三角形拉普拉斯算子谱分析(Ⅳ)

Spectrum of Laplician on Triangles(Ⅳ)

几何上具有对称性的三角形,它的本征函数,也具有相应的对称性,即对称性群的一维表示.文章利用D4群的4个一维表示,构造出了等腰直角三角形上4种边界条件的本征函数.本征波矢量由边界条件确定,得到了Zeta函数和热核的表达式.由热核展开系数,验证了本征函数的完备性.同时也讨论了一些遗留问题.

The eigenfunctions of some triangles with symmetric group, such as right angle isosceles, transform as one dimensional representation of this point group. In this paper, the eigenfunctions are constructed from the four one dimensional representation of dihedral group （n=4）, which corresponding to four boundaries conditions （BC）. The eigenvectors are set by the four BC. The zeta functions and heat kernels are also derived. The completeness of eigenfunctions is confirmed by the first two heat kernel coefficients. Some legacy problems are also discussed.