The Galerkin method for perturbed self-adjoint operators and applications

We consider the Galerkin method for approximating the spectrum of an operator T+A where T is semi-bounded self-adjoint and A satisfies a relative compactness condition. We show that the method is reliable in all regions where it is reliable for the unperturbed problem - which always contains C∖R. The results lead to a new technique for identifying eigenvalues of T, and for identifying spectral pollution which arises from applying the Galerkin method directly to T. The new technique benefits from being applicable on the form domain