A periodic Schrödinger operator with two degenerate spectral gaps (Q2923223)

The author considers a \(2\pi\)-periodic Schrödinger operator on \(\mathbb R\) with domain \(W^{2,2}(\mathbb R\setminus\Gamma)\) and with four point interactions in the periodic cell \([0,2\pi)\), where {\parindent=0.5cm\begin{itemize}\item[{\(\bullet\)}] \(\Gamma=\Gamma_1\cup\Gamma_2\cup\Gamma_3\cup\Gamma_4\), \item[{\(\bullet\)}] \(\Gamma_j=\{k_j\}+2\pi\mathbb Z\), \item[{\(\bullet\)}] \(0<k_1<k_2<k_3<k_4=2\pi\), \item[{\(\bullet\)}] the four point interactions are given in terms of a boundary condition NEWLINE\[NEWLINE \left (y(x+0),y'(x+0))^T =e^{i \theta_j}A_j(y(x-0),y'(x-0)\right)^T, \quad x\in\Gamma_j,\, \theta_j\in\mathbb R,\, A_j\in\mathrm{SO}(2)\setminus\{\pm\mathrm{id}\}.NEWLINE\]NEWLINENEWLINENEWLINE\end{itemize}} Under these assumptions, the author discusses the coexistence problem for the band spectrum of the operator. In particular, he shows that there exist rotation matrices \(A_j\) such that the band spectrum of the operator contains exactly two degenerate gaps.