Properties of almost-bosonic coherent states We first show the following rigorous properties of the Weyl operator which was introduced in (3.3):

Lemma 5.1

Let \(\eta ,\xi \in \bigoplus _l^(\mathcal _)\) and \(W_(\eta )e^e^(\eta )-\sigma c(\eta )}\) for all \(\sigma \in [0,1]\), then it holds

$$\begin W_(\eta )^c(\xi )W_(\eta )&=c(\xi )+\sigma \langle \xi ,\eta \rangle +\langle \xi ,\mathcal ^\rangle _,\\ W_(\eta )^c^(\xi )W_(\eta )&=c^(\xi )+\sigma \langle \eta ,\xi \rangle +\langle \mathcal ^,\xi \rangle _, \end$$

with \(\langle \cdot , \cdot \rangle _: \bigoplus _l^(\mathcal _) \times \bigoplus _l^(\mathcal _) \rightarrow \mathbb \) given by

$$\begin \langle \xi ,\mathcal ^\rangle _\sum _\sum __}\xi _(k)^\mathcal ^_(k) \end$$

(5.1)

and a \(\sigma \)-dependent error term \(\mathcal ^_(k)\int _^\text \tau \ e^\left( \sum _\eta _(l)\mathcal _(l,k)\right) e^\).

Remark 5.2

We will give an estimate for \(\mathcal \) to show that this term corresponds indeed to a small error. Note that it holds \(\mathcal _(k,l)=\mathcal _(l,k)^\) for all \(l,k\in \Gamma \) and \(\alpha \in \mathcal _\cap \mathcal _\) from Lemma A.3 and therefore

$$\begin \mathcal ^_(l)^\xi _(l)&=\int _^\text \tau \ e^\left( \sum _\mathcal _(l,k)\overline(k)}\xi _(l)\right) e^, \end$$

(5.2)

$$\begin \overline(l)}\mathcal ^_(l)&=\int _^\text \tau \ e^\left( \sum _\overline(l)}\eta _(k)\mathcal _(k,l)\right) e^. \end$$

(5.3)

For \(\xi =\eta \) the above equations coincide, i.e.,

$$\begin \langle \eta ,\mathcal ^\rangle _=\langle \mathcal ^,\eta \rangle _\end$$

(5.4)

from which it follows immediately that \(\big (c^(\eta )-c(\eta )\big )W_(\eta )=W_(\eta )\big (c^(\eta )-c(\eta )\big )\), i.e., \([B,W_(\eta )]=0\).

Proof

We observe that \(W_(\eta )= e^\) defines a strongly continuous one-parameter semigroup. Thus, we can define a derivative and make use of Duhamel’s formula of the form

$$\begin e^c_(l)e^ = c_(l)+ \int _^\text \tau \ e^[c_(l),B]e^. \end$$

(5.5)

The interested reader is referred to [10, 21] where definitions and properties of operator derivatives are discussed. The desired statement follows with the CCR as stated in (2.22)

$$\begin c_(l),e^]&=e^\int _^\text \tau \ e^[c_(l),B]e^\nonumber \\&=e^\int _^\text \tau \ e^\left( \eta _(l)+\sum _\eta _(k)\mathcal _(k,l)\right) e^\nonumber \\&=\sigma \eta _(l)e^+e^\mathcal ^_(l) \end$$

(5.6)

Since \(c(\xi )\equiv \sum _\sum __}\overline(k)}c_(k)\) is linear the result follows from the above identity. \(\square \)

The following statement shows that the number of particles in the state \(W(\eta )\phi \otimes \Omega \) corresponds to a random variable with expectation approximately being \(2\Vert \eta \Vert ^\).

Proposition 5.3

. Let \(\zeta =\phi \otimes \Omega \in L^(\Lambda ,\text y)\otimes \mathcal _^\), then it holds for all \(\eta \in \bigoplus _l^(\mathcal _)\)

$$\begin \langle W(\eta )\zeta ,\mathcal W(\eta )\zeta \rangle =2\Vert \eta \Vert ^+4\int _^\text \sigma \langle \zeta ,\langle \eta ,\mathcal ^\rangle _\zeta \rangle . \end$$

Proof

Using \(W^W=\text \) yields for all \(\zeta \in L^(\Lambda ,\text y)\otimes \mathcal _^\) with \(\Vert \zeta \Vert =1\)

$$\begin \langle W(\eta )\zeta ,\mathcal W(\eta )\zeta \rangle =\langle W(\eta )\zeta ,[\mathcal ,W(\eta )]\zeta \rangle +\langle \zeta ,\mathcal \zeta \rangle . \end$$

(5.7)

We use Duhamel’s formula to calculate

$$\begin \mathcal ,e^]&=e^\int _^\text \tau \ e^[\mathcal ,B]e^\nonumber \\&=2e^\int _^\text \tau \ e^\left( c^(\eta )+c(\eta )\right) e^\nonumber \\&=2e^\int _^\text \tau \ e^\left( B+2c(\eta )\right) e^\nonumber \\&=2e^B+4e^\int _^\text \tau \ e^c(\eta )e^ \end$$

(5.8)

where we used (2.24). Therefore,

$$\begin \langle W(\eta )\zeta ,[\mathcal ,W(\eta )]\zeta \rangle&=2\langle \zeta ,B\zeta \rangle +4\int _^\text \tau \ \langle e^\zeta ,c(\eta )e^\zeta \rangle \nonumber \\&=2\langle \zeta ,B\zeta \rangle +2\Vert \eta \Vert ^+4\langle \zeta ,c(\eta )\zeta \rangle +4\int _^\text \tau \langle \zeta ,\langle \eta ,\mathcal ^\rangle _\zeta \rangle \end$$

(5.9)

where we used the shift property Lemma 5.1 and that \(e^\) is unitary

$$\begin \langle e^\zeta ,c(\eta )e^\zeta \rangle&=\langle e^\zeta ,[c(\eta ),e^]\zeta \rangle +\langle \zeta ,c(\eta )\zeta \rangle \nonumber \\&=\tau \Vert \eta \Vert ^+\langle \zeta ,\langle \eta ,\mathcal ^\rangle _\zeta \rangle +\langle \zeta ,c(\eta )\zeta \rangle . \end$$

(5.10)

