Completeness theorems for Dirac-type operators with boundary conditions of general form that depend on the spectral parameter. (Q1889537)

Suppose that \(a<0<b\) and \(B=\text{diag}(a^{-1},b^{-1})\) is a diagonal \(2\times 2\)-matrix. In the space \(L_2[0,1]\oplus L_2[0,1]\), the authors consider boundary value problems for a system of integro-differential equations of the form \[ \tfrac{1}{i}By'+Q(x)y+\int_0^xM(x,t)y(t)\,dt=\lambda y, \] with \(Q(x)=\left(\begin{smallmatrix} 0 & q_1 \\ q_2 & 0\end{smallmatrix}\right)\), \(M(x,t)=\left(\begin{smallmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{smallmatrix}\right)\), \(y(x)=\left(\begin{smallmatrix} y_1(x) \\ y_2(x)\end{smallmatrix}\right)\), \(q_j\in L_1[0,1]\), \(M_{ij}\in L_{\infty}(\Omega)\), \(\Omega=\{ 0\leq t\leq x\leq 1\}\) and with linear or quadratic boundary conditions of general form depending on the spectral parameter. Completeness theorems for the system of eigen- and adjoint functions of these boundary value problems are formulated and short proofs are given.