Structure of spectrum of Schrödinger operator with quasiperiodic potential near the ground state. The discrete and continuous cases (Q1175810)

The quasi-periodic Schrödinger operator \[ Ly\equiv-d^ 2y/dt^ 2+\lambda U(x)y,\;dx/dt=\omega,\;x=(x_ 1,\dots,x_ n)\in\mathbb{R}^ n\backslash Z^ n \] (discrete or continuous) is considered. The spectrum structure at the locality of its left edge, in the case when the ground state of the operator \((\epsilon=0)\) has some definite properties is investigated. The measure of the set \(\Lambda(\epsilon_ 1)=\{\epsilon:\;\epsilon\in(0,\epsilon_ 1),\;\hbox {the equation } Ly=\epsilon y, \epsilon>0\;\hbox{has two independent Bloch solutions}\}\) is estimated in dependence of the approximation velocity of the rotation number by the rational numbers.