The Resolvent Algebra of Non-relativistic Bose Fields: Sectors, Morphisms, Fields and Dynamics

Abstract

It was recently shown that the resolvent algebra of a non-relativistic Bose field determines a gauge invariant (particle number preserving) kinematical algebra of observables which is stable under the automorphic action of a large family of interacting dynamics involving pair potentials. In the present article, this observable algebra is extended to a field algebra by adding to it isometries, which transform as tensors under gauge transformations and induce particle number changing morphisms of the observables. Different morphisms are linked by intertwiners in the observable algebra. It is shown that such intertwiners also induce time translations of the morphisms. As a consequence, the field algebra is stable under the automorphic action of the interacting dynamics as well. These results establish a concrete C*-algebraic framework for interacting non-relativistic Bose systems in infinite space. It provides an adequate basis for studies of long range phenomena, such as phase transitions, stability properties of equilibrium states, condensates, and the breakdown of symmetries.

Appendix

In this appendix we give the proof of Theorem 5.3, carrying out the various steps outlined prior to its statement. We begin by introducing the notation used in what follows.

1.1 Fields and particle picture

Since we will freely alternate between the field theoretic approach and the particle picture, based on the interpretation of Fock space, let us recall some standard formulas. Given $f_1, \dots , f_n \in {\mathcal {D}}({{\mathbb {R}}}^s)$ one has for the symmetric tensor product of the corresponding single particle vectors the relation

$$\begin{aligned} |f_1 \rangle \otimes _s \cdots \otimes _s | f_n \rangle = (1 / n!)^{1/2} \ a^*(f_1) \cdots a^*(f_n) \, {\varvec{\varOmega }}\in {\mathcal {F}}_n. \end{aligned}$$

Next, let $O_1$ be an operator on the single particle space ${\mathcal {F}}_1$ with (distributional) kernel ${\varvec{x}}, {\varvec{y}}\mapsto \langle {\varvec{x}}| O_1 | {\varvec{y}}\rangle$. Its canonical lift to ${\mathcal {F}}_n$, $n \in {{\mathbb {N}}}_0$, obtained by forming symmetrized tensor products with the unit operator and amplifying it with the appropriate weight factor n, is given by

$$\begin{aligned} n \ (O_1 \otimes _s \underbrace{1 \otimes _s \cdots \otimes _s 1}_{n-1}) = \int \! d{\varvec{x}}\! \int \! d{\varvec{y}}\, a^*({\varvec{x}}) \, \langle {\varvec{x}}| O_1 | {\varvec{y}}\rangle \, a({\varvec{y}}) \upharpoonright {\mathcal {F}}_n. \end{aligned}$$

The field theoretic operator on the right hand side of this equality will be called second quantization of $O_1$. Similarly, if $O_2$ is a two-particle operator acting on ${\mathcal {F}}_2$ with kernel ${\varvec{x}}_1, {\varvec{x}}_2, {\varvec{y}}_1, {\varvec{y}}_2 \mapsto \langle {\varvec{x}}_1, {\varvec{x}}_2 | O_2 | {\varvec{y}}_1, {\varvec{y}}_2 \rangle$, one has, $n \in {{\mathbb {N}}}_0$,

$$\begin{aligned}&n(n-1) \ (O_2 \otimes _s \underbrace{1 \otimes _s \cdots \otimes _s 1}_{n-2}) \\&\quad = \int \! d{\varvec{x}}_1 \! \! \int \! d{\varvec{x}}_2 \! \! \int \! d{\varvec{y}}_1 \! \! \int \! d{\varvec{y}}_2 \, a^*({\varvec{x}}_1) a^*({\varvec{x}}_2) \, \langle {\varvec{x}}_1 , {\varvec{x}}_2 | O_2 | {\varvec{y}}_1, {\varvec{y}}_2 \rangle \, a({\varvec{y}}_1) a({\varvec{y}}_2) \upharpoonright {\mathcal {F}}_n. \end{aligned}$$

The operator on the right hand side will be called second quantization of $O_2$.

The Hamiltonians of interest here, given in Eq. (5.1), have the form

$$\begin{aligned} H&= \int \! d{\varvec{x}}\, \big ( {{\varvec{\partial }}}a^*({\varvec{x}}) \, {{\varvec{\partial }}}a({\varvec{x}}) + \kappa ^2 {\varvec{x}}^2 \, a^*({\varvec{x}}) \, a({\varvec{x}}) \big ) \nonumber \\&\quad + \int \! d{\varvec{x}}\! \! \int \! d{\varvec{y}}\ a^*({\varvec{x}}) a^*({\varvec{y}}) \, V({\varvec{x}}- {\varvec{y}}) \, a({\varvec{x}}) a({\varvec{y}}). \end{aligned}$$

The first integral is the second quantization of the operator ${\varvec{P}}_\kappa ^2 \doteq {\varvec{P}}^2 + \kappa ^2 \, {\varvec{Q}}^2$ on ${\mathcal {F}}_1$, where ${\varvec{P}}$ is the momentum , ${\varvec{Q}}$ the position operator and $\kappa \ge 0$; the second integral is the second quantization of the two-particle potential V on ${\mathcal {F}}_2$. Note that the kernel of proper pair potentials on ${\mathcal {F}}_2$ has the singular form

$$\begin{aligned} {\varvec{x}}_1, {\varvec{x}}_2, {\varvec{y}}_1, {\varvec{y}}_2 \mapsto (1/2) \, (\delta ({\varvec{x}}_1 - {\varvec{y}}_1) \delta ({\varvec{x}}_2 - {\varvec{y}}_2) + \delta ({\varvec{x}}_1 - {\varvec{y}}_2) \delta ({\varvec{x}}_2 - {\varvec{y}}_1) ) \, V({\varvec{y}}_1 - {\varvec{y}}_2), \end{aligned}$$

which reduces the second quantization of V to a double integral. We will have occasion to consider also less singular potentials whose second quantization requires more integrations. Given $n \in {{\mathbb {N}}}_0$, the restriction $H_n \doteq H \upharpoonright {\mathcal {F}}_n$ can thus be presented in the form

$$\begin{aligned} H_n = \ n \, ({\varvec{P}}_\kappa ^2 \otimes _s \underbrace{1 \otimes _s \cdots \otimes _s 1}_{n-1}) + \ n(n-1) \, (V \otimes _s \underbrace{1 \otimes _s \cdots \otimes _s 1}_{n-2}) . \end{aligned}$$

(A.1)

This version will be useful in our subsequent analysis, where we need to decompose the operators ${\varvec{P}}_\kappa ^2$ and V into different pieces in order to relate them to elements of the algebras ${{\mathfrak {K}}}_n$, cf. Eq. (3.1)

We will also make use of the second quantization $N_f$ of the one-particle operator $E_{f,1}$, the projection onto the ray of $|f \rangle$ in ${\mathcal {F}}_1$. The restriction of this number operator to ${\mathcal {F}}_n$ is given by $N_{f,n} \doteq N_f \upharpoonright {\mathcal {F}}_n = n \, (E_{f,1} \otimes _s \underbrace{1 \otimes _s \cdots \otimes _s 1}_{n-1})$. Hence all bounded functions of $N_{f,n}$ are elements of ${{\mathfrak {K}}}_n$, cf. Eq. (3.1). We also note that the projection $E_{f,n} = E_f \upharpoonright {\mathcal {F}}_n$ is a function of this kind and can be expressed in terms of $E_{f,1}$ by the formula

$$\begin{aligned} E_{f,n} = \mathbf 1 _n - \underbrace{(1-E_{f,1}) \otimes _s \cdots \otimes _s (1 - E_{f,1}) }_n, \end{aligned}$$

where $\mathbf 1 _n$ is the unit operator on ${\mathcal {F}}_n$.

1.2 Comparison of Hamiltonians

As outlined in the main text, we need to consider the difference of Hamiltonians $(H_n - W_f H_{n-1} W_f^*) \, E_{f,n}$, cf. Eq. (5.3). In our first technical lemma we focus on the second term in this difference and compute lifts of operators on ${\mathcal {F}}_{n-1}$ to $E_{f,n} \, {\mathcal {F}}_n \subset {\mathcal {F}}_n$, which are induced by the adjoint action of $W_f$; recall that $E_{f,n} \, {\mathcal {F}}_n = |f\rangle \otimes _s {\mathcal {F}}_{n-1}$, $n \in {{\mathbb {N}}}$. In the statement of the lemma there appear similarity transformations $\sigma _f$ of gauge invariant operators O on ${\mathcal {F}}$, given by

$$\begin{aligned} \sigma _f(O) \doteq (1 + N_f)^{-1/2} \, O \, (1 + N_f)^{1/2}. \end{aligned}$$

(A.2)

We put $\sigma _{f,n}$ for the restriction of $\sigma _f$ to gauge invariant operators on ${\mathcal {F}}_n$, $n \in {{\mathbb {N}}}_0$.

Lemma A.1

Let $n \in {{\mathbb {N}}}$ and let $O_{n-1}$ be an operator with domain ${\mathcal {D}}_{n-1} \subset {\mathcal {F}}_{n-1}$ which is stable under the action of the spectral projections of $N_{f,n-1}$. Then $\sigma _{f, n-1}(O_{n-1})$ and $\sigma _{f, n-1}^{-1}(O_{n-1})$ are defined on ${\mathcal {D}}_{n-1}$. Moreover, one has for any $\, {\varvec{\varPhi }}_{n-1} \in {\mathcal {D}}_{n-1}$ the equalities

• (i)    $W_f O_{n-1} W_f^* \ \big ( | f \rangle \otimes _s {\varvec{\varPhi }}_{n-1} \big ) \ = \ | f \rangle \otimes _s \sigma _{f,n-1}(O_{n-1}) \, {\varvec{\varPhi }}_{n-1}$

• (ii)    $| f \rangle \otimes _s O_{n-1} {\varvec{\varPhi }}_{n-1} \ = \ W_f \, \sigma _{f,n-1}^{-1}(O_{n-1}) \, W_f^* \ \big ( | f \rangle \otimes _s {\varvec{\varPhi }}_{n-1} \big )$.

Proof

Noticing that the spectral decompositions of $(\mathbf 1 _{n-1} + N_{f,n-1})^{\pm 1/2}$ are finite linear combinations of the spectral projections of $N_{f,n-1}$, the statement concerning the domains of the similarity transformed operators follow. For the proof of (i) we note that $a(f) a^*(f) \, {\varvec{\varPhi }}_{n-1} = (\mathbf 1 _{n-1} + N_{f,n-1}) \, {\varvec{\varPhi }}_{n-1}$, hence

$$\begin{aligned} a(f) \ \big ( | f \rangle \otimes _s {\varvec{\varPhi }}_{n-1} \big ) = a(f) \ n^{-1/2} a^*(f) \, {\varvec{\varPhi }}_{n-1} = n^{-1/2} \, (\mathbf 1 _{n-1} + N_{f,n-1}) \, {\varvec{\varPhi }}_{n-1}. \end{aligned}$$

Thus, by the spectral properties of $N_{f,n-1}$, the vector $W_f^* \ \big ( | f \rangle \otimes _s {\varvec{\varPhi }}_{n-1} \big )$ is also an element of ${\mathcal {D}}_{n-1}$. So one has

$$\begin{aligned} W_f \, O_{n-1} \,&W_f^* \ \big ( | f \rangle \otimes _s {\varvec{\varPhi }}_{n-1} \big ) \, \, = \, n^{-1/2} \, W_f \, O_{n-1} \, (\mathbf 1 _{n-1} + N_{f,n-1})^{1/2} \, {\varvec{\varPhi }}_{n-1} \\&= n^{-1/2} \, a^*(f) \, \sigma _{f,n-1}(O_{n-1}) \, {\varvec{\varPhi }}_{n-1} = | f \rangle \otimes _s \sigma _{f,n-1}(O_{n-1}) \, {\varvec{\varPhi }}_{n-1}, \end{aligned}$$

proving the first statement. Statement (ii) follows from (i) if one replaces the operator $O_{n-1}$ by $\sigma _{f,n-1}^{-1}(O_{n-1})$, completing the proof. $\quad \square$

We consider now the Hamiltonians $H_{n-1}$, $n \in {{\mathbb {N}}}$. For them the spaces

$$\begin{aligned} {\mathcal {D}}_{n-1} \doteq \underbrace{{\mathcal {D}}({{\mathbb {R}}}^s) \otimes _s \cdots \otimes _s {\mathcal {D}}({{\mathbb {R}}}^s)}_{n-1} \subset {\mathcal {F}}_{n-1} \end{aligned}$$

are domains of essential selfadjointness. In view of the choice of the function f, it is also evident that these spaces are stable under the action of the spectral projections of $N_{f,n-1}$. So the first part of the preceding lemma applies to $W_f \, H_{n-1} \, W_f^*$, giving the equality

$$\begin{aligned} W_f \, H_{n-1} \, W_f^* \upharpoonright | f \rangle \otimes _s {\mathcal {D}}_{n-1} = | f \rangle \otimes _s \big (\sigma _{f,n-1}(H_{n-1}) \upharpoonright {\mathcal {D}}_{n-1} \big ) . \end{aligned}$$

We compare now the operators $H_{n-1}$ and $\sigma _{f,n-1}(H_{n-1})$.

