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

TL;DR

This paper develops quantitative inductive methods to estimate Green's functions of non-self-adjoint matrices with decaying off-diagonal entries, leading to explicit bounds on spectral properties and large deviation estimates.

Contribution

It introduces a novel inductive approach for Green's function estimates in higher dimensions, applicable to non-self-adjoint matrices with low complexity perturbations.

Findings

Explicit bounds for large deviation theorems of Toeplitz operators

Modulus of continuity results for the integrated density of states

Proof of pure point spectrum for certain quasi-periodic operators

Abstract

We provide quantitative inductive estimates for Green's functions of matrices with (sub)expoentially decaying off diagonal entries in higher 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 quasi-periodic operators. Moreover, our inductions are self-improved and work for perturbations with low complexity interactions.