定义 Neumann 域的谱位置
Defining the spectral position of a Neumann domain
AI总结：
本文研究了二维 Riemann 流形上 Neumann 域的谱位置问题，证明了 Neumann Laplacian 的自伴性与纯离散谱，并指出受限特征函数性质不同于节点域上的 Dirichlet Laplacian。
摘要：
二维 Riemann 流形上的 Laplacian 特征函数提供了一种向 Neumann 域（又称 Morse--Smale 复形）的自然划分。该划分由特征函数的梯度流线生成，这些流线界定了所谓的 Neumann 域。我们证明了定义在 Neumann 域上的 Neumann Laplacian 是自伴的，且具有纯离散谱。此外，我们证明将特征函数限制在其任一 Neumann 域上时，它是 Neumann Laplacian 的特征函数。相比之下，关于特征函数节点域上 Dirichlet Laplacian 的类似陈述是基础且广为人知的。此处的困难在于，Neumann 域的边界可能存在尖点和裂缝，因此无法应用关于 Sobolev 空间的标准结论。另一个非常有用的常见事实是，节点域上的受限特征函数是 Dirichlet Laplacian 的第一特征函数。这对于 Neumann 域不再成立。我们的结果使得能够研究 Neumann 域所引发的谱位置问题，该问题比其节点域的对应问题复杂得多。
英文摘要：
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.
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