The Landau-Pekar equations: Adiabatic theorem and accuracy
Nikolai Leopold, Simone Rademacher, Benjamin Schlein, Robert Seiringer
aa r X i v : . [ m a t h - ph ] A p r The Landau-Pekar equations: Adiabatic theorem andaccuracy
Nikolai Leopold, Simone Rademacher, Benjamin Schlein and Robert SeiringerApril 30, 2019
Abstract
We prove an adiabatic theorem for the Landau-Pekar equations. This allows us toderive new results on the accuracy of their use as effective equations for the time evolutiongenerated by the Fröhlich Hamiltonian with large coupling constant α . In particular, weshow that the time evolution of Pekar product states with coherent phonon field andthe electron being trapped by the phonons is well approximated by the Landau-Pekarequations until times short compared to α . I Introduction
We are interested in the dynamics of an electron in a ionic crystal. For situations in which theextension of the electron is much larger than the lattice spacing, Fröhlich [8] derived a modelwhich treats the crystal as a continuous medium and describes the polarization of the latticeas the excitations of a quantum field, called phonons. If the coupling between the electronand the phonons is large it is expected that the dynamics of the system can be approximatedby the Landau-Pekar equations, a set of nonlinear differential equations which model thephonons by means of a classical field. The coupling parameter of the Fröhlich model entersinto the Landau-Pekar equations and causes the speed of the electron to be much larger thanthe group velocity of the phonon field. This separation of scales, often referred to as adiabaticdecoupling [20], is believed to be responsible for the classical behaviour of the radiation field.The physical picture one has in mind is that the electron is trapped in a cloud of slowerphonons which increase the effective mass of the electron [16].The goal of this paper is to compare the time evolution generated by the Fröhlich Hamil-tonian with the Landau-Pekar equations and to give a quantitative justification of the appliedapproximation. In particular, we will consider the evolution of factorized initial data, witha coherent phonon field and an electron trapped by the phonons and minimizing the corre-sponding energy. For such initial data, we show that the Landau-Pekar equations provide agood approximation of the dynamics, up to times short compared to α , with α denoting thecoupling between electron and phonons. This result improves previous bounds in [5, 6], whichonly holds up to times of order α (but for more general initial data). Also, it extends the find-inds of [10], which show a result similar to ours but only for initial data minimizing the Pekarenergy functional (in this case, the solution of the Landau-Pekar equations remains constant).To prove our bound, we establish an adiabatic theorem for the solution of the Landau-Pekarequations. The idea of considering states with the electron trapped by the phonon field and toshow an adiabatic theorem was first proposed in [4, 7], where an adiabatic theorem is provedfor a one-dimensional version of the Landau-Pekar equations.1 I Model and Results
We consider the Fröhlich model which describes the interaction between an electron and aquantized phonon field. The state of the phonon field is represented by an element of thebosonic Fock space F := L n ≥ L ( R ) ⊗ ns , where the subscript s indicates symmetry underthe interchange of variables. The system is described by elements Ψ t ∈ H of the Hilbert space H := L ( R ) ⊗ F . (1)Its time evolution is governed by the Schrödinger equation i∂ t Ψ t = H α Ψ t (2)with the Fröhlich Hamiltonian H α := − ∆ + ˆ d k | k | − (cid:0) e ik · x a k + e − ik · x a ∗ k (cid:1) + ˆ d k a ∗ k a k . (3)Here, a ∗ k and a k are the creation and annihilation operators in the Fock space F , satisfyingthe commutation relations [ a k , a ∗ k ′ ] = α − δ ( k − k ′ ) , [ a k , a k ′ ] = [ a ∗ k , a ∗ k ′ ] = 0 for all k, k ′ ∈ R , (4)for a coupling constant α > . One should note that the Hamiltonian is written in the strongcoupling units, which gives rise to the α dependence in the commutation relations. Theseunits are related to the usual ones by rescaling all lengths by α , see [5, Appendix A]. We willbe interested in the limit α → ∞ . Motivated by Pekar’s Ansatz, we consider the evolution ofinitial states of product form ψ ⊗ W ( α ϕ )Ω . (5)Here Ω denotes the vacuum of the Fock space F and W ( f ) for f ∈ L ( R ) denotes the Weyloperator given by W ( f ) = exp (cid:20) ˆ d k (cid:16) f ( k ) a ∗ k − f ( k ) a k (cid:17)(cid:21) . (6)Note that the Weyl operator is unitary and satisfies the shifting property with respect to thecreation and annihilation operator, i.e. W ∗ ( f ) a k W ( f ) = a k + α − f ( k ) , W ∗ ( f ) a ∗ k W ( f ) = a ∗ k + α − f ( k ) (7)for all f ∈ L ( R ) . Due to the interaction the system will develop correlations between theelectron 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 approximatedup to times short compared to α (in the limit of large α ) by a product state ψ t ⊗ W ( α ϕ t )Ω ,with ( ψ t , ϕ t ) being a solution of the Landau-Pekar equations ( i∂ t ψ t = h − ∆ + ´ d k | k | − (cid:16) e ik · x ϕ t ( k ) + e − ik · x ϕ t ( k ) (cid:17)i ψ t ( x ) ,iα ∂ t ϕ t ( k ) = ϕ t ( k ) + | k | − ´ d x e − ik · x | ψ t ( x ) | (8)with initial data ( ψ , ϕ ) . For later convenience, we define for ϕ ∈ L ( R ) the potential V ϕ ( x ) = 2 / π − / Re (cid:0) | · | − ∗ ˇ ϕ (cid:1) ( x ) , (9)2here ˇ ϕ denotes the inverse Fourier transform defined for ϕ ∈ L ( R ) through ˇ ϕ ( x ) = (2 π ) − / ˆ d k e ik · x ϕ ( k ) . (10)We are interested, in particular, in initial data of the form (5) where the phonon field ϕ issuch that the Schrödinger operator h ϕ := − ∆ + V ϕ (11)has a non-degenerate eigenvalue at the bottom of its spectrum separated from the rest of thespectrum by a gap, and the electron wave function ψ is a ground state vector of (11). Assumption II.1.
