A rigidity result for effective Hamiltonians with 3-mode periodic potentials

A main result in this paper is that, if the dimension n=2, and each of V1,V2 contains exactly 3 mutually non-parallel Fourier modes, thenH‾1≡H‾2⇔V1(x)=V2(xc+x0) for all x∈T2=R2/Z2, for some c∈Q∖{0} and x0∈T2. When n≥3, the scenario is slightly more subtle, and a complete description is provided for any dimension. These resolve partially a conjecture stated in [14]. Some other related results and open problems are also discussed.