Zubkova O. Inverse scattering problem for the non-Hermitian systems of the differential equations

The thesis is devoted to the solving of the inverse scattering problems (ISP) for Schrodinger systems with the triangular matrix-valued potentials on the semi-axis and on the axis and also with other classes of non-Hermitian matrix-valued potentials which are constructed by means of introducing concept of phase-equivalent matrix-valued potentials. Methods are methods of functional analysis, complex analysis and linear algebra. All results of the thesis are new and consist in the following: the characteristic properties of the scattering data (SD) for the problem with the triangular matrix-valued nxn potential which have first moment on the semi-axis, is real on the diagonal and without the virtual level, are established; the necessary and with a small distinction sufficient conditions for given values would SD of the problem with the triangular matrix-valued 2x2 potential which have second moment on the axis, is real on the diagonal and without the virtual level are obtained; вy means of the introducedconcepts and the obtained results for them the Marchenko-Agranovich theorem on the characteristic properties of SD for the Hermitian matrix-valued potential is extended to the characteristic properties of SD for the introduced classes of the non-Hermitian problems. Results may be to employ at investigations of problems of the scattering theory, and also at solving of nonlinear equations of the mathematical physics by ISP Method.