On the completeness of general boundary value problems for 2 × 2 first-order systems of ordinary differential equations - Oridoroga, L. L.; Malamud, M. M.; Agibalova, A. V. (Methods of Functional Analysis and Topology, 2012)
Let B=diag (b_1^-1, b_2^-1) ot = B^* be a 2× 2 diagonal matrix with b_1^-1b_2 otinR and let Q be a smooth 2× 2 matrix function. Consider the system -i B y’+Q(x)y=λ y, y= col(y_1,y_2), x[0,1], of ordinary differential equations subject to general linear boundary conditions U_1(y) = U_2(y) = 0. We find sufficient conditions on Q and U_j that guaranty completeness of root vector system of the boundary value problem. Moreover, we indicate a condition on Q that leads to a completeness criterion in terms of the linear boundary forms U_j, j{1,2}.