Letters in Mathematical Physics (2021) 111:45 https://doi.org/10.1007/s11005-021-01380-7 A note on the Fröhlich dynamics in the strong coupling limit David Mitrouskas1,2 Received: 26 March 2020 / Revised: 25 February 2021 / Accepted: 3 March 2021 / Published online: 5 April 2021 © The Author(s) 2021 Abstract We revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in Griesemer (Rev Math Phys 29(10):1750030, 2017). In the latter it was shown that the Fröhlich time evolution applied to the initial state ϕ0 ⊗ ξα , where ϕ0 is the electron ground state of the Pekar energy functional and ξα the associated coherent state of the phonons, can be approximated by a global phase for times small compared to α2. In the present note we prove that a similar approximation holds for t = O(α2) if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to α−2 and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order α2, while the phonon fluctuations around the coherent state ξα can be described by a time-dependent Bogoliubov transformation. Keywords Fröhlich polaron · Strong coupling limit · Effective dynamics · Quantum corrections Mathematics Subject Classification 81Q05 · 81Q15 · 82C10 1 Introduction andmain result 1.1 Themodel The Fröhlich polaron is a quantum model for a large polaron which describes an electron in an ionic lattice interacting with the excitations (phonons) of this lattice [1,12]. Large refers to the assumption that the extension of the electron is much larger compared to the lattice spacing which can thus be approximated by a continuum. In B David Mitrouskas mitrouskas@mathematik.uni-stuttgart.de ; david.mitrouskas@ist.ac.at 1 Fachbereich Mathematik, Universität Stuttgart, Stuttgart, Germany 2 Present Address: Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria 123 http://crossmark.crossref.org/dialog/?doi=10.1007/s11005-021-01380-7&domain=pdf 45 Page 2 of 24 D. Mitrouskas this model, the energy and the dynamics of the electron and the phonons are described by the Fröhlich Hamiltonian HF phys,α = p2 ⊗ 1 + 1 ⊗ N + √ αφ(Gx ) (1) that acts on theHilbert spaceH = L2(R3, dx)⊗F . HereF = ⊕∞ n=0 L 2(R3, dk)⊗ n sym is the bosonic Fock space, x and p = −i∇x denote the position and momentum operator of the electron, respectively, and N is the number operator onF . The interac- tion between the electron and the phonons is described by φ(Gx ) = a(Gx ) + a∗(Gx ) with a( f ) and a∗( f ) the usual annihilation and creation operators on F and Gx the bounded multiplication operator defined for any x ∈ R 3 by the function Gx (k) = e−ikx 2π |k| . (2) The creation and annihilation operators satisfy the canonical commutation relations [a( f ), a∗(g)] = 〈 f , g〉L2 , [a( f ), a(g)] = [a∗( f ), a∗(g)] = 0 ∀ f , g ∈ L2(R3, dk). (3) Finally the number α > 0 is a dimensionless coupling parameter that models the strength of the interaction. The regime α → ∞ is called the strong coupling limit. By a change of units which corresponds to rescaling all lengths by a factor α−1, the Fröhlich Hamiltonian HF phys, is unitarily equivalent to the operator α2HF α with1 HF α = p2 ⊗ 1 + 1 ⊗ α−2N + α−1φ(Gx ). (4) In the analysis of the strong coupling limit it is more convenient to work in strong coupling units, i.e., to use HF α instead of the original Fröhlich Hamiltonian HF phys,α and then consider rescaled values of energy E = α−2Ephys and time t = α2tphys. This explains why t = O(α2) is the time scale we are interested in for the dynamics generated by HF α . In this work we study the large α limit of the time-evolved state�α(t) = e−i HF α t�α for a special initial state, namely the Pekar product state �α = ϕ0 ⊗ ξα where ϕ0 ∈ H1(R3, dx) is the self-trapped electron ground state of the Pekar energy functional (to be defined below) and ξα = W (α f0)∗�0 is the corresponding coherent phonon state. That is to say, �0 = (1, 0, 0, . . .) is the normalized vacuum state in F and W (α f0) = exp ( a∗(α f0) − a(α f0) ) (5) denotes the Weyl operator w.r.t. the function α f0(k) = α 〈 ϕ0,G ·(k)ϕ0 〉 L2 = α 2π |k| ∫ R3 e−ikx |ϕ0(x)|2 dx . (6) 1 See [13, Appendix B]. 123 A note on the Fröhlich dynamics in the strong coupling… Page 3 of 24 45 We recall that the Weyl operator is unitary and satisfies the shift relation W (α f0) ∗a(g)W (α f0) = a(g) + α 〈 g, f0 〉 L2 (7) for any g ∈ L2(R3, dk). The Pekar energy functional is defined by EP(ϕ) = ∫ R3 |∇ϕ(x)|2dx − 1 2 ∫ R3 ∫ R3 |ϕ(x)|2|ϕ(y)|2 |x − y| dxdy (8) with constraint ||ϕ||L2 = 1. It was shown in [20] that EP(ϕ) admits a uniqueminimizer (unique up to spatial translations) ϕ0 ∈ H1(R3, dx) ∩ { ϕ ∈ L2(R3, dx) : ||ϕ||L2 = 1 } (9) that can be chosen positively. The minimizer further solves the Euler–Lagrange equa- tion (hϕ0 − λ)ϕ0 = 0 where hϕ0 = p2 + V ϕ0 , V ϕ0(x) = −2Re〈Gx , f0〉L2 , (10) and λ = EP(ϕ0) − || f0||2L2 . By its positivity, it follows that ϕ0 is the unique ground state of the Schrödinger operator hϕ0 and that λ = inf σ(hϕ0) belongs to the discrete spectrum of hϕ0 , see [25, Sec. 12]. Introducing the orthogonal projector Q = 1 − P with P = |ϕ0〉〈ϕ0|, we hence know that hϕ0 − λ is a positive operator when restricted to the closed subspace ranQ ⊆ L2(R3, dx). This allows the definition of the restricted resolvent R = Q(hϕ0 − λ)−1Q (11) as a bounded operator in L2(R3, dx). The fact that R is independent of α and thus bounded uniformly as α → ∞ is a crucial ingredient in the analysis of the strong coupling limit of �α(t). In a nutshell, it ensures a separation of scales as α → ∞ of the different parts of the Fröhlich Hamiltonian HF α when the latter is applied to states of the form ϕ ⊗ W (α f0)∗η for suitable ϕ ∈ ranQ and η ∈ F . That the scale