Quantifying the Size of Spectra

Motivated by aperiodic operators with Cantor-type spectra, Chapter 7 develops algorithms to quantify spectral size. We study five notions central in physics-driven numerics and conjecture formation: number of connected components, Lebesgue measure, logarithmic capacity, box-counting dimension, and Hausdorff dimension. Each is computed from spectral covers (finite unions of intervals enclosing the spectrum), where adaptive refinement is essential for fractal dimensions. This approach is dual to Chapter 3’s approximations 'from below'. We consider three increasingly general information models and their Solvability Complexity Index (SCI) classifications, pinpointing exactly what data are needed for each notion of size. When only distance-to-spectrum estimates are available, we introduce Swiss-cheese covers that exclude gaps. Sharp lower bounds (impossibility results) are proved in several regimes, including limit-periodic discrete Schrödinger classes. The algorithms are demonstrated in state-of-the-art computations for the almost Mathieu operator (Hofstadter butterfly), Fibonacci Hamiltonians (canonical 1D quasicrystal), and Penrose tiling models (canonical 2D quasicrystal).