Asymptotic behaviour of the non-real pair-eigenvalues of a two parameter eigenvalue problem (Q6102010)

The author considers a special class of two-parameter eigenvalue problems in a block-operator setting. Let \(L\), \(V_1\), and \(V_2\) be self-adjoint linear operators in a Hilbert space \(H\). Then a two-parameter eigenvalue problem is defined as \(M(\alpha, \beta)x = (L-\alpha V_1-\beta V_2)x = 0\), where \(\alpha\) and \(\beta\) are complex numbers. The author focuses on the case \(H = H_1 \oplus H_2\) and \[ L = \left[ \begin{array}{cc} A & C \\ C^* & D \end{array} \right], \qquad V_1 = \left[ \begin{array}{cc} I & 0 \\ 0 & 0 \end{array} \right], \qquad V_2 = \left[ \begin{array}{cc} 0 & 0 \\ 0 & I \end{array} \right]. \] The pair \((\alpha, \beta)\) is said to be a pair-eigenvalue of \(M = M(\alpha, \beta)\) if \(M x = 0\) has a non-zero solution. The aim of this paper is to investigate the \((\alpha, \beta)\) pairs-eigenvalues of \(M\).