Riesz basis property of Hill operators with potentials in weighted spaces

Abstract:

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 $v(x)= \sum \nolimits _{\kern 1pt 2\mathbb {Z}} V(k) e^{ikx}$ 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 \[ V(-2n) \asymp V(2n) \quad \text {as}~n \in 2\mathbb {N}~(\text {or}~n \in 1+ 2\mathbb {N}),\quad n \to \infty .\] For such potentials we prove that $\lambda _n^+ - \lambda _n^- \sim \pm 2\sqrt {V(-2n)V(2n)}$ and\[ \mu _n - \frac {1}{2}(\lambda _n^+ + \lambda _n^-) \sim -\frac {1}{2} (V(-2n) + V(2n)).\]