Relation Between the Resonance and the Scattering Matrix in the Massless Spin-Boson Model

Abstract

We establish the precise relation between the integral kernel of the scattering matrix and the resonance in the massless Spin-Boson model which describes the interaction of a two-level quantum system with a second-quantized scalar field. For this purpose, we derive an explicit formula for the two-body scattering matrix. We impose an ultraviolet cut-off and assume a slightly less singular behavior of the boson form factor of the relativistic scalar field but no infrared cut-off. The purpose of this work is to bring together scattering and resonance theory and arrive at a similar result as provided by Simon (Ann Math Sect Ser 97(2):247–274, 1973), where it was shown that the singularities of the meromorphic continuation of the integral kernel of the scattering matrix are located precisely at the resonance energies. The corresponding problem has been open in quantum field theory ever since. To the best of our knowledge, the presented formula provides the first rigorous connection between resonance and scattering theory in the sense of (Simon 1973) in a model of quantum field theory.

Appendices

Standard Estimates

In the following we shall use the well-known standard inequalities

$$\begin{aligned} \begin{aligned} \Vert a(h)\Psi \Vert&\le \Vert h/\sqrt{\omega }\Vert _2 \, \Vert H_f^{1/2}\Psi \Vert \\ \Vert a(h)^*\Psi \Vert&\le \Vert h/\sqrt{\omega }\Vert _2 \, \Vert H_f^{1/2}\Psi \Vert + \Vert h \Vert _2 \, \Vert \Psi \Vert \end{aligned} \end{aligned}$$

(A.1)

which hold for all $h , h/\sqrt{\omega }\in {\mathfrak {h}}$ and $\Psi \in {\mathcal {H}}$ such that the left- and right-hand side are well-defined; see [40, Eq. (13.70)].

Lemma A.1

Let $h, h/\sqrt{\omega }\in \mathfrak {h}$. Then, we have the following estimates:

$$\begin{aligned} \left||a(h)^* (H_f+1)^{-\frac{1}{2}}\right||&\le \left||h\right||_2 + \left||h/\sqrt{\omega }\right||_2 , \end{aligned}$$

(A.2)

$$\begin{aligned} \left||a(h) (H_f+1)^{-\frac{1}{2}}\right||&\le \left||h/\sqrt{\omega }\right||_2 , \end{aligned}$$

(A.3)

$$\begin{aligned} \left|| V \left( H_f +1 \right) ^{-\frac{1}{2}}\right||&\le \left||f\right||_2 +2\left||f/\sqrt{\omega }\right||_2 . \end{aligned}$$

(A.4)

Proof

Let $\Psi \in {\mathcal {F}} [\mathfrak {h}]$ with $\Vert \Psi \Vert _{{\mathcal {H}}}=1$. Applying (A.1) and the spectral theorem, we find

$$\begin{aligned} \Vert a(h)^* (H_f+1)^{-\frac{1}{2}}\Psi \Vert&\le \Vert h\Vert _2 \Vert (H_f+1)^{ -\frac{1}{2}}\Psi \Vert + \Vert h/\sqrt{\omega }\Vert _2 \Vert H_f^{\frac{1}{2}}(H_f+1)^{ -\frac{1}{2}}\Psi \Vert \nonumber \\&\le \Vert h\Vert _2+\Vert h/\sqrt{\omega }\Vert _2, \end{aligned}$$

(A.5)

$$\begin{aligned} \Vert a(h) (H_f+1)^{-\frac{1}{2}}\Psi \Vert&\le \Vert h/\sqrt{\omega }\Vert _2 \Vert H_f^{\frac{1}{2}}(H_f+1)^{ - \frac{1}{2}}\Psi \Vert \le \Vert h/\sqrt{\omega }\Vert _2. \end{aligned}$$

(A.6)

The inequality (A.4) is implied by the boundedness of $\sigma _1$ and the triangle inequality:

$$\begin{aligned}&\left|| V \left( H_f +1 \right) ^{-\frac{1}{2}}\right|| \le \left|| \sigma _1 \otimes a(f)\left( H_f +1 \right) ^{-\frac{1}{2}} \right||+ \left||\sigma _1 \otimes a(f)^*\left( H_f +1 \right) ^{-\frac{1}{2}} \right|| \nonumber \\&\le \left|| a(f) \left( H_f +1 \right) ^{-\frac{1}{2}} \right|| +\left|| a(f)^*\left( H_f +1 \right) ^{-\frac{1}{2}} \right|| \le \left||f\right||_2 +2\left||f/\sqrt{\omega }\right||_2. \end{aligned}$$

