Spectral Structure of a Finite-Energy Electron Core

This work presents a numerical investigation of the spectral structure of a stationary finite-energy electron core obtained within a nonlinear density–vector field framework. The study focuses on the linear stability properties of the stationary configuration by analyzing the spectrum of the discrete Hessian operator associated with the system's energy functional. The stability problem is formulated as a large-scale eigenvalue problem for the Hessian operator evaluated around a numerically computed stationary state. For the highest grid resolution considered in this work (N = 96), the perturbation space contains more than 3.5 million degrees of freedom. The lowest portion of the spectrum is computed using a matrix-free shift–invert Krylov eigensolver, which allows the extraction of the smallest eigenvalues without explicitly constructing the full Hessian matrix. The numerical results reveal that the low-lying spectrum of the linearized operator is not diffuse but instead organizes into a compact cluster of five negative eigenvalues located near λ ≈ −4.97. This cluster is clearly separated from the rest of the spectrum by a substantial spectral gap. These findings indicate that the unstable sector of the linearized operator is confined to a finite-dimensional subspace spanned by five collective deformation modes. The structure suggests that the stationary electron core occupies a well-defined saddle point of the energy landscape, with instability restricted to a small number of collective deformation channels. The results provide quantitative insight into the spectral organization and stability structure of finite-energy localized electron configurations within nonlinear field models.