。这个提议旨在解决大部分源于数学物理的数学问题。上述非线性薛定谔方程出现在各种应用中，例如，光学.这样一个非线性方程的束缚态（孤立子）代表一个粒子或光束，它的运动不会分裂。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。也就是说，它们在小扰动下是否持续存在？显然，光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是非常困难的，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.