Spectral Approximation for Quasiperiodic Jacobi Operators

Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals and in more general studies of structures exhibiting aperiodic order. The spectra of these self-adjoint operators can be quite exotic, such as Cantor sets, and their fine properties yield insight into the associated quantum dynamics, that is, the one-parameter unitary group that solves the time-dependent Schrödinger equation. Quasiperiodic operators can be approximated by periodic ones, the spectra of which can be computed via two finite dimensional eigenvalue problems. Since long periods are necessary for detailed approximations, both computational efficiency and numerical accuracy become a concern. We describe a simple method for numerically computing the spectrum of a period- K Jacobi operator in O ( K 2 ) operations, then use the algorithm to investigate the spectra of Schrödinger operators with Fibonacci, period doubling, and Thue–Morse potentials.