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

Abstract

For $N\ge 3$ 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 $\phi \in \mathbb{R}$, $A_N(\phi ,\vartheta )=e^{i\phi }(S+S^{-1})/2+\mathrm{diag}(\cos (2\pi j/N+\vartheta ){)}_{j\in \mathbb{Z}/N\mathbb{Z}}$, where $S{e}_j=e_{j+1}$ is the periodic shift on ${\mathbb{C}}^N$. We derive a Chambers formula and isolate the part $Q_{N,\phi }$ of the characteristic polynomial that depends only on $N$ and $\phi$, but not on $\vartheta$ or on a change of boundary conditions for the shift operator. We then show, for every $N$, that the zeros of $Q_{N,\phi }$ lie on the two perpendicular lines $e^{i\phi /2}\mathbb{R}\cup e^{i(\phi /2+\pi /2)}\mathbb{R}$. For even $N$, the same property holds for the matrices $A_N(\phi ,\vartheta )$ with $\vartheta \in 2\pi \mathbb{Z}/N$, and we compute their limiting eigenvalue measure explicitly. For $\phi \in [-\pi ,\pi ]$, the eigenvalue distribution approximates elliptic-integral densities with masses $1-|\phi |/\pi$ and $|\phi |/\pi$, and maximal radii $2|\cos (\phi /2)|$ and $2|\sin (\phi /2)|$, respectively. At $\phi =\pi /2$, the central polynomial $Q_{N,\phi }$ factors into positive quartic factors. This proves that the Scottish flag matrix, after Trefethen and Chapman, has its spectrum on the two diagonal lines of the saltire.