Spectra of Non-Self-Adjoint Almost Mathieu Matrices and the Scottish Flag Operator

For $N\geq 3$ and a potential phase $\vartheta\in\mathbb{R}$, we study the non-self-adjoint almost Mathieu matrix obtained by multiplying the discrete Laplacian by a complex phase with angle $φ\in\mathbb{R}$, $A_N(φ,\vartheta)=e^{iφ}(S+S^{-1})/2+\operatorname{diag}(\cos(2πj/N+\vartheta))_{j\in\mathb