Mathematics – Spectral Theory
Scientific paper
2009-05-28
Methods Funct. Anal. Topology 17 (2011), no. 3, 235-243
Mathematics
Spectral Theory
9 pages
Scientific paper
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-Schr\"{o}dinger operators $$S(q)u=-u"+q(x)u,\quad x\in\mathbb{R},$$ with 1-periodic real-valued distribution potentials $$q(x)=\sum_{k\in \mathbb{Z}}\hat{q}(k) e^{i k 2\pi x}\in H^{-1}(\mathbb{T}),\quad\text{and}\quad\hat{q}(k)=\bar{\hat{q}(-k)}, k\in \mathbb{Z},$$ in dependence on the weight $\omega$ of the H\"ormander 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 $$ \{\hat{q}(\cdot)\}\in h^{\omega}(\mathbb{N})\Leftrightarrow\{\gamma_{q}(\cdot)\}\in h^{\omega}(\mathbb{N}) \leqno(\ast) $$ 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).
Mikhailets Vladimir
Molyboga Volodymyr
No associations
LandOfFree
Hill's potentials in Hörmander spaces and their spectral gaps does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Hill's potentials in Hörmander spaces and their spectral gaps, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hill's potentials in Hörmander spaces and their spectral gaps will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-294099