Inserting (5.9) and (5.8) into (5.7), we obtain

$$\begin \langle W(\eta )\zeta ,\mathcal W(\eta )\zeta \rangle= & 2\Vert \eta \Vert ^+2\langle \zeta ,\left( c^(\eta )+c(\eta )\right) \zeta \rangle +\langle \zeta ,\mathcal \zeta \rangle \nonumber \\ & +4\int _^\text \tau \langle \zeta ,\langle \eta ,\mathcal ^\rangle _\zeta \rangle . \end$$

(5.11)

The desired result holds for \(\zeta =\phi \otimes \Omega \) since \(c(\eta )\phi \otimes \Omega =0\). \(\square \)

For later purposes, we can bound the expectation of the number operator in the following way:

Proposition 5.4

. Let \(\eta \in \bigoplus _l^(\mathcal _)\). There exists a constant \(C>0\) such that it holds for all \(\tau \in [-1,1],n\in \mathbb \) and \(\zeta \in L^(\Lambda ,\text y)\otimes \mathcal _^\)

$$\begin \langle e^\zeta ,(\mathcal +1)^e^\zeta \rangle \le e^\langle \zeta ,(\mathcal +3)^\zeta \rangle . \end$$

Proof

The proof works analogously to the proof of Proposition 4.1 with a Grönwall argument and B instead of \(\mathbb ^}\) which is given later. Note that

$$\begin \mathcal ,B]=[\mathcal ,c^(\eta )-c(\eta )]=2\sum _\sum __}\eta _(k)\big (c_^(k)+c_(k)\big ) \end$$

(5.12)

where we again used (2.24). The result is than obtained by using the same estimates with \(\Vert \eta \Vert \) taking the role of \(\Vert h_\Vert \). \(\square \)

Lemma 5.5

Let \(\eta _\in \bigoplus _l^(\mathcal _)\) be differentiable in t with derivative \(\dot}\in \bigoplus _l^(\mathcal _)\) for all \(t\in \mathbb \). Then it holds for all \(t\in \mathbb \)

$$\begin \partial _W(\eta _)= & \left( c^(\dot_)-c(\dot_)+i\text \langle \dot_,\eta _\rangle \right) W(\eta _)\\ & +2i\int _^\text \tau \ W_(\eta _)\text \langle \dot_,\mathcal ^\rangle _W_(\eta _) \end$$

with the short-hand notation \(\text \langle A,B\rangle _-\frac\sum _\sum __}(A_^(k)B_(k)-B_^(k)A_(k))\).

Proof

For arbitrary \(s\in \mathbb \) it holds

$$\begin W(\eta _)^\partial _W(\eta _)&=\left. e^}\partial _e^}\right| _^=\int _^\text \tau \ \partial _\left( e^}\partial _e^}\right) \nonumber \\&=\int _^\text \tau \left( -B_e^}\partial _e^}+e^}\partial _\partial _e^}\right) \end$$

(5.13)

$$\begin&=\int _^\text \tau \left( -B_e^}\partial _e^}+e^}\partial _\left( B_e^}\right) \right) \end$$

(5.14)

$$\begin&=\int _^\text \tau \left( -B_e^}\partial _e^}+e^}\left( \partial _B_\right) e^}+e^}B_\partial _e^}\right) \end$$

(5.15)

$$\begin&=\int _^\text \tau \ e^}\left( \partial _B_\right) e^}. \end$$

(5.16)

Thus,

$$\begin \partial _W(\eta _)&=\int _^\text \tau \ e^}\left( \partial _B_\right) e^}\nonumber \\&=\int _^\text \tau \ \left( \partial _B_\right) e^}e^}+\int _^\text \tau \ \left[ e^},\partial _B_\right] e^}\end$$

(5.17)

$$\begin&=\left( \partial _B_\right) e^}+\int _^\text \tau \ \left[ e^},\partial _B_\right] e^}. \end$$

(5.18)

With

$$\begin \partial _B_=\partial _\left\(\eta _)-c(\eta _)\right\} =c^(\dot_)-c(\dot_) \end$$

(5.19)

from the linearity of \(c(\eta _)\), it follows

$$\begin&\left[ e^},\partial _B_\right] e^}\nonumber \\&=\left( [W_(\eta _),c^(\dot_)]-[W_(\eta _),c(\dot_)]\right) W_(\eta _)\end$$

(5.20)

$$\begin&=W_(\eta _)\bigg ((1-\tau )\langle \dot_,\eta _\rangle -(1-\tau )\langle \eta _,\dot_\rangle \end$$

(5.21)

$$\begin&+\langle \dot_,\mathcal ^\rangle _-\langle \mathcal ^,\dot_\rangle _\bigg ) W_(\eta _)\end$$

(5.22)

$$\begin&=W(\eta _)(1-\tau )2i\text \langle \dot_,\eta _\rangle +2iW_(\eta _)\text \langle \dot_,\mathcal ^\rangle _W_(\eta _). \end$$

(5.23)

Inserting the above identity yields

$$\begin \partial _W(\eta _)= & \left( c^(\dot_)-c(\dot_)+i\text \langle \dot_,\eta _\rangle \right) W(\eta _)\nonumber \\ & +2i\int _^\text \tau \ W_(\eta _)\text \langle \dot_,\mathcal ^\rangle _W_(\eta _). \end$$

(5.24)

\(\square \)

Proof of the main theorem First, we collect some useful observations on the function \(\eta _\) in the form of the following two lemmata. We postpone the proofs to the end of the section in order to concentrate on presenting the proof of the main result.

Lemma 5.6

Let \(\eta _\) be defined as in (3.15) for \(N^\ll M\ll N^-2\delta }\), then it holds for all \(s\in \mathbb \)

$$\begin \Vert \eta _\Vert ^&=\pi \lambda ^\sum _\frac(k)^}\left( \log (2k_}|k|s)-\text (2k_}|k|s)+\gamma \right) \\&\quad \times \left\}|k|s)\ \mathcal \left( M^}N^+\delta }+N^\right) \right\} . \end$$

where \(\gamma \) is the Euler–Mascheroni constant and \(g:\mathbb \rightarrow \mathbb _\) is a function independent of \(k_}\) and monotonically increasing.

