Smoothness of Hill's potential and lengths of spectral gaps

Abstract

Let ${\gamma_q(n)}_{n \in \mathbb{N}}$ be the lengths of spectral gaps in a continuous spectrum of the Hill-Schr\"odinger operators S(q)u=-u''+q(x)u,\quad x\in \mathbb{R}, with 1-periodic real-valued potentials $q \in L^{2}(\mathbb{T})$. Let weight function $\omega:\;[1,\infty)\to (0,\infty)$. We prove that under the condition \exists s\in [0,\infty):\quad k^{s}\ll\omega(k)\ll k^{s+1},\; k\in \mathbb{N}, the map $\gamma:\, q \mapsto \{\gamma_{q}(n)}_{n \in \mathbb{N}}$ satisfies the equalities: \verb"i")\quad \gamma(H^{\omega}) = h_{+}^{\omega}, \verb"ii")\quad \gamma^{-1}(h_{+}^{\omega}) = H^{\omega}, where the real function space H^{\omega} & ={f=\sum_{k\in \mathbb{Z}}\hat{f}\,(k)e^{i k2\pi x}\in L^{2}(\mathbb{T})| \sum_{k\in \mathbb{N}} \omega^{2}(k)|\hat{f}(k)|^{2}<\infty,\; \hat{f}(k)=\bar{\hat{f}(-k)},\;k\in \mathbb{Z}.}, and h^{\omega} = {a=\{a(k)\}_{k\in \mathbb{N}}|\sum_{k\in \mathbb{N}}\omega^{2}(k)|a(k)|^{2}<\infty.}, h_{+}^{\omega} = {a=\{a(k)\}_{k\in \mathbb{N}}\in h^{\omega}| a(k)\geq 0.}. If the weight $\omega$ is such that \exists a>1,c>1:\qquad c^{-1}\leq \frac{\omega(\lambda t)}{\omega (t)}\leq c\quad\forall t\geq 1,\;\lambda\in [1,a] then the function class $H^{\omega}$ is a real H\"ormander space $H_{2}^{\omega}(\mathbb{T},\mathbb{R})$ with the weight $\omega(\sqrt{1+\xi^{2}})$.