I am looking for specific references regarding the fact that there are quasiperiodic and, also, limit-periodic potentials $q(x)$ such that the $L^2(\mathbb{R})$-spectrum $\sigma(H)$ of the operator \begin{equation*} Hu = -\dfrac{d^2u}{dx^2} + q(x) u \end{equation*} is a closed interval, i.e. $\sigma(H) = [\lambda_0, \infty)$.
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