Lemma A.2

Let $n \in {{\mathbb {N}}}$. Then

$$\begin{aligned} H_{n-1} - \sigma _{f,n-1}(H_{n-1}) = {\check{A}}_{f, n-1} + {\check{B}}_{f, n-1}. \end{aligned}$$

Here ${\check{A}}_{f. n-1} = {\check{A}}_f \upharpoonright {\mathcal {F}}_{n-1}$, where ${\check{A}}_f = \big ({\check{O}}_f - \sigma _f({\check{O}}_f) \big )$ and ${\check{O}}_f$ is the second quantization of one- and two-particle operators of finite rank; so ${\check{A}}_{f, n-1} \in {{\mathfrak {K}}}_{n-1}$. If $n \ge 3$ one has ${\check{B}}_{f,n-1} = {\check{B}}_f \upharpoonright {\mathcal {F}}_{n-1}$, where ${\check{B}}_f = \big ( {\check{V}}_f - \sigma _f({\check{V}}_f) \big )$ and ${\check{V}}_f$ is the second quantization of the localized pair potential V on ${\mathcal {F}}_2$. This localized potential is given by

$$\begin{aligned} {\check{V}}_{f,2} = 2 \, (E_{f,1} \otimes _s 1) \, V \, (E_{f,1}^\perp \otimes _s 1) + 2 \, (E_{f,1}^\perp \otimes _s 1) \, V \, (E_{f,1} \otimes _s 1), \end{aligned}$$

where $E_{f,1}^\perp \doteq (1 - E_{f,1})$. So the restriction of the corresponding second quantized operator ${\check{V}}_{f, n-1} = {\check{V}}_f \upharpoonright {\mathcal {F}}_{n-1}$ is

$$\begin{aligned} {\check{V}}_{f, n-1} = (n-1)(n-2) \, ({\check{V}}_{f,2} \otimes _s \underbrace{1 \otimes _s \cdots \otimes _s 1}_{n-3}), \end{aligned}$$

and the resulting operator ${\check{B}}_{f, n-1}$ is bounded.

Remark

Since the operator ${\check{V}}_{f,2}$ is not an element of ${\mathcal {K}}_2$, it has to be treated separately. It will be crucial in the subsequent analysis that ${\check{V}}_{f,2}$ is effectively localized by the factor $(E_{f,1} \otimes _s 1)$ next to V.

Proof

According to relation (A.1) we have

$$\begin{aligned} H_n = n \, ({\varvec{P}}_\kappa ^2 \otimes _s \underbrace{1 \otimes _s \cdots \otimes _s 1}_{n-1}) + n(n-1) \, (V \otimes _s \underbrace{1 \otimes _s \cdots \otimes _s 1}_{n-2}). \end{aligned}$$

We decompose the operator ${\varvec{P}}_\kappa ^2$, acting in ${\mathcal {F}}_1$, into

$$\begin{aligned} {\varvec{P}}_\kappa ^2 = E_{f,1}^\perp \, {\varvec{P}}_\kappa ^2 \, E_{f,1}^\perp + E_{f,1} \, {\varvec{P}}_\kappa ^2 \, E_{f,1}^\perp + E_{f,1}^\perp \, {\varvec{P}}_\kappa ^2 \, E_{f, 1} + E_{f,1} \, {\varvec{P}}_\kappa ^2 \, E_{f, 1} \, . \end{aligned}$$

This decomposition is meaningful since $| f \rangle$ lies in the domain of ${\varvec{P}}_\kappa ^2$. The first operator on the right hand side of this equality maps the orthogonal complement of the ray of $| f \rangle$ into itself; the three remaining operators are of rank one. Similarly, we decompose the pair potential V on ${\mathcal {F}}_2$ into

$$\begin{aligned} V&= (E_{f,1}^\perp \otimes _s E_{f,1}^\perp ) \, V \, (E_{f,1}^\perp \otimes _s E_{f,1}^\perp ) - (E_{f,1} \otimes _s E_{f,1}) \, V \, (E_{f,1} \otimes _s E_{f,1}) \\&\quad - \, (E_{f,1} \otimes _s E_{f,1}) \, V \, ((1 - 2 E_{f,1}) \otimes _s 1) - ((1 - 2 E_{f,1}) \otimes _s 1) \, V \, (E_{f,1} \otimes _s E_{f,1}) \\&\quad + \, 2 \, (E_{f,1} \otimes _s 1) \, V \, (E_{f,1}^\perp \otimes _s 1) + 2 \, (E_{f,1}^\perp \otimes _s 1) \, V \, (E_{f,1} \otimes _s 1) \, . \end{aligned}$$

The first operator on the right hand side of this equality maps the orthogonal complement of $| f\rangle \otimes _s {\mathcal {F}}_1 \subset {\mathcal {F}}_2$ into itself. The second up to the fourth terms are operators of finite rank due to the appearance of the factor ($E_{f,1} \otimes E_{f,1}$). The two terms in the last line form the localized pair potential ${\check{V}}_{f,2}$ in the statement of the lemma.

Tensoring the resulting operators with unit operators 1 and multiplying them with appropriate factors of n, we obtain on ${\mathcal {F}}_n$ a corresponding decomposition of the operator $\vartheta _{n-1} \doteq H_{n-1} - \sigma _{f,n-1}(H_{n-1})$. Since the operators

$$\begin{aligned}&E_{f,1}^\perp \, {\varvec{P}}^2 \, E_{f,1}^\perp \otimes _s \underbrace{1 \otimes _s \cdots \otimes _s 1}_{n-2}, \\&(E_{f,1}^\perp \otimes _s E_{f,1}^\perp ) \, V \, (E_{f,1}^\perp \otimes _s E_{f,1}^\perp ) \otimes _s \underbrace{1 \otimes _s \cdots \otimes _s 1}_{n-3} \end{aligned}$$

commute with $N_{f,n-1}$ and consequently stay fixed under the action of the similarity transformation $\sigma _{f,n-1}$, they do not contribute to $\vartheta _{n-1}$ and can be omitted from $H_{n-1}$. The remaining terms in $H_{n-1}$ consist of two types. The first one is, for any $n \in {{\mathbb {N}}}$, a sum of fixed one- and two-particle operators of finite rank which are tensored with unit operators and multiplied by appropriate factors of n. Denoting by ${\check{O}}_f$ the second quantization of these one- and two-particle operators, it follows from Eq. (3.5) that ${\check{O}}_{f, n-1} = {\check{O}}_f \upharpoonright {\mathcal {F}}_{n-1} \in {{\mathfrak {K}}}_{n-1}$. Since $(\mathbf 1 _{n-1} + N_{f,n-1})^{\pm 1/2} \in {{\mathfrak {K}}}_{n-1}$ it is also clear that $\sigma _{f,n-1}({\check{O}}_{f, n-1}) \in {{\mathfrak {K}}}_{n-1}$.

The second type of terms in $H_{n-1}$, contributing to $\vartheta _{n-1}$, arise from the second quantization ${\check{V}}_f$ of the pair potential ${\check{V}}_{f,2}$. The resulting operators ${\check{V}}_{f,n-1} = {\check{V}}_f \upharpoonright {\mathcal {F}}_{n-1}$ and their similarity transformed images $\sigma _{f,n-1}({\check{V}}_{f,n-1})$ are bounded since the pair potential V and the operators $(\mathbf 1 _{n-1} + N_{f,n-1})^{\pm 1/2}$ are bounded. $\quad \square$

Next, we compare the operators $H_n \upharpoonright |f\rangle \otimes _s {\mathcal {D}}_{n-1}$ and $|f \rangle \otimes _s \big ( H_{n-1} \upharpoonright {\mathcal {D}}_{n-1} \big )$.

Lemma A.3

Let $n \in {{\mathbb {N}}}$. One has on ${\mathcal {F}}_n$ the equality (pointwise on ${\mathcal {D}}_{n-1}$)

$$\begin{aligned} H_n \upharpoonright |f\rangle \otimes _s {\mathcal {D}}_{n-1} - | f \rangle \otimes _s \big ( H_{n-1} \upharpoonright {\mathcal {D}}_{n-1} \big ) = {\hat{A}}_{f,n} + {\hat{B}}_{f,n}. \end{aligned}$$

Here ${\hat{A}}_{f,n} = {\hat{A}}_f \upharpoonright {\mathcal {F}}_n \in {{\mathfrak {K}}}_n$, where ${\hat{A}}_f = {\hat{O}}_f N_f^{-1} E_f$ and ${\hat{O}}_f$ is the second quantization of a one-particle operator of rank one. If $n \ge 2$ one has ${\hat{B}}_{f,n} = {\hat{B}}_f \upharpoonright {\mathcal {F}}_n$, where ${\hat{B}}_f = {\hat{V}}_f N_f^{-1} E_f$ and ${\hat{V}}_f$ is the second quantization of the localized pair potential $\, {\hat{V}}_{f,2} = V \, (E_{f,1} \otimes _s 1)$. Its restriction $\ {\hat{V}}_{f,n} = {\hat{V}}_{f} \upharpoonright {\mathcal {F}}_n$ is given by

$$\begin{aligned}{\hat{V}}_{f,n} = n(n-1) \, \big ( {\hat{V}}_{f,2} \otimes _s \underbrace{1 \otimes _s \cdots \otimes _s 1}_{n-2} \big ) , \end{aligned}$$

so the operator ${\hat{B}}_{f,n} = {\hat{V}}_{f,n} \, N_{f,n}^{-1} E_{f,n}$ is bounded.

Proof

It suffices to establish the statement for vectors of the special form

$$\begin{aligned} {\varvec{\varPhi }}_{n-1} = | f_1 \rangle \otimes _s \cdots \otimes _s | f_{n-1} \rangle \, \in \, {\mathcal {D}}_{n-1}, \end{aligned}$$

where $f_1, \dots , f_{n-1} \in {\mathcal {D}}({{\mathbb {R}}}^s)$ are members of some orthonormal basis in $L^2({{\mathbb {R}}}^s)$ which includes f. Making use of the fact that the Hamiltonians are symmetrized sums of the one- and two-particle operators, given above, we obtain

$$\begin{aligned}&H_n \, \big ( |f \rangle \otimes _s | f_1 \rangle \otimes _s \cdot \! \cdot \otimes _s | f_{n-1} \rangle \big ) \, - \, |f \rangle \otimes _s H_{n-1} \, \big ( | f_1 \rangle \otimes _s \cdot \! \cdot \otimes _s | f_{n-1} \rangle \big ) \\&= |{\varvec{P}}_\kappa ^2 f \rangle \otimes _s | f_1 \rangle \otimes _s \cdot \! \cdot \otimes _s | f_{n-1} \rangle \! + \! \sum _{i=1}^{n-1} \, ( V \, |f \rangle \otimes _s |f_i\rangle ) \otimes _s |f_1 \rangle \otimes _s \cdot \! \cdot \overset{i}{\vee } \cdot \! \cdot \otimes _s |f_{n-1} \rangle , \end{aligned}$$

where the symbol $\overset{i}{\vee }$ indicates the omission of the single particle component $|f_i \rangle$. So we must exhibit an operator which maps the initial vectors $\,|f \rangle \otimes _s | f_1 \rangle \otimes _s \cdot \cdot \otimes _s | f_{n-1} \rangle \in {\mathcal {F}}_n$ to the image vectors on the right hand side of the preceding equality. Recalling that $f, f_1, \dots , f_{n-1}$ are members of some orthonormal basis, we have

$$\begin{aligned} ({\varvec{P}}_\kappa ^2 E_{f,1} \otimes _s \underbrace{1 \otimes _s \cdot \cdot \otimes _s 1}_{n-1} ) \, \big ( |f \rangle&\otimes _s | f_1 \rangle \otimes _s \cdots \otimes _s | f_{n-1} \rangle \big ) \\&= n_f/n \, |{\varvec{P}}_\kappa ^2 f \rangle \otimes _s | f_1 \rangle \otimes _s \cdots \otimes _s | f_{n-1} \rangle , \end{aligned}$$