(A.7)

This completes the proof. $\quad \square$

As preparation of the proof of Lemma 4.1 (in “Appendix C” below) we recall that the Hamiltonians H, c.f. (1.7), as well as $H_f$, c.f. (1.3), are self-adjoint on the common domain $D(H)={\mathcal {K}}\otimes {\mathcal {D}}(H_f)$ and bounded below by the constant $b\in {\mathbb {R}}$; c.f. Proposition 1.1 and (1.22). By spectral calculus we can define the operators $H_f^{1/2}$, $(H-b+1)^{1/2}$ and $(H_f+1)^{-1/2}$, $(H-b+1)^{-1/2}$ which are closed and densely defined and the latter two are even bounded by 1. For the proof Lemma 4.1 we shall need the following lemma.

Lemma A.2

The following operators are bounded:

$$\begin{aligned}&H_f^{\frac{1}{2}}(H-b+1)^{-\frac{1}{2}} , \end{aligned}$$

(A.8)

$$\begin{aligned}&(H-b+1)^{\frac{1}{2}}(H_f+1)^{-\frac{1}{2}}. \end{aligned}$$

(A.9)

Proof

Let $\Psi \in \mathcal H$ with $\Vert \Psi \Vert =1$. The boundedness of (A.8) follows from the equality

$$\begin{aligned} \Vert H_f^{\frac{1}{2}}(H-b+1)^{-\frac{1}{2}}\Psi \Vert ^2&= \langle (H-b+1)^{-\frac{1}{2}}\Psi ,H_f(H-b+1)^{-\frac{1}{2}}\Psi \rangle \nonumber \\&= \langle (H-b+1)^{-\frac{1}{2}}\Psi ,(H-K-gV)(H-b+1)^{-\frac{1}{2}}\Psi \rangle \end{aligned}$$

(A.10)

and the fact that K is bounded by $|e_1|$ and that for all $\epsilon >0$

$$\begin{aligned}&|\langle (H-b+1)^{-\frac{1}{2}}\Psi ,gV(H-b+1)^{-\frac{1}{2}}\Psi \rangle | \le \Vert (H-b+1)^{-\frac{1}{2}}\Psi \Vert \,\Vert gV(H-b+1)^{-\frac{1}{2}}\Psi \Vert \nonumber \\&\le \frac{g}{\epsilon } 2\Vert f/\sqrt{\omega }\Vert _2 \, \epsilon \Vert H_f^{\frac{1}{2}}(H-b+1)^{-\frac{1}{2}}\Psi \Vert +\Vert f\Vert _2 \Vert \Psi \Vert \nonumber \\&\le \left( \frac{g}{\epsilon } 2\Vert f/\sqrt{\omega }\Vert _2\right) ^2 + \epsilon ^2 \Vert H_f^{\frac{1}{2}}(H-b+1)^{-\frac{1}{2}}\Psi \Vert ^2 +\Vert f\Vert _2 \end{aligned}$$

(A.11)

holds, which is a consequence of (A.1). Choosing $0<\epsilon <1$ an explicit bound is

$$\begin{aligned} \Vert H_f^{\frac{1}{2}}(H-b+1)^{-\frac{1}{2}}\Psi \Vert ^2 \le \frac{1 + |e_1| + \left( \frac{g}{\epsilon } 2\Vert f/\sqrt{\omega }\Vert _2\right) ^2 +\Vert f\Vert _2 }{1-\epsilon ^2} <\infty . \end{aligned}$$

(A.12)

The boundedness of (A.9) is implied by

$$\begin{aligned} \Vert (H-b+1)^{\frac{1}{2}}(H_f+1)^{-\frac{1}{2}}\Psi \Vert ^2 = \langle (H_f+1)^{-\frac{1}{2}}\Psi ,(K+H_f+gV-b+1)(H_f+1)^{-\frac{1}{2}}\Psi \rangle \end{aligned}$$

(A.13)

and, again as a consequence of (A.1),

$$\begin{aligned} |\langle (H_f+1)^{-\frac{1}{2}}\Psi ,gV(H_f+1)^{-\frac{1}{2}}\Psi \rangle |&\le g 2\Vert f/\sqrt{\omega }\Vert _2 \, \Vert H_f^{\frac{1}{2}}(H_f+1)^{-\frac{1}{2}}\Psi \Vert +\Vert f\Vert _2 \\ \nonumber&\le \Vert f\Vert _2+ 2 \Vert f/\sqrt{\omega }\Vert _2. \end{aligned}$$

(A.14)

$\square$

Proofs for Section 1.2

It is well-known that there is a dense domain of analytic vectors; for example

$$\begin{aligned} {\mathcal {D}} = \left\{ \chi _{[-R,R]}(A)\Psi : \Psi \in {\mathcal {H}} , R>0 \right\} \end{aligned}$$

with A being the generator of $U_\theta$ and $\chi$ the corresponding spectral projection (c.f. [4, 32]).

