NLS and KdV Hamiltonian linearized operators: A priori bounds on the spectrum and optimal [formula omitted] estimates for the semigroups

Abstract

Motivated by the NLS and KdV linearizations near traveling waves, we study general forms of such operators. We prove a priori bounds on the unstable spectrum, by showing that if any unstable spectrum exists, it is contained in a strip around the real axis, with an explicit estimate of its width in terms of the potentials. To the best of our knowledge, this is the first result of this nature in the literature. We show that all sufficiently large (relative to the potential) pure imaginary eigenvalues are necessarily simple. In the case of spectral stability, we show optimal, at most polynomial in time, L2 bounds for the associated semigroups generated such linearized operators. As it is for finite matrices, the power rate matches the maximal size of any Jordan block minus one.