Eigenfunction localization for the 2D periodic Schrödinger operator

We prove that for any {\it fixed} trigonometric polynomial potential satisfying a genericity condition, the spectrum of the two dimension 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 \cite{TZ}.

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