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
$$ V(-2n)\asymp V(2n) \text{ as } n\in2\mathbb{N}\ (\text{or } n\in1+2\mathbb{N}), \quad n\to\infty. $$
For such potentials we prove that $\lambda_n^+-\lambda_n^-\sim\pm 2\sqrt{V(-2n)V(2n)}$ and
$$ \mu_n-\frac12(\lambda_n^++\lambda_n^-)\sim-\frac12(V(-2n)+V(2n)). $$

References: 32 entries.