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

TL;DR

This paper constructs a specific two-dimensional periodic Schr"odinger operator that shares the same Fermi variety as the zero potential, providing a counterexample to previous conjectures about spectral rigidity.

Contribution

It presents the first explicit counterexample to Fermi isospectral rigidity in two dimensions, disproving longstanding conjectures.

Findings

Existence of a nontrivial potential Fermi isospectral to zero

Disproof of the irreducibility conjecture for Fermi varieties in 2D

Numerical certification of the counterexample

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.