A note on the Fröhlich dynamics in the strong coupling limit

Abstract

We revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in Griesemer (Rev Math Phys 29(10):1750030, 2017). In the latter it was shown that the Fröhlich time evolution applied to the initial state $\varphi _0 \otimes \xi _\alpha$, where $\varphi _0$ is the electron ground state of the Pekar energy functional and $\xi _\alpha$ the associated coherent state of the phonons, can be approximated by a global phase for times small compared to $\alpha ^2$. In the present note we prove that a similar approximation holds for $t=O(\alpha ^2)$ if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to $\alpha ^{-2}$ and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order $\alpha ^2$, while the phonon fluctuations around the coherent state $\xi _\alpha$ can be described by a time-dependent Bogoliubov transformation.

1 Introduction and main result

1.1 The model

The Fröhlich polaron is a quantum model for a large polaron which describes an electron in an ionic lattice interacting with the excitations (phonons) of this lattice [1, 12]. Large refers to the assumption that the extension of the electron is much larger compared to the lattice spacing which can thus be approximated by a continuum. In this model, the energy and the dynamics of the electron and the phonons are described by the Fröhlich Hamiltonian

$$\begin{aligned} H^\mathrm{F}_{\mathrm{phys},\alpha } = p^2 \otimes 1 + 1 \otimes N + \sqrt{\alpha }\phi (G_x) \end{aligned}$$

(1)

that acts on the Hilbert space ${\mathscr {H}} = L^2({\mathbb {R}}^3,{\text {d}}x)\otimes {\mathcal {F}}$. Here ${\mathcal {F}} = \bigoplus _{n=0}^\infty L^2({\mathbb {R}}^3,{\text {d}}k)^{\otimes _\mathrm{sym}^n}$ is the bosonic Fock space, x and $p= -i\nabla _x$ denote the position and momentum operator of the electron, respectively, and N is the number operator on ${\mathcal {F}}$. The interaction between the electron and the phonons is described by $\phi (G_x) = a( G_x ) + a^*(G_x)$ with a(f) and $a^*(f)$ the usual annihilation and creation operators on ${\mathcal {F}}$ and $G_x$ the bounded multiplication operator defined for any $x\in {\mathbb {R}}^3$ by the function

$$\begin{aligned} G_x(k) = \frac{e^{ - ikx}}{ 2\pi \vert k \vert }. \end{aligned}$$

(2)

The creation and annihilation operators satisfy the canonical commutation relations

$$\begin{aligned}&[ a(f) , a^*(g) ] = \langle f,g \rangle _{L^2},\nonumber \\&\quad [ a(f) , a(g) ] = [ a^*(f) , a^*(g) ] = 0\quad \ \forall \ f,g\in L^2(\mathbb R^3,{\text {d}}k). \end{aligned}$$

(3)

Finally the number $\alpha >0$ is a dimensionless coupling parameter that models the strength of the interaction. The regime $\alpha \rightarrow \infty$ is called the strong coupling limit.

By a change of units which corresponds to rescaling all lengths by a factor $\alpha ^{-1}$, the Fröhlich Hamiltonian $H^\mathrm{F}_\mathrm{phys,\alpha }$ is unitarily equivalent to the operator $\alpha ^{2} H^\mathrm{F}_{\alpha }$ with¹

$$\begin{aligned} H^\mathrm{F}_{\alpha } = p^2 \otimes 1 + 1 \otimes \alpha ^{-2}N + \alpha ^{-1} \phi (G_x) . \end{aligned}$$

(4)

In the analysis of the strong coupling limit it is more convenient to work in strong coupling units, i.e., to use $H^\mathrm{F}_{\alpha }$ instead of the original Fröhlich Hamiltonian $H^\mathrm{F}_{\text {phys},\alpha }$ and then consider rescaled values of energy $E = \alpha ^{-2} E_\mathrm{phys}$ and time $t = \alpha ^2 t_\mathrm{phys}$. This explains why $t=O(\alpha ^2)$ is the time scale we are interested in for the dynamics generated by $H^\mathrm{F}_\alpha$.

In this work we study the large $\alpha$ limit of the time-evolved state $\Psi _\alpha (t) = e^{-iH^\mathrm{F}_\alpha t }\Psi _\alpha$ for a special initial state, namely the Pekar product state $\Psi _\alpha = \varphi _0 \otimes \xi _\alpha$ where $\varphi _0\in H^1(\mathbb R^3,{\text {d}}x)$ is the self-trapped electron ground state of the Pekar energy functional (to be defined below) and $\xi _\alpha = W(\alpha f_0 )^* \Omega _0$ is the corresponding coherent phonon state. That is to say, $\Omega _0 = (1,0,0,\ldots )$ is the normalized vacuum state in ${\mathcal {F}}$ and

$$\begin{aligned} W(\alpha f_0 ) = \exp \big ( a^*(\alpha f_0 ) - a( \alpha f_0 ) \big ) \end{aligned}$$

(5)

denotes the Weyl operator w.r.t. the function

$$\begin{aligned} \alpha f_0(k) = \alpha \big \langle \varphi _0, G_\cdot (k) \varphi _0 \big \rangle _{L^2} = \frac{\alpha }{2\pi \vert k \vert } \int _{{\mathbb {R}}^3} e^{-ikx} \vert \varphi _0(x)\vert ^2 \, {\text {d}}x . \end{aligned}$$

(6)

We recall that the Weyl operator is unitary and satisfies the shift relation

$$\begin{aligned} W(\alpha f_0)^* a(g) W(\alpha f_0) = a(g) + \alpha \big \langle g,f_0\big \rangle _{L^2} \end{aligned}$$

(7)

for any $g\in L^2({\mathbb {R}}^3,{\text {d}}k)$.

The Pekar energy functional is defined by

$$\begin{aligned} {\mathcal {E}}^\mathrm{P}(\varphi ) = \int _{{\mathbb {R}}^3} \vert \nabla \varphi (x)\vert ^2 {\text {d}}x - \frac{1}{2}\int _{{\mathbb {R}}^3} \int _{{\mathbb {R}}^3} \frac{\vert \varphi (x)\vert ^2 \vert \varphi (y)\vert ^2}{\vert x-y\vert } {\text {d}}x {\text {d}}y \end{aligned}$$

(8)

with constraint $| | \varphi | |_{L^2} =1$. It was shown in [20] that ${\mathcal {E}}^\mathrm{P}(\varphi )$ admits a unique minimizer (unique up to spatial translations)

$$\begin{aligned} \varphi _0 \in H^1({\mathbb {R}}^3, {\text {d}}x ) \cap \big \{\varphi \in L^2({\mathbb {R}}^3,{\text {d}}x )\, : \, | | \varphi | |_{L^2}=1\big \} \end{aligned}$$

(9)

that can be chosen positively. The minimizer further solves the Euler–Lagrange equation $(h^{\varphi _0} - \lambda ) \varphi _0 =0$ where

$$\begin{aligned} h^{\varphi _0} = p^2 + V^{\varphi _0} , \quad V^{\varphi _0}(x) = - 2{\text {Re}}\langle G_x , f_0 \rangle _{L^2}, \end{aligned}$$

(10)

and $\lambda = {\mathcal {E}}^\mathrm{P}(\varphi _0) - | | f_0 | |_{L^2}^2$. By its positivity, it follows that $\varphi _0$ is the unique ground state of the Schrödinger operator $h^{\varphi _0}$ and that $\lambda = \inf \sigma (h^{\varphi _0})$ belongs to the discrete spectrum of $h^{\varphi _0}$, see [25, Sec. 12]. Introducing the orthogonal projector $Q = 1- P$ with $P = \vert \varphi _0 \rangle \langle \varphi _0 \vert$, we hence know that $h^{\varphi _0} - \lambda$ is a positive operator when restricted to the closed subspace $\text {ran}Q \subseteq L^2({\mathbb {R}}^3,{\text {d}}x)$. This allows the definition of the restricted resolvent

$$\begin{aligned} R = Q ( h^{\varphi _0} - \lambda )^{-1} Q \end{aligned}$$

(11)

as a bounded operator in $L^2({\mathbb {R}}^3,{\text {d}}x)$. The fact that R is independent of $\alpha$ and thus bounded uniformly as $\alpha \rightarrow \infty$ is a crucial ingredient in the analysis of the strong coupling limit of $\Psi _\alpha (t)$. In a nutshell, it ensures a separation of scales as $\alpha \rightarrow \infty$ of the different parts of the Fröhlich Hamiltonian $H^\mathrm{F}_\alpha$ when the latter is applied to states of the form $\varphi \otimes W(\alpha f_0)^* \eta$ for suitable $\varphi \in \text {ran}Q$ and $\eta \in {\mathcal {F}}$.

That the scale separation of the different parts in $H_\alpha ^\mathrm{F}$ allows an effective description of the Fröhlich dynamics for times $t=o(\alpha ^2)$ was first observed in [13]. There, it was shown that the wave function $\Psi _\alpha (t) = e^{-iH^\mathrm{F }_\alpha t} \varphi _0 \otimes W(\alpha f_0 )^* \Omega _0$ remains close to its initial state up to a global phase factor, i.e.,

$$\begin{aligned} \big | \big | \Psi _\alpha (t) - e^{-i {\mathcal {E}}^\mathrm{P}(\varphi _0) t} \varphi _0 \otimes W(\alpha f_0 )^* \Omega _0 \big | \big |_{\mathscr {H}} \le C \, \vert t\vert ^{1/2} \alpha ^{-1} \end{aligned}$$

(12)

for some $C>0$. Since the initial state is normalized to one, the upper bound is meaningful for $t \ll \alpha ^2$. A similar approximation was obtained in [19] for more general initial states, namely Pekar product states in which the electron is initially trapped in the classical field produced by a given coherent state of the phonons. Modulo a global phase factor, the effective dynamics is then described by the Pekar product state $\varphi _{\text {\tiny LP}}(t) \otimes W(\alpha f_\text {\tiny LP}(t))^*\Omega _0$ with $(\varphi _\text {\tiny LP}(t), f_\text {\tiny LP}(t))$ solving the time-dependent Landau–Pekar equations, cf. [19, Eqn. (8)]. In fact, the effective dynamics in (12) can be understood as the special case in which $(\varphi _0, f_0)$ are the stationary ground state solutions of the Landau–Pekar equations. The proof of the nonstationary problem, however, is technically more demanding as it is based on a nonlinear adiabatic theorem for the solution of the Landau–Pekar equations, see [19, Theorem II.1]. Loosely speaking, the latter shows that the scale separation of the different parts in the Fröhlich Hamiltonian remains valid on some suitable time scale also in the nonstationary case. An adiabatic theorem for the Landau–Pekar equations in one spatial dimension has been derived in [9, 10]. Earlier results about the Fröhlich dynamics in the strong coupling limit provide approximations for $t=o(\alpha )$ but for much more general initial Pekar product states $\varphi \otimes W(\alpha f)^*\Omega _0$ with no particular assumption about the relation between $\varphi$ and f, see [7, 8]. To our knowledge, there are no results available to date that provide an approximation for the Fröhlich dynamics for $t=O(\alpha ^2)$.²

Remark 1

The particular choice of our initial state $\Psi _\alpha = \varphi _0 \otimes \xi _\alpha$ is motivated by Pekar’s approximation of the ground state energy of the Fröhlich Hamiltonian [24]. Taking the expectation value of $H^\mathrm{F}_\alpha$ for general Pekar states $\varphi \otimes W(\alpha f)^*\Omega _0$ and minimizing over the phonon mode $f \in L^2({\mathbb {R}},{\text {d}}k )$ leads to the Pekar functional ${\mathcal {E}}^\mathrm{P}(\varphi )$. That Pekar’s approximations is accurate in the strong coupling limit was rigorously proved in [3] and later, using a different approach which provided in addition a quantitative error estimate, in [21]. They showed

$$\begin{aligned} \inf \sigma ( H^\mathrm{F}_\alpha ) = {\mathcal {E}}^\mathrm{P}(\varphi _0 ) + o(1) \end{aligned}$$

(13)

as $\alpha \rightarrow \infty$. The physical picture behind this result is that the electron creates a classical phonon field which in turn leads to an effective trapping of the electron. This self-trapping mechanism is described by the ground state of (8). Let us also mention that the rigorous derivation of the next order contribution in (13) is still an open problem that was recently solved in [11] for a model in which the Fröhlich polaron is assumed to be confined to a suitably bounded region $\Lambda \subset {\mathbb {R}}^3$.

