Spectral asymptotics of all the eigenvalues of Schrödinger operators on flat tori

We study Schrödinger operators with Floquet boundary conditions on flat tori obtaining a spectral result giving an asymptotic expansion of all the eigenvalues. The is in λ−δ δ∈(0,1) for most eigenvalues λ (stable eigenvalues), while it “directional expansion” remaining (unstable eigenvalues). proof based structure theorem which variant one proved [31], [32] and new iterative quasimode argument.