Resonance Tongues and Instability Pockets in the Quasi–Periodic Hill–Schrödinger Equation

Abstract

This paper concerns Hill’s equation with a (parametric) forcing that is real analytic and quasi-periodic with frequency vector ωℝd and a ‘frequency’ (or ‘energy’) parameter a and a small parameter b. The 1-dimensional Schrodinger equation with quasi-periodic potential occurs as a particular case. In the parameter plane ℝ2={a, b}, for small values of b we show the following. The resonance ‘‘tongues’’ with rotation number \({{\frac{{1}}{{2}}\langle{{\bf{ k}}},\omega\rangle,{{\bf{ k}}}\in\mathbb{{Z}}^d}}\) have C∞-boundary curves. Our arguments are based on reducibility and certain properties of the Schrodinger operator with quasi-periodic potential. Analogous to the case of Hill’s equation with periodic forcing (i.e., d=1), several further results are obtained with respect to the geometry of the tongues. One result regards transversality of the boundaries at b=0. Another result concerns the generic occurrence of instability pockets in the tongues in a reversible near-Mathieu case, that may depend on several deformation parameters. These pockets describe the generic opening and closing behaviour of spectral gaps of the Schrodinger operator in dependence of the parameter b. This result uses a refined averaging technique. Also consequences are given for the behaviour of the Lyapunov exponent and rotation number in dependence of a for fixed b.