Let ϕ ∈ L ( R ) such that e ( ϕ ) := inf {h ψ, h ϕ ψ i : ψ ∈ H ( R ) , k ψ k = 1 } < . (12)This assumption ensures the existence of a unique positive ground state vector ψ ϕ for h ϕ with corresponding eigenvalue separated from the rest of the spectrum by a gap of size Λ(0) > . If we then consider solutions of (8) with initial data ( ψ ϕ , ϕ ) the spectral gap canbe shown to persist at least for times of order α . Lemma II.1.
Let ϕ satisfy Assumption II.1 and let ( ψ t , ϕ t ) ∈ H ( R ) × L ( R ) denote thesolution of the Landau-Pekar equations with initial value ( ψ ϕ , ϕ ) .Then, for all Λ with < Λ < Λ(0) there is a constant C Λ > such that, for all | t | ≤ C Λ α ,the Hamiltonian h ϕ t has a unique positive and normalized ground state ψ ϕ t with eigenvalue e ( ϕ t ) < , which is separated from the rest of the spectrum by a gap of size Λ( t ) := inf λ ∈ spec( h ϕt ) λ = e ( ϕ t ) | e ( ϕ t ) − λ | ≥ Λ . (13)The Lemma is proven in Subsection IV.1. Using the persistence of the spectral gap, wecan 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 adiabatictheorem for the Landau-Pekar equations in one dimension is proved. Theorem II.1.
Let
T > , Λ > and ( ψ t , ϕ t ) ∈ H ( R ) × L ( R ) denote the solution ofthe Landau-Pekar equations with initial value ( ψ ϕ , ϕ ) ∈ H ( R ) × L ( R ) . Assume that theHamiltonian h ϕ t has a unique positive and normalized ground state ψ ϕ t and a spectral gap ofsize Λ( t ) > Λ for all | t | ≤ T . Then, k ψ t − e − i ´ t du e ( ϕ u ) ψ ϕ t k ≤ C Λ − (1 + Λ − ) α − (1 + α − | t | ) , ∀| t | ≤ T. (14) Remark
II.1 . One also has k ψ t − e − i ´ t du e ( ϕ u ) ψ ϕ t k ≤ Cα − Λ − | t | for all | t | ≤ T . Remark
II.2 . Note that the proof of the theorem only requires the existence of the spectralgap Λ > . Assuming Λ to be of order one for times of order α , the theorem shows that ψ t is well approximated by the ground state ψ ϕ t for any | t | ≪ α . Remark
II.3 . Lemma II.1 shows that the existence of the ground state and the spectral gapfor all times | t | ≤ C Λ α can be inferred from Assumption II.1. In this case, (16) is valid forall | t | ≤ C Λ α without any assumptions on h ϕ t and Λ( t ) at times t > .Using Theorem II.1 we can show that the Landau-Pekar equations (8) provide a goodapproximation to the solution of the Schrödinger equation (2), for initial data of the form (5),with ϕ satisfying Assumption II.1 and with ψ = ψ ϕ being the ground state of the operator h ϕ defined as in (11). 3 heorem II.2. Let ϕ satisfy Assumption II.1 and α > . Let ( ψ t , ϕ t ) ∈ H ( R ) × L ( R ) denote the solution of the Landau-Pekar equations with initial data ( ψ ϕ , ϕ ) ∈ H ( R ) × L ( R ) and ω ( t ) := α Im (cid:10) ϕ t , ∂ t ϕ t (cid:11) + k ϕ t k . (15) Then, there exists a constant
C > such that (cid:13)(cid:13)(cid:13) e − iH α t ψ ϕ ⊗ W ( α ϕ )Ω − e − i ´ t du ω ( u ) ψ t ⊗ W ( α ϕ t )Ω (cid:13)(cid:13)(cid:13) ≤ Cα − | t | / (16) for all α ≥ α .Remark II.4 . Theorem II.2 shows that the Pekar ansatz is a good approximation for timessmall compared to α . Note that even though ( ψ t , ϕ t ) stay close to their initial values for thesetimes (as shown in Theorem II.1), it is essential to use the time-evolved version in (16). Thisis due to the large factor α in the Weyl operator W ( α ϕ t ) , which leads to a very sensitivebehavior of the state on ϕ t .It remains an open problem to decided whether the Pekar product ansatz remains validalso for times of order α and larger.A first rigorous result concerning the evolution of the Fröhlich polaron was obtained in[5], where the product ψ t ⊗ W ( α ϕ )Ω , with ψ t solving the linear equation i∂ t ψ t = − ∆ ψ t + ˆ dk | k | − (cid:16) e ik · x ϕ ( k ) + e − ik · x ϕ ( k ) (cid:17) ψ t ( x ) (17)was proven to give a good approximation for the solution e − iH α t ψ ⊗ W ( α ϕ )Ω of theSchrödinger equation (2), up to times of order one . This result was improved in [6], whereconvergence towards (8) was established for all times | t | ≪ α (the analysis in [6] also givesmore detailed information on the solution of Schrödinger equation (2); in particular, it