Unique continuation theorems for systems of partial differential equations and their applications

首席研究员Okaji关注一阶椭圆系统，特别是数学物理中最重要的算子之一狄拉克算子。我们感兴趣的课题是光谱问题。他特别研究了特征值的缺失、绝对连续谱和共振。他还研究了非紧曲面上的狄拉克算子。首先，Okaji与H.Kalf和O.Yamada合作，得到了在无穷远处具有发散势的Dirac算子特征值不存在的一个很好的结果。此外，Okaji还证明了这类狄拉克算子的极限吸收原理是有效的。由这一结果可知，光谱在整个实线上是绝对连续的。与这一性质相反，众所周知，薛定谔算子在非相对论极限下具有发散势的狄拉克算子的极限具有纯粹离散的特征值。为了澄清这些现象，Okaji研究了狄拉克型算子的共振行为，表明共振在非相对论极限下收敛于薛定谔算子的特征值。他还处理了非紧曲面上的狄拉克算子，证明了谱在整个实线上是纯连续的。这是一个非常有意义的结果，因为拉普拉斯-贝尔特拉米算子的谱取决于高斯曲率的符号。我们的一位研究员Yamada和他的学生Ikoma合作证明了具有Aharonov-Bohm磁场的狄拉克算子具有独特的延拓性。他还成功地推广了Veselic （Glasnik Mat. 4, 1969）关于非相对论极限下狄拉克算子谱集中的结果。

The head investigator Okaji is concerned with elliptic systems of first order, especially Dirac operators that are one of the most important operators in mathematical physics. The subjects in which we are interested are spectral problems. In particular he has studied absence of eigenvalues, absolutely continuous spectrum, and resonances. He has also studied Dirac operators on non-compact surfaces.First of all, Okaji has collaborated with H.Kalf and O.Yamada in obtaining a nice result on absence of eigenvalues of Dirac operators with diverging potentials at infinity. Moreover, Okaji has proved that the limiting absorption principle for such Dirac operators was valid. From this result, it follows that the spectrum is absolutely continuous in the whole real line. Contrary to this property it is well known that Schrodinger operators which is the limit of Dirac operators with diverging potential in the non-relativistic limit has purely discrete eigenvalues. To clarify these phenomena Okaji has investigated the behavior of resonances of Dirac type operators to show that the resonances converged to the eigenvalues of Schrodinger operators in the non-relativistic limit. He has also treated Dirac operators in a non-compact surface to show that the spectrum is purely continuous in the whole real line. This is a very meaningful result because spectrum of Laplace-Beltrami operators depends on the sign of the Gaussian curvature.One of our investigators Yamada has collaborated with his student Ikoma in proving that Dirac aperator with Aharonov-Bohm magnetic fields has unique continuation property. He also succeeded in extending a result by Veselic (Glasnik Mat. 4, 1969) on spectral concentration of Dirac operators in non-relativistiv limit in collaboration with H.Ito.