Self-Adjointness: Part 2. The Kato-Rellich Theorem

Abstract

The Kato-Rellich theorem and the Kato inequality form the basic tools for proving self-adjointness of Schrödinger operators H = -△ + V. The Kato inequality allows us to consider positive potentials that grow at infinity. These Schrödinger operators typically have a spectrum consisting only of eigenvalues. The semiclas-sical behavior of these eigenvalues near the bottom of the spectrum was studied in Chapters 11 and 12. The Kato-Rellich theorem will permit us to consider the other extreme. Families of such potentials, which might not necessarily be small in operator norm, will be shown to be small perturbations relative to the Laplacian -△, in a sense to be defined ahead. We will see in the next chapter that the essential spectrum of such operators H = -△ + V is usually, that is, completely characterized by the essential spectrum of the Laplacian. We will first discuss the general theory of relatively bounded perturbations and then its application to Schrödinger operators.