Zero measure spectrum for the almost Mathieu operator

DOI10.1007/BF02101708zbMath0814.11040OpenAlexW4251989922MaRDI QIDQ1335250

Yoram Last

Publication date: 27 September 1994

Published in: Communications in Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/bf02101708

zbMATH Keywords

spectrumHausdorff dimensionLebesgue measureupper and lower estimatesalmost Mathieu operatorHarper's operator

Mathematics Subject Classification ID

Metric theory of other algorithms and expansions; measure and Hausdorff dimension (11K55) General spectral theory of ordinary differential operators (34L05) Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems (37J99) Hamiltonian and Lagrangian mechanics (70H99)

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Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Scaling for the discrete Mathieu equation
• Purely absolutely continuous spectrum for almost Mathieu operators
• Localization for a class of one dimensional quasi-periodic Schrödinger operators
• On the measure of the spectrum for the almost Mathieu operator
• A metal-insulator transition for the almost Mathieu model
• Almost periodic Schrödinger operators. II: The integrated density of states
• Almost periodic Schrödinger operators: A review
• Cantor spectrum for the almost Mathieu equation
• Gauss polynomials and the rotation algebra
• A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants
• Theory of nonlinear lattices.
• Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential
• Absence of localisation in the almost Mathieu equation
• A sum rule for the dispersion relations of the rational Harper equation
• Single Band Motion of Conduction Electrons in a Uniform Magnetic Field