Absence of embedded eigenvalues for the linearized operator around NLS solitons

Prove that the linearized vector Schrödinger operator H_ω around a solitary wave Φ_ω for the one-dimensional nonlinear Schrödinger equation i ∂_t v − ∂_x^2 v − F'(|v|^2) v = 0 has no embedded eigenvalues in its essential spectrum (−∞, −ω] ∪ [ω, ∞) for general admissible nonlinearities F, beyond the currently established pure-power case.

Background

The spectral properties of the non-self-adjoint linearized operator H_ω are central to asymptotic stability analyses. Embedded eigenvalues—eigenvalues lying within the essential spectrum—would obstruct dispersive and Strichartz estimates and are a major spectral pathology to exclude.

The review notes a conjecture that such embedded eigenvalues do not occur for H_ω in the setting considered, but confirms that rigorous proofs currently exist only in the pure-power case F(z) = z^{σ+1}. Establishing their absence for general nonlinearities would solidify the spectral foundation required by many stability approaches.