Modern analysis of fundamental structures of solutions to partial differential equations

首席研究员Okaji研究了两个基本性质，一阶椭圆系统的特征值的缺乏和Shrodinger方程解的奇异性的传播。关于第一个性质，他已经处理了物理上重要的系统，狄拉克算子和稳态麦克斯韦算子。在与H。卡尔夫和0。Yamada等人的工作，得到了一个很好的条件，保证了在无穷远处位势发散的Dirac算子在连续谱中不存在嵌入本征值。此外，他还将结果推广到Dirac型算子以及非各向同性介质中的麦克斯韦算子。他还获得了Stokes方程的强唯一连续性。至于第二个主题，他发明了一种新的方法来研究传播的奇异性的解决方案薛定谔方程。这种方法是基于一个微局部守恒定律的维格纳变换的解决方案服从。它与解的波包变换密切相关。作为应用，他可以澄清一个精细结构的传播微局部奇性的解决方案薛定谔方程与矢量势以及电势可能增长的无穷大。它包括平滑效果，重建的奇异性和创造的奇异性从振荡的初始数据。

The head investigator Okaji has investigated two fundamental properties, absence of eigenvalues for first order elliptic systems and propagation of singularities of solutions to Shrodinger equations. As for the first property, he has treated physically important systems, Dirac operator and stationary Maxwell operators. In a joint work with H. Kalf and 0. Yamada, he obtained a nice condition which assures the nonexistence of embedded eigenvalues in continuous spectrum for the Dirac operators with potential diverging at infinity. Furthermore, he has extended the result to Dirac type operators as well as Maxwell operators in a non-isotropic media. He has obtained also strong unique continuation property for Stokes equations. As for the second topic, he has invented a new approach to the study of propagation of singularities of solutions to Schrodinger equations. This approach is based on a microlocal conservation law that the Wigner transformation of the solution obeys. It is strongly connected to the wave packet transform of the solutions. As applications, he can clarify a fine structure of propagation of microlocal singularities of solutions to Schrodinger equations with vector potential as well as electric potential which may grow at the infinity. It includes smoothing effects, reconstruction of singularities and creation of singularities from oscillatory initial data.