The Discrete Self-Adjoint Dirac Systems of General Type: Explicit Solutions of Direct and Inverse Problems, Asymptotics of Verblunsky-Type Coefficients and the Stability of Solving of the Inverse Problem

摘要

We consider discrete self-adjoint Dirac systems determined by the potentials(sequences) fCkg such that the matrices Ck are positive de nite andj-unitary, where j is a diagonal m m matrix which has m1 entries 1 andm2 entries ??1 (m1 +m2 = m) on the main diagonal. We construct systemswith the rational Weyl functions and explicitly solve the inverse problem torecover systems from the contractive rational Weyl functions. Moreover, westudy the stability of this procedure. The matrices Ck (in the potentials)are the so-called Halmos extensions of the Verblunsky-type coe cients k.We show that in the case of the contractive rational Weyl functions the coe cients k tend to zero and the matrices Ck tend to the identity matrixIm.