Perturbation Theory: Relatively Bounded Perturbations

Abstract

We have so far discussed two aspects of perturbation theory: perturbations of self-adjoint operators which preserve self-adjointness, and perturbations of self-adjoint operators which preserve the essential spectrum. This latter subject is part of what we call spectral stability. In this chapter, we begin a discussion of spectral stability for the discrete spectrum. The typical situation can be described as follows. Most quantum mechanical systems are described by a Hamiltonian of the form H = -△ + V. Different systems are distinguished by the potential V. For example, the Coulomb potential, where e is the unit electric charge, describes a hydrogen atom. In most situations, any given V is an idealization; there are always other effects that contribute small “corrections” to V. Hence, we may begin with a model Hamiltonian H 0, about which we have a lot of information concerning the eigenvalues, and consider the effects on H 0 caused by adding a small correction to the potential, say V 1. We are then led to consider a family of operators H(K) = H 0 + KV 1, as K varies from K=0 (where we have a lot of information) to KK K= K 0 (where we desire information). Exactly how the spectrum of H(K) varies with K is one of the topics of perturbation theory. We will give a criteria for the “smallness” of a perturbation V 1 relative to H 0 and, in certain cases (for example, for k sufficiently small), give very exact information about σ(H(K)). We know from Weyl’s theorem, Theorem 14.6, that relatively compact perturbations preserve the essential spectrum. Hence, our focus in this chapter will be on how the discrete eigenvalues change under a perturbation. We will return to more general questions of spectral stability in Chapter 19. Another aspect of perturbation theory, which we will not consider in this book, concerns the stability of spectral type, in particular, the absolutely continuous spectrum. This is the topic of scattering theory, and we refer the reader to Kato [K] and to Volume 3 of Reed and Simon [RS3].