论文标题
三角棱镜的宽词特征值的渐近学
Asymptotics of sloshing eigenvalues for a triangular prism
论文作者
论文摘要
我们考虑了三角棱镜上的三维荡妇问题，该棱镜的角度与晃动的表面的角度为$ \fracπ{2q} $，其中$ q $是整数。我们有兴趣找到特征值计数函数的两项渐近扩张。当两个角度为$ \fracπ{4} $时，我们计算第二项的确切值。至于一般情况，我们通过为问题构造准例状并计算相关准元素值的计数函数来猜测渐近扩展。这些准烟象来自倾斜海滩问题的解决方案，对应于两种波浪，边缘波和表面波。我们表明，准元素价值与荡妇问题的真实特征值成倍接近。其计数函数的渐近膨胀与在扰动椭圆内的晶格计数问题密切相关，其中扰动是随机的。然后可以通过扰动检测角度的贡献。
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.
