Numerical verification of a gap condition for linearized NLS

We make a detailed numerical study of the spectrum of two Schroedinger operators L_- and L_+ arising in the linearization of the supercritical nonlinear Schroedinger equation (NLS) about the standing wave, in three dimensions. This study was motivated by a recent result of the second author on conditional asymptotic stability of solitary waves in the case of a cubic nonlinearity. Underlying the validity of this result is a spectral condition on the operators L_- and L_+, namely that they have no eigenvalues nor resonances in the gap (a region of the positive real axis between zero and the continuous spectrum,) which we call the gap property. The present numerical study verifies this spectral condition, and further shows that the gap property holds for NLS exponents of the form 2*beta + 1, as long as beta_* < beta <= 1, where beta_* = 0.913958905 +- 1e-8. Our strategy consists of rewriting the original eigenvalue problem via the Birman-Schwinger method. From a numerical analysis viewpoint, our main contribution is an efficient quadrature rule for the kernel 1/|x-y| in R^3, i.e., provably spectrally accurate. As a result, we are able to give similar accuracy estimates for all our eigenvalue computations. We also propose an improvement of the Petviashvili's iteration for the computation of standing wave profiles which automatically chooses the radial solution.