Remark 2

We note that (4), and equally (1), is somewhat formal since $G_x\notin L^2({\mathbb {R}}^3,{\text {d}}k)$ and hence $\phi (G_x)$ is not a densely defined operator. However, by a well-known argument that goes back to Lieb and Yamazaki [22], the right side of (4) defines a closed semi-bounded quadratic form with domain given by the form domain of $p^2\otimes 1 + 1\otimes N$. The Hamiltonian $H^\mathrm{F}_{\alpha }$ is then defined as the unique self-adjoint operator associated with this quadratic form, cf. [26, Thm. VIII.15]. For the purpose of this work, it is sufficient to use the form representation given in (4). Alternative approaches to define the Fröhlich Hamiltonian with an explicit characterization of its domain have been discussed more recently in [14, 17].

1.2 Effective dynamics

Our goal is to derive an approximation similar to (12) for times $t= O(\alpha ^2)$. To achieve this, we compare $\Psi _\alpha (t)$ with an effective time evolution that is generated by the Hamiltonian

$$\begin{aligned} H_{\alpha }^{\varphi _0} = 1 \otimes \big \langle \varphi _0 , \big ( H^\mathrm{F}_\alpha - (\alpha ^{-1} \phi (G_\cdot )-V^{\varphi _0}) (R \otimes 1) (\alpha ^{-1} \phi (G_\cdot )-V^{\varphi _0} ) \big ) \varphi _0 \big \rangle _{L^2} \end{aligned}$$

(14)

acting on ${\mathscr {H}} = L^2({\mathbb {R}}^3,{\text {d}}x)\otimes {\mathcal {F}}$. This operator acts nontrivially only on the Fock space. The expectation value in the second factor of the tensor product is taken with respect to the inner product in $L^2({\mathbb {R}}^3,{\text {d}}x)$. Equivalently we could write the second factor as a partial trace,

$$\begin{aligned} \text {Tr}_{L^2({\mathbb {R}}^3)} \big ( \vert \varphi _0 \rangle \langle \varphi _0 \vert \big ( H^\mathrm{F}_\alpha - (\alpha ^{-1} \phi (G_\cdot )-V^{\varphi _0}) (R \otimes 1) (\alpha ^{-1} \phi (G_\cdot )-V^{\varphi _0} ) \big ) \vert \varphi _0 \rangle \langle \varphi _0 \vert \big ). \end{aligned}$$

(15)

In the following proposition we clarify the difference compared to the ansatz in (12) and, more importantly, we obtain the existence of a unitary time evolution generated by $H_{\alpha }^{\varphi _0}$.

Proposition 1.1

For any $\alpha >0$ we have

$$\begin{aligned} W(\alpha f_0) H^{\varphi _0}_{ \alpha } W(\alpha f_0)^* - {\mathcal {E}}^\mathrm{P}(\varphi _0) = 1\otimes \alpha ^{-2}(N-A^{\varphi _0}) \end{aligned}$$

(16)

with the operator $A^{\varphi _0}: {\mathcal {F}} \rightarrow {\mathcal {F}}$ defined by

$$\begin{aligned} A^{\varphi _0}&= \big \langle \varphi _0 , \phi (G_{\cdot } ) (R \otimes 1) \phi (G_{\cdot } ) \varphi _0 \big \rangle _{L^2}\nonumber \\&= \int _{{\mathbb {R}}^3}\int _{{\mathbb {R}}^3}\, \big \langle \varphi _0, G_\cdot (k) R G_\cdot (l) \varphi _0 \big \rangle _{L^2} \, ( a_k^* + a_{-k} )(a_{ l}^* + a_{-l}) \, {\text {d}}k {\text {d}}l \end{aligned}$$

(17)

Moreover, ${\mathscr {D}}(N) \subseteq {\mathscr {D}}(N-A^{\varphi _0})$ and $N-A^{\varphi _0}$ is essentially self-adjoint on ${\mathcal {F}}$. (We denote its closure again by $N-A^{\varphi _0}$.)

We prove this proposition in Sect. 2.4. By unitarity of the Weyl operator, it follows that $H_{\alpha }^{\varphi _0}$ is self-adjoint on ${\mathscr {H}}$ and thus $\exp ( -iH_{\alpha }^{\varphi _0} t )$ defines a unitary time evolution.

Let us emphasize that the effective Hamiltonian acts nontrivially only on the phonons. This implies in particular that the time evolved state $\exp ( -iH_{\alpha }^{\varphi _0} t ) \varphi _0 \otimes \xi _\alpha$ is still an exact product. Because of the operator $A^{\varphi _0}$ in (16), however, the coherent state structure of the initial state $\xi _\alpha$ is not conserved. In this regard, our effective dynamics is different compared to the known results discussed in the previous section.

Remark 3

As a motivation of our ansatz in (14) let us mention its analogy to the well-known second-order perturbation formula

$$\begin{aligned} E_\varepsilon = \big \langle u_0, \big ( H_\varepsilon -\varepsilon V R_0 \varepsilon V \big ) u_0 \big \rangle + O(\varepsilon ^3)\quad ( \varepsilon \ll 1) \end{aligned}$$

(18)

for the nondegenerate ground state energy $E_\varepsilon$ of a suitable Hamiltonian $H_\varepsilon = H_0 + \varepsilon V$ by means of the ground state vector $u_0$ of $H_0$ and the reduced resolvent $R_0 = (1- \vert u_0 \rangle \langle u_0 \vert ) (H_0 - \langle u_0, H_0 u_0 \rangle )^{-1} (1- \vert u_0 \rangle \langle u_0 \vert )$. Despite this analogy, we emphasize that the expectation value in (14) is taken only w.r.t. to the electron wave function $\varphi _0 \in L^2(\mathbb R^3,{\text {d}}x)$ and not w.r.t. to the full Pekar product $\varphi _0 \otimes \xi _\alpha$. The reason why the expectation value w.r.t. $\varphi _0 \otimes \xi _\alpha$ would not lead to a good ansatz for the effective dynamics is the appearance of the factor $\alpha ^{-2}$ in front of the number operator N.

1.3 Main results

We are now ready to state our main results.

Theorem 1.2

Let $\varphi _0 \in H^1({\mathbb {R}}^3, {\text {d}}x)$ be the unique minimizer of the Pekar functional (8) with $| | \varphi _0 | |_{L^2} = 1$ and let $f_0 \in L^2({\mathbb {R}}^3, {\text {d}}k)$ be defined as in (6). Let further $\eta _0 \in {\mathcal {F}}$ satisfy $| | \eta _0 | |_{\mathcal {F}}=1$ and $\sup _{\alpha >0} | | (N+1)^{5/2}\eta _0 | |_{\mathcal {F}}<\infty$. Then, there are constants $c,C>0$ such that

$$\begin{aligned} \big | \big | \big ( e^{-iH^\mathrm{F }_\alpha t} - e^{-i H^{\varphi _0}_{ \alpha } t } \big ) \varphi _0 \otimes W(\alpha f_0 )^* \eta _0 \big | \big |_{\mathscr {H}} \le C \alpha ^{ -1} \exp ( c \vert t \vert \alpha ^{-2} ) \end{aligned}$$

(19)

for all $t\in {\mathbb {R}}$ and $\alpha >0$.

Since the initial state is normalized to one, the approximation is accurate for $t=O(\alpha ^2)$ (indeed, it is accurate for $t \ll \alpha ^2 \ln \alpha$). As a direct consequence of (19) together with $[e^{-i H_{\alpha }^{\varphi _0} t} , P\otimes 1 ]=0$, we obtain the following statement that shows that the reduced density of the electron remains approximately constant.

Corollary 1.3

Under the same assumptions as in Theorem 1.2 there exist constants $c,C>0$ such that

$$\begin{aligned} \text {Tr}_{L^2}\Big \vert \text {Tr}_{{\mathcal {F}}} \big \vert \Psi _{\alpha }(t) \big \rangle \big \langle \Psi _{\alpha } (t) \big \vert - \big \vert \varphi _0 \big \rangle \big \langle \varphi _0 \big \vert \Big \vert \le C \alpha ^{-1} \exp ( c \vert t \vert \alpha ^{-2} ) \end{aligned}$$

(20)

with $\Psi _{\alpha }(t) = e^{-iH^\mathrm{F}_\alpha t} \varphi _0 \otimes W(\alpha f_0)^* \eta _0$.

Theorem 1.2 shows that on the time scale $t=O(\alpha ^2)$ it is important to include the creation and annihilation of noncoherent phonons in the effective time evolution. In earlier findings which provided approximations for $t=o(\alpha )$ [7, 8] and $t=o(\alpha ^2)$ [13, 19], respectively, it was not necessary to take such noncoherent phonons into account as the effective dynamics was still described by exact Pekar product states. In our next corollary, we use the fact that the operator $N-A^{\varphi _0}$ is quadratic in creation and annihilation operators in order to describe the fluctuations around the coherent phonons by means of a time-dependent Bogoliubov transformation.

To make the last statement precise we need to introduce some well-known notions related to the Bogoliubov transformation. The generalized annihilation and creation operators are defined by $A(F) = a(f) + a^*(g)$ and $A^*(F)= a^*(f) + a(g)$, respectively, for any $F = f\oplus Jg \in L^2({\mathbb {R}}^3,{\text {d}}k)\oplus L^2(\mathbb R^3,{\text {d}}k)$ where J denotes the complex conjugation map $(J g)(x) = \overline{g(x)}$. A bounded invertible map ${\mathcal {V}}$ on $L^2({\mathbb {R}}^3,{\text {d}}k)\oplus L^2({\mathbb {R}}^3, {\text {d}}k)$ is called a Bogoliubov map if it satisfies

$$\begin{aligned} A^*({\mathcal {V}} F) = A({\mathcal {V}} {\mathcal {J}} F) , \quad \big [ A({\mathcal {V}} F), A^*({\mathcal {V}} G) \big ] = \big \langle F,{\mathcal {S}} G\big \rangle _{L^2\oplus L^2} \end{aligned}$$

(21)

for all $F,G \in L^2({\mathbb {R}}^3,{\text {d}}k)\oplus L^2({\mathbb {R}}^3,{\text {d}}k)$ where

$$\begin{aligned} {\mathcal {J}} = \begin{pmatrix} 0 &{} J\\ J &{}\quad 0 \end{pmatrix} , \quad {\mathcal {S}} = \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad -1 \end{pmatrix}. \end{aligned}$$

(22)

In case that the Bogoliubov map ${\mathcal {V}}$ is a Hilbert–Schmidt operator, i.e., if ${\mathcal {V}}^*{\mathcal {V}}$ is trace class, it can be implemented as a unitary operator on ${\mathcal {F}}$. This is the content of the Shale–Stinespring condition which states that there exists a unitary operator $U_{{\mathcal {V}}} : {\mathcal {F}} \rightarrow \mathcal F$ such that

$$\begin{aligned} U_{{\mathcal {V}}} A(F) U_{{\mathcal {V}}}^* = A( \mathcal VF) \end{aligned}$$

(23)

for any $F\in L^2({\mathbb {R}}^3, {\text {d}}k)\oplus L^2({\mathbb {R}}^3,{\text {d}}k)$ if and only if $\text {Tr} {\mathcal {V}}^*{\mathcal {V}} <\infty$, see, e.g., [27, Thm. 9.5]. We call the operator $U_{\mathcal V}$ the Bogoliubov transformation associated with the Bogoliubov map ${\mathcal {V}}$. Finally we need the concept of (pure bosonic) quasi-free states in ${\mathcal {F}}$. A quasi-free state $\eta \in {\mathcal {F}}$ is defined by the property that there is a Bogoliubov map ${\mathcal {V}}_\eta$ such that $\eta$ can be written as the transformed vacuum $\eta = U_{{\mathcal {V}}_\eta }\Omega _0$ (in particular, $\Omega _0$ is quasi-free). For a detailed introduction to Bogoliubov transformations and quasi-free states, we refer to [27, Sec. 9 and 10].