where $n_f$ is the number of factors $| f \rangle$ appearing in the initial vector. This equality holds for arbitrary components ${\varvec{\varPhi }}_{n-1}$ in $| f \rangle \otimes _s {\varvec{\varPhi }}_{n-1}$ if one replaces the number $n_f$ by the operator $N_{f, n}$. Moreover, since the initial vector is an element of the space $| f \rangle \otimes _s {\mathcal {F}}_{n-1}$, it stays constant if one multiplies it by the projection $E_{f, n}$. This gives

$$\begin{aligned} |{\varvec{P}}_\kappa ^2 f \rangle&\otimes _s | f_1 \rangle \otimes _s \cdots \otimes _s | f_{n-1} \rangle \\&= n \, ({\varvec{P}}^2 E_{f,1} \otimes _s \underbrace{1 \otimes _s \cdot \cdot \otimes _s 1}_{n-1}) \ N_{f,n}^{-1} E_{f,n} \ \big ( |f \rangle \otimes _s | f_1 \rangle \otimes _s \cdots \otimes _s | f_{n-1} \rangle \big ) \\&= {\hat{O}}_{f,n} \, N_{f,n}^{-1} E_{f,n} \, \big ( |f \rangle \otimes _s | f_1 \rangle \otimes _s \cdots \otimes _s | f_{n-1} \rangle \big ), \end{aligned}$$

where ${\hat{O}}_{f,n} = {\hat{O}}_f \upharpoonright {\mathcal {F}}_n$ and ${\hat{O}}_f$ is the second quantization of the one-particle operator ${\varvec{P}}_\kappa ^2 E_{f,1}$ on ${\mathcal {F}}_1$, having rank one. So the operator appearing on the right hand side of the second equality is the restriction of ${\hat{A}}_f \doteq {\hat{O}}_f \, N_f^{-1} E_f$ to ${\mathcal {F}}_n$, as stated in the lemma. In a similar manner

$$\begin{aligned}&\sum _{i=1}^{n-1} \, (V \, |f \rangle \otimes _s |f_i\rangle ) \otimes _s |f_1 \rangle \otimes _s \cdot \cdot \overset{i}{\vee } \cdot \cdot \otimes _s |f_{n-1} \rangle \\&\quad = n(n-1) (V (E_{f,1} \otimes _s 1) \otimes _s \underbrace{1 \otimes _s \cdots \otimes _s 1}_{n - 2} ) \, N_{f,n}^{-1} E_{f,n} \, (|f \rangle \otimes _s | f_1 \rangle \otimes _s \cdot \cdot \otimes _s | f_{n-1} \rangle ) \\&\quad = {\hat{V}}_{f, n} \, N_{f,n}^{-1} E_{f,n} \, \big ( |f \rangle \otimes _s | f_1 \rangle \otimes _s \cdot \cdot \otimes _s | f_{n-1} \rangle \big ). \end{aligned}$$

The operator appearing on the right hand side of the second equality is the restriction ${\hat{B}}_{f,n}$ of ${\hat{B}}_f \doteq {\hat{V}}_f \, N_f^{-1} E_f$ to ${\mathcal {F}}_n$. Since the two-body potential is bounded, ${\hat{B}}_{f,n}$ is bounded, completing the proof. $\square$

In the last technical lemma of this subsection, which will also be used further below, we consider the adjoint action $\beta _g \doteq \text {Ad} \, \, W_g$ on the algebra of bounded operators on ${\mathcal {F}}$, which is induced by the isometries $W_g \in \overline{{{\varvec{\mathfrak {F}}}}}$ for arbitrary normalized $g \in L^2({{\mathbb {R}}}^s)$, cf. Lemma 5.1. The restrictions of these maps to the algebras of bounded operators ${\mathcal {B}}({\mathcal {F}}_{n-1})$ on ${\mathcal {F}}_{n-1}$, having range in ${\mathcal {B}}({\mathcal {F}}_n)$, $n \in {{\mathbb {N}}}$, are denoted by

$$\begin{aligned} \beta _{g,n} \doteq \beta _g \upharpoonright {\mathcal {B}}({\mathcal {F}}_{n-1}) = \text {Ad} \, W_f \upharpoonright {\mathcal {B}}({\mathcal {F}}_{n-1}). \end{aligned}$$

(A.3)

The norm of arbitrary linear maps $\beta _n : {\mathcal {B}}({\mathcal {F}}_{n-1}) \rightarrow {\mathcal {B}}({\mathcal {F}}_n)$ is denoted by $\Vert \beta _n \Vert$, $n \in {{\mathbb {N}}}$.

Lemma A.4

Let $n \in {{\mathbb {N}}}$ and let $g \in L^2({{\mathbb {R}}}^s)$ be normalized. Then one has the inclusion $\beta _{g,n}({{\mathfrak {K}}}_{n-1}) \subset {{\mathfrak {K}}}_n$. Moreover, there exists some constant $c_n$ such that for any pair of normalized elements $g_1, g_2 \in L^2({{\mathbb {R}}}^s)$

$$\begin{aligned} \Vert \beta _{g_1,n} - \beta _{g_2,n} \Vert \le c_n \, \Vert g_1 - g_2 \Vert _2 \, . \end{aligned}$$

Proof

According to [2, Lemma 3.3] there exists for given $K_{n-1} \in {{\mathfrak {K}}}_{n-1}$ some observable $A_{n-1} \in \overline{{{\varvec{\mathfrak {A}}}}}$ such that $A_{n-1} \upharpoonright {\mathcal {F}}_{n-1} = K_{n-1}$. It follows from Lemmas 5.1 and 3.2 that $W_g A_{n-1} W_g^* \in \overline{{{\varvec{\mathfrak {A}}}}}$. Hence, applying the results in [2, Lem. 3.3] another time, one obtains

$$\begin{aligned} \beta _{g,n}(K_{n-1}) = (W_g A_{n-1} W_g^*) \upharpoonright {\mathcal {F}}_n \, \in \, \overline{{{\varvec{\mathfrak {A}}}}}\upharpoonright {\mathcal {F}}_n = {{\mathfrak {K}}}_n, \end{aligned}$$

as claimed. The continuity of the maps is a consequence of Lemma 5.1, which leads to the estimate

$$\begin{aligned} \Vert \beta _{g_1,n} - \beta _{g_2,n} \Vert \le 2 \, \Vert \big ( W_{g_1}^* - W_{g_2}^* \big ) P_n \Vert \le c_n \, \Vert g_1 - g_2\Vert _2, \end{aligned}$$

completing the proof. $\quad \square$

We have gathered now the information needed for the description of the structure of the operator $\big (H - \beta _f(H)\big ) \, E_f$.

Proposition A.5

Let $n \in {{\mathbb {N}}}_0$, then

$$\begin{aligned} \big (H - \beta _f(H)\big ) \, E_f \upharpoonright {\mathcal {F}}_n = A_{f, n} + B_{f, n}. \end{aligned}$$

Here $A_{f, n} = A_f \upharpoonright {\mathcal {F}}_n$, where $A_f = {\hat{A}}_f + \beta _f \, \circ \, \sigma _f^{-1}({\check{A}}_f)$ and the operators ${\check{A}}_f$, ${\hat{A}}_f$ were defined in Lemmas A.2 and A.3, respectively. One has $A_{f, n} \in {{\mathfrak {K}}}_n$. In a similar manner, $B_{f, n} = B_f \upharpoonright {\mathcal {F}}_n$, where $B_f = {\hat{B}}_f + \beta _f \, \circ \, \sigma _f^{-1}({\check{B}}_f)$ and the operators ${\check{B}}_f$, ${\hat{B}}_f$ were likewise defined in these two lemmas. The operator $B_{f, n}$ is bounded.

Proof

Recalling that $E_f {\mathcal {F}}_n = | f \rangle \otimes _s {\mathcal {F}}_{n-1}$, one obtains for ${\varvec{\varPhi }}_{n-1} \in {\mathcal {D}}_{n-1}$

$$\begin{aligned} \big (H_n - \beta _f(H_{n-1})\big ) \, ( | f\rangle \otimes _s {\varvec{\varPhi }}_{n-1} )&= \big (H_n \, (|f \rangle \otimes _s {\varvec{\varPhi }}_{n-1}) - |f \rangle \otimes _s H_{n-1} \, {\varvec{\varPhi }}_{n-1} \big ) \\&\quad + |f\rangle \otimes _s\big (H_{n-1}-\sigma _{f,n-1}(H_{n-1})\big ) {\varvec{\varPhi }}_{n-1}. \end{aligned}$$

The first term on the right hand side of this equality coincides according to Lemma A.3 with $({\hat{A}}_{f,n} + {\hat{B}}_{f,n}) \, ( | f \rangle \otimes _s {\varvec{\varPhi }}_{n-1})$, where ${\hat{A}}_{f,n} \in {{\mathfrak {K}}}_n$ and ${\hat{B}}_{f,n}$ is bounded. In the second term we made use of Lemma A.1(i) according to which

$$\begin{aligned} \beta _f(H_{n-1}) \ \big ( | f\rangle \otimes _s {\varvec{\varPhi }}_{n-1} \big ) = | f\rangle \otimes _s \sigma _{f,n-1}(H_{n-1}) \, {\varvec{\varPhi }}_{n-1} \, . \end{aligned}$$

Hence this second term can be presented in the form $| f \rangle \otimes _s ({\check{A}}_{f,n-1} + {\check{B}}_{f,n-1}) {\varvec{\varPhi }}_{n-1}$, as was shown in Lemma A.2. According to Lemma A.1(ii), the latter vector coincides with the image of $| f \rangle \otimes {\varvec{\varPhi }}_{n-1}$ under the action of $\beta _f \, \circ \, \sigma _f^{-1}({\check{A}}_f + {\check{B}}_f) \upharpoonright {\mathcal {F}}_n$.

Turning to the proof that $A_{f,n} \in {{\mathfrak {K}}}_n$, we note that

$$\begin{aligned} \sigma _{f,n-1}^{-1}({\check{A}}_{f,n-1}) = (\mathbf 1 _{n-1} + N_{f,n-1})^{1/2} {\check{A}}_{f,n-1} (\mathbf 1 _{n-1} + N_{f,n-1})^{-1/2} \in {{\mathfrak {K}}}_{n-1}. \end{aligned}$$

It therefore follows from the preceding lemma that

$$\begin{aligned} \beta _f \, \circ \, \sigma _f^{-1}({\check{A}}_f) \upharpoonright {\mathcal {F}}_n = \beta _{f,n} \, \circ \, \sigma _{f,n-1}^{-1}({\check{A}}_{f,n-1}) \in {{\mathfrak {K}}}_n. \end{aligned}$$

Since also ${\hat{A}}_{f,n} = {\hat{A}}_f \upharpoonright {\mathcal {F}}_n \in {{\mathfrak {K}}}_n$, we obtain $A_{f,n} \in {{\mathfrak {K}}}_n$. That $B_{f,n}$ is bounded is apparent, completing the proof. $\quad \square$

1.3 Dyson expansions with values in ${{\mathfrak {K}}}_n$

We turn now to the analysis of the operator function $t \mapsto \varGamma _f(t) E_f$, defined in Eq. (5.2). It is differentiable in t in the sense of sesquilinear forms between vectors in the domains of H, respectively $W_f H W_f^*$. The derivatives are given by

$$\begin{aligned} \frac{d}{dt} \, \varGamma _f(t) \, E_f&= i \, e^{itH}(H - W_f H W_f^*) \, e^{-it W_f H W_f^*} \, E_f \\&= i \, e^{itH}(H - W_f H W_f^*) E_f \, e^{-it W_f H W_f^*} \, E_f \\&= i \, e^{itH}(H - W_f H W_f^*) E_f \, e^{-itH} \ \varGamma _f(t) \, E_f, \end{aligned}$$

where the second equality holds since $W_f H W_f^*$ commutes with $E_f$. We restrict this equality to ${\mathcal {F}}_n$ and put $\varGamma _{f,n}(t) \doteq \varGamma _f(t) \, E_f \upharpoonright {\mathcal {F}}_n$, hence $\varGamma _{f,n}(t) \upharpoonright (\mathbf 1 _n - E_{f,n}) \, {\mathcal {F}}_n = 0, n \in {{\mathbb {N}}}_0$. By Proposition A.5 we have

$$\begin{aligned} (H - W_f H W_f^*) E_f \upharpoonright {\mathcal {F}}_n = A_{f,n} + B_{f,n}, \end{aligned}$$

where $A_{f,n} \in {\mathcal {K}}_n$ and $B_{f,n}$ is a bounded operator. Putting

$$\begin{aligned} C_{f,n}(s) \doteq \text {Ad} \, e^{isH_n} (A_{f,n} + B_{f,n}), \quad s \in {{\mathbb {R}}}, \end{aligned}$$

we can solve the above differential equation for $t \mapsto \varGamma _{f,n}(t)$ on $E_{f,n} \, {\mathcal {F}}_n$ by the Dyson series of time ordered integrals, defined in the strong operator topology,

$$\begin{aligned} \varGamma _{f,n}(t) = \big (E_{f,n} + \sum _{k=1}^\infty i^k \, \! \! \! \int _0^t \! ds_k \! \! \int _0^{s_k} \! ds_{k-1} \dots \! \! \int _0^{s_2} \! ds_1 \, C_{f,n}(s_k) \cdots C_{f,n}(s_1) \big ). \end{aligned}$$

(A.4)

This series converges absolutely in norm since the operators $C_{f,n}$ are bounded.

