Cherenkov Radiation with Massive Bosons and Quantum Friction

Abstract

This work is devoted to several translation-invariant models in nonrelativistic quantum field theory (QFT), describing a nonrelativistic quantum particle interacting with a quantized relativistic field of bosons. In this setting, we aim at the rigorous study of Cherenkov radiation or friction effects at small disorder, which amounts to the metastability of the embedded mass shell of the bare nonrelativistic particle when the coupling to the quantized field is turned on. Although this problem is naturally approached by means of Mourre’s celebrated commutator method, important regularity issues are known to be inherent to QFT models and restrict the application of the method. In this perspective, we introduce a novel non-standard procedure to construct Mourre conjugate operators, which differs from second quantization and allows to circumvent many regularity issues. To show its versatility, we apply this construction to the Nelson model with massive bosons, to Fröhlich’s polaron model, and to a quantum friction model with massless bosons introduced by Bruneau and De Bièvre: for each of those examples, we improve on previous results.

Appendix A: Mourre’s Commutator Method

In this “Appendix,” we briefly recall for convenience standard definitions and statements from Mourre’s theory that we use in this work; we refer, for example, to [2, 27] for more details. We start with the notion of regularity with respect to a self-adjoint operator, which is crucial to define commutators and deal with domain issues.

Definition A.1

(Regularity). Let A be a self-adjoint operator on a Hilbert space ${\mathcal {H}}$.

• A bounded operator B on ${\mathcal {H}}$ is said to be of class $C^k(A)$ if for all $\phi \in {\mathcal {H}}$ the function $t\mapsto e^{-itA} B e^{itA}\phi$ is k-times continuously differentiable.

• A self-adjoint operator H on ${\mathcal {H}}$ is said to be of class $C^k(A)$ if its resolvent $(H-z)^{-1}$ is of class $C^k(A)$ for some $z\in {\mathbb {C}}{\setminus }{\mathbb {R}}$.$\diamond$

We recall the following characterization: a bounded operator B is of class $C^1(A)$ if and only if it maps ${\mathcal {D}}(A)$ into itself and if the commutator ${{\text {ad}}}_{iA}(B):=[B,iA]$ extends uniquely from ${\mathcal {D}}(A)$ to a bounded operator on ${\mathcal {H}}$. Therefore, if H is a self-adjoint operator of class $C^1(A)$, we may use the resolvent identity $[(H-z)^{-1},iA]=-(H-z)^{-1}[H,iA](H-z)^{-1}$ in the sense of forms on ${\mathcal {D}}(A)$, and we infer that the commutator ${{\text {ad}}}_{iA}(H):=[H,iA]$ extends uniquely from ${\mathcal {D}}(H)\cap {\mathcal {D}}(A)$ to a bounded form on ${\mathcal {D}}(H)$. Equivalently, this means for all $\phi ,\psi \in {\mathcal {D}}(H)\cap {\mathcal {D}}(A)$,

$$\begin{aligned} |\langle \phi ,[H,iA]\psi \rangle _{\mathcal {H}}|\,\lesssim \,\Vert (|H|+1)\phi \Vert _{\mathcal {H}}\Vert (|H|+1)\psi \Vert _{\mathcal {H}}. \end{aligned}$$

(A.1)

In fact, we state that the converse is also true under a technical assumption; see, e.g., [2, Theorem 6.3.4].

Lemma A.2

(Characterization of regularity; [2]). Let A and H be self-adjoint operators on a Hilbert space ${\mathcal {H}}$, and assume that the unitary group generated by A leaves the domain of H invariant,

$$\begin{aligned} e^{itA}{\mathcal {D}}(H)\subset {\mathcal {D}}(H)\quad \text {for all}\,\, t\in {\mathbb {R}}. \end{aligned}$$

(A.2)

Then, the domain ${\mathcal {D}}(H)\cap {\mathcal {D}}(A)$ is a core for H. In addition, H is of class $C^1(A)$ if and only if (A.1) holds.

