Hadamard-type formulas for real eigenvalues of canonically symplectic operators

TL;DR

This paper develops first-order asymptotic and Hadamard-type formulas for eigenvalue curves of symplectic operators, incorporating boundary condition dependence and perturbations, with applications to nonlinear Schrödinger equations on star graphs.

Contribution

It introduces new formulas for eigenvalue derivatives of symplectic operators considering boundary and perturbation effects, using two distinct analytical methods.

Findings

Derived explicit formulas for eigenvalue derivatives.

Applied results to spectral index theorem in nonlinear Schrödinger context.

Provided methods for handling boundary condition dependence in eigenvalue analysis.

Abstract

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 Schr\"odinger equation on a compact star graph.