Dirichlet eigenvalues of asymptotically flat triangles
Dirichlet eigenvalues of asymptotically flat triangles
复制标题
渐近平坦三角形的狄利克雷特征值
作者：
T. Ourmières
本文研究了一类三角形的狄利克雷拉普拉斯特征对，其中两个顶点是固定的，第三个顶点的高度趋于零。我们研究了特征值对这个高度的依赖性。对于第一特征值和特征函数，由于Airy算子的影响，我们在该高度的尺度立方根处得到了任意阶的渐近展开式。提供了特征对的渐近展开式，当高度趋于零时显示出两个不同的尺度。此外，我们将我们的分析推广到收缩对称多边形的情况，并量化了相应的隧道效应。
This paper is devoted to the study of the eigenpairs of the Dirichlet Laplacian on a family of triangles where two vertices are fixed and the altitude associated with the third vertex goes to zero. We investigate the dependence of the eigenvalues on this altitude. For the first eigenvalues and eigenfunctions, we obtain an asymptotic expansion at any order at the scale cube root of this altitude due to the influence of the Airy operator. Asymptotic expansions of the eigenpairs are provided, exhibiting two distinct scales when the altitude tends to zero. In addition, we generalize our analysis to the case of a shrinking symmetric polygon and we quantify the corresponding tunneling effect.