Spectral and Scattering Theory and its Application

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The present project has been devoted to the study on the following three subjects related to the spectral and scattering theory for Schr_dinger operators.(1) For exponential product formula(Lie-Trotter-Kato product formula), the convergence in operator norm has been proved and the error estimate has been also established. The 9btained results have been applied to Schr_dinger semi-groups or propagators with singular or time-dependent potentials.(2) The unperturbed Pauli operator without electric potentials has zero eigenvalue with infinite multiplicities as its bottom of essential spectrum. When the operators are perturbed by potentials falling off at infinity, the asymptotic distribution of discrete eigenvalues near the origin has been studied. The special emphasis is placed on the case that Pauli operators do not necessarily have constant magnetic fields.(3) The asymptotic behavior at low energy of scattering amplitudes has been analysed for scattering by two dimensional magnetic fields and the relation to scattering by magnetic fields with small support has been also discussed.