Modern analysis of fundamental structures of solutions to partial differential equations

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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.