抽象希尔伯特空间设定下变双线性形式特征值的稳定性及其应用
On the stability of eigenvalues of varying bilinear forms in abstract Hilbertian settings and applications
AI总结：
本文发展抽象希尔伯特空间下变双线性形式特征值的谱扰动理论，量化收敛速率并给出扰动特征值渐近展开首项的变分刻画，还将其应用于拉普拉斯相关算子的特征值问题研究。
AI中文摘要：
本文旨在从更高视角发展谱扰动理论。更准确地说，我们考虑一个单参数的变双线性形式族，每个形式定义在（可能不同的）希尔伯特空间上。假设对应谱的稳定性，我们的第一个主要结果确立了收敛速率的量化。一个关键特征是扰动特征值渐近展开中首项的显式变分刻画，该刻画仅通过求解极小化问题依赖于“数据”，即极限特征空间和扰动的大小。值得注意的是，我们未对扰动特征元作出额外假设（除了自然的谱稳定性），并且全面覆盖了简单和多重极限特征值的情形。在第二部分，我们探索抽象结果的若干具体应用：首先，考虑带有变测度权重的拉普拉斯-贝尔特拉米算子的特征值问题（也受谱几何优化的启发）；其次，研究拉普拉斯算子的斯捷克洛夫特征值的诺伊曼逼近；最后，聚焦于当一个小的第二流形粘合在黎曼流形的一小部分上时，该流形上拉普拉斯-贝尔特拉米算子的谱如何变化。
英文摘要：
The aim of the present paper is to develop a spectral perturbation theory from a higher perspective. More precisely, we consider a one-parameter family of varying bilinear forms, each of them defined on a (possibly) different Hilbert space. Assuming the stability of the corresponding spectra, our first main result establishes a quantification of the rate of convergence. A key feature is the explicit variational characterization of the first term in the asymptotic expansion of the perturbed eigenvalues, which only depends on the ``data'', i.e. the limit eigenspace and the magnitude of the perturbation, through the resolution of a minimization problem. Remarkably, we make no assumptions on the perturbed eigenelements (besides, naturally, the spectral stability). Moreover, we cover both the cases of simple and multiple limit eigenvalues in full generality. In the second part, we explore some concrete applications of our abstract results. First, we consider eigenvalue problems for the Laplace-Beltrami operator with varying measure weights (also motivated by optimization in spectral geometry); secondly, we investigate the Neumann approximation of the Steklov eigenvalues of the Laplacian; finally, we focus on how the spectrum of the Laplace-Beltrami operator on a Riemannian manifold changes when a second small manifold is glued on a small portion of it.
