Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

Abstract

We consider the Hill operator $$Ly=-y^{\prime \prime }+v(x)y,0\le x\le \pi ,$$ subject to periodic or antiperiodic boundary conditions, with potentials $v$ which are trigonometric polynomials with nonzero coefficients, of the form (i) $a{e}^{-2ix}+b{e}^{2ix};$ (ii) $a{e}^{-2ix}+B{e}^{4ix};$ (iii) $a{e}^{-2ix}+A{e}^{-4ix}+b{e}^{2ix}+B{e}^{4ix}.$ 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 $|a|\ne |b|$ in the case (i), if $|A|\ne |B|$ 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.

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