Proof of Lemma 1.5

Let $\theta \in {\mathbb {C}}$. Definition in (1.3) implies that $H^\theta _0=K \otimes \mathbb {1}_{{\mathcal {F}}[{\mathfrak {h}}]} + \mathbb {1}_{{\mathcal {K}}} \otimes H^\theta _f$ is a sum of commuting self-adjoint operators and $\sigma (K)=\left\{ e_0,e_1 \right\}$. As shown in [36], we have $\sigma (H_f)={\mathbb {R}}_0^+$ and it follows from the definition of $H_f^\theta =e^{-\theta }H_f$ in (1.28) that $\sigma (H^\theta _f)= \left\{ e^{-\theta } r : r\ge 0 \right\}$. The claim then follows from the spectral theorem for two commuting normal operators. $\quad \square$

Asymptotic Creation/Annihilation Operators

Proof of Lemma 4.1

Let $h,l\in {\mathfrak {h}}_0$ and $\Psi \in {\mathcal {K}} \otimes {\mathcal {D}}(H_f^{1/2})$. Thanks to Lemma A.2 we have ${\mathcal {K}} \otimes {\mathcal {D}}(H_f^{\frac{1}{2}})=\mathcal D((H - b + 1 )^{\frac{1}{2}})$. We prove claims (i)–(vi) separately:

• (ii) The subspace of $\mathcal H_0$, defined in (4.18), is dense in the domain of $(H-b+1)^{1/2}$ w.r.t. the graph norm $\Vert \cdot \Vert _{(H-b+1)^{1/2}}$ of $(H-b+1)^{\frac{1}{2}}$ so that there is a sequence $(\Psi _n)_{n\in {\mathbb {N}}}$ in ${\mathcal {K}} \otimes \mathcal F_{\text {fin}}[{\mathfrak {h}}_0]$ with $\Psi _n\rightarrow \Psi$ in this norm as $n\rightarrow \infty$. For all $n\in {\mathbb {N}}$, the definition in (1.34) together with the group properties $(e^{-itH})_{t\in {\mathbb {R}}}$, in particularly, the strong continuous differentiability on D(H), justify

  $$\begin{aligned} a_t(h)\Psi _n&= e^{itH}a(h_t)e^{-itH} = a(h)\Psi _n + \int _0^t ds\, \frac{d}{ds} e^{isH} a(h_s) e^{-isH}\Psi _n \nonumber \\&= a(h)\Psi _n -ig \int _0^t ds\, \langle h_s,f\rangle _2 e^{isH} \sigma _1 e^{-isH}\Psi _n, \end{aligned}$$

  (C.1)

  where the last integrand was computed by observing the CCR (c.f. (1.19))

  $$\begin{aligned}{}[V ,a(h_s)]&= \sigma _1 \otimes [a(f)+a(f)^*,a(h_s)]=-\sigma _1 \left\langle h_s, f\right\rangle _2. \end{aligned}$$

  (C.2)

  We may now take the limit $n\rightarrow \infty$ of identity (C.1) and find

  $$\begin{aligned} a_t(h)\Psi&= a(h)\Psi -ig \int _0^t ds\, \langle h_s,f\rangle _2 \, e^{isH} \sigma _1 e^{-isH}\Psi \end{aligned}$$

  (C.3)

  because of the following two ingredients: First, by definition (1.34), the standard estimate (A.1) and Lemma A.2, for all $m\in \mathfrak h_0$, there is a finite constant $C_{(\hbox {C}.4)}$ such that

  $$\begin{aligned} \Vert a_t(m)(\Psi -\Psi _n)\Vert&= \Vert a(m_t)(H-b+1)^{-\frac{1}{2}}e^{-itH}(H-b+1)^{\frac{1}{2}}(\Psi -\Psi _n)\Vert \nonumber \\&\le \Vert m/ \sqrt{\omega }\Vert _2 \, \Vert H_f^{\frac{1}{2}}(H-b+1)^{-\frac{1}{2}}\Vert \, \Vert (H-b+1)^{\frac{1}{2}}(\Psi -\Psi _n)\Vert \nonumber \\&= C_{(\hbox {C}.4)} \Vert \Psi -\Psi _n\Vert _{(H-b+1)^{1/2}}, \end{aligned}$$

