论文标题
定义诺伊曼域的光谱位置
Defining the spectral position of a Neumann domain
论文作者
论文摘要
二维Riemannian歧管上的拉普拉斯征函数为Neumann域（又称Morse-Smale-Complex）提供了自然分区。该分区是由特征性函数的梯度流线生成的，该特征功能结合了所谓的Neumann域。我们证明，在诺伊曼（Neumann）域上定义的诺伊曼·拉普拉斯（Neumann Laplacian）是自我相邻的，并且具有纯粹的离散频谱。此外，我们证明对其任何一个neumann域的特征功能的限制是Neumann Laplacian的特征功能。相比之下，关于本征函数的淋巴结域上关于dirichlet laplacian的类似陈述是基本的和众所周知的。这里的困难是，诺伊曼域的边界可能具有尖和裂纹，因此无法提供有关Sobolev空间的标准结果。 另一个非常有用的共同事实是，淋巴结域上受限制的征函数是dirichlet laplacian的第一个特征功能。对于诺伊曼（Neumann）域而言，这不再是正确的。我们的结果使Neumann结构域的结果光谱位置问题能够研究，这比其淋巴结类似物更重要。
A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition into Neumann domains (a.k.a. a Morse--Smale complex). This partition is generated by gradient flow lines of the eigenfunction, which bound the so-called Neumann domains. We prove that the Neumann Laplacian defined on a Neumann domain is self-adjoint and has a purely discrete spectrum. In addition, we prove that the restriction of an eigenfunction to any one of its Neumann domains is an eigenfunction of the Neumann Laplacian. By comparison, similar statements about the Dirichlet Laplacian on a nodal domain of an eigenfunction are basic and well-known. The difficulty here is that the boundary of a Neumann domain may have cusps and cracks, so standard results about Sobolev spaces are not available. Another very useful common fact is that the restricted eigenfunction on a nodal domain is the first eigenfunction of the Dirichlet Laplacian. This is no longer true for a Neumann domain. Our results enable the investigation of the resulting spectral position problem for Neumann domains, which is much more involved than its nodal analogue.
