Hörmander空间中的Hill势及其谱隙
Hill's potentials in Hörmander spaces and their spectral gaps
摘要：
本文研究了具有1-周期实值分布势$$q(x)=\sum_{k\in \mathbb{Z}}\hat{q}(k) e^{i k 2πx}\in H^{-1}(\mathbb{T}),\quad\text{and}\quad\hat{q}(k)=\bar{\hat{q}(-k)}, k\in \mathbb{Z},$$的Hill-Schrödinger算子$$S(q)u=-u"+q(x)u,\quad x\in\mathbb{R},$$的连续谱中谱隙$\{γ_{q}(n)\}_{n\in \mathbb{N}}$的长度依赖于Hörmander空间$H^ω(\mathbb{T})\ni q$，$\mathbb{T}=\mathbb{R}/\mathbb{Z}$的权$ω$的行为.设$h^ω(\mathbb{N})$是加权序列的Hilbert空间。证明了如果正的、一般非单调的权$ω=\{ω(k)\}_{k\in \mathbb{N}}$是幂间权，则$$ \{\hat{q}(\cdot)\}\in h^ω(\mathbb{N})\Leftrightarrow\{γ_{q}(\cdot)\}\in h^ω(\mathbb{N}) \leqno(\ast) $$是幂间权.在$q\in L^{2}(\mathbb{T})$和$ω(k)=(1+2k)^{s}$、$s\in \mathbb{Z}_{+}$的例子中，$(\ast)$的陈述是由Marchenko和Ostrovskii（1975）提出的。
英文摘要：
In the paper we study the behaviour of the lengths of spectral gaps $\{γ_{q}(n)\}_{n\in \mathbb{N}}$ in a continuous spectrum of the Hill-Schrödinger operators $$S(q)u=-u"+q(x)u,\quad x\in\mathbb{R},$$ with 1-periodic real-valued distribution potentials $$q(x)=\sum_{k\in \mathbb{Z}}\hat{q}(k) e^{i k 2πx}\in H^{-1}(\mathbb{T}),\quad\text{and}\quad\hat{q}(k)=\bar{\hat{q}(-k)}, k\in \mathbb{Z},$$ in dependence on the weight $ω$ of the Hörmander spaces $H^ω(\mathbb{T})\ni q$, $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Let $h^ω(\mathbb{N})$ 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 $ω=\{ω(k)\}_{k\in \mathbb{N}}$ is inter-power one. In the case $q\in L^{2}(\mathbb{T})$, and $ω(k)=(1+2k)^{s}$, $s\in \mathbb{Z}_{+}$, the statement $(\ast)$ is due to Marchenko and Ostrovskii (1975).
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