The linearized cubic NLS has no embedded eigenvalue

Abstract The linearized operator ℒ associated with the three-dimensional cubic nonlinear Schrödinger equation is shown to possess no embedded eigenvalues in the essential spectrum. This result rigorously confirms the key spectral hypothesis underpinning the construction of center-stable manifolds by Schlag (Ann. Math. (2) 169(1):139–227, 2009), and thereby provides the final analytical link for an unconditional stability theory near the ground state soliton. Unlike the one-dimensional case, the 3D model lacks integrability; the non-self-adjointness of ℒ and the non-explicit profile of the ground state render classical techniques insufficient to exclude eigenvalues in the continuum. The proof rests on the introduction of a weight-modulated positivity trap to exploit the hyperbolic instability of the fundamental mode, combined with a novel constrained shooting method and delicate comparison arguments for higher angular momenta. Beyond the cubic model, the machinery introduced here provides a robust analytical framework for addressing the spectral coercivity conjectures central to the Merle–Raphaël’s log-log blow-up on mass-critical NLS.