General-type discrete self-adjoint Dirac systems: explicit solutions of direct and inverse problems, asymptotics of Verblunsky-type coefficients and stability of solving inverse problem

2018/02/28 by I. Ya. Roitberg, A. L. Sakhnovich
Mathematics · #math.SP #math.CA #math.OC #msc:34B20 #msc:39A12 #msc:39A30 #msc:47A57

paper · pdf · doi:10.15407/mag14.04.532

published as Zh. Mat. Fiz. Anal. Geom. 14 (2018), issue 4 dedicated to V.A. Marchenko, pp. 532--548 · This paper is a generalization and further development of the topics discussed in arXiv:math/0703369, arXiv:1206.2915, arXiv:1508.07954, arXiv:1510.00793

arxiv created 2018/02/28 · arxiv updated 2020/07/03

Abstract

We consider discrete self-adjoint Dirac systems determined by the potentials (sequences) \C_k\ such that the matrices C_k are positive definite and j-unitary, where j is a diagonal m× m matrix and has m₁ entries 1 and m₂ entries -1 (m₁+m₂=m) on the main diagonal. We construct systems with rational Weyl functions and explicitly solve inverse problem to recover systems from the contractive rational Weyl functions. Moreover, we study the stability of this procedure. The matrices C_k (in the potentials) are so called Halmos extensions of the Verblunsky-type coefficients ρ_k. We show that in the case of the contractive rational Weyl functions the coefficients ρ_k tend to zero and the matrices C_k tend to the indentity matrix I_m.

Citations

• Weyl matrix functions and inverse problems for discrete Dirac type self-adjoint system: explicit and general solutions
• Discrete Dirac system: rectangular Weyl functions, direct and inverse problems
• Inverse problems for self-adjoint Dirac systems: explicit solutions and stability of the procedure
• Skew-selfadjoint Dirac systems: stability of the procedure of explicit solving the inverse problem