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.
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- 1.
T is a finite rank operator if and only if \(\mathrm{Ran} T\) is a finite dimensional space.
- 2.
D is uniformly equicontinuous if and only if for \(\forall \varepsilon >0\) there exists \(\delta >0\) such that \(d(x, y)<\delta \) implies \(\sup _{f\in D}|f(x)-f(y)|<\varepsilon \).
- 3.
X is totally bounded if and only if for given any \(\varepsilon >0\) there is a finite covering of X by balls of radius \(\varepsilon \).
- 4.
Note that \(F\subset C(X)\) is equicontinuous on metric space (X, d) if and only if for every \(\varepsilon >0\) there exists \(\delta >0\) such that for \(x, y\in X\), \(d(x, y)<\delta \) implies that \(\Vert f(x)-f(y)\Vert <\varepsilon \) for arbitrary \(f\in F\).
- 5.
Let \(\{U_t:t\in {\mathbb R}\}\) be a strongly continuous one-parameter unitary group on a Hilbert space \({\mathscr {H}}\). Then there exists a unique selfadjoint operator A such that \(U_{t}=e^{itA}\) for \(t\in {\mathbb R}\). The domain of A is given by \(D(A)=\{ \psi \in {\mathscr {H}}| \lim _{\epsilon \rightarrow 0} \frac{1}{\epsilon } \left( U_{\epsilon }(\psi )-\psi \right) {\text {exists}} \}\).
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© 2019 The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd.
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Hiroshima, F. (2019). Preliminaries. In: Ground States of Quantum Field Models. SpringerBriefs in Mathematical Physics, vol 35. Springer, Singapore. https://doi.org/10.1007/978-981-32-9305-2_2
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DOI: https://doi.org/10.1007/978-981-32-9305-2_2
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Print ISBN: 978-981-32-9304-5
Online ISBN: 978-981-32-9305-2
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