关于等边三角形的Robin谱
On the Robin spectrum for the equilateral triangle
摘要：
等边三角形是为数不多的平面域的狄利克雷和诺依曼本征值问题明确确定，由拉梅在1833年，尽管不承认分离的变量。本文研究了McCartin在2004年用超越耦合久期方程组确定的等边三角形的Robin谱。我们给出了Robin-Neumann间隙的统一上界，表明它们是由极限均值所限定的，因此这是一个几乎必然的界.光谱承认一个系统的双重多重性，并删除它后，我们研究的差距所产生的desymmetrized频谱。我们显示的频谱间隙的属性，有任意大的差距，也任意小的，而且，最近邻间距分布的desymmetrized频谱是一个δ函数在原点。我们表明，足够小的罗宾参数，去对称化的频谱是简单的。
英文摘要：
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.
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