On characteristic matrices and eigenfunction expansions of two singular point symmetric systems

We study general (not necessarily Hamiltonian) first‐order symmetric systems J y ′ − B ( t ) y = Δ ( t ) f ( t )on an interval I = ( a , b ) with both singular endpoints a and b . For such a system we give a criterion of existence and description of self‐adjoint separated boundary conditions. We prove the Titchmarsh type formula for the characteristic matrix Ω ( λ ) of the self‐adjoint linear relation inL Δ 2 ( I )generated by separated boundary conditions as well as by a certain type of mixed boundary conditions. This formula enables one to express Ω ( λ ) in terms of the m ‐functions at the endpoints of I . By using the Titchmarsh type formula we parametrize all spectral functions corresponding to boundary problems with the mentioned boundary conditions immediately in terms of self‐adjoint boundary parameters at the end points a and b .