Weyl Solutions and J-selfadjointness for Dirac operators

Abstract

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 $Q\in L_{loc}^1(\mathcal{I},{\mathbb{C}}^{n\times n})$ and a signature matrix $J_n=-J_n^{-1}=-J_n^{\ast }\in {\mathbb{C}}^{n\times n}$. 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{ö}dinger equation.

31 pages, typos fixed and Proposition 2.16 strengthened