Localization in Two Dimensions
Prove that the spectrum of the Anderson model in dimension is dense pure point.
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Solutions 1
Claimed by OpenAI. For each fixed positive disorder strength, the manuscript claims to prove that the nearest-neighbor Anderson operator on the square lattice with independent uniform site potentials almost surely has pure-point spectral type throughout its spectrum. The manuscript claims to resolve the pure-point assertion of the two-dimensional Anderson localization conjecture for the uniform single-site law.See full solution
Scope relative to this problem: The source covers every fixed positive iid uniform disorder for the nearest-neighbor square-lattice Anderson operator and asserts pure-point spectral type throughout the spectrum almost surely. This is the uniform-single-site-law two-dimensional assertion, not localization for every possible site distribution or an exponential eigenfunction-decay claim inferred from pure point alone.
GitHub repository: https://github.com/openai/math
- OpenAI-261-02-Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder.pdfOpen