Resolvent estimates for Helmholtz equations in an exterior domain and their applications to scattering problems

研究了在经典物理中以波动方程形式出现的Helmholtz方程和在量子力学中以Schr-odinger方程形式出现的Helmholtz方程.作为应用，我们建立了包括二维散射问题在内的一致预解估计.特别是在二维情况下，我们发现可以用与辐射条件有关的Hardy型不等式来补偿仍为负项的项.因此，我们可以证明在二维情形下的一致预解估计，这在$N(Geq3)$D情形中已被证明.

We studied the Helmholtz equation, which is appeared in classical physics as wave equations, and in quantum mechanics as Schr\"odinger equations. We established the uniform resolvent estimate including the case of a two-dimensional, which is important in the scattering problem, corresponding as the application. In particular, in 2-D case, we find that we can compensate the term which remain negative term using the Hardy type inequalities related to radiation conditions. As a result, we can prove the uniform resolvent estimate in 2D case, which was already proved in $N(\geqq3)$ D case.