The Spectrum of a Schr\"odinger Operator With Small Quasi-Periodic Potential is Homogeneous
The Spectrum of a Schr\"odinger Operator With Small Quasi-Periodic Potential is Homogeneous
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小准周期势薛定谔算子的谱是齐次的
DOI：
10.4171/jst/128
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Milivoje Lukic
中科院分区：
文献类型：
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作者：
D. Damanik;Michael Goldstein;Milivoje Lukic
本文考虑L^2中的拟周期Schr\“odinger算子[H \psi]（x）= -\psi\”（x）+ V（x）\psi（x）（\mathbb{R}）$，其中电势由$$ V（x）= \sum_{m \in \mathbb{Z}^\nu \setminus \{ 0 \}} c（m）\exp给出（2\pi i m \omega x）$$具有丢番图频率向量$\omega =（\omega_1，\dots，\omega_\nu）\in \mathbb{R}^\nu$和指数衰减傅立叶系数$|c（m）|\le \varepad\exp（-\kappa_0| M|)$.在小$\vareps> 0$的制度下，我们证明了算子$H$的谱在Carleson意义下是齐次的。
We consider the quasi-periodic Schr\"odinger operator $$ [H \psi](x) = -\psi"(x) + V(x) \psi(x) $$ in $L^2(\mathbb{R})$, where the potential is given by $$ V(x) = \sum_{m \in \mathbb{Z}^\nu \setminus \{ 0 \}} c(m)\exp (2\pi i m \omega x) $$ with a Diophantine frequency vector $\omega = (\omega_1, \dots, \omega_\nu) \in \mathbb{R}^\nu$ and exponentially decaying Fourier coefficients $|c(m)| \le \varepsilon \exp(-\kappa_0|m|)$. In the regime of small $\varepsilon > 0$ we show that the spectrum of the operator $H$ is homogeneous in the sense of Carleson.