On the characterization of the smoothness of skew-adjoint potentials in periodic Dirac operators
Let \[ L(\varphi)u=i\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix} u'+ \begin{pmatrix} 0 & \varphi_1 \\ \varphi_2 & 0 \end{pmatrix} u \] be the linear operator acting on \(u\in H^1_{\text{loc}}([0,2],{\mathbb C}^2)\) subject to periodic boundary conditions, where the potential \(\varphi=(\varphi_1,\varphi_2)\) is skew symmetric in the sense that \(\varphi_2=-\overline{\varphi_1}\). In the literature, \(L(\varphi)\) is often referred to either as a `Dirac' or a `Zakharov-Shabat' operator. The main objective of this paper is to establish precise connections between asymptotics of the periodic spectrum of \(L(\varphi)\) and regularity properties (in suitable weighted Sobolev spaces) of \(\varphi\).
scientific article; zbMATH DE number 5553024