Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

Details Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

2009-11-17

arxiv.org/abs/0911.3218v1

Mathematics

Spectral Theory

Scientific paper

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.

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