We want to show that $\varGamma _{f,n}(t) \in {{\mathfrak {K}}}_n$, $n \in {{\mathbb {N}}}_0$. As we shall see, it is sufficient to prove that the functions $t \mapsto \int _0^t \! ds \, C_{f,n}(s)$ have range in ${{\mathfrak {K}}}_n$ and are norm continuous, $t \in {{\mathbb {R}}}$. For the summand $A_{f,n} \in {{\mathfrak {K}}}_n$ of $C_{f,n}$ this property follows from the fact that the time evolution acts pointwise norm continuously on ${{\mathfrak {K}}}_n$, cf. Proposition 4.4 and the appendix in [2]. The argument for the second summand $B_{f,n}$ is more involved since these operators are not contained in ${{\mathfrak {K}}}_n$. We begin with a technical lemma about integrals of functions having values in operators, respectively linear maps between C*-algebras. In order to avoid repetitions of technicalities, we make the following standing assertion.

Statement: In the subsequent analysis all integrals are defined in the strong operator (s.o.) topology of the underlying Hilbert spaces, unless otherwise stated.

Lemma A.6

For $k = 1,2$, let ${\mathcal {H}}_k$ be a Hilbert space and let ${{\mathfrak {B}}}_k \subset {\mathcal {B}}({\mathcal {H}}_k)$ be a C*-algebra. Furthermore, let $s \mapsto B_1(s) \in {\mathcal {B}}({\mathcal {H}}_1)$, $s \in {{\mathbb {R}}}$, be a s.o. continuous operator function such that $\int _0^t \! ds \, B_1(s) \in {{\mathfrak {B}}}_1$, $t \in {{\mathbb {R}}}$. Finally, let $s \mapsto \lambda (s)$ be a norm continuous function with values in linear maps from ${\mathcal {B}}({\mathcal {H}}_1)$ into ${\mathcal {B}}({\mathcal {H}}_2)$, which, for fixed $s \in {{\mathbb {R}}}$, are normal (s.o. continuous) and whose restrictions to ${{\mathfrak {B}}}_1$ have values in the algebra ${{\mathfrak {B}}}_2$, i.e. $\lambda (s)({{\mathfrak {B}}}_1) \subset {{\mathfrak {B}}}_2$.

Under these conditions the function $s \mapsto \lambda (s)\big (B_1(s)\big ) \in {\mathcal {B}}({\mathcal {H}}_2)$ is s.o. continuous. Its integral $\, t \mapsto \int _0^t \! ds \, \lambda (s)\big (B_1(s)\big )$ is norm continuous and has values in ${{\mathfrak {B}}}_2$, $t \in {{\mathbb {R}}}$. For fixed t, it can be approximated in norm in the limit $m \rightarrow \infty$ by the sums

$$\begin{aligned} \sum _{l = 1}^m \lambda (lt/m)\Big (\int _{(l-1)t/m}^{lt/m} \! \! ds \, B_1(s) \Big ) \in {{\mathfrak {B}}}_2, \quad m \in {{\mathbb {N}}}. \end{aligned}$$

(Note that the functions in this lemma are not necessarily defined by the action of some dynamics.)

Proof

Let $s_0 \in {{\mathbb {R}}}$. Then one has on ${\mathcal {H}}_2$ the equality

$$\begin{aligned} \lambda (s) \big (B_1(s) \big ) - \lambda (s_0) \big (B_1(s_0) \big ) = \lambda (s_0) \big (B_1(s) - B_1(s_0) \big ) + \big (\lambda (s) - \lambda (s_0)\big ) \big (B_1(s) \big ). \end{aligned}$$

Since the map $\lambda (s_0)$ is normal on ${\mathcal {B}}({\mathcal {H}}_1)$, the first term on the right hand side of this equality vanishes in the s.o. topology in the limit $s \rightarrow s_0$. The second term vanishes in this limit as well, since $\lambda (s) \rightarrow \lambda (s_0)$ in the norm topology of ${\mathcal {B}}({\mathcal {H}}_2)$, uniformly on bounded subsets of ${\mathcal {B}}({\mathcal {H}}_1)$. Thus $s \mapsto \lambda (s) \big (B_1(s) \big )$ is continuous in the s.o. topology and the integrals exist. Assuming without loss of generality that $t \ge 0$, we partition [0, t] into $m \in {{\mathbb {N}}}$ intervals, giving the estimate in ${\mathcal {B}}({\mathcal {H}}_2)$

$$\begin{aligned} \Vert \int _0^t \! ds \, \lambda (s)&\big (B_1(s)\big ) - \sum _{l = 1}^m \lambda (lt/m)\Big (\int _{(l-1)t/m}^{lt/m} \! \! ds \, B_1(s) \Big ) \Vert \\&= \Vert \sum _{l = 1}^m \int _{(l-1)t/m}^{lt/m} \! ds \, \big (\lambda (s) - \lambda (lt/m) \big ) \big (B_1(s)\big ) \Vert \\&\le \Vert B_1 \Vert _\infty \sum _{l = 1}^m \int _{(l-1)t/m}^{lt/m} \! ds \, \Vert \lambda (s) - \lambda (lt/m) \Vert , \end{aligned}$$

where $\Vert B_1 \Vert _\infty \doteq \sup _{0 \le s \le t} \Vert B(s) \Vert$. Because of the norm continuity of $s \mapsto \lambda (s)$, this shows that the expression on the first line tends to 0 in the limit $m \rightarrow \infty$. Since, by assumption, $\int _{(l-1)t/m)}^{lt/m} \! ds \, B_1(s) \in {{\mathfrak {B}}}_1$ and $\lambda (lt/m)$ maps the C*-algebra ${{\mathfrak {B}}}_1$ into ${{\mathfrak {B}}}_2$, $1 \le l \le m$, it follows that $\int _0^t \! ds \, \lambda (s)\big (B_1(s)\big ) \in {{\mathfrak {B}}}_2$. Moreover, the integral can be approximated in norm by the finite sums given in the lemma.

The statement about the continuity properties follows from the estimate

$$\begin{aligned}&\Vert \int _0^{t_2} \! ds \, \lambda (s) \big (B(s) \big ) - \int _0^{t_1} \! ds \, \lambda (s) \big (B(s) \big ) \Vert \\&\quad = \Vert \int _{t_1}^{t_2} \! ds \, \lambda (s) \big (B(s) \big ) \Vert \le \Vert B \Vert _\infty \Vert \lambda \Vert _\infty |t_2 - t_1|, \end{aligned}$$

where $\Vert B_1 \Vert _\infty$, $\Vert \lambda \Vert _\infty$ are the suprema of $s \mapsto \Vert B_1(s) \Vert$, respectively $s \mapsto \Vert \lambda (s) \Vert$, on any given bounded subset of ${{\mathbb {R}}}$, containing the integration intervals. $\quad \square$

This lemma will be applied to various types of functions and has therefore been formulated in general terms. It will allow us to determine the properties of integrals involving the localized potentials, cf. (A.4). In the subsequent lemma we consider integrals of operators evolving under the non-interacting time evolution $e^{isH_0}$, $s \in {{\mathbb {R}}}$. Here $H_0$ is the second quantization of the single particle operator ${\varvec{P}}_\kappa ^2 = {\varvec{P}}^2 + \kappa ^2 \, {\varvec{Q}}^2$, where $\kappa \ge 0$ is kept fixed. We make use of the short hand notation $B(s)_{0} = \text {Ad} \, e^{isH_0}(B)$ for arbitrary bounded operators B on ${\mathcal {F}}$. If B is gauge invariant (preserves the particle number), we denote its restriction to the n-particle space by $B_n(s)_{0} \doteq \text {Ad} \, e^{isH_0,n}(B_n) = B(s)_{0} \upharpoonright {\mathcal {F}}_n$, $n \in {{\mathbb {N}}}_0$. Note that the functions $s \mapsto B(s)_{0}$ are continuous in the s.o. topology.

Lemma A.7

Let $B_n \in {\mathcal {B}}({\mathcal {F}}_n)$ such that the functions $t \mapsto \int _0^t \! ds \, B_n(s)_0$, $t \in {{\mathbb {R}}}$, have values in ${{\mathfrak {K}}}_n$, $n \in {{\mathbb {N}}}_0$.

• (i) The functions $t \mapsto \int _0^t \! ds \, \big (K_n' B_n K_n'' \big )(s)_0$ have values in ${{\mathfrak {K}}}_n$ and are norm continuous for any choice of $K_n', K_n'' \in {{\mathfrak {K}}}_n$.

• (ii) Let $\beta _{g,n} : {\mathcal {B}}({\mathcal {F}}_{n-1}) \rightarrow {\mathcal {B}}({\mathcal {F}}_n)$ be the map defined in Eq. (A.3) for normalized $g \in L^2({{\mathbb {R}}}^s)$. The functions $t \mapsto \int _0^t \! ds \, \big (\beta _{g,n}(K_{n-1}' B_{n-1} K_{n-1}'') \big )(s)_0$ have values in ${{\mathfrak {K}}}_n$ and are norm continuous for any choice of $K_{n-1}', K_{n-1}'' \in {{\mathfrak {K}}}_{n-1}$.

Proof

(i) Consider the function $\lambda _n : {{\mathbb {R}}}\times {\mathcal {B}}({\mathcal {F}}_n) \rightarrow {\mathcal {B}}({\mathcal {F}}_n)$, which is given by

$$\begin{aligned} \lambda _n(s)(B_n') \doteq K_n'(s)_0 \, B_n' \, K_n''(s)_0, \quad s \in {{\mathbb {R}}}, \ B_n' \in {\mathcal {B}}({\mathcal {F}}_n). \end{aligned}$$

For fixed s the linear map $\lambda _n(s)$ is clearly normal. Moreover, since the time translations leave the algebra of observables $\overline{{{\varvec{\mathfrak {A}}}}}$ invariant and act pointwise norm continuously on ${{\mathfrak {K}}}_n = \overline{{{\varvec{\mathfrak {A}}}}}\upharpoonright {\mathcal {F}}_n$, cf. Proposition 4.4 and the appendix in [2], the function is norm continuous and its restriction to ${{\mathfrak {K}}}_n$ maps this subalgebra into itself. The function $s \mapsto B_n(s)_0$ is continuous in the s.o. topology and, by assumption, $t \mapsto \int _0^t \! ds \, B_n(s)_0 \in {{\mathfrak {K}}}_n$, $t \in {{\mathbb {R}}}$, so the first statement follows from Lemma A.6.

(ii) Since we are dealing with the non-interacting time evolution, we have

$$\begin{aligned} \big (\beta _{g,n}(K_{n-1}' B_{n-1} K_{n-1}'') \big )(s)_0 = \beta _{g(s)_0 ,n} \, (K_{n-1}'(s)_0 \, B_{n-1}(s)_0 \, K_{n-1}''(s)_0 ) , \end{aligned}$$

where $g(s)_0 \doteq e^{is {\varvec{P}}_\kappa ^2} \, g \in L^2({{\mathbb {R}}}^s)$ is normalized. We consider now the function of maps $\lambda _n : {{\mathbb {R}}}\times {\mathcal {B}}({\mathcal {F}}_{n-1}) \rightarrow {\mathcal {B}}({\mathcal {F}}_n)$, given by

$$\begin{aligned} \lambda _n(s)(B_{n-1}') \doteq \beta _{g(s)_0 ,n}(K_{n-1}'(s)_0 \, B_{n-1}' \, K_{n-1}''(s)_0 ) \, , \ \ s \in {{\mathbb {R}}}, \ B_{n-1}' \in {\mathcal {B}}({\mathcal {F}}_{n-1}). \end{aligned}$$

For fixed s the linear map $\lambda _n(s)$ is normal. Moreover, it follows from the norm continuity of $s \mapsto \beta _{g(s)_0 ,n}$, established in the second part of Lemma A.4, and the arguments in step (i) that $s \mapsto \lambda _n(s)$ is norm continuous. Making use of the first part of Lemma A.4, it is also clear that $\lambda _n(s)$ maps ${{\mathfrak {K}}}_{n-1}$ into ${{\mathfrak {K}}}_n$, $s \in {{\mathbb {R}}}$. Furthermore, the function $s \mapsto B_{n-1}(s)_0$ is continuous in the s.o. topology and, by assumption, $t \mapsto \int _0^t \! ds \, B_{n-1} (s)_0 \in {{\mathfrak {K}}}_{n-1}$, $t \in {{\mathbb {R}}}$. So the second statement follows likewise from Lemma A.6. $\quad \square$

In the next lemma, we consider the non-interacting time evolution of localized pair potentials, defined above, and study their integrals.

