Preliminaries

In this chapter we introduce fundamental tools used throughout this book. Compact operators on Banach spaces and compact embeddings of Sobolev spaces of the form (Formula presented) are reviewed, which can be applied to study perturbations of eigenvalues embedded in the continuous spectrum of selfadjoint operators which describe Hamiltonians in quantum field theory. The boson Fock space F(W) over Hilbert space W is defined. Creation operators a(f), annihilation operators (Formula presented), second quantization Γ (T) and differential second quantization dΓ (h) are introduced as operators in F(W). We also define operator dΓ (k, h) being an extension of dΓ (h) and discuss localizations in F(W) via the canonical identification (Formula presented). Finally we review compact operators of the form (Formula presented) in (Formula presented) and (Formula presented) in (Formula presented), and demonstrate their applications.