Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

Abstract

We consider the Hill operator $$Ly = - y^{\prime \prime} + v(x)y, \quad0 \leq x \leq \pi,$$ subject to periodic or antiperiodic boundary conditions, with potentials v which are trigonometric polynomials with nonzero coefficients, of the form 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| ≠ |b| in the case (i), if |A| ≠ |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.