On the gap property of a linearized NLS operator

We consider a pair of linear operators corresponding to the linearization around the ground state soliton of the cubic nonlinear Schrödinger equation in dimension three. We introduce a new comparison-based approach and rigorously prove that the interval ( 0 , 1 ] does not contain any eigenvalues of these operators. Furthermore we show the absence of resonances at the bottom of the essential spectrum. All the obtained results are for the fully non-radial case. The method can be adapted to many other spectral problems.