论文标题
晶体学组，严格的镶嵌多型和分析本征函数
Crystallographic groups, strictly tessellating polytopes, and analytic eigenfunctions
论文作者
论文摘要
晶体结构的数学连接分析，几何，代数和数理论。平面晶体学组分为19世纪后期。一百年后，贝拉德（Bérard）证明，所有此类群体的基本领域都满足了一个非常特殊的分析特性：拉普拉斯特征值方程的Dirichlet本征函数都是三角函数。 2008年，麦卡汀证明，在两个维度上，这种特殊的分析特性既具有等效的代数配方，也具有等效的几何表述。在这里，我们将Bérard和McCartin的结果推广到所有维度。我们证明了以下等效的：在多层室上拉普拉斯特征值方程的第一个dirichlet本征功能是真实的分析性，严格镶嵌空间，而多层人士是晶体学交换器组的基本结构域。此外，我们证明在任何等效条件下，所有本征函数都是三角函数。总之，我们将这些主题连接到fuglede和Goldbach的猜想，并纯粹对Goldbach的猜想进行了几何表述。
The mathematics of crystalline structures connects analysis, geometry, algebra, and number theory. The planar crystallographic groups were classified in the late 19th century. One hundred years later, Bérard proved that the fundamental domains of all such groups satisfy a very special analytic property: the Dirichlet eigenfunctions for the Laplace eigenvalue equation are all trigonometric functions. In 2008, McCartin proved that in two dimensions, this special analytic property has both an equivalent algebraic formulation, as well as an equivalent geometric formulation. Here we generalize the results of Bérard and McCartin to all dimensions. We prove that the following are equivalent: the first Dirichlet eigenfunction for the Laplace eigenvalue equation on a polytope is real analytic, the polytope strictly tessellates space, and the polytope is the fundamental domain of a crystallographic Coxeter group. Moreover, we prove that under any of these equivalent conditions, all of the eigenfunctions are trigonometric functions. In conclusion, we connect these topics to the Fuglede and Goldbach conjectures and give a purely geometric formulation of Goldbach's conjecture.
