Weyl solutions and j-selfadjointness for Dirac operators

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:480

Article

Brown, B. Malcolm¹  Klaus, Martin²  Malamud, Mark³  Mogilevskii, Vadim⁴  Wood, Ian⁵ 

[1] Cardiff Univ, Cardiff Sch Comp Sci & Informat, Queens Bldg,5 Parade, Cardiff CF24 3AA, S Glam, Wales

[2] Virginia Tech, Math Dept, Blacksburg, VA 24061 USA

[3] RUDN Univ, Peoples Friendship Univ Russia, 6 Miklukho Maklaya St, Moscow 117198, Russia

[4] Poltava VG Korolenko Natl Pedag Univ, Ostrogradski St 2, UA-36000 Poltava, Ukraine

[5] Univ Kent, Sch Math Stat & Actuarial Sci, Sibson Bldg, Canterbury CT2 7FS, Kent, England

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

DOI  :  10.1016/j.jmaa.2019.07.034

来源: Elsevier

【 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.

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