The Landau-Pekar Equations: Adiabatic Theorem and Accuracy

Home , Adiabatic theorem

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II Model and Results

We consider the Fröhlich model which describes the interaction between an electron and a quantized phonon ﬁeld. The state of the phonon ﬁeld is represented by an element of the n 2 R3 ⊗s bosonic Fock space F := n≥0 L ( ) , where the subscript s indicates symmetry under the interchange of variables. The system is described by elements Ψ ∈H of the Hilbert space L t H := L2(R3) ⊗ F. (1)

Its time evolution is governed by the Schrödinger equation

i∂tΨt = HαΨt (2) with the Fröhlich Hamiltonian

H := −∆+ d3k |k|−1 eik·xa + e−ik·xa∗ + d3k a∗a . (3) α ˆ k k ˆ k k

∗ Here, ak and ak are the creation and annihilation operators in the Fock space F, satisfying the commutation relations

∗ −2 ′ ′ ∗ ∗ ′ R3 [ak, ak′ ]= α δ(k − k ), [ak, ak ] = [ak, ak′ ] = 0 for all k, k ∈ , (4) for a coupling constant α> 0. One should note that the Hamiltonian is written in the strong coupling units, which gives rise to the α dependence in the commutation relations. These units are related to the usual ones by rescaling all lengths by α and time by α2, see [5, Appendix A]. We will be interested in the limit α →∞. Motivated by Pekar’s Ansatz, we consider the evolution of initial states of product form

2 ψ0 ⊗ W (α ϕ0)Ω. (5)

Here Ω denotes the vacuum of the Fock space F and W (f) for f ∈ L2(R3) denotes the Weyl operator given by

W (f) = exp d3k f(k)a∗ − f(k)a . (6) ˆ k k Note that the Weyl operator is unitary and satisﬁes the shifting property with respect to the creation and annihilation operator, i.e.

∗ −2 ∗ ∗ ∗ −2 W (f)akW (f)= ak + α f(k), W (f)akW (f)= ak + α f(k) (7) for all f ∈ L2(R3). Due to the interaction the system will develop correlations between the electron and the radiation ﬁeld and the solution of (2) will no longer be of product form. However, for an appropriate class of initial states we will show that it can be approximated

2 2 2 up to times short compared to α (in the limit of large α) by a product state ψt ⊗ W (α ϕt)Ω, with (ψt, ϕt) being a solution of the Landau-Pekar equations

3 −1 ik·x −ik·x i∂tψt = −∆+ d k |k| e ϕt(k)+ e ϕt(k) ψt(x), (8) 2 ´ −1 3 −ik·x 2 (iα ∂tϕt(k) = ϕh t(k)+ |k| d x e |ψt(x)| i ´ 2 3 with initial data (ψ0, ϕ0). We deﬁne for ϕ ∈ L (R ) the potential

V (x)= d3k |k|−1 ϕ(k)eik·x + ϕ(k)e−ik·x . (9) ϕ ˆ We are interested, in particular, in initial data of the form (5) where the phonon ﬁeld ϕ is such that the Schrödinger operator

hϕ := −∆+ Vϕ (10) has a non-degenerate eigenvalue at the bottom of its spectrum separated from the rest of the spectrum by a gap, and the electron wave function ψ is a ground state vector of (10).

2 3 Assumption II.1. Let ϕ0 ∈ L (R ) such that 1 R3 e(ϕ0) := inf{hψ, hϕ0 ψi : ψ ∈ H ( ), kψk2 = 1} < 0. (11)

This assumption ensures the existence of a unique positive ground state vector ψϕ0 for hϕ0 with corresponding eigenvalue separated from the rest of the spectrum by a gap of size

Λ(0) > 0. If we then consider solutions of (8) with initial data (ψϕ0 , ϕ0) the spectral gap can be shown to persist at least for times of order α2.

1 3 2 3 Lemma II.1. Let ϕ0 satisfy Assumption II.1 and let (ψt, ϕt) ∈ H (R ) × L (R ) denote the solution of the Landau-Pekar equations with initial value (ψϕ0 , ϕ0). Moreover, let

Λ(t) := inf |e(ϕt) − λ|. (12) λ∈spec(hϕt ) λ6=e(ϕt)

2 Then, for all Λ with 0 < Λ < Λ(0) there is a constant CΛ > 0 such that, for all |t| ≤ CΛα , the Hamiltonian hϕt has a unique positive and normalized ground state ψϕt with eigenvalue e(ϕt) < 0, which is separated from the rest of the spectrum by a gap of size Λ(t) ≥ Λ. The Lemma is proven in Subsection IV.1. Using the persistence of the spectral gap, we can prove the following adiabatic theorem for the solution of the Landau-Pekar equations (8). As mentioned in the introduction, the idea of such result is based on [4, 7], where an adiabatic theorem for the Landau-Pekar equations in one dimension is proved.

1 3 2 3 Theorem II.1. Let T > 0, Λ > 0 and (ψt, ϕt) ∈ H (R ) × L (R ) denote the solution of 1 R3 2 R3 the Landau-Pekar equations with initial value (ψϕ0 , ϕ0) ∈ H ( ) × L ( ). Assume that the

Hamiltonian hϕt has a unique positive and normalized ground state ψϕt and a spectral gap of size Λ(t) > Λ for all |t|≤ T . Then,

t 2 kψ − e−i 0 du e(ϕu)ψ k2 ≤ CΛ−4α−4 1+ 1 + Λ−1 α−4|t|2 , ∀|t|≤ T. (13) t ´ ϕt 2 t Remark II.1. One also has kψ − e−i 0 du e(ϕu)ψ k2 ≤ Cα−2Λ−1 |t| for all |t|≤ T . t ´ ϕt 2

3 Remark II.2. Note that the proof of the theorem only requires the existence of the spectral 4 gap Λ > 0. Assuming Λ to be of order one for times of order α , the theorem shows that ψt 4 is well approximated by the ground state ψϕt for any |t|≪ α . Remark II.3. Lemma II.1 shows that the existence of the ground state and the spectral gap 2 for all times |t| ≤ CΛα can be inferred from Assumption II.1. In this case, (16) is valid for 2 all |t|≤ CΛα without any assumptions on hϕt and Λ(t) at times t> 0. Theorem II.1 implies that there exists CΛ, CΛ > 0 such that

t kψ − e−i 0 du e(ϕu)ψ k2 ≤ C α−4 (14) e t ´ ϕt 2 Λ 2 for all |t|≤ CΛα . f Using Theorem II.1 we can show that the Landau-Pekar equations (8) provide a good approximation to the solution of the Schrödinger equation (2), for initial data of the form (5), with ϕ0 satisfying Assumption II.1 and with ψ0 = ψϕ0 being the ground state of the operator hϕ0 deﬁned as in (10).

