Weyl Solutions and \(j\)-selfadjointness for Dirac Operators

We consider a non-selfadjoint Dirac-type differential expression

$$(0.1)D(Q)y := J_n {dy \over dx} +Q(x)y$$

with a non-selfadjoint potential matrix \(Q\) \(\epsilon\) \(L\) \(^{1}_{loc}(\scr J, \Bbb C^{n \times n})\) and a signature matrix \(J_n=-J^{-J}_n=-J^*_n\) \(\epsilon\) \(\Bbb C ^{n \times n}\). Here \(\scr J\) denotes either the line \(\Bbb R\) or the half-line \(\Bbb R_+\). With this differential expression one associates in \(L^2(\scr J, \Bbb C^n)\) the (closed) maximal and minimal operators \(D_{max}(Q)\) and \(D_{min}(Q)\), respectively. One of our main results for the whole line case states that \(D_{max}(Q)=D_{min}(Q)\) in \(L^2\) \(\Bbb R, \Bbb C^n\). Moreover, we show that if the minimal operator \(D_{min}(Q)\) in \(L^2(\Bbb R, \Bbb C^n)\) 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 \(\Bbb R_+\). In particular, we show that if \(n=2p\) and the minimal operator \(D^+_{min}(Q)\) in \(L^2(\Bbb R_+,\Bbb C^{2p})\) is (\j\)-symmetric, then there exists a \(2p \times p-\)Weyl-type matrix solution.

\(\Psi(z,\cdot) \) \(\epsilon\) \(L^2(\Bbb R_+, \Bbb C^{2p \times p})\) of the equation \(D^+_{max}(Q)\Psi(z,\cdot)=z \Psi(z,\cdot)\). A similar result is valid for the expression (0.1) whenever there exists a proper extension \(\tilde A\) with dim (dom \(\tilde A\)/dom \(D^+_{min}(Q))=p\) and nonempty resolvent set. In particular, it holds if a potential matrix (\Q\) has a bounded imaginary part. This leads to the existence of a unique Weyl function for the express (0.1). The main results are proven by means of a reduction to the self-adjoint case by using the technique of dual pairs of operators. The differential expression (0.1) is of significance as it appears in the Lax formulation of the vector valued nonlinear Schrödinger equation.