TY - CHAP

T1 - Preliminaries

AU - Hiroshima, Fumio

N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2019.

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

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U2 - 10.1007/978-981-32-9305-2_2

DO - 10.1007/978-981-32-9305-2_2

M3 - Chapter

AN - SCOPUS:85082334119

T3 - SpringerBriefs in Mathematical Physics

SP - 15

EP - 40

BT - SpringerBriefs in Mathematical Physics

PB - Springer

ER -