标题： 波导中远处扰动的谱的渐近行为 显示英文标题

标题： Asymptotic behaviour of the spectrum of a waveguide with distant perturbations

摘要： 我们考虑由$n$维无限管建模的波导。 我们研究的算子是受两个远距离扰动影响的狄利克雷拉普拉斯算子。 扰动由某种意义上“局域化”的任意抽象算子描述，它们的“支撑”之间的距离趋于无穷大。 我们研究此类系统的离散谱的渐近行为。 主要结果是一个收敛定理和特征值的渐近展开式。 还描述了相关特征函数的渐近行为。 我们还提供了一些远距离扰动的特定例子。 这些例子包括势能、二阶微分算子、磁薛定谔算子、弯曲和变形波导、δ相互作用和积分算子。 显示英文摘要

摘要： We consider the waveguide modelled by a $n$-dimensional infinite tube. The operator we study is the Dirichlet Laplacian perturbed by two distant perturbations. The perturbations are described by arbitrary abstract operators ''localized'' in a certain sense, and the distance between their ''supports'' tends to infinity. We study the asymptotic behaviour of the discrete spectrum of such system. The main results are a convergence theorem and the asymptotics expansions for the eigenvalues. The asymptotic behaviour of the associated eigenfunctions is described as well. We also provide some particular examples of the distant perturbations. The examples are the potential, second order differential operator, magnetic Schroedinger operator, curved and deformed waveguide, delta interaction, and integral operator.