Our next goal is to show that the dynamics of the noncoherent phonons in $\Psi _\alpha (t) = e^{-iH^\mathrm{F}_\alpha t }\varphi _0 \otimes W(\alpha f_0)^* \eta _0$ can be described by a time-dependent Bogoliubov transformation $U_{{\mathcal {V}}_\alpha (t)}$ associated with the Bogoliubov map

$$\begin{aligned} {\mathcal {V}}_\alpha (t) = \exp \left[ -\frac{it}{\alpha ^2} \, \begin{pmatrix} 1- {\mathcal {G}} &{}\quad {\mathcal {K}} \\ -\overline{ {\mathcal {K}} } &{}\quad -1 + \overline{ {\mathcal {G}} } \end{pmatrix} \right] {\mathcal {V}}_{\alpha }(0) , \quad {\mathcal {V}}_{\alpha }(0) = \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad 1 \end{pmatrix}, \end{aligned}$$

(24)

where ${\mathcal {K}}$, ${\mathcal {G}}$ denote integral operators in $L^2({\mathbb {R}}^3,{\text {d}}k)$ defined by the kernels

$$\begin{aligned} {\mathcal {K}}(k,l)&= (2\pi \vert k \vert )^{-1} (2\pi \vert l \vert )^{-1} \big \{ \big \langle \varphi _0, e^{- ikx} R e^{- ilx} \varphi _0 \big \rangle _{L^2} + \big \langle \varphi _0, e^{- i l x} R e^{- i k x} \varphi _0 \big \rangle _{L^2}\big \} , \end{aligned}$$

(25)

$$\begin{aligned} {\mathcal {G}}(k,l)&= (2\pi \vert k \vert )^{-1} (2\pi \vert l \vert )^{-1} \big \{ \big \langle \varphi _0, e^{+ ikx} R e^{- ilx} \varphi _0 \big \rangle _{L^2} + \big \langle \varphi _0, e^{- i k x} R e^{ + il x} \varphi _0 \big \rangle _{L^2}\big \} , \end{aligned}$$

(26)

and where $\overline{{\mathcal {K}} }$, $\overline{{\mathcal {G}} }$ are to be understood as the integral operators with kernels $\overline{{\mathcal {K}} }(k,l) = \overline{{\mathcal {K}} (k,l)}= {\mathcal {K}}(-k,-l)$ and $\overline{{\mathcal {G}} }(k,l) = \overline{{\mathcal {G}} (k,l)} = {\mathcal {G}}(l,k)$, respectively.

Corollary 1.4

Under the same assumptions as in Theorem 1.2 with the additional requirement that $\eta _0 \in {\mathcal {F}}$ is quasi-free, there exist constants $c,C>0$ such that

$$\begin{aligned}&\text {Tr}_{{\mathcal {F}}}\Big \vert \text {Tr}_{ L^2} \big \vert W(\alpha f_0) \Psi _{\alpha }(t) \big \rangle \big \langle W(\alpha f_0) \Psi _{\alpha } (t) \big \vert - \big \vert U_{{\mathcal {V}}_\alpha (t)} \eta _0 \big \rangle \big \langle U_{{\mathcal {V}}_\alpha (t)} \eta _0 \big \vert \Big \vert \nonumber \\&\quad \le C \alpha ^{-1} \exp ( c \vert t \vert \alpha ^{-2} ) \end{aligned}$$

(27)

with $\Psi _{\alpha }(t) = e^{-iH^\mathrm{F}_\alpha t} \varphi _0 \otimes W(\alpha f_0)^* \eta _0$ and $U_{{\mathcal {V}}_\alpha (t)}$ the Bogoliubov transformation associated with the time-dependent Bogoliubov map ${\mathcal {V}}_\alpha (t)$ defined in (24).

The remainder of this note is organized as follows. We conclude section one with a short remark about the notation and a sketch of the proof of Theorem 1.2. In the second section we begin by stating two preliminary lemmas which are useful for the proof of Theorem 1.2. The latter is given in Sect. 2.2, whereas the preliminary lemmas are proved in Sect. 2.3. Finally we prove Proposition 1.1 together with Corollaries 1.3 and 1.4 in Sect. 2.4.

1.4 Notation

From now on, we omit the tensor product with the identity in operators of the form $h^{\varphi _0} = h^{\varphi _0}\otimes 1$ and $N = 1 \otimes N$. Moreover we make use of the abbreviation

$$\begin{aligned} \delta G_x = G_x - f_0, \end{aligned}$$

(28)

with $f_0$ defined as in (6) and by $\varphi _0\in H^1({\mathbb {R}}^3,{\text {d}}x)$ we always denote the ground state of the Pekar functional (8) satisfying $| | \varphi _0 | |_{L^2}=1$. The letter C is used for positive constants that are independent of t and $\alpha$. The exact value of C may vary from line to line.

1.5 Sketch of the proof

The proof of Theorem 1.2 is motivated mainly by the proof of inequality (12) given in [13]. To demonstrate our main idea it is instructive to start with a sketch of the derivation of (12) (in slightly different way compared to [13]). To this end, we use the shift relation (7) to verify

$$\begin{aligned} W(\alpha f_0 ) H^\mathrm{F}_\alpha W(\alpha f_0 )^* - {\mathcal {E}}^\mathrm{P}(\varphi _0) = h^{\varphi _0} - \lambda + \alpha ^{-2} N + \alpha ^{-1} \phi (\delta G_\cdot ). \end{aligned}$$

(29)

With $W(\alpha f_0 ) e^{ -iH^\mathrm{F}_\alpha t } W(\alpha f_0 )^* = \exp (-i W(\alpha f_0 ) H^\mathrm{F}_\alpha W(\alpha f_0 )^* t )$ and by Duhamel’s principle, one then obtains

$$\begin{aligned}&\big | \big | \big ( e^{-i H^\mathrm{F}_\alpha t } - e^{-i {\mathcal {E}}^\mathrm{P}(\varphi _0) t} \big ) \varphi _0 \otimes W(\alpha f_0)^* \Omega _0 \big | \big |_{\mathscr {H}}^2 \nonumber \\&\quad = - 2 \alpha ^{-1} {\text {Re}}\ \int _0^t i \big \langle e^{ - i ( h^{\varphi _0} - \lambda + \alpha ^{-2} N + \alpha ^{-1} \phi (\delta G_\cdot ) ) s } \varphi _0 \otimes \Omega _0 , Q \phi ( \delta G_\cdot ) \varphi _0 \otimes \Omega _0 \big \rangle _{{\mathscr {H}}} {\text {d}}s . \end{aligned}$$

(30)

Note that we further used $(h^{\varphi _0}-\lambda )\varphi _0 =0$ and $\phi ( \delta G_x) \varphi _0 \otimes \Omega _0 = Q \phi ( \delta G_x ) \varphi _0 \otimes \Omega _0$ which holds because of $\langle \varphi _0, \delta G_\cdot \varphi _0 \rangle _{L^2} =0$ (recall $P= \vert \varphi _0 \rangle \langle \varphi _0\vert$ and $Q=1-P$). A rough estimate of the right side would now lead to an upper bound proportional to $\vert t \vert \alpha ^{-1}$. The reason why the right side behaves actually better than this is a phase inside the integral which oscillates with nonzero ($\alpha$-independent) frequency.³ To take advantage of this phase we rewrite the integrand as

$$\begin{aligned} \big \langle e^{i(h^{\varphi _0 } -\lambda ) s } e^{ - i ( h^{\varphi _0} - \lambda + \alpha ^{-2} N + \alpha ^{-1} \phi (\delta G_\cdot ) ) s } \varphi _0 \otimes \Omega _0 , \left( \frac{\mathrm{d}}{\mathrm{d}s}e^{i(h^{\varphi _0 } -\lambda ) s } R \right) \phi ( \delta G_\cdot ) \varphi _0 \otimes \Omega _0 \big \rangle _{{\mathscr {H}}} \end{aligned}$$

(31)

and then integrate by parts. This leads to a perturbation like expansion of (30) which among other contributions (e.g., the boundary terms which are of order $\alpha ^{-1}$) includes the term

$$\begin{aligned} 2 \alpha ^{-2} {\text {Re}}\int _0^t i \big \langle e^{ - i ( h^{\varphi _0} - \lambda + \alpha ^{-2} N + \alpha ^{-1} \phi (\delta G_\cdot ) ) s } \varphi _0 \otimes \Omega _0 , \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot ) \varphi _0 \otimes \Omega _0 \big \rangle _{{\mathscr {H}}}\, {\text {d}}s. \end{aligned}$$

(32)

Apart from some technical difficulties being related to $G_x\notin L^2({\mathbb {R}}^3, {\text {d}}k)$, one then applies the estimate (here we use that R is uniformly bounded)

$$\begin{aligned}&\big \vert \big \langle e^{ - i ( h^{\varphi _0} - \lambda + \alpha ^{-2} N + \alpha ^{-1} \phi (\delta G_\cdot ) ) s } \varphi _0 \otimes \Omega _0 , \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot ) \varphi _0 \otimes \Omega _0 \big \rangle _{{\mathscr {H}}}\vert \nonumber \\&\quad \le C \big | \big | (N+1) \Omega _0 \big | \big |_{\mathcal {F}} \end{aligned}$$

(33)

in order to arrive at $\vert$ (32) $\vert \le C \alpha ^{-2}\vert t \vert$. This bound is indeed the reason why (12) is limited to $t=o(\alpha ^2)$. Our idea to improve upon this is to use the oscillating phase in (32) a second time. Inserting the identity $1= P+Q$ on the left of $\phi (\delta G_x)$ we obtain two contributions,

$$\begin{aligned}&(32.\text {a}) \\&\quad = 2\alpha ^{-2} \int _0^t {\text {Re}}i \big \langle e^{ - i ( h^{\varphi _0} - \lambda ) + \alpha ^{-2} N + \alpha ^{-1} \phi (\delta G_\cdot ) ) s } \varphi _0 \otimes \Omega _0, Q \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot ) \varphi _0 \otimes \Omega _0 \big \rangle _{{\mathscr {H}}}\, {\text {d}}s,\\&(32.\text {b}) \\&\quad = 2\alpha ^{-2} \int _0^t {\text {Re}}i \big \langle e^{ - i ( h^{\varphi _0} - \lambda ) + \alpha ^{-2} N + \alpha ^{-1} \phi (\delta G_\cdot ) ) s } \varphi _0 \otimes \Omega _0 , P \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot ) \varphi _0 \otimes \Omega _0 \big \rangle _{{\mathscr {H}}}\, {\text {d}}s. \end{aligned}$$

In the first one we can proceed similarly as before and improve the bound through integration by parts to $\vert$ (32.a) $\vert \le C( \alpha ^{-2} + \vert t\vert \alpha ^{-3})$. In the second line, however, integration by parts is not applicable since $(h^{\varphi _0}-\lambda )P=0$. In other words, there is no fast oscillating phase in this term and thus (32.b) seems to be really of order $\vert t \vert \alpha ^{-2}$. To avoid this term in the first place we include the operator $W(\alpha f_0)^* \alpha ^{-2} (N-A^{\varphi _0}) W(\alpha f_0 )$ into the effective dynamics, see (16). Starting over again with the new effective dynamics we now obtain an additional term in the first-order Duhamel expansion which cancels exactly the contribution from (32.b), cf. (51) and (57). Because of the nontrivial dynamics of the phonons we now have to take into account the number of excitations in the effective time evolution. Using a Gronwall argument, this is shown to be bounded by a constant times $\exp (c\vert t \vert \alpha ^{-2})$ which leads to the exponential factor in (19). This already explains much of our proof and aside from the technical details, it would lead to an upper bound in (19) that is proportional to $\alpha ^{-1/2} \exp (\vert t \vert \alpha ^{-2})$. By a third integration by parts we can improve the accuracy of this upper bound further and finally arrive at the stated bound in (19).

In principle there is no obstacle for continuing the explained strategy by additional integration by parts. In the present work, however, we do not pursue this idea further for the following two reasons. First, the effective Hamiltonian would become more complicated and involve terms with more than two creation and annihilation operators (hence the advantage of having a quadratic Hamiltonian would be lost). Second, an extension of our result to even longer times (e.g., $t= O(\alpha ^3)$) would not follow as an immediate consequence. This is due to the difficulty of controlling the number of phonon excitations produced in the effective time evolution. In Lemma 2.2 we shall see that our bound would be insufficient for times $t \gg \alpha ^2$.