1 3 2 3 Theorem II.2. Let ϕ0 satisfy Assumption II.1 and α0 > 0. Let (ψt, ϕt) ∈ H (R ) × L (R ) 1 R3 denote the solution of the Landau-Pekar equations with initial data (ψϕ0 , ϕ0) ∈ H ( ) × L2(R3) and

2 2 ω(t) := α Im ϕt, ∂tϕt + kϕtk2 . (15)

Then, there exists a constant C > 0 such that

t e−iHαtψ ⊗ W (α2ϕ )Ω − e−i 0 du ω(u)ψ ⊗ W (α2ϕ )Ω ≤ Cα−1 |t|1/2 (16) ϕ0 0 ´ t t

for all α ≥ α0. The constant C > 0 depends only on α0 > 0 and the initial condition, i.e. 1 R3 2 R3 (ψϕ0 , ϕ0) ∈ H ( ) × L ( ) and the spectral gap Λ(0) of hϕ0 . Remark II.4. Theorem II.2 shows that the Pekar ansatz is a good approximation for times 2 small compared to α . Note that even though (ψt, ϕt) stay close to their initial values for these times (as shown in Theorem II.1), it is essential to use the time-evolved version in (16). This 2 2 is due to the large factor α in the Weyl operator W (α ϕt), which leads to a very sensitive behavior of the state on ϕt. It remains an open problem to decided whether the Pekar product ansatz remains valid also for times of order α2 and larger. A ﬁrst rigorous result concerning the eﬀective evolution of the Fröhlich polaron was ob- 2 tained in [5], where the product ψt ⊗ W (α ϕ0)Ω, with ψt solving the linear equation

i∂ ψ = −∆ψ + dk |k|−1 eik·xϕ (k)+ e−ik·xϕ (k) ψ (x) (17) t t t ˆ 0 0 t −iHαt 2 was proven to give a good approximation for the solution e ψ0 ⊗ W (α ϕ0)Ω of the Schrödinger equation (2), up to times of order one1. This result was improved in [6], where convergence towards (8) was established for all times |t| ≪ α (the analysis in [6] also gives more detailed information on the solution of Schrödinger equation (2); in particular, it implies convergence of reduced density matrices). Notice that the results of [5, 6] hold for general initial data of the form (5), with no assumption on the relation between the initial electron wave function ψ0 and the initial phonon ﬁeld ϕ0. Theorem II.2 shows therefore that, under

1 In fact, a simple modiﬁcation of the Gronwall argument in [5] leads to convergence for times |t|≪ α.

4 the additional assumption that ψ0 = ψϕ0 the Landau-Pekar equations (8) provide a good approximation to (2) for longer times (times short compared to α2). In this sense, Theorem II.2 also extends the result of [10], where the validity of the Landau-Pekar equations was 2 established for times short compared to α , but only for initial data (ψ0, ϕ0) minimizing the Pekar energy functional (in this case, the solution of (8) is stationary, i.e. (ψt, ϕt) = (ψ0, ϕ0) for all t). In fact, similarly to the analysis in [10], we use the observation that the spectral gap above the ground state energy of hϕt allows us to obtain bounds that are valid on longer time scales (it allows us to integrate by parts, after (60) and after (97); this step is crucial to save a factor of t). The classical behaviour of a quantum field does not only appear in the strong coupling limit of the Fröhlich polaron but has also been studied in other situations. In [9] it was shown in case of the Nelson model that a quantum scalar field behaves classically in a certain limit where the number of field bosons becomes infinite while the coupling constant tends to zero. The emergence of classical radiation was also proven for the Nelson model with ultraviolet cutoff [3, 12], the renormalized Nelson model [1] and the Pauli-Fierz Hamiltonian [13] in situations in which a large number of particles weakly couple to the radiation field. The articles [2, 11, 19] revealed in addition that quantum fields can sometimes be replaced by two-particle interactions if the particles are much slower than the bosons of the quantum field.

III Preliminaries

In this section, we collect properties of the Landau-Pekar equations that are used in the proofs of Theorem II.1 and Theorem II.2. For ψ ∈ L2(R3) we deﬁne the function

σ (k)= |k|−1 d3x e−ik·x|ψ(x)|2. (18) ψ ˆ 2 Next, we notice that the ﬁrst and second derivative of the potential Vϕt are given by

∂ V (x)= V (x)= −α−2 d3k |k|−1 eik·xiϕ (k)+ e−ik·xiϕ (k) t ϕt ϕ˙ t ˆ t t − iα−2 d3k |k|−1 eik·xσ (k) − e−ik·xσ (−k) = −α−2V (x) (19) ˆ ψt ψt iϕt and

−2 −2 ∂tViϕ (x)= Viϕ˙ (x)= V −2 (x)= α Vϕ (x)+ α Vσ (x). (20) t t α (ϕt+σψt ) t ψt Then we deﬁne the energy functional E : H1(R3) × L2(R3) → R

2 E(ψ, ϕ)= hψ, hϕψi + kϕk2. (21) Using standard methods one can show that the Landau-Pekar equations are well posed and that the energy E(ψt, ϕt) is conserved if (ψt, ϕt) is a solution of (8). For a proof of the following Lemma see [6, Appendix C].

1 3 2 3 Lemma III.1 ([6], Lemma 2.1). For any (ψ0, ϕ0) ∈ H (R ) × L (R ), there is a unique global solution (ψt, ϕt) of the Landau-Pekar equations (8). The following conservation laws hold true

3 kψtk2 = kψ0k2 and E(ψt, ϕt)= E(ψ0, ϕ0) ∀t ∈ R . (22)

2 ˙ We use the notation f to denote the derivative of a function f with respect to time.

5 Moreover, there exists a constant C such that

kψtkH1(R3) ≤ C, kϕtk2 ≤ C (23) for all α> 0 and all t ∈ R.

The next Lemma (also proven in [6, Appendix B,C]) collects some properties of quantities occurring in the Landau-Pekar equations.

Lemma III.2. For Vϕ being deﬁned as in (9) there exists a constant C > 0 such that for every ψ ∈ H1(R3) and ϕ ∈ L2(R3)

kVϕk6 ≤ Ckϕk2 and kVϕψk2 ≤ Ckϕk2 kψkH1(R3). (24)

Furthermore, for every δ > 0 there exists Cδ > 0 such that

±Vϕ ≤−δ∆+ Cδ, (25) thus there exists C > 0 such that 1 − ∆ − C ≤ h ≤−2∆ + C. (26) 2 ϕ

Let σψ be deﬁned as in (18). Then, there exists C > 0 such that

2 kσψk2 ≤ CkψkH1(R3). (27)

1 3 2 3 Remark III.1. Let T > 0, Λ > 0 and (ψt, ϕt) ∈ H (R ) × L (R ) denote the solution of the 1 R3 2 R3 Landau-Pekar equations with initial value (ψϕ0 , ϕ0) ∈ H ( ) × L ( ). Assume that the

Hamiltonian hϕt has a unique positive and normalized ground state ψϕt and a spectral gap of size Λ(t) > Λ for all t ≤ T . Lemma III.2 then implies the existence of constant such that

1 3 kψϕt kH (R ) ≤ C ∀ |t|≤ T. (28)

Proof of Lemma III.2. Recall the deﬁnition (9) of the potential Vϕ. We write

d3y V (x) = 23/2π−1/2 Re ϕˇ(y), (29) ϕ ˆ |x − y|2 where ϕˇ denotes the inverse Fourier transform deﬁned for ϕ ∈ L1(R3) through

ϕˇ(x) = (2π)−3/2 d3k eik·xϕ(k). (30) ˆ

The ﬁrst inequality follows directly from the Hardy-Littlewood-Sobolev inequality

kVϕk6 ≤ Ckϕk2. (31)

In order to prove the second inequality we use the ﬁrst one and the Hölder inequality

kVϕψk2 ≤ kVϕk6 kψk3 ≤ Ckϕk2 kψk3. (32)

Since the interpolation inequality together with the Sobolev inequality implies

1/2 1/2 1/2 1/2 kψk3 ≤ kψk2 kψk6 ≤ kψk2 k∇ψk2 , (33)

6 we obtain

1/2 1/2 kVϕψk2 ≤ Ckϕk2 k∇ψk2 kψk2 ≤ Ckϕk2kψkH1 (R3). (34)

The second operator inequality follows again from the Sobolev inequality. For this let ψ ∈ H1(R3), then for ε> 0

2 2 −1 2 hψ, Vϕψi≤ CkVϕk6kψk12/5 ≤ CkVϕk6 εk∇ψk2 + ε kψk2 , (35) where we used the interpolation inequality. The ﬁrst inequa lity of the Lemma implies

±hψ, Vϕψi ≤ hψ, (−ε∆+ Cε) ψi, (36) and (26) follows. The last inequality of the Lemma follows from the observation that dk kσ k2 = dxdy |ψ(x)|2|ψ(y)|2eik·(x−y) ψ 2 ˆ |k|2 ˆ |ψ(x)|2|ψ(y)|2 = 2π2 dxdy ≤ Ck|ψ|2k2 , (37) ˆ |x − y| 6/5 where we used again the Hardy-Littlewood-Sobolev inequality. As before, the interpolation and the Sobolev inequality imply (27).

