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Session S30 - Mathematical Methods in Quantum Mechanics

Thursday, July 15, 20:00 ~ 20:25 UTC-3

Large numerators of quasiperiodic operators

Wencai Liu

Texas A&M University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We initiate an approach to simultaneously treat numerators and denominators arising from quasi-periodic operators, which allows us to obtain (possibly) sharp estimates of Green's functions. In particular, we can study the completely resonant phases of the almost Mathieu operators. Let \[ (H_{\lambda,\alpha,\theta}u) (n)=u(n+1)+u(n-1)+ 2\lambda \cos2\pi(\theta+n\alpha)u(n) \] be the almost Mathieu operator on $\ell^2(\mathbb{Z})$, where $\lambda, \alpha, \theta\in \mathbb{R}$. Let \[ \beta(\alpha)=\limsup_{k\rightarrow \infty}-\frac{\ln ||k\alpha||_{\mathbb{R}/\mathbb{Z}}}{|k|}. \] We prove that for any $\theta$ with $2\theta\in \alpha \mathbb{Z}+\mathbb{Z}$, $H_{\lambda,\alpha,\theta}$ satisfies Anderson localization if $|\lambda|>e^{2\beta(\alpha)}$. This confirms a conjecture of Avila and Jitomirskaya [The Ten Martini Problem. Ann. of Math. (2) 170 (2009), no. 1, 303-342].

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