Research Paper#Mathematics/Physics (Quantum Mechanics, Spectral Theory)🔬 ResearchAnalyzed: Jan 3, 2026 17:10
Arithmetic Localization for the Unitary Almost Mathieu Operator
Analysis
This paper extends previous work on the Anderson localization of the unitary almost Mathieu operator (UAMO). It establishes an arithmetic localization statement, providing a sharp threshold in frequency for the localization to occur. This is significant because it provides a deeper understanding of the spectral properties of this quasi-periodic operator, which is relevant to quantum walks and condensed matter physics.
Key Takeaways
- •Establishes an arithmetic localization statement for the UAMO.
- •Provides a sharp threshold in frequency for Anderson localization.
- •Extends previous results on Diophantine frequencies.
- •Relevant to quantum walks and condensed matter physics.
Reference
“For every irrational ω with β(ω) < L, where L > 0 denotes the Lyapunov exponent, and every non-resonant phase θ, we prove Anderson localization, i.e. pure point spectrum with exponentially decaying eigenfunctions.”