Multi-Linear Singular Integrals

在过去的两年里，我一直致力于多线性奇异积分算子。最近，在与T.Tao和C.Thiele的合作工作中，我们将以前关于双线性Hilbert变换的结果推广到具有更多奇异符号的多线性算子的情况。关于这些发展的惊人事实是，似乎这些技术可以有一个非常好的和重要的应用到薛定谔算子的谱理论。本课题的最终目的是证明对于具有平方可积势的经典Schroedinger算子，当其本征值几乎为正时，其相应的本征函数是有界的。上述猜想的答案将意味着更好地理解薛定谔算子的行为，这是量子力学的核心。

Over the past two years I have been working on multi-linear singular integral operators. Recently, in a joint work with T.Tao andC.Thiele we generalized the previous results on the bilinear Hilberttransform to the case of multilinear operators with much more singularsymbols. The striking fact about these developments is that it seems that thesetechniques could have a very nice and important application to thespectral theory of Schroedinger operators. The final goal of our project, is to show that for the classicalSchroedinger operators with square integrable potential and for almostevery positive eigenvalue, their corresponding eigenfunctions are bounded. The answer to the above conjecture would imply a better understandingof the behaviour of the Schroedinger operators, which lie in the heart ofQuantum Mechanics.