Furthermore, define \(f:\mathbb \times \mathbb \rightarrow \mathbb \) by

$$\begin (y,x)\mapsto f_(x)\min \\Vert \hat(\cdot )^\Vert _ yx}, e^\Vert \hat\Vert _ (\log (18)+\frac) y} e^}\Vert \hat\Vert _ yx}\}. \end$$

Then there exists a \(C>0\) independent of \(k_}\) such that for \(c_>0\) and \(k_}\) sufficiently large

$$\begin \Vert \eta _\Vert&\le \log \left( f_( \lambda k_}s)\right) , \end$$

(5.25)

$$\begin e^\Vert \eta _\Vert }&\le f_ }( \lambda k_}s). \end$$

(5.26)

Remark 5.7

Note that \(f_\) is for all \(y\ge 0\) monotonically increasing with \(f_(0)=1\).

The previous statement is useful when combined with the following estimate:

Lemma 5.8

There exists a constant \(C>0\) only depending on V such that it holds for all \(s\in \mathbb \), \(n\in \mathbb \), \(\psi \in \mathcal \)

(i)

\(\sum _\Vert |k|^\eta _(k)\Vert _}\le C\Vert \eta _\Vert \),

(ii)

\(\langle \eta _,|k|^\eta _\rangle \le C\Vert \eta _\Vert ^\),

(iii)

\(\Vert c^(|k|^\eta _)\psi \Vert \le C\Vert \eta _\Vert \ \Vert (\mathcal +1)^\psi \Vert \).

We will now give the proof of the second main theorem.

Proof of Theorem 3.6

We use the approach as sketched in Remark 3.9. Since the bosonic property holds only with an error, the equality (3.21) holds only approximately:

$$\begin&\Vert e^^}t}\psi -e^W(\eta _)\psi \Vert \nonumber \\&\quad =\Vert \psi -e^^}t}e^^}t}e^(\nu _)}e^\int _^\text s\langle \dot_,\eta _\rangle }W(\eta _)\psi \Vert \nonumber \\&\quad \le \int _^\text s\Vert \big (\mathbb ^}-E_^}+2\text (\dot_)-\text \langle \dot_,\eta _\rangle \big )W(\eta _)\psi -i\partial _W(\eta _)\psi \Vert \nonumber \\&\quad \le \int _^\text s\Vert h_W(\eta _)\phi \otimes \Omega \Vert +\Vert \text _\Vert +\Vert \text _\Vert . \end$$

(5.27)

We will first estimate the error terms and then subsequently treat the \(h_\) term in a separate lemma. We give an explicit expression for the first error term using Lemma 5.1 on the approximate shift property applied to \(c_(k)\) in the \(c^c(\epsilon ),c(h_)\) and \(c(i\dot_)\) terms. Thus, the error of (3.21) is given by

$$\begin \text _\int _^\text s\sum _\sum __}\left( \epsilon _(k)c_^(k)+(1-e^(k)})\overline)_(k)}\right) W(\eta _)\mathcal ^_(k)\psi .\nonumber \\ \end$$

(5.28)

The second error term is given by Lemma 5.5 on the time derivative of the almost-bosonic Weyl operator and therefore

$$\begin \text _&- 2i\int _^\text s\int _^\text \tau \ W_(\eta _)\text \langle \dot_,\mathcal ^\rangle _W_(\eta _)\psi \nonumber \\&\equiv 2\int _^\text s\int _^\text \tau e^\bigg (\sum _\sum __}e^(k)}\overline)_(k)}\mathcal ^_(k)\nonumber \\&\quad -\sum _\sum __}\mathcal ^_(k)^e^(k)}(h_)_(k)\bigg ) e^\psi . \end$$

(5.29)

Firstly, we show that the term \(\mathcal ^_(k)\psi =\int _^\text \tau \ e^\left( \sum _\eta _(l)\mathcal _(l,k)\right) e^\psi \) as defined in Lemma 5.1 constitutes indeed a small error. We estimate

$$\begin&\sum _\sum __}\Vert \mathcal ^_(l)\psi \Vert ^\nonumber \\&\quad \le \sum _\sum __\cap \mathcal _}\bigg (\sum _|\eta _(k)|\int _^\text \tau \Vert \mathcal _(k,l)e^\psi \Vert \bigg )^\nonumber \\&\quad \le C\left( MN^+\delta }(e^\Vert }-1)\Vert (\mathcal +3)\psi \Vert \right) ^ \end$$

(5.30)

where we used \(\sum __}\bigg (\sum _|\eta _(k)|\bigg )^\le \sum _\Vert \eta (k)\Vert _}\Vert \eta (k')\Vert _}\le C\Vert \eta \Vert ^\) by Lemma 5.6 and

$$\begin&\int _^\text \tau \Vert \mathcal _(k,l)e^\psi \Vert \end$$

(5.31)

$$\begin&\le \int _^\text \tau \langle e^\psi ,|\mathcal _(k,l)|^e^\psi \rangle ^\le CMN^+\delta }\int _^\text \tau \langle e^\psi ,\mathcal ^e^\psi \rangle ^ \end$$

(5.32)

$$\begin&\le CMN^+\delta }\int _^\text \tau e^\Vert \tau }\Vert (\mathcal +3)\psi \Vert \le C\Vert \eta _\Vert ^(e^\Vert }-1)MN^+\delta }\Vert (\mathcal +3)\psi \Vert \end$$

(5.33)

which follows from \(e^\) is unitary in the first inequality, Lemma A.3 in the second inequality and Proposition 5.4 in the third inequality.

