Multiparameter spectral problem for some weakly coupled systems of Hamiltonian equations
The authors are concerned with the constructive solution of multiparameter spectral problems for Hamiltonian systems of the type \[ J \; y_i' = \left( q_i(t) + \sum_{j=1}^n \lambda_i a_{ij}(t) \right) y_i, \qquad i=1, \dots ,n,\quad t \in [0,1], \] where \( q_i, a_{ij}\) are continuous function from \( [0,1] \to \mathbb{C}^{2 \times 2},\;\) \( \lambda_j\) are real numbers and \(J\) is the structure matrix of the Hamiltonian system. This differential system is supplemented with linear boundary conditions \(\varphi_i y_i(0) = 0,\) \(\psi_i y_i(1) = 0,\) where \(\varphi_i, \psi_i \in \mathbb{C}^{1 \times 2}\) are rank one matrices with \(\varphi_i J \varphi_i^* = 0\) and \( \psi_i J \psi_i^* = 0\). In this context, assuming the strong right definiteness of the problem, the authors propose to take suitable \( t_j^0 \in [0,1], j=1, \dots,n, \) and non zero columns \(v_j^0 \in \mathbb{C}^2, j=1, \dots,n,\) so that an auxiliary differential system as above with \((a_{ij}(t))\) substituted by \(a_{ij}^0 I = (v_i^0)^* \; a_{ij}(t_i^0) \; v_i^0\) with the same boundary conditions reduces to \(n\) uncoupled boundary value problems and can be easily solved. After that, a well defined continuation process is defined so that leads to the solution of the original problem. The paper ends with some remarks on the practical solution with the proposed approach.
scientific article; zbMATH DE number 5611103