Spectral and Quantum Dynamical Properties of the Weakly Coupled Fibonacci Hamiltonian
Spectral and Quantum Dynamical Properties of the Weakly Coupled Fibonacci Hamiltonian
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DOI：
10.1007/s00220-011-1220-2
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发表时间：
2010-01
影响因子：
2.4
通讯作者：
D. Damanik;A. Gorodetski
中科院分区：
文献类型：
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作者：
D. Damanik;A. Gorodetski
我们考虑的耦合常数的小值的斐波那契哈密顿量的频谱。已知这个集合是一个勒贝格测度为零的康托集合。在这里，我们研究的限制，作为耦合常数的值接近零，其厚度和Hausdorff维数。我们证明，厚度趋于无穷大，因此，Hausdorff维数的频谱趋于1。我们还表明，在小耦合，所有的间隙允许的差距标记定理是开放的，每个间隙的长度趋于零线性。此外，对于足够小的耦合，频谱与自身的和是一个间隔。这最后一个结果提供了一个严格的解释现象的斐波那契平方格发现的数字埃文达尔曼德尔和Lifshitz。最后，我们为差分方程的解提供了显式的上界和下界，并使用它们来研究谱测量和传输指数。
We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We prove that the thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. We also show that at small coupling, all gaps allowed by the gap labeling theorem are open and the length of every gap tends to zero linearly. Moreover, for a sufficiently small coupling, the sum of the spectrum with itself is an interval. This last result provides a rigorous explanation of a phenomenon for the Fibonacci square lattice discovered numerically by Even-Dar Mandel and Lifshitz. Finally, we provide explicit upper and lower bounds for the solutions to the difference equation and use them to study the spectral measures and the transport exponents.