On a quadratic eigenvalue problem and its applications
Let \(C\) and \(R\) be real diagonal matrices with diagonals \((c_0,\dots,c_{N-1})\) and \((r_0,\dots,r_{N-1})\), respectively, let \(J\) be a real tridiagonal matrix with main diagonal \((b_0,\dots,b_{N-2},b_{N-1}-ha_{N-1})\) and super- and subdiagonal \((a_0,\dots,a_{N-2})\), and let \(f,\tilde{f}\in\mathbb{C}^N\). Consider the system of linear differential equations \[ Cu''(t)=iJu'(t)-Ru(t),\quad t\geq 0, \] with initial conditions \(u(0)=f\), \(u'(0)=\tilde{f}\). Seeking its nontrivial solution \(u=e^{i\lambda t}y\), \(y\in\mathbb{C}^N\), leads to the quadratic eigenvalue problem \[ (\lambda^2C-\lambda J-R)y=0. \] The second author and \textit{M.~Terziler} [Linear Algebra Appl.~259, 329--346 (1997; Zbl 0877.15009)] studied this problem assuming that \(a_n\neq0\) and \(c_n,r_n>0\) for all \(n=0,\dots,N-1\). The present authors allow that \(r_0=0\).
scientific article; zbMATH DE number 6147174