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

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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.