separation of the different parts in HF α allows an effective description of the Fröhlich dynamics for times t = o(α2) was first observed in [13]. There, it was shown that the wave function �α(t) = e−i HF α tϕ0 ⊗ W (α f0)∗�0 remains close to its initial state up to a global phase factor, i.e., ∣ ∣ ∣ ∣�α(t) − e−iEP(ϕ0)tϕ0 ⊗ W (α f0) ∗�0 ∣ ∣ ∣ ∣ H ≤ C |t |1/2α−1 (12) for some C > 0. Since the initial state is normalized to one, the upper bound is mean- ingful for t � α2. A similar approximation was obtained in [19] for more general initial states, namely Pekar product states in which the electron is initially trapped in the classical field produced by a given coherent state of the phonons. Modulo 123 45 Page 4 of 24 D. Mitrouskas a global phase factor, the effective dynamics is then described by the Pekar prod- uct state ϕLP(t) ⊗ W (α fLP(t))∗�0 with (ϕLP(t), fLP(t)) solving the time-dependent Landau–Pekar equations, cf. [19, Eqn. (8)]. In fact, the effective dynamics in (12) can be understood as the special case in which (ϕ0, f0) are the stationary ground state solutions of the Landau–Pekar equations. The proof of the nonstationary problem, however, is technically more demanding as it is based on a nonlinear adiabatic theo- rem for the solution of the Landau–Pekar equations, see [19, Theorem II.1]. Loosely speaking, the latter shows that the scale separation of the different parts in the Fröh- lich Hamiltonian remains valid on some suitable time scale also in the nonstationary case. An adiabatic theorem for the Landau–Pekar equations in one spatial dimension has been derived in [9,10]. Earlier results about the Fröhlich dynamics in the strong coupling limit provide approximations for t = o(α) but for much more general initial Pekar product states ϕ ⊗W (α f )∗�0 with no particular assumption about the relation between ϕ and f , see [7,8]. To our knowledge, there are no results available to date that provide an approximation for the Fröhlich dynamics for t = O(α2).2 Remark 1 The particular choice of our initial state �α = ϕ0 ⊗ ξα is motivated by Pekar’s approximation of the ground state energy of the Fröhlich Hamiltonian [24]. Taking the expectation value of HF α for general Pekar states ϕ⊗W (α f )∗�0 and mini- mizing over the phononmode f ∈ L2(R, dk) leads to the Pekar functional EP(ϕ). That Pekar’s approximations is accurate in the strong coupling limit was rigorously proved in [3] and later, using a different approach which provided in addition a quantitative error estimate, in [21]. They showed inf σ(HF α ) = EP(ϕ0) + o(1) (13) as α → ∞. The physical picture behind this result is that the electron creates a classical phonon field which in turn leads to an effective trapping of the electron. This self-trapping mechanism is described by the ground state of (8). Let us also mention that the rigorous derivation of the next order contribution in (13) is still an open problem that was recently solved in [11] for a model in which the Fröhlich polaron is assumed to be confined to a suitably bounded region � ⊂ R 3. Remark 2 We note that (4), and equally (1), is somewhat formal since Gx /∈ L2(R3, dk) and hence φ(Gx ) is not a densely defined operator. However, by a well- known argument that goes back to Lieb and Yamazaki [22], the right side of (4) defines a closed semi-bounded quadratic form with domain given by the form domain of p2 ⊗ 1 + 1 ⊗ N . The Hamiltonian HF α is then defined as the unique self-adjoint operator associated with this quadratic form, cf. [26, Thm. VIII.15]. For the purpose of this work, it is sufficient to use the form representation given in (4). Alternative approaches to define the Fröhlich Hamiltonian with an explicit characterization of its domain have been discussed more recently in [14,17]. 2 After the submission of this article new results have been reported about the derivation of the Landau– Pekar equations (and the corresponding quantum fluctuations) for times of order α2 [6,18]. 123 A note on the Fröhlich dynamics in the strong coupling… Page 5 of 24 45 1.2 Effective dynamics Our goal is to derive an approximation similar to (12) for times t = O(α2). To achieve this, we compare �α(t) with an effective time evolution that is generated by the Hamiltonian Hϕ0 α = 1 ⊗ 〈 ϕ0, ( HF α − (α−1φ(G ·) − V ϕ0)(R ⊗ 1)(α−1φ(G ·) − V ϕ0) ) ϕ0 〉 L2 (14) acting on H = L2(R3, dx) ⊗ F . This operator acts nontrivially only on the Fock space. The expectation value in the second factor of the tensor product is taken with respect to the inner product in L2(R3, dx). Equivalently we could write the second factor as a partial trace, TrL2(R3) (|ϕ0〉〈ϕ0| ( HF α − (α−1φ(G ·) − V ϕ0)(R ⊗ 1)(α−1φ(G ·) − V ϕ0) )|ϕ0〉〈ϕ0| ) . (15) In the following proposition we clarify the difference compared to the ansatz in (12) and, more importantly, we obtain the existence of a unitary time evolution generated by Hϕ0 α . Proposition 1.1 For any α > 0 we have W (α f0)H ϕ0 α W (α f0) ∗ − EP(ϕ0) = 1 ⊗ α−2(N − Aϕ0) (16) with the operator Aϕ0 : F → F defined by Aϕ0 = 〈 ϕ0, φ(G ·)(R ⊗ 1)φ(G ·)ϕ0 〉 L2 = ∫ R3 ∫ R3 〈 ϕ0,G ·(k)RG ·(l)ϕ0 〉 L2 (a∗ k + a−k)(a ∗ l + a−l) dkdl (17) Moreover, D(N ) ⊆ D(N − Aϕ0) and N − Aϕ0 is essentially self-adjoint on F . (We denote its closure again by N − Aϕ0 .) We prove this proposition in Sect. 2.4. By unitarity of the Weyl operator, it follows that Hϕ0 α is self-adjoint onH and thus exp(−i Hϕ0 α t) defines a unitary time evolution. Let us emphasize that the effective Hamiltonian acts nontrivially only on the phonons. This implies in particular that the time evolved state exp(−i Hϕ0 α t)ϕ0 ⊗ ξα is still an exact product. Because of the operator Aϕ0 