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arXiv · 1802.08149

An homogenization approach for the inverse spectral problem of periodic Schrödinger operators

Lorenzo Zanelli
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Abstract

We study the inverse spectral problem for periodic Schrödinger opera\-tors of kind $- \frac{1}{2} \hbar^2 Δ_x + V(x)$ on the flat torus $\Bbb T^n := (\Bbb R / 2 π\Bbb Z)^n$ with potentials $V \in C^{\infty} (\Bbb T^n)$. We show that if two operators are isospectral for any $0 < \hbar \le 1$ then they have the same effective Hamiltonian given by the periodic homogenization of Hamilton-Jacobi equation. This result provides a necessary condition for the isospectrality of these Schrödinger operators. We also provide a link between our result and the spectral limit of quantum integrable systems.

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Lorenzo Zanelli. 2018-02-26. An homogenization approach for the inverse spectral problem of periodic Schrödinger operators. https://arxiv.org/abs/1802.08149

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