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Resonance tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation. (English) Zbl 1098.37014

Summary: This paper concerns Hill’s equation with a (parametric) forcing that is real analytic and quasi-periodic with frequency vector \(\omega\in \mathbb{R}^d\) and a ‘frequency’ (or ‘energy’) parameter \(a\) and a small parameter \(b\). The 1-dimensional Schrödinger equation with quasi-periodic potential occurs as a particular case. In the parameter plane \(\mathbb{R}^2=a,b\), for small values of \(b\) we show the following. The resonance “tongues” with rotation number \(\frac12\langle k,\omega\rangle,k\in \mathbb{Z}^d\) have \(C^{\infty}\)-boundary curves. Our arguments are based on reducibility and certain properties of the Schrödinger 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 Schrödinger 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 \(b\).

MSC:

37B55 Topological dynamics of nonautonomous systems
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C28 Complex behavior and chaotic systems of ordinary differential equations
34C29 Averaging method for ordinary differential equations
37E45 Rotation numbers and vectors
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