Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

Abstract

We consider the Hill operator $$ Ly = - y^{\prime \prime} + v(x)y, \quad 0 \leq x \leq \pi, $$ subject to periodic or antiperiodic boundary conditions, with potentials $v$ which are trigonometric polynomials with nonzero coefficients, of the form (i) $ ae^{-2ix} +be^{2ix}; $ (ii) $ ae^{-2ix} +Be^{4ix}; $ (iii) $ ae^{-2ix} +Ae^{-4ix} + be^{2ix} +Be^{4ix}. $ Then the system of eigenfunctions and (at most finitely many) associated functions is complete but it is not a basis in $L^2 ([0,\pi], \mathbb{C})$ if $|a| \neq |b| $ in the case (i), if $|A| \neq |B| $ and neither $-b^2/4B$ nor $-a^2/4A$ is an integer square in the case (iii), and it is never a basis in the case (ii) subject to periodic boundary conditions.