The discrete periodic Schrödinger operator on a lattice maps a potential function to a spectrum — a set of eigenvalues at each quasimomentum. The Fermi variety at a given energy is the set of quasimomenta where that energy is an eigenvalue. Knowing the Fermi variety at every energy is the strongest spectral information you can extract from the operator. The question: does this information determine the potential? If two potentials have the same Fermi variety at every energy, must they be the same potential?
This is Fermi isospectral rigidity. In one dimension, the answer is yes — the Fermi variety determines the potential up to translation. The question in two dimensions, posed by Gieseker, Knörrer, and Trubowitz in the 1990s, was whether the same rigidity holds.
Brysiewicz, Faust, and Liu (arXiv:2603.11183, 2026) prove it does not. They construct a nontrivial real-valued periodic potential in two dimensions whose Fermi variety at every energy is identical to that of the zero potential. The potential is there — it deforms the local physics at every lattice site — but no spectral measurement at any energy can detect it. The operator with the potential and the operator without are Fermi isospectral.
The counterexample also disproves a secondary conjecture: that the Fermi variety of any nontrivial potential is irreducible at all energies. The constructed potential produces a reducible Fermi variety, which is what enables the isospectrality — the variety factors into components that can rearrange without changing the total set.
The structural point: the spectral data that determines a one-dimensional potential completely fails to determine a two-dimensional one. The additional spatial dimension creates enough algebraic room for distinct potentials to produce identical spectra. Dimensionality does not merely increase the difficulty of the inverse problem — it changes whether the problem has a unique solution.