The Spectrum of a Schrödinger Operator With Small Quasi-Periodic Potential is Homogeneous

Abstract

We consider the quasi-periodic Schrödinger operator $$[H\psi ](x)=-\psi "(x)+V(x)\psi (x)$$ in $L^2(\mathbb{R})$, where the potential is given by $$V(x)=\sum _{m\in {\mathbb{Z}}^{\nu }\setminus \{0\}}c(m)\exp (2\pi im\omega x)$$ with a Diophantine frequency vector $\omega =({\omega }_1,\dots ,{\omega }_{\nu })\in {\mathbb{R}}^{\nu }$ and exponentially decaying Fourier coefficients $|c(m)|\le \epsilon \exp (-{\kappa }_0|m|)$. In the regime of small $\epsilon >0$ we show that the spectrum of the operator $H$ is homogeneous in the sense of Carleson.

arXiv comment10 pages, to appear in J. Spectr. Theory