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We study the inverse spectral problem for periodic Schr\"odinger opera\-torsof kind $- \frac{1}{2} \hbar^2 \Delta_x + V(x)$ on the flat torus $\Bbb T^n :=(\Bbb R / 2 \pi \Bbb Z)^n$ with potentials $V \in C^{\infty} (\Bbb T^n)$. Weshow that if two operators are isospectral for any $0 < \hbar \le 1$ then theyhave the same effective Hamiltonian given by the periodic homogenization ofHamilton-Jacobi equation. This result provides a necessary condition for theisospectrality of these Schr\"odinger operators. We also provide a link betweenour result and the spectral limit of quantum integrable systems.

An homogenization approach for the inverse spectral problem of periodic Schr\"odinger operators