Harmonic Analysis with Applications to Mathematical Physics

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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.