Periodic potentials for which all gaps are nontrivial (Q1392324)

The authors consider the selfadjoint periodic Schrödinger operator \(H\) (i.e., Hill operator) defined as the closure in \(L^2 (\mathbb{R})\) of the symmetric operator \(-y''+ U(x)y\) on \(C_0^{\infty}(\mathbb{R})\), where \(U(x)\) is a \(2T\)-periodic square integrable function on the closed interval \([-T, T]\). The operator \(H\) has an absolutely continuous spectrum consisting of a sequence of closed intervals separated by gaps. It is proved that all gaps in the spectrum of \(H\) are nontrivial if the nonconstant \(2T\)-periodic function \(U\) defined on \([-T, T]\) extends to be an even real-analytic convex function on the whole real axis.