Preliminaries

Research output: Chapter in Book/Report/Conference proceedingChapter

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 (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.

Original languageEnglish
Title of host publicationSpringerBriefs in Mathematical Physics
PublisherSpringer
Pages15-40
Number of pages26
DOIs
Publication statusPublished - Jan 1 2019

Publication series

NameSpringerBriefs in Mathematical Physics
Volume35
ISSN (Print)2197-1757
ISSN (Electronic)2197-1765

All Science Journal Classification (ASJC) codes

  • Mathematical Physics
  • Physics and Astronomy (miscellaneous)

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    Hiroshima, F. (2019). Preliminaries. In SpringerBriefs in Mathematical Physics (pp. 15-40). (SpringerBriefs in Mathematical Physics; Vol. 35). Springer. https://doi.org/10.1007/978-981-32-9305-2_2