Continuous quasiperiodic Schrödinger operators with Gordon type potentials

Abstract

Let us concern the quasi-periodic Schrödinger operator in the continuous case, \begin{equation*} (Hy)(x)=-y^{\prime\prime}(x)+V(x,ωx)y(x), \end{equation*} where $V:(\mathbb{R}/\mathbb{Z}{)}^2\to \mathbb{R}$ is piecewisely $\gamma$-Hölder continuous with respect to the second variable. Let $L(E)$ be the Lyapunov exponent of $Hy=Ey$. Define $\beta (\omega )$ as \begin{equation*} β(ω)= \limsup_{k\to \infty}\frac{-\ln ||kω||}{k}. \end{equation*} We prove that $H$ admits no eigenvalue in regime $\{E\in \mathbb{R}:L(E)<\gamma \beta (\omega )\}$.

References in corpus (9)

• Schrödinger Operators with Dynamically Defined Potentials: A Survey
• Sharp Phase transitions for the almost Mathieu operator
• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Arithmetic Spectral Transitions for the Maryland Model
• Full measure reducibility and localization for Jacobi operators: a topological criterion
• Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators
• Singular continuous spectrum for singular potential
• Dynamical bounds for quasiperiodic Schrödinger operators with rough potentials
• Non-Perturbative Localization with Quasiperiodic Potential in Continuous Time

Cited by in corpus (1)

• Absence of eigenvalues of analytic quasi-periodic Schrodinger operators on ${\mathbb{R}}^d$