On the number of eigenvalues of a rational eigenproblem

Abstract

where A,B,Cj are linear, continuous and symmetric operators on a real Hilbert space H. A and B are positive definite, B is completely continuous and the operators Cj are positive semidefinite and have finite dimensional range. The poles σj of problem (1.1) are assumed to be positive and ordered by magnitude: 0 < σ1 < σ2 < · · · < σp. Problems of this type govern eigenvibrations of plates with elastically attached loads, and mechanical vibrations of fluid-solid structures, e.g. In [1], [6], [7] the second author studied iterative projection methods of JacobiDavidson and of Arnoldi type for the rational sparse matrix eigenproblem (1.1). These methods determine eigenvalues quite efficiently, however, it was an open question whether all eigenvalues in a given interval (in particular between consecutive poles) had been found or not. The question can be easily answered for intervals (μ1, μ2] such that no pole σk is contained in [μ1, μ2] considering the parameter dependent linear eigenproblem