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

Abstract

We study the inverse spectral problem for periodic Schrödinger opera\-tors of kind $-\frac{1}{2}{\hslash }^2{\Delta }_x+V(x)$ on the flat torus ${\mathbb{T}}^n:=(\mathbb{R}/2\pi \mathbb{Z}{)}^n$ with potentials $V\in C^{\infty }({\mathbb{T}}^n)$. We show that if two operators are isospectral for any $0<\hslash \le 1$ 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.

References in corpus (1)

• Lecture notes on Mather's theory for Lagrangian systems