Hill's potentials in Hörmander spaces and their spectral gaps

Abstract

In the paper we study the behaviour of the lengths of spectral gaps $\{{\gamma }_q(n){\}}_{n\in \mathbb{N}}$ in a continuous spectrum of the Hill-Schrödinger operators $$S(q)u=-u"+q(x)u,x\in \mathbb{R},$$ with 1-periodic real-valued distribution potentials $$q(x)=\sum _{k\in \mathbb{Z}}\hat{q}(k)e^{ik2\pi x}\in H^{-1}(\mathbb{T}),\text{and}\hat{q}(k)=\overline{\hat{q}(-k)},k\in \mathbb{Z},$$ in dependence on the weight $\omega$ of the Hörmander spaces $H^{\omega }(\mathbb{T})\ni q$, $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Let $h^{\omega }(\mathbb{N})$ be a Hilbert space of weighted sequences. It is proved that $$ \{\hat{q}(\cdot)\}\in h^ω(\mathbb{N})\Leftrightarrow\{γ_{q}(\cdot)\}\in h^ω(\mathbb{N}) \leqno(\ast) $$ if a positive, in general non-monotonic, weight $\omega =\{\omega (k){\}}_{k\in \mathbb{N}}$ is inter-power one. In the case $q\in L^2(\mathbb{T})$, and $\omega (k)=(1+2k{)}^s$, $s\in {\mathbb{Z}}_{+}$, the statement $(\ast )$ is due to Marchenko and Ostrovskii (1975).

arXiv comment9 pages

Cited by in corpus (1)

• Elliptic operators on refined Sobolev scales on vector bundles