论文标题
对称操作员扩展的一阶渐近扰动理论
First-order asymptotic perturbation theory for extensions of symmetric operators
论文作者
论文摘要
这项工作为渐近扰动理论提供了一种新的前瞻性，用于改变对称操作员的自我偶像扩展。采用自相关性的符合性表述，我们获得了一个新版本的孔林公式，以解决分解差异，该公式通过一阶扩展与与扰动操作员相关的拉格朗日平面家族的一阶扩展来促进分辨率运算符的渐近分析。具体而言，我们得出了一个riccati-Type微分方程，以及一阶渐近扩展，用于通过拉格朗日平面的平滑一参数家族确定的自我偶会扩展的分解。这种渐近扰动理论产生了抽象的Kato选择定理和Hadamard-rellich型变异公式的符合性版本，用于从不受干扰的操作员的特征值分叉的多个特征值曲线的斜率。反过来，后者给出了一个著名公式的一般无限版本，将自相会扩展路径的光谱和拉格朗日平面相应路径的Maslov索引等同。应用于量子图，周期性的kronig-penney模型，具有罗宾边界条件的椭圆二阶偏差算子以及具有导热率的物理相关的热方程。
This work offers a new prospective on asymptotic perturbation theory for varying self-adjoint extensions of symmetric operators. Employing symplectic formulation of self-adjointness we obtain a new version of Krein formula for resolvent difference which facilitates asymptotic analysis of resolvent operators via first order expansion for the family of Lagrangian planes associated with perturbed operators. Specifically, we derive a Riccati-type differential equation and the first order asymptotic expansion for resolvents of self-adjoint extensions determined by smooth one-parameter families of Lagrangian planes. This asymptotic perturbation theory yields a symplectic version of the abstract Kato selection theorem and Hadamard-Rellich-type variational formula for slopes of multiple eigenvalue curves bifurcating from an eigenvalue of the unperturbed operator. The latter, in turn, gives a general infinitesimal version of the celebrated formula equating the spectral flow of a path of self-adjoint extensions and the Maslov index of the corresponding path of Lagrangian planes. Applications are given to quantum graphs, periodic Kronig-Penney model, elliptic second order partial differential operators with Robin boundary conditions, and physically relevant heat equations with thermal conductivity.
