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Abstract. A non-classical Weyl theory is developed for Dirac systems with rectangular matrix potentials. The notion of the Weyl function is introduced and the corresponding direct problem is solved. Furthermore, explicit solutions of the direct and inverse problems are obtained for the case of rational Weyl matrix functions.Mathematics subject classification (2010): 34B20, 34L40, 15A15, 93B15.

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.

A non-classical Weyl theory is developed for skew-self-adjoint Dirac systems with rectangular matrix potentials. The notion of the Weyl function is introduced and direct and inverse problems are solved. A Borg-Marchenko type uniqueness result and the evolution of the Weyl function for the corresponding focusing nonlinear Schrödinger equation are also derived. MSC(2010): 34B20, 34L40, 37K15.