Multiparameter spectral problem for some weakly coupled systems of Hamiltonian equations
type: Item
instance of: scholarly article
title: Multiparameter spectral problem for some weakly coupled systems of Hamiltonian equations
zbMATH Open document ID: 1178.65090
DOI: 10.1134/S0012266108070069
author: A. A. Abramov
author: V. I. Ul'yanova
author: L. F. Yukhno
published in: Differential Equations
publication date: 2009-10-08T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65L15
Mathematics Subject Classification ID: 34L16
Mathematics Subject Classification ID: 37J05
zbMATH DE Number: 5611103
zbMATH Keywords: eigenvalue problems
zbMATH Keywords: linear differential systems
zbMATH Keywords: boundary value problems
zbMATH Keywords: multiparameter spectral problems
zbMATH Keywords: Hamiltonian systems
reviewed by: Manuel Calvo
MaRDI profile type: Publication
cites work: Multiparameter eigenvalue problems and expansion theorems
description: scientific article; zbMATH DE number 5611103