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Quantitative inductive estimates for Green's functions of non-self-adjoint matrices

Wencai Liu

Vol. 15 (2022), No. 8, 2061–2108

We provide quantitative inductive estimates for Green’s functions of matrices with (sub-)exponentially decaying off-diagonal entries in arbitrary dimensions. Together with Cartan’s estimates and discrepancy estimates, we establish explicit bounds for the large-deviation theorem for non-self-adjoint Toeplitz operators. As applications, we obtain the modulus of continuity of the integrated density of states with explicit bounds and the pure point spectrum property for analytic quasiperiodic operators. Moreover, our inductions are self-improved and work for perturbations with low-complexity interactions.

multiscale analysis, large-deviation theorem, discrepancy, semialgebraic sets, Cartan's techniques, Anderson localization
Mathematical Subject Classification
Primary: 81Q10
Secondary: 37C55, 82B44
Received: 24 September 2020
Revised: 29 March 2021
Accepted: 19 May 2021
Published: 10 February 2023
Wencai Liu
Department of Mathematics
Texas A&M University
College Station, TX
United States