  (C.4)

  and likewise

  $$\begin{aligned} \Vert a(m)(\Psi -\Psi _n)\Vert&= \Vert a(m)(H-b+1)^{-\frac{1}{2}}(H-b+1)^{\frac{1}{2}}(\Psi -\Psi _n)\Vert \nonumber \\&\le \Vert m/ \sqrt{\omega }\Vert _2 \, \Vert H_f^{\frac{1}{2}}(H-b+1)^{-\frac{1}{2}}\Vert \, \Vert (H-b+1)^{\frac{1}{2}}(\Psi -\Psi _n)\Vert \nonumber \\&= C_{(\hbox {C}.4)} \Vert \Psi -\Psi _n\Vert _{(H-b+1)^{1/2}}. \end{aligned}$$

  (C.5)

  Second, the integrand in (C.1) is continuous in s and, for sufficiently large n, fulfills an n-independent bound

  $$\begin{aligned} \Vert e^{isH} \sigma _1 e^{-isH}(\Psi -\Psi _n)\Vert \le \Vert \sigma _1 \Vert \,\Vert \Psi -\Psi _n\Vert \le 1 \end{aligned}$$

  (C.6)

  so dominated convergence can be applied to interchanging the integral and the $n\rightarrow \infty$ limit to prove (C.3).

  Finally, a stationary phase argument in $\omega (k)=|k|$ as well as the facts that $h\in {\mathfrak {h}}_0$ and $f\in \mathcal C^\infty ({\mathbb {R}}{\setminus }\{0\})$, c.f. (1.5), provide the estimate

  $$\begin{aligned} \langle h_s,f\rangle = C \frac{1}{1 + |s|^2} \end{aligned}$$

  (C.7)

  for all $s\in {\mathbb {R}}$, thanks to a two-fold partial integration. Hence, me way finally carry out the limit $t\rightarrow \pm \infty$ to find

  $$\begin{aligned} a_\pm (h)\Psi = \lim _{t\rightarrow \pm \infty } a_t(h)\Psi = a(h)\Psi -ig \int _0^{\pm \infty } ds\, \langle h_s,f\rangle _2 e^{isH} \sigma _1 e^{-isH}\Psi \end{aligned}$$

  (C.8)

  as the indefinite integral exists thanks to (C.7) and the continuity of the integrand in s. We omit the proof for the asymptotic creation operator $a^*_\pm$ as the argument is almost the same.

• (i) This follows from (ii).

• (iii) Next, we calculate

  $$\begin{aligned} e^{-isH} a_- (h)^*\psi&=\lim \limits _{t\rightarrow -\infty } e^{-isH} e^{itH}a(h_t)^*e^{-itH} \psi \nonumber \\&=\lim \limits _{t\rightarrow -\infty } e^{i(t-s)H}a(h_{(t-s)+s})^* e^{-i(t-s)H} e^{-isH} \psi \nonumber \\&=\lim \limits _{t'\rightarrow -\infty } e^{it'H} a(h_{t'+s})^* e^{-it'H} e^{-isH} \psi = a_- (h_s)^* e^{-isH}\psi \end{aligned}$$

  (C.9)

  which proves the pull-through formula in (iii).

• (iv) First, for all $t\in {\mathbb {R}}$ we observe

  $$\begin{aligned} \Vert a_t(h)\Psi _{\lambda _0}\Vert = \Vert e^{itH}a(h_t)e^{-itH}\Psi _{\lambda _0}\Vert = \Vert a(h_t)\Psi _{\lambda _0}\Vert \end{aligned}$$

  (C.10)

  due to the ground state property in (1.25). Second, for $\Psi =\Psi _{\lambda _0}\in {\mathcal {D}}(H)\subset {\mathcal {K}} \otimes \mathcal D(H_f^{1/2})$, we employ the same sequence $(\Psi _n)_{n\in {\mathbb {N}}}$ as in (ii) to compute

  $$\begin{aligned} \Vert a(h_t)\Psi _n\Vert ^2=\sum _{l\in {\mathbb {N}}}\sqrt{l+1}\int d^3k_1\ldots d^3k_l \left| \int d^3k \, e^{it\omega (k)}\overline{h(k)}\psi _n^{(l+1)}(k,k_1,\ldots ,k_l) \right| ^2, \end{aligned}$$

