Weyl functions of generalized dirac systems: Integral representation, the inverse problem and discrete interpolation

Abstract

Self-adjoint Dirac systems and subclasses of canonical systems, which generalize Dirac systems are studied. Explicit and global solutions of direct and inverse problems are obtained. A local Borg-Marchenko-type theorem, integral representation of the Weyl function, and results on interpolation of Weyl functions are also derived. MSC(2010): 34B20; 34L40; 47G10; 65D05.

Skew-Self-Adjoint Dirac System with a Rectangular Matrix Potential: Weyl Theory, Direct and Inverse Problems

“…we get that Φ 1 (x) does not depend on l for l > x. Compare this with the proof of Proposition 4.1 in[15], where the fact that E(x, t) (and so Φ 1 ) does not depend on l follows from the uniqueness of the factorizations of operators S −1 l . See also Section 3 in[4] on the uniqueness of the accelerant.…”

“…Step 3. Because of (3.3) and (3.42) we get Next, using (3.14) and (3.21) we easily obtain (see, e.g., [15,39])…”

Recovery of the Dirac system from the rectangular Weyl matrix function

“…Compare this with the proof of Proposition 4.1 in [17], where the fact that E(x, t) (and so Φ 1 ) does not depend on l follows from the uniqueness of the factorizations of operators S −1 l . See also Section 3 in [5] on the uniqueness of the accelerant.…”

Initial Value Problems for Integrable Systems on a Semi-Strip

“…We note that the theory of Weyl functions (Weyl-Titchmarsh theory) is actively developing in recent years (see, e.g., [14,15,21,23,24,35,48,52,54,56] and references therein) and its applications to initial-boundary value problems are of growing interest.…”