Remark 4

The described idea of improving the approximation to longer times by changing the effective Hamiltonian as in (16) was similarly used also in [15, 16]. These works treat very different models, namely the dynamics of a single tracer particle resp. two tracer particles interacting with an ideal Fermi gas in the high density limit. The used approximations and the proofs of their accuracy, however, are analogous to the one we apply to the Fröhlich Polaron. The scale separation in these models comes from the large momenta of the gas modes that are close to the Fermi surface. (For the ideal Fermi gas, high density is equivalent to a large Fermi momentum.)

2 Proofs

2.1 Preliminary lemmas

Before we start with the proof of Theorem 1.2, let us state two lemmas with several helpful estimates. Their proofs are postponed to Sect. 2.3.

Lemma 2.1

Let $P=\vert \varphi _0 \rangle \langle \varphi _0 \vert$ and R as defined in (11). There is a constant $C>0$ such that for any $\Psi = \varphi _0 \otimes \eta \in {\mathscr {H}}$ with $\eta \in {\mathscr {D}}(N^{5/2})$, the following bounds hold.

$$\begin{aligned}&\big | \big | R \phi (\delta G_\cdot ) \Psi \big | \big |_{\mathscr {H}} + \big | \big | \big [N, R \phi (\delta G_\cdot ) \big ] \Psi \big | \big |_{\mathscr {H}} \le C \big | \big | ( N + 1 )^{ 1 / 2 } \eta \big | \big |_{{\mathcal {F}} }, \end{aligned}$$

(34)

$$\begin{aligned}&\big | \big | ( R \phi (\delta G_\cdot ))^2 \Psi \big | \big |_{\mathscr {H}} + \big | \big | \big [N, ( R \phi (\delta G_\cdot ))^2 \big ] \Psi \big | \big |_{\mathscr {H}} \le C \big | \big | ( N + 1 ) \eta \big | \big |_{{\mathcal {F}} }, \end{aligned}$$

(35)

$$\begin{aligned}&\big | \big | ( R \phi (\delta G_\cdot ) )^3 \Psi \big | \big |_{\mathscr {H}}+ \big | \big | P \phi (\delta G_\cdot ) ( R \phi (\delta G_\cdot ) )^2 \Psi \big | \big |_{\mathscr {H}} \nonumber \\&+ \big | \big | R \phi (\delta G_\cdot ) P \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) \Psi \big | \big |_{\mathscr {H}} + \big | \big | \big [ N, ( R \phi (\delta G_\cdot ) )^3 \big ] \Psi \big | \big |_{\mathscr {H}} \le C \big | \big | ( N + 1 )^{3 / 2 } \eta \big | \big |_{{\mathcal {F}} }, \end{aligned}$$

(36)

$$\begin{aligned}&\big | \big | ( R \phi (\delta G_\cdot ))^2 P \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) \Psi \big | \big |_{\mathscr {H}} \le C \big | \big | ( N + 1 )^{ 2 } \eta \big | \big |_{{\mathcal {F}} }, \end{aligned}$$

(37)

$$\begin{aligned}&\big | \big | ( R \phi (\delta G_\cdot ))^3 P \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) \Psi \big | \big |_{\mathscr {H}} \le C \big | \big | ( N + 1 )^{ 5/2 } \eta \big | \big |_{{\mathcal {F}} } . \end{aligned}$$

(38)

Moreover for $\Phi \in H^1({\mathbb {R}}^3,{\text {d}}x) \otimes {\mathcal {F}}$ we have

$$\begin{aligned} \big \vert \big \langle \Phi , \phi ( \delta G_\cdot ) ( R \phi ( \delta G_\cdot ))^3 \Psi \big \rangle _{{\mathscr {H}}} \big \vert&\le C \big ( 1 + \big | \big | p\Phi \big | \big |_{\mathscr {H}}\big ) \big | \big | ( N+1)^{2} \eta \big | \big |_{{\mathcal {F}} }. \end{aligned}$$

(39)

Lemma 2.2

Let $\eta \in {\mathscr {D}} (N^{5/2})$ with $\vert \vert \eta \vert \vert _{{\mathcal {F}}}=1$ and $\sup _{\alpha >0}| | (N+1)^{5/2} \eta _0 | |_{{\mathcal {F}} } < \infty$. Then there are constants $c,C>0$ such that

$$\begin{aligned} \sum _{j=0}^5 \big | \big | ( N+1)^{j/2} \exp ( -i \alpha ^{-2}(N-A^{\varphi _0 }) t ) \eta \big | \big |_{\mathcal {F}}^2&\le C \exp (c \vert t \vert \alpha ^{-2} ), \end{aligned}$$

(40)

$$\begin{aligned} \big | \big | p\, e^{-i H^\mathrm{F}_\alpha t} \varphi _0 \otimes W(\alpha f_0)^* \eta \big | \big |_{{\mathscr {H}} }&\le C \end{aligned}$$

(41)

for all $t\in {\mathbb {R}}$ and $\alpha >0$.

2.2 Proof of Theorem 1.2

We recall the relations

$$\begin{aligned} W(\alpha f_0 ) H^\mathrm{F}_\alpha W(\alpha f_0 )^* - {\mathcal {E}}^\mathrm{P}(\varphi _0)&= h^{\varphi _0} -\lambda + \alpha ^{-2}N + \alpha ^{-1}\phi (\delta G_x ), \end{aligned}$$

(42)

$$\begin{aligned} W(\alpha f_0 ) H^{\varphi _0 }_\alpha W(\alpha f_0 )^* - \mathcal E^\mathrm{P}(\varphi _0)&= \alpha ^{-2} (N-A^{\varphi _0}), \end{aligned}$$

(43)

which are verified by the commutation relations

$$\begin{aligned} W(\alpha f_0 ) \alpha ^{-2} N W(\alpha f_0 )^*&= \alpha ^{-2} N - \alpha ^{-1} \phi (f_0) + | | f_0 | |_{L^2}^2, \end{aligned}$$

(44)

$$\begin{aligned} W(\alpha f_0 ) \alpha ^{-1} \phi ( G_x ) W(\alpha f_0 )^*&= \alpha ^{-1}\phi ( G_x ) + V^{\varphi _0} , \end{aligned}$$

(45)

which in turn are easily obtained via (7). Using the unitarity of the Weyl operator we thus shall estimate

$$\begin{aligned}&\big | \big | \big ( e^{-iH^\mathrm{F}_\alpha t } - e^{-i H^{\varphi _0 }_\alpha t} \big ) \varphi _0 \otimes W(\alpha f_0 )^* \eta _0 \big | \big |_{\mathscr {H}} \nonumber \\&\quad = \big | \big | \big ( e^{- i ( h^{\varphi _0} -\lambda + \alpha ^{-2}N + \alpha ^{-1}\phi (\delta G_\cdot ) ) t } - e^{-i \alpha ^{-2} (N-A^{\varphi _0}) t} \big ) \varphi _0 \otimes \eta _0 \big | \big |_{\mathscr {H}}. \end{aligned}$$

(46)

For notational convenience let us abbreviate

$$\begin{aligned} \psi _\alpha (t ) = e^{- i ( h^{\varphi _0} -\lambda + \alpha ^{-2}N + \alpha ^{-1}\phi (\delta G_x ) ) t } \varphi _0 \otimes \eta _0, \quad \xi _\alpha (t) = \varphi _0 \otimes e^{-i \alpha ^{-2} (N-A^{\varphi _0}) t} \eta _0 . \end{aligned}$$

(47)

By the fundamental theorem of calculus we have

$$\begin{aligned} \big | \big | \psi _\alpha (t ) - \xi _\alpha (t ) \big | \big |_{\mathscr {H}}^2&= 2{\text {Re}}\small \langle \psi _\alpha (0) , \psi _\alpha (0) - e^{ i ( h^{\varphi _0} -\lambda + \alpha ^{-2}N + \alpha ^{-1}\phi (\delta G_\cdot ) ) t } \xi _\alpha (t)\small \rangle _{{\mathscr {H}} } \nonumber \\&= - 2{\text {Re}}\big \langle \psi _\alpha (0 ) , \int _0^t \frac{\mathrm{d}}{\mathrm{d}s} \big ( e^{ i ( h^{\varphi _0} -\lambda + \alpha ^{-2}N + \alpha ^{-1}\phi (\delta G_\cdot ) ) s } \xi _\alpha (s) \big ) {\text {d}}s \big \rangle _{{\mathscr {H}} }. \end{aligned}$$

(48)

Hence we get

$$\begin{aligned} \big | \big | \psi _\alpha (t ) - \xi _\alpha (t ) \big | \big |_{\mathscr {H}}^2 = 2{\text {Re}}f_\alpha (t)+ 2 {\text {Re}}g_\alpha (t) \end{aligned}$$

(49)

with

$$\begin{aligned} f_\alpha (t)&= - i \alpha ^{-1}\int _0^t \big \langle \psi _\alpha (s ) , \phi ( \delta G_\cdot ) \xi _\alpha (s) \big \rangle _{{\mathscr {H}}} \, {\text {d}}s , \end{aligned}$$

(50)

$$\begin{aligned} g_\alpha (t)&= - i \alpha ^{-2} \int _0^t \big \langle \psi _\alpha (s ) , P \phi ( \delta G_\cdot ) R \phi (\delta G_\cdot ) \xi _\alpha (s) \big \rangle _{{\mathscr {H}}} \, {\text {d}}s . \end{aligned}$$

(51)

Note that here we have used $[N-A^{\varphi _0} , P ]=0$, $P\xi _\alpha (s) = \xi _\alpha (s)$ and $(h^{\varphi _0} - \lambda ) P = 0$. With $1= P + Q$ and $\langle \varphi _0, \delta G_\cdot \varphi _0 \rangle _{L^2} = 0$ one further obtains

$$\begin{aligned} f_\alpha (t)&= - i\alpha ^{-1} \int _0^t \big \langle \psi _\alpha (s) , Q \phi ( \delta G_\cdot ) \xi _\alpha (s) \big \rangle _{{\mathscr {H}}} \, {\text {d}}s . \end{aligned}$$

(52)

In the first part of the proof we do three integrations by part w.r.t. the time variable s. This leads to a perturbation like expansion of (52) into different contributions. In particular, after the first partial integration, we obtain one term that equals $-g_\alpha ( t )$. Since this term would contribute an error of order $\vert t\vert \alpha ^{-2}$, it is crucial that we included the second-order correction in the effective dynamics. All remaining contributions will be estimated separately in the second part of the proof and finally lead to the error in (19).

