Spectral asymptotics of all the eigenvalues of Schr\"odinger operators on flat tori

We study Schr"odinger 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 \cite{PS10,PS12} and on a new iterative quasimode argument.