Riesz basis property of Hill operators with potentials in weighted spaces

Consider the Hill operator $L(v) = - d^2/dx^2 + v(x) $ on $[0,\pi]$ with Dirichlet, periodic or antiperiodic boundary conditions; then for large enough $n$ close to $n^2 $ there are one Dirichlet eigenvalue $\mu_n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $\lambda_n^-, \, \lambda_n^+ $ (counted with multiplicity). We describe classes of complex potentials $v(x)= \sum_{2\mathbb{Z}} V(k) e^{ikx}$ in weighted spaces (defined in terms of the Fourier coefficients of $v$) such that the periodic (or antiperiodic) root function system of $L(v) $ contains a Riesz basis if and only if $$V(-2n) \asymp V(2n) \quad \text{as} \;\; n \in 2\mathbb{N}\;\; (\text{or} \; n \in 1+ 2\mathbb{N}), \;\; n \to \infty.$$ For such potentials we prove that $\lambda_n^+ - \lambda_n^- \sim \pm 2\sqrt{V(-2n)V(2n)} $ and $$\mu_n - \frac{1}{2}(\lambda_n^+ + \lambda_n^-) \sim -\frac{1}{2} (V(-2n) + V(2n)).$$