To prepare the first integration by parts we use the restricted resolvent $R = Q ( h^{\varphi _0} - \lambda )^{-1} Q$ in order to write

$$\begin{aligned} f_\alpha (t) = - \alpha ^{-1} \int _0^t \big \langle e^{ i ( h^{\varphi _0} -\lambda ) s } \psi _\alpha (s) , \left( \frac{\mathrm{d}}{\mathrm{d}s} e^{ i ( h^{\varphi _0} - \lambda ) s } R \right) \phi ( \delta G_\cdot ) \xi _\alpha (s) \big \rangle _{{\mathscr {H}}} \, {\text {d}}s. \end{aligned}$$

(53)

Using

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}s} e^{ i ( h^{\varphi _0} -\lambda ) s } \psi _\alpha (s)&= - i e^{ i ( h^{\varphi _0} - \lambda ) s } \big ( \alpha ^{-2}N + \alpha ^{-1}\phi ( \delta G_x ) \big ) \psi _\alpha (s) , \end{aligned}$$

(54)

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}s} \xi _\alpha (s)&= -i \alpha ^{-2} ( N - A^{\varphi _0} ) \xi _\alpha (s) , \end{aligned}$$

(55)

together with $R \psi _\alpha (0) = R\varphi _0 \otimes \eta _0 = 0$, one finds by integration by parts

$$\begin{aligned} f_\alpha (t)&= - \alpha ^{-1} \big \langle \psi _\alpha (t ) , R \phi ( \delta G_\cdot ) \xi _\alpha (t) \big \rangle _{{\mathscr {H}}} \end{aligned}$$

(56a)

$$\begin{aligned}&\quad + i \alpha ^{-3} \int _0^t \big \langle \psi _\alpha (s) , R \big ( \big [N,\phi ( \delta G_\cdot )\big ] + \phi (\delta G_\cdot ) A^{\varphi _0} \big ) \xi _\alpha (s) \big \rangle _{{\mathscr {H}}}\, {\text {d}}s \end{aligned}$$

(56b)

$$\begin{aligned}&\quad + i \alpha ^{-2} \int _0^t \big \langle \psi _\alpha (s) , \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot ) \xi _\alpha (s) \big \rangle _{{\mathscr {H}}}\, {\text {d}}s. \end{aligned}$$

(56c)

In the last line the prefactor $\alpha ^{-2}$ is not sufficient and we need to do a second integration by parts. For that, we insert again the identity $1 = P + Q$ on the left of $\phi (\delta G_x )$. The term containing P equals

$$\begin{aligned} i \alpha ^{-2} \int _0^t \big \langle \psi _\alpha (s) , P \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot )\xi _\alpha (s) \big \rangle _{{\mathscr {H}}}\, {\text {d}}s = - g_\alpha ( t ) , \end{aligned}$$

(57)

and thus

$$\begin{aligned} (56\text {c}) + g_\alpha (t) = i \alpha ^{-2} \int _0^t \big \langle \psi _\alpha (s) , Q \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot ) \xi _\alpha (s) \big \rangle _{{\mathscr {H}}} \, {\text {d}}s . \end{aligned}$$

(58)

In this term we can integrate by parts similarly as in (53) which leads to

$$\begin{aligned} (56\text {c}) + g_\alpha (t)&= \alpha ^{-2} \big \langle \psi _\alpha ( t ) , R \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot ) \xi _\alpha ( t ) \big \rangle _{{\mathscr {H}}} \end{aligned}$$

(59a)

$$\begin{aligned}&\quad - i \alpha ^{-4} \int _0^t \big \langle \psi _\alpha (s) , \big [N, R \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot ) \big ] \xi _\alpha (s) \big \rangle _{{\mathscr {H}}}\, {\text {d}}s \end{aligned}$$

(59b)

$$\begin{aligned}&\quad - i \alpha ^{-4} \int _0^t \big \langle \psi _\alpha (s) , R \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot ) A^{\varphi _0} \xi _\alpha (s) \big \rangle _{{\mathscr {H}}}\, {\text {d}}s \end{aligned}$$

(59c)

$$\begin{aligned}&\quad - i \alpha ^{-3} \int _0^t \big \langle \psi _\alpha (s) , P \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot ) \xi _\alpha (s) \big \rangle _{{\mathscr {H}}}\, {\text {d}}s \end{aligned}$$

(59d)

$$\begin{aligned}&\quad - i \alpha ^{-3} \int _0^t \big \langle \psi _\alpha (s) , Q \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot ) \xi _\alpha (s) \big \rangle _{{\mathscr {H}}}\, {\text {d}}s . \end{aligned}$$

(59e)

In the last line we do a third integration by parts, i.e.,

$$\begin{aligned} (59\text {e})&= - \alpha ^{-3} \big \langle \psi _\alpha (t) , R \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) \xi _\alpha (t) \big \rangle _{{\mathscr {H}}} \end{aligned}$$

(60a)

$$\begin{aligned}&\quad + i \alpha ^{-5} \int _0^t \big \langle \psi _\alpha (s) , \big [N, R \phi ( \delta G_\cdot ) R \phi (\delta G_\cdot )R \phi ( \delta G_\cdot ) \big ] \xi _\alpha (s) \big \rangle _{{\mathscr {H}}}\, {\text {d}}s \end{aligned}$$

(60b)

$$\begin{aligned}&\quad + i \alpha ^{-5} \int _0^t \big \langle \psi _\alpha (s) , R \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) A^{\varphi _0} \xi _\alpha (s) \big \rangle _{{\mathscr {H}}}\, {\text {d}}s \end{aligned}$$

(60c)

$$\begin{aligned}&\quad + i \alpha ^{-4} \int _0^t \big \langle \psi _\alpha (s), \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) \xi _\alpha (s) \big \rangle _{{\mathscr {H}}}\, {\text {d}}s . \end{aligned}$$

(60d)

Summing up the above expansion we arrive at

$$\begin{aligned}&f_\alpha (t) + g_\alpha (t) \nonumber \\&\quad = (56\text {a}) + (56\text {b}) + (59\text {a}) + (59\text {b}) + (59\text {c}) + (59\text {d}) + (60\text {a}) + (60\text {b}) + (60\text {c}) (60\text {d}). \end{aligned}$$

In the remainder of the proof we separately estimate each summand on the right side. This is readily done using basic inequalities in combination with Lemmas 2.1 and 2.2. At the end, we conclude by applying Gronwall’s inequality.

Term (56a). In the first boundary term from the integration by parts we have

$$\begin{aligned} (56\text {a})&= - i \alpha ^{-1} \big \langle \psi _\alpha (t) - \xi _\alpha (t) , R \phi (\delta G_\cdot ) \xi _\alpha (t) \big \rangle _{{\mathscr {H}}} \end{aligned}$$

(61)

since $R \xi _\alpha (t) = 0$. Using the Cauchy–Schwarz inequality we obtain

$$\begin{aligned} \vert (56\text {a}) \vert&\le \frac{1}{4} \big | \big | \psi _\alpha (t) - \xi _\alpha (t) \big | \big |_{\mathscr {H}}^2 + \alpha ^{-2} \big | \big | R \phi ( \delta G_\cdot ) \xi _\alpha (t) \big | \big |_{{\mathscr {H}}}^2, \end{aligned}$$

(62)

and with (34) and (40),

$$\begin{aligned} \big | \big | R \phi ( \delta G_\cdot ) \xi _\alpha (t) \big | \big |_{{\mathscr {H}}}^2 \le C \big | \big | (N+1)^{1/2} \xi _\alpha (t) \big | \big |_{\mathscr {H}}^2 \le C \exp (c \vert t \vert \alpha ^{-2} ). \end{aligned}$$

(63)

Terms (59a) and (60a). For the other two boundary terms we proceed similarly and find

$$\begin{aligned} \vert (59\text {a}) \vert&\le \frac{1}{4} \big | \big | \psi _\alpha (t) - \xi _\alpha (t) \big | \big |_{\mathscr {H}}^2 + \alpha ^{-4} \big | \big | R \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot ) \xi _\alpha (t) \big | \big |_{\mathscr {H}} \nonumber \\&\le \frac{1}{4} \big | \big | \psi _\alpha (t) - \xi _\alpha (t) \big | \big |_{\mathscr {H}}^2 +C\alpha ^{ - 4} \exp (c\vert t\vert \alpha ^{-2}) \end{aligned}$$

(64)

as well as

$$\begin{aligned} \vert (60\text {a}) \vert&\le \frac{1}{4} \big | \big | \psi _\alpha (t) - \xi _\alpha (t) \big | \big |_{\mathscr {H}}^2 + \alpha ^{-6} \big | \big | R \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot ) R \phi ( \delta G_\cdot ) \xi _\alpha (t) \big | \big |_{\mathscr {H}} \nonumber \\&\le \frac{1}{4} \big | \big | \psi _\alpha (t) - \xi _\alpha (t) \big | \big |_{\mathscr {H}}^2 +C\alpha ^{ - 6} \exp (c\vert t\vert \alpha ^{-2}) \end{aligned}$$

(65)

where we have used (35) and (36) in combination with (40).

Term (56b). In this term we have

$$\begin{aligned} (56\text {b}) = i \alpha ^{-3} \int _0^t \big \langle \psi _\alpha (s) - \xi _\alpha (s) , R \big ( \big [ N,\phi ( \delta G_\cdot )\big ] + \phi (\delta G_\cdot ) A^{\varphi _0} \big ) \xi _\alpha (s) \big \rangle _{{\mathscr {H}}}\, {\text {d}}s. \end{aligned}$$

(66)

Using (34), the third line of (36) and (40) we estimate

$$\begin{aligned} \vert (56\text {b})\vert&\le \alpha ^{-2} \int _0^t \big | \big | \psi _\alpha (s) - \xi _\alpha (s) \big | \big |_{\mathscr {H}}^2 \, {\text {d}}s \nonumber \\&\quad + \frac{1}{2} \alpha ^{-4} \int _0^t \big ( \big | \big | \big [N,R \phi (\delta G_\cdot ) \big ] \xi _\alpha (s) \big | \big |_{\mathscr {H}}^2 + \big | \big | R \phi (\delta G_\cdot ) A^{\varphi _0} \xi _\alpha (s) \big | \big |_{\mathscr {H}}^2 \big ) {\text {d}}s \nonumber \\&\le \alpha ^{-2} \int _0^t \big | \big | \psi _\alpha (s) - \xi _\alpha (s) \big | \big |_{\mathscr {H}}^2 \, {\text {d}}s + C \alpha ^{-2} (\exp (c \vert t\vert \alpha ^{-2})-1) . \end{aligned}$$

(67)

Terms (59b) and (59c). Similarly as in the previous term,

$$\begin{aligned}&\vert (59\text {b}) \vert + \vert (59\text {c}) \vert \le \alpha ^{-2} \int _0^t \big | \big | \psi _\alpha (s) - \xi _\alpha (s) \big | \big |_{\mathscr {H}}^2\, {\text {d}}s \nonumber \\&\quad + \frac{1}{2} \alpha ^{-6} \int _0^t \big ( \big | \big | \big [ N, R \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) \big ] \xi _\alpha (s) \big | \big |_{\mathscr {H}}^2 \nonumber \\&\quad + \big | \big | R \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) A^{\varphi _0} \xi _\alpha (s) \big | \big |_{\mathscr {H}}^2 \big ) {\text {d}}s , \end{aligned}$$

(68)

and thus by means of (35), (37) and (40) we obtain

$$\begin{aligned} \vert (59\text {b}) \vert + \vert (59\text {c}) \vert \le \alpha ^{-2} \int _0^t \big | \big | \psi _\alpha (s) - \xi _\alpha (s) \big | \big |_{\mathscr {H}}^2 \, {\text {d}}s + C\alpha ^{-4} ( \exp (c\vert t \vert \alpha ^{-2}) -1 ) . \end{aligned}$$

(69)

Term (59d). In this line we keep the real part (cf. (49)) and have

$$\begin{aligned}&{\text {Re}}(59\text {d}) = \alpha ^{-3} \int _0^t {\text {Im}}\big \langle \psi _{\alpha }(s) - \xi _\alpha (s) , P \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) \xi _\alpha (s) \big \rangle _{{\mathscr {H}}}\, {\text {d}}s \end{aligned}$$

(70)

(the imaginary part of the added expectation value is zero). The absolute value of the right side is bounded from above by

$$\begin{aligned} \vert {\text {Re}}(59\text {d}) \vert&\le \frac{1}{2}\alpha ^{-2} \int _0^t \big ( \big | \big | \psi _{\alpha }(s) - \xi _\alpha (s) \big | \big |_{\mathscr {H}}^2 \nonumber \\&\quad + \frac{1}{2} \alpha ^{-4} \int _0^t \big | \big | P \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) \xi _\alpha (s) \big | \big |_{\mathscr {H}}^2 \big ) {\text {d}}s \nonumber \\&\le \alpha ^{-2} \int _0^t \big | \big | \psi _{\alpha }(s) - \xi _\alpha (s) \big | \big |_{\mathscr {H}}^2 \, {\text {d}}s + C \alpha ^{-2} ( \exp (c\vert t \vert \alpha ^{-2}) - 1 ), \end{aligned}$$

(71)

where one uses (36) and (40) in the second step.

