三棱柱晃动特征值的渐近性
Asymptotics of sloshing eigenvalues for a triangular prism
摘要：
本文考虑与晃荡面成$\fracπ{2q}$形式的三角柱上的三维晃荡问题，其中$q$为整数。我们感兴趣的是找到本征值计数函数的两项渐近展开。当两个角度都是$\fracπ{4}$时，我们计算第二项的精确值。对于一般情况，我们通过构造问题的拟模和计算相关拟特征值的计数函数，推测了一个渐近展开。这些准模态来自于斜滩问题的解，对应于两种波：边缘波和表面波。我们表明，拟特征值是指数接近真实的晃动问题的特征值。他们的计数函数的渐近展开与扰动椭圆内的格子计数问题密切相关，其中扰动在某种意义上是随机的。然后可以通过该扰动来检测角度的贡献。
英文摘要：
We consider the three-dimensional sloshing problem on a triangular prism whose angles with the sloshing surface are of the form $\fracπ{2q}$, where $q$ is an integer. We are interested in finding a two-term asymptotic expansion of the eigenvalue counting function. When both angles are $\fracπ{4}$, we compute the exact value of the second term. As for the general case, we conjecture an asymptotic expansion by constructing quasimodes for the problem and computing the counting function of the related quasi-eigenvalues. These quasimodes come from solutions of the sloping beach problem and correspond to two kinds of waves, edge waves and surface waves. We show that the quasi-eigenvalues are exponentially close to real eigenvalues of the sloshing problem. The asymptotic expansion of their counting function is closely related to a lattice counting problem inside a perturbed ellipse where the perturbation is in a sense random. The contribution of the angles can then be detected through that perturbation.
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