Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators
Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators
复制标题
DOI：
10.1137/110836420
复制
发表时间：
2011-04
期刊：
影响因子：
--
通讯作者：
J. Sylvester
中科院分区：
文献类型：
--
作者：
J. Sylvester
传输特征值是内部传输算子谱中的点，是一个椭圆偏微分方程的耦合 $2 \times 2$ 系统，其中一个未知函数必须满足两个边界条件，而另一个则必须不满足任何边界条件。通过证明内部传输算子具有上三角紧解算子，并且这些算子的谱具有紧解算子的许多性质，我们证明了内部传输特征值是离散的，并且连续地依赖于对比度。特别是，谱是离散的，广义特征空间是有限维的。我们的主要假设是对比度的矫顽力条件必须仅在边界附近成立。
Transmission eigenvalues are points in the spectrum of the interior transmission operator, a coupled $2 \times 2$ system of elliptic partial differential equations, where one unknown function must satisfy two boundary conditions and the other must satisfy none. We show that the interior transmission eigenvalues are discrete and depend continuously on the contrast by proving that the interior transmission operator has upper triangular compact resolvent, and that the spectrum of these operators share many of the properties of operators with compact resolvent. In particular, the spectrum is discrete and the generalized eigenspaces are finite-dimensional. Our main hypothesis is a coercivity condition on the contrast that must hold only in a neighborhood of the boundary.