Riesz basis property of Hill operators with potentials in weighted spaces

Abstract

Consider the Hill operator $L(v)=-d^2/d{x}^2+v(x)$ 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 $$V(-2n)\asymp V(2n)\text{as}n\in 2\mathbb{N}(\text{or}n\in 1+2\mathbb{N}),n\to \infty .$$ For such potentials we prove that ${\lambda }_n^{+}-{\lambda }_n^{-}\sim \pm 2\sqrt{V(-2n)V(2n)}$ and $${\mu }_n-\frac{1}{2}({\lambda }_n^{+}+{\lambda }_n^{-})\sim -\frac{1}{2}(V(-2n)+V(2n)).$$