  (C.11)

  where we used the Fock vector representation $\Psi _n=(\psi _n^{(l)})_{l\in {\mathbb {N}}_0}$. We observe that $\Psi _n\in \mathcal H_0$ implies $\psi ^{(l)}_n\in {\mathcal {K}} \otimes C_0^{\infty } ( \mathbb {R}^{3l} {\setminus } \{ 0\} )$ and, by definition of ${\mathcal {H}}_0$, c.f. (4.18), there is a constant L such that $\psi ^{(l)}_{n}=0$ for $l\ge L$ . A stationary phase argument in $\omega (k)=|k|$ and a partial integration in k gives

  $$\begin{aligned}&\left| \int d^3k \, e^{it\omega (k)}\overline{ h(k)} \psi _n^{(l+1)}(k,k_1,\ldots ,k_{l}) \right| \nonumber \\&\le \frac{1}{t} \int d^3k \, |k|^{-2} | \partial _{|k|} ( |k|^2 \overline{h( |k|,\Sigma )}\psi _n^{(l+1)}( |k|,\Sigma , |k_1|,\Sigma _1,\ldots , |k_l|,\Sigma _l)) | , \end{aligned}$$

  (C.12)

  where we use spherical coordinates $k=(|k|,\Sigma )$ and $k_i=(|k_i|,\Sigma _i)$. Here, $\Sigma$ and $\Sigma _i$ denote the solid angles. Then, we find

  $$\begin{aligned} (\hbox {C.}11)\le&\frac{1}{t} \sum _{0\le l< L} \sqrt{l+1} \int d^3k_1\ldots d^3k_l \nonumber \\&\times \left( \int d^3k \, |k|^{-2} | \partial _{|k|} ( |k|^{2} \overline{h(|k|,\Sigma )} \Psi _n^{(l+1)}( |k|,\Sigma , |k_1|,\Sigma _1,\ldots , |k_l|,\Sigma _l) |\right) ^2 \end{aligned}$$

  (C.13)

  which converges to zero for $t\rightarrow \pm \infty$. In conclusion, for all $n\in {\mathbb {R}}$ we have

  $$\begin{aligned} \lim _{t\rightarrow \pm \infty }a(h_t)\Psi _n = 0. \end{aligned}$$

  (C.14)

  Moreover, there is a t-independent, finite constant $C_{(\hbox {C}.15)}(h)$ such that

  $$\begin{aligned} \Vert a_t(h)(\Psi _{\lambda _0}-\Psi _n)\Vert&= \Vert e^{itH}a(h_t)e^{-itH}(\Psi _{\lambda _0}-\Psi _n)\Vert \nonumber \\&= \Vert a(h_t)(H-b+1)^{-\frac{1}{2}}e^{-itH}(H-b+1)^{\frac{1}{2}}(\Psi -\Psi _n)\Vert \nonumber \\&\le \Vert |h|/ \sqrt{\omega }\Vert _2 \, \Vert H_f^{\frac{1}{2}}(H-b+1)^{-\frac{1}{2}}\Vert \Vert \Psi -\Psi _n\Vert _{(H-b+1)^{1/2}} \nonumber \\&=C_{(\hbox {C}.15)}(h) \Vert \Psi -\Psi _n\Vert _{(H-b+1)^{1/2}} \end{aligned}$$

  (C.15)

  and

  $$\begin{aligned} \Vert a_\pm (h)\Psi _{\lambda _0} \Vert&\le \lim _{t\rightarrow \pm \infty } \left( \Vert a_t(h)(\Psi _{\lambda _0}-\Psi _n)\Vert + \Vert a_t(h)\Psi _n\Vert \right) \nonumber \\&\le C_{(\hbox {C}.15)}(h) \Vert \Psi -\Psi _n\Vert _{(H-b+1)^{1/2}} \end{aligned}$$

  (C.16)

  holds true for all $n\in {\mathbb {N}}$, where we have use the standard inequalities (A.1), Lemma A.2 and (C.14). Taking the limit $n\rightarrow \infty$ proves the claim (iv).

• (v) We consider the same sequence $(\Psi _n)_{n\in {\mathbb {N}}}$ as in (iv) and, for all $n\in {\mathbb {N}}$, we observe that, by (i) and definition in (1.34), it holds

  $$\begin{aligned} \langle a(h)^*_\pm \Psi _{\lambda _0}, a(l)^*_\pm \Psi _{\lambda _0}\rangle&= \lim _{t\rightarrow \pm \infty } \langle a(h_t)^*\Psi _{\lambda _0}, a(l_t)^*\Psi _{\lambda _0}\rangle . \end{aligned}$$

  (C.17)