Term (60b). By means of (36) and (40) one obtains

$$\begin{aligned} \vert (60\text {b}) \vert&\le \alpha ^{-5} \int _0^t \big | \big | \big [ N, R \phi ( \delta G_\cdot ) R \phi (\delta G_\cdot )R \phi ( \delta G_\cdot ) \big ] \xi _\alpha (s) \big | \big |_{}\, {\text {d}}s \nonumber \\&\le C \alpha ^{-3} (\exp (c\vert t \vert \alpha ^{-2})-1). \end{aligned}$$

(72)

Term (60c). In this term one can use (38) and (40) to find

$$\begin{aligned} \vert (60\text {c}) \vert&\le \alpha ^{-5} \int _0^t \big | \big | R \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) A^{\varphi _0} \xi _\alpha (s) \big | \big |_{}\, {\text {d}}s \nonumber \\&\le C \alpha ^{-3} ( \exp (c\vert t \vert \alpha ^{-2}) - 1) . \end{aligned}$$

(73)

Term (60d). For the last term we apply (39) in combination with

$$\begin{aligned} | | p \, \psi _\alpha (s) | |_{\mathscr {H}} = | | p \, e^{ - i H^\mathrm{F}_ \alpha s}\varphi _0 \otimes W(\alpha f_0)^* \eta _0 | |_{\mathscr {H}}, \end{aligned}$$

(74)

see (42), as well as (40) and (41). This leads to

$$\begin{aligned} \vert (60\text {d}) \vert&\le \alpha ^{-4} \int _0^t \big \vert \big \langle \psi _\alpha (s) , \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) R \phi (\delta G_\cdot ) \xi _\alpha (s) \big \rangle _{{\mathscr {H}}}\big \vert \, {\text {d}}s \nonumber \\&\le C \alpha ^{-4} \int _0^t \big (1+\big | \big | p \, e^{ - i H^\mathrm{F}_\alpha s} \varphi _0 \otimes W(\alpha f_0)^* \eta _0 \big | \big |_{\mathscr {H}} \big ) \big | \big | (N+1)^2 \xi _\alpha (s) \big | \big |_{\mathscr {H}}\, {\text {d}}s \nonumber \\&\le C \alpha ^{-2}( \exp ( c \vert t \vert \alpha ^{-2}) - 1) . \end{aligned}$$

(75)

Conclusion. In total, we have shown

$$\begin{aligned} \big | \big | \psi _{\alpha }(s) - \xi _\alpha (s) \big | \big |_{\mathscr {H}}^2&\le C \alpha ^{-2} \exp (c \vert t\vert \alpha ^{-2} ) + C \alpha ^{-2} \int _0^t \big | \big | \psi _{\alpha }(s) - \xi _\alpha (s) \big | \big |_{\mathscr {H}}^2 \, {\text {d}}s, \end{aligned}$$

(76)

from which the claimed bound follows by the integral version of Gronwall’s inequality.$\square$

2.3 Proofs of Lemmas 2.1 and 2.2

The main tool of the proof of Lemma 2.1 is the commutator method by Lieb and Yamazaki [22] by which one improves the behavior of the interaction at large momenta using the regularity of the electron wave function. More precisely one writes

$$\begin{aligned} G_x (k) = {\tilde{G}}_{x} (k) - p \cdot K_{x}(k) + K_{x} (k) \cdot p \end{aligned}$$

(77)

with

$$\begin{aligned} {\tilde{G}}_{x}(k) = G_x(k) \chi _{[0,1]}(\vert k \vert ), \quad K_{ x}(k) = \frac{k}{\vert k \vert ^2}G_x(k) \chi _{(1,\infty )}(\vert k \vert ), \end{aligned}$$

(78)

where $\chi$ denotes the characteristic function, i.e., $\chi _A(r)=1$ for all $r \in A\subseteq {\mathbb {R}}$ and $\chi _A(r)=0$ otherwise. The advantage of rewriting $G_x$ this way is that $\tilde{G}_{x}$ and $K_x$ are square-integrable,

$$\begin{aligned} \sup _{x\in {\mathbb {R}}^3}( \vert \vert {\tilde{G}}_{x}(\cdot ) \vert \vert _{L^2} + \vert \vert K_{ x} (\cdot ) \vert \vert _{L^2} ) <\infty . \end{aligned}$$

(79)

For the next proof let us recall the common bounds for the annihilation and creation operators,

$$\begin{aligned} | | a(g) \Psi | |_{\mathscr {H}} \le | | g | |_{L^2}\, | | N^{1/2}\Psi | |_{\mathscr {H}},\quad | | a^*(g) \Psi | |_{\mathscr {H}} \le | | g | |_{L^2}\, | | (N+1)^{1/2}\Psi | |_{\mathscr {H}} \end{aligned}$$

(80)

for any $g\in L^2({\mathbb {R}}^3,{\text {d}}k)$.

Proof of Lemma 2.1

For the proof of (34), we set $a^{\#} \in \{ a,a^*\}$ and use (77), (79), $| | f_0 | |_{L^2}<\infty$ and (80) to estimate

$$\begin{aligned} \big | \big | R a^{\#} (\delta G_\cdot ) P \Psi \big | \big |_{\mathscr {H}}&\le \big | \big | R a^{\#} ({\tilde{G}}_\cdot -f_0 ) P \Psi \big | \big |_{\mathscr {H}} \nonumber \\&\quad + \big | \big | R p \cdot a^{\#} (K_\cdot ) P \Psi \big | \big |_{\mathscr {H}} +\big | \big | R a^{\#} ( K_\cdot ) \cdot p P \Psi \big | \big |_{\mathscr {H}} \nonumber \\&\le C \big ( | | R | |_{} + | | R p | |_{} + | | R | |_{} \, | | p P | |_{} \big ) \big | \big | (N+1)^{1/2} \eta \big | \big |_{\mathcal {F}} \nonumber \\&\le C \big | \big | (N+1)^{1/2} \eta \big | \big |_{{\mathcal {F}} }, \end{aligned}$$

(81)

where $| | \cdot | |_{} = | | \cdot | |_{\mathscr {L}}$ denotes the norm on the space of bounded operators ${\mathscr {L}}(L^2(\mathbb R^3,{\text {d}}x))$. That $| | R | |_{} + | | p P | |_{L^2} <\infty$ is clear. To show $| | R p | |_{} <\infty$ we compute

$$\begin{aligned} \big | \big | p R \psi \big | \big |_{L^2}^2&= \big \langle \psi , R (h^{\varphi _0} - \lambda ) R \psi \big \rangle _{L^2} + \big \langle \psi , R ( \lambda - V^{\varphi _0}) R \psi \big \rangle _{L^2}\nonumber \\&\le \big \langle \psi , R \psi \big \rangle _{L^2} + \frac{1}{2} \big \langle \psi , R p^2 R \psi \big \rangle _{L^2} + C \big | \big | R \psi \big | \big |_{L^2}^2 \end{aligned}$$

(82)

where we used $\pm V^{\varphi _0} \le \frac{1}{2} p^2 + C$ as shown, e.g., in [19, Lemma III.2].⁴ Since the bound (81) holds equally if R is replaced by P and since

$$\begin{aligned} \big [ N, \phi (\delta G_x ) \big ] = a^*(\delta G_x ) - a(\delta G_x ), \end{aligned}$$

(83)

this proves (34).

In order to prove (35) we derive the bound for $\big | \big | R a^{\#_1}(\delta G_\cdot ) R a^{\#_2}(\delta G_\cdot ) \Psi \big | \big |_{}$ with $a^{\#_i} \in \{ a,a^*\}$. Proceeding similarly as in (81), we find

$$\begin{aligned} \big | \big | R a^{\#_1} (\delta G_\cdot ) R a^{\#_2}(\delta G_\cdot ) \Psi \big | \big |_{\mathscr {H}}&\le C \big | \big | (N+1)^{1/2} R^{1/2} a^{\#_2}(\delta G_\cdot ) \Psi \big | \big |_{\mathscr {H}} . \end{aligned}$$

(84)

From here we use

$$\begin{aligned} (N+1)^{1/2} R^{1/2} a(\delta G_x )\Psi&= R^{1/2} a(\delta G_x ) N^{1/2}\Psi , \end{aligned}$$

(85)

$$\begin{aligned} (N+1)^{1/2} R^{1/2} a^*(\delta G_x )\Psi&= R^{1/2} a^*(\delta G_x ) (N+2)^{1/2}\Psi , \end{aligned}$$

(86)

together with

$$\begin{aligned} \big | \big | R^{1/2} a^{\#_2} (\delta G_\cdot ) \Psi \big | \big |_{\mathscr {H}} \le C \big | \big | (N+1)^{1/2} \eta \big | \big |_{\mathcal {F}} . \end{aligned}$$

(87)

The latter is obtained in complete analogy to (81). The bounds for the other terms on the l.h.s. of (35) are derived the same way. Since the derivation of (36) and (37) is also very similar, we omit further details.

To prove (38) we proceed again as in (81) and find

$$\begin{aligned}&\big \vert \big \langle \Phi , a^{\#_1} (\delta G_\cdot )R a^{\#_2}( \delta G_\cdot ) R a^{\#_3}(\delta G_\cdot ) R a^{\#_4} (\delta G_\cdot ) \Psi \big \rangle _{{\mathscr {H}}} \big \vert \nonumber \\&\quad \le \big | \big | \Phi \big | \big |_{\mathscr {H}}\big | \big | \big ( a^{\#_1}( {\tilde{G}}_{\cdot }-f_0 ) - a^{\#_1}(K_{\cdot } ) \cdot p \big ) R a^{\#_2} ( \delta G_\cdot ) R a^{\#_3}( \delta G_\cdot ) R a^{\#_4} (\delta G_\cdot ) \Psi \big | \big |_{\mathscr {H}} \nonumber \\&\qquad + \Big \vert \big \langle \Phi , p\cdot a^{\#_1}(K_\cdot ) R a^{\#_2} (\delta G_\cdot ) R a^{\#_3}( \delta G_\cdot ) R a^{\#_4} (\delta G_\cdot ) \Psi \big \rangle _{{\mathscr {H}}} \Big \vert \nonumber \\&\quad \le C \big (1 + | | p\Phi | |_{\mathscr {H}} \big ) \big | \big | (N+1)^{1/2} R^{1/2} a^{\#_2}( \delta G_\cdot ) R a^{\#_3}(\delta G_\cdot ) R^{1/2} a^{\#_4}( \delta G_\cdot ) \Psi \big | \big |_{\mathscr {H}}. \end{aligned}$$

(88)

By estimating the last factor similarly as the right hand side of (84) we obtain (38). $\square$

Proof of Lemma 2.2

We start by verifying the following bound,

$$\begin{aligned} \big \vert \big \langle \eta , (N+1)^{j-1} [N,A^{\varphi _0} ] (N+1)^{m-j} \eta \big \rangle _{{\mathcal {F}}}\big \vert&\le C \big | \big | (N+1)^{m/2} \eta \big | \big |_{\mathcal {F}}^2 \end{aligned}$$

(89)

for $1\le j\le m$. To do so, use (83) to write

$$\begin{aligned}{}[N,A^{\varphi _0}]&= \big \langle \varphi _0, (a^*( G_\cdot ) - a(G_\cdot ) ) R (a^*( G_\cdot ) + a(G_\cdot ) ) \varphi _0 \big \rangle + \text {h.c.}, \end{aligned}$$

(90)

and then estimate each term separately. We illustrate the argument for the term $A_{++}^{\varphi _0} = \langle \varphi _0, a^*(G_\cdot ) R a^*(G_\cdot ) \varphi _0 \rangle _{L^2}$ for which we have

$$\begin{aligned}&\big \langle \eta , (N+1)^{j-1} A^{\varphi _0}_{++} (N+1)^{m-j} \eta \big \rangle _{{\mathcal {F}}} \\&\quad = {\left\{ \begin{array}{ll} \big \langle \eta , (N+1)^{j-1} (N-1)^{\frac{m}{2}+1-j} A^{\varphi _0}_{++} (N+1)^{\frac{m}{2}-1} \eta \big \rangle _{{\mathcal {F}}}&{}\quad \left( {\frac{m}{2}+1}\ge j \right) ,\\ \big \langle \eta , (N+1)^{\frac{j+i}{2}-1} A^{\varphi _0}_{++} (N+3)^{\frac{j-i}{2}} (N+1)^{m-j} \eta \big \rangle _{{\mathcal {F}}},&{}\quad \text {with}\quad i = m+2-j\quad \left( j\ge \frac{m}{2}+1 \right) . \end{array}\right. } \end{aligned}$$

Taking the absolute value and using the Cauchy–Schwarz inequality we can bound the first line from above by

$$\begin{aligned} \big | \big | (N+1)^{j-1} (N-1)^{\frac{m}{2}+1-j}\eta \big | \big |_{{\mathcal {F}} } \big | \big | A^{\varphi _0}_{++} (N+1)^{\frac{m}{2}-1} \eta \big | \big |_{\mathcal {F}} \le C \big | \big | (N+1)^{\frac{m}{2}}\eta \big | \big |_{\mathcal {F}}^2 , \end{aligned}$$

(91)

where we used $| | A^{\varphi _0}_{++} \eta | |_{\mathcal {F}}\le C | | (N+1)\eta | |_{\mathcal {F}}$, $\eta \in {\mathcal {F}}$, which is proved the same way as the bound for the left side of (84). Similarly we find the following upper bound for the second line,

$$\begin{aligned} \big | \big | (N+1)^{\frac{j+i}{2}-1} \eta \big | \big |_{\mathcal {F}} \big | \big | A_{++}^{\varphi _0} (N+3)^{\frac{j-i}{2}} (N+1)^{m-j} \eta \big | \big |_{\mathcal F} \le C \big | \big | (N+1)^{\frac{m}{2}}\eta \big | \big |_{\mathcal {F}}^2 . \end{aligned}$$

(92)

Repeating the same argument for the other terms in (90) leads to the stated bound in (89).

