Almost Mathieu operators with completely resonant phases uri icon

abstract

  • Let $unicode[STIX]{x1D6FC}in mathbb{R}\backslash mathbb{Q}$ and $unicode[STIX]{x1D6FD}(unicode[STIX]{x1D6FC})=limsup _{n
    ightarrow infty }(ln q_{n+1})/q_{n}    , where $p_{n}/q_{n}$ is the continued fraction approximation to $unicode[STIX]{x1D6FC}$. Let $(H_{unicode[STIX]{x1D706},unicode[STIX]{x1D6FC},unicode[STIX]{x1D703}}u)(n)=u(n+1)+u(n-1)+2unicode[STIX]{x1D706}cos 2unicode[STIX]{x1D70B}(unicode[STIX]{x1D703}+nunicode[STIX]{x1D6FC})u(n)$ be the almost Mathieu operator on $ell ^{2}(mathbb{Z})$, where $unicode[STIX]{x1D706},unicode[STIX]{x1D703}in mathbb{R}$. Avila and Jitomirskaya [The ten Martini problem. Ann. of Math. (2), 170(1) (2009), 303342] conjectured that, for $2unicode[STIX]{x1D703}in unicode[STIX]{x1D6FC}mathbb{Z}+mathbb{Z}$, $H_{unicode[STIX]{x1D706},unicode[STIX]{x1D6FC},unicode[STIX]{x1D703}}$ satisfies Anderson localization if $|unicode[STIX]{x1D706}|>e^{2unicode[STIX]{x1D6FD}(unicode[STIX]{x1D6FC})}$. In this paper, we develop a method to treat simultaneous frequency and phase resonances and obtain that, for $2unicode[STIX]{x1D703}in unicode[STIX]{x1D6FC}mathbb{Z}+mathbb{Z}$, $H_{unicode[STIX]{x1D706},unicode[STIX]{x1D6FC},unicode[STIX]{x1D703}}$ satisfies Anderson localization if $|unicode[STIX]{x1D706}|>e^{3unicode[STIX]{x1D6FD}(unicode[STIX]{x1D6FC})}$.

published proceedings

  • ERGODIC THEORY AND DYNAMICAL SYSTEMS

altmetric score

  • 1.75

author list (cited authors)

  • Liu, W.

citation count

  • 2

complete list of authors

  • Liu, Wencai

publication date

  • July 2020