1D Dirac Operators with Special Periodic Potentials
Volume 60 / 2012
Bulletin Polish Acad. Sci. Math. 60 (2012), 59-75
MSC: Primary 47E05; Secondary 34L40.
DOI: 10.4064/ba60-1-5
Abstract
For one-dimensional Dirac operators of the form $$ Ly= i \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \frac{dy}{dx} + v y, \quad\ v= \begin{pmatrix} 0 & P \\ Q & 0 \end{pmatrix}, \, y=\begin{pmatrix} y_1 \\ y_2 \end{pmatrix}, \, x \in \mathbb{R}, $$ we single out and study a class $X$ of $\pi$-periodic potentials $v$ whose smoothness is determined only by the rate of decay of the related spectral gaps $\gamma_n = |\lambda_n^+ - \lambda_n^-|, $ where $ \lambda_n^\pm$ are the eigenvalues of $L=L(v)$ considered on $[0,\pi]$ with periodic (for even $n$) or antiperiodic (for odd $n$) boundary conditions.