Secondly, we estimate

$$\begin&\sum _\sum __}\Vert c_^(k)e^\mathcal ^_(k)\psi \Vert \nonumber \\&\quad \le \sum _\sum __}\int _^\text \tau \Vert c_^(k)e^\left( \sum _\eta _(l)\mathcal _(l,k)\right) e^\psi \Vert \nonumber \\&\quad \le \int _^\text \tau \sum _\sum __\cap \mathcal _}|\eta _(l)|\ \Vert c_^(k)\mathcal _(l,k)e^\psi \Vert \nonumber \\&\qquad +\int _^\text \tau \sum _\sum __\cap \mathcal _}|\eta _(l)|\ \Vert [c_^(k),e^]\mathcal _(l,k)e^\psi \Vert \nonumber \\&\quad \le C\Vert \eta _\Vert MN^+\delta }\int _^\text \tau \bigg (\Vert (\mathcal +1)^}e^\psi \Vert ^\bigg )^\nonumber \\&\qquad +C\Vert \eta _\Vert ^MN^+\delta }\int _^\text \tau \ (1-\tau )\Vert \mathcal e^\psi \Vert \nonumber \\&\qquad +\sum _\Vert \eta _\Vert \int _^\text \tau \bigg (\sum _\sum __\cap \mathcal _}\Vert \mathcal ^_(k)\mathcal _(l,k)e^\psi \Vert ^\bigg )^\nonumber \\&\quad \le CMN^+\delta }(e^\Vert }-1)\Vert (\mathcal +3)^}\psi \Vert \nonumber \\&\qquad +CMN^+\delta }(e^\Vert }-C\Vert \eta _\Vert -1)\Vert (\mathcal +3)\psi \Vert \nonumber \\&\qquad +CM^}N^+\delta }(e^\Vert }-1)^\Vert (\mathcal +3)^\psi \Vert \nonumber \\&\quad \le CM^}N^+\delta }(e^\Vert }-1)\Vert (\mathcal +3)^\psi \Vert . \end$$

(5.34)

where we used \([c_^(k),e^]=-\lambda \eta _(k)e^-e^\mathcal ^_(k)\) from Lemma 5.1, the Cauchy–Schwarz inequality for the \(\alpha \)-summation, Lemma A.3 in the third inequality and in the fourth inequality we used Proposition 5.4 and (5.30).

In total by combining (5.30) and (5.34), we end up with the following estimate

$$\begin \Vert \text _\Vert&\le \int _^\text s\sum _\sum __}\Vert \left\(k)c_^(k)+(1-e^(k)})\overline)_(k)}\right\} e^\mathcal ^_(k)\psi \Vert \nonumber \\&\le Ck_}\int _^\text s\sum _\sum __}\Vert c_^(k)e^\mathcal ^_(k)\psi \Vert \nonumber \\&\quad +C\lambda \int _^\text s\sum _|\hat(k)|\sum __}\Vert n_(k)e^\mathcal ^_(k)\psi \Vert \nonumber \\&\le Ck_}\int _^\text s\sum _\sum __}\Vert c_^(k)e^\mathcal ^_(k)\psi \Vert \nonumber \\&\quad +C\lambda k_}\int _^\text s\Vert \hat\Vert _\bigg (\sum _\sum __}\Vert \mathcal ^_(k)\psi \Vert ^\bigg )^\nonumber \\&\le Ck_}MN^+\delta }\int _^\text s\ (e^\Vert }-1)\Vert (\mathcal +3)^\psi \Vert \nonumber \\&\quad +C\lambda k_}MN^+\delta }\int _^\text s\ (e^\Vert }-1)\Vert (\mathcal +3)\psi \Vert \nonumber \\&\le C\ (f_( \lambda k_}t)-1)(\lambda +1)k_}tMN^+\delta }\Vert (\mathcal +3)^\psi \Vert . \end$$

(5.35)

where we used (A.2) and \(e^\) unitary in the third inequality and Lemma 5.6 in the last line. Using \(\psi =\phi \otimes \Omega \) we obtain the desired bound.

Similarly, we obtain an estimate for the second error term using Cauchy–Schwarz, (5.30) and Proposition 5.4

$$\begin&\Vert \text _\Vert \nonumber \\&\quad \le 2\int _^\text s\int _^\text \tau \sum _\sum __}\bigg (\Vert \overline)_(k)}\mathcal ^_(k)e^\psi \Vert +\Vert \mathcal ^_(k)^(h_)_(k)e^\psi \Vert \bigg )\nonumber \\&\quad \le 2\lambda \int _^\text s\int _^\text \tau \sum _\sum __}|\hat(k)n_(k)|\bigg (\Vert \mathcal ^_(k)e^\psi \Vert +\Vert \mathcal ^_(k)^e^\psi \Vert \bigg )\nonumber \\&\quad \le C\int _^\text s\int _^\text \tau \ \lambda k_}\Vert \hat\Vert _\bigg (\bigg (\sum _\sum __}\Vert \mathcal ^_(k)e^\psi \Vert ^\bigg )^+\bigg (\sum _\sum __}\Vert \mathcal ^_(k)^e^\psi \Vert ^\bigg )^\bigg )\nonumber \\&\quad \le C\lambda k_}MN^+\delta }\int _^\text s\int _^\text \tau \ (e^\Vert (1-\tau )}-1)\Vert (\mathcal +3)e^\psi \Vert \nonumber \\&\quad \le C\lambda k_}MN^+\delta }\int _^\text s\int _^\text \tau \ (e^\Vert }-e^\Vert \tau })\Vert (\mathcal +5)\psi \Vert \nonumber \\&\quad \le C(f_( \lambda k_}t)-1)\lambda k_}tMN^+\delta }\Vert (\mathcal +5)\psi \Vert . \end$$

(5.36)

Again, by using \(\psi =\phi \otimes \Omega \) we obtain the desired bound.

With the subsequent Lemma 5.9, we can conclude with a bound on \(h_=-\beta \Delta _y \) of the form

$$\begin&\int _^\text s\Vert h_W(\eta _)\phi \otimes \Omega \Vert \le C\beta \int _^\text s\left\\Vert +\Vert \eta _\Vert ^)(e^\Vert }+1)\right\} \nonumber \\&\quad \le C\beta t\left\( \lambda k_}t)\right] +\log \left[ f_( \lambda k_}t)\right] ^\right\} \left\(\lambda k_}t)+1\right\} \end$$

(5.37)

where we used (5.25) and (5.26) in the second inequality. Together with (5.35) and (5.36) inserted in (5.27), we obtain the desired result. \(\square \)

Lemma 5.9

Under the assumptions of Theorem 3.6, it holds that for all \(t\ge 0\)

$$\begin \Vert \Delta _W(\eta _)\phi \otimes \Omega \Vert \le C(\Vert \eta _\Vert +\Vert \eta _\Vert ^)(e^\Vert }+1) \end$$

for \(C>0\) independent of \(k_}\).

