# Weyl solutions and j-selfadjointness for Dirac operators

We consider a non-selfadjoint Dirac-type differential expression D(Q)y:=Jn [Formula presented] +Q(x)y, with a non-selfadjoint potential matrix Q∈Lloc 1(I,Cn×n) and a signature matrix Jn=−Jn −1=−Jn ⁎∈Cn×n. Here I denotes either the line R or the half-line R+. With this differential expression one associates in L2(I,Cn) the (closed) maximal and minimal operators Dmax(Q) and Dmin(Q), respectively. One of our main results for the whole line case 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 Dmax +(Q)Ψ(z,⋅)=zΨ(z,⋅). A similar result is valid for the expression (0.1) whenever there exists a proper extension A˜ with dim(domA˜/domDmin +(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 expression (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. © 2019 Elsevier Inc.

Authors
Brown B.M.1 , Klaus M.2 , Malamud M. 3 , Mogilevskii V.4 , Wood I.5
Publisher
Number of issue
2
Language
English
Status
Published
Number
123344
Volume
480
Year
2019
Organizations
• 1 Cardiff School of Computer Science and Informatics, Cardiff University, Queen's Buildings, 5 The Parade, Cardiff, CF24 3AA, United Kingdom
• 2 Mathematics Department, Virginia Tech, Blacksburg, VA 24061-0123, United States
• 3 Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation
• 4 Poltava V.G. Korolenko National Pedagogical University, Ostrogradski Street 2, Poltava, 36000, Ukraine
• 5 School of Mathematics, Statistics and Actuarial Science, University of Kent, Sibson Building, Canterbury, CT2 7FS, United Kingdom
Keywords
Dirac-type operator; Dual pair of operators; j-selfadjointness; Weyl function; Weyl solution
24.12.2019
Date of change
04.03.2021
https://repository.rudn.ru/en/records/article/record/54803/
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