IV Proof of the adiabatic theorem

IV.1 The ground state ψϕt

Before proving the adiabatic theorem, we show that the spectral gap of hϕt does not close for 2 times of order α . In particular, we show the existence of the ground state ψϕt .

Lemma IV.1. Let ϕ0 satisfy Assumption II.1. Then, there exists a unique positive and 1 R3 2 R3 normalized ground state ψϕ0 of hϕ0 = −∆+ Vϕ0 . Let (ψt, ϕt) ∈ H ( ) × L ( ) denote the solution of the Landau-Pekar equations (8) with initial value (ψϕ0 , ϕ0). There exists C > 0 2 such that for all |t|≤ Cα , there exists a unique, positive and normalized ground state ψϕt of hϕt = −∆+ Vϕt with corresponding eigenvalue e(ϕt) < 0. It satisﬁes

−2 −1 ∂tψϕt = α RtViϕt ψϕt with Rt = qt(hϕt − e(ϕt)) qt, (38)

2 R3 where qt = 1 − |ψϕt ihψϕt | denotes the projection onto the subspace of L ( ) orthogonal to the span of ψϕt . 6 R3 R Proof. Lemma III.1 and Lemma III.2 imply that Vϕt ∈ L ( ) for all t ∈ . The existence of the ground state ψϕt at time t = 0 then follows from the negativity of the inﬁmum of the spectrum (see [15, Theorem 11.5]). In order to prove the existence of the ground state ψϕt of hϕt at later times, it suﬃces to show that e(ϕt) is negative. For this we pick the ground state ψϕ0 at time t = 0 and estimate

t −2 e(ϕt) ≤ hψϕ0 , (−∆+ Vϕt )ψϕ0 i = e(ϕ0) − α ds hψϕ0 ,Viϕs ψϕ0 i ˆ0 t −2 2 −2 ≤ e(ϕ0)+ C ds α kϕsk2 kψϕ0 kH1(R3) ≤ e(ϕ0)+ C|t|α , (39) ˆ0

7 by means of (19), Lemma III.1 and Lemma III.2. Thus if we restrict our consideration to −1 2 times |t|

0 = (hϕt − e(ϕt)) ψϕt (40)

Diﬀerentiating both sides of the equality with respect to the time variable leads to

˙ ˙ 0= hϕt − e˙(ϕt) ψϕt + (hϕt − e(ϕt)) ψϕt . (41) ˙ −2 On the one hand hϕt = Vϕ˙ t = −α Viϕt by means of (19) and on the other hand, the Hellmann-Feynman theorem implies

˙ −2 e˙(ϕt)= hψϕt , hϕt ψϕt i = −α hψϕt ,Viϕt ψϕt i (42) so that (41) becomes

−2 ˙ 0= −α qtViϕt ψϕt + (hϕt − e(ϕt)) ψϕt . (43)

R ˙ Since ψϕt is chosen to be real and normalized for all t ∈ , it follows that ψϕt , ψϕt = 0 for all t ∈ R. Hence,

˙ −2 −1 −2 ψϕt = α qt(hϕt − e(ϕt)) qtViϕt ψϕt = α RtViϕt ψϕt . (44)

Using the Lemma above, we prove Lemma II.1.

Proof of Lemma II.1. By the min-max principle [15, Theorem 12.1], the ﬁrst excited eigenvalue of hϕt (or the bottom of the essential spectrum) is given by

e (t)= inf sup hψ, h ψi. (45) 1 2 3 ϕt A⊂L (R ) ψ∈A dimA=2 kψk2=1

2 3 For any ψ ∈ L (R ) with kψk2 = 1 we have by Lemma III.2

t −2 hψ, hϕt ψi =hψ, hϕ0 ψi− α ds hψ, Viϕs ψi ˆ0 −2 2 −2 2 ≥hψ, hϕ0 ψi− C|t|α sup kϕsk2k∇ψk2 − C|t|α sup kϕsk2kψk2 |s|≤|t| |s|≤|t| −2 −2 ≥(1 − 2C|t|α )hψ, hϕ0 ψi− C|t|α (46)

Inserting in (45), we conclude that

−2 −2 e1(t) ≥ (1 − 2C|t|α )e1(0) − C|t|α . (47)

−2 With e(ϕt) ≤ e(ϕ0)+ C|t|α (see (39)), we obtain

Λ(t) ≥ Λ(0) − C|t|α−2. (48)

−1 Using the persistence of the spectral gap, the resolvent Rt = qt (hϕt − e(ϕt)) qt can be estimated as follows.

8 1 3 2 3 Lemma IV.2. Let T > 0, Λ > 0 and (ψt, ϕt) ∈ H (R ) × L (R ) denote the solution of 1 R3 2 R3 the Landau-Pekar equations with initial value (ψϕ0 , ϕ0) ∈ H ( ) × L ( ). Assume that the

Hamiltonian hϕt has a unique positive and normalized ground state ψϕt with e(ϕt) < 0 and a spectral gap of size Λ(t) > Λ for all t ≤ T . Then, for all |t|≤ T

−1 1/2 1/2 −1 1/2 kRtk≤ Λ , k(−∆ + 1) Rt k≤ C(1 + Λ ) , (49) and

−3/2 −2 −1 1/2 kR˙ tk≤ CΛ α (1 + Λ ) (50) with C > 0 depending only on ϕ0. Proof. Since the spectral gap is at least of size Λ > 0 for times |t|≤ T , it follows that

−1 kRtk≤ Λ . (51)

To prove the second inequality, we estimate for arbitrary ψ ∈ L2(R3)

1/2 1/2 2 1/2 1/2 1/2 1/2 k(−∆ + 1) Rt ψk2 = hψ, Rt (−∆ + 1)Rt ψi≤ Chψ, Rt (hϕt + 1) Rt ψi, (52) where the last inequality follows from Lemma III.2. Thus,

1/2 1/2 2 1/2 1/2 k(−∆ + 1) Rt ψk2 ≤ Chψ, Rt (hϕt − e(ϕt)+ e(ϕt) + 1) Rt ψi = Chψ, qt + (e(ϕt) + 1)Rt ψi ≤ Chψ, 1+ Rt ψi (53) since e(ϕt) < 0 for all |t|≤ T by assumption. The gap condition then implies

1/2 1/2 2 −1 k(−∆ + 1) Rt ψk2 ≤ C(1 + Λ ). (54)

In order to prove the third bound of the Lemma we calculate (with pt = 1 − qt)

˙ −2 2 −2 2 −1 Rt = −α ptViϕt Rt − α Rt Viϕt pt + qt ∂t(hϕt − e(ϕt)) qt (55) by means of the Leibniz rule and Lemma IV.1. With the resolvent identities, (19) and (42) this becomes

˙ −2 2 −2 2 −1 ˙ −1 Rt = −α ptViϕt Rt − α Rt Viϕt pt − qt(hϕt − e(ϕt)) hϕt − e˙(ϕt) (hϕt − e(ϕt)) qt −2 2 −2 2 −2 = −α ptViϕt Rt − α Rt Viϕt pt + α Rt (Viϕt − hψϕt ,Viϕt ψϕt i) Rt. (56)

Hence, Lemma III.2 leads to

˙ −2 −2 2 2 kRtk≤Cα kViϕt Rtk kRtk + Cα kRtk kψϕt kH1(R3) −2 1/2 −2 2 2 ≤Cα k(−∆ + 1) Rtk kRtk + α kRtk kψϕt kH1(R3) ≤CΛ−3/2α−2(1 + Λ−1)1/2 + Cα−2Λ−2, (57) for all |t|≤ T , where we used (28) and the second bound of the Lemma.