Proof of Lemma 5.9

We explicitly calculate the action of the Laplacian on the coupled coherent state \(W(\eta _)\psi =e^\psi \), i.e.,

$$\begin -\Delta _W(\eta _)\psi =-\big (\Delta _W(\eta _)\big )\psi -2\beta \nabla _W(\eta _)\cdot \nabla _\psi -W(\eta _)\Delta _\psi .\nonumber \\ \end$$

(5.38)

In total we expect, that all terms can be bounded by assumption on the initial condition on \(\phi \). We will first focus on the term \(\Delta _W(\eta _)\). Recall that it holds

$$\begin W(\eta _)^\partial _}W(\eta _) =\int _^\text \tau \ e^\left( \partial _}B_\right) e^ \end$$

(5.39)

due to the same calculation as in the proof of Lemma 5.5. With

$$\begin \partial _}B_&= \partial _}\left\(\eta _)-c(\eta _)\right\} =c^(\partial _}\eta _)-c(\partial _}\eta _), \end$$

(5.40)

$$\begin \partial _}\eta _&= \frac(k)}-1}(k)}\lambda \hat(k)n_(k)ik_e^=ik_\eta _ \end$$

(5.41)

it follows analogously to Lemma 5.5 that

$$\begin \partial _}W(\eta _)&=\int _^\text \tau \ e^}\left( \partial _}B_\right) e^}\nonumber \\&=\left( c^(ik_\eta _)-c(ik_\eta _)+i\text \langle \eta _,ik_\eta _\rangle \right) W(\eta _)\nonumber \\&\quad +2i\int _^\text \tau \ W_(\eta _)\text \langle ik_\eta _,\mathcal ^\rangle _W_(\eta _). \end$$

(5.42)

And repeating the differentiation with

$$\begin \partial _}W_(\eta _)&=\tau \left( c^(ik_\eta _)-c(ik_\eta _)+i\tau \text \langle \eta _,ik_\eta _\rangle \right) W_(\eta _)\nonumber \\&+2i\int _^\text \sigma \ W_(\eta _)\text \langle ik_\eta _,\mathcal ^\rangle _W_(\eta _) \end$$

(5.43)

yields

$$\begin&\Delta W(\eta _)=\sum _^\partial _}^W(\eta _)\nonumber \\ =&\left( c^(k^\eta _)-c(k^\eta _)+i\text \langle \eta _,k^\eta _\rangle \right) W(\eta _)+2i\int _^\text \tau \ W_(\eta _)\text \langle k^\eta _,\mathcal ^\rangle _W_(\eta _)\nonumber \\&+\sum _^\left( c^(ik_\eta _)-c(ik_\eta _)+i\text \langle \eta _,ik_\eta _\rangle \right) \partial _}W(\eta _)\nonumber \\&+2i\sum _^\int _^\text \tau \ \text \langle ik_\eta _,\partial _}\left\(\eta _)\mathcal ^_(k)W_(\eta _)\right\} \rangle _\end$$

(5.44)

$$\begin =&\left( c^(k^\eta _)-c(k^\eta _)+i\text \langle \eta _,k^\eta _\rangle \right) W(\eta _)+2i\int _^\text \tau \ \text \langle k^\eta _,\mathcal ^\rangle _W_(\eta _)\nonumber \\&+\sum _^\left( c^(ik_\eta _)-c(ik_\eta _)+i\text \langle \eta _,ik_\eta _\rangle \right) \times \nonumber \\&\times \left\(ik_\eta _)-c(ik_\eta _)+i\text \langle \eta _,ik_\eta _\rangle \right) W(\eta _)+2i\int _^\text \tau \ \text \langle ik_\eta _,\mathcal ^\rangle _W_(\eta _)\right\} \nonumber \\&+2i\sum _^\int _^\text \tau \ \text \langle ik_\eta _,\partial _}\left\(\eta _)\mathcal ^_(k)W_(\eta _)\right\} \rangle _\nonumber \\ &I_+I_+I_+I_ \end$$

(5.45)

with

$$\begin I_&\left( c^(k^\eta _)-c(k^\eta _)+i\text \langle \eta _,k^\eta _\rangle \right) W(\eta _)\nonumber \\&\quad -\sum _^\left( c^(ik_\eta _)-c(ik_\eta _)+i\text \langle \eta _,ik_\eta _\rangle \right) \nonumber \\&\quad \times \left( c^(ik_\eta _)-c(ik_\eta _)+i\text \langle \eta _,ik_\eta _\rangle \right) W(\eta _), \end$$

(5.46)

$$\begin I_&2i\int _^\text \tau \ W_(\eta _)\text \langle k^\eta _,\mathcal ^\rangle _W_(\eta _), \end$$

(5.47)

$$\begin I_&2\sum _^\left( c^(ik_\eta _)-c(ik_\eta _)+i\text \langle \eta _,ik_\eta _\rangle \right) \nonumber \\&\quad \int _^\text \tau \ iW_(\eta _)\text \langle ik_\eta _,\mathcal ^\rangle _W_(\eta _), \end$$

(5.48)

$$\begin I_&2i\sum _^\int _^\text \tau \ \text \langle ik_\eta _,\partial _}\left\(\eta _)\mathcal ^_(k)W_(\eta _)\right\} \rangle _. \end$$

(5.49)

We show that each term can be bounded here by a constant at most of order 1.

For \(I_\), we can treat all \(c^(\cdots )\) and \(c(\cdots )\) terms with Lemma 5.6 and Lemma 5.8. Furthermore, we use that \(c_(k)\mathcal =(\mathcal +2)c_(k)\) to estimate

$$\begin \Vert I_\psi \Vert&\le C\Vert \eta _\Vert \ \Vert (\mathcal +1)^W(\eta _)\psi \Vert +C(\Vert \eta _\Vert ^+\Vert \eta _\Vert ^)\Vert W(\eta _)\psi \Vert \nonumber \\&\quad +C\Vert \eta _\Vert ^\Vert (\mathcal +3)W(\eta _)\psi \Vert \nonumber \\&\le C(\Vert \eta _\Vert +\Vert \eta _\Vert ^)e^\Vert }\Vert (\mathcal +5)\psi \Vert +C(\Vert \eta _\Vert ^+\Vert \eta _\Vert ^) \end$$

(5.50)

where we used Proposition 5.4.