  Furthermore, using the CCR in (1.19), we find for all $n\in {\mathbb {N}}$ that

  $$\begin{aligned}&\langle a(h_t)^*\Psi _{\lambda _0}, a(l_t)^*\Psi _n\rangle = \langle \Psi _{\lambda _0}, a(h_t)a(l_t)^*\Psi _n\rangle \nonumber \\&= \langle \Psi _{\lambda _0}, \left( a(l_t)^*a(h_t)+[a(h_t), a(l_t)^*]\right) \Psi _n\rangle \nonumber \\&= \langle a(l_t)\Psi _{\lambda _0}, a(h_t)\Psi _n\rangle + \langle \Psi _{\lambda _0},\Psi _n\rangle \, \langle h, l\rangle _2 \end{aligned}$$

  (C.18)

  holds. We may control the limit $n\rightarrow \infty$ of this identity by

  $$\begin{aligned}&|\langle a(h_t)^*\Psi _{\lambda _0}, a(l_t)^*(\Psi _{\lambda _0}-\Psi _n)\rangle | \le \Vert a(h_t)^*\Psi _{\lambda _0}\Vert \, \Vert a(l_t)^*(\Psi _{\lambda _0}-\Psi _n)\Vert \nonumber \\&\le (\Vert h\Vert _2 + \Vert h/\sqrt{\omega }\Vert _2) \Vert \Psi _{\lambda _0}\Vert _{(H-b+1)^{1/2}} (\Vert l\Vert _2 + \Vert l/\sqrt{\omega }\Vert _2) \Vert \Psi _{\lambda _0}-\Psi _n\Vert _{(H-b+1)^{1/2}}, \end{aligned}$$

  (C.19)

  and likewise,

  $$\begin{aligned}&|\langle a(l_t)\Psi _{\lambda _0}, a(h_t)(\Psi _{\lambda _0}-\Psi _n)\rangle | \le \Vert a(l_t)\Psi _{\lambda _0}\Vert \, \Vert a(h_t)(\Psi _{\lambda _0}-\Psi _n)\Vert \nonumber \\&\le (\Vert l\Vert _2 + \Vert l/\sqrt{\omega }\Vert _2) \Vert \Psi _{\lambda _0}\Vert _{(H-b+1)^{1/2}} (\Vert h\Vert _2 + \Vert h/\sqrt{\omega }\Vert _2) \Vert \Psi _{\lambda _0}-\Psi _n\Vert _{(H-b+1)^{1/2}}, \end{aligned}$$

  (C.20)

  which are ensured by the standard estimates (A.1) and Lemma A.2. These bounds allow to take the limit $n\rightarrow \infty$ of identity (C.18) which yields

  $$\begin{aligned} \langle a(h_t)^*\Psi _{\lambda _0}, a(l_t)^*\Psi _{\lambda _0}\rangle&= \langle a(l_t)\Psi _{\lambda _0}, a(h_t)\Psi _{\lambda _0}\rangle + \langle \Psi _{\lambda _0},\Psi _{\lambda _0}\rangle \, \langle h, l\rangle _2. \end{aligned}$$

  Finally, recalling (C.17) and exploiting (iv) that states $a_\pm (h)\Psi _{\lambda _0}=0$, we find

  $$\begin{aligned} \langle a(h)^*_\pm \Psi _{\lambda _0}, a(l)^*_\pm \Psi _{\lambda _0}\rangle&= \lim _{t\rightarrow \pm \infty }\langle a(h_t)^*\Psi _{\lambda _0} a(l_t)^*\Psi _{\lambda _0}\rangle = \langle \Psi _{\lambda _0},\Psi _{\lambda _0}\rangle \, \langle h, l\rangle _2 \end{aligned}$$

  which concludes the proof of (v).

• (vi) Let $t\in {\mathbb {R}}$. Thanks to the standard estimate (A.1), we find

  $$\begin{aligned}&\Vert a_t(h)(H_f+1)^{-\frac{1}{2}}\Vert = \Vert e^{itH}a(h_t)(H-b+1)^{-\frac{1}{2}}e^{-itH}(H-b+1)^{\frac{1}{2}}(H_f+1)^{-\frac{1}{2}}\Vert \nonumber \\&\le \Vert a(h_t)(H-b+1)^{-\frac{1}{2}}\Vert \, \Vert (H-b+1)^{\frac{1}{2}}(H_f+1)^{-\frac{1}{2}}\Vert \nonumber \\&\le \Vert h/\sqrt{\omega }\Vert _2\, \Vert H_f^{\frac{1}{2}}(H-b+1)^{-\frac{1}{2}}\Vert \, \Vert (H-b+1)^{\frac{1}{2}}(H_f+1)^{-\frac{1}{2}}\Vert . \end{aligned}$$

  (C.21)

  Lemma A.2 ensures that the right-hand side of (C.21) is bounded by a finite constant C(h) which depends only on h. This proves the first inequality of (vi). The proof of the second is omitted here as it is almost identical. $\quad \square$

The Principle Term $T_p(h,l)$

In the section, we prove that if $G \equiv G(h, l)$ is positive and strictly positive at ${\text {Re}}\,\lambda _1 - \lambda _0$ then the absolute of the principal term $T_P(h,l)$ can be bounded by a strictly positive constant times $g^2$.