Lemma A.8

Let ${\check{V}}_{f,2}$ and ${\hat{V}}_{f,2}$ be the localized pair potentials, defined in Lemmas A.2 and A.3, respectively. Denoting by $V_{f,2}$ either one of these potentials, one has

• (i) the function $t \mapsto \int _0^t \! ds \, V_{f,2}(s)_0$ on ${\mathcal {F}}_2$ is norm continuous and has values in the compact operators;

• (ii) for any $n \in {{\mathbb {N}}}$, $n \ge 2$, the function $t \mapsto \int _0^t \! ds \, V_{f, n}(s)_0$ on ${\mathcal {F}}_n$ is norm continuous and has values in ${{\mathfrak {K}}}_n$, $t \in {{\mathbb {R}}}$, where

  $$\begin{aligned} s \mapsto V_{f,n}(s)_0 = n(n-1) \, (V_{f,2}(s)_0 \otimes _s \underbrace{1 \otimes _s \cdots \otimes _s 1}_{n-2}). \end{aligned}$$

Proof

We give the proof for the potential ${\hat{V}}_{f,2} = V \, (E_{f,1} \otimes _s 1)$. Since ${\check{V}}_{f,2}$ also contains the localizing factor $(E_{f,1} \otimes _s 1)$, the corresponding argument is similar and therefore omitted.

(i) In a first step we consider potentials V, having compact support. Picking a smooth function ${\varvec{x}}\mapsto \chi ({\varvec{x}})$ which is equal to 1 for ${\varvec{x}}\in \text {supp} \, f \cup (\text {supp} \, f + \text {supp} \, V)$ and has compact support, we can proceed to ${\hat{V}}_{f,2} = V_{f, \chi } \, (E_{f,1} \otimes _s 1)$, where the potential

$$\begin{aligned}{\varvec{x}}, {\varvec{y}}\mapsto V_{f, \chi }({\varvec{x}},{\varvec{y}}) \doteq V({\varvec{x}}- {\varvec{y}}) \, \chi ({\varvec{x}}) \chi ({\varvec{y}})\end{aligned}$$

is symmetric in ${\varvec{x}}, {\varvec{y}}$, continuous, and compactly supported on the two-particle configuration space ${{\mathbb {R}}}^s \times {{\mathbb {R}}}^s$. The function $s \mapsto V_\chi (s)_0$ is continuous in the s.o. topology and the resulting integral $t \mapsto \int _0^t \! ds \, V_\chi (s)_0$ depends norm continuously on $t \in {{\mathbb {R}}}$ and has values in the compact operators on ${\mathcal {F}}_2$. Similar results were established in previous work, cf. for example the appendix of [5]. We briefly sketch here the argument for the case at hand. Consider the functions, $\kappa \ge 0$,

$$\begin{aligned} s \mapsto c_\kappa (s) \doteq \cos (2\kappa s), \quad s \mapsto s_\kappa (s) \doteq \sin (2\kappa s)/\kappa \, \quad s \in {{\mathbb {R}}}, \end{aligned}$$

where we put $s_0(s) = 2s$. The non-interacting time translations act on the two-particle operator $V_{f, \chi } = V_{f, \chi }({\varvec{Q}}_1, {\varvec{Q}}_2)$ according to

$$\begin{aligned} V_{f, \chi }({\varvec{Q}}_1,{\varvec{Q}}_2)(s)_0 = V_{f, \chi }(c_\kappa (s) {\varvec{Q}}_1 + s_\kappa (s) {\varvec{P}}_1, \ c_\kappa (s) {\varvec{Q}}_2 + s_\kappa (s) {\varvec{P}}_2) \end{aligned}$$

in an obvious notation. For any $(s', s'') \in {{\mathbb {R}}}^2$ the operators $\big ( c_\kappa (s') {\varvec{Q}}_1 + s_\kappa (s') {\varvec{P}}_1 \big )$ commute with $\big ( c_\kappa (s'') {\varvec{Q}}_2 + s_\kappa (s'') {\varvec{P}}_2 \big )$ and one has, $l = 1,2$,

$$\begin{aligned}{}[ \big ( c_\kappa (s') {\varvec{Q}}_l + s_\kappa (s') {\varvec{P}}_l \big ), \big ( c_\kappa (s'') {\varvec{Q}}_l + s_\kappa (s'') {\varvec{P}}_l \big )] = i s_\kappa (s'' - s') \, \mathbf 1 . \end{aligned}$$

Thus for almost all $(s', s'') \in {{\mathbb {R}}}^2$ the operators in the latter commutator do not commute and are canonically conjugate (with rescaled Planck constant). Since ${\varvec{x}},{\varvec{y}}\mapsto V_{f,\chi }({\varvec{x}},{\varvec{y}})$ is continuous and has compact support it follows from standard arguments that the function

$$\begin{aligned} s', s'' \mapsto V_{f, \chi }({\varvec{Q}}_1,{\varvec{Q}}_2)(s')_0 \, V_{f, \chi }({\varvec{Q}}_1,{\varvec{Q}}_2)(s'')_0 \end{aligned}$$

has values in compact operators on ${\mathcal {F}}_2$ for almost all $(s', s'') \in {{\mathbb {R}}}^2$. Since it is also uniformly bounded, this implies that

$$\begin{aligned} | \int _0^t \! ds \, V_{f, \chi }({\varvec{Q}}_1,{\varvec{Q}}_2)(s)_0 |^2 = \int _0^t \! ds' \! \! \int _0^t \! ds'' V_{f, \chi }({\varvec{Q}}_1,{\varvec{Q}}_2)(s')_0 \, V_{f, \chi }({\varvec{Q}}_1,{\varvec{Q}}_2)(s'')_0 \end{aligned}$$

is a compact operator. Taking its square root and making use of polar decomposition, we conclude that $\int _0^t \! ds \, V_{f, \chi }(s)_0 = \int _0^t \! ds \, V_{f, \chi }({\varvec{Q}}_1,{\varvec{Q}}_2)(s)_0$ is compact on ${\mathcal {F}}_2$. (It is this result where we made use of the special form of the external single particle potential; but we expect that it holds more generally.)

In order to see that this conclusion holds also for the original localized potential $V_f = V (E_{f,1} \otimes _s 1)$, we introduce the function $\lambda : {{\mathbb {R}}}\times {\mathcal {B}}({\mathcal {F}}_2) \rightarrow {\mathcal {B}}({\mathcal {F}}_2)$ given by

$$\begin{aligned} \lambda (s)(B_2) \doteq B_2 \, (E_{f,1} \otimes _s 1)(s)_0 = B_2 \, (E_{f,1}(s)_0 \otimes _s 1), \quad s \in {{\mathbb {R}}}, \, B_2 \in {\mathcal {B}}({\mathcal {F}}_2). \end{aligned}$$

It is linear, normal, continuous in norm (recall that $E_{f,1}$ is a one-dimensional projection), and it maps compact operators on ${\mathcal {F}}_2$ to compact operators. It therefore follows from Lemma A.6 that

$$\begin{aligned} t \mapsto \int _0^t \! ds \, \lambda (s)(V_{f,\chi }(s)_0) = \! \int _0^t \! ds \, \big (V_{f,\chi } \, (E_{f,1} \otimes _s 1)\big )(s)_0 = \! \int _0^t \! ds \, \big (V \, (E_{f,1} \otimes _s 1)\big )(s)_0 \end{aligned}$$

is norm continuous and has values in the compact operators on ${\mathcal {F}}_2$ for pair potentials with compact support. The last integral in the preceding equality is norm continuous on ${\mathcal {F}}_2$ with regard to $V \in C_0({{\mathbb {R}}}^s)$, equipped with the supremum topology. Since the algebra of compact operators is norm-closed, the preceding result for pair potentials with compact support therefore extends to all potentials in $C_0({{\mathbb {R}}}^s)$.

(ii) By the very definition of the spaces ${\mathcal {K}}_n$, any compact operator C on ${\mathcal {F}}_2$ gives rise to elements $C \otimes _s \underbrace{1 \otimes \cdots \otimes 1}_{n-2} \in {\mathcal {K}}_n$, $n \in {{\mathbb {N}}}$. Moreover, for the non-interacting dynamics one has

$$\begin{aligned} \big ( C \otimes _s \underbrace{1 \otimes \cdots \otimes 1}_{n-2} \big ) (s)_0 = C(s)_0 \otimes _s \underbrace{1 \otimes \cdots \otimes 1}_{n-2}. \end{aligned}$$

So the second statement follows from the preceding one. As has been mentioned, analogous arguments apply to the localized pair potential ${\check{V}}_{f,2}$, completing the proof. $\quad \square$

Next, we show that the assumptions in Lemma A.7 imply that the statements (i) and (ii) still hold if one replaces the non-interacting time evolution in the respective integrals by the interacting one. This fact will enable us to show that the functions $t \mapsto \int _0^t \! ds \, C_{f,n}(s)$ in the Dyson expansion (A.4) are norm continuous and have values in ${{\mathfrak {K}}}_n$, $n \in {{\mathbb {N}}}$. We recall the short hand notation $B_n(s) \doteq \text {Ad} \, e^{isH_{n}} (B_n)$ for the adjoint action of the interacting dynamics and $B_n(s)_0 \doteq \text {Ad} \, e^{isH_{0,n}} (B_n)$ in case of no interaction, $n \in {{\mathbb {N}}}_0$.

Lemma A.9

Let $n \in {{\mathbb {N}}}_0$ and let $B_n \in {\mathcal {B}}({\mathcal {F}}_n)$ be an operator such that the function $t \mapsto \int _0^t \! ds \, B_n(s)_0$, defined in terms of the non-interacting time evolution, has values in ${{\mathfrak {K}}}_n$. Then the function $t \mapsto \int _0^t \! ds \, B_n(s)$, involving the interacting time evolution, has values in ${{\mathfrak {K}}}_n$ and is norm continuous, $t \in {{\mathbb {R}}}$.

Proof

Let $\varLambda _n(s)^{-1} \doteq e^{isH_{0,n}} e^{-isH_n}$ and put $\lambda _n(s)^{-1} \doteq \text {Ad} \, \varLambda _n(s)^{-1}$, $s \in {{\mathbb {R}}}$. Given $B_n \in {\mathcal {B}}({\mathcal {F}}_n)$, one obtains by the familiar Dyson expansion the equality

$$\begin{aligned}&\lambda _n(s)^{-1}(B_n) = B_n \\&\quad + \! \sum _{k = 1}^\infty (-i)^k \! \! \int _0^s \! \! du_k \! \! \int _0^{u_k} \! \! \! du_{k-1} \dots \! \! \int _0^{u_2} \! \! \! \! \! du_1 \, [V_{\! n}(u_k)_0, [V_{\!n}(u_{k-1})_0, [ \cdots [V_{\!n}(u_1)_0, B_n]]] \cdots ] , \end{aligned}$$

where $V_{\!n}$ is the restriction of the interaction potential to ${\mathcal {F}}_n$. Since the underlying pair potential is bounded, this series converges absolutely in the norm topology. Moreover, for $s_2 \ge s_1$ one has

$$\begin{aligned} \Vert (\lambda _n(s_2)^{-1} - \lambda _n(s_1)^{-1})(B_n) \Vert \le \Vert B_n \Vert \, \sum _{k=1}^\infty 2^k/k! \ \Vert V_{\!n} \Vert ^k \, \int _{s_1}^{s_2} \! du \, |u|^{k-1}, \end{aligned}$$

proving that the function $s \mapsto \lambda _n(s)^{-1}$ of linear maps on ${\mathcal {B}}({\mathcal {H}}_n)$ is norm continuous. It is also apparent from the dominated convergence theorem that these maps are normal for fixed s. Finally, it was shown in Proposition 4.4 and the appendix of [2] that the interacting and non-interacting time evolutions map the algebra ${{\mathfrak {K}}}_n$ onto itself, hence this is also true for $\lambda _n(s)^{-1}$, $s \in {{\mathbb {R}}}$.