For \(I_\), we use a similar approach to (5.30) to obtain

$$\begin \Vert I_\psi \Vert&\le C\Vert \eta _\Vert MN^+\delta }\int _^\text \tau \ (e^\Vert (1-\tau )}-1)\Vert (\mathcal +3)W_(\eta _)\psi \Vert \nonumber \\&\le C\Vert \eta _\Vert MN^+\delta }\int _^\text \tau \ (e^\Vert }e^-C)\Vert \eta _\Vert \tau }-e^\Vert \eta _\Vert \tau })\Vert (\mathcal +5)\psi \Vert \nonumber \\&\le CMN^+\delta }e^\Vert }(e^\Vert }-1)\Vert (\mathcal +5)\psi \Vert . \end$$

(5.51)

For \(I_\), we first observe that for \(n\in \mathbb \) and \(\mu \in [0,1]\)

$$\begin \Vert \mathcal ^\mathcal ^_(k)\psi \Vert&\le \int _^\text \tau \ \Vert \mathcal ^e^\left( \sum _\eta _(l)\mathcal _(l,k)\right) e^\psi \Vert \nonumber \\&\le \int _^\text \tau \ e^\Vert \tau }\sum _|\eta _(l)|\ \Vert \mathcal _(l,k)(\mathcal +3)^e^\psi \Vert \nonumber \\&\le C\int _^\text \tau \ e^\Vert \tau }\sum _|\eta _(l)|\ MN^+\delta }\Vert \mathcal (\mathcal +3)^e^\psi \Vert \nonumber \\&\le C\Vert \eta _\Vert ^(e^\Vert \mu }-1)MN^+\delta }\sum _|\eta _(l)|\ \Vert (\mathcal +5)^\psi \Vert . \end$$

(5.52)

We use a similar approach to (5.34) and insert the above inequality (5.52) to obtain

$$\begin \Vert I_\psi \Vert&\le C\Vert \eta _\Vert \sum _\sum __}\int _^\text \tau \ \Vert \left( \Vert \eta _\Vert (\mathcal +1)^+C\Vert \eta _\Vert ^\right) W_(\eta _)\mathcal ^_(k)W_(\eta _)\psi \Vert \nonumber \\&\le C\Vert \eta _\Vert ^\sum _\sum __}\int _^\text \tau \ \Vert \mathcal W_(\eta _)\mathcal ^_(k)W_(\eta _)\psi \Vert +C\Vert \eta _\Vert ^\Vert I_\psi \Vert \nonumber \\&\le C\Vert \eta _\Vert ^\sum _\sum __}\int _^\text \tau \ e^\Vert (1-\tau )}\Vert \mathcal \mathcal ^_(k)W_(\eta _)\psi \Vert +C\Vert \eta _\Vert ^\Vert I_\psi \Vert \nonumber \\&\le C\Vert \eta _\Vert ^MN^+\delta }\int _^\text \tau \ e^\Vert }(e^\Vert (1-\tau )}-1)\Vert (\mathcal +5)^\psi \Vert +C\Vert \eta _\Vert ^\Vert I_\psi \Vert \nonumber \\&\le C\Vert \eta _\Vert MN^+\delta }e^\Vert }(e^\Vert }-1)\Vert (\mathcal +5)^\psi \Vert \nonumber \\&\quad +CMN^+\delta }\Vert \eta _\Vert ^e^\Vert }(e^\Vert }-1)\Vert (\mathcal +5)\psi \Vert \nonumber \\&\le CMN^+\delta }(\Vert \eta _\Vert +\Vert \eta _\Vert ^)e^\Vert }(e^\Vert }-1)\Vert (\mathcal +5)^\psi \Vert . \end$$

(5.53)

For \(I_\), we first calculate

$$\begin&\partial _}\left\(\eta _)\mathcal ^_(k)W_(\eta _)\right\} \nonumber \\&\quad =\int _^\text \sigma \ \partial _}e^\left( \sum _\eta _(l)\mathcal _(l,k)\right) e^\nonumber \\&\qquad +\int _^\text \sigma \ e^\left( \sum _ik_\eta _(l)\mathcal _(l,k)\right) e^\nonumber \\&\qquad +\int _^\text \sigma \ e^\left( \sum _\eta _(l)\mathcal _(l,k)\right) \partial _}e^ \nonumber \\&I_+I_+I_+I_+I_ \end$$

(5.54)

with

$$\begin I_&=\int _^\text \sigma \ (1-\sigma )\left( c^(ik_\eta _)-c(ik_\eta _)+i(1-\sigma )\text \langle \eta _,ik_\eta _\rangle \right) \nonumber \\&\quad \times e^\left( \sum _\eta _(l)\mathcal _(l,k)\right) e^, \end$$

(5.55)

$$\begin I_&=2i\int _^\text \sigma \int _^\text a\ e^\text \langle ik_\eta _,\mathcal ^\rangle _e^\left( \sum _\eta _(l)\mathcal _(l,k)\right) e^, \end$$

(5.56)

$$\begin I_&=\int _^\text \sigma \ e^\left( \sum _ik_\eta _(l)\mathcal _(l,k)\right) e^, \end$$

(5.57)

$$\begin I_&=\int _^\text \sigma \ e^\left( \sum _\eta _(l)\mathcal _(l,k)\right) \sigma \left( c^(ik_\eta _)-c(ik_\eta _)+i\sigma \text \langle \eta _,ik_\eta _\rangle \right) e^, \end$$

(5.58)

$$\begin I_&=2i\int _^\text \sigma \ e^\left( \sum _\eta _(l)\mathcal _(l,k)\right) \int _^\text a\ e^\text \langle ik_\eta _,\mathcal ^\rangle _e^. \end$$

(5.59)

We approach each term similarly to (5.30).

