Transformation of a class of non-self-adjoint systems into self -adjoint Hamiltonian systems (Q2367820)

Let us consider the linear differential operator \(Ly=P(t)y'+Q(t)y\), where \(y \in \mathbb{R}^ n\), \(P(t)\), \(Q(t)\) are matrix functions, \(t \in[a,b]\) and consider BVP \(Ly=\lambda R(t)y\), \(My(a)+Ny(b)=0\), \(R(t)\) is a matrix function and \(M\), \(N\) are constant matrices. The authors investigate the necessary and sufficient conditions under which the general non- selfadjoint BVP can be transformed into a self-adjoint eigenvalue problem and such a way all the spectral theory that is known for self-adjoint problems can be used. The second section of the paper contains a brief, excellent presentation of some background material. In the closing section, the algorithm for determining whether the eigenvalue problem \(Ly=\lambda R(t)y\) can be converted to a selfadjoint problem, is presented on two examples. This interesting paper is very well written and organized.