Isospectral periodic Torii in dimension 2

Waters, Alden

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 6, pp. 1173-1188.

Nous considérons les potentiels analytiques à valeurs réelles en dimension deux pour l'equation de Schrödinger qui sont périodiques sur un réseau $𝕃$. Sous certaines hypothèses sur la forme du potentiel et du réseau $𝕃$, nous montrons qu' il y a une grande classe de potentiels analytiques Floquet rigides et denses dans l'ensemble de $C^{\infty }({ℝ}^2/𝕃)$ potentiels. Ce résultat prolonge le travail de Eskin et al., dans “Les potentiels périodiques isospectraux dans ${ℝ}^n$, II.”

We consider two dimensional real-valued analytic potentials for the Schrödinger equation which are periodic over a lattice $𝕃$. Under certain assumptions on the form of the potential and the lattice $𝕃$, we can show there is a large class of analytic potentials which are Floquet rigid and dense in the set of $C^{\infty }({ℝ}^2/𝕃)$ potentials. The result extends the work of Eskin et al., in “On isospectral periodic potentials in ${ℝ}^n$, II.”

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