Persistence of embedded eigenvalues

Abstract

We consider conditions under which an embedded eigenvalue of a self-adjoint operator remains embedded under small perturbations. In the case of a simple eigenvalue embedded in continuous spectrum of multiplicity m < ∞ we show that in favorable situations, the set of small perturbations of a suitable Banach space which do not remove the eigenvalue form a smooth submanifold of codimension m. We also have results regarding the cases when the eigenvalue is degenerate or when the multiplicity of the continuous spectrum is infinite.