具有奇异周期势的一维薛定谔算子的谱隙
Spectral gaps of the one-dimensional Schrödinger operators with singular periodic potentials
摘要：
研究了具有实值1-周期分布势$q(x)\in H_{1{-}per}^{-1}(\mathbb{R})$的Hill-Schr dinger算子S（q）u=-u“+q（x）u，\quad u \in \mathrm{Dom}（S（q））的谱隙长度$\{γ_{n}(q)\}_{n=1}^{\infty}$的性质.我们表明，他们表现出相同的行为的傅立叶系数$\{\widehat{q}(n)\}_{n=-\infty}^{\infty}$的潜力$q(x)$相对于加权序列空间$h^{s,φ}$，$s>-1$，$φ\in \mathrm{SV}$。情况$q(x)\in L_{1{-}per}^{2}(\mathbb{R})$、$s\in \mathbb{Z}_{+}$、$φ\equiv 1$对应于马琴科-奥斯特洛夫斯基定理。
英文摘要：
The behaviour of the lengths of spectral gaps $\{γ_{n}(q)\}_{n=1}^{\infty}$ of the Hill-Schrödinger operators S(q)u=-u''+q(x)u,\quad u\in \mathrm{Dom}(S(q)) with real-valued 1-periodic distributional potentials $q(x)\in H_{1{-}per}^{-1}(\mathbb{R})$ is studied. We show that they exhibit the same behaviour as the Fourier coefficients $\{\widehat{q}(n)\}_{n=-\infty}^{\infty}$ of the potentials $q(x)$ with respect to the weighted sequence spaces $h^{s,φ}$, $s>-1$, $φ\in \mathrm{SV}$. The case $q(x)\in L_{1{-}per}^{2}(\mathbb{R})$, $s\in \mathbb{Z}_{+}$, $φ\equiv 1$ corresponds to the Marchenko-Ostrovskii Theorem.
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