Instabilities in the Bogoliubov Spectrum of a condensate in a 1-D periodic potential

We study the stability of standing wave solutions to a one-dimensional Gross-Pitaevsky equation with a periodic potential. We use some simple complex analysis and the Hamiltonian structure of the problem to give a simple rigorous criterion which guarantees the existence of non-real spectrum, which corresponds to exponential instability of the standing wave solution. This criterion can be stated simply in terms of the spectrum of one of these self-adjoint operators. When the standing wave has small amplitude this criterion simplifies further, and agrees with arguments based on the effective mass in the periodic potential.