Thus the maps $\lambda _n(s)^{-1}$ are automorphisms, both, of ${\mathcal {B}}({\mathcal {F}}_n)$ and of ${{\mathfrak {K}}}_n$. So all of the preceding statements hold also for their inverse $\lambda _n(s)$, given by the adjoint action of $\varLambda _n(s) = e^{isH_n} e^{-isH_{0, n}}$, $s \in {{\mathbb {R}}}$. Hence the maps $s \mapsto \lambda _n(s)$ comply with all conditions given in Lemma A.6. Moreover, the function $s \mapsto B_n(s)_0$ is s.o. continuous and, by assumption, $\int _0^t \! ds \, B_n(s)_0 \in {{\mathfrak {K}}}_n$, $t \in {{\mathbb {R}}}$. It therefore follows from Lemma A.6 that

$$\begin{aligned} t \mapsto \int _0^t \! ds \, B_n(s) = \int _0^t \! ds \, \lambda _n(s)(B_n(s)_0) \end{aligned}$$

is norm continuous and has values in ${{\mathfrak {K}}}_n$, completing the proof. $\quad \square$

With the help of the preceding three lemmas we can establish now the main result of this subsection.

Proposition A.10

Let $n \in {{\mathbb {N}}}_0$. The function $t \mapsto \varGamma _{f,n}(t)$, given in Eq. (A.4), is norm-continuous and has values in ${{\mathfrak {K}}}_n$, $t \in {{\mathbb {R}}}$.

Proof

We begin by studying the properties of the function $t \mapsto \int _0^t \! ds \, C_{f,n}(s)$, which appears in lowest non-trivial order of the series expansion (A.4). According to Proposition A.5 one has $C_{f,n} = A_{f,n} + B_{f,n}$, where $A_{f,n} \in {{\mathfrak {K}}}_n$ and $B_{f,n}$ is bounded.

It was shown in [2, Prop. 4.4] and the appendix of that reference that the function $s \mapsto A_{f,n}(s)$, involving the interacting dynamics, is norm continuous, $s \in {{\mathbb {R}}}$. Its integrals $t \mapsto \int _0^t \! ds \, A_{f,n}(s)$ are therefore defined in the norm topology, whence have values in ${{\mathfrak {K}}}_n$, and depend norm continuously on $t \in {{\mathbb {R}}}$.

Turning to the operators $B_{f,n}$, we recall their form $B_{f} = {\hat{B}}_{f} + \beta _{f} \, \circ \, \sigma _{f}^{-1}({\check{B}}_{f})$, established in Proposition A.5. Plugging into this equation the operators ${\check{B}}_f, {\hat{B}}_f$, given in Lemmas A.2 and A.3, as well as the maps $\sigma _f$, $\beta _f$, defined in Eqs. (A.2) and (A.3), respectively, we obtain for $B_{f,n} = B_f \upharpoonright {\mathcal {F}}_n$, $n \in {{\mathbb {N}}}$,

$$\begin{aligned} B_{f,n} = {\hat{V}}_{f,n} N_{f,n}^{-1} E_{f,n} + \beta _{f,n} \big ((1 + N_{f,n-1})^{1/2} {\check{V}}_{f,n-1} (1 + N_{f,n-1})^{-1/2} - {\check{V}}_{f,n-1} \big ). \end{aligned}$$

It was shown in Lemma A.8 that the integrals of the localized potentials with regard to the non-interacting dynamics, $\int _0^t \! ds \, {\hat{V}}_{f,n}(s)_0$, $\int _0^t \! ds \, {\check{V}}_{f,n}(s)_0$, are elements of ${{\mathfrak {K}}}_n$, $t \in {{\mathbb {R}}}$. Since the operators $N_{f,n}^{-1} E_{f,n}$ and $(1 + N_{f,n})^{\pm 1/2}$ are elements of ${{\mathfrak {K}}}_n$, Lemma A.7 implies that this is also true for the integral $\int _0^t \! ds \, B_{f,n}(s)_0$. It then follows from Lemma A.9 that the integral with regard to the interacting dynamics, $\int _0^t \! ds \, B_{f,n}(s)$ has values in ${{\mathfrak {K}}}_n$. Combining the preceding results, we see that the function $t \mapsto \int _0^t \! ds \, C_{f,n}(s)$ has values in ${{\mathfrak {K}}}_n$. Since $C_{f,n}$ is bounded, it is also clear that it is norm continuous, $t \in {{\mathbb {R}}}$.

The proof that also the contributions of higher order in the series expansion (A.4) are contained in ${{\mathfrak {K}}}_n$ is accomplished by induction. Putting

$$\begin{aligned} t \mapsto D_{n,k}(t) \doteq \int _0^t \! \! ds_k \! \! \int _0^{s_k} \! \! \! ds_{k-1} \! \dots \! \! \int _0^{s_2} \! \! ds_1 \, C_{f,n}(s_k) \cdots C_{f,n}(s_1), \ \ k \in {{\mathbb {N}}}, \end{aligned}$$

(A.5)

we will show that these functions have values in ${\mathcal {K}}_n$ and are norm continuous, $t \in {{\mathbb {R}}}$. For $k=1$ this was shown in the preceding step.

For the induction step from k to $k+1$, we make use of the fact that relation (A.5) implies that $D_{n,k+1}(t) = \int _0^t \! ds \, C_{f,n}(s) \, D_{n,k}(s)$. According to the induction hypothesis, the function $s \mapsto D_{n,k}(s)$ is norm continuous and has values in ${{\mathfrak {K}}}_n$. Thus the function $s \mapsto \lambda _{k,n}(s)$ of normal linear maps on ${\mathcal {B}}({\mathcal {F}}_n)$ given by $\lambda _{k,n}(s)(B_n) \doteq B_n D_{n,k}(s)$, $B_n \in {\mathcal {B}}({\mathcal {F}}_n)$, is norm continuous and maps ${{\mathfrak {K}}}_n$ into itself. The function $s \mapsto C_{f,n}(s)$ is s.o. continuous and $\int _0^t \! ds \, C_{f,n}(s) \in {{\mathfrak {K}}}_n$, $t \in {{\mathbb {R}}}$, as was shown in the initial step. Hence, according to Lemma A.6, the function

$$\begin{aligned}t \mapsto D_{n,k+1}(t) = \int _0^t \! ds \, C_{f,n}(s) \, D_{n,k}(s) = \int _0^t \! ds \, \lambda _{k,n}(s)\big (C_{f,n}(s)\big )\end{aligned}$$

has the desired properties, completing the induction.

So each term in the Dyson expansion A.4 is an element of ${{\mathfrak {K}}}_n$. Moreover, this series converges absolutely in the norm topology, which implies $\varGamma _{f,n}(t) \in {{\mathfrak {K}}}_n$. Since the operators $C_{f,n}$ are bounded, it is also clear that the function $t \mapsto \varGamma _{f,n}(t)$ is norm continuous, $t \in {{\mathbb {R}}}$, cf. the argument in Lemma A.9. This completes the proof of the proposition. $\quad \square$

1.4 Verification of the coherence condition

Having seen that the operators $\varGamma _{f,n}(t)$, defined in Eq. (A.4), are elements of ${\mathcal {K}}_n$, $t \in {{\mathbb {R}}}$, we will show next that these operators form coherent sequences, $n \in {{\mathbb {N}}}_0$. At this point the inverse maps $\kappa _n: {\mathcal {K}}_n \rightarrow {\mathcal {K}}_{n-1}$, defined in Eq. (3.2), enter. We recall that these maps are homomorphisms, mapping ${\mathcal {K}}_n$ onto ${\mathcal {K}}_{n-1}$, and that a sequence of operators ${\varvec{K}}= \{ K_n \in {{\mathfrak {K}}}_n \}_{n \in {{\mathbb {N}}}_0}$ is said to be coherent if $\kappa _n(K_n) = K_{n-1}$, $n \in {{\mathbb {N}}}_0$.

In order to establish the desired result, we make use again of the Eq. (A.5), relating subsequent terms in the Dyson expansion (A.4). The essential step in our argument consists of proving the equality

$$\begin{aligned} \kappa _n \Big ( \int _0^t \! ds \, C_{f,n}(s) D_n(s) \Big ) = \int _0^t \! ds \, C_{f,n-1}(s) \, \kappa _n\big (D_n(s)\big ) \end{aligned}$$

for any norm continuous function $s \mapsto D_n(s)$ with values in ${{\mathfrak {K}}}_n$, $n \in {{\mathbb {N}}}_0$. Since the values of the functions $s \mapsto C_{f,n}(s)$ are not contained in ${{\mathfrak {K}}}_n$, this requires some further arguments. We begin with a statement, involving the non-interacting time evolution.

Lemma A.11

Let $m = 1$ or 2, let $O_m$ be a bounded m-particle operator on ${\mathcal {F}}_m$ such that $\ \int _0^t \! ds \, O_m(s)_0 \,$ is a compact operator, $t \in {{\mathbb {R}}}$, and let O be the second quantization of $O_m$. Putting $O_n \doteq O \upharpoonright {\mathcal {F}}_n$, one has $\, \int _0^t \! ds \, O_n(s)_0 \in {{\mathfrak {K}}}_n \,$, $t \in {{\mathbb {R}}}$, and

$$\begin{aligned} \kappa _n \Big ( \int _0^t \! ds \, O_n(s)_0 \Big ) = \int _0^t \! ds \, O_{n-1}(s)_0, \quad n \in {{\mathbb {N}}}_0. \end{aligned}$$

Proof

The second quantizations of one- and two-particle operators and their restrictions to ${\mathcal {F}}_n$ were explained in subsection A.1. Since the non-interacting time evolution does not mix tensor factors, one has

Applying $\kappa _n$, cf. relation (3.3), one obtains

completing the proof. $\quad \square$

In the next step we extend this result to operators $O_n$, which are sandwiched between elements of ${{\mathfrak {K}}}_n$ and are acted upon by the maps $\beta _{g,n}$, defined in Eq. (A.3).

Lemma A.12

Let $n \in {{\mathbb {N}}}_0$ and let $O_n \doteq O \upharpoonright {\mathcal {F}}_n$, $n \in {{\mathbb {N}}}_0$, be the restriction of a second quantized one- or two-particle operator with properties given in the preceding lemma. Then

(i)  for $K_n', K_n'' \in {{\mathfrak {K}}}_n$ one has

$$\begin{aligned} \kappa _n\Big ( \int _0^t \! ds \, \big (K_n' O_n K_n''\big )(s)_0 \Big ) = \int _0^t \! ds \, \big ( \kappa _n(K_n') \, O_{n-1} \, \kappa _n(K_n'') \big )(s)_0, \end{aligned}$$

(ii)  for $K_{n-1}', K_{n-1}'' \in {{\mathfrak {K}}}_n$ and any normalized $g \in L^2({{\mathbb {R}}}^s)$ one has

$$\begin{aligned} \kappa _n \Big (&\int _0^t \! ds \, \beta _{g,n}(K_{n-1}' O_{n-1} K_{n-1}''\big )(s)_0 \Big ) \\&= \int _0^t \! ds \, \beta _{g,n-1}\big ( \kappa _{n-1}(K_{n-1}') \, O_{n-2} \, \kappa _n(K_{n-1}'') \big )(s)_0, \quad t \in {{\mathbb {R}}}. \end{aligned}$$

The integrals in (i) and (ii) are elements of ${{\mathfrak {K}}}_n$, respectively ${{\mathfrak {K}}}_{n-1}$, cf. Lemmas (A.11) and (A.7).

