Abstract
In this paper, we establish quantitative Green’s function estimates for some higher-dimensional lattice quasi-periodic (QP) Schrödinger operators. The resonances in the estimates can be described via a pair of symmetric zeros of certain functions and the estimates apply to the sub-exponential-type non-resonance conditions. As the application of quantitative Green’s function estimates, we prove both the arithmetic version of Anderson localization and the finite volume version of (\(({1 \over 2} - )\))-Hölder continuity of the integrated density of states (IDS) for such QP Schrödinger operators. This gives an affirmative answer to Bourgain’s problem in Bourgain (2000).
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Acknowledgements
Yunfeng Shi was supported by National Natural Science Foundation of China (Grant No. 12271380) and National Key R&D Program (Grant No. 2021YFA1001600). Zhifei Zhang was supported by National Natural Science Foundation of China (Grant Nos. 12171010 and 12288101). The authors are grateful to the referees for their helpful suggestions.
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Cao, H., Shi, Y. & Zhang, Z. Quantitative Green’s function estimates for lattice quasi-periodic Schrödinger operators. Sci. China Math. 67, 1011–1058 (2024). https://doi.org/10.1007/s11425-022-2126-8
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DOI: https://doi.org/10.1007/s11425-022-2126-8
Keywords
- Hölder continuity of the IDS
- quantitative Green’s function estimates
- quasi-periodic Schrödinger operators
- arithmetic Anderson localization
- multi-scale analysis