in (16), however, the coherent state structure of the initial state ξα is not conserved. In this regard, our effective dynamics is different compared to the known results discussed in the previous section. Remark 3 As a motivation of our ansatz in (14) let us mention its analogy to the well-known second-order perturbation formula Eε = 〈 u0, ( Hε − εV R0εV ) u0 〉 + O(ε3) (ε � 1) (18) 123 45 Page 6 of 24 D. Mitrouskas for the nondegenerate ground state energy Eε of a suitable Hamiltonian Hε = H0 + εV by means of the ground state vector u0 of H0 and the reduced resolvent R0 = (1 − |u0〉〈u0|)(H0 − 〈u0, H0u0〉)−1(1 − |u0〉〈u0|). Despite this analogy, we emphasize that the expectation value in (14) is taken only w.r.t. to the electron wave function ϕ0 ∈ L2(R3, dx) and not w.r.t. to the full Pekar product ϕ0 ⊗ ξα . The rea- son why the expectation value w.r.t. ϕ0 ⊗ ξα would not lead to a good ansatz for the effective dynamics is the appearance of the factor α−2 in front of the number operator N . 1.3 Main results We are now ready to state our main results. Theorem 1.2 Let ϕ0 ∈ H1(R3, dx) be the unique minimizer of the Pekar functional (8) with ||ϕ0||L2 = 1 and let f0 ∈ L2(R3, dk) be defined as in (6). Let further η0 ∈ F satisfy ||η0||F = 1 and supα>0 ||(N + 1)5/2η0||F < ∞. Then, there are constants c,C > 0 such that ∣ ∣ ∣ ∣ ( e−i HF α t − e−i H ϕ0 α t)ϕ0 ⊗ W (α f0) ∗η0 ∣ ∣ ∣ ∣ H ≤ Cα−1 exp(c|t |α−2) (19) for all t ∈ R and α > 0. Since the initial state is normalized to one, the approximation is accurate for t = O(α2) (indeed, it is accurate for t � α2 ln α). As a direct consequence of (19) together with [e−i H ϕ0 α t , P ⊗1] = 0, we obtain the following statement that shows that the reduced density of the electron remains approximately constant. Corollary 1.3 Under the same assumptions as in Theorem 1.2 there exist constants c,C > 0 such that TrL2 ∣ ∣ ∣TrF ∣ ∣�α(t) 〉〈 �α(t) ∣ ∣ − ∣ ∣ϕ0 〉〈 ϕ0 ∣ ∣ ∣ ∣ ∣ ≤ Cα−1 exp(c|t |α−2) (20) with �α(t) = e−i HF α tϕ0 ⊗ W (α f0)∗η0. Theorem 1.2 shows that on the time scale t = O(α2) it is important to include the creation and annihilation of noncoherent phonons in the effective time evolution. In earlier findings which provided approximations for t = o(α) [7,8] and t = o(α2) [13,19], respectively, it was not necessary to take such noncoherent phonons into account as the effective dynamics was still described by exact Pekar product states. In our next corollary, we use the fact that the operator N − Aϕ0 is quadratic in creation and annihilation operators in order to describe the fluctuations around the coherent phonons by means of a time-dependent Bogoliubov transformation. To make the last statement precise we need to introduce some well-known notions related to the Bogoliubov transformation. The generalized annihilation and creation operators are defined by A(F) = a( f ) + a∗(g) and A∗(F) = a∗( f ) + a(g), respectively, for any F = f ⊕ Jg ∈ L2(R3, dk) ⊕ L2(R3, dk) where J denotes 123 A note on the Fröhlich dynamics in the strong coupling… Page 7 of 24 45 the complex conjugation map (Jg)(x) = g(x). A bounded invertible map V on L2(R3, dk) ⊕ L2(R3, dk) is called a Bogoliubov map if it satisfies A∗(VF) = A(VJ F), [ A(VF), A∗(VG) ] = 〈 F,SG〉 L2⊕L2 (21) for all F,G ∈ L2(R3, dk) ⊕ L2(R3, dk) where J = ( 0 J J 0 ) , S = ( 1 0 0 −1 ) . (22) In case that the Bogoliubov map V is a Hilbert–Schmidt operator, i.e., if V∗V is trace class, it can be implemented as a unitary operator onF . This is the content of the Shale– Stinespring condition which states that there exists a unitary operator UV : F → F such that UV A(F)U∗ V = A(VF) (23) for any F ∈ L2(R3, dk) ⊕ L2(R3, dk) if and only if TrV∗V < ∞, see, e.g., [27, Thm. 9.5]. We call the operator UV the Bogoliubov transformation associated with the Bogoliubov map V . Finally we need the concept of (pure bosonic) quasi-free states inF . A quasi-free state η ∈ F is defined by the property that there is a Bogoliubovmap Vη such that η can be written as the transformed vacuum η = UVη �0 (in particular,�0 is quasi-free). For a detailed introduction toBogoliubov transformations and quasi-free states, we refer to [27, Sec. 9 and 10]. Our next goal is to show that the dynamics of the noncoherent phonons in �α(t) = e−i HF α tϕ0 ⊗ W (α f0)∗η0 can be described by a time-dependent Bogoliubov transformation UVα(t) associated with the Bogoliubov map Vα(t) = exp [ − i t α2 ( 1 − G K −K −1 + G )] Vα(0), Vα(0) = ( 1 0 0 1 ) , (24) where K, G denote integral operators in L2(R3, dk) defined by the kernels K(k, l) = (2π |k|)−1(2π |l|)−1{〈ϕ0, e −ikx Re−ilxϕ0 〉 L2 + 〈 ϕ0, e −ilx Re−ikxϕ0 〉 L2 } , (25) G(k, l) = (2π |k|)−1(2π |l|)−1{〈ϕ0, e +ikx Re−ilxϕ0 〉 L2 + 〈 ϕ0, e −ikx Re+ilxϕ0 〉 L2 } , (26) and where K, G are to be understood as the integral operators with kernels K(k, l) = K(k, l) = K(−k,−l) and G(k, l) = G(k, l) = G(l, k), respectively. Corollary 1.4 Under the same assumptions as in Theorem 1.2 with the additional requirement that η0 ∈ F is quasi-free, there exist constants c,C > 0 such that TrF ∣ ∣ ∣TrL2 ∣ ∣W (α f0)�α(t) 〉〈 W (α f0)�α(t) ∣ ∣ − ∣ ∣UVα(t)η0 〉〈 UVα(t)η0 ∣ ∣ ∣ ∣ ∣ 123 45 Page 8 of 24 D. Mitrouskas ≤ Cα−1 exp(c|t |α−2) (27) with �α(t) = e−i HF α tϕ0 ⊗ W (α f0)∗η0 and UVα(t) the Bogoliubov transformation associated with the time-dependent Bogoliubov map Vα(t) defined in (24). The remainder of this note is organized as follows. We conclude section one with a short remark about the notation and a sketch of the proof of Theorem 1.2. In the second section we begin by stating two preliminary lemmas which are useful for the proof of Theorem 1.2. The latter is given in Sect. 2.2, whereas the preliminary lemmas are proved in Sect. 2.3. Finally we prove Proposition 1.1 together with Corollaries 1.3 and 1.4 in Sect. 2.4. 