The spectrum of a Schrödinger operator with small quasi-periodic potential is homogeneous (Q290735)

The authors consider the Schrödinger operator on the torus \(T^{\nu}\) NEWLINE\[NEWLINE[H\Psi](x)=-\Psi''(x)+V(x)\Psi(x),NEWLINE\]NEWLINE with NEWLINE\[NEWLINEV(x)=\sum_{n\in\mathbb Z^\nu\backslash\{0\}}c(n)e^{2\pi inxw},NEWLINE\]NEWLINE where \(\omega\in\mathbb R^{\nu}\), NEWLINE\[NEWLINE| n\omega|\geq a_0|n|^{-b_0},\quad n\in\mathbb Z^{\nu}\backslash\{0\}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE0<a_0<1,\quad \nu<b_0<\infty.NEWLINE\]NEWLINE Recall that a closed set NEWLINE\[NEWLINE\mathcal E=\mathbb R\backslash\bigcup\limits_n(E_{n}^-,E_{n}^+)NEWLINE\]NEWLINE is called homogeneous if there is \(\tau>0\) such that for any \(E\in\mathcal E\) and any \(\sigma>0\) one has \(|(E-\sigma,E+\sigma)\cap\mathcal E|>\tau\sigma\). Here, the authors assume that \(c(n)=c(-n)\), \(n\in\mathbb Z^{\nu}\backslash\{0\}\), NEWLINE\[NEWLINE|n(c)|\leq\varepsilon\exp(-\kappa_0|n|),\quad n\in\mathbb Z^{\nu}\backslash\{0\},NEWLINE\]NEWLINE where \(\varepsilon>0\) and \(0<\kappa_0\leq 1\). Then, they obtain that there exists \(\varepsilon_0>0\) such that \(\varepsilon_0=\varepsilon_0(\kappa_0,a_0,b_0)\) such that for \(0<\varepsilon<\varepsilon_0\), the spectrum of the operator \(H\) is homogeneous with \(\tau=1/2\).