Endpoints of the spectrum of periodic operators are generically simple.
The paper deals with the Schrödinger operator in \({\mathbb R}^n\), \[ H(V)=-\Delta+V, \] where the potential \(V\) is a real measurable \({\mathbb Z}^n\)-periodic bounded function. The spectrum of \(H(V)\) is a union of bands \[ B_j=\{\lambda_j(k,V)\mid k\in{\mathbb T}^n\}, \] where \(\lambda_j(k,V)\) are the eigenvalues (counted with multiplicities, and in increasing order) of the reduced operators \[ H(k,V)=-(\partial+2\pi\text{i}k)^2+V \] in \(L^2({\mathbb T}^n)\). A point \(\lambda\) in the spectrum of \(H(V)\) is called simple if there exists a unique \(j\geq1\) such that \(\lambda_j(k,V)=\lambda\) for some \(k\in{\mathbb T}^n\). Let \({\mathcal V}_l\) be the set of bounded \({\mathbb Z}^n\)-periodic potentials \(V\) for which \(H(V)\) has at least \(l\) open gaps. Then, the main result of the paper under review shows that the set of potentials \(V\) for which the endpoints of at least \(l\) gaps in the spectrum of \(H(V)\) are simple is a dense open subset of \({\mathcal V}_l\) with respect to the \(L^\infty({\mathbb T}^n)\)-topology.
scientific article; zbMATH DE number 1757312