impliesconvergence of reduced density matrices). Notice that the results of [5, 6] hold for generalinitial data of the form (5), with no assumption on the relation between the initial electronwave function ψ and the initial phonon field ϕ . Theorem II.2 shows therefore that, underthe additional assumption that ψ = ψ ϕ the Landau-Pekar equations (8) provide a goodapproximation to (2) for longer times (times short compared to α ). In this sense, TheoremII.2 also extends the result of [10], where the validity of the Landau-Pekar equations wasestablished for times short compared to α , but only for initial data ( ψ , ϕ ) minimizing thePekar energy functional (in this case, the solution of (8) is stationary, i.e. ( ψ t , ϕ t ) = ( ψ , ϕ ) for all t ). In fact, similarly to the analysis in [10], we use the observation that the spectralgap above the ground state energy of h ϕ t allows us to obtain bounds that are valid on longertime scales (it allows us to integrate by parts, after (59) and after (93); this step is crucial tosave a factor of t ). The classical behaviour of a quantum field does not only appear in thestrong 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 ina certain limit where the number of field bosons becomes infinite while the coupling constanttends to zero. The emergence of classical radiation was also proven for the Nelson model withultraviolet 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 replacedby two-particle interactions if the particles are much slower than the bosons of the quantumfield. In fact, a simple modification of the Gronwall argument in [5] leads to convergence for times | t | ≪ α . II Preliminaries
In this section, we collect properties of the Landau-Pekar equations that are used in the proofsof Theorem II.1 and Theorem II.2. For ψ ∈ L ( R ) we define the function σ ψ ( k ) = | k | − ˆ d x e − ik · x | ψ ( x ) | . (18)Next, we notice that the first and second derivative of the potential V ϕ t are given by ∂ t V ϕ t ( x ) = V ˙ ϕ t ( x ) = − α − ˆ d k | k | − (cid:0) e ik · x iϕ t ( k ) + e − ik · x iϕ t ( k ) (cid:1) − iα − ˆ d k | k | − (cid:0) e ik · x σ ψ t ( k ) − e − ik · x σ ψ t ( − k ) (cid:1) = − α − V iϕ t ( x ) (19)and ∂ t V iϕ t ( x ) = V i ˙ ϕ t ( x ) = V α − ( ϕ t + σ ψt ) ( x ) = α − V ϕ t ( x ) + α − V σ ψt ( x ) . (20)Then we define the energy functional E : H ( R ) × L ( R ) → R E ( ψ, ϕ ) = h ψ, h ϕ ψ i + k ϕ k . (21)Using standard methods one can show that the Landau-Pekar equations are well posed andthat the energy E ( ψ t , ϕ t ) is conserved if ( ψ t , ϕ t ) is a solution of (8). For a proof of the followingLemma see [6, Appendix C]. Lemma III.1 ([6], Lemma 2.1) . For any ( ψ , ϕ ) ∈ H ( R ) × L ( R ) , there is a uniqueglobal solution ( ψ t , ϕ t ) of the Landau-Pekar equations (8) . The following conservation lawshold true k ψ t k = k ψ k and E ( ψ t , ϕ t ) = E ( ψ , ϕ ) ∀ t ∈ R . (22) Moreover, there exists a constant C such that k ψ t k H ( R ) ≤ C, k ϕ t k ≤ C (23) for all α > and all t ∈ R . The next Lemma (also proven in [6, Appendix B,C]) collects some properties of quantitiesoccurring in the Landau-Pekar equations.
Lemma III.2.
For V ϕ being defined as in (9) there exists a constant C > such that forevery ψ ∈ H ( R ) and ϕ ∈ L ( R ) k V ϕ k ≤ C k ϕ k and k V ϕ ψ k ≤ C k ϕ k k ψ k H ( R ) . (24) Furthermore, for every δ > there exists C δ > such that ± V ϕ ≤ − δ ∆ + C δ , (25) thus there exists C > such that −
12 ∆ − C ≤ h ϕ ≤ −
2∆ + C. (26) Let σ ψ be defined as in (18) . Then, there exists C > such that k σ ψ k ≤ C k ψ k H ( R ) . (27) We use the notation ˙ f to denote the derivative of a function f with respect to time. emark III.1 . Let
T > , Λ > and ( ψ t , ϕ t ) ∈ H ( R ) × L ( R ) denote the solution of theLandau-Pekar equations with initial value ( ψ ϕ , ϕ ) ∈ H ( R ) × L ( R ) . Assume that theHamiltonian h ϕ t has a unique positive and normalized ground state ψ ϕ t and a spectral gapof size Λ( t ) > Λ for all t ≤ T . Lemma III.2 then implies the existence of constant such that k ψ ϕ t k H ( R ) ≤ C ∀ | t | ≤ T. (28) Proof of Lemma III.2.
