Preliminaries

Abstract

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 $W^{1,p}(U)\subset \subset L^q(U)$ 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 ${\mathscr {F}}({\mathscr {W}})$ over Hilbert space ${\mathscr {W}}$ is defined. Creation operators a(f), annihilation operators $a^\dagger (f)$, second quantization $\varGamma (T)$ and differential second quantization ${\mathrm{d}}\varGamma (h)$ are introduced as operators in ${\mathscr {F}}({\mathscr {W}})$. We also define operator ${\mathrm{d}}\varGamma (k, h)$ being an extension of ${\mathrm{d}}\varGamma (h)$ and discuss localizations in ${\mathscr {F}}({\mathscr {W}})$ via the canonical identification ${\mathscr {F}}({\mathscr {W}}_1\oplus {\mathscr {W}}_2)\cong {\mathscr {F}}({\mathscr {W}}_1)\otimes {\mathscr {F}}({\mathscr {W}}_2)$. Finally we review compact operators of the form $Q(x)P(-i\nabla )$ in ${L^2({{\mathbb R}^d})}$ and $({\mathrm{d}}\varGamma (|k|)+\mathbbm {1})^{-1}\varGamma (F(-i\nabla _k))$ in ${\mathscr {F}}({L^2({{\mathbb R}^d})})$, and demonstrate their applications.