Lemma D.1

Suppose that $G \equiv G(h, l)$ is positive and strictly positive at ${\text {Re}}\,\lambda _1 - \lambda _0$, then, for small enough g (depending on G), there is a constant $C(h,l)>0$ (independent of g) such that

$$\begin{aligned} |T_P(h,l)|\ge C(h,l) g^2. \end{aligned}$$

(D.1)

Proof

We set

$$\begin{aligned} I:=\int \mathrm {d}r \frac{ G(r) }{ ( r + \lambda _0 - {\text {Re}}\,\lambda _1- i g^2 E_1 ) ( r - \lambda _0 + \overline{\lambda _1} ) }, \end{aligned}$$

(D.2)

and take small enough g. Recalling (2.4), we observe that

$$\begin{aligned} T_P(h,l)=g^2 E_1 M I. \end{aligned}$$

(D.3)

We recall from the discussion below Definition 2.1 that $E_1=E_I +g^{ a }\Delta$, where $a>0$, $\Delta \equiv \Delta (g)$ is uniformly bounded and $E_I$ is a strictly negative constant that does not depend on g, see (3.11). Additionally, it follows from (3.25) together with $\left||\varphi _0\otimes \Omega \right||=1$ that $\left||\Psi _{\lambda _0}\right||\ge C>0$, for some constant C that does not depend on g. Moreover, we conclude from (3.28) that ${\text {Re}}\,\lambda _1 -\lambda _0 \ge C>0$ for some constant C (independent of g). Consequently, (2.6) guarantees that there is a constant C (independent of g) such that $|M|\ge C>0$.

This together with (D.3) implies that it suffices to show that there is a constant $C(h,l)>0$ such that

$$\begin{aligned} | I |\ge C(h,l), \end{aligned}$$

(D.4)

in order to conclude (D.1).

For $\alpha \equiv \alpha _g:={\text {Re}}\,\lambda _1-\lambda _0$ and recalling (1.2), we observe

$$\begin{aligned} I=\int \mathrm {d}r \frac{ G(r) }{ ( r -\alpha - i g^2 E_1 )( r +\alpha - i g^2 E_1 ) } =\int \mathrm {d}r \frac{ G(r) \left( r^2 -\alpha ^2 -g^4E_1^2 + 2i g^2 E_1r \right) }{ ( r^2 -\alpha ^2 -g^4E_1^2)^2 +4g^4 E_1^2r^2 }. \end{aligned}$$

(D.5)

Let $c > 0$ be such that G is supported in the complement of the ball or radius c and center 0. Then, we have

$$\begin{aligned} |{\text {Im}}\,(I)|\ge |E_1| \int \mathrm {d}r G(r) \frac{ 2 g^2 r }{ ( r^2 -\alpha ^2 -g^4E_1^2)^2 +4g^4 E_1^2c^2 }. \end{aligned}$$

(D.6)

Substituting $s=r^2$, yields

$$\begin{aligned} |{\text {Im}}\,(I)|\ge |E_1| \int \mathrm {d}s G( \sqrt{s} ) \frac{ g^2 }{ ( s -\alpha ^2 -g^4E_1^2)^2 +4g^4 E_1^2c^2 }. \end{aligned}$$

(D.7)

Since $G(\alpha ) \ne 0$, then for small enough g there is a constant $r_0$, that does not depend on g and a constant $C > 0$ (independent of g) such that $G(\sqrt{s}) \ge C$, for every $s \in [ \alpha ^2 + g^4E_1^2 - r_0, -\alpha ^2 -g^4E_1^2 + r_0 ]$. We apply the change of variables $u = s - \alpha ^2 - g^4E_1^2$ and obtain

$$\begin{aligned} |{\text {Im}}\,(I)|\ge C |E_1| \int _{- r_0}^{r_0} \mathrm {d}s \frac{ g^2 }{ s^2 +4g^4 E_1^2c^2 }. \end{aligned}$$

(D.8)