Recall the definition (9) of the potential V ϕ V ϕ ( x ) = 2 / π − / Re ˆ d y | x − y | ˇ ϕ ( y ) . (29)The first inequality follows directly from the Hardy-Littlewood-Sobolev inequality k V ϕ k ≤ C k ϕ k . (30)In order to prove the second inequality we use the first one and the Hölder inequality k V ϕ ψ k ≤ k V ϕ k k ψ k ≤ C k ϕ k k ψ k . (31)Since the interpolation inequality together with the Sobolev inequality implies k ψ k ≤ k ψ k / k ψ k / ≤ k ψ k / k∇ ψ k / , (32)we obtain k V ϕ ψ k ≤ C k ϕ k k∇ ψ k / k ψ k / ≤ C k ϕ k k ψ k H ( R ) . (33)The second operator inequality follows again from the Sobolev inequality. For this let ψ ∈ H ( R ) , then for ε > h ψ, V ϕ ψ i ≤ C k V ϕ k k ψ k / ≤ C k V ϕ k (cid:0) ε k∇ ψ k + ε − k ψ k (cid:1) , (34)where we used the interpolation inequality. The first inequality of the Lemma implies ±h ψ, V ϕ ψ i ≤ h ψ, ( − ε ∆ + C ε ) ψ i , (35)and (26) follows.The last inequality of the Lemma follows from the observation that k σ ψ k = ˆ dk | k | ˆ dxdy | ψ ( x ) | | ψ ( y ) | e ik · ( x − y ) = 2 π ˆ dxdy | ψ ( x ) | | ψ ( y ) | | x − y | ≤ C k| ψ | k / , (36)where we used again the Hardy-Littlewood-Sobolev inequality. As before, the interpolationand the Sobolev inequality imply (27). IV Proof of the adiabatic theorem
IV.1 The ground state ψ ϕ t Before proving the adiabatic theorem, we show that the spectral gap of h ϕ t does not close fortimes of order α . In particular, we show the existence of the ground state ψ ϕ t .6 emma IV.1. Let ϕ satisfy Assumption II.1. Then, there exists a unique positive andnormalized ground state ψ ϕ of h ϕ = − ∆ + V ϕ . Let ( ψ t , ϕ t ) ∈ H ( R ) × L ( R ) denote thesolution of the Landau-Pekar equations (8) with initial value ( ψ ϕ , ϕ ) . There exists C > such that for all | t | ≤ Cα , there exists a unique, positive and normalized ground state ψ ϕ t of h ϕ t = − ∆ + V ϕ t with corresponding eigenvalue e ( ϕ t ) < . It satisfies ∂ t ψ ϕ t = α − R t V iϕ t ψ ϕ t with R t = q t ( h ϕ t − e ( ϕ t )) − q t , (37) where q t = 1 − | ψ ϕ t ih ψ ϕ t | denotes the projection onto the subspace of L ( R ) orthogonal tothe span of ψ ϕ t .Proof. Lemma III.1 and Lemma III.2 imply that V ϕ t ∈ L ( R ) for all t ∈ R . The existenceof the ground state ψ ϕ t at time t = 0 then follows from the negativity of the infimum of thespectrum (see [15, Theorem 11.5]). In order to prove the existence of the ground state ψ ϕ t of h ϕ t at later times, it suffices to show that e ( ϕ t ) is negative. For this we pick the ground state ψ ϕ at time t = 0 and estimate e ( ϕ t ) ≤ h ψ ϕ , ( − ∆ + V ϕ t ) ψ ϕ i = e ( ϕ ) − α − ˆ t ds h ψ ϕ , V iϕ s ψ ϕ i≤ e ( ϕ ) + C ˆ t ds α − k ϕ s k k ψ ϕ k H ( R ) ≤ e ( ϕ ) + C | t | α − , (38)by means of (19), Lemma III.1 and Lemma III.2. Thus if we restrict our consideration totimes | t | < C − | e ( ϕ ) | α , we conclude that e ( ϕ t ) < . The ground state ψ ϕ t satisfies h ϕ t − e ( ϕ t )) ψ ϕ t (39)Differentiating both sides of the equality with respect to the time variable leads to (cid:16) ˙ h ϕ t − ˙ e ( ϕ t ) (cid:17) ψ ϕ t + ( h ϕ t − e ( ϕ t )) ˙ ψ ϕ t . (40)On the one hand ˙ h ϕ t = V ˙ ϕ t = − α − V iϕ t by means of (19) and on the other hand, theHellmann-Feynman theorem implies ˙ e ( ϕ t ) = h ψ ϕ t , ˙ h ϕ t ψ ϕ t i = − α − h ψ ϕ t , V iϕ t ψ ϕ t i (41)so that (39) becomes − α − q t V iϕ t ψ ϕ t + ( h ϕ t − e ( ϕ t )) ˙ ψ ϕ t . (42)Since ψ ϕ t is chosen to be real and normalized for all t ∈ R , it follows that (cid:10) ψ ϕ t , ˙ ψ ϕ t (cid:11) = 0 forall t ∈ R . Hence, ˙ ψ ϕ t = α − q t ( h ϕ t − e ( ϕ t )) − q t V iϕ t ψ ϕ t = α − R t V iϕ t ψ ϕ t . (43)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 eigen-value of h ϕ t (or the bottom of the essential spectrum) is given by e ( t ) = inf A ⊂ L ( R )dim A =2 sup ψ ∈ A k ψ k =1 h ψ, h ϕ t ψ i . (44)7or any ψ ∈ L ( R ) with k ψ k = 1 we have by Lemma III.2 h ψ, h ϕ t ψ i = h ψ, h ϕ ψ i − α − ˆ t ds h ψ, V iϕ s ψ i≥h ψ, h ϕ ψ i − C | t | α − sup | s |≤| t | k ϕ s k k∇ ψ k − C | t | α − sup | t |≤| s | k ϕ s k k ψ k ≥ (1 − C | t | α − ) h ψ, h ϕ ψ i − C | t | α − (45)where we used Lemma III.2. Inserting in (44), we conclude that e ( t ) ≥ (1 − C | t | α − ) e (0) − C | t | α − . (46)With e ( ϕ t ) ≤ e ( ϕ ) + C | t | α − (see (38)), we obtain Λ( t ) ≥ Λ(0) − C | t | α − . (47)Using the persistence of the spectral gap, the resolvent R t = q t ( h ϕ t − e ( ϕ t )) − q t can beestimated as follows. Lemma IV.2.
Let
T > , Λ > and ( ψ t , ϕ t ) ∈ H ( R ) × L ( R ) denote the solution ofthe Landau-Pekar equations with initial value ( ψ ϕ , ϕ ) ∈ H ( R ) × L ( R ) . Assume that theHamiltonian h ϕ t has a unique positive and normalized ground state ψ ϕ t with e ( ϕ t ) < and aspectral gap of size Λ( t ) > Λ for all t ≤ T . Then, for all | t | ≤ T k R t k ≤ Λ − , k ( − ∆ + 1) / R / t k ≤ C (1 + Λ − ) / , (48) and k ˙ R t k ≤ C Λ − / α − (1 + Λ − ) / (49) with C > depending only on ϕ .Proof. Since the spectral gap is at least of size Λ > for times | t | ≤ T , it follows that k R t k ≤ Λ − . (50)To prove the second inequality, we estimate for arbitrary ψ ∈ L ( R ) k ( − ∆ + 1) / R / t ψ k = h ψ, R / t ( − ∆ + 1) R / t ψ i ≤ C h ψ, R / t ( h ϕ t + 1) R / t ψ i , (51)where the last inequality follows from Lemma III.2. Thus, k ( − ∆ + 1) / R / t ψ k ≤ C h ψ, R / t ( h ϕ t − e ( ϕ t ) + e ( ϕ t ) + 1) R / t ψ i = C h ψ, (cid:0) q t + ( e ( ϕ t ) + 1) R t (cid:1) ψ i≤ C h ψ, (cid:0) R t (cid:1) ψ i (52)since e ( ϕ t ) < for all | t | ≤ T by assumption. The gap condition then implies k ( − ∆ + 1) / R / t ψ k ≤ C (1 + Λ − ) . (53)In order to prove the third bound of the Lemma we calculate (with p t = 1 − q t ) ˙ R t = − α − p t V iϕ t R t − α − R t V iϕ t p t + q t (cid:0) ∂ t ( h ϕ t − e ( ϕ t )) − (cid:1) q t (54)8y means of the Leibniz rule and Lemma IV.1. With the resolvent identities, (19) and (41)this becomes ˙ R t = − α − p t V iϕ t R t − α − R t V iϕ t p t − q t ( h ϕ t − e ( ϕ t )) − (cid:16) ˙ h ϕ t − ˙ e ( ϕ t ) (cid:17) ( h ϕ t − e ( ϕ t )) − q t = − α − p t V iϕ t R t − α − R t V iϕ t p t + α − R t ( V iϕ t − h ψ ϕ t , V iϕ t ψ ϕ t i ) R t . (55)Hence, Lemma III.2 leads to k ˙ R t k ≤ Cα − k V iϕ t R t k k R t k + Cα − k R t k k ψ ϕ t k H ( R ) ≤ Cα − k ( − ∆ + 1) / R t k k R t k + α − k R t k k ψ ϕ t k H ( R ) ≤ C Λ − / α − (1 + Λ − ) / + Cα − Λ − , (56)for all | t | ≤ T , where we used (28) and the second bound of the Lemma. IV.2 Proof of Theorem II.1
