We prove that for any { \it fixed} trigonometric polynomial potential satisfying a genericity condition, the spectrum of the two dimension periodic Schr\"odinger 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 \cite{ TZ}.
Categories: math.AP math.SP