Spectral Rigidity for Periodic Schr\"odinger Operators in Dimension 2

We consider two dimensional real-valued analytic potentials for the Schr\"odinger equation which are periodic over a lattice $L$. Under certain assumptions on the form of the potential and the lattice $L$, we can show there is a large class of analytic potentials which are Floquet rigid and dense in the set of $C^\infty(R^2/L)$ potentials. The result extends the work of Eskin et. al, in "On isospectral periodic potentials in $R^n$, II."