二维区域中点相互作用的几何摄动近似
Approximation of point interactions by geometric perturbations in two-dimensional domains
摘要：
本文给出了二阶椭圆算子在点相互作用平面区域上的一种新的逼近。它是一个几何性质的逼近族，由具有相同符号和正则系数的算子组成。在其边界处施加Robin条件，其系数取决于孔洞的线性尺寸。我们表明，作为孔收缩到一个点和边界条件中的参数以适当的方式缩放，非线性和奇异的，所示的家庭收敛在范数预解式意义上的操作与点的相互作用。这个预解式的收敛性是建立在几个运营商的规范和顺序夏普估计的收敛速度。
英文摘要：
We present a new type of approximation of a second-order elliptic operator in a planar domain with a point interaction. It is of a geometric nature, the approximating family consists of operators with the same symbol and regular coefficients on the domain with a small hole. At the boundary of it Robin condition is imposed with the coefficient which depends on the linear size of a hole. We show that as the hole shrinks to a point and the parameter in the boundary condition is scaled in a suitable way, nonlinear and singular, the indicated family converges in the norm-resolvent sense to the operator with the point interaction. This resolvent convergence is established with respect to several operator norms and order-sharp estimates of the convergence rates are provided.
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