On the completeness of general boundary value problems for \(2\times 2\) first order systems of ordinary differential equations. (English) Zbl 1249.34249

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, M. M. Malamud and 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)], M. M. Malamud and 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)].
In 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.


34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
47E05 General theory of ordinary differential operators
34B15 Nonlinear boundary value problems for ordinary differential equations