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

We study Schrodinger operators with Floquet boundary conditions on flat tori obtaining a spectral result giving an asymptotic expansion of all the eigenvalues. The expansion is in $\lambda^{-\delta}$, with $\delta$ in (0,1), for most of the eigenvalues $\lambda$ (stable eigenvalues), while it is a "directional expansion" for the remaining eigenvalues (unstable eigenvalues). The proof is based on a structure theorem which is a variant of the one proved in [31,32] and on a new iterative quasimode argument.