Next let $\eta _\alpha (t) = \exp ( -i \alpha ^{-2}(N-A^{\varphi _0 }) t ) \eta$ and compute the time derivative

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \big \langle \eta _\alpha (t) , N^m \eta _\alpha (t)\big \rangle _{{\mathcal {F}}}&= - \alpha ^{-2} \sum _{j=1}^m \big \langle \eta _\alpha (t) , N^{j-1} i [N, A^{\varphi _0}] N^{m-j} \eta _\alpha (t)\big \rangle _{{\mathcal {F}}} \end{aligned}$$

(93)

which for $m \in \{1,2,3,4,5\}$ is easily checked explicitly. Setting $z(t) = \sum _{j=1}^5 \big | \big | (N+1)^{j/2} \eta _\alpha (t) \big | \big |_{{\mathcal {F}}}^2$, we have by (89) and (93), $\vert \frac{\mathrm{d}}{\mathrm{d}t} z(t)\vert \le C \alpha ^{-2} z(t)$. Since $\sup _{\alpha >0} z(0) <\infty$ by assumption, it follows from Gronwall’s inequality that $z(t) \le C \exp (c\vert t \vert \alpha ^{-2})$.

For a proof of (41), let $\psi _\alpha (t) = e^{-i H^\mathrm{F}_\alpha t} \varphi _0 \otimes W(\alpha f_0)^* \eta$ and estimate

$$\begin{aligned} \big | \big | p\, \psi _\alpha (t) \big | \big |_{{\mathscr {H}} }^2&\le C \big \langle \psi _\alpha (t), (H_\alpha ^\mathrm{F} +1) \psi _\alpha (t) \big \rangle _{{\mathscr {H}}} = C \big \langle \psi _\alpha (0), (H_\alpha ^\mathrm{F} +1) \psi _\alpha (0) \big \rangle _{{\mathscr {H}}} \nonumber \\&= C \big ( 1 + {\mathcal {E}}^\mathrm{P}(\varphi _0) + \alpha ^{-2} \big \langle \eta , N \eta \big \rangle _{{\mathcal {F}}} \big ) \end{aligned}$$

(94)

for some constant $C>0$. Here we used $N\ge 0$ and $p^2 + \alpha ^{-2}N \le C ( H_\alpha ^\mathrm{F}+1)$ in the first step (see, e.g., [13, Lemma A.5]) and the commutation relation (42) together with $\langle \varphi _0 , \delta G_x \varphi _0 \rangle _{L^2}=0$ in the third step. $\square$

2.4 Proofs of Proposition 1.1 and Corollaries 1.3 and 1.4

Proof of proposition 1.1

The identity in (16) follows from ${\mathcal {E}}^\mathrm{P}(\varphi _0) = \lambda + | | f_0 | |_{L^2}^2$ together with the commutation relations (44) and (45). That ${\mathscr {D}}(N) \subseteq {\mathscr {D}}(N-A^{\varphi _0})$ follows from

$$\begin{aligned} \big | \big | A^{\varphi _0}\eta \big | \big |_{\mathcal {F}} = \big | \big | P\phi (G_\cdot ) R \phi (G_\cdot ) \varphi _0 \otimes \eta \big | \big |_{\mathscr {H}} \le C \big | \big | (N+1)\eta \big | \big |_{\mathcal {F}} \end{aligned}$$

(95)

which is proven the same way as the bound for the l.h.s. of (84). Using (83) one further finds

$$\begin{aligned} \big \vert \big \langle \eta ,\big [ A^{\varphi _0} , N \big ] \eta \big \rangle _{{\mathcal {F}}} \big \vert \le C \big \langle \eta , N \eta \big \rangle _{\mathcal F} \end{aligned}$$

(96)

for all $\eta \in {\mathcal {F}}_0$ with ${\mathcal {F}}_0\subseteq \mathcal F$ denoting the dense subspace of all Fock space vectors that have only finitely many nonzero components. Since ${\mathcal {F}}_0$ is a core of the number operator N, we can infer that $N-A^{\varphi _0}$ is essentially self-adjoint by a variant of Nelson’s commutator theorem [5, Corollary 1.1]. Alternatively one could conclude self-adjointness of $N-A^{\varphi _0}$ from the criteria for self-adjointness of Fock space operators found in [4]. $\square$

In the following two proofs we make use of the bound

$$\begin{aligned} \text {Tr}_{{\mathscr {H}}_1} \big \vert \text {Tr}_{\mathscr {H}_2} \vert \Psi \rangle \langle \Phi \vert \big \vert \le | | \Psi | |_{{\mathscr {H}}_1 \otimes {\mathscr {H}}_2} \, | | \Phi | |_{\mathscr {H}_1 \otimes {\mathscr {H}}_2} \end{aligned}$$

(97)

where ${\mathscr {H}}_1,{\mathscr {H}}_2$ are two separable Hilbert spaces and $\Psi ,\Phi \in {\mathscr {H}}_1\otimes {\mathscr {H}}_2$. The inequality follows from the variational characterization of the trace. For a proof see [8, Appendix D].

Proof of Corollary 1.3

We recall $\Psi _{\alpha }(t) = e^{-iH^\mathrm{F}_\alpha t} \varphi _0 \otimes W(\alpha f_0)^* \eta _0$ and write

$$\begin{aligned} \big \vert \Psi _{\alpha }(t) \big \rangle \big \langle \Psi _{\alpha } (t) \big \vert&= \big \vert e^{-i H^{\varphi _0}_\alpha t } \Psi _{\alpha }(0) \big \rangle \big \langle e^{-i H^{\varphi _0}_\alpha t } \Psi _{\alpha }(0) \big \vert \nonumber \\&\qquad + \big \vert \Psi _{\alpha }(t) \big \rangle \big \langle ( e^{-iH^\mathrm{F}_\alpha t} - e^{-i H^{\varphi _0}_\alpha t } ) \Psi _{\alpha }(0) \big \vert \nonumber \\&\qquad + \big \vert ( e^{-iH^\mathrm{F}_\alpha t} - e^{-i H^{\varphi _0}_\alpha t } ) \Psi _{\alpha }(0) \big \rangle \big \langle e^{-i H^{\varphi _0}_\alpha t } \Psi _{\alpha }(0) \big \vert . \end{aligned}$$

(98)

Since $e^{-i H^{\varphi _0}_\alpha t }$ acts nontrivially only on the Fock space,

$$\begin{aligned} \text {Tr}_{{\mathcal {F}}} \Big \vert e^{-i H^{\varphi _0}_\alpha t } \Psi _\alpha (0) \Big \rangle \Big \langle e^{-i H^{\varphi _0}_\alpha t } \Psi _\alpha (0) \Big \vert = \vert \varphi _0 \rangle \langle \varphi _0 \vert . \end{aligned}$$

(99)

Applying (97) to the two last terms in (98) thus leads to

$$\begin{aligned} \text {Tr}_{L^2}\Big \vert \text {Tr}_{{\mathcal {F}}} \big \vert \Psi _{\alpha }(t) \big \rangle \big \langle \Psi _{\alpha } (t) \big \vert - \big \vert \varphi _0 \big \rangle \big \langle \varphi _0 \big \vert \Big \vert \le 2 \big | \big | \big ( e^{-i H^\mathrm{F }_\alpha t } - e^{-i H^{\varphi _0}_\alpha t } \big ) \Psi _{\alpha }(0) \big | \big |_{\mathscr {H}}. \end{aligned}$$

(100)

Together with Theorem 1.2 this proves the corollary. $\square$

Proof of Corollary 1.4

For $E\in {\mathbb {R}}$ we write

$$\begin{aligned}&\big \vert W(\alpha f_0) \Psi _{\alpha }(t) \big \rangle \big \langle W(\alpha f_0) \Psi _{\alpha } (t) \big \vert \nonumber \\&\quad = \big \vert \varphi _0\otimes e^{-i \alpha ^{-2}(N-A^{\varphi _0 } ) t } \eta _0 \big \rangle \big \langle \varphi _0 \otimes e^{-i \alpha ^{-2}(N-A^{\varphi _0 } ) t } \eta _0 \big \vert \nonumber \\&\qquad + \big \vert W(\alpha f_0) \Psi _{\alpha }(t) \big \rangle \big \langle W(\alpha f_0) \Psi _{\alpha }(t) - \varphi _0\otimes e^{-i \alpha ^{-2}(N-A^{\varphi _0 } + E ) t } \eta _0 \big \vert \nonumber \\&\qquad + \big \vert W(\alpha f_0) \Psi _{\alpha }(t) - \varphi _0\otimes e^{-i \alpha ^{-2}(N-A^{\varphi _0 } + E ) t } \eta _0 \big \rangle \big \langle \varphi _0\otimes e^{-i \alpha ^{-2}(N-A^{\varphi _0 } + E ) t } \eta _0 \big \vert . \end{aligned}$$

(101)

From here we proceed similarly as in the proof of Corollary 1.3. Taking the partial trace in the first line gives

$$\begin{aligned}&\text {Tr}_{L^2} \Big \vert \varphi _0 \otimes e^{-i \alpha ^{-2}(N-A^{\varphi _0 } ) t } \eta _0 \Big \rangle \Big \langle \varphi _0 \otimes e^{-i \alpha ^{-2}(N-A^{\varphi _0 } ) t } \eta _0 \Big \vert \nonumber \\&\quad = \big \vert e^{-i \alpha ^{-2}(N-A^{\varphi _0 } ) t } \eta _0 \big \rangle \big \langle e^{-i \alpha ^{-2}(N-A^{\varphi _0 } ) t } \eta _0 \big \vert . \end{aligned}$$

(102)

Using (97) for the second and third line then leads to

$$\begin{aligned}&\text {Tr}_{{\mathcal {F}}}\Big \vert \text {Tr}_{L^2 } \big \vert W(\alpha f_0) \Psi _{\alpha }(t) \big \rangle \big \langle W(\alpha f_0) \Psi _{\alpha } (t) \big \vert \nonumber \\&\qquad - \big \vert e^{-i \alpha ^{-2}(N-A^{\varphi _0 } ) t } \eta _0 \big \rangle \big \langle e^{-i \alpha ^{-2}(N-A^{\varphi _0 } + ) t } \eta _0 \big \vert \Big \vert \nonumber \\&\quad \le 2 \big | \big | W(\alpha f_0) \Psi _\alpha (t) - \varphi _0\otimes e^{-i \alpha ^{-2}(N-A^{\varphi _0 } + E ) t } \eta _0 \big | \big |_{\mathscr {H}} . \end{aligned}$$

(103)

Setting $E = {\mathcal {E}}^\mathrm{P}(\varphi _0)$ we can employ (16) and use the unitarity of $W(\alpha f_0)$ to see that the last line equals

$$\begin{aligned} 2 \big | \big | W(\alpha f_0)( \Psi _\alpha (t) - e^{-i H_\alpha ^{\varphi _0} t } \Psi _\alpha (0) ) \big | \big |_{\mathscr {H}} = 2 \big | \big | \big ( e^{-i H^\mathrm{F }_\alpha t } - e^{-i H^{\varphi _0}_\alpha t } \big ) \Psi _{\alpha }(0) \big | \big |_{\mathscr {H}}. \end{aligned}$$

(104)

The corollary now follows from Theorem 1.2 together with the identity

$$\begin{aligned} \big \vert e^{-i \alpha ^{-2}(N-A^{\varphi _0 } ) t } \eta _0 \big \rangle \big \langle e^{-i \alpha ^{-2}(N-A^{\varphi _0 } ) t } \eta _0 \big \vert = \big \vert U_{{\mathcal {V}}_\alpha (t)} \eta _0 \big \rangle \big \langle U_{{\mathcal {V}}_\alpha (t)} \eta _0 \big \vert , \end{aligned}$$

(105)

which we shall prove below.

