The asymptotics of spectral gaps of a 1D Dirac operator with cosine potential
The authors investigate the asymptotic behaviour of the spectral gaps of the one-dimensional Dirac operator with a cosine potential \(Ly = L_{0}y + Vy\), where \[ L_{0}y = i \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}{dy \over dx} ,\quad Vy = \begin{pmatrix} 0 & P(x) \\ Q(x) & 0 \end{pmatrix} y ,\quad a\in \mathbb{R},\;y = \begin{pmatrix} y_{1}\\ y_{2}\end{pmatrix}, \] and with \(P(x) = Q(x) = 2a \cos 2\pi x\). This expression gives rise to a selfadjoint operator on \(L^2(\mathbb{R},\mathbb{C}^2)\) whose spectrum on the real line has countably many gaps. Using the eigenfunctions of the free Dirac operator \(L_{0}\) on \([0,1]\) with periodic and antiperiodic boundary conditions and Fourier methods, it is shown that the length of the gaps \(\gamma_{n},\;n\in\mathbb{Z}\), satisfies \(\gamma_{-n}= \gamma_{n}\). Furthermore, for large enough \(| n| \) it is proven that \(\gamma_{n}=0\) for \(n\) even and that the following asymptotic behaviour holds \[ \gamma_{n}= 2a\left({a\over 4\pi}\right)^{n-1}\left[\left({n-1\over 2}\right)!\right]^{-2}\left[1+O\left(\ln n\over n\right)\right] . \] From this result and its proof, the authors derive the asymptotic behaviour of the spectral lengths for perturbations of the free Dirac operator perturbed by potentials like \(P(x) = ae^{-2\pi i x} + b e^{2\pi ix} = \overline{Q(x)}\) or \(P(x) = a + be^{2\pi ix} = \overline{Q(x)}\).
scientific article; zbMATH DE number 2021977