Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials (Q2042891)

The author considers the ordinary differential equation \[ -y''+u(\omega t)y=Ey, \] where \(y=y(t)\), \(t\in\mathbb{R}\), \(u\) is a real-analytic quasi-periodic function such that \[ |\ell \, \omega|\geq M(\alpha/(|\ell|^\tau)), \] where \(\alpha>0\), \(\tau>d-1\), \(\ell\in\mathbb{Z}^{d}\backslash 0\). It is proved that for any \(a>0\), \(r>0\) and \(\alpha>0\) there exists \(M^*=M^*(d,r,a,\alpha)\) such that if \(M\geq M^*\) for the interval \([a,\infty)\) there exists a Cantor subset \(\Delta_\alpha\subset\Delta\) such that for any \(\sqrt{E}\in\Delta_\alpha\), the above equation has two linearly independent solutions.