Dirac operators with periodic \(\delta\)-interactions: spectral gaps and inhomogeneous Diophantine approximation (Q1045739)

The author considers the Dirac operator \[ (Hf)(x)=-i\sigma_1f'(x)+m\sigma_3f(x),\quad x\in \mathbb R\setminus \Gamma, \] where \(f=\begin{pmatrix} f_1\\ f_2\end{pmatrix}\), \(\sigma_1=\begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}\), \(\sigma_3=\begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}\), \(m\geq 0\), \(\Gamma =\{ 0,\kappa \} +2\pi \mathbb Z\), \(\kappa \in (0,2\pi )\), with the interface conditions \(f_1\in H^1(\mathbb R)\), \(f_2(x+0)-f_2(x-0)=-i\beta f_1(x)\) (\(\beta \in \mathbb R\setminus \{0\}\)), \(x\in \Gamma\). The operator \(H\) is selfadjoint, and its spectrum has a band structure. The author studies asymptotic properties of the lengths of the gaps in the spectrum, depending on Diophantine properties of all the involved parameters. For the solution of a similar problem in the case of the Schrödinger operator see the author's [Math. Proc. Camb. Philos. Soc. 143, No.~1, 185--199 (2007; Zbl 1123.34068)].