On a singular boundary-value problem for linear Hamiltonian systems of ordinary differential equations (Q1264628)

The authors investigate spectral properties of a singular differential operator associated with the linear Hamiltonian system \[ \mathcal Jy'=A(t)y,\qquad \mathcal J=\begin{pmatrix} 0 &-I\\ I &0\end{pmatrix}, \qquad A=\begin{pmatrix} A_{11} &A_{12}\\ A_{21} &A_{22}\end{pmatrix}, \tag{\(*\)} \] where \(I\) is the identity \(n\times n\) matrix and \(A\) is a symmetric \(2n\times 2n\) matrix, i.e. \(A_{11},A_{22}\) are symmetric and \(A_{21}=A^T_{12}\). Let \(M(t;t_0)\) denote the solution of the Riccati equation corresponding to \((*)\): \[ M'=A_{11}(t)+A_{12}(t)M+MA_{21}(t)+MA_{22}(t)M=0, \tag{\(**\)} \] satisfying the initial condition \(M(t_0;t_0)=3M_0\), \(M_0\) a given symmetric matrix. In the first part of the paper conditions on \(A\) are given which guarantee that \(\lim_{t_0\to \infty} M(t;t_0)=:N(t)\) exists, is finite, and \(N\) solves \((**)\). In the second part of the paper the authors use the above result to investigate the spectral properties of the singular boundary value problem \[ \mathcal Jy'=A(t,\lambda)y,\qquad \Psi\mathcal Jy(0)=0,\qquad y'\in L^2(0,\infty), \] where \(\Psi\) is a \(n\times 2n\) matrix satisfying \(\Psi\mathcal J\Psi^*=3D_0\) (\(^*\) stands for the conjugate transpose). Some numerical computations where the half-line \([0,\infty)\) is replaced by a compact interval are also given.