Weyl matrix functions and inverse problems for discrete Dirac-type self-adjoint systems: explicit and general solutions

Abstract

Abstract. It is shown that the discrete Dirac-type self-adjoint system is equivalent to the block Szegö recurrence. A representation of the fundamental solution is obtained, inverse problems on the interval and semiaxis are solved. A Borg-Marchenko type result is obtained, too. Connections with block Toeplitz matrices are treated Mathematics subject classification (2000): 39A12, 37K35, 47B35.

On a new class of structured matrices related to the discrete skew-self-adjoint Dirac systems

“…An analogue of Proposition 6.1 for the self-adjoint discrete Dirac system and block Toeplitz matrices S follows from the proof of Theorem 5.2 in [11]. Proposition 6.3.…”

“…We refer also to [20]- [24] and references therein for the general method of the operator identities. The analogs of various results on the Toeplitz matrices and j-theory from [6]- [11] can be obtained for the class Ω n , too. with the p × p entries s kj , which satisfies the identity…”

“…See, for instance, [1]- [5], [12,25] and numerous references therein. The connections between block Toeplitz matrices and Weyl theory for the self-adjoint discrete Dirac system were treated in [11]. (See [26] for the Weyl theory of the discrete analog of the Schrödinger equation.)…”

Semiseparable Integral Operators and Explicit Solution of an Inverse Problem for a Skew-Self-Adjoint Dirac-Type System

“…The inverse problem to recover a self-adjoint Dirac-type system from its Weyl or spectral function is closely related to the inversion of integral operators with difference kernels, see [9,26,32,36,37] and various references therein. For the discrete analogues of Dirac systems, Toeplitz matrices appear instead of the operators with difference kernels [7,10,15,38]. (Various results on Toeplitz matrices and related j-theory one can find, for instance, in [5,8,13,14].…”

“…One of the approaches to solve the inverse problem explicitly is connected with a version of the Bäcklund-Darboux transformation and some notions from system theory [20,22]. (See also [15,16,24,27] for this approach, and see [39] and the Vol. 66 (2010) Semiseparable Operators and an Inverse Problem 233 references therein for explicit formulas for the radial Dirac equation.)…”