For \(I_\), it holds

$$\begin&\Vert I_\psi \Vert \nonumber \\ &\quad \le C\sum _|\eta _(l)|\int _^\text\sigma \ (1-\sigma ) \nonumber \\ &\qquad \left( \Vert \eta _\Vert \ \Vert (\mathcal +1)e^\mathcal _(l,k)e^\psi \Vert +C\Vert \eta _\Vert ^\Vert \mathcal _(l,k)e^\psi \Vert \right) \nonumber \\ &\quad \le C\Vert \eta _\Vert \sum _|\eta _(l)|\int _^\text\sigma \ (1-\sigma ) \nonumber \\ &\qquad \left( e^\Vert (1-\sigma )}\Vert (\mathcal +3)\mathcal _(l,k)e^\psi \Vert +C\Vert \eta _\Vert \,\Vert \mathcal _(l,k)e^\psi \Vert \right) \nonumber \\ &\quad \le C\Vert \eta _\Vert MN^+\delta }\sum _|\eta _(l)|\int _^\text\sigma \ \left( (1-\sigma )e^\Vert (1-\sigma )}+C\Vert \eta _\Vert \right) \Vert (\mathcal +3)^e^\psi \Vert \nonumber \\ &\quad \le C\Vert \eta _\Vert MN^+\delta }\sum _|\eta _(l)|\int _^\text\sigma \ \left( (1-\sigma )e^\Vert (1-\sigma )}+C\Vert \eta _\Vert \right) e^\Vert \eta _\Vert \sigma }\Vert (\mathcal +5)^\psi \Vert \end$$

(5.60)

$$\begin&\quad \le CMN^+\delta }\sum _|\eta _(l)|\left( (1-\tau )e^\Vert (1-\tau )}+C\Vert \eta _\Vert e^\Vert }\right) \Vert (\mathcal +5)^\psi \Vert \end$$

(5.61)

where we used Lemma A.4, Lemma 5.6 and Lemma 5.8 in the first inequality, Lemma A.3 in the third inequality and Proposition 5.4 in the second and forth inequality. Therefore, using \(\int _^(1-\tau )e^\text \tau =y^\left( (y-1)e^+1\right) \) yields

$$\begin&\int _^\text \tau \ \Vert \eta _\Vert \sqrt\sum __\cap \mathcal _}\Vert I_\psi \Vert ^} \end$$

(5.62)

$$\begin&\le CMN^+\delta }\left( (\Vert \eta _\Vert ^+\Vert \eta _\Vert -1)e^\Vert }+1\right) \Vert (\mathcal +5)^\psi \Vert \end$$

(5.63)

where we used \(\sum __}\bigg (\sum _|\eta _(l)|\bigg )^\le \sum _\Vert \eta (l)\Vert _}\Vert \eta (l')\Vert _}\le C\Vert \eta \Vert ^\) by Lemma 5.6 and Lemma 5.8.

For \(I_\), it holds

$$\begin&\Vert I_\psi \Vert \nonumber \\&\le C\Vert \eta _\Vert MN^+\delta }\sum _|\eta _(l)|\int _^\text \sigma \int _^\text a\ \Vert (\mathcal +3)e^\mathcal _(l,k)e^\psi \Vert \nonumber \\&\le C\Vert \eta _\Vert \left( MN^+\delta }\right) ^\sum _|\eta _(l)|\int _^\text \sigma \int _^\text a\ e^\Vert a}\Vert (\mathcal +5)^e^\psi \Vert \nonumber \\&\le C\left( MN^+\delta }\right) ^\sum _|\eta _(l)|\int _^\text \sigma (e^\Vert (1-\sigma )}-1)e^\Vert \eta _\Vert \sigma }\Vert (\mathcal +8)^\psi \Vert \nonumber \\&\le C\Vert \eta _\Vert ^\left( MN^+\delta }\right) ^\sum _|\eta _(l)|e^\Vert (1-\tau )}\Vert (\mathcal +8)^\psi \Vert \end$$

(5.64)

where we used Cauchy–Schwarz with (5.30), Lemma 5.6 and Lemma 5.8 in the first inequality, Proposition 5.4 and Lemma A.3 in the second inequality. Therefore, it follows

$$\begin & \int _^\text \tau \ \Vert \eta _\Vert \sqrt\sum __\cap \mathcal _}\Vert I_\psi \Vert ^}\\ & \quad \le CM^N^+2\delta }(e^\Vert }-1)\Vert (\mathcal +8)^\psi \Vert . \end$$

The term \(I_\) is estimated similarly to (5.30) by

$$\begin \Vert I_\psi \Vert \le C\Vert \eta _\Vert ^MN^+\delta }\sum _|\eta _(l)|(e^\Vert }-e^\Vert \tau })\Vert (\mathcal +3)\psi \Vert \end$$

(5.65)

and therefore

$$\begin&\int _^\text \tau \ \Vert \eta _\Vert \sqrt\sum __\cap \mathcal _}\Vert I_\psi \Vert ^}\le CMN^+\delta }\Vert (\mathcal +3)\psi \Vert . \end$$

(5.66)

Similarly for \(I_\), we estimate

$$\begin&\Vert I_\psi \Vert \nonumber \\&\le CMN^+\delta }\sum _|\eta _(l)|\int _^\text \sigma \ \sigma \Vert \mathcal \left( c^(ik_\eta _)-c(ik_\eta _)+i\sigma \text \langle \eta _,ik_\eta _\rangle \right) e^\psi \Vert \nonumber \\&\le C(\Vert \eta _\Vert +\Vert \eta _\Vert ^)MN^+\delta }\sum _|\eta _(l)|\int _^\text \sigma \ \sigma \Vert (\mathcal +1)^e^\psi \Vert \nonumber \\&\le C(\Vert \eta _\Vert ^+1)MN^+\delta }\sum _|\eta _(l)|\ \left( e^\Vert }+(1-C\Vert \eta _\Vert \tau )e^\Vert \tau }\right) \Vert (\mathcal +3)^\psi \Vert \nonumber \\&\le C(\Vert \eta _\Vert ^+1)MN^+\delta }\sum _|\eta _(l)|\ e^\Vert }\Vert (\mathcal +3)^\psi \Vert \end$$

(5.67)

using \(\int _^\text \sigma \ \sigma e^=y^\left( (y-1)e^-(\tau y-1)e^\right) \) and therefore

$$\begin&\int _^\text \tau \ \Vert \eta _\Vert \sqrt\sum __\cap \mathcal _}\Vert I_\psi \Vert ^}\nonumber \\&\quad \le CMN^+\delta }(\Vert \eta _\Vert +\Vert \eta _\Vert ^)e^\Vert }\Vert (\mathcal +1)^\psi \Vert . \end$$

(5.68)

