A counterexample to Fermi isospectral rigidity for two dimensional discrete periodic Schr\"odinger operators

Abstract

Using numerical certification, we prove the existence of a nontrivial real-valued two dimensional periodic potential whose associated discrete Schr\"odinger operator is Fermi isospectral to the zero potential. This provides a negative answer to a question posed by the third author concerning the rigidity of Fermi isospectrality in dimension two. This example also disproves a conjecture of Gieseker, Kn\"orrer, and Trubowitz in the 1990s stating that for any nontrivial real-valued periodic potential in dimension two, the Fermi variety is irreducible at all energy levels.