Finite Fermi Isospectrality

Given a discrete operator L on a Z^d-periodic graph with a periodic potential,
Floquet theory refines the spectrum of L in terms of the unitary characters of
Z^d.  This gives the Floquet Variety whose level sets at a fixed energy are Fermi
varieties.  A natural question is how much do the resulting Floquet and Fermi
varieties determine the potential and parameters of the graph?  Potentials with
the same Floquet (Fermi) varieties are Floquet (Fermi) isospectral.  For the
Schroediniger operator on the grid graph on Z^d acted on by the free abelian
subgroup q_1Z + ... + q_d Z (q_i are pairwise coprime), Kappeler showed that
there are only finitely many potentials Floquet isospectral to a given
potential.  Liu introduced the term Fermi isospectrality and considered it for
separable potentials when d=2. 

This talk will discuss this history and present continuing work with Faust and
Liu studying Fermi isospectrality for the grid graph when d=2.