We study fractal dimension properties of singular Jacobi operators. We prove quantitative lower spectral/quantum dynamical bounds for general operators with strong repetition properties and controlled singularities. For analytic quasiperiodic Jacobi operators in the positive Lyapunov exponent regime, we obtain a sharp arithmetic criterion of full spectral dimensionality. The applications include the extended Harper’s model where we obtain arithmetic results on spectral dimensions and quantum dynamical exponents.

中文翻译：

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β-概周期奇异Jacobi算子的谱维和扩展的Har​​per模型

我们研究奇异Jacobi算子的分形维数性质。我们证明了具有强重复特性和受控奇异性的一般算子的定量较低的频谱/量子动力学界。对于正Lyapunov指数系统中的解析拟周期Jacobi算子，我们获得了完整光谱维数的清晰算术准则。应用包括扩展的Har​​per模型，在该模型中我们获得了光谱尺寸和量子动力学指数的算术结果。