对称算子扩张的一阶渐近扰动理论
First-order asymptotic perturbation theory for extensions of symmetric operators
AI总结：
本文利用辛几何与Krein公式建立对称算子自伴扩张的一阶渐近扰动理论，推导Riccati方程、特征值变分公式及谱流-Maslov指数关系，并应用于量子图等模型。
摘要：
这项工作为对称算子的变化自伴扩张的渐近扰动理论提供了一个新视角。利用自伴性的辛表述，我们获得了预解式之差的新版本Krein公式，该公式通过与受扰算子相关的Lagrangian平面族的一阶展开，促进了预解算子的渐近分析。具体而言，我们推导了Riccati型微分方程，以及由Lagrangian平面的光滑单参数族确定的自伴扩张预解式的一阶渐近展开。这一渐近扰动理论产生了抽象Kato选择定理的辛版本，以及用于从无扰算子的一个特征值分叉出的多重特征值曲线斜率的Hadamard-Rellich型变分公式。后者反过来给出了著名公式的一般无穷小版本，该公式将自伴扩张路径的谱流与相应Lagrangian平面路径的Maslov指数等同起来。文中给出了在量子图、周期Kronig-Penney模型、具有Robin边界条件的椭圆二阶偏微分算子以及与物理相关的具有热导率的热方程中的应用。
英文摘要：
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.
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