Lipshitz continuity of gap boundaries for Hofstadter-like spectra
Lipshitz continuity of gap boundaries for Hofstadter-like spectra
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类 Hofstadter 光谱的间隙边界的 Lipshitz 连续性
DOI：
10.1007/bf02173432
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发表时间：
1994
影响因子：
2.4
通讯作者：
J. Bellissard
中科院分区：
文献类型：
--
作者：
J. Bellissard
本文考虑一个表示单带二维电子在均匀磁场中运动的有效哈密顿H。H属于旋转代数，即非交换2-环面上的连续函数代数。我们定义了一个非交换模拟的光滑函数的类Cl，n的元素，其中l和n分别表征的程度，相对于磁场和环面变量的可微性。我们证明了，如果H是C_（1，3+ε）类，则H的能谱的差距边界是磁场在每个差距开放点处的Lipshitz连续函数.
We consider an effective HamiltonianH representing the motion of a single-band-two-dimensional electron in a uniform magnetic field. ThenH belongs to the rotation algebra, namely the algebra of continuous functions over a non-commutative 2-torus. We define a non-commutative analog of smooth functions by mean of elements of classCl,n, wherel andn characterize respectively the degree of differentiability with respect to the magnetic field and the torus variables. We show that ifH is of classC1,3+ε, the gap boundaries of the spectrum ofH are Lipshitz continuous functions of the magnetic field at each point for which the gap is open.