1.4 Notation From now on, we omit the tensor product with the identity in operators of the form hϕ0 = hϕ0 ⊗ 1 and N = 1 ⊗ N . Moreover we make use of the abbreviation δGx = Gx − f0, (28) with f0 defined as in (6) and by ϕ0 ∈ H1(R3, dx) we always denote the ground state of the Pekar functional (8) satisfying ||ϕ0||L2 = 1. The letter C is used for positive constants that are independent of t and α. The exact value of C may vary from line to line. 1.5 Sketch of the proof The proof of Theorem 1.2 is motivated mainly by the proof of inequality (12) given in [13]. To demonstrate our main idea it is instructive to start with a sketch of the derivation of (12) (in slightly different way compared to [13]). To this end, we use the shift relation (7) to verify W (α f0)H F α W (α f0) ∗ − EP(ϕ0) = hϕ0 − λ + α−2N + α−1φ(δG ·). (29) With W (α f0)e−i HF α tW (α f0)∗ = exp(−iW (α f0)HF α W (α f0)∗t) and by Duhamel’s principle, one then obtains ∣ ∣ ∣ ∣ ( e−i HF α t − e−iEP(ϕ0)t ) ϕ0 ⊗ W (α f0) ∗�0 ∣ ∣ ∣ ∣2 H = −2α−1 Re ∫ t 0 i 〈 e−i(hϕ0−λ+α−2N+α−1φ(δG·))sϕ0 ⊗ �0, Qφ(δG ·)ϕ0 ⊗ �0 〉 H ds. (30) Note that we further used (hϕ0 −λ)ϕ0 = 0 and φ(δGx )ϕ0 ⊗�0 = Qφ(δGx )ϕ0 ⊗�0 which holds because of 〈ϕ0, δG ·ϕ0〉L2 = 0 (recall P = |ϕ0〉〈ϕ0| and Q = 1 − P). A rough estimate of the right side would now lead to an upper bound proportional 123 A note on the Fröhlich dynamics in the strong coupling… Page 9 of 24 45 to |t |α−1. The reason why the right side behaves actually better than this is a phase inside the integral which oscillates with nonzero (α-independent) frequency.3 To take advantage of this phase we rewrite the integrand as 〈 ei(h ϕ0−λ)se−i(hϕ0−λ+α−2N+α−1φ(δG·))sϕ0 ⊗ �0, ( d ds ei(h ϕ0−λ)s R ) φ(δG ·)ϕ0 ⊗ �0 〉 H (31) and then integrate by parts. This leads to a perturbation like expansion of (30) which among other contributions (e.g., the boundary terms which are of order α−1) includes the term 2α−2 Re ∫ t 0 i 〈 e−i(hϕ0−λ+α−2N+α−1φ(δG·))sϕ0 ⊗ �0, φ(δG ·)Rφ(δG ·)ϕ0 ⊗ �0 〉 H ds. (32) Apart from some technical difficulties being related to Gx /∈ L2(R3, dk), one then applies the estimate (here we use that R is uniformly bounded) ∣ ∣ 〈 e−i(hϕ0−λ+α−2N+α−1φ(δG·))sϕ0 ⊗ �0, φ(δG ·)Rφ(δG ·)ϕ0 ⊗ �0 〉 H | ≤ C ∣ ∣ ∣ ∣(N + 1)�0 ∣ ∣ ∣ ∣F (33) in order to arrive at | (32) | ≤ Cα−2|t |. This bound is indeed the reason why (12) is limited to t = o(α2). Our idea to improve upon this is to use the oscillating phase in (32) a second time. Inserting the identity 1 = P + Q on the left of φ(δGx ) we obtain two contributions, (32.a) = 2α−2 ∫ t 0 Re i 〈 e−i(hϕ0−λ)+α−2N+α−1φ(δG·))sϕ0 ⊗ �0, Qφ(δG ·)Rφ(δG ·)ϕ0 ⊗ �0 〉 H ds, (32.b) = 2α−2 ∫ t 0 Re i 〈 e−i(hϕ0−λ)+α−2N+α−1φ(δG·))sϕ0 ⊗ �0, Pφ(δG ·)Rφ(δG ·)ϕ0 ⊗ �0 〉 H ds. In the first one we can proceed similarly as before and improve the bound through integration by parts to | (32.a) | ≤ C(α−2 +|t |α−3). In the second line, however, inte- gration by parts is not applicable since (hϕ0 −λ)P = 0. In other words, there is no fast oscillating phase in this termand thus (32.b) seems to be really of order |t |α−2. To avoid this term in the first place we include the operatorW (α f0)∗α−2(N−Aϕ0)W (α f0) into the effective dynamics, see (16). Starting over again with the new effective dynamics we now obtain an additional term in the first-order Duhamel expansion which can- cels exactly the contribution from (32.b), cf. (51) and (57). Because of the nontrivial 3 One should think of the improved t-dependence in ∫ t 0 ie ibsds = b−1(eibt − 1) compared to ∫ t 0 1ds = t . 123 45 Page 10 of 24 D. Mitrouskas dynamics of the phonons we now have to take into account the number of excita- tions in the effective time evolution. Using a Gronwall argument, this is shown to be bounded by a constant times exp(c|t |α−2) which leads to the exponential factor in (19). This already explains much of our proof and aside from the technical details, it would lead to an upper bound in (19) that is proportional to α−1/2 exp(|t |α−2). By a third integration by parts we can improve the accuracy of this upper bound further and finally arrive at the stated bound in (19). In principle there is no obstacle for continuing the explained strategy by additional integration by parts. In the present work, however, we do not pursue this idea further for the following two reasons. First, the effective Hamiltonian would become more complicated and involve terms with more than two creation and annihilation operators (hence the advantage of having a quadratic Hamiltonian would be lost). Second, an extension of our result to even longer times (e.g., t = O(α3)) would not follow as an immediate consequence. This is due to the difficulty of controlling the number of phonon excitations produced in the effective time evolution. In Lemma 2.2 we shall see that our bound would be insufficient for times t � α2. Remark 4 The described idea of improving the approximation to longer times by changing the effective Hamiltonian as in (16) was similarly used also in [15,16]. These works treat very different models, namely the dynamics of a single tracer particle resp. two tracer particles interacting with an ideal Fermi gas in the high density limit. The used approximations and the proofs of their accuracy, however, are analogous to the one we apply to the Fröhlich Polaron. The scale separation in these models comes from the large momenta of the gas modes that are close to the Fermi surface. (For the ideal Fermi gas, high density is equivalent to a large Fermi momentum.) 