Perturbations of Self-adjoint Operators

Abstract

Chapter 8 begins with differential operators with constant coefficients on ℝ^d. The Fourier transform allows us to give an elegant approach to these operators and their spectral properties. Then the self-adjointness of the sum A+B of self-adjoint operators under relatively bounded perturbations is studied, and the theorems of Kato–Rellich and Wüst are derived. The essential spectrum of a self-adjoint operator is investigated, and versions of Weyl’s theorem on the invariance of the essential spectrum under relatively compact perturbations are proved. The main motivation for these investigations stems from quantum mechanics, where A+B=−Δ+V is a Schrödinger operator. The operator-theoretic results of this chapter are applied for studying the self-adjointness and the essential spectrum of Schrödinger operators.