Eigenfunction localization for the 2D periodic Schr\"odinger operator

TL;DR

This paper proves that for a generic class of 2D periodic Schrödinger operators, eigenfunctions are exponentially localized around finitely many frequencies, with bounded $L^p$ norms, advancing understanding of spectral properties.

Contribution

It establishes exponential localization of eigenfunctions and bounded $L^p$ norms for a broad class of 2D periodic Schrödinger operators, answering a longstanding question.

Findings

Eigenfunctions are exponentially localized around finitely many frequencies.

The spectrum has finite multiplicity.

Eigenfunctions have bounded $L^p$ norms for all p > 0.

Abstract

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}.