On the completeness of general boundary value problems for \(2\times 2\) first order systems of ordinary differential equations (Q2896659)

The paper is devoted to general \(2\times 2\) first order systems of ordinary differential equations with general linear boundary conditions. While the problem of completeness of root vectors has been studied thoroughly for a single higher order equation, for systems of differential equations much less is known. However, in several recent papers by the authors, the class of systems and boundary conditions, for which the set of root vectors is known to be complete or incomplete is being gradually extended. See, in particular, \textit{M. M. Malamud} and \textit{L. L. Oridoroga} [Funct. Anal. Appl. 34, No. 4, 308-310 (2000); translation from Funkts. Anal. Prilozh. 34, No. 4, 88--90 (2000; Zbl 0979.34058)], \textit{M. M. Malamud} and \textit{L. L. Oridoroga} [Dokl. Math. 82, No. 3, 899--904 (2010); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 435, No. 3, 298--304 (2010; Zbl 1231.34152)].NEWLINENEWLINEIn the paper under review, this process is continued. The authors study various classes of systems not covered by earlier results. Here, the completeness or incompleteness depend on lower order terms. In the case of a holomorphic potential matrix and some nondegeneracy assumptions, even necessary and sufficient conditions are obtained.