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

TL;DR

This paper investigates spectral rigidity of two-dimensional periodic Schrödinger operators, demonstrating that under specific conditions, a large class of potentials are Floquet rigid and dense among smooth potentials, extending previous isospectral results.

Contribution

It establishes Floquet rigidity for a broad class of 2D periodic Schrödinger potentials, extending prior isospectral work to a more general setting.

Findings

Large class of Floquet rigid potentials identified

Rigidity results extend to dense subsets of smooth potentials

Builds upon and generalizes previous isospectral findings

Abstract

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."