We could write down similar characterizations for higher regularity, but we shall only need the following sufficient criterion in case of H-bounded commutators. Note that this H-boundedness condition is much stronger than (A.1) and is not always satisfied; see in particular our setting in Sect. 3.2.

Lemma A.3

(Sufficient criterion for higher regularity; [2]). Let A and H be self-adjoint operators on a Hilbert space ${\mathcal {H}}$, and assume that the unitary group generated by A leaves the domain of H invariant, cf. (A.2). Given $\nu \ge 1$, assume iteratively for all $0\le s\le \nu$, starting with ${{\text {ad}}}_{iA}^0(H):=H$, that the iterated commutator ${{\text {ad}}}_{iA}^s(H)$ is defined as a form on ${\mathcal {D}}(H)\cap {\mathcal {D}}(A)$ and satisfies

$$\begin{aligned} \Vert |\langle \phi ,{{\text {ad}}}_{iA}^s(H)\psi \rangle |\,\lesssim \,\Vert \phi \Vert _{\mathcal {H}}\Vert (|H|+1)\psi \Vert _{\mathcal {H}}, \end{aligned}$$

(A.3)

which entails that ${{\text {ad}}}_{iA}^s(H)$ extends uniquely to the form of an H-bounded operator and that the next commutator ${{\text {ad}}}_{iA}^{s+1}(H):=[{{\text {ad}}}_{iA}^{s}(H),iA]$ is also well defined as a form on ${\mathcal {D}}(H)\cap {\mathcal {D}}(A)$. Then, H is of class $C^\nu (A)$.

With these regularity assumptions at hand, we may now turn to Mourre commutator estimates, which constitute a key tool for spectral analysis.

Definition A.4

(Mourre estimates). Let A be a self-adjoint operator on a Hilbert space ${\mathcal {H}}$, let H be a self-adjoint operator of class $C^1(A)$, and let $J\subset {\mathbb {R}}$ be a bounded open interval. The operator H is said to satisfy a Mourre estimate on J with respect to the conjugate operator A if there exist a constant $c_0>0$ and a compact operator K such that there holds in the sense of forms,

$$\begin{aligned} \mathbb {1}_J(H)[H,iA]\mathbb {1}_J(H)\,\ge \, c_0\mathbb {1}_J(H)+K. \end{aligned}$$

The Mourre estimate is said to be strict if it holds with $K=0$, and the constant $c_0$ is referred to as the Mourre constant.

The main motivation for these commutator estimates is that they lead to precise information on the nature of the spectrum of H; see [2, 39].

Theorem A.5

(Mourre’s theory; [2, 39]). Let A be a self-adjoint operator on a Hilbert space ${\mathcal {H}}$, let H be a self-adjoint operator of class $C^1(A)$, and assume that H satisfies a Mourre estimate with respect to A on a bounded open interval $J\subset {\mathbb {R}}$. Then, the following properties hold:

• H has at most a finite number of eigenvalues in J (counting multiplicities);

• if H is of class $C^2(A)$, then H has no singular continuous spectrum in J;

• if the Mourre estimate is strict, then H has no eigenvalue in J. $\diamond$

Next, we adapt these developments to the setting of perturbation theory. First, the following standard lemma states that, if H satisfies a Mourre estimate and if a perturbation V is sufficiently regular, then the perturbed operators $H_g:=H+gV$ also satisfy a corresponding Mourre estimate for g small enough. In view of Sect. 3.2, care is taken not to assume that [H, iA] be H-bounded; the outline of the proof is included for convenience.

Lemma A.6

(Mourre estimates under perturbations). Let A be a self-adjoint operator on a Hilbert space ${\mathcal {H}}$, let H be a self-adjoint operator of class $C^1(A)$, let V be a symmetric $|H|^{1/2}$-bounded operator, and assume that:

• the commutator [H, iA] satisfies the following strengthened version of (A.1),

  $$\begin{aligned} |\langle \phi ,[H,iA]\psi \rangle _{\mathcal {H}}|\,\lesssim \,\Vert (|H|+1)^\frac{1}{2}\phi \Vert _{\mathcal {H}}\Vert (|H|+1)\psi \Vert _{\mathcal {H}}; \end{aligned}$$

  (A.4)

• the commutator [V, iA] extends as an H-bounded operator, in the sense that

  $$\begin{aligned} |\langle \phi ,[V,iA]\psi \rangle _{\mathcal {H}}|\,\lesssim \,\Vert \phi \Vert _{\mathcal {H}}\Vert (|H|+1)\psi \Vert _{\mathcal {H}}. \end{aligned}$$

  (A.5)

Then, the following properties hold.