Proof of (105). For $\varepsilon = \int _{{\mathbb {R}}^3} (2\pi \vert k \vert )^{-2} | | R^{1/2} e^{-ikx}\varphi _0 | |_{L^2}^2 \text {d}k$ it holds that

$$\begin{aligned} \exp \big ( -i \alpha ^{-2}(N-A^{\varphi _0 } + \varepsilon ) t \big ) \eta _0 = U_{{\mathcal {V}}_\alpha (t)} \eta _0. \end{aligned}$$

(106)

To show this we follow closely the argument from [2, Lem. 2.8 and App. B] where a similar identity was proven in the context of the dynamics of weakly interacting bosons. The argument is based on well-known facts about Bogoliubov transformations and quasi-free states and a general result about the dynamics generated by quadratic Hamiltonians [23, Prop. 7].

At this point it is useful to introduce the pointwise annihilation and creation operators $a_k,a_k^*$ defined by the requirement that

$$\begin{aligned} a(g) = \int _{{\mathbb {R}}^3} \overline{g(k)}\, a_k\, \text {d}k, \quad a^*(g) = \int _{{\mathbb {R}}^3} \, g(k)\, a_k^*\, \text {d}k \end{aligned}$$

(107)

for any $g\in L^2({\mathbb {R}}^3, \text {d}k )$. The commutation relations (3) now read

$$\begin{aligned}{}[a_k,a_l^*]= \delta (k-l), \quad [a_k,a_l] = [a^*_k,a^*_l]= 0\ \ \ \forall \, k,l \in {\mathbb {R}}^3. \end{aligned}$$

(108)

Using (17), (107) and (108) a short computation leads to

$$\begin{aligned} N - A^{\varphi _0} + \varepsilon = \text {d}\Gamma (1 - {\mathcal {G}}) - \frac{1}{2}\int _{{\mathbb {R}}^3} \int _{{\mathbb {R}}^3} \big ( \mathcal K(k,l) \, a^*_k a_l^* + \overline{{\mathcal {K}}(k,l)} \, a_k a_l \big ) \text {d}k \text {d}l \end{aligned}$$

(109)

where $\text {d}\Gamma (1 - {\mathcal {G}})$ denotes the second quantization of the one-body operator $1-{\mathcal {G}}$, i.e.,

$$\begin{aligned} \text {d}\Gamma (1 - {\mathcal {G}}) = \int _{{\mathbb {R}}^3} \int _{{\mathbb {R}}^3} \big ( \delta (k-l) - {\mathcal {G}}(k,l) \big ) \, a_k^* a_l\, \text {d}k \text {d}l, \end{aligned}$$

(110)

see (25) and (26) for a definition of ${\mathcal {K}}(k,l)$ and ${\mathcal {G}}(k,l)$, respectively. To the operator on the right side of (109) we can apply [23, Prop. 7]. The requirements of this proposition are satisfied since $1 - {\mathcal {G}} : L^2({\mathbb {R}}^3,{\text {d}}k) \rightarrow L^2({\mathbb {R}}^3, {\text {d}}k)$ is bounded and ${\mathcal {K}}:L^2({\mathbb {R}}^3, {\text {d}}k) \rightarrow L^2({\mathbb {R}}^3, {\text {d}}k)$ is a Hilbert–Schmidt operator which can be verified by means of (77). By part (iii) of [23, Prop. 7] it follows in particular that for any quasi-free state $\eta _0 \in {\mathcal {F}}$, the time-evolved state $\eta _\alpha (t) = \exp (-i \alpha ^{-2} (N - A^{\varphi _0} + \varepsilon )t )\eta _0$ is again quasi-free (the bound $\langle \eta _\alpha (t), N \eta _\alpha (t) \rangle \le C \exp (c\vert t \vert \alpha ^{-2})$ can be checked directly by means of Gronwall’s inequality). It is further not difficult to verify that the state $U_{\mathcal V_\alpha (t)}\eta _0$ is also quasi-free ($\eta _0 = U_{\mathcal W}\Omega _0$ for some Bogoliubov map ${\mathcal {W}}$ and thus $U_{\mathcal V_\alpha (t)}\eta _0 = U_{{\mathcal {V}}_\alpha (t) \circ \mathcal W}\Omega _0$ with Bogoliubov map ${\mathcal {V}}_\alpha (t) \circ \mathcal W$). To show equality between the quasi-free states $\eta _\alpha (t)$ and $U_{{\mathcal {V}}_\alpha (t)}\eta _0$ we compare their reduced one-body density matrices. This is sufficient because of the well-known fact that quasi-free states are uniquely determined by their reduced one-body density matrices. For $\xi \in {\mathcal {F}}$ the reduced one-body density matrices $\gamma _\xi : L^2({\mathbb {R}}^3, {\text {d}}k) \rightarrow L^2( {\mathbb {R}}^3, {\text {d}}k )$ and $\alpha _\xi : L^2(\mathbb R^3, {\text {d}}k) \rightarrow L^2( {\mathbb {R}}^3, {\text {d}}k )$ are defined by

$$\begin{aligned} \big \langle f, \gamma _\xi g\big \rangle _{L^2} = \big \langle \xi , a^*(g)a(f) \xi \big \rangle _{{\mathcal {F}}}, \quad \big \langle f, \alpha _\xi {\overline{g}}\big \rangle _{L^2} = \big \langle \xi , a(g)a(f) \xi \big \rangle _{{\mathcal {F}}} \end{aligned}$$

(111)

for all $f,g\in L^2({\mathbb {R}}^3, {\text {d}}k)$. In order to show $\gamma _{\eta _\alpha (t)} = \gamma _{U_{{{\mathcal {V}}}_\alpha (t)} \eta _0 }$ and $\alpha _{\eta _\alpha (t)} = \alpha _{U_{{\mathcal V}_\alpha (t)} \eta _0}$ we argue that they solve the same pair of differential equations with the same initial condition $\gamma _{\eta _0}$ and $\alpha _{\eta _0}$, respectively, and then use that the solution to this pair of differential equations is unique (the latter was shown in [23, Prop. 7]).

Instead of computing the time derivative of $\gamma _{\eta _\alpha (t)}$ and $\alpha _{\eta _\alpha (t)}$, and similarly for $U_{{\mathcal {V}}_\alpha (t)}\eta _0$, it is more convenient to determine the time derivative of $\big \langle \eta _\alpha (t), A(F_1) A(F_2) \eta _\alpha (t) \big \rangle _{{\mathcal {F}}}$ with A(F) the generalized annihilation operator as defined above (21). For $F_1,F_2 \in L^2(\mathbb R^3, {\text {d}}k)\oplus L^2({\mathbb {R}}^3,{\text {d}}k)$ we have

$$\begin{aligned} i\frac{\mathrm{d}}{\mathrm{d}t}\big \langle \eta _\alpha (t), A(F_1) A(F_2) \xi _\alpha (t) \big \rangle _{{\mathcal {F}}}&= \alpha ^{-2} \big \langle \eta _\alpha (t) , \big [ N-A^{\varphi _0}, A(F_1) A(F_2) \big ] \xi _\alpha (t) \big \rangle _{{\mathcal {F}}} \end{aligned}$$

(112)

and it follows by a straightforward computation that

$$\begin{aligned} \big [ N - A^{\varphi _0}, A(F_1) A(F_2) \big ] = A( \mathcal AF_1 ) A( F_2) + A( F_1) A( {\mathcal {A}} F_2) \end{aligned}$$

(113)

with

$$\begin{aligned} {\mathcal {A}} = \begin{pmatrix} 1- {\mathcal {G}} &{}\quad {\mathcal {K}} \\ -\overline{ {\mathcal {K}} } &{} \quad -1 + \overline{ {\mathcal {G}} } \end{pmatrix}. \end{aligned}$$

(114)

Next we use $U_ {{\mathcal {V}}}^* A(F) U_{{\mathcal {V}}} = A( \mathcal V^{-1} F)$, cf. (23), to obtain

$$\begin{aligned} \big \langle U_{ {{\mathcal {V}}}_\alpha (t)} \eta _0 , A(F_1) A(F_2) U_{ {{\mathcal {V}}}_\alpha (t)} \eta _0 \big \rangle _{{\mathcal {F}}}&= \big \langle \eta _0 , A({\mathcal {V}}_\alpha ^{-1 }(t) F_1) A( \mathcal V_\alpha ^{-1 }(t) F_2) \eta _0 \big \rangle _{{\mathcal {F}}} . \end{aligned}$$

(115)

By means of $( i\partial _t {\mathcal {V}}_\alpha ^{-1}(t)) \mathcal V_\alpha (t) = - {\mathcal {V}}_\alpha ^{-1}(t) (i\partial _t \mathcal V_\alpha (t))$ together with $i\partial _t {\mathcal {V}}_\alpha (t) = \alpha ^{-2} {\mathcal {A}} {\mathcal {V}}_\alpha (t)$, we can compute the time derivative

$$\begin{aligned}&i \frac{\mathrm{d}}{\mathrm{d}t} \big \langle \eta _0 , A({\mathcal {V}}_\alpha ^{-1 }(t) F_1) A( {\mathcal {V}}_\alpha ^{-1 }(t) F_2) \eta _0 \big \rangle _{{\mathcal {F}}} \nonumber \\&\quad = \big \langle \eta _0 , \big ( A(-i \partial _t {\mathcal {V}}_\alpha ^{-1 }(t) F_1) A( {\mathcal {V}}_\alpha ^{-1 }(t) F_2) + A({\mathcal {V}}_\alpha ^{-1 }(t) F_1) A( -i \partial _t {\mathcal {V}}_\alpha ^{-1 }(t) F_2) \big ) \eta _0 \big \rangle _{{\mathcal {F}}} \nonumber \\&\quad = \alpha ^{-2} \big \langle \eta _0 , \big ( A( {\mathcal {V}}_\alpha ^{-1 }(t){\mathcal {A}} F_1) A( {\mathcal {V}}_\alpha ^{-1 }(t) F_2) + A({\mathcal {V}}_\alpha ^{-1 }(t) F_1) A( {\mathcal {V}}_\alpha ^{-1 }(t) {\mathcal {A}} F_2) \big ) \eta _0 \big \rangle _{{\mathcal {F}}} \nonumber \\&\quad = \alpha ^{-2}\big \langle U_{{\mathcal {V}}_\alpha (t)} \eta _0 , \big ( A( {\mathcal {A}} F_1) A( F_2) + A( F_1) A( {\mathcal {A}} F_2) \big ) U_{{\mathcal {V}}_\alpha (t)} \eta _0 \big \rangle _{{\mathcal {F}}} . \end{aligned}$$

(116)

Comparing (112) and (113) with (115) and (116) we see that the pairs of reduced one-body density matrices $(\gamma _{\eta _\alpha (t)} , \alpha _{\eta _\alpha (t)})$ and $(\gamma _{U_{{{\mathcal {V}}}_\alpha (t)} \eta _0 } , \alpha _{U_{{{\mathcal {V}}}_\alpha (t)} \eta _0})$ solve the same differential equation. Since the solution to this equation is unique, see [23, Prop. 7], and since $\eta _\alpha (0) = U_{{{\mathcal {V}}}_\alpha (0)} \eta _0 = \eta _0$, we conclude their equality. This implies $\eta _\alpha (t) = U_{{{\mathcal {V}}}_\alpha (t)} \eta _0$ and hence proves the claimed identity. $\square$