Isospectral periodic Torii in dimension 2
The focus of this paper is the class of Schrödinger operators \[ -\sum_{j=1}^2\frac{\partial^2}{\partial x_j^2}+q(x), \] where \(q(x):\mathbb{R}^2\to\mathbb{R}\) is a real-valued periodic potential over a lattice \(\mathbb{L}\subset\mathbb{R}^2\). It is assumed that \(\mathbb{L}\) has the property that \(|d|=|d'|\) implies \(d=\pm d'\) for all \(d,d'\) in \(\mathbb{L}\). The Floquet spectrum of the Schrödinger operator is the set of \(\lambda \in {\mathbb R}^1\) such that the eigenvalue problem \((-\Delta +q)u=\lambda u\) has a solution which satisfy \(u(x+d)=e^{2\pi ik\cdot d}u(x)\), \(x\in {\mathbb R}^2\), \(d\in {\mathbb L}\), where \(k\) ranges over \({\mathbb R}^2\). A potential is called Floquet rigid if there are only a finite number of potentials modulo translations which are Floquet isospectral to it. The author considers analytic periodic potentials which break down into a finite number of one-dimensional finite gap potentials \[ q(x)=\sum_{j=1}^S |\delta_j|^2 q_j(\delta_j\cdot x). \] It is proved that potentials from this class are Floquet rigid and dense in the set of \(C^\infty(\mathbb{R}^2 / \mathbb{L})\) potentials. The result extends the work of \textit{G. Eskin} et al. [Commun. Pure Appl. Math. 37, 715--753 (1984; Zbl 0582.35031)].
scientific article; zbMATH DE number 6520568
scientific article