On the completeness of general boundary value problems for $2 \times 2$ first-order systems of ordinary differential equations

Abstract

Let $B={\rm diag} (b_1^{-1}, b_2^{-1}) \not = B^*$ be a $2\times 2$ diagonal matrix with \break $b_1^{-1}b_2 \notin{\Bbb R}$ and let $Q$ be a smooth $2\times 2$ matrix function. Consider the system $$-i B y'+Q(x)y=\lambda y, \; y= {\rm col}(y_1,y_2), \; x\in[0,1],$$ 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 $U_j,\ j\in \{1,2\}.$