Finally, we change to the variable $\tau = s / g^2$ to obtain:

$$\begin{aligned} |{\text {Im}}\,(I)|\ge C |E_1| \int _{- r_0/g^2}^{r_0/g^2} \mathrm {d}\tau \frac{ 1 }{ \tau ^2 +4 E_1^2c^2 } \ge C |E_1| , \end{aligned}$$

(D.9)

for small enough g (depending on G). $\quad \square$

List of Main Notations

In this section we provide of list of main notations and their place of definition used in this

Symbol	Place of definition
$E_1$	Below (1.1)
$H_0$, K, $H_f$	(1.3)
$e_0$, $e_1$	Below (1.3)
$\omega$	Below (1.3)
V, $\sigma _1$	(1.4)
f	(1.5)
$\mu$	(1.6)
H	(1.7)
g	Below (1.7), see also Definition 3.1, (3.31) and Definition 4.3 in [14]
${\mathcal {H}}$, ${\mathcal {K}}$	(1.8)
${\mathcal {F}}[\mathfrak {h}]$, $\mathfrak {h}$	(1.9)
$\odot$	Below (1.9)
$\Omega$	(1.10)
${\mathcal {F}}_0$	(1.11)
$S ({\mathbb {R}}^3,{\mathbb {C}})$	Below (1.11)
a(h)	(1.12)
$a(h)^*$	(1.13)
a(k)	(1.14)
$a(k)^*$	(1.15)
$\varphi _0$, $\varphi _1$	(1.20)
${\mathcal {D}}(\bullet )$	Below (1.20)
$\sigma (\bullet )$	Below (1.20)
$\theta$, $u_\theta$, $U_\theta$	Definition 1.3
$H^\theta$	(1.27)
$H_f^\theta$, $V^\theta$	(1.28)
$\omega ^\theta$, $f^\theta$	(1.29)
$D(\bullet ,\bullet )$	(1.30)
$\lambda _0$, $\lambda _1$	Below Lemma 1.5
$\Psi _{\lambda _0}$, $\Psi _{\lambda _1}$	(1.32)
$\mathfrak {h}_0$	(1.33)
$a_\pm (h)$	(1.34)
$a_\pm (h)^*$	Below (1.34)
${\mathcal {K}}^\pm$, ${\mathcal {H}}^\pm$	(1.35)
$\Omega _\pm$	(1.36)
S(h, l)	(1.37)

Symbol	Place of definition
T(h, l)	(1.38)
G	(2.1)
$\iota$	(3.31) and (3.32)
$T_P(h,l)$	(2.4)
R(h, l)	(2.5)
$\nu$	(3.1)
${\mathcal {S}}$	(3.2)
$\varvec{\nu }$	Below (3.2)
$\rho _0$, $\rho$	(3.3) and (3.31)
A	(3.4)
$B_0^{(1)}$, $B_1^{(1)}$	(3.8)
${\mathcal {C}}_m(z)$	(3.9)
$E_I$	(3.11)
$\rho _n$	(3.14)
$H^{(n),\theta }$	(3.15)
$H_f^{(n),\theta }$	(3.16)
$V^{(n),\theta }$	(3.17)
${\mathcal {H}}^{(n)}$, $\mathfrak {h}^{(n)}$	(3.19)
${\tilde{H}}^{(n)}$	(3.20)
$\mathfrak {h}^{(n, \infty )}$	(3.21)
$\Omega ^{(n, \infty )}$, $P_{\Omega ^{(n, \infty )}}$	Below (3.21)
$\lambda _0^{(n)}$, $\lambda _1^{(n)}$	Above (3.22)
$P_0^{(n),\theta }$, $P_1^{(n),\theta }$	(3.22)
$P_0^{\theta }$, $P_1^{\theta }$	(3.23)
${\varvec{C}}$	Below (3.32)
$\mathfrak {F}$, $\mathfrak {F}^{-1}$	Definition 4.2
W	(4.9)
$\Sigma$	Below (4.9)
${\mathcal {H}}_0$	(4.18)
${\mathcal {F}}_\text {fin}[\mathfrak {h}_0]$	(4.19)
$\left||\bullet \right||_{\bullet }$	Below (4.19)
$\Gamma (\epsilon ,R)$	Above (4.30)
$\Gamma _-(\epsilon ,R)$	(4.30)
$\Gamma _d(R)$	(4.30)
$\Gamma _c(\epsilon )$	(4.30)
$\epsilon _n$	(4.48)
$R_1(q,Q)$	(4.79)
$P_1(q,Q)$	(4.86)
${\tilde{P}}_1(q,Q)$	(4.96)