The Quantum Adiabatic Theorem

Abstract

This chapter is devoted to the quantum adiabatic theorem, and to its connection with the classical cases examined in the preceding chapters. In this section, we present the proof of the quantum theorem in a slightly informal way, without emphasizing the regularity hypotheses nor the domains of definition of the operators involved. The first proof of the theorem appears in an article by M. Born and V. Fock ([Bor]) but it is incomplete in several respects, for example, they consider only discrete spectra, which rarely occur in quantum mechanics. Here we follow the much later proof due to T. Kato ([Kat]).