2017/12/29 by Brown, B. Malcolm, Klaus, Martin, Malamud, Mark +2 · 3 citations
#34B20 #34L10 (Primary) #47B25 (Secondary) #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1712.10140
We consider a non-selfadjoint Dirac-type differential expression D(Q)y:= Jn (dy)/(dx) + Q(x)y, (1) with a non-selfadjoint potential matrix Q ∈ L1loc(\mathcal I,ℂn× n) and a signature matrix Jn =-Jn-1 = -Jn^*∈ ℂn× n. Here \mathcal I denotes either the line ℝ or the half-line ℝ+. With this differential expression one associates in L2(\mathcal I,ℂn) the (closed) maximal and minimal operators Dmax(Q) and Dmin(Q), respectively. One of our main results states that Dmax(Q) = Dmin(Q) in L2(ℝ,ℂn). Moreover, we show that if the minimal operator Dmin(Q) in L2(ℝ,ℂn) 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 ℝ+. In particular, we show that if n=2p and the minimal operator Dmin(Q) in L2(ℝ+,ℂ2p) is j-symmetric, then there exists a 2p× p-Weyl-type matrix solution Ψ(z, ⋅)∈ L2(ℝ+,ℂ2p× p) of the equation D+max(Q)Ψ(z, ⋅)= zΨ(z, ⋅). 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 Schrödinger equation.