Spectral analysis of θ-periodic Schrödinger operators and applications to periodic waves

Assuming a symmetric matrix-valued potential, we consider the associated θ-periodic Schrödinger operators Hθ on intervals [0,P], where in our applications P denotes the period of a stationary periodic solution to a nonlinear evolutionary PDE. We relate the Morse index of Hθ to certain Maslov indices, and apply our theory to operators obtained when Allen-Cahn equations and systems are linearized about stationary periodic solutions.