• (i) The perturbed operator $H_g=H+gV$ is self-adjoint on ${\mathcal {D}}(H)$ and is of class $C^1(A)$ for all $g\in {\mathbb {R}}$.

• (ii) Further assume that H satisfies a Mourre estimate with respect to A on a bounded interval (a, b), with constant $c_0$. Then $H_g$ satisfies a Mourre estimate with respect to A on the restricted interval

  $$\begin{aligned} (a+\eta ,b-\eta ),\qquad \eta :=\tfrac{gC}{c_0} (1+|a|+|b|)^{\frac{3}{2}}, \end{aligned}$$

  for some constant C only depending on the multiplicative constants in (A.4)–(A.5). If in addition [H, iA] is H-bounded, in the sense that $(|H|+1)^{1/2}\phi$ can be replaced by $\phi$ in the right-hand side of (A.4), then the same holds with $\eta =\frac{gC}{c_0}(1+|a|+|b|)$. Finally, if the Mourre estimate for H is strict, then the one for $H_g$ is strict too.

• (iii) Further assume that H is of class $C^2(A)$ and that [[V, iA], iA] extends as an H-bounded operator. Then, $H_g$ is of class $C^2(A)$ for all $g\in {\mathbb {R}}$.$\diamond$

Proof

As the perturbation V is $|H|^\frac{1}{2}$-bounded, the perturbed operator $H_g=H+gV$ is self-adjoint and has the same domain as H for all $g\in {\mathbb {R}}$. The proof of items (i) and (iii) is standard, following the same lines as, for example, [37, proof of Proposition 2.5], starting from identities

$$\begin{aligned} (H_g-z)^{-1}= & {} (H-z)^{-1}(1+gV(H-z)^{-1})^{-1},\\ {}[(H_g-z)^{-1},iA]= & {} [(H-z)^{-1},iA](1+gV(H-z)^{-1})^{-1}\\{} & {} -g(H_g-z)^{-1}V[(H-z)^{-1},iA](1+gV(H-z)^{-1})^{-1}\\{} & {} -g(H_g-z)^{-1}[V,iA](H_g-z)^{-1}, \end{aligned}$$

where $\Im z$ is chosen large enough so that $\Vert gV(H-z)^{-1}\Vert <1$. We skip the detail and turn to item (ii). Assume that H satisfies a Mourre estimate with respect to H on a bounded interval $J=(a,b)$. Let $\eta \in (0,1)$, let $J_\eta :=(a+\eta ,b-\eta )$, and choose $h_\eta \in C_c^\infty ({\mathbb {R}})$ such that $\mathbb {1}_{J_\eta }\le h_\eta \le \mathbb {1}_J$ and $|\nabla h_\eta |\lesssim \frac{1}{\eta }$. Multiplying both sides of the Mourre estimate for H with $h_\eta (H)$, we get for some compact operator K,

$$\begin{aligned} h_\eta (H)[H,iA]h_\eta (H)\,\ge \,c_0h_\eta (H)+h_\eta (H)Kh_\eta (H), \end{aligned}$$

hence, as [V, iA] is H-bounded,

$$\begin{aligned} h_\eta (H)[H_g,iA]h_\eta (H)\,\ge \,\Big (c_0-gC(1+|a|+|b|)\Big )h_\eta (H)+h_\eta (H)Kh_\eta (H). \nonumber \\ \end{aligned}$$

(A.6)

Next, we decompose

$$\begin{aligned}{} & {} h_\eta (H_g)[H_g,iA]h_\eta (H_g)\,=\,h_\eta (H)[H_g,iA]h_\eta (H)\nonumber \\{} & {} \quad +(h_\eta (H_g)-h_\eta (H))[H_g,iA]h_\eta (H_g)+h_\eta (H)[H_g,iA](h_\eta (H_g)-h_\eta (H)). \nonumber \\ \end{aligned}$$

(A.7)

Recalling (A.4), the $|H|^\frac{1}{2}$-boundedness of V, and the H-boundedness of [V, iA], and noting that $\Vert h_\eta (H_g)-h_\eta (H)\Vert \lesssim \frac{1}{\eta }g$, we easily find that the last two right-hand side terms in (A.7) have operator norm bounded by $\frac{gC}{\eta }(1+|a|+|b|)^{3/2}$. Combined with (A.6), this yields

$$\begin{aligned} h_\eta (H_g)[H_g,iA]h_\eta (H_g)\,\ge \,\Big (c_0-\tfrac{gC}{\eta }(1+|a|+|b|)^{\frac{3}{2}}\Big )h_\eta (H_g)+h_\eta (H)Kh_\eta (H). \end{aligned}$$

Now multiplying both sides with $\mathbb {1}_{J_\eta }(H_g)$, the conclusion (ii) follows. $\square$

An important question concerns the perturbation of an eigenvalue embedded in continuous spectrum [46]. In view of formal second-order perturbation theory, Fermi’s golden rule is expected to provide an instability criterion, cf. (A.8), and various works have shown how Mourre’s theory can be used to establish it rigorously, e.g., [1, 20, 31]. Revisiting [31, Theorem 8.8], we can derive for instance the following statement, where care is taken again not to assume that [H, iA] is H-bounded; the outline of the proof is included for convenience.

Theorem A.7

(Instability of embedded bound states). Let A be a self-adjoint operator on a Hilbert space ${\mathcal {H}}$, let H be a self-adjoint operator of class $C^2(A)$, let V be a symmetric $|H|^{1/2}$-bounded operator, and assume that:

• the commutator [H, iA] satisfies (A.4);

• the commutators [V, iA] and [[V, iA], iA] extend as H-bounded operators;

• H satisfies a Mourre estimate with respect to A on a bounded open interval $J\subset {\mathbb {R}}$.

In addition, assume that H has an eigenvalue $E_0\in J$, denote by $\Pi _0$ the associated eigenprojector, let ${\bar{\Pi }}_0:=1-\Pi _0$, assume that the eigenspace satisfies ${{\text {Ran}}}(\Pi _0)\subset {\mathcal {D}}(A^2)$ and ${{\text {Ran}}}(A\Pi _0)\subset {\mathcal {D}}(V)$, and assume that Fermi’s condition holds, that is, there exists $\gamma _0>0$ such that

$$\begin{aligned} \lim _{\varepsilon \downarrow 0}\,\Im \Big \{\Pi _0V{\bar{\Pi }}_0(H-E_0-i\varepsilon )^{-1}{\bar{\Pi }}_0V\Pi _0\Big \}\,\ge \,\gamma _0\Pi _0. \end{aligned}$$

(A.8)

Then, there exists $g_0>0$ and a neighborhood $J_0\subset J$ of $E_0$ such that the perturbed operator $H_g=H+gV$ satisfies

$$\begin{aligned} \sigma _{{\text {pp}}}(H_g)\cap J_0\,=\,\varnothing \qquad \text {for all}\,\, 0<|g|\le g_0. \end{aligned}$$

$\diamond$

Proof

Note that all assumptions of Lemma A.6 are satisfied, hence the perturbed operator $H_g$ is of class $C^2(A)$ and satisfies a Mourre estimate on $J'$ with respect to A for all $J'\Subset J$ and g small enough. Consider the reduced perturbed operator ${\bar{H}}_g:={\bar{\Pi }}_0 H_g{\bar{\Pi }}_0$ on the range ${{\text {Ran}}}({\bar{\Pi }}_0)$, and set also ${\bar{H}}:={\bar{\Pi }}_0H{\bar{\Pi }}_0$, $\bar{V}:={\bar{\Pi }}_0V{\bar{\Pi }}_0$, ${\bar{A}}:={\bar{\Pi }}_0A{\bar{\Pi }}_0$. We follow the approach in [31, Theorem 8.8] and split the proof into three steps.

Step 1. Proof that ${\bar{H}}_g$ is of class $C^2({\bar{A}})$ for all g and that there exists $g_0>0$ and an open interval $J_0\subset J$ with $E_0\in J_0$ such that for all $|g|\le g_0$ the operator $\bar{H}_g$ satisfies a strict Mourre estimate on $J_0$ with respect to ${\bar{A}}$. In particular, in view of Theorem A.5(iii), this entails that ${\bar{H}}_g$ has no eigenvalue in $J_0$ for any $|g|\le g_0$.

It is easily checked that reduced operators ${\bar{A}},\bar{H},{\bar{V}}$ satisfy all the assumptions of Lemma A.6 on ${{\text {Ran}}}({\bar{\Pi }}_0)$. In particular, in order to ensure that $[\bar{V},i{\bar{A}}]$ and $[[{\bar{V}},i{\bar{A}}],i{\bar{A}}]$ are ${\bar{H}}$-bounded, it suffices to decompose

$$\begin{aligned} {}[{\bar{V}},i{\bar{A}}]= & {} {\bar{\Pi }}_0[V,iA]{\bar{\Pi }}_0-{\bar{\Pi }}_0V\Pi _0iA{\bar{\Pi }}_0+{\bar{\Pi }}_0iA\Pi _0V{\bar{\Pi }}_0,\\ {}[[{\bar{V}},i{\bar{A}}],i{\bar{A}}]= & {} {\bar{\Pi }}_0[[V,iA], iA]{\bar{\Pi }}_0 +{\bar{\Pi }}_0ViA\Pi _0 iA{\bar{\Pi }}_0+{\bar{\Pi }}_0 iA\Pi _0iAV{\bar{\Pi }}_0\\{} & {} -\,{\bar{\Pi }}_0V\Pi _0(iA)^2{\bar{\Pi }}_0-{\bar{\Pi }}_0(i A)^2\Pi _0V{\bar{\Pi }}_0\\{} & {} +\,{\bar{\Pi }}_0V\Pi _0iA\Pi _0 iA{\bar{\Pi }}_0+{\bar{\Pi }}_0i A\Pi _0iA\Pi _0V{\bar{\Pi }}_0\\{} & {} +\,2{\bar{\Pi }}_0iA\Pi _0[V,iA]{\bar{\Pi }}_0-2{\bar{\Pi }}_0[V,i A]\Pi _0iA{\bar{\Pi }}_0-2{\bar{\Pi }}_0i A\Pi _0V\Pi _0iA{\bar{\Pi }}_0, \end{aligned}$$

and to note that our assumptions precisely ensure that the different right-hand side terms are all ${\bar{H}}$-bounded. Applying Lemma A.6, we then deduce that ${\bar{H}}_g$ is of class $C^2({\bar{A}})$ for all g and satisfies a Mourre estimate on $J'$ with respect to ${\bar{A}}$ for all $J'\Subset J$ and g small enough. Next, multiplying both sides of this estimate with $\mathbb {1}_L({\bar{H}})$ and using the fact that $\mathbb {1}_L({\bar{H}})$ converges strongly to 0 as $L\rightarrow \{E_0\}$, we deduce that there is a neighborhood $L_0$ of $E_0$ on which ${\bar{H}}$ satisfies a strict Mourre estimate. The claimed strict Mourre estimate for ${\bar{H}}_g$ then follows from Lemma A.6(ii) for any $J_0\Subset L_0$ and g small enough.

Step 2. Proof that, if for some $|g|\le g_0$ the perturbed operator $H_g$ has an eigenvalue $E\in J_0$ with eigenvector $\psi$, then it satisfies

$$\begin{aligned} \lim _{\varepsilon \downarrow 0}~\Im \Big \langle \psi ,\,\Pi _0 W{\bar{\Pi }}_0({\bar{H}}_g-E-i\varepsilon )^{-1}{\bar{\Pi }}_0 W\Pi _0\psi \Big \rangle _{\mathcal {H}}\,=\,0. \end{aligned}$$

(A.9)

This observation is found, for example, in [31, Lemma 8.10], but we repeat the proof for convenience. Decomposing $1=\Pi _0+{\bar{\Pi }}_0$ and using $\Pi _0H\Pi _0=E_0\Pi _0$ and $\Pi _0H{\bar{\Pi }}_0=0$, the eigenvalue equation $H_g\psi =E\psi$ is equivalent to the system

$$\begin{aligned} \left\{ \begin{array}{l} g\Pi _0 W\Pi _0\psi +g\Pi _0 W{\bar{\Pi }}_0\psi =(E-E_0)\Pi _0\psi ,\\ {\bar{H}}_g{\bar{\Pi }}_0\psi +g{\bar{\Pi }}_0 W\Pi _0\psi =E{\bar{\Pi }}_0\psi . \end{array}\right. \end{aligned}$$

(A.10)

For all $\varepsilon >0$, the second equation entails

$$\begin{aligned} {\bar{\Pi }}_0\psi =-g({\bar{H}}_g-E-i\varepsilon )^{-1}{\bar{\Pi }}_0 W\Pi _0\psi -i\varepsilon ({\bar{H}}_g-E-i\varepsilon )^{-1}{\bar{\Pi }}_0\psi . \end{aligned}$$

By Step 1, we know that $E\in J_0$ cannot be an eigenvalue of $\bar{H}_g$, hence the last right-hand side term converges strongly to 0 as $\varepsilon \downarrow 0$ and we get

$$\begin{aligned} {\bar{\Pi }}_0\psi =-g\lim _{\varepsilon \downarrow 0}({\bar{H}}_g-E-i\varepsilon )^{-1}{\bar{\Pi }}_0 W\Pi _0\psi . \end{aligned}$$

inserting this into the first equation of (A.10), taking the scalar product with $\psi$, and taking the imaginary part of both sides, the claim (A.9) follows.

Step 3. Conclusion.

In view of Step 1, as ${\bar{H}}_g$ is of class $C^2({\bar{A}})$ and satisfies a strict Mourre estimate on $J_0$ for all $|g|\le g_0$, Mourre’s theory entails the validity of the following strong limiting absorption principle, cf. [2, 45]: for all $s>\frac{1}{2}$ and $J_0'\Subset J_0$, the limit $\lim _{\varepsilon \downarrow 0}\langle {\bar{A}}\rangle ^{-s}(\bar{H}_g-E-i\varepsilon )^{-1}\langle {\bar{A}}\rangle ^{-s}$ exists in the weak operator topology, uniformly for $E\in J_0'$ and $|g|\le g_0$. (Note that we could not find a reference for the uniformity with respect to g, but it is easily checked to follow from [2, 45] by further making use of the H-boundedness of V and [V, iA].) Decomposing $iA{\bar{\Pi }}_0V\Pi _0=ViA\Pi _0-[V,iA]\Pi _0-iA\Pi _0V\Pi _0$ and noting that our assumptions ensure that the different right-hand side terms are all bounded, we find that $\langle A\rangle {\bar{\Pi }}_0W\Pi _0$ is bounded (and finite-rank), hence the limiting absorption principle entails that the limit

$$\begin{aligned} F_g(E):= & {} \lim _{\varepsilon \downarrow 0}~\Pi _0V{\bar{\Pi }}_0({\bar{H}}_g-E-i\varepsilon )^{-1}{\bar{\Pi }}_0V\Pi _0\\= & {} \lim _{\varepsilon \downarrow 0}~\big (\Pi _0V{\bar{\Pi }}_0\langle {\bar{A}}\rangle \big )\Big (\langle {\bar{A}}\rangle ^{-1}({\bar{H}}_g-E-i\varepsilon )^{-1}\langle {\bar{A}}\rangle ^{-1}\Big )\big (\langle {\bar{A}}\rangle {\bar{\Pi }}_0V\Pi _0\big ) \end{aligned}$$

exists, uniformly for $E\in J_0'$ and $|g|\le g_0$. This ensures in particular that the limit in (A.8) exists. Assumption (A.8) takes the form $\Im F_0(E_0)\ge \gamma _0\Pi _0$, and therefore by uniformity there exists $g_0'>0$ and a neighborhood $J''_0$ of $E_0$ such that

$$\begin{aligned} \Im F_g(E)\ge \tfrac{1}{2}\gamma _0\Pi _0\qquad \text {for all }\,\,E\in J_0'' \textrm{and} |g|\le g_0'. \end{aligned}$$

In view of Step 2, this implies that for $|g|\le g_0'$ any eigenvalue of $H_g$ in $J_0''$ must have eigenvector in ${{\text {Ran}}}({\bar{\Pi }}_0)$. However, this would entail that it is actually an eigenvalue of the reduced operator ${\bar{H}}_g$, which is excluded by Step 1. $\square$

Moreover, if there is enough analyticity for the analytic continuation of the resolvent, the perturbed embedded eigenvalue is actually expected to become a complex resonance when dissolving in the absolutely continuous spectrum [46]. This resonance then describes the metastability of the bound state and the quasi-exponential decay of the system away from this state. While this is not guaranteed in the general framework of Mourre’s theory, the following result by Cattaneo et al. [8] shows how additional regularity allows to develop an approximate dynamical resonance theory. We emphasize that $C^2$-regularity is no longer enough here.

Theorem A.8

(Approximate dynamical resonances; [8]). Let A and H be self-adjoint operators on a Hilbert space ${\mathcal {H}}$, let V be symmetric and $|H|^{1/2}$-bounded, and assume that for some $\nu \ge 0$,

• the unitary group generated by A leaves the domain of H invariant, cf. (A.2);

• for all $0\le j\le 5+\nu$, the iterated commutators ${{\text {ad}}}^j_{iA}(H)$ and ${{\text {ad}}}^j_{iA}(V)$ extend as H-bounded operators;

• H satisfies a Mourre estimate with respect to A on a bounded open interval $J\subset {\mathbb {R}}$.

In addition, assume that H has a simple eigenvalue $E_0\in J$ with normalized eigenvector $\psi _0$, denote by ${\bar{\Pi }}_0$ the orthogonal projection on $\{\psi _0\}^\bot$, and assume that Fermi’s condition is satisfied, that is,

$$\begin{aligned} \gamma _0:=\,\lim _{\varepsilon \downarrow 0}\Im \Big \langle {\bar{\Pi }}_0(V\psi _0),\,(H-E_0-i\varepsilon )^{-1}{\bar{\Pi }}_0(V\psi _0)\Big \rangle \,>\,0. \end{aligned}$$

(A.11)

Then, the perturbed operator $H_g=H+gV$ satisfies the following quasi-exponential decay law: for all smooth cutoff functions h supported in J and equal to 1 in a neighborhood of $E_0$, and for all g small enough, there holds for all $t\ge 0$,

$$\begin{aligned} \Big |\big \langle \psi _0,e^{-iH_gt}h(H_g)\psi _0\big \rangle -e^{-iz_gt}\Big |\,\lesssim _{h,\gamma _0}\,\left\{ \begin{array}{ll} g^2|\!\log g|\langle t\rangle ^{-\nu },&{}\quad \text {if}\quad \nu \ge 0;\\ g^2\langle t\rangle ^{-(\nu -1)},&{}\quad \text {if}\quad \nu \ge 1; \end{array}\right. \end{aligned}$$

where the dynamical resonance $z_g$ is given by Fermi’s golden rule,

$$\begin{aligned} z_g\,=\,E_0+g\langle \psi _0,V\psi _0\rangle -g^2\lim _{\varepsilon \downarrow 0}\Big \langle {\bar{\Pi }}_0(V\psi _0), \,\big (H-E_0-i\varepsilon \big )^{-1}{\bar{\Pi }}_0(V\psi _0)\Big \rangle . \end{aligned}$$

In particular, in view of (A.11), this satisfies $\Im z_g<0$. $\diamond$