论文标题
在罗宾频谱上的等边三角形
On the Robin spectrum for the equilateral triangle
论文作者
论文摘要
等边三角形是少数平面领域之一，尽管不承认变量分离，但通过Lamé明确确定了Dirichlet和Neumann特征值问题。在本文中，我们研究了等边三角形的罗宾光谱，这是由麦卡汀在2004年根据先验耦合的世俗方程式确定的。 我们为罗宾·尼曼（Robin-Neumann）的间隙提供了统一的上限，表明它们受其限制的平均值界定，因此几乎可以肯定地结合。频谱允许系统的双重多样性，在去除它之后，我们研究了所得的降低频谱中的差距。我们显示了一个频谱间隙属性，即任意差距很大，而且是任意的差距，此外，Desmorgatization Spectrum的最近的邻居间距分布是原点上的Delta函数。我们表明，对于足够小的robin参数，对称光谱很简单。
The equilateral triangle is one of the few planar domains where the Dirichlet and Neumann eigenvalue problems were explicitly determined, by Lamé in 1833, despite not admitting separation of variables. In this paper, we study the Robin spectrum of the equilateral triangle, which was determined by McCartin in 2004 in terms of a system of transcendental coupled secular equations. We give uniform upper bounds for the Robin-Neumann gaps, showing that they are bounded by their limiting mean value, which is hence an almost sure bound. The spectrum admits a systematic double multiplicity, and after removing it we study the gaps in the resulting desymmetrized spectrum. We show a spectral gap property, that there are arbitrarily large gaps, and also arbitrarily small ones, moreover that the nearest neighbour spacing distribution of the desymmetrized spectrum is a delta function at the origin. We show that for sufficiently small Robin parameter, the desymmetrized spectrum is simple.
