Eigenfunction Localization for the 2D Periodic Schrödinger Operator

We prove that for any fixed trigonometric polynomial potential satisfying a genericity condition, the spectrum of the two-dimensional periodic Schrödinger operator has finite multiplicity and the Fourier series of the eigenfunctions are uniformly exponentially localized about a finite number of frequencies. As a corollary, the L^p norms of the eigenfunctions are bounded for all p > 0, which answers a question of Toth and Zelditch [15].