Proof

(i) Let $s \mapsto \lambda _n(s)$, $s \in {{\mathbb {R}}}$, be the function, having values in normal linear maps on ${\mathcal {B}}({\mathcal {F}}_n)$, which is given by

$$\begin{aligned} \lambda _n(s)(B_n) \doteq (K_n')(s)_0 \, B_n \, (K_n'')(s)_0, \quad B_n \in {\mathcal {B}}({\mathcal {F}}_n). \end{aligned}$$

In view of the norm continuous action of the time translations on ${{\mathfrak {K}}}_n$, this function is norm continuous and maps ${{\mathfrak {K}}}_n$ into itself. Next, the function $s \mapsto O_n(s)_0$ is s.o. continuous and $\int _0^t \! ds \, O_n(s)_0 \in {{\mathfrak {K}}}_n$ according to the preceding lemma, $t \in {{\mathbb {R}}}$. Thus, by Lemma (A.6), the function $t \mapsto \int _0^t \! ds \, \big ( K_n' \, O_n \, K_n'' \big )(s)_0$ has values in ${{\mathfrak {K}}}_n$. Moreover, it can be approximated in the limit of large $m \in {{\mathbb {N}}}$ by finite sums of the form

$$\begin{aligned}&\sum _{l = 1}^m \lambda (lt/m)\Big ( \int _{(l-1)t/m}^{lt/m} \! ds \, O_n(s)_0 \Big ) \\&= \sum _{l = 1}^m K_n'(lt/m)_0 \Big ( \int _{(l-1)t/m}^{lt/m} \! ds \, O_n(s)_0 \Big ) K_n''(lt/m)_0. \end{aligned}$$

We apply to this equality the homomorphism $\kappa _n$. Taking into account that the non-interacting dynamics does not mix tensor factors of the operators $K_n \in {{\mathfrak {K}}}_n$, we have $\kappa _n \big (K_n(s)_0 \big ) = \big (\kappa _n (K_n)\big )(s)_0$, $s \in {{\mathbb {R}}}$. So by Lemma A.11, we obtain

$$\begin{aligned} \kappa _n&\Big ( \sum _{l = 1}^m \lambda (lt/m)\Big (\int _{(l-1)t/m}^{lt/m} \! \! ds \, O_n(s)_0 \Big ) \Big ) \\&= \sum _{l = 1}^m \kappa _n(K_n')(lt/m)_0 \, \Big ( \int _{(l-1)t/m}^{lt/m} \! \! ds \, O_{n-1}(s)_0 \Big ) \, \kappa _n(K_n'')(lt/m)_0. \end{aligned}$$

Since $\kappa _n$ is norm continuous, we can proceed in the latter equality to the limit $m \rightarrow \infty$, giving the first statement of the lemma.

(ii) For the proof of the second statement, we make use of the fact that for any $K_{n-1} \in {{\mathfrak {K}}}_{n-1}$ there exists some operator $A \in \overline{{{\varvec{\mathfrak {A}}}}}$ such that $A \upharpoonright {\mathcal {F}}_{n-1} = K_{n-1}$, cf. [2, Lem. 3.3]. Conversely, given an observable A and any $l \in {{\mathbb {N}}}_0$, there exists some operator $K_l \in {{\mathfrak {K}}}_l$ such that $K_l = A \upharpoonright {\mathcal {F}}_l$, cf. [2, Lem. 3.2]. Moreover, the operators $K_l$ satisfy the coherence condition $\kappa _l(K_l) = K_{l-1}$, cf. [2, Lem. 3.4]. Since $\beta _{g}(A) \in \overline{{{\varvec{\mathfrak {A}}}}}$, the latter fact implies $L_l \doteq \beta _{g,l}(K_{l-1}) = \beta _{g}(A) \upharpoonright {\mathcal {F}}_l \in {{\mathfrak {K}}}_l$. Thus

$$\begin{aligned} \kappa _l\big (\beta _{g,l}(K_{l-1})\big ) = \kappa _l\big (L_l) = L_{l-1} = \beta _{g,l-1}(K_{l-2}\big ) = \beta _{g,l-1}\big (\kappa _l(K_{l-1})\big ), \end{aligned}$$

leading to the intertwining relations $\kappa _l \, \circ \, \beta _{g,l} = \beta _{g,l-1} \, \circ \, \kappa _l$, $l \in {{\mathbb {N}}}_0$.

Bearing in mind that we are dealing with the non-interacting time evolution, we can proceed now as in the proof of Lemma A.7, giving

$$\begin{aligned} \big ( \beta _{g,n}(K'_{n-1} \, O_{n-1} \, K''_{n-1}) \big )(s)_0 = \beta _{g(s),n} \big ((K'_{n-1} \, O_{n-1} \, K''_{n-1})(s)_0 \big ) \, , \end{aligned}$$

where $g(s) = e^{i s {\varvec{P}}^{\, 2}_{\kappa }} \, g$, $\, s \in {{\mathbb {R}}}$. The function $s \mapsto \beta _{g(s),n}$ of normal linear maps on ${\mathcal {B}}({\mathcal {F}}_{n-1})$ is norm continuous and maps ${{\mathfrak {K}}}_{n-1}$ into ${{\mathfrak {K}}}_n$, cf. Lemma A.4. Moreover, $\int _0^t \! ds \, \big ( K_{n-1}' O_{n-1} K_{n-1}'' \big )(s)_0 \in {{\mathfrak {K}}}_{n-1}$ according to the preceding step. Hence, by Lemma A.6, the integrals $\int _0^t \! ds \, \big (\beta _{g,n} (K_{n-1}' O_{n-1} K_{n-1}'') \big )(s)_0$ are elements of ${{\mathfrak {K}}}_n$, $t \in {{\mathbb {R}}}$. They can be approximated in norm in the limit of large $m \in {{\mathbb {N}}}$ by the sums

$$\begin{aligned} \sum _{l = 1}^m \beta _{g \, (lt/m), \, n} \Big ( \int _{(l-1)t/m}^{lt/m} \! ds \, (K'_{n-1} \, O_{n-1} \, K''_{n-1})(s)_0 \Big ). \end{aligned}$$

Applying to this relation the homomorphism $\kappa _n$, the initial remarks imply

$$\begin{aligned}&\kappa _n \Big (\sum _{l = 1}^m \beta _{g \, (lt/m), \, n} \Big ( \int _{(l-1)t/m}^{lt/m} \! ds \, (K'_{n-1} \, O_{n-1} \, K''_{n-1})(s)_0 \Big ) \Big ) \\&\quad = \sum _{l = 1}^m \, \beta _{g \, (lt/m), \, n-1} \Big ( \int _{(l-1)t/m}^{lt/m} \! ds \, (\kappa _n(K'_{n-1}) \, O_{n-2} \, \kappa _n(K''_{n-1}) \Big )(s)_0. \end{aligned}$$

Proceeding to the limit of large m, this establishes the second part of the lemma. $\square$

The statements of the preceding two lemmas remain true if one replaces the non-interacting dynamics by the interacting one. The proof of this assertion is accomplished by the following result.

Lemma A.13

Let $B_n \in {\mathcal {B}}({\mathcal {F}}_n)$, $B_{n-1} \in {\mathcal {B}}({\mathcal {F}}_{n-1})$, such that $\int _0^t \! ds \, B_n(s)_0 \in {{\mathfrak {K}}}_n$ and $\kappa _n \big (\int _0^t \! ds \, B_n(s)_0 \big ) = \int _0^t \! ds \, B_{n-1}(s)_0$, $n \in {{\mathbb {N}}}_0$. Then one has for the interacting dynamics $\int _0^t \! ds \, B_n(s) \in {{\mathfrak {K}}}_n$ and $\kappa _n \big (\int _0^t \! ds \, B_n(s) \big ) = \int _0^t \! ds \, B_{n-1}(s)$, $t \in {{\mathbb {R}}}$.

Proof

As was shown in Lemma A.9, the condition $\int _0^t \! ds \, B_n(s)_0 \in {{\mathfrak {K}}}_n$ implies that

$$\begin{aligned} \int _0^t \! ds \, B_n(s) = \int _0^t \! ds \, \lambda _n(s) \big ( B_n(s)_0 \big ) \in {{\mathfrak {K}}}_n, \quad t \in {{\mathbb {R}}}, \end{aligned}$$

where $\lambda _n(s) = \text {Ad} \, e^{isH_n} e^{-isH_{0,n}} = \alpha _n(s) \, \circ \, \alpha _{0,n}(-s)$, $s \in {{\mathbb {R}}}$. In view of the properties of the function $s \mapsto \lambda _n(s)$, established in the proof of Lemma A.9, and the anticipated properties of $s \mapsto B_n(s)_0$, we can apply again Lemma A.6. It implies that the integral on the right hand side of the above equality can be approximated in the limit of large $m \in {{\mathbb {N}}}$ in norm by the sums

$$\begin{aligned} \sum _{l = 1}^m \lambda _n(lt/m)\Big (\! \! \int _{(l-1)t/m}^{lt/m} \! ds \, B_n(s)_0 \Big ). \end{aligned}$$

We apply to these sums the homomorphism $\kappa _n$, making use of Lemma 4.5 and the appendix in [2] according to which

$$\begin{aligned} \kappa _n \, \circ \, \lambda _n(s) = \kappa _n \, \circ \, \alpha _n(s) \, \circ \, \alpha _{0,n}(-s) = \alpha _{n-1}(s) \, \circ \, \alpha _{0,n-1}(-s) \, \circ \, \kappa _n = \lambda _{n-1}(s) \, \circ \, \kappa _n. \end{aligned}$$

Thus we obtain

$$\begin{aligned} \kappa _n&\Big (\! \sum _{l = 1}^m \lambda _n(lt/m)\Big (\! \int _{(l-1)t/m}^{lt/m} \! ds \, B_n(s)_0 \Big ) \Big ) \\&= \sum _{l= 1}^m \lambda _{n-1}(lt/m) \, \circ \, \kappa _n \Big (\! \int _{(l-1)t/m}^{lt/m} \! ds \, B_n(s)_0 \Big ) \\&= \sum _{l = 1}^m \lambda _{n-1}(lt/m) \Big (\! \int _{(l-1)t/m}^{lt/m} \! ds \, B_{n-1}(s)_0 \Big ). \end{aligned}$$

Bearing in mind that $\kappa _n$ is norm continuous, the statement then follows in the limit of large m by the norm convergence of the sums. $\quad \square$

With the help of the preceding three lemmas we can determine now the action of $\kappa _n$ on integrals involving the function $s \mapsto C_{f,n}(s)$ in the Dyson expansion (A.4), $n \in {{\mathbb {N}}}$. Recall that $C_{f,n} = A_{f,n} + B_{f,n}$, cf. Proposition A.5, where

$$\begin{aligned} A_{f,n} = {\hat{O}}_{f,n} \, N_{f,n}^{-1} E_{f,n} + \beta _{f,n}\big ({\check{O}}_{f,n-1} - \sigma _{f,n-1}({\check{O}}_{f,n-1})\big ). \end{aligned}$$

(A.6)

Here ${\hat{O}}_{f,n}, {\check{O}}_{f,n-1}$ are the restrictions to ${\mathcal {F}}_n$, respectively ${\mathcal {F}}_{n-1}$, of sums of second quantizations of compact one- and two-particle operators. Furthermore,

$$\begin{aligned} B_{f,n} = {\hat{V}}_{f,n} \, N_{f,n}^{-1} E_{f,n} + \beta _{f,n}\big ({\check{V}}_{f,n-1} - \sigma _{f,n-1}({\check{V}}_{f,n-1})\big ), \end{aligned}$$

(A.7)

where ${\hat{V}}_{f,n}, {\check{V}}_{f,n-1}$ are the restrictions to ${\mathcal {F}}_n$, respectively ${\mathcal {F}}_{n-1}$, of the second quantizations of the localized pair potential V, cf. Lemmas A.2 and A.3. We then have the following result involving the interacting dynamics.

Lemma A.14

Let $n \in {{\mathbb {N}}}$, let $C_{f,n} = A_{f,n} + B_{f,n}$ be the operator given above, and let $s \mapsto D_n(s)$ be a norm continuous function with values in ${{\mathfrak {K}}}_n$. Then

$$\begin{aligned} \kappa _n \Big (\int _0^t \! ds \, C_{f,n}(s) \, D_n(s) \Big ) = \int _0^t \! ds \, C_{f,n-1}(s) \, \kappa _n\big (D_n(s)\big ), \quad t \in {{\mathbb {R}}}, \end{aligned}$$

where the integrals have values in ${{\mathfrak {K}}}_n$, respectively ${{\mathfrak {K}}}_{n-1}$.

Proof

For the proof that the integrals in this lemma have values in ${{\mathfrak {K}}}_n$, respectively ${{\mathfrak {K}}}_{n-1}$, we make use of Lemma A.6: the function $s \mapsto \lambda _n(s)$ of normal linear maps on ${\mathcal {B}}({\mathcal {F}}_n)$, given by $\lambda _n(s)(B_n) \doteq B_n \, D_n(s)$, $B_n \in {\mathcal {B}}({\mathcal {F}}_n)$, is continuous in norm and maps ${{\mathfrak {K}}}_n$ into itself; and, as was shown in the proof of Proposition A.10, $\int _0^t \! ds \, C_{f,n}(s) \in {{\mathfrak {K}}}_n$. Thus the first integral in the statement is an element of ${{\mathfrak {K}}}_n$. The same argument applies to the second integral since $s \mapsto \kappa _n\big (D_n(s)\big ) \in {{\mathfrak {K}}}_{n-1}$ is norm continuous and $\int _0^t \! ds \, C_{f,n-1}(s) \in {{\mathfrak {K}}}_{n-1}$.

