Weyl Solutions and J-selfadjointness for Dirac operators

We consider a non-selfadjoint Dirac-type differential expression \begin{equation} D(Q)y:= J_n \frac{dy}{dx} + Q(x)y, \quad\quad\quad (1) \end{equation} with a non-selfadjoint potential matrix $Q \in L^1_{loc}({\mathcal I},\mathbb{C}^{n\times n})$ and a signature matrix $J_n =-J_n^{-1} = -J_n^*\in \mathbb{C}^{n\times n}$. Here ${\mathcal I}$ denotes either the line $\mathbb{R}$ or the half-line $\mathbb{R}_+$. With this differential expression one associates in $L^2(\mathcal I,\mathbb{C}^{n})$ the (closed) maximal and minimal operators $D_{\max}(Q)$ and $D_{\min}(Q)$, respectively. One of our main results states that $D_{\max}(Q) = D_{\min}(Q)$ in $L^2(\mathbb{R},\mathbb{C}^{n})$. Moreover, we show that if the minimal operator $D_{\min}(Q)$ in $L^2(\mathbb{R},\mathbb{C}^{n})$ is $j$-symmetric with respect to an appropriate involution $j$, then it is $j$-selfadjoint. Similar results are valid in the case of the semiaxis $\mathbb{R}_+$. In particular, we show that if $n=2p$ and the minimal operator $D_{\min}(Q)$ in $L^2(\mathbb{R}_+,\mathbb{C}^{2p})$ is $j$-symmetric, then there exists a $2p\times p$-Weyl-type matrix solution $\Psi(z, \cdot)\in L^2(\mathbb{R}_+,\mathbb{C}^{2p\times p})$ of the equation $D^+_{\max}(Q)\Psi(z, \cdot)= z\Psi(z, \cdot)$. A similar result is valid for the expression (1) with a potential matrix having a bounded imaginary part. This leads to the existence of a unique Weyl function for the expression (1). The differential expression (1) is of significance as it appears in the Lax formulation of the vector-valued nonlinear Schr{\"o}dinger equation.