9 IV.2 Proof of Theorem II.1 t In the following we denote ψ = e−i 0 du e(ϕu)ψ . The fundamental theorem of calculus ϕt ´ ϕt implies that e t 2 d kψt − ψϕt k2 = − ds 2 Rehψs, ψϕs i ˆ0 ds t e e = − 2 ds Reh−ihϕs ψs, ψϕs i ˆ0 t e −2 − 2 ds Rehψs, −ie(ϕs)+ α RsViϕs ψϕs i ˆ0 t −2 e = − 2α ds Rehψs,RsViϕs ψϕs i, (58) ˆ0 e where we used that hϕs ψϕs = e(ϕs)ψϕs and Lemma IV.1 to compute the derivative of the ground state ψϕs . Using the Cauchy-Schwarz inequality, Lemma II.1 and Lemma III.2 together with Lemma IV.2e and (28), wee obtain the inequality from Remark II.1, i.e. a bound s of order α−2|t|. In the following we shall improve this bound. We deﬁne ψ := ei 0 dτ e(ϕτ )ψ s ´ s satisfying e

i∂sψs = (hϕs − e(ϕs)) ψs (59) and write (58) as e e

t 2 −2 kψt − ψϕt k2 = − 2α ds Rehqsψs,RsViϕs ψϕs i. (60) ˆ0 e e Then, we exploit that the time derivative of ψs is of order one while the time derivatives of −2 Rs, Viϕs and ψϕs are of order α (compare also with [20, p.9]). We observe that e −2 i∂s ψs − ψϕs = (hϕs − e(ϕs)) ψs + iα RsViϕs ψϕs (61) Hence, e e

−2 2 qsψs = Rsi∂s ψs − ψϕs − iα RsViϕs ψϕs . (62) e e 3 Plugging this identity into (60) and using that Im ψϕs ,Viϕs RsViϕs ψϕs = 0, we obtain

t 2 −2 2 kψt − ψϕt k2 = − 2α ds Imh∂s(ψs − ψϕs ),RsViϕs ψϕs i. (63) ˆ0 e e Integrating by parts and using the initial condition ψ0 = ψϕ0 lead to

t 2 −2 2 kψt − ψϕt k2 =2α ds ImhRs ψs − ψϕs , ∂s RsViϕs ψϕs i ˆ0 −2 2 e − 2α Imh ψt − ψϕet ,Rt Viϕt ψϕt i. (64) e

10 The Leibniz rule with (20) and Lemma IV.1 leads to

2 −2 kψt − ψϕt k2 = − 2α ImhRt ψt − ψϕt ,RtViϕt ψϕt i (65a) t −2 2 e + 2α ds ImhRs eψs − ψϕs , e∂sRs Viϕs ψϕs i (65b) ˆ0 t α−4 ds R ψ ψe ,R V ψe + 2 Imh s s − ϕs s ϕs+σψs ϕs i (65c) ˆ0 t −4 e 2 e + 2α ds ImhRs ψs − ψϕs , (RsViϕs ) ψϕs i. (65d) ˆ0 Using Lemma III.2 the ﬁrst term can be estimated by e e

−2 2 |(65a)|≤ Cα kRtk kViϕt ψϕt k2 ψt − ψϕt 2

−2 2 1 3 ≤ Cα kRtk kψϕt keH (R ) kϕtk2 eψt − ψϕt . (66) 2

1 3 On the one hand Lemma III.1 and (28)e show that kϕtk2 and keψϕ t kH (R ) are uniformly bounded in time. On the other hand Lemma IV.2 implies that the resolvent Rt is bounded for all times |t|≤ T , so that we obtain

2 −2 −2 1 −4 −4 |(65a)|≤ CΛ α ψt − ψϕt ≤ ψt − ψϕt + CΛ α , ∀ |t|≤ T. (67) 2 2 2

Similarly, we bound the second terme by e t −2 ˙ |(65b)|≤ Cα ds kRsk kViϕs ψϕs k2 kRsk ψs − ψϕs ˆ0 2 t −4 −5/2 −1 1/2e e ≤ Cα Λ (1 + Λ ) ds ψs − ψϕs (68) ˆ0 2

for all |t|≤ T , using Lemma III.1, Lemma III.2 and Lemma IV.2.e The third term (65c) can 2 be bounded using kσψt k2 ≤ CkψtkH1(R3) ≤ C by Lemma III.1. We ﬁnd

t −2 −4 |(65c)|≤ CΛ α ds ψs − ψϕs , ∀ |t|≤ T. (69) ˆ0 2

Using the same ideas we estimate the last term by e

t −4 2 |(65d)|≤ Cα ds kRsk kViϕs Rsk ψs − ψϕs ˆ0 2 t −1 −4 2 e ≤ CΛ α ds kViϕs Rsk ψ s − ψϕs . (70) ˆ0 2

Lemma III.2 implies that for all ψ ∈ H1(R3) e

1 3 kViϕs ψk2 ≤ Ckϕsk2 kψkH (R ) (71)

−1/2 and therefore that kViϕs (−∆ + 1) k≤ C for all |t|≤ T by Lemma III.2. Hence

1/2 −1/2 −1 1/2 kViϕs Rsk≤ Ck(−∆ + 1) Rsk≤ CΛ (1 + Λ ) , (72)

11 for all |t|≤ T where we used Lemma IV.2 for the last inequality. Thus,

t −4 −2 −1 |(65d)|≤ Cα Λ (1 + Λ ) ds ψs − ψϕs , ∀ |t|≤ T, (73) ˆ0 2

In total, this leads to the estimate e

t 2 −4 −4 −2 −1 −4 kψt − ψϕt k2 ≤ CΛ α + CΛ 1 + Λ α ds kψs − ψϕs k2. (74) ˆ0 A Gronwall type argumente then leads to e

2 −4 −4 −4 −1 2 −8 2 kψt − ψϕt k2 ≤ CΛ α + CΛ 1 + Λ α |t| . (75) e

V Accuracy of the Landau-Pekar equations

V.1 Preliminaries For notational convenience we deﬁne

Φ = d3k |k|−1 eik·xa + e−ik·xa∗ =Φ+ +Φ− (76) x ˆ k k x x with

Φ+ = d3k |k|−1eik·xa , and Φ− = d3k |k|−1e−ik·xa∗. (77) x ˆ k x ˆ k

In addition we introduce for f ∈ L2(R3) the creation operator a∗(f) and the annihilation operator a(f) which are given through

a∗(f)= d3k f(k)a∗, a(f)= d3k f(k)a (78) ˆ k ˆ k

3 ∗ and bounded with respect to the number of particles operator N = d k akak, i.e. ´ 1/2 ∗ −2 1/2 ka(f)ξk ≤ kfk2 kN ξk, ka (f)ξk ≤ kfk2k(N + α ) ξk (79)

∗ for all ξ ∈ F. Moreover, recall the deﬁnition (6) of the Weyl operator W (f) = ea (f)−a(f). 2 3 For a time dependent function ft ∈ L (R ) the time derivative of the Weyl operator is given by

α−2 ∂ W (f )= (hf , ∂ f i − h∂ f ,f i) W (f ) + (a∗(∂ f ) − a(∂ f )) W (f ). (80) t t 2 t t t t t t t t t t t t The proof of this formula can be found in [6, Lemma A.3].

12 V.2 Proof of Theorem II.2 It should be noted that (16) is valid for all times which are at least of order α2 because both states in the inequality have norm one. To show its validity for shorter times we split the norm diﬀerence by the triangle inequality into two parts and use Remark II.1 to estimate

t 2 e−iHαtψ ⊗ W (α2ϕ )Ω − e−i 0 du ω(u)ψ ⊗ W (α2ϕ )Ω ϕ0 0 ´ t t t 2 ≤ 2 e−iHαtψ ⊗ W (α2ϕ )Ω − e−i 0 du ω(u)ψ ⊗ W ( α2ϕ )Ω ϕ0 0 ´ ϕt t 2 ψ W α2ϕ ψ W α2ϕ + 2 ϕt ⊗ ( t)Ω − t ⊗ ( t)Ω e t 2 ≤ Cα− 2 |t| + 2 e−iHαtψ ⊗ W (α2ϕ )Ω − e−i 0 du ω(u)ψ ⊗ W (α2ϕ )Ω (81) e ϕ0 0 ´ ϕt t

t for all |t| ≤ C α2 where we used the notation ψ = e−i 0 du e(eϕu)ψ . Therefore it remains Λ ϕt ´ ϕt to estimate the second term. For this, we introduce

s e ξ = ei 0 du ω(u)W ∗(α2ϕ )e−iHαsψ ⊗ W (α2ϕ )Ω (82) s ´ s ϕ0 0 to shorten the notation and compute

∗ 2 2 2 ∗ W (α ϕt)HαW (α ϕt)= hϕt + kϕtk2 + N +Φx + a(ϕt)+ a (ϕt).

Using (7), this leads to

∗ 2 2 ∗ 2 2 i∂sξs = i∂sW (α ϕs) W (α ϕs)ξs + W (α ϕs)HαW (α ϕs) − ω(s) ξs ∗ = hϕs +Φx − a(σψs ) − a (σψs )+ N ξs. (83) Note that ξ0 = ψϕ0 ⊗ Ω. We apply Duhamel’s formula and use the unitarity of the Weyl 2 operator W (α ϕt) to compute

t 2 e−iHαtψ ⊗ W (α2ϕ )Ω − e−i 0 du ω(u)ψ ⊗ W (α2ϕ )Ω ϕ0 0 ´ ϕt t t t 2 2 = ξt − ψϕt ⊗ Ω = ds ∂s ξs − ψϕse⊗ Ω = −2Re ds ∂s ξs, ψϕs ⊗ Ω . (84) ˆ0 ˆ0

e 3 −1 ik·x −ike·x ∗ e Recall that Φx = d k |k| e ak + e ak . We obtain ´ t 2 e−iHαtψ ⊗ W (α2ϕ )Ω − e−i 0 du ω(u)ψ ⊗ W (α2ϕ )Ω ϕ0 0 ´ ϕt t t

= 2Im ds i∂sξs, ψϕs ⊗ Ω − ξs, i∂seψϕs ⊗ Ω ˆ0 t e e ∗ = 2Im ds ξs, hϕs − e(ϕs)+Φx − a(σψs ) − a (σψs )+ N ψϕs ⊗ Ω ˆ0 t −2 e − 2α Re ds ξs,RsViϕs ψϕs ⊗ Ω ˆ0 t t − ∗ e −2 = 2Im ds ξs, Φx − a (σψs ) ψϕs ⊗ Ω − 2α Re ds ξs,RsViϕs ψϕs ⊗ Ω . (85) ˆ0 ˆ0 e e

13 −1 Here we used the deﬁnition Rs = qs(hϕs − e(ϕs)) qs and Lemma IV.1. Thus if we insert the ∗ identity 1= ps + qs and note that qsa (σψs )ψϕs ⊗ Ω = 0 and qsψϕs = 0, we get

t 2 e−iHαtψ ⊗ W (α2ϕ )Ω − e−i 0 du ω(u)ψ ⊗ W (α2ϕ )Ω ϕ0 0 e ´ ϕt e t t −2 = −2α Re ds ξs,RsViϕs ψϕse⊗ Ω (86a) ˆ0 t − ∗e + 2Im ds ξs,ps Φx − a (σψs ) ψϕs ⊗ Ω (86b) ˆ0 t − e + 2Im ds ξs − ψϕs ⊗ Ω,qsΦx ψϕs ⊗ Ω . (86c) ˆ0

We observe that the ﬁrst term (86a) is alreadye of the righte order, namely α−2t. To be more precise, t −2 |(86a)|≤ 2α ds kξsk kRsk Viϕs ψϕs ˆ0 2 t −2 e −2 ≤ 2α dskϕsk2 kRsk ψϕ ≤ Cα |t| (87) s 1 R3 ˆ0 H ( )

2 for all |t| ≤ CΛα where we used Lemma III.2, Lemma e IV.2, Lemma III.1 and (28). We estimate the remaining two terms (86b) and (86c) separately.

The term (86b) We have t |(86b)|≤ 2 ds ξ , d3k a∗ |k|−1 ψ , e−ik·ψ − ψ , e−ik·ψ ψ ⊗ Ω ˆ s ˆ k ϕs ϕs s s ϕs 0 t ≤ 2 ds d3k a∗ |k|−1 ψ ,e e−ik·ψ e− ψ , e−ik·ψ ψ e⊗ Ω ˆ ˆ k ϕs ϕs s s ϕs 0 t = 2 ds d3k a∗ |k|−1 ψe , e−ik· eψ − ψ + ψ −eψ , e− ik·ψ ψ ⊗ Ω . ˆ ˆ k ϕs ϕs s ϕs s s ϕs 0 (88) e e e e

∗ −1 2 3 Since ka (f)ψ ⊗ Ωk = α kfk2 kψk for all f ∈ L (R ), we ﬁnd 1/2 t 2 2 −1 3 −2 −ik· −ik· |(86b)|≤ Cα ds d k |k| ψϕs , e ψϕs − ψs + ψϕs − ψs , e ψs . ˆ0 " ˆ # (89) e e e [ With the help of | · |−2(x)= π−1|x|−1 and the inequalities of Hardy-Littlewood-Sobolev and Hölder we obtain 2 d3k |k|−2 ψ , e−ik· ψ − ψ ˆ ϕs ϕs s =eC d3x ed3y |x − y |−1 (ψ − ψ )(x)ψ (x)ψ (y)(ψ − ψ )(y) ˆ ˆ ϕs s ϕs ϕs ϕs s 2 2 2 ≤ C ψϕs (ψϕs − ψs) ≤ Ce ψϕs − ψs e ψϕs e e 6/5 2 3 2 2 2 e e e e ≤ C ψϕs − ψs ψ ϕs ≤ C ψϕs − ψs (90) 2 H1(R3) 2

e e e 14 2 for all |t|≤ CΛα by (28). Similarly,

2 2 2 3 −2 −ik· 2 d k |k| ψϕs − ψs , e ψs ≤ C kψskH1(R3) ψs − ψϕs ≤ C ψs − ψϕs (91) ˆ 2 2

by Lemma III.1. Hence,e e e

t −1 |(86b)|≤ Cα ds ψs − ψϕs (92) ˆ0 2

2 e for all |t|≤ CΛα . Applying Theorem II.1 leads to

−2 2 |(86b)|≤ Cα |t| for all |t|≤ CΛα . (93)

The term (86c) In order to continue we note that [18, Theorem X.71], whose assumptions can easily shown to be satisﬁed by Lemma III.2, guarantees the existence of a two parameter group Uh(s; τ) on L2(R3) such that

d U (s; τ)ψ = −ih U (s; τ)ψ, U (τ; τ)ψ = ψ for all ψ ∈ H1(R3). (94) ds h ϕs h h Moreover, we deﬁne

i s du e(ϕ ) U (s; τ)= e τ u U (s; τ). (95) h ´ h

We then have for all s ∈ R e d U ∗(s; τ)= U ∗(s; τ)i h − e(ϕ ) (96) ds h h ϕs s and e e d U ∗(s; τ)q f = −i U ∗(s; τ)R f + iU ∗(s; τ)R˙ f + iU ∗(s; τ)R ∂ f , (97) h s s ds h s s h s s h s s s h i 2 3 for fs ∈ L (eR ). This allows us toe express e e

t − (86c) = 2Im ds ξs − ψϕs ⊗ Ω,qsΦx ψϕs ⊗ Ω ˆ0 t ∗ e e ∗ − = 2Im ds Uh(s; 0) ξs − ψϕs ⊗ Ω , Uh(s; 0)qsΦx ψϕs ⊗ Ω (98) ˆ0 by three integrals which contain a derivative withe respecte to the the time variable. Note that ∗ we absorbed the phase factor of ψϕs in the dynamics Uh(s; 0). Thus, t d |(86c)|≤ 2 ds U ∗(es; 0) ξ − ψ ⊗ Ω , e U ∗(s; 0)R Φ−ψ ⊗ Ω ˆ h s ϕs ds h s x ϕs 0 t + 2 ds U ∗(s; 0) ξ −e ψ ⊗ Ω , U ∗(s; 0)R˙ Φ−ψ ⊗ Ω ˆ h s ϕs h s x ϕs 0 t + 2 ds U ∗(s; 0) ξ − ψe ⊗ Ω , Ue∗(s; 0)R Φ−∂ ψ ⊗ Ω . (99) ˆ h s ϕs h s x s ϕs 0 e e

15 In the ﬁrst term we integrate by parts and we use ξ0 = ψϕ0 ⊗ Ω. We ﬁnd

∗ ∗ − |(86c)|≤ 2 Uh (t; 0) ξt − ψϕt ⊗ Ω ,Uh (t; 0)RtΦex ψϕt ⊗ Ω t d + 2 ds Ue∗(s; 0) ξ − ψ ⊗ Ω ,Ue ∗(s; 0)R Φ−ψ ⊗ Ω ˆ ds h s ϕs h s x ϕs 0 t h i + 2 ds ξ − ψ ⊗ Ω , R˙ eΦ−ψ ⊗ Ω e ˆ s ϕs s x ϕs 0 t + 2 ds ξ ,R Φe−∂ ψ ⊗ Ω . e (100) ˆ s s x s ϕs 0

In order to compute the time derivative occurring in the second summand, we use (83) and the notation

∗ δHs =Φx − a(σψs ) − a (σψs )+ N (101) and get d U ∗(s; 0) ξ − ψ ⊗ Ω = −iU ∗(s; 0)δH ξ − α−2U ∗(s; 0)R V ψ ⊗ Ω (102) ds h s ϕs h s s h s iϕs ϕs h i as well as e e

t t ∗ ∗ −2 ∗ Uh(t; 0) ξt − ψϕt ⊗ Ω = −i ds Uh (s; 0)δHs ξs − α ds Uh(s; 0)RsViϕs ψϕs ⊗ Ω. (103) ˆ0 ˆ0 Applying (103)e to the ﬁrst term of the r.h.s. of (100) and (102) to the second,e we obtain

t |(86c)|≤ 2 ds δH ξ ,R Φ−ψ ⊗ Ω (104a) ˆ s s s x ϕs 0 t + 2 ds δH ξ ,U ∗(te; s)R Φ− ψ ⊗ Ω (104b) ˆ s s h t x ϕt 0 t + 2α −2 ds R V ψ ⊗ Ω,ReΦ−ψ ⊗ Ω (104c) ˆ s iϕs ϕs s x ϕs 0 t + 2α−2 ds R V ψe ⊗ Ω,U ∗(t; se)R Φ−ψ ⊗ Ω (104d) ˆ s iϕs ϕs h t x ϕ t 0 t + 2 ds ξ − ψ ⊗eΩ , R˙ Φ−ψ ⊗ Ω e (104e) ˆ s ϕs s x ϕs 0 t + 2 ds ξ ,R Φe−∂ ψ ⊗ Ω . e (104f) ˆ s s x s ϕs 0

The term (104a): According to the deﬁnition of δH, we decompose (104a) as

t − (104a) ≤ 2 ds ξs, N RsΦx ψϕs ⊗ Ω (105a) ˆ0 t e ∗ − + 2 ds ξs, a(σψs )+ a (σψ s ) RsΦx ψϕs ⊗ Ω (105b) ˆ0 t − e + 2 ds ξs, ΦxRsΦx ψϕs ⊗ Ω . (105c) ˆ0

e 16 −2 We notice that N ,Rs = 0 and that N Ψ= α Ψ if Ψ ∈H is a one-phonon state and write the ﬁrst line as t −2 − (105a) = 2α ds ξs,RsΦx ψϕs ⊗ Ω ˆ0

t = 2α−2 ds ξ ,R de3k |k|−1 e−ikxa∗ψ ⊗ Ω . (106) ˆ s s ˆ k ϕs 0

e By means of Lemma IV.2 and Lemma A.1 this becomes t −3 1/2 −3 (105a) ≤ Cα ds kξsk Rs(−∆ + 1) kψϕs k2 ≤ Cα |t| (107) ˆ0

2 ∗ −2 R3 for all |t|≤ CΛα . In a similar way, we calculate a(σψs ), ak = α σψs (k) for all k ∈ and estimate t (105b) = 2 ds ξ , a(σ )+ a∗(σ ) R d3k |k|−1 e−ik·xa∗ ψ ⊗ Ω ˆ s ψs ψs s ˆ k ϕs 0 t ≤ 2 ds ξ ,R d3k |k|−1 e−ik·xa∗(σ )a∗ψ ⊗ Ω e ˆ s s ˆ ψs k ϕs 0 t + 2α−2 ds ξ ,R d3k |k|−1 e−ik·xσ (ek) ψ ⊗ Ω . (108) ˆ s s ˆ ψs ϕs 0

e Applying Lemma A.1, Lemma III.2 and Lemma IV.2 to the ﬁrst line and using the same arguments as in Lemma A.1 for the second line this becomes

t −2 1/2 (105b) ≤ Cα ds kξsk Rs(−∆ + 1) kσψs k2 kψϕs k2 ˆ0

≤ Cα−2 |t| for all |t|≤ C α2. (109) Λ + − Since Φx =Φx +Φx we have

t − (105c) = 2 ds ξs, ΦxRsΦx ψϕs ⊗ Ω ˆ0 t + −e ≤ 2 ds ξs, Φx RsΦx ψϕs ⊗ Ω (110a) ˆ0 t + e− + 2 ds Φx ξs,RsΦx ψϕs ⊗ Ω . (110b) ˆ0

Making use of Lemma A.2 the ﬁrst line can be estimatede by

t 1/2 1/2 − (110a) ≤ C ds (−∆ + 1) N RsΦx ψϕs ⊗ Ω ˆ0 t 1/2 1/2 − e = C ds (−∆ + 1) RsN Φx ψϕs ⊗ Ω . (111) ˆ0

1/2 1/2 2 e 1/2 −1 Since k(−∆+1) Rs(−∆+1) k≤ C for all |t|≤ CΛα by Lemma IV.2 and N Ψ= α Ψ if Ψ ∈H is a one-phonon state, we ﬁnd

t −1 −1/2 − (110a) ≤ Cα ds (−∆ + 1) Φx ψϕs ⊗ Ω (112) ˆ0

e 17 2 for all |t|≤ CΛα . With Lemma A.1 we arrive at t −2 −2 (110a) ≤ Cα ds kψϕs k2 ≤ Cα |t| (113) ˆ0

2 − for all |t|

2 for all |t|

t −2 1/2 −iHαs 2 (110b) ≤ Cα ds (Hα + 1) e ψϕ0 ⊗ W (α ϕ0)Ω ˆ0 t −2 1/2 2 = Cα ds (Hα + 1) ψϕ0 ⊗ W (α ϕ0)Ω ˆ0 t −2 2 1/2 −2 = Cα ds e(ϕ0)+ kϕ0k + 1 = Cα |t| (115) ˆ0 2 −2 −2 for all |t|

The term (104b): For the next estimate, we recall the notation (101) to write (104b) as

t ∗ − (104b) ≤ 2 ds ξs, N Uh (t; s)RtΦx ψϕt ⊗ Ω ˆ0 t ∗ e ∗ − + 2 ds ξs, a(σψs )+ a (σψs Uh(t; s)RtΦx ψϕt ⊗ Ω ˆ0 t ∗ − e + 2 ds ξs, ΦxUh (t; s)RtΦx ψϕt ⊗ Ω . (116) ˆ0

∗ ∗ ∗ ∗ e Using [N ,Uh (t; s)] = [a(σψs ),Uh ( t; s)] = [a (σψs ),Uh (t; s)] = 0 allows us to estimate the ﬁrst two lines in exactly the same way as (105a) and (105b) and leaves us with

t −2 ∗ − (104b) ≤ Cα |t| + 2 ds ξs, ΦxUh (t; s))RtΦx ψϕt ⊗ Ω (117) ˆ0

e 18 2 ∗ for all |t| < CΛα . The diﬃculty of this term is the fact that the operators Φx and Uh (t; s) + − do not commute. Nevertheless, we can use Φx =Φx +Φx to get t −2 + ∗ − (104b) ≤ Cα |t| + 2 ds Φx ξs,Uh (t; s)RtΦx ψϕt ⊗ Ω (118a) ˆ0 t + ∗ − e + 2 ds ξs, Φx U h(t; s)RtΦx ψϕt ⊗ Ω . (118b) ˆ0

Using the same estimates as in (114) and (115) we bounde the ﬁrst integral by t + ∗ − ˜ 2 ds Φx ξs,Uh (t; s)RtΦx ψϕt ⊗ Ω ˆ0 t −2 −1/2 + ∗ −2 1/2 − ˜ = 2 ds (N + α ) Φx ξs,Uh (t; s)(N + α ) RtΦx ψϕt ⊗ Ω ˆ0 t −2 −1/2 + −2 1/2 − ˜ ≤ 2 ds (N + α ) Φx ξs (N + α ) RtΦx ψϕt ⊗ Ω ˆ0

≤ Cα−2 |t| for all |t|≤ C α2. (119) Λ ∗ For the second term Lemma A.2 and Uh(t; s)= Uh(s; t) imply t 1/2 1/2 − ˜ (118b) ≤ C ds (−∆ + 1) N Uh(s; t)RtΦx ψϕt ⊗ Ω . (120) ˆ0

2 3 It follows from Lemma III.2 and (19) that for ξ ∈ L (R ) ⊗ F ∗ ∗ hξ, Uh (s; τ)(−∆ + 1)Uh(s; τ)ξi≤Chξ, Uh (s; τ)(hϕs + 1)Uh(s; τ)ξi s −2 ′ ∗ ′ ′ =Chξ, (hϕτ + 1)ξi− α dτ hξ, Uh (τ ; τ)Viϕτ′ Uh(τ ; τ)ξi ˆτ s −2 ′ ∗ ′ ′ ≤Chξ, (−∆ + 1)ξi + α dτ hξ, Uh (τ ; τ)(−∆ + 1)Uh(τ ; τ)ξi. ˆτ (121) The Gronwall inequality yields

1/2 α−2|s−τ| 1/2 1/2 k(−∆ + 1) Uh(s; τ)ξk≤ Ce k(−∆ + 1) ξk≤ Ck(−∆ + 1) ξk (122) 2 for all |s − τ|≤ CΛα . Thus t 1/2 1/2 − −2 (118b) ≤ C ds (−∆ + 1) N RtΦx ψϕt ⊗ Ω ≤ Cα |t| (123) ˆ0

−2 for all |t| ≤ CΛα , where we concluded by Lemma A.1e and Lemma IV.2 as for the term (110a).

The terms (104c) and (104d): With the help of Lemma III.2, Lemma A.1, Lemma IV.2 and (28) one obtains t (104c) = 2α−2 ds R V ψ ⊗ Ω,R d3k |k|−1 e−ik·xa∗ψ ⊗ Ω ˆ s iϕs ϕs s ˆ k ϕs 0 t −2 e 1/2 e −1/2 3 −1 −ikx ∗ ≤ 2α ds kR k kψ k 1 3 R (−∆ + 1) (−∆ + 1) d k |k| e a ψ ⊗ Ω ˆ s ϕs H (R ) s ˆ k ϕs 0 −3 ≤ Cα |t| (124)e

−3 2 and (104d) ≤ Cα |t| for all |t|

19 The term (104e): Applying Lemma A.1 once more we estimate

t (104e) = 2 ds ξ − ψ ⊗ Ω , R˙ d3k |k|−1 e−ik·xa∗ ψ ⊗ Ω ˆ s ϕs s ˆ k ϕs 0 t ≤ 4 ds R˙ − e∆ + 1 1/2 (−∆ + 1)−1/2 d3k |ek|−1 e−ik ·xa∗ ψ ⊗ Ω . (125) ˆ s ˆ k ϕs 0 From (28), (56) and Lemma IV.2 we get e

−3 2 (104e) ≤ Cα |t| for all |t|

The term (104f): With the help of Lemma IV.1 and Lemma III.2 we get

t (104f) = 2α−2 ds ξ ,R Φ−R V ψ ⊗ Ω ˆ s s x s iϕs ϕs 0 t −2 1/2 −1/2 − ≤ 2α ds Rs(−∆ + 1) (−∆ + 1) Φx RsViϕs ψϕs ⊗ Ω ˆ0 t −3 1/2 −3 ≤ Cα ds Rs(−∆ + 1) kRsViϕs k kψϕs k2 ≤ Cα |t|. (127) ˆ0

Here we used again Lemma A.1 and Lemma IV.2. In total, we obtain −2 2 |(86c)|≤ Cα |t| for all |t|

t 2 e−iHαtψ ⊗ W (α2ϕ )Ω − e−i 0 du ω(u)ψ ⊗ W (α2ϕ )Ω ≤ Cα−2|t|, (129) ϕ0 0 ´ ϕt t

2 for all |t|≤ CΛα . e

A Auxiliary estimates

Lemma A.1. There exists a constant C > 0 such that for all u ∈ L2(R3) and f ∈ L2(R3)

(−∆ + 1)−1/2 d3k |k|−1 e−ik·xa∗u ⊗ Ω ≤ Cα−1 kuk , (130) ˆ k 2

(−∆ + 1) −1/2 d3k |k|−1 e−ik·xa∗(f)a∗u ⊗ Ω ≤ Cα −2 kuk kfk . (131) ˆ k 2 2

Proof. The commutation relations imply

k(−∆ + 1)−1/2 d3k |k|−1e−ik·xa∗u ⊗ Ωk2 ˆ k

3 3 ′ −1 ′ −1 −ik·x ∗ −1 −ik′·x ∗ = d k d k |k| |k | he a u ⊗ Ω, (−∆ + 1) e a ′ u ⊗ Ωi ˆ ˆ k k =α−2 d3k |k|−2he−ik·xu, (−∆ + 1)−1e−ik·xui ˆ =α−2 d3k |k|−2hu, ((−i∇− k)2 + 1)−1ui ˆ 1 =α−2 d3p |uˆ(p)|2 d3k . (132) ˆ ˆ ((p − k)2 + 1)|k|2

20 Since | · |−2 and (| · |2 + 1)−1 are radial symmetric and decreasing functions we have

3 1 3 1 sup d k 2 2 = d k 2 2 < ∞ (133) p∈R3 ˆ ((p − k) + 1)|k| ˆ (k + 1)|k| by the rearrangement inequality. Hence,

k(−∆ + 1)−1/2 d3|k|−1e−ik·xa∗u ⊗ Ωk2 ≤ Cα−2 d3p |uˆ(p)|2 = Cα−2kuk . (134) ˆ k ˆ 2 The second bound of the Lemma follows from the ﬁrst one and the bounds (79) for the creation and annihilation operators.

+ 3 −1 ik·x 3 ∗ Lemma A.2 (Lemma 4, Lemma 10 in [5]). Let Φx = d k |k| e ak and N = d k akak. Then ´ ´

+ 1/2 1/2 −2 −1/2 + 1/2 Φx Ψ ≤ C (−∆ + 1) N Ψ and (N + α ) Φx Ψ ≤ C (−∆ + 1) Ψ . (135)

+ +,> +,< Proof. We split the operator Φx =Φx +Φx , where 3 3 +,> d k ik·x +,< d k ik·x Φx = e ak, Φx = e ak (136) ˆ|k|>κ |k| ˆ|k|<κ |k| for a constant κ> 0 of order one. Then, we deduce from [5, Lemma 10] that

+,> 1/2 1/2 −2 −1/2 +,> 1/2 Φx Ψ ≤ C (−∆ + 1) N Ψ and (N + α ) Φx Ψ ≤ C (−∆ + 1) Ψ , (137)

3 where the constant C > 0 depends only on κ. Since the function f< : R → R given through 2 R3 f<(k)= |k|χ|k|≤κ is in L ( ), [5, Lemma 4] implies

+,< 1/2 −2 −1/2 +,< Φx Ψ ≤ C N Ψ and (N + α ) Φx Ψ ≤ C kΨk . (138)

C > κ for a constant 0 depending only on .

Lemma A.3 ([5], p.7). Let α0 > 0. Let Hα denote the Fröhlich Hamiltonian deﬁned in (3) and ε> 0. There exists a constant Cε (depending on α0) , such that

(1 − ε)(−∆+ N ) − Cε ≤ Hα ≤ (1 + ε)(−∆+ N )+ Cε. (139) for all α ≥ α0. The proof is given in [5] (see Lemma 7 and p.7) and relies on arguments of Lieb and Yamazaki [17] (see [14, p.12] for a concise explanation).

Acknowledgments

N. L. and R. S. gratefully acknowledge ﬁnancial support by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 694227). B. S. acknowledges support from the Swiss National Science Foun- dation (grant 200020_172623) and from the NCCR SwissMAP. N. L. would like to thank Andreas Deuchert and David Mitrouskas for interesting discussions. B. S. and R. S. would like to thank Rupert Frank for stimulating discussions about the time-evolution of a polaron.

21 References

[1] Z. Ammari and M. Falconi, Bohr’s correspondence principle for the renormalized Nelson model. SIAM J. Math. Anal. 49 (6), 5031–5095 (2017).

[2] E. B. Davies, Particle-boson interactions and the weak coupling limit. J. Math. Phys. 20 (3), 345–351 (1979).

[3] M. Falconi, Classical limit of the Nelson model with cutoﬀ. J. Math. Phys. 54 (1), 012303 (2013).

[4] R. L. Frank, A non-linear adiabatic theorem for the Landau-Pekar equations. Oberwolfach Reports, DOI: 10.4171, OWR/2017/27 (2017).

[5] R. L. Frank and B. Schlein, Dynamics of a strongly coupled polaron. Lett. Math. Phys. 104 (8), 911–929 (2014).

[6] R. L. Frank and Z. Gang, Derivation of an eﬀective evolution equation for a strongly coupled polaron. Anal. PDE 10 (2), 379–422 (2017).

[7] R. L. Frank, Z. Gang, A non-linear adiabatic theorem for the one-dimensional Landau- Pekar equations, arXiv:1906.07908v1.

[8] H. Fröhlich, Theory of electrical breakdown in ionic crytals. Proc. R. Soc. Lond. A 160 (901), 230–241 (1937).

[9] J. Ginibre, F. Nironi, and G. Velo, Partially classical limit of the Nelson model. Ann. H. Poincaré 7 (1), 21–43 (2006).

[10] M. Griesemer, On the dynamics of polarons in the strong-coupling limit. Rev. Math. Phys. 29 (10), 1750030 (2017).

[11] F. Hiroshima, Weak coupling limit with a removal of an ultraviolet cutoff for a Hamilto- nian of particles interacting with a massive scalar field. Infin. Dimens. Anal. Qu. 1 (3), 407–423 (1998).

[12] N. Leopold and S. Petrat, Mean-ﬁeld Dynamics for the Nelson Model with Fermions. Ann. H. Poincaré 20(10), 3471–3508 (2019).

[13] N. Leopold and P. Pickl, Derivation of the Maxwell-Schrödinger equations from the Pauli- Fierz Hamiltonian. Preprint, arXiv:1609.01545v2 (2016).

[14] E.H. Lieb and L.E. Thomas, Exact ground state energy of the strong-coupling polaron. Comm. Math. Phys. 183 (3), 511–519 (1997), Erratum: ibid. 188, no.2, 499 – 500 (1997).

[15] E.H. Lieb and M. Loss, Analysis, second edition, American Mathematical Society, 2001.

[16] E.H. Lieb and R. Seiringer, Divergence of the eﬀective mass of a polaron in the strong coupling limit. Preprint, arXiv:1902.04025 (2019).

[17] E.H. Lieb and K. Yamazaki. Ground-state energy and eﬀective mass of polaron. Phys. Rev. 111, 728 – 722 (1958).

[18] M. Reed and B. Simon, Methods of Modern Mathematical Physics II,Fourier Analysis, Self-adjointness, Academic Press, 1975.

22 [19] S. Teufel, Eﬀective N-body dynamics for the massless Nelson model and adiabatic decoupling without spectral gap. Ann. H. Poincaré 3 (5), 939–965 (2002).

[20] S. Teufel, Adiabatic perturbation theory in quantum dynamics, Lecture Notes in Mathe- matics 1821, Springer-Verlag, 2003.

(Nikolai Leopold) Institute of Science and Technology Austria (IST Austria) Am Campus 1, 3400 Klosterneuburg, Austria current address: University of Basel, Department of Mathematics and Computer Science Spiegelgasse 1, 4051 Basel, Switzerland E-mail address: [email protected]

(Simone Rademacher) Institute of Mathematics, University of Zurich Winterthurerstrasse 190, 8057 Zurich, Switzerland current address: Institute of Science and Technology Austria (IST Austria) Am Campus 1, 3400 Klosterneuburg, Austria E-mail address: [email protected]

(Benjamin Schlein) Institute of Mathematics, University of Zurich Winterthurerstrasse 190, 8057 Zurich, Switzerland E-mail address: [email protected]

(Robert Seiringer) Institute of Science and Technology Austria (IST Austria) Am Campus 1, 3400 Klosterneuburg, Austria E-mail address: [email protected]