晶体群、严格镶嵌多面体和解析本征函数
Crystallographic groups, strictly tessellating polytopes, and analytic eigenfunctions
摘要：
晶体结构的数学联系了分析、几何、代数和数论。平面晶体群在19世纪后期被分类。一百年后，贝拉尔证明了所有这些群的基本区都满足一个非常特殊的解析性质：拉普拉斯本征方程的狄利克雷本征函数都是三角函数。在2008年，McCartin证明了在二维空间中，这种特殊的解析性质既有一个等价的代数公式，也有一个等价的几何公式。这里我们把Bérard和McCartin的结果推广到所有维数。证明了以下等价性：多面体上Laplace本征方程的第一Dirichlet本征函数是实解析的，多面体严格镶嵌空间，多面体是晶体学Coxeter群的基本区.此外，我们还证明了在任何一个等价条件下，所有的特征函数都是三角函数。最后，我们将这些问题与Fuglede和Goldbach猜想联系起来，给出了Goldbach猜想的一个纯几何形式。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
