Hadamard-type formulas for real eigenvalues of canonically symplectic operators

TL;DR

This work develops a unified first-order perturbation theory for real eigenvalues of canonically symplectic, off-diagonal operator blocks under parameter-dependent boundary data. It yields Hadamard-type formulas for eigenvalue curves via two routes: a symplectic resolvent-difference approach and a Lyapunov–Schmidt reduction, both expressed through Maslov crossing forms. The abstract results are then specialized to the linearization of standing waves for nonlinear Schrödinger equations on compact star graphs, culminating in a spectral index theorem that relates the count of unstable real eigenvalues to crossing data and boundary-condition curvature (including a VK-type criterion). The framework connects Rayleigh–Hadamard–Rellich theory with Arnold–Maslov–Keller index theory and provides tools for stability analysis of networked PDE systems. Practical impact lies in enabling local and global eigenvalue counts and stability assessments for PDEs on graphs and similar networks.

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 Schrödinger equation on a compact star graph.

Figures (3)

• Figure 1: (a) Phase curves (in bold) in the phase plane for \ref{['SWE']} describing a standing wave solution $\phi = (\phi_1, \phi_2, \phi_3)^\top$ on a 3-star, where $\phi_1(0)=\phi_2(0)=\phi_3(0) = 1$ and $\phi_1'(0) = -a-b, \phi_2'(0) = -a-3,\phi_3'(0) = -\phi_1'(0) -\phi_2'(0) = 2a+3+b$, with $a\approx 0.8660$; (b) a schematic of the standing wave. In (c), (d) and (e) we give the eigenvalue curves for the standing wave solution described in (a) with $b$ as indicated. The standing waves all have one bump and two tails, and are non-negative on $\mathcal{G}$.
• Figure 2: Eigenvalue curves for $\mathcal{N}_t+V_t$ where (a), (b) and (c) correspond to items (1), (2) and (3) described in the text.
• Figure 3: Schematic of the Lagrangian path (solid black) through the top left corner crossing $(\lambda,t)=(0,1)$ when the eigenvalue curve $t(\lambda)$ (blue) satisfies (a) $t"(0)<0$ and (c) $t"(0)>0$, and the new path (dashed black) to which we homotope the original path when (b) $t"(0)<0$ and (d) $t"(0)>0$. The new path has only regular crossings.

Theorems & Definitions (35)

• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Proposition 2.2
• proof
• Theorem 2.3
• proof
• Lemma 3.2
• proof