We prove that for Diophantine ω and almost every θ, the almost Mathieu operator, (Hω,λ,θΨ)(n) = Ψ(n+ 1) + Ψ(n− 1) +λ cos 2π(ωn+ θ)Ψ(n), exhibits localization for λ > 2 and purely absolutely continuous spectrum for λ < 2. This completes the proof of (a correct version of) the Aubry-André conjecture.

We will express Daubechies localization operators in Bargmann Fock space. We will prove that the Hermite functions are eigenfunctions of Daubechies localization operator. By making use of generating function of eigenvalues of Daubechies localization operator, we will show some reconstruction formulas for symbol function of Daubechies localization operator with rotational invariant symbol.

In this work‎, ‎an iterative method based on a matrix form of LSQR algorithm is constructed for solving the linear operator equation $mathcal{A}(X)=B$‎ ‎and the minimum Frobenius norm residual problem $||mathcal{A}(X)-B||_F$‎ ‎where $Xin mathcal{S}:={Xin textsf{R}^{ntimes n}~|~X=mathcal{G}(X)}$‎, ‎$mathcal{F}$ is the linear operator from $textsf{R}^{ntimes n}$ onto $textsf{R}^{rtimes s}$‎, ‎$ma...

We study an operator norm localization property and its applications to the coarse Novikov conjecture in operator K-theory. A metric space X is said to have operator norm localization property if there exists 0 < c ≤ 1 such that for every r > 0, there is R > 0 for which, if ν is a positive locally finite Borel measure on X, H is a separable infinite dimensional Hilbert space and T is a bounded ...

In the new agricultural scenarios, the interaction between autonomous tractors and a human operator is important when they jointly perform a task. Obtaining and exchanging accurate localization information between autonomous tractors and the human operator, working as a team, is a critical to maintaining safety, synchronization, and efficiency during the execution of a mission. An advanced loca...

in this work‎, ‎an iterative method based on a matrix form of lsqr algorithm is constructed for solving the linear operator equation $mathcal{a}(x)=b$‎ ‎and the minimum frobenius norm residual problem $||mathcal{a}(x)-b||_f$‎ ‎where $xin mathcal{s}:={xin textsf{r}^{ntimes n}~|~x=mathcal{g}(x)}$‎, ‎$mathcal{f}$ is the linear operator from $textsf{r}^{ntimes n}$ onto $textsf{r}^{rtimes s}$‎, ‎$ma...

We study the operator growth problem and its complexity in many-body localization (MBL) system from Lanczos algorithm perspective. Using Krylov basis, can be viewed as a single-particle hopping on semi-infinite chain with amplitudes given by coefficients. find that, MBL systems, coefficients scale \sim n/\ln(n) ...