Smoothness of Hill's potential and lengths of spectral gaps

TL;DR

This paper characterizes the relationship between spectral gap lengths of Hill-Schrödinger operators with periodic potentials and specific weighted function spaces, establishing a precise correspondence under certain conditions.

Contribution

It proves a bijective mapping between spectral gap lengths and weighted function spaces for Hill operators, extending understanding of spectral properties in relation to potential smoothness.

Findings

Spectral gap lengths correspond exactly to weighted function space elements.

The mapping between potentials and spectral gaps is bijective under specified conditions.

The function spaces involved are identified as real Hörmander spaces with particular weights.

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 $\omega(\sqrt{1+\xi^{2}})$.