In the following we denote e ψ ϕ t = e − i ´ t du e ( ϕ u ) ψ ϕ t . The fundamental theorem of calculusimplies that k ψ t − e ψ ϕ t k = − ˆ t ds dds h ψ s , e ψ ϕ s i = − ˆ t ds Re h− ih ϕ s ψ s , e ψ ϕ s i− ˆ t ds Re h ψ s , (cid:0) − ie ( ϕ s ) + α − R s V iϕ s (cid:1) e ψ ϕ s i = − α − ˆ t ds Re h ψ s , R s V iϕ s e ψ ϕ s i , (57)where we used that h ϕ s e ψ ϕ s = e ( ϕ s ) e ψ ϕ s and Lemma IV.1 to compute the derivative of theground state ψ ϕ s . Using the Cauchy-Schwarz inequality, Lemma II.1 and Lemma III.2 to-gether with Lemma IV.2 and (28), we obtain the inequality from Remark II.1, i.e. a boundof order α − | t | . In the following we shall improve this bound. We define e ψ s := e i ´ s dτ e ( ϕ τ ) ψ s satisfying i∂ s e ψ s = ( h ϕ s − e ( ϕ s )) e ψ s (58)and write (57) as k ψ t − e ψ ϕ t k = − α − ˆ t ds Re h q s e ψ s , R s V iϕ s ψ ϕ s i . (59)Then, we exploit that the time derivative of e ψ s is of order one while the time derivatives of R s , V iϕ s and ψ ϕ s are of order α − (compare also with [20, p.9]). We observe that ∂ s (cid:16) iR s e ψ s (cid:17) − i ˙ R s e ψ s = R s ( h ϕ s − e ( ϕ s )) e ψ s = q s e ψ s . (60)9lugging this identity into (59) and integrating by parts, we obtain k ψ t − e ψ ϕ t k = − α − ˆ t ds Im h ∂ s ( R s e ψ s ) , R s V iϕ s ψ ϕ s i + 2 α − ˆ t ds Im h ˙ R s e ψ s , R s V iϕ s ψ ϕ s i =2 α − ˆ t ds Im h R s e ψ s , ∂ s ( R s V iϕ s ψ ϕ s ) i− α − Im h R t e ψ t , R t V iϕ t ψ ϕ t i + 2 α − Im h R e ψ , R V iϕ ψ ϕ i + 2 α − ˆ t ds Im h ˙ R s e ψ s , R s V iϕ s ψ ϕ s i . (61)The Leibniz rule with (20) and Lemma IV.1 leads together with the initial condition R e ψ = R ψ ϕ = 0 to k ψ t − e ψ ϕ t k = − α − Im h R t e ψ t , R t V iϕ t ψ ϕ t i (62a) + 2 α − ˆ t ds Im h R s e ψ s , ˙ R s V iϕ s ψ ϕ s i (62b) + 2 α − ˆ t ds Im h ˙ R s e ψ s , R s V iϕ s ψ ϕ s i (62c) + 2 α − ˆ t ds Im h R s e ψ s , R s V ϕ s + σ ψs ψ ϕ s i (62d) + 2 α − ˆ t ds Im h R s e ψ s , ( R s V iϕ s ) ψ ϕ s i . (62e)Using Lemma III.2 the first term can be estimated by | (62a) | ≤ Cα − k R t k k V iϕ t ψ ϕ t k ≤ Cα − k R t k k ψ ϕ t k H ( R ) k ϕ t k . (63)On the one hand Lemma III.1 and (28) show that k ϕ t k resp. k ψ ϕ t k H ( R ) are uniformlybounded in time. On the other hand Lemma IV.2 implies that the resolvent R t is boundedfor all times | t | ≤ T , so that we obtain | (62a) | ≤ C Λ − α − , ∀ | t | ≤ T. (64)Similarly, we bound the second and the third term by | (62b) | + | (62c) | ≤ Cα − ˆ t ds k R s k k V iϕ s ψ ϕ s k k ˙ R s k ≤ Cα − Λ − / (1 + Λ − ) / | t | (65)for all | t | ≤ T , using Lemma III.1, Lemma III.2 and Lemma IV.2. The forth term (62d) canbe bounded using k σ ψ t k ≤ C k ψ t k H ( R ) ≤ C by Lemma III.1. We find | (62d) | ≤ C Λ − α − | t | , ∀ | t | ≤ T. (66)Using the same ideas we estimate the last term by | (62e) | ≤ Cα − ˆ t ds k R s k k V iϕ s R s k ≤ C Λ − α − ˆ t ds k V iϕ s R s k . (67)10emma III.2 implies that for all ψ ∈ H ( R ) k V iϕ s ψ k ≤ C k ϕ s k k ψ k H ( R ) (68)and therefore that k V iϕ s ( − ∆ + 1) − / k ≤ C for all | t | ≤ T by Lemma III.2. Hence k V iϕ s R s k ≤ C k ( − ∆ + 1) / R s k ≤ C Λ − / (1 + Λ − ) / , (69)for all | t | ≤ T where we used Lemma IV.2 for the last inequality. Thus, | (62e) | ≤ Cα − Λ − (1 + Λ − ) | t | , ∀ | t | ≤ T, (70)and we finally obtain (14). V Accuracy of the Landau-Pekar equations
V.1 Preliminaries
For notational convenience we define Φ x = ˆ d k | k | − (cid:16) e ik · x a k + e − ik · x a ∗ k (cid:17) = Φ + x + Φ − x (71)with Φ + x = ˆ d k | k | − e ik · x a k , and Φ − x = ˆ d k | k | − e − ik · x a ∗ k . (72)In addition we introduce for f ∈ L ( R ) the creation operator a ∗ ( f ) and the annihilationoperator a ( f ) which are given through a ∗ ( f ) = ˆ d k f ( k ) a ∗ k , a ( f ) = ˆ d k f ( k ) a k (73)and bounded with respect to the number of particles operator N = ´ d k a ∗ k a k , i.e. k a ( f ) ξ k ≤ k f k kN / ξ k , k a ∗ ( f ) ξ k ≤ k f k k ( N + α − ) / ξ k (74)for all ξ ∈ F . Moreover, recall the definition (6) of the Weyl operator W ( f ) = e a ∗ ( f ) − a ( f ) .For a time dependent function f t ∈ L ( R ) the time derivative of the Weyl operator is givenby ∂ t W ( f t ) = α − h f t , ∂ t f t i − h ∂ t f t , f t i ) W ( f t ) + ( a ∗ ( ∂ t f t ) − a ( ∂ t f t )) W ( f t ) . (75)The proof of this formula can be found in [6, Lemma A.3]. V.2 Proof of Theorem II.2
It should be noted that (16) is valid for all times which are at least of order α because bothstates in the inequality have norm one. To show its validity for shorter times we split thenorm difference by the triangle inequality into two parts and use Remark II.1 to estimate (cid:13)(cid:13)(cid:13) e − iH α t ψ ϕ ⊗ W ( α ϕ )Ω − e − i ´ t du ω ( u ) ψ t ⊗ W ( α ϕ t )Ω (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) e − iH α t ψ ϕ ⊗ W ( α ϕ )Ω − e − i ´ t du ω ( u ) e ψ ϕ t ⊗ W ( α ϕ t )Ω (cid:13)(cid:13)(cid:13) + 2 (cid:13)(cid:13)(cid:13) e ψ ϕ t ⊗ W ( α ϕ t )Ω − ψ t ⊗ W ( α ϕ t )Ω (cid:13)(cid:13)(cid:13) ≤ Cα − | t | + 2 (cid:13)(cid:13)(cid:13) e − iH α t ψ ϕ ⊗ W ( α ϕ )Ω − e − i ´ t du ω ( u ) e ψ ϕ t ⊗ W ( α ϕ t )Ω (cid:13)(cid:13)(cid:13) (76)11or all | t | ≤ C Λ α where we used the notation e ψ ϕ t = e − i ´ t du e ( ϕ u ) ψ ϕ t . Therefore it remainsto estimate the second term. We apply Duhamel’s formula and use the unitarity of the Weyloperator W ( α ϕ t ) to compute (cid:13)(cid:13)(cid:13) e − iH α t ψ ϕ ⊗ W ( α ϕ )Ω − e − i ´ t du ω ( u ) e ψ ϕ t ⊗ W ( α ϕ t )Ω (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) e i ´ t du ω ( u ) W ∗ ( α ϕ t ) e − iH α t ψ ϕ ⊗ W ( α ϕ )Ω − e ψ ϕ t ⊗ Ω (cid:13)(cid:13)(cid:13) = ˆ t ds ∂ s (cid:13)(cid:13)(cid:13) e i ´ s du ω ( u ) W ∗ ( α ϕ s ) e − iH α s ψ ϕ ⊗ W ( α ϕ )Ω − e ψ ϕ s ⊗ Ω (cid:13)(cid:13)(cid:13) = − ˆ t ds ∂ s (cid:10) e i ´ s du ω ( u ) W ∗ ( α ϕ s ) e − iH α s ψ ϕ ⊗ W ( α ϕ )Ω , e ψ ϕ s ⊗ Ω (cid:11) . (77)We introduce ξ s = e i ´ s du ω ( u ) W ∗ ( α ϕ s ) e − iH α s ψ ϕ ⊗ W ( α ϕ )Ω (78)to shorten the notation and compute W ∗ ( α ϕ t ) H α W ( α ϕ t ) = h ϕ t + k ϕ t k + N + Φ x + a ( ϕ t ) + a ∗ ( ϕ t ) . (79)by means of (7). Using (7) again and (75) we get i∂ s ξ s = (cid:0) i∂ s W ∗ ( α ϕ s ) (cid:1) W ( α ϕ s ) ξ s + (cid:16) W ∗ ( α ϕ s ) H α W ( α ϕ s ) − ω ( s ) (cid:17) ξ s = (cid:16) h ϕ s + Φ x − a ( σ ψ s ) − a ∗ ( σ ψ s ) + N (cid:17) ξ s . (80)Recall that Φ x = ´ d k | k | − (cid:0) e ik · x a k + e − ik · x a ∗ k (cid:1) . We obtain (cid:13)(cid:13)(cid:13) e − iH α t ψ ϕ ⊗ W ( α ϕ )Ω − e − i ´ t du ω ( u ) e ψ ϕ t ⊗ W ( α ϕ t )Ω (cid:13)(cid:13)(cid:13) = 2Im ˆ t ds (cid:16)(cid:10) i∂ s ξ s , e ψ ϕ s ⊗ Ω (cid:11) − (cid:10) ξ s , i∂ s e ψ ϕ s ⊗ Ω (cid:11)(cid:17) = 2Im ˆ t ds (cid:10) ξ s , (cid:16) h ϕ s − e ( ϕ s ) + Φ x − a ( σ ψ s ) − a ∗ ( σ ψ s ) + N (cid:17) e ψ ϕ s ⊗ Ω (cid:11) − α − Re ˆ t ds (cid:10) ξ s , R s V iϕ s e ψ ϕ s ⊗ Ω (cid:11) = 2Im ˆ t ds (cid:10) ξ s , (cid:0) Φ − x − a ∗ ( σ ψ s ) (cid:1) e ψ ϕ s ⊗ Ω (cid:11) − α − Re ˆ t ds (cid:10) ξ s , R s V iϕ s e ψ ϕ s ⊗ Ω (cid:11) . (81)Here we used the definition R s = q s ( h ϕ s − e ( ϕ s )) − q s and Lemma IV.1. Thus if we insert theidentity p s + q s and note that q s a ∗ ( σ ψ s ) e ψ ϕ s ⊗ Ω = 0 and q s e ψ ϕ s = 0 , we get (cid:13)(cid:13)(cid:13) e − iH α t ψ ϕ ⊗ W ( α ϕ )Ω − e − i ´ t du ω ( u ) e ψ ϕ t ⊗ W ( α ϕ t )Ω (cid:13)(cid:13)(cid:13) = − α − Re ˆ t ds (cid:10) ξ s , R s V iϕ s e ψ ϕ s ⊗ Ω (cid:11) (82a) + 2Im ˆ t ds (cid:10) ξ s , p s (cid:0) Φ − x − a ∗ ( σ ψ s ) (cid:1) e ψ ϕ s ⊗ Ω (cid:11) (82b) + 2Im ˆ t ds (cid:10) ξ s − e ψ ϕ s ⊗ Ω , q s Φ − x e ψ ϕ s ⊗ Ω (cid:11) . (82c)12e observe that the first term (82a) is already of the right order, namely α − t . To be moreprecise, | (82a) | ≤ α − ˆ t ds k ξ s k k R s k (cid:13)(cid:13)(cid:13) V iϕ s e ψ ϕ s (cid:13)(cid:13)(cid:13) ≤ α − ˆ t ds k ϕ s k k R s k (cid:13)(cid:13)(cid:13) e ψ ϕ s (cid:13)(cid:13)(cid:13) H ( R ) ≤ Cα − | t | (83)for all | t | ≤ C Λ α where we used Lemma III.2, Lemma IV.2, Lemma III.1 and (28). Weestimate the remaining two terms (82b) and (82c) separately. The term (82b)
We have | (82b) | ≤ ˆ t ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:10) ξ s , ˆ d k a ∗ k | k | − (cid:16)(cid:10) e ψ ϕ s , e − ik · e ψ ϕ s (cid:11) − (cid:10) ψ s , e − ik · ψ s (cid:11)(cid:17) e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ˆ t ds (cid:13)(cid:13)(cid:13)(cid:13) ˆ d k a ∗ k | k | − (cid:16)(cid:10) e ψ ϕ s , e − ik · e ψ ϕ s (cid:11) − (cid:10) ψ s , e − ik · ψ s (cid:11)(cid:17) e ψ ϕ s ⊗ Ω (cid:13)(cid:13)(cid:13)(cid:13) = 2 ˆ t ds (cid:13)(cid:13)(cid:13)(cid:13) ˆ d k a ∗ k | k | − (cid:16)(cid:10) e ψ ϕ s , e − ik · (cid:0) e ψ ϕ s − ψ s (cid:1)(cid:11) + (cid:10)(cid:0) e ψ ϕ s − ψ s (cid:1) , e − ik · ψ s (cid:11)(cid:17) e ψ ϕ s ⊗ Ω (cid:13)(cid:13)(cid:13)(cid:13) . (84)Since k a ∗ ( f ) ψ ⊗ Ω k = α − k f k k ψ k for all f ∈ L ( R ) , we find | (82b) | ≤ Cα − ˆ t ds " ˆ d k | k | − (cid:16) (cid:12)(cid:12)(cid:12)(cid:10) e ψ ϕ s , e − ik · (cid:0) e ψ ϕ s − ψ s (cid:1)(cid:11)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:10)(cid:0) e ψ ϕ s − ψ s (cid:1) , e − ik · ψ s (cid:11)(cid:12)(cid:12)(cid:12) (cid:17) / . (85)With the help of [ | · | − ( x ) = π − | x | − and the inequalities of Hardy-Littlewood-Sobolev andHölder we obtain ˆ d k | k | − (cid:12)(cid:12)(cid:12)(cid:10) e ψ ϕ s , e − ik · (cid:0) e ψ ϕ s − ψ s (cid:1)(cid:11)(cid:12)(cid:12)(cid:12) = C ