  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.
