关于 canonical 对称辛算子实特征值的 Hadamard 型公式
Hadamard-type formulas for real eigenvalues of canonically symplectic operators
AI总结：
本文提出了一类 canonical 对称辛算子的 Hadamard 型公式，并应用于非线性 Schrödinger 方程的谱指数定理推导。
摘要：
我们为一类 canonical 对称辛算子的一参数族的特征值曲线给出了首次阶渐近展开式和 Hadamard 型公式。我们允许边界条件、与每个非对角算子相关的有界扰动和迹算子的参数依赖性，并给出了离散特征值的未扰动算子的特征值曲线导数的公式，以 Maslov 交叉形式表示。我们通过两种不同的方法推导出 Hadamard 型公式：通过辛残差差分公式和残差的渐近展开，以及通过 Lyapunov-Schmidt 减缩和隐函数定理。后者方法在将特征值曲线视为谱参数函数时有助于导数公式。我们将抽象结果应用于推导非线性 Schrödinger 方程在紧星图上驻波线性化算子的谱指数定理。
英文摘要：
We give first-order asymptotic expansions for the resolvent and Hadamard-type formulas for the eigenvalue curves of one-parameter families of canonically symplectic operators. We allow for parameter dependence in the boundary conditions, bounded perturbations and trace operators associated with each off-diagonal operator, and give formulas for derivatives of eigenvalue curves emanating from the discrete eigenvalue of the unperturbed operator in terms of Maslov crossing forms. We derive the Hadamard-type formulas using two different methods: via a symplectic resolvent difference formula and asymptotic expansions of the resolvent, and using Lyapunov-Schmidt reduction and the implicit function theorem. The latter approach facilitates derivative formulas when the eigenvalue curves are viewed as functions of the spectral parameter. We apply our abstract results to derive a spectral index theorem for the linearised operator associated with a standing wave in the nonlinear Schrödinger equation on a compact star graph.
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