On the density of certain spectral points for a class of $ C^{2} $ quasiperiodic Schrödinger cocycles

Abstract

<p style='text-indent:20px;'>For <inline-formula><tex-math id="M2">\begin{document}$ C^2 $\end{document}</tex-math></inline-formula> cos-type potentials, large coupling constants, and fixed <inline-formula><tex-math id="M3">\begin{document}$ Diophantine $\end{document}</tex-math></inline-formula> frequency, we show that the density of the spectral points associated with the Schrödinger operator is larger than 0. In other words, for every fixed spectral point <inline-formula><tex-math id="M4">\begin{document}$ E $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \liminf\limits_{\epsilon\to 0}\frac{|(E-\epsilon,E+\epsilon)\bigcap\Sigma_{\alpha,\lambda\upsilon}|}{2\epsilon} = \beta $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ \beta\in [\frac{1}{2},1] $\end{document}</tex-math></inline-formula>. Our approach is a further improvement on the papers [<xref ref-type="bibr" rid="b15">15</xref>] and [<xref ref-type="bibr" rid="b17">17</xref>].</p>