On the spectral properties of L_± in three dimensions

This paper is part of the radial asymptotic stability analysis of the ground state soliton for either the cubic nonlinear Schrödinger or Klein–Gordon equations in three dimensions. We demonstrate by a rigorous method that the linearized scalar operators which arise in this setting, traditionally denoted by L_±, satisfy the gap property, at least over the radial functions. This means that the interval (0, 1] does not contain any eigenvalues of L_± and that the threshold 1 is neither an eigenvalue nor a resonance. The gap property is required in order to prove scattering to the ground states for solutions starting on the centre-stable manifold associated with these states. This paper therefore provides the final instalment in the proof of this scattering property for the cubic Klein–Gordon equation in the radial case, see the recent theory of Nakanishi and the third author. The method developed here is quite general, and applicable to other spectral problems which arise in the theory of nonlinear equations.