In order to determine the action of $\kappa _n$, we first restrict attention to the constant function $s \mapsto D_n(s) \doteq 1 \upharpoonright {\mathcal {F}}_n$, i.e. the integral $\int _0^t \! ds \, C_{f,n}(s)$. In the contributions (A.6) and (A.7) to this integral, there appear the operators $N_{f,n}^{-1} E_{f,n}$ and $(\mathbf 1 _{n-1} + N_{f,n-1})^{\pm 1/2}$. Putting $l=n, n-1$, these are bounded functions $b(N_{f,l}) \in {{\mathfrak {K}}}_l$ of the operators $N_{f,l} \in {{\mathfrak {K}}}_l$, which in turn are restrictions to ${\mathcal {F}}_l$ of the second quantization $N_f$ of the one-dimensional projection $E_{f,1}$ on ${\mathcal {F}}_1$. But the maps $\kappa _l$ are homomorphisms, so $\kappa _l\big (b(N_{f,l})\big ) = b(N_{f,l-1})$. Furthermore, as was shown in Lemma A.11, one has $\kappa _n \, \circ \, \beta _{f,n} = \beta _{f,n-1} \, \circ \, \kappa _n$. Finally, the operators ${\hat{O}}_{f,n}, {\check{O}}_{f,n-1}$ and ${\hat{V}}_{f,n}, {\check{V}}_{f,n-1}$ in relations (A.6) and (A.7) are of the type of operators O considered in Lemma A.11, cf. also Lemma A.8. Thus, Lemmas A.12 and A.13 apply to the function $s \mapsto C_{f,n}(s) = A_{f,n}(s) + B_{f,n}(s)$. Whence, making also use of the preceding relations, we arrive at

$$\begin{aligned} \kappa _n \Big (\int _0^t \! ds \, C_{f,n}(s)\Big ) = \int _0^t \! ds \, C_{f,n-1}(s), \quad t \in {{\mathbb {R}}}. \end{aligned}$$

Let us turn now to the case of arbitrary norm continuous functions $s \mapsto D_n(s)$ with values in ${{\mathfrak {K}}}_n$. Adopting the notation in the beginning of this proof, we have

$$\begin{aligned} \int _0^t \! ds \, C_{f,n}(s) \, D_n(s) = \int _0^t \! ds \, \lambda _n(s)\big (C_{f,n}(s)\big ). \end{aligned}$$

According to Lemma A.6, the latter integral can be approximated in norm in the limit of large $m \in {{\mathbb {N}}}$ by

$$\begin{aligned} \sum _{l = 1}^m \lambda _n(lt/m)\Big (\int _{(l-1)t/m}^{lt/m} \! ds \, C_n(s) \Big ) = \sum _{l = 1}^m \Big ( \int _{(l-1)t/m}^{lt/m} \! ds \, C_n(s) \Big ) D_n(lt/m). \end{aligned}$$

Applying to the expression on the right hand side of this equality the homomorphism $\kappa _n$, we obtain

$$\begin{aligned} \sum _{l = 1}^m \Big ( \int _{(l-1)t/m}^{lt/m} \! ds \, C_{n-1}(s) \Big ) \, \kappa _n\big (D_n(lt/m)\big ), \end{aligned}$$

where we made use of the result obtained in the preceding step. Since the function $s \mapsto \kappa _n\big (D_n(s)\big ) \in {{\mathfrak {K}}}_{n-1}$ is norm continuous and $\int _0^t \! ds \, C_{n-1}(s) \in {{\mathfrak {K}}}_{n-1}$, we can proceed in the latter sum again to the limit of large m. By Lemma A.6, we thereby arrive at the integral on the right hand side of the equality in the statement of the lemma, completing its proof. $\quad \square$

With the help of the preceding lemma we can establish now the coherence condition for the operators $\varGamma _{f,n}(t)$, which, according to Proposition A.10, are elements of ${{\mathfrak {K}}}_n$, $n \in {{\mathbb {N}}}$.

Proposition A.15

Let $n \in {{\mathbb {N}}}$ and let $\varGamma _{f,n}(t) \in {{\mathfrak {K}}}_n$ be the operators, given in Eq. (A.4). Then $\kappa _n\big (\varGamma _{f,n}(t)\big ) = \varGamma _{f,n-1}(t)$.

Proof

We make use again of the Dyson expansion (A.4) and show that the multiple integrals $D_{n,k}(s)$, $k \in {{\mathbb {N}}}$, involving the operator $C_{f,n}$, cf. Eq. (A.5), are mapped by $\kappa _n$ into corresponding integrals, where $C_{f,n}$ is replaced by $C_{f,n-1}$ and $D_{n,k}(s)$ by $D_{n-1,k}(s)$, $s \in {{\mathbb {R}}}$. The statement then follows from the norm convergence of the series. For its proof we make use of the inductive argument given in the proof of Proposition A.10. We have shown in the preceding lemma that

$$\begin{aligned} \kappa _n\big (D_{n,1}(t)\big ) = \kappa _n \Big ( \int _0^t \! ds \, C_n(s) \Big ) = \int _0^t \! ds \, C_{n-1}(s) = D_{n-1,1}(t), \quad n \in {{\mathbb {N}}}. \end{aligned}$$

Assuming that the analogous relation holds for the k-fold integrals of $C_n$, we represent the $(k+1)$-fold integral in the form $t \mapsto D_{n,k+1}(t) = \int _0^t \! ds \, C_n(s) \, D_{n,k}(s)$, where $s \mapsto D_{n,k}(s) \in {{\mathfrak {K}}}_n$ is norm continuous. Thus it follows from the preceding lemma that

$$\begin{aligned} \kappa _{n}\big (D_{n,k+1}(t)\big )&= \kappa _{n} \Big ( \int _0^t \! ds \, C_n(s) \, D_{n,k}(s) \Big ) \\&= \int _0^t \! ds \, C_{n-1}(s) \, \kappa _n\big (D_{n,k}(s)\big ) = \int _0^t \! ds \, C_{n-1}(s) \, D_{n-1,k}(s), \end{aligned}$$

where in the last equality we made use of the induction hypothesis. This establishes the coherence condition. $\quad \square$

Let us summarize the results of this appendix. In order to prove Theorem 5.3, we have analyzed the properties of the operators (intertwiners between morphisms) $\varGamma _f(t) E_f$, $t \in {{\mathbb {R}}}$, which were defined in Eq. (5.2). Since these operators commute with the particle number operator N, we could proceed to their restrictions $\varGamma _{f,n}(t) = \varGamma _f(t) E_f \upharpoonright {\mathcal {F}}_n$, $n \in {{\mathbb {N}}}_0$. We have shown in Proposition A.10 that $\varGamma _{f,n}(t) \in {{\mathfrak {K}}}_n$, and from Proposition A.15 we know that $\kappa _n\big (\varGamma _{f,n}(t)\big ) = \varGamma _{f,n-1}(t) \in {{\mathfrak {K}}}_{n-1}$. Since $\varGamma _f(t) E_f$ is a bounded operator on ${\mathcal {F}}$, this implies $\varGamma _f(t) E_f \in \overline{{{\varvec{\mathfrak {A}}}}}$. Hence

$$\begin{aligned} \text {Ad} \, e^{itH} (W_f) = \varGamma _f(t) \, W_f \in \overline{{{\varvec{\mathfrak {F}}}}}, \quad t \in {{\mathbb {R}}}, \end{aligned}$$

cf. also Lemma 4.1. Since the observable algebra $\overline{{{\varvec{\mathfrak {A}}}}}$ is stable under the time translations [2, Thm. 4.6] and the field algebra $\overline{{{\varvec{\mathfrak {F}}}}}$ is generated by $\overline{{{\varvec{\mathfrak {A}}}}}$ and the tensors $W_f$, $W_f^*$, this proves that $\text {Ad} \, e^{itH} (\overline{{{\varvec{\mathfrak {F}}}}}) = \overline{{{\varvec{\mathfrak {F}}}}}$, $t \in {{\mathbb {R}}}$.

It also follows from Proposition A.10 that the functions $t \mapsto \varGamma _{f,n}(t) \in {{\mathfrak {K}}}_n$ are norm continuous, $n \in {{\mathbb {N}}}_0$. Hence $t \mapsto \varGamma _f(t) E_f$ is lct-continuous. It implies that the time translated tensors $t \mapsto \text {Ad} \, e^{itH} (W_f) = \varGamma _f(t) \, W_f$ are lct-continuous, $t \in {{\mathbb {R}}}$. Since the time translations act lct-continuously on the observable algebra $\overline{{{\varvec{\mathfrak {A}}}}}$, cf. [2, Thm. 4.6], and the finite polynomials in the basic tensors, multiplied with observables, are norm dense in the field algebra $\overline{{{\varvec{\mathfrak {F}}}}}$, this establishes the lct-continuity of the time translations $t \mapsto \text {Ad} \, e^{itH}$ on $\overline{{{\varvec{\mathfrak {F}}}}}$.

For the proof of existence of an lct-dense sub-C*-algebra $\overline{{{\varvec{\mathfrak {F}}}}}_0 \subset \overline{{{\varvec{\mathfrak {F}}}}}$ on which the time translations act norm-continuously, we proceed as in Theorem 5.2. Let $F_m \in \overline{{{\varvec{\mathfrak {F}}}}}$ be any tensor, $m \in {{\mathbb {Z}}}$, let $t \mapsto k(t)$ be any continuous function on ${{\mathbb {R}}}$ with compact support, and consider the integral $F_m(k) \doteq \int \! dt \, k(t) \, \text {Ad} \, e^{itH}(F_m)$. The resulting function $t \mapsto \text {Ad} \, e^{itH} \big (F_m(k)\big )$ is, due to the regularization, norm-continuous on the full Fock space ${\mathcal {F}}$. It has values in $\overline{{{\varvec{\mathfrak {F}}}}}$ and the C*-algebra $\overline{{{\varvec{\mathfrak {F}}}}}_0$ generated by the operators $F_m(k)$ is lct-dense in $\overline{{{\varvec{\mathfrak {F}}}}}$. For the proof of the latter assertions, we make use of the fact that for any $l \in {{\mathbb {N}}}_0$ the restrictions of the gauge invariant operators, $m \le 0$,

$$\begin{aligned} G_l(t) \doteq W_f^m \, \text {Ad} \, e^{itH}(F_m) \upharpoonright {\mathcal {F}}_l = \text {Ad} \, e^{itH}\big ( \big (\text {Ad} \, e^{-itH}(W_f) \big )^m \, F_m \big ) \, \upharpoonright {\mathcal {F}}_l \in {{\mathfrak {K}}}_l \end{aligned}$$

are norm continuous with regard to $t \in {{\mathbb {R}}}$ and satisfy $\kappa _l\big (G_l(t)\big ) = G_{l-1}(t)$. So their integrals $G_l(k) = \int \! dt \, k(t) \, G_l(t)$ exist in the norm topology, hence are elements of ${{\mathfrak {K}}}_l$, and $\kappa _l\big (G_l(k)\big ) = G_{l-1}(k)$. The coherent sequence $\{ G_l(k) \}_{l \in {{\mathbb {N}}}_0}$ defines some bounded element $G(k) \in \overline{{{\varvec{\mathfrak {A}}}}}$ and $F_m(k) = W_f^{* \, m} \, G(k) \in \overline{{{\varvec{\mathfrak {F}}}}}$, as claimed. A similar argument applies to tensors with $m \ge 0$.

The assertion that the C*-algebra $\overline{{{\varvec{\mathfrak {F}}}}}_0$ is lct-dense in $\overline{{{\varvec{\mathfrak {F}}}}}$ follows from the fact that one can proceed in the integrals $G_l(k) = \int \! dt \, k(t) \, G_l(t)$ with k to the Dirac measure, whereby the sequence $G_l(k) \in {{\mathfrak {K}}}_l$ converges in norm to $G_l$, $l \in {{\mathbb {N}}}_0$. This establishes the lct-density of $\overline{{{\varvec{\mathfrak {F}}}}}_0$ in $\overline{{{\varvec{\mathfrak {F}}}}}$ and completes the proof of Theorem 5.3.