Smoothness of Hill's potential and lengths of spectral gaps

Details Smoothness of Hill's potential and lengths of spectral gaps Smoothness of Hill's potential and lengths of spectral gaps

2010-03-25

arxiv.org/abs/1003.5000v1

Mathematics

Spectral Theory

8 pages

Scientific paper

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

Affiliated with

Also associated with

No associations

Rating

Smoothness of Hill's potential and lengths of 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 Smoothness of Hill's potential and lengths of spectral gaps, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Smoothness of Hill's potential and lengths of spectral gaps will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-181561