For \(I_\), we estimate

$$\begin \Vert I_\psi \Vert&=\Vert 2i\int _^\text \sigma \ e^\left( \sum _\eta _(l)\mathcal _(l,k)\right) \int _^\text a\ e^\text \langle ik_\eta _,\mathcal ^\rangle _e^\psi \Vert \nonumber \\&\le CMN^+\delta }\sum _|\eta _(l)|\int _^\text \sigma \int _^\text a\ \Vert \mathcal e^\text \langle ik_\eta _,\mathcal ^\rangle _e^\psi \Vert \nonumber \\&\le C\Vert \eta _\Vert MN^+\delta }\sum _|\eta _(l)|\int _^\text \sigma \int _^\text a\ e^\Vert (1-a)}\nonumber \\ &\quad \bigg (\sum __}\Vert (\mathcal +3)\mathcal ^_(k)e^\psi \Vert ^\bigg )^\nonumber \\&\le C\Vert \eta _\Vert M^N^+2\delta }\sum _|\eta _(l)|\int _^\text \sigma \int _^\text a\ e^\Vert (1-a)}\nonumber \\ &\quad (e^\Vert (\sigma -a)}-1)\Vert (\mathcal +8)^e^\psi \Vert \nonumber \\&\le C\Vert \eta _\Vert M^N^+2\delta }\sum _|\eta _(l)|\int _^\text \sigma \int _^\text a\ e^\Vert }e^\Vert \sigma }e^\Vert a}\nonumber \\ &\quad \Vert (\mathcal +10)^\psi \Vert \nonumber \\&\le C\Vert \eta _\Vert ^M^N^+2\delta }\sum _|\eta _(l)|\ (e^\Vert }-e^\Vert \tau })\Vert (\mathcal +10)^\psi \Vert \end$$

(5.69)

where we used (5.52) in the third inequality. Therefore, we obtain

$$\begin \int _^\text \tau \ \Vert \eta _\Vert \sqrt\sum __\cap \mathcal _}\Vert I_\psi \Vert ^}\le CM^N^+2\delta }\Vert (\mathcal +10)^\psi \Vert . \end$$

(5.70)

Taking the five bounds together, we finally obtain

$$\begin \Vert I_\psi \Vert&=\Vert 2i\sum _^\int _^\text \tau \ \text \langle ik_\eta _,\partial _}\left\(\eta _)\mathcal ^_(k)W_(\eta _)\right\} \rangle _\psi \Vert \nonumber \\&\le C\int _^\text \tau \sum _^\Vert \eta _\Vert \bigg (\sum _\sum __\cap \mathcal _}\Vert I_\psi \Vert ^\bigg )^\nonumber \\&\le CMN^+\delta }\left( (\Vert \eta _\Vert ^+\Vert \eta _\Vert -1)e^\Vert }+1\right) \Vert (\mathcal +8)^\psi \Vert . \end$$

(5.71)

Combining (5.50), (5.51), (5.53) and (5.71) with \(\psi \equiv \phi \otimes \Omega \) yields the desired final result of

$$\begin \Vert \big (\Delta _W(\eta _)\big )\psi \Vert \le C(\Vert \eta _\Vert +\Vert \eta _\Vert ^)(e^\Vert }+1). \end$$

(5.72)

Similarly to (5.50) and (5.51), we further estimate the second term of (5.38)

$$\begin&\Vert \nabla _W(\eta _)\cdot \nabla _\psi \Vert \nonumber \\&\quad = \Vert \sum _^\Big (\left( c^(ik_\eta _)-c(ik_\eta _)+i\text \langle \eta _,ik_\eta _\rangle \right) W(\eta _)\nonumber \\&\qquad +2i\int _^\text \tau \ \text \langle ik_\eta _,\mathcal ^\rangle _W_(\eta _)\Big )\partial _}\psi \Vert \nonumber \\&\quad \le C\left( (\Vert \eta _\Vert +\Vert \eta _\Vert ^)e^\Vert }+(\Vert \eta _\Vert ^+\Vert \eta _\Vert ^)\right) \sum _^\Vert \partial _}\phi \Vert \nonumber \\&\qquad +CMN^+\delta }e^\Vert }(e^\Vert }-1)\sum _^\Vert \partial _}\phi \Vert \end$$

(5.73)

$$\begin&\quad \le C(\Vert \eta _\Vert +\Vert \eta _\Vert ^)(e^\Vert }+1)\sum _^\Vert \partial _}\phi \Vert . \end$$

(5.74)

Therefore, in total for all \(t\in \mathbb \)

$$\begin&\Vert \Delta _W(\eta _)\phi \otimes \Omega \Vert \end$$

(5.75)

$$\begin&\le C\left\\Vert +\Vert \eta _\Vert ^)(e^\Vert }+1)+\sum _^\big ((\Vert \eta _\Vert +\Vert \eta _\Vert ^)(e^\Vert }+1)\Vert \partial _}\phi \Vert +\Vert \partial _}^\phi \Vert \big )\right\} . \end$$

(5.76)

\(\square \)

Proofs of properties of \(\eta \)

Proof of Lemma 5.6

Recall that by definition it holds

$$\begin \Vert \eta _\Vert ^=\sum _\sum __}|\big (\eta _\big )_(k)|^=\lambda ^\sum _|\hat(k)|^\underbrace_}\bigg |n_(k)\frac(k)s/2\big )}(k)/2}\bigg |^}_S_}\nonumber \\ \end$$

(5.77)

with \(\epsilon _(k)=2 k_|k\cdot \omega _|\). We will first approximate \(S_\) and then give a upper bound.

Firstly, we approximate the \(\alpha \)-sum \(S_\) by an integral by identifying

$$\begin n_(k)^=k_}^|k|\sigma (p_)u_(k)^\left( 1+\mathcal (M^}N^+\delta })\right) \end$$

(5.78)

with \(\cos \theta _|\hat\cdot \hat_|\equiv u_(k)^\) analogously to Lemma A.1. Thus, we calculate the half-sphere integral as approximation for \(S_\)

$$\begin S_&=\frac\int _^\text \varphi \int _^\text \theta \frac}|k|s\cos \theta )^}\sin \theta +\mathcal \nonumber \\&=\frac\int _^\text u\frac}|k|su)^}+\mathcal \nonumber \\&=\frac\left\}|k|s)-\text (2k_}|k|s)+\gamma \right\} +\mathcal \end$$

(5.79)

with total error \(\mathcal =\mathcal _+\mathcal _+\mathcal _\) consisting three terms: the error \(\mathcal _\) from (5.78), the error \(\mathcal _\) from approximating the discrete va