Research Paper#Mathematics/Physics (Quantum Mechanics, Spectral Theory)🔬 ResearchAnalyzed: Jan 3, 2026 17:10
Arithmetic Localization for the Unitary Almost Mathieu Operator
Published:Dec 31, 2025 04:19
•1 min read
•ArXiv
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.”