Spectral analysis of tridiagonal Fibonacci Hamiltonians

Inverse Spectral Problems for Tridiagonal N by N Complex Hamiltonians

In this paper, the concept of generalized spectral function is introduced for finite-order tridiagonal symmetric matrices (Jacobi matrices) with complex entries. The structure of the generalized spectral function is described in terms of spectral data consisting of the eigenvalues and normalizing numbers of the matrix. The inverse problems from generalized spectral function as well as from spectral data are investigated. In this way, a procedure for construction of complex tridiagonal matrices having real eigenvalues is obtained.

Hölder Continuity of the Integrated Density of States for the Fibonacci Hamiltonian

We prove Hölder continuity of the integrated density of states for the Fibonacci Hamiltonian for any positive coupling, and obtain the asymptotics of the Hölder exponents for large and small couplings.

Tridiagonal substitution Hamiltonians

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice with Laplacian and potential terms modulated by a primitive invertible two-letter substitution. We investigate the spectrum and the spectral type, the fractal structure and fractal dimensions of the spectrum, exact dimensionality of the integrated density of states, and the gap structure. We present a review of previous results, some applications, and open problems. Our investigation is based largely on the dynamics of trace maps. This work is an extension of similar results on Schroedinger operators, alth…

Hyperbolicity of the Trace Map for the Weakly Coupled Fibonacci Hamiltonian

We consider the trace map associated with the Fibonacci Hamiltonian as a diffeomorphism on the invariant surface associated with a given coupling constant and prove that the non-wandering set of this map is hyperbolic if the coupling is sufficiently small. As a consequence, for these values of the coupling constant, the local and global Hausdorff dimension and the local and global box counting dimension of the spectrum of the Fibonacci Hamiltonian all coincide and are smooth functions of the coupling constant.

The Fibonacci Hamiltonian

We consider the Fibonacci Hamiltonian, the central model in the study of electronic properties of one-dimensional quasicrystals, and provide a detailed description of its spectrum and spectral characteristics (namely, the optimal Hölder exponent of the integrated density of states, the dimension of the density of states measure, the dimension of the spectrum, and the upper transport exponent) for all values of the coupling constant (in contrast to all previous quantitative results, which could be established only in the regime of small or large coupling). In particular, we show that the spectr…

The Density of States Measure of the Weakly Coupled Fibonacci Hamiltonian

We consider the density of states measure of the Fibonacci Hamiltonian and show that, for small values of the coupling constant $V$, this measure is exact-dimensional and the almost everywhere value $d_V$ of the local scaling exponent is a smooth function of $V$, is strictly smaller than the Hausdorff dimension of the spectrum, and converges to one as $V$ tends to zero. The proof relies on a new connection between the density of states measure of the Fibonacci Hamiltonian and the measure of maximal entropy for the Fibonacci trace map on the non-wandering set in the $V$-dependent invariant surf…

Quantum Dynamics via Complex Analysis Methods: General Upper Bounds Without Time-Averaging and Tight Lower Bounds for the Strongly Coupled Fibonacci Hamiltonian

We develop further the approach to upper and lower bounds in quantum dynamics via complex analysis methods which was introduced by us in a sequence of earlier papers. Here we derive upper bounds for non-time averaged outside probabilities and moments of the position operator from lower bounds for transfer matrices at complex energies. Moreover, for the time-averaged transport exponents, we present improved lower bounds in the special case of the Fibonacci Hamiltonian. These bounds lead to an optimal description of the time-averaged spreading rate of the fast part of the wavepacket in the large…

The Spectrum of the Weakly Coupled Fibonacci Hamiltonian

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 announce the following results and explain some key ideas that go into their proofs. The thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. Moreover, the length of every gap tends to zero linearly. Finally, for sufficiently small coupling, the sum of the s…

On Anomalous Lieb-Robinson Bounds for the Fibonacci XY Chain

We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for the isotropic XY chain with Fibonacci external magnetic field at arbitrary coupling. It is anomalous in that the usual exponential decay in $x-vt$ is replaced by exponential decay in $x-vt^α$ with $0<α<1$. In fact, we can characterize the values of $α$ for which such a bound holds as those exceeding $α_u^+$, the upper transport exponent of the one-body Fibonacci Hamiltonian. Following the approach of \cite{HSS11}, we relate Lieb-Robinson bounds to dynamical bounds for the one-body Hamiltonian corresponding t…

Spectral and Quantum Dynamical Properties of the Weakly Coupled Fibonacci Hamiltonian

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 sufficiently small coupling, the …

平衡态的傅里叶衰减与斐波那契哈密顿量

Fourier decay of equilibrium states and the Fibonacci Hamiltonian

研究曲面上非线性保面积公理 A 微分同胚平衡态的傅里叶衰减，利用和积现象及推广 Tsujii - Zhang 方法，得到自共形测度下傅里叶维数为正等结果，还证明了关于圆扩张和斐波那契哈密顿量的推论。

关于筛分 Schrödinger 算子的谱

On the Spectra of Sieved Schrödinger Operators

该研究基于 Fibonacci Hamiltonian 构造离散 Schrödinger 算子，证明谱维数在筛分下可不不变，并揭示筛分参数趋于无穷时谱局部 Hausdorff 维数趋于零的现象。