2 Proofs 2.1 Preliminary lemmas Before we start with the proof of Theorem 1.2, let us state two lemmas with several helpful estimates. Their proofs are postponed to Sect. 2.3. Lemma 2.1 Let P = |ϕ0〉〈ϕ0| and R as defined in (11). There is a constant C > 0 such that for any � = ϕ0 ⊗ η ∈ H with η ∈ D(N 5/2), the following bounds hold. ∣ ∣ ∣ ∣Rφ(δG ·)� ∣ ∣ ∣ ∣ H + ∣ ∣ ∣ ∣ [ N , Rφ(δG ·) ] � ∣ ∣ ∣ ∣ H ≤ C ∣ ∣ ∣ ∣(N + 1)1/2η ∣ ∣ ∣ ∣F , (34) ∣ ∣ ∣ ∣(Rφ(δG ·))2� ∣ ∣ ∣ ∣ H + ∣ ∣ ∣ ∣ [ N , (Rφ(δG ·))2 ] � ∣ ∣ ∣ ∣ H ≤ C ∣ ∣ ∣ ∣(N + 1)η ∣ ∣ ∣ ∣F , (35) ∣ ∣ ∣ ∣(Rφ(δG ·))3� ∣ ∣ ∣ ∣ H + ∣ ∣ ∣ ∣Pφ(δG ·)(Rφ(δG ·))2� ∣ ∣ ∣ ∣ H + ∣ ∣ ∣ ∣Rφ(δG ·)Pφ(δG ·)Rφ(δG ·)� ∣ ∣ ∣ ∣ H + ∣ ∣ ∣ ∣ [ N , (Rφ(δG ·))3 ] � ∣ ∣ ∣ ∣ H ≤ C ∣ ∣ ∣ ∣(N + 1)3/2η ∣ ∣ ∣ ∣F , (36) ∣ ∣ ∣ ∣(Rφ(δG ·))2Pφ(δG ·)Rφ(δG ·)� ∣ ∣ ∣ ∣ H ≤ C ∣ ∣ ∣ ∣(N + 1)2η ∣ ∣ ∣ ∣F , (37) ∣ ∣ ∣ ∣(Rφ(δG ·))3Pφ(δG ·)Rφ(δG ·)� ∣ ∣ ∣ ∣ H ≤ C ∣ ∣ ∣ ∣(N + 1)5/2η ∣ ∣ ∣ ∣F . (38) 123 A note on the Fröhlich dynamics in the strong coupling… Page 11 of 24 45 Moreover for � ∈ H1(R3, dx) ⊗ F we have ∣ ∣ 〈 �,φ(δG ·)(Rφ(δG ·))3� 〉 H ∣ ∣ ≤ C ( 1 + ∣ ∣ ∣ ∣p� ∣ ∣ ∣ ∣ H )∣ ∣ ∣ ∣(N + 1)2η ∣ ∣ ∣ ∣F . (39) Lemma 2.2 Let η ∈ D(N 5/2) with ||η||F = 1 and supα>0 ||(N + 1)5/2η0||F < ∞. Then there are constants c,C > 0 such that 5∑ j=0 ∣ ∣ ∣ ∣(N + 1) j/2 exp(−iα−2(N − Aϕ0)t)η ∣ ∣ ∣ ∣2F ≤ C exp(c|t |α−2), (40) ∣ ∣ ∣ ∣p e−i HF α tϕ0 ⊗ W (α f0) ∗η ∣ ∣ ∣ ∣ H ≤ C (41) for all t ∈ R and α > 0. 2.2 Proof of Theorem 1.2 We recall the relations W (α f0)H F α W (α f0) ∗ − EP(ϕ0) = hϕ0 − λ + α−2N + α−1φ(δGx ), (42) W (α f0)H ϕ0 α W (α f0) ∗ − EP(ϕ0) = α−2(N − Aϕ0), (43) which are verified by the commutation relations W (α f0)α −2NW (α f0) ∗ = α−2N − α−1φ( f0) + || f0||2L2 , (44) W (α f0)α −1φ(Gx )W (α f0) ∗ = α−1φ(Gx ) + V ϕ0 , (45) which in turn are easily obtained via (7). Using the unitarity of the Weyl operator we thus shall estimate ∣ ∣ ∣ ∣ ( e−i HF α t − e−i H ϕ0 α t)ϕ0 ⊗ W (α f0) ∗η0 ∣ ∣ ∣ ∣ H = ∣ ∣ ∣ ∣ ( e−i(hϕ0−λ+α−2N+α−1φ(δG·))t − e−iα−2(N−Aϕ0 )t)ϕ0 ⊗ η0 ∣ ∣ ∣ ∣ H . (46) For notational convenience let us abbreviate ψα(t) = e−i(hϕ0−λ+α−2N+α−1φ(δGx ))tϕ0 ⊗ η0, ξα(t) = ϕ0 ⊗ e−iα−2(N−Aϕ0 )tη0. (47) By the fundamental theorem of calculus we have ∣ ∣ ∣ ∣ψα(t) − ξα(t) ∣ ∣ ∣ ∣2 H = 2Re〈ψα(0), ψα(0) − ei(h ϕ0−λ+α−2N+α−1φ(δG·))tξα(t)〉H = −2Re 〈 ψα(0), ∫ t 0 d ds ( ei(h ϕ0−λ+α−2N+α−1φ(δG·))sξα(s) ) ds 〉 H . (48) 123 45 Page 12 of 24 D. Mitrouskas Hence we get ∣ ∣ ∣ ∣ψα(t) − ξα(t) ∣ ∣ ∣ ∣2 H = 2Re fα(t) + 2Re gα(t) (49) with fα(t) = −iα−1 ∫ t 0 〈 ψα(s), φ(δG ·)ξα(s) 〉 H ds, (50) gα(t) = −iα−2 ∫ t 0 〈 ψα(s), Pφ(δG ·)Rφ(δG ·)ξα(s) 〉 H ds. (51) Note that here we have used [N − Aϕ0 , P] = 0, Pξα(s) = ξα(s) and (hϕ0 −λ)P = 0. With 1 = P + Q and 〈ϕ0, δG ·ϕ0〉L2 = 0 one further obtains fα(t) = −iα−1 ∫ t 0 〈 ψα(s), Qφ(δG ·)ξα(s) 〉 H ds. (52) In the first part of the proof we do three integrations by part w.r.t. the time variable s. This leads to a perturbation like expansion of (52) into different contributions. In particular, after the first partial integration, we obtain one term that equals −gα(t). Since this term would contribute an error of order |t |α−2, it is crucial that we included the second-order correction in the effective dynamics. All remaining contributions will be estimated separately in the second part of the proof and finally lead to the error in (19). To prepare the first integration by parts we use the restricted resolvent R = Q(hϕ0 − λ)−1Q in order to write fα(t) = −α−1 ∫ t 0 〈 ei(h ϕ0−λ)sψα(s), ( d ds ei(h ϕ0−λ)s R ) φ(δG ·)ξα(s) 〉 H ds. (53) Using d ds ei(h ϕ0−λ)sψα(s) = −iei(h ϕ0−λ)s(α−2N + α−1φ(δGx ) ) ψα(s), (54) d ds ξα(s) = −iα−2(N − Aϕ0)ξα(s), (55) together with Rψα(0) = Rϕ0 ⊗ η0 = 0, one finds by integration by parts fα(t) = −α−1〈ψα(t), Rφ(δG ·)ξα(t) 〉 H (56a) + iα−3 ∫ t 0 〈 ψα(s), R ([ N , φ(δG ·) ] + φ(δG ·)Aϕ0 ) ξα(s) 〉 H ds (56b) + iα−2 ∫ t 0 〈 ψα(s), φ(δG ·)Rφ(δG ·)ξα(s) 〉 H ds. (56c) 123 A note on the Fröhlich dynamics in the strong coupling… Page 13 of 24 45 In the last line the prefactor α−2 is not sufficient andwe need to do a second integration by parts. For that, we insert again the identity 1 = P + Q on the left of φ(δGx ). The term containing P equals iα−2 ∫ t 0 〈 ψα(s), Pφ(δG ·)Rφ(δG ·)ξα(s) 〉 H ds = −gα(t), (57) and thus (56c) + gα(t) = iα−2 ∫ t 0 〈 ψα(s), Qφ(δG ·)Rφ(δG ·)ξα(s) 〉 H ds. (58) In this term we can integrate by parts similarly as in (53) which leads to (56c) + gα(t) = α−2〈ψα(t), Rφ(δG ·)Rφ(δG ·)ξα(t) 〉 H (59a) − iα−4 ∫ t 0 〈 ψα(s), [ N , Rφ(δG ·)Rφ(δG ·) ] ξα(s) 〉 H ds (59b) − iα−4 ∫ t 0 〈 ψα(s), Rφ(δG ·)Rφ(δG ·)Aϕ0ξα(s) 〉 H ds (59c) − iα−3 ∫ t 0 〈 ψα(s), Pφ(δG ·)Rφ(δG ·)Rφ(δG ·)ξα(s) 〉 H ds (59d) − iα−3 ∫ t 0 〈 ψα(s), Qφ(δG ·)Rφ(δG ·)Rφ(δG ·)ξα(s) 〉 H ds. (59e) In the last line we do a third integration by parts, i.e., (59e) = −α−3〈ψα(t), Rφ(δG ·)Rφ(δG ·)Rφ(δG ·)ξα(t) 〉 H (60a) + iα−5 ∫ t 0 〈 ψα(s), [ N , Rφ(δG ·)Rφ(δG ·)Rφ(δG ·) ] ξα(s) 〉 H ds (60b) + iα−5 ∫ t 0 〈 ψα(s), Rφ(δG ·)Rφ(δG ·)Rφ(δG ·)Aϕ0ξα(s) 〉 H ds (60c) + iα−4 ∫ t 0 〈 ψα(s), φ(δG ·)Rφ(δG ·)Rφ(δG ·)Rφ(δG ·)ξα(s) 〉 H ds. (60d) Summing up the above expansion we arrive at fα(t) + gα(t) = (56a) + (56b) + (59a) + (59b) + (59c) + (59d) + (60a) + (60b) + (60c) + (60d). In the remainder of the proof we separately estimate each summand on the right side. This is readily done using basic inequalities in combination with Lemmas 2.1 and 2.2. At the end, we conclude by applying Gronwall’s inequality. 123 45 Page 14 of 24 D. Mitrouskas Term (56a). In the first boundary term from the integration by parts we have (56a) = −iα−1〈ψα(t) − ξα(t), Rφ(δG ·)ξα(t) 〉 H (61) since Rξα(t) = 0. Using the Cauchy–Schwarz inequality we obtain |(56a)| ≤ 1 4 ∣ ∣ ∣ ∣ψα(t) − ξα(t) ∣ ∣ ∣ ∣2 H + α−2 ∣ ∣ ∣ ∣Rφ(δG ·)ξα(t) ∣ ∣ ∣ ∣2 H , (62) and with (34) and (40), ∣ ∣ ∣ ∣Rφ(δG ·)ξα(t) ∣ ∣ ∣ ∣2 H ≤ C ∣ ∣ ∣ ∣(N + 1)1/2ξα(t) ∣ ∣ ∣ ∣2 H ≤ C exp(c|t |α−2). (63) Terms (59a) and (60a). For the other two boundary terms we proceed similarly and find |(59a)| ≤ 1 4 ∣ ∣ ∣ ∣ψα(t) − ξα(t) ∣ ∣ ∣ ∣2 H + α−4 ∣ ∣ ∣ ∣Rφ(δG ·)Rφ(δG ·)ξα(t) ∣ ∣ ∣ ∣ H ≤ 1 4 ∣ ∣ ∣ ∣ψα(t) − ξα(t) ∣ ∣ ∣ ∣2 H + Cα−4 exp(c|t |α−2) (64) as well as |(60a)| ≤ 1 4 ∣ ∣ ∣ ∣ψα(t) − ξα(t) ∣ ∣ ∣ ∣2 H + α−6 ∣ ∣ ∣ ∣Rφ(δG ·)Rφ(δG ·)Rφ(δG ·)ξα(t) ∣ ∣ ∣ ∣ H ≤ 1 4 ∣ ∣ ∣ ∣ψα(t) − ξα(t) ∣ ∣ ∣ ∣2 H + Cα−6 exp(c|t |α−2) (65) where we have used (35) and (36) in combination with (40). Term (56b). In this term we have (56b) = iα−3 ∫ t 0 〈 ψα(s) − ξα(s), R ([ N , φ(δG ·) ] + φ(δG ·)Aϕ0 ) ξα(s) 〉 H ds. (66) Using (34), the third line of (36) and (40) we estimate |(56b)| ≤ α−2 ∫ t 0 ∣ ∣ ∣ ∣ψα(s) − ξα(s) ∣ ∣ ∣ ∣2 H ds + 1 2 α−4 ∫ t 0 (∣ ∣ ∣ ∣ [ N , Rφ(δG ·) ] ξα(s) ∣ ∣ ∣ ∣2 H + ∣ ∣ ∣ ∣Rφ(δG ·)Aϕ0ξα(s) ∣ ∣ ∣ ∣2 H ) ds ≤ α−2 ∫ t 0 ∣ ∣ ∣ ∣ψα(s) − ξα(s) ∣ ∣ ∣ ∣2 H ds + Cα−2(exp(c|t |α−2) − 1). (67) Terms (59b) and (59c). Similarly as in the previous term, |(59b)| + |(59c)| ≤ α−2 ∫ t 0 ∣ ∣ ∣ ∣ψα(s) − ξα(s) ∣ ∣ ∣ ∣2 H ds 123 A note on the Fröhlich dynamics in the strong coupling… Page 15 of 24 45 + 1 2 α−6 ∫ t 0 (∣ ∣ ∣ ∣ [ N , Rφ(δG ·)Rφ(δG ·) ] ξα(s) ∣ ∣ ∣ ∣2 H + ∣ ∣ ∣ ∣Rφ(δG ·)Rφ(δG ·)Aϕ0ξα(s) ∣ ∣ ∣ ∣2 H ) ds, (68) and thus by means of (35), (37) and (40) we obtain |(59b)| + |(59c)| ≤ α−2 ∫ t 0 ∣ ∣ ∣ ∣ψα(s) − ξα(s) ∣ ∣ ∣ ∣2 H ds + Cα−4(exp(c|t |α−2) − 1). (69) Term (59d). In this line we keep the real part (cf. (49)) and have Re (59d) = α−3 ∫ t 0 Im 〈 ψα(s) − ξα(s), Pφ(δG ·)Rφ(δG ·)Rφ(δG ·)ξα(s) 〉 H ds (70) (the imaginary part of the added expectation value is zero). The absolute value of the right side is bounded from above by |Re (59d)| ≤ 1 2 α−2 ∫ t 0 (∣ ∣ ∣ ∣ψα(s) − ξα(s) ∣ ∣ ∣ ∣2 H + 1 2 α−4 ∫ t 0 ∣ ∣ ∣ ∣Pφ(δG ·)Rφ(δG ·)Rφ(δG ·)ξα(s) ∣ ∣ ∣ ∣2 H ) ds ≤ α−2 ∫ t 0 ∣ ∣ ∣ ∣ψα(s) − ξα(s) ∣ ∣ ∣ ∣2 H ds + Cα−2(exp(c|t |α−2) − 1), (71) where one uses (36) and (40) in the second step. Term (60b). By means of (36) and (40) one obtains |(60b)| ≤ α−5 ∫ t 0 ∣ ∣ ∣ ∣ [ N , Rφ(δG ·)Rφ(δG ·)Rφ(δG ·) ] ξα(s) ∣ ∣ ∣ ∣ ds ≤ Cα−3(exp(c|t |α−2) − 1). (72) Term (60c). In this term one can use (38) and (40) to find |(60c)| ≤ α−5 ∫ t 0 ∣ ∣ ∣ ∣Rφ(δG ·)Rφ(δG ·)Rφ(δG ·)Aϕ0ξα(s) ∣ ∣ ∣ ∣ ds ≤ Cα−3(exp(c|t |α−2) − 1). (73) Term (60d). For the last term we apply (39) in combination with ||pψα(s)||H = ||p e−i HF α sϕ0 ⊗ W (α f0) ∗η0||H , (74) 123 45 Page 16 of 24 D. Mitrouskas see (42), as well as (40) and (41). This leads to |(60d)| ≤ α−4 ∫ t 0 ∣ ∣ 〈 ψα(s), φ(δG ·)Rφ(δG ·)Rφ(δG ·)Rφ(δG ·)ξα(s) 〉 H ∣ ∣ ds ≤ Cα−4 ∫ t 0 ( 1 + ∣ ∣ ∣ ∣p e−i HF α sϕ0 ⊗ W (α f0) ∗η0 ∣ ∣ ∣ ∣ H )∣ ∣ ∣ ∣(N + 1)2ξα(s) ∣ ∣ ∣ ∣ H ds ≤ Cα−2(exp(c|t |α−2) − 1). (75) Conclusion. In total, we have shown ∣ ∣ ∣ ∣ψα(s) − ξα(s) ∣ ∣ ∣ ∣2 H ≤ Cα−2 exp(c|t |α−2) + Cα−2 ∫ t 0 ∣ ∣ ∣ ∣ψα(s) − ξα(s) ∣ ∣ ∣ ∣2 H ds, (76) from which the claimed bound follows by the integral version of Gronwall’s inequal- ity. � 2.3 Proofs of Lemmas 2.1 and 2.2 The main tool of the proof of Lemma 2.1 is the commutator method by Lieb and Yamazaki [22] bywhich one improves the behavior of the interaction at largemomenta using the regularity of the electron wave function. More precisely one writes Gx (k) = G̃x (k) − p · Kx (k) + Kx (k) · p (77) with G̃x (k) = Gx (k)χ[0,1](|k|), Kx (k) = k |k|2Gx (k)χ(1,∞)(|k|), (78) where χ denotes the characteristic function, i.e., χA(r) = 1 for all r ∈ A ⊆ R and χA(r) = 0 otherwise. The advantage of rewriting Gx this way is that G̃x and Kx are square-integrable, sup x∈R3 (||G̃x (·)||L2 + ||Kx (·)||L2) < ∞. (79) For the next proof let us recall the common bounds for the annihilation and creation operators, ||a(g)�||H ≤ ||g||L2 ||N 1/2�||H , ||a∗(g)�||H ≤ ||g||L2 ||(N + 1)1/2�||H (80) for any g ∈ L2(R3, dk). 123 A note on the Fröhlich dynamics in the strong coupling… Page 17 of 24 45 Proof of Lemma 2.1 For the proof of (34), we set a# ∈ {a, a∗} and use (77), (79), || f0||L2 < ∞ and (80) to estimate ∣ ∣ ∣ ∣Ra#(δG ·)P� ∣ ∣ ∣ ∣ H ≤ ∣ ∣ ∣ ∣Ra#(G̃ · − f0)P� ∣ ∣ ∣ ∣ H + ∣ ∣ ∣ ∣Rp · a#(K·)P� ∣ ∣ ∣ ∣ H + ∣ ∣ ∣ ∣Ra#(K·) · pP� ∣ ∣ ∣ ∣ H ≤ C (||R|| + ||Rp|| + ||R|| ||pP||)∣∣∣∣(N + 1)1/2η ∣ ∣ ∣ ∣F ≤ C ∣ ∣ ∣ ∣(N + 1)1/2η ∣ ∣ ∣ ∣F , (81) where || · || = || · ||L denotes the norm on the space of bounded operators L (L2(R3, dx)). That ||R||+||pP||L2 < ∞ is clear. To show ||Rp|| < ∞we compute ∣ ∣ ∣ ∣pRψ ∣ ∣ ∣ ∣2 L2 = 〈 ψ, R(hϕ0 − λ)Rψ 〉 L2 + 〈 ψ, R(λ − V ϕ0)Rψ 〉 L2 ≤ 〈 ψ, Rψ 〉 L2 + 1 2 〈 ψ, Rp2Rψ 〉 L2 + C ∣ ∣ ∣ ∣Rψ ∣ ∣ ∣ ∣2 L2 (82) where we used ±V ϕ0 ≤ 1 2 p 2 + C as shown, e.g., in [19, Lemma III.2].4 Since the bound (81) holds equally if R is replaced by P and since [ N , φ(δGx ) ] = a∗(δGx ) − a(δGx ), (83) this proves (34). In order to prove (35) we derive the bound for ∣ ∣ ∣ ∣Ra#1(δG ·)Ra#2(δG ·)� ∣ ∣ ∣ ∣ with a#i ∈ {a, a∗}. Proceeding similarly as in (81), we find ∣ ∣ ∣ ∣Ra#1(δG ·)Ra#2(δG ·)� ∣ ∣ ∣ ∣ H ≤ C ∣ ∣ ∣ ∣(N + 1)1/2R1/2a#2(δG ·)� ∣ ∣ ∣ ∣ H . (84) From here we use (N + 1)1/2R1/2a(δGx )� = R1/2a(δGx )N 1/2�, (85) (N + 1)1/2R1/2a∗(δGx )� = R1/2a∗(δGx )(N + 2)1/2�, (86) together with ∣ ∣ ∣ ∣R1/2a#2(δG ·)� ∣ ∣ ∣ ∣ H ≤ C ∣ ∣ ∣ ∣(N + 1)1/2η ∣ ∣ ∣ ∣F . (87) The latter is obtained in complete analogy to (81). The bounds for the other terms on the l.h.s. of (35) are derived the same way. Since the derivation of (36) and (37) is also very similar, we omit further details. To prove (38) we proceed again as in (81) and find ∣ ∣ 〈 �, a#1(δG ·)Ra#2(δG ·)Ra#3(δG ·)Ra#4(δG ·)� 〉 H ∣ ∣ ≤ ∣ ∣ ∣ ∣� ∣ ∣ ∣ ∣ H ∣ ∣ ∣ ∣ ( a#1(G̃ · − f0) − a#1(K·) · p)Ra#2(δG ·)Ra#3(δG ·)Ra#4(δG ·)� ∣ ∣ ∣ ∣ H 4 Note that our potential V ϕ0 coincides (up to a factor) with Vϕ for ϕ = f0 ∈ L2(R3, dk) in [19]. 123 45 Page 18 of 24 D. Mitrouskas + ∣ ∣ ∣ 〈 �, p · a#1(K·)Ra#2(δG ·)Ra#3(δG ·)Ra#4(δG ·)� 〉 H ∣ ∣ ∣ ≤ C ( 1 + ||p�||H )∣ ∣ ∣ ∣(N + 1)1/2R1/2a#2(δG ·)Ra#3(δG ·)R1/2a#4(δG ·)� ∣ ∣ ∣ ∣ H . (88) By estimating the last factor similarly as the right hand side of (84) we obtain (38). �� Proof of Lemma 2.2 We start by verifying the following bound, ∣ ∣ 〈 η, (N + 1) j−1[N , Aϕ0 ](N + 1)m− jη 〉 F ∣ ∣ ≤ C ∣ ∣ ∣ ∣(N + 1)m/2η ∣ ∣ ∣ ∣2F (89) for 1 ≤ j ≤ m. To do so, use (83) to write [N , Aϕ0 ] = 〈 ϕ0, (a ∗(G ·) − a(G ·))R(a∗(G ·) + a(G ·))ϕ0 〉 + h.c., (90) and then estimate each term separately.We illustrate the argument for the term Aϕ0++ = 〈ϕ0, a∗(G ·)Ra∗(G ·)ϕ0〉L2 for which we have 〈 η, (N + 1) j−1Aϕ0++(N + 1)m− jη 〉 F = {〈 η, (N + 1) j−1(N − 1) m 2 +1− j Aϕ0++(N + 1) m 2 −1η 〉 F (m 2 + 1 ≥ j ) , 〈 η, (N + 1) j+i 2 −1Aϕ0++(N + 3) j−i 2 (N + 1)m− jη 〉 F , with i = m + 2 − j ( j ≥ m 2 + 1 ) . Taking the absolute value and using the Cauchy–Schwarz inequality we can bound the first line from above by ∣ ∣ ∣ ∣(N + 1) j−1(N − 1) m 2 +1− jη ∣ ∣ ∣ ∣F ∣ ∣ ∣ ∣Aϕ0++(N + 1) m 2 −1η ∣ ∣ ∣ ∣F ≤ C ∣ ∣ ∣ ∣(N + 1) m 2 η ∣ ∣ ∣ ∣2F , (91) where we used ||Aϕ0++η||F ≤ C ||(N + 1)η||F , η ∈ F , which is proved the same way as the bound for the left side of (84). Similarly we find the following upper bound for the second line, ∣ ∣ ∣ ∣(N + 1) j+i 2 −1η ∣ ∣ ∣ ∣F ∣ ∣ ∣ ∣Aϕ0++(N + 3) j−i 2 (N + 1)m− jη ∣ ∣ ∣ ∣F ≤ C ∣ ∣ ∣ ∣(N + 1) m 2 η ∣ ∣ ∣ ∣2F . (92) Repeating the same argument for the other terms in (90) leads to the stated bound in (89). Next let ηα(t) = exp(−iα−2(N − Aϕ0)t)η and compute the time derivative d dt 〈 ηα(t), Nmηα(t) 〉 F = −α−2 m∑ j=1 〈 ηα(t), N j−1i[N , Aϕ0 ]Nm− jηα(t) 〉 F (93) which form ∈ {1, 2, 3, 4, 5} is easily checked explicitly. Setting z(t) = ∑5 j=1 ∣ ∣ ∣ ∣(N + 1) j/2ηα(t) ∣ ∣ ∣ ∣2F , we have by (89) and (93), | ddt z(t)| ≤ Cα−2z(t). Since supα>0 z(0) < ∞ by assumption, it follows from Gronwall’s inequality that z(t) ≤ C exp(c|t |α−2). 123 A note on the Fröhlich dynamics in the strong coupling… Page 19 of 24 45 For a proof of (41), let ψα(t) = e−i HF α tϕ0 ⊗ W (α f0)∗η and estimate ∣ ∣ ∣ ∣pψα(t) ∣ ∣ ∣ ∣2 H ≤ C 〈 ψα(t), (HF α + 1)ψα(t) 〉 H = C 〈 ψα(0), (HF α + 1)ψα(0) 〉 H = C ( 1 + EP(ϕ0) + α−2〈η, Nη 〉 F ) (94) for some constant C > 0. Here we used N ≥ 0 and p2 + α−2N ≤ C(HF α + 1) in the first step (see, e.g., [13, Lemma A.5]) and the commutation relation (42) together with 〈ϕ0, δGxϕ0〉L2 = 0 in the third step. �� 2.4 Proofs of Proposition 1.1 and Corollaries 1.3 and 1.4 Proof of proposition 1.1 The identity in (16) follows from EP(ϕ0) = λ + || f0||2L2 together with the commutation relations (44) and (45). That D(N ) ⊆ D(N − Aϕ0) follows from ∣ ∣ ∣ ∣Aϕ0η ∣ ∣ ∣ ∣F = ∣ ∣ ∣ ∣Pφ(G ·)Rφ(G ·)ϕ0 ⊗ η ∣ ∣ ∣ ∣ H ≤ C ∣ ∣ ∣ ∣(N + 1)η ∣ ∣ ∣ ∣F (95) which is proven the sameway as the bound for the l.h.s. of (84). Using (83) one further finds ∣ ∣ 〈 η, [ Aϕ0 , N ] η 〉 F ∣ ∣ ≤ C 〈 η, Nη 〉 F (96) for all η ∈ F0 withF0 ⊆ F denoting the dense subspace of all Fock space vectors that have only finitelymany nonzero components. SinceF0 is a core of the number operator N , we can infer that N − Aϕ0 is essentially self-adjoint by a variant of Nelson’s com- mutator theorem [5, Corollary 1.1]. Alternatively one could conclude self-adjointness of N − Aϕ0 from the criteria for self-adjointness of Fock space operators found in [4]. �� In the following two proofs we make use of the bound TrH1 ∣ ∣TrH2 |�〉〈�|∣∣ ≤ ||�||H1⊗H2 ||�||H1⊗H2 (97) whereH1,H2 are two separable Hilbert spaces and�,� ∈ H1⊗H2. The inequality follows from the variational characterization of the trace. For a proof see [8, Appendix D]. Proof of Corollary 1.3 We recall �α(t) = e−i HF α tϕ0 ⊗ W (α f0)∗η0 and write ∣ ∣�α(t) 〉〈 �α(t) ∣ ∣ = ∣ ∣e−i H ϕ0 α t�α(0) 〉〈 e−i H ϕ0 α t�α(0) ∣ ∣ + ∣ ∣�α(t) 〉〈 (e−i HF α t − e−i H ϕ0 α t )�α(0) ∣ ∣ + ∣ ∣(e−i HF α t − e−i H ϕ0 α t )�α(0) 〉〈 e−i H ϕ0 α t�α(0) ∣ ∣. (98) 123 45 Page 20 of 24 D. Mitrouskas Since e−i H ϕ0 α t acts nontrivially only on the Fock space, TrF ∣ ∣ ∣e−i H ϕ0 α t�α(0) 〉〈 e−i H ϕ0 α t�α(0) ∣ ∣ ∣ = |ϕ0〉〈ϕ0|. (99) Applying (97) to the two last terms in (98) thus leads to TrL2 ∣ ∣ ∣TrF ∣ ∣�α(t) 〉〈 �α(t) ∣ ∣ − ∣ ∣ϕ0 〉〈 ϕ0 ∣ ∣ ∣ ∣ ∣ ≤ 2 ∣ ∣ ∣ ∣ ( e−i HF α t − e−i H ϕ0 α t)�α(0) ∣ ∣ ∣ ∣ H . (100) Together with Theorem 1.2 this proves the corollary. �� Proof of Corollary 1.4 For E ∈ R we write ∣ ∣W (α f0)�α(t) 〉〈 W (α f0)�α(t) ∣ ∣ = ∣ ∣ϕ0 ⊗ e−iα−2(N−Aϕ0 )tη0 〉〈 ϕ0 ⊗ e−iα−2(N−Aϕ0 )tη0 ∣ ∣ + ∣ ∣W (α f0)�α(t) 〉〈 W (α f0)�α(t) − ϕ0 ⊗ e−iα−2(N−Aϕ0+E)tη0 ∣ ∣ + ∣ ∣W (α f0)�α(t) − ϕ0 ⊗ e−iα−2(N−Aϕ0+E)tη0 〉〈 ϕ0 ⊗ e−iα−2(N−Aϕ0+E)tη0 ∣ ∣. (101) From here we proceed similarly as in the proof of Corollary 1.3. Taking the partial trace in the first line gives TrL2 ∣ ∣ ∣ϕ0 ⊗ e−iα−2(N−Aϕ0 )tη0 〉〈 ϕ0 ⊗ e−iα−2(N−Aϕ0 )tη0 ∣ ∣ ∣ = ∣ ∣e−iα−2(N−Aϕ0 )tη0 〉〈 e−iα−2(N−Aϕ0 )tη0 ∣ ∣. (102) Using (97) for the second and third line then leads to TrF ∣ ∣ ∣TrL2 ∣ ∣W (α f0)�α(t) 〉〈 W (α f0)�α(t) ∣ ∣ − ∣ ∣e−iα−2(N−Aϕ0 )tη0 〉〈 e−iα−2(N−Aϕ0+)tη0 ∣ ∣ ∣ ∣ ∣ ≤ 2 ∣ ∣ ∣ ∣W (α f0)�α(t) − ϕ0 ⊗ e−iα−2(N−Aϕ0+E)tη0 ∣ ∣ ∣ ∣ H . (103) Setting E = EP(ϕ0) we can employ (16) and use the unitarity of W (α f0) to see that the last line equals 2 ∣ ∣ ∣ ∣W (α f0)(�α(t) − e−i H ϕ0 α t�α(0)) ∣ ∣ ∣ ∣ H = 2 ∣ ∣ ∣ ∣ ( e−i HF α t − e−i H ϕ0 α t)�α(0) ∣ ∣ ∣ ∣ H . (104) The corollary now follows from Theorem 1.2 together with the identity ∣ ∣e−iα−2(N−Aϕ0 )tη0 〉〈 e−iα−2(N−Aϕ0 )tη0 ∣ ∣ = ∣ ∣UVα(t)η0 〉〈 UVα(t)η0 ∣ ∣, (105) 123 A note on the Fröhlich dynamics in the strong coupling… Page 21 of 24 45 which we shall prove below. Proof of (105). For ε = ∫ R3(2π |k|)−2||R1/2e−ikxϕ0||2L2dk it holds that exp ( − iα−2(N − Aϕ0 + ε)t ) η0 = UVα(t)η0. (106) To show this we follow closely the argument from [2, Lem. 2.8 and App. B] where a similar identity was proven in the context of the dynamics of weakly interacting bosons. The argument is based onwell-known facts about Bogoliubov transformations and quasi-free states and a general result about the dynamics generated by quadratic Hamiltonians [23, Prop. 7]. At this point it is useful to introduce the pointwise annihilation and creation oper- ators ak, a∗ k defined by the requirement that a(g) = ∫ R3 g(k) ak dk, a∗(g) = ∫ R3 g(k) a∗ k dk (107) for any g ∈ L2(R3, dk). The commutation relations (3) now read [ak, a∗ l ] = δ(k − l), [ak, al ] = [a∗ k , a ∗ l ] = 0 ∀ k, l ∈ R 3. (108) Using (17), (107) and (108) a short computation leads to N − Aϕ0 + ε = d�(1 − G) − 1 2 ∫ R3 ∫ R3 (K(k, l) a∗ k a ∗ l + K(k, l) akal ) dkdl (109) where d�(1−G) denotes the second quantization of the one-body operator 1−G, i.e., d�(1 − G) = ∫ R3 ∫ R3 ( δ(k − l) − G(k, l) ) a∗ k al dkdl, (110) see (25) and (26) for a definition of K(k, l) and G(k, l), respectively. To the opera- tor on the right side of (109) we can apply [23, Prop. 7]. The requirements of this proposition are satisfied since 1 − G : L2(R3, dk) → L2(R3, dk) is bounded and K : L2(R3, dk) → L2(R3, dk) is a Hilbert–Schmidt operator which can be verified by means of (77). By part (iii) of [23, Prop. 7] it follows in particular that for any quasi- free state η0 ∈ F , the time-evolved state ηα(t) = exp(−iα−2(N − Aϕ0 + ε)t)η0 is again quasi-free (the bound 〈ηα(t), Nηα(t)〉 ≤ C exp(c|t |α−2) can be checked directly by means of Gronwall’s inequality). It is further not difficult to verify that the state UVα(t)η0 is also quasi-free (η0 = UW�0 for some Bogoliubov map W and thus UVα(t)η0 = UVα(t)◦W�0 with Bogoliubov map Vα(t) ◦ W). To show equality between the quasi-free states ηα(t) and UVα(t)η0 we compare their reduced one-body densitymatrices. This is sufficient because of thewell-known fact that quasi- free states are uniquely determined by their reduced one-body density matrices. For ξ ∈ F the reduced one-body density matrices γξ : L2(R3, dk) → L2(R3, dk) and αξ : L2(R3, dk) → L2(R3, dk) are defined by 123 45 Page 22 of 24 D. Mitrouskas 〈 f , γξ g 〉 L2 = 〈 ξ, a∗(g)a( f )ξ 〉 F , 〈 f , αξ g 〉 L2 = 〈 ξ, a(g)a( f )ξ 〉 F (111) for all f , g ∈ L2(R3, dk). In order to show γηα(t) = γUVα(t)η0 and αηα(t) = αUVα(t)η0 we argue that they solve the same pair of differential equations with the same initial condition γη0 and αη0 , respectively, and then use that the solution to this pair of differential equations is unique (the latter was shown in [23, Prop. 7]). Instead of computing the time derivative of γηα(t) and αηα(t), and similarly for UVα(t)η0, it is more convenient to determine the time derivative of 〈 ηα(t), A(F1)A(F2)ηα(t) 〉 F with A(F) the generalized annihilation operator as defined above (21). For F1, F2 ∈ L2(R3, dk) ⊕ L2(R3, dk) we have i d dt 〈 ηα(t), A(F1)A(F2)ξα(t) 〉 F = α−2〈ηα(t), [ N − Aϕ0 , A(F1)A(F2) ] ξα(t) 〉 F (112) and it follows by a straightforward computation that [ N − Aϕ0 , A(F1)A(F2) ] = A(AF1)A(F2) + A(F1)A(AF2) (113) with A = ( 1 − G K −K −1 + G ) . (114) Next we use U∗ V A(F)UV = A(V−1F), cf. (23), to obtain 〈 UVα(t)η0, A(F1)A(F2)UVα(t)η0 〉 F = 〈 η0, A(V−1 α (t)F1)A(V−1 α (t)F2)η0 〉 F . (115) By means of (i∂tV−1 α (t))Vα(t) = −V−1 α (t)(i∂tVα(t)) together with i∂tVα(t) = α−2AVα(t), we can compute the time derivative i d dt 〈 η0, A(V−1 α (t)F1)A(V−1 α (t)F2)η0 〉 F = 〈 η0, ( A(−i∂tV−1 α (t)F1)A(V−1 α (t)F2) + A(V−1 α (t)F1)A(−i∂tV−1 α (t)F2) ) η0 〉 F = α−2〈η0, ( A(V−1 α (t)AF1)A(V−1 α (t)F2) + A(V−1 α (t)F1)A(V−1 α (t)AF2) ) η0 〉 F = α−2〈UVα(t)η0, ( A(AF1)A(F2) + A(F1)A(AF2) ) UVα(t)η0 〉 F . (116) Comparing (112) and (113) with (115) and (116) we see that the pairs of reduced one-body density matrices (γηα(t), αηα(t)) and (γUVα(t)η0 , αUVα(t)η0) solve the same differential equation. Since the solution to this equation is unique, see [23, Prop. 7], and since ηα(0) = UVα(0)η0 = η0, we conclude their equality. This implies ηα(t) = UVα(t)η0 and hence proves the claimed identity. �� 123 A note on the Fröhlich dynamics in the strong coupling… Page 23 of 24 45 Acknowledgements I thankMarcel Griesemer for many interesting discussions about the Fröhlich polaron and also for valuable comments on this manuscript. Helpful discussions with Nikolai Leopold and Robert Seiringer are also gratefully acknowledged. This work was partially supported by theDeutsche Forschungs- gemeinschaft (DFG) through theResearchTrainingGroup 1838: Spectral Theory andDynamics ofQuantum Systems. Funding Open Access funding enabled and organized by Projekt DEAL. OpenAccess This article is licensedunder aCreativeCommonsAttribution 4.0 InternationalLicense,which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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Lecture notes. https://www.mathematik.uni-muenchen. de/~sorensen/Lehre/SoSe2013/MQM2/skript.pdf (2007) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 123 http://arxiv.org/abs/2005.02098 http://arxiv.org/abs/1904.12532 https://www.mathematik.uni-muenchen.de/~sorensen/Lehre/SoSe2013/MQM2/skript.pdf https://www.mathematik.uni-muenchen.de/~sorensen/Lehre/SoSe2013/MQM2/skript.pdf A note on the Fröhlich dynamics in the strong coupling limit Abstract 1 Introduction and main result 1.1 The model 1.2 Effective dynamics 1.3 Main results 1.4 Notation 1.5 Sketch of the proof 2 Proofs 2.1 Preliminary lemmas 2.2 Proof of Theorem 1.2 2.3 Proofs of Lemmas 2.1 and 2.2 2.4 Proofs of Proposition 1.1 and Corollaries 1.3 and 1.4 Acknowledgements References