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

TL;DR

This paper demonstrates that for periodic Schrödinger operators, isospectrality across all small Planck constants implies identical effective Hamiltonians obtained via homogenization, linking spectral properties to classical limits.

Contribution

It establishes a homogenization-based criterion for the inverse spectral problem of periodic Schrödinger operators, connecting quantum spectra to classical effective Hamiltonians.

Findings

Isospectral operators share the same effective Hamiltonian.

Homogenization provides a necessary condition for spectral equivalence.

Results relate quantum spectral limits to classical Hamilton-Jacobi solutions.

Abstract

We study the inverse spectral problem for periodic Schr\"odinger opera\-tors of 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 $V \in C^{\infty} (\Bbb T^n)$. 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 Schr\"odinger operators. We also provide a link between our result and the spectral limit of quantum integrable systems.