This is off-topic: But powerful techniques to show that gap exist, can be found in the following papers:

* Michael Goldstein, Wilhelm Schlag, _[On resonances and the formation of gaps in the spectrum of quasi-periodic Schroedinger equations](https://arxiv.org/abs/math/0511392)_

* Artur Avila, Jairo Bochi, David Damanik, _[Opening Gaps in the Spectrum of Strictly Ergodic Schrödinger Operators](https://arxiv.org/abs/0903.2281)_

of course. The setting is somewhat different, since they consider discrete Schroedinger operators ...