Weyl solutions and j-selfadjointness for Dirac operators

[摘要] We consider a non-selfadjoint Dirac-type differential expression D(Q)y := J(n) dy/dx + Q(x)y, (0.1) with a non-selfadjoint potential matrix Q is an element of L-loc(1)(I, C-n x n) and a signature matrix J(n) = -J(n)(-1) = - J(n)* is an element of C-n (x) (n). Here I denotes either the line R or the half-line R+. With this differential expression one associates in L-2 (I, 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 (R, C-n). Moreover, we show that if the minimal operator D-mi(n)(Q) in L-2 (R, 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 R+. In particular, we show that if n = 2p and the minimal operator D-min(+)(Q) in L-2 (R+, C-2P) is j-symmetric, then there exists a 2p x p-Weyl-type matrix solution Psi(z, .) is an element of L-2 (R+, C-2p x p) of the equation D-min(+)(Q)Psi(z, .) = ,z Psi(z, .). A similar result is valid for the expression (0.1) whenever there exists a proper extension (A) over tilde with dim (dom (A) over tilde /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 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 Schrodinger equation. (C) 2019 Elsevier Inc. All rights reserved.

[发布日期] 2019-12-15

[关键词] Dirac-type operator;j-selfadjointness;Weyl solution;Weyl function;Dual pair of operators