A new Canonical Center Bundle framework for long-range operators coincides with finite-range approximations and yields absolute continuity of the IDS plus Anderson localization for analytic quasi-periodic Schrödinger operators with Diophantine frequency.
Anderson localization and H\"older continuity of the integrated density of states for analytic quasiperiodic Schr\"odinger operators
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abstract
We establish both Anderson localization and H\"older continuity of the integrated density of states for quasiperiodic Schr\"odinger operators on $\mathbb{Z}^d$ with any non-constant analytic potential and any Diophantine frequency in the perturbative regime. Our proof is based on a new method for controlling Green's functions and eliminating double resonances, in the spirit of multi-scale analysis. To the best of our knowledge, this is the first multi-scale analysis approach that works for fixed Diophantine frequencies and potentials beyond the cosine type.
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2026 1verdicts
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Transfer Operators, Canonical Center Dynamics, and Spectral Applications for Long-Range Operators
A new Canonical Center Bundle framework for long-range operators coincides with finite-range approximations and yields absolute continuity of the IDS plus Anderson localization for analytic quasi-periodic Schrödinger operators with Diophantine frequency.