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We study Schr\"odinger operators with Floquet boundary conditions on flattori obtaining a spectral result giving an asymptotic expansion of all theeigenvalues. The expansion is in $\lambda^{-\delta}$ with $\delta\in(0,1)$ formost 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 provedin \cite{PS10,PS12} and on a new iterative quasimode argument.

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