薄壁曲管节点集的位置
Location of the nodal set for thin curved tubes
摘要：
研究了任意维欧几里得空间中任意等截面曲管沿有界曲线随Tang标架旋转时的Dirichlet Laplacian，在截面体积减小的极限下.我们表明，谱性质的拉普拉斯算子，在这个限制，近似的Dirichlet拉普拉斯算子的横截面和一维薛定谔算子，其潜在的表示仅在第一曲率的参考曲线的总和。特别地，我们建立了特征值的收敛性，特征函数的一致收敛性以及Dirichlet Laplacian算子在一维Schrodinger算子节点附近的节点集.作为结果，我们证明了一类非凸和可能的多连通域的“切线猜想”。这些结果是基于一个扰动理论发展的薛定谔型运营商在一个直管的截面缩小。
英文摘要：
The Dirichlet Laplacian in curved tubes of arbitrary constant cross-section rotating together with the Tang frame along a bounded curve in Euclidean spaces of arbitrary dimension is investigated in the limit when the volume of the cross-section diminishes. We show that spectral properties of the Laplacian are, in this limit, approximated well by those of the sum of the Dirichlet Laplacian in the cross-section and a one-dimensional Schrodinger operator whose potential is expressed solely in terms of the first curvature of the reference curve. In particular, we establish the convergence of eigenvalues, the uniform convergence of eigenfunctions and locate the nodal set of the Dirichlet Laplacian in the tube near nodal points of the one-dimensional Schrodinger operator. As a consequence, we prove the "nodal-line conjecture" for a class of non-convex and possibly multiply connected domains. The results are based on a perturbation theory developed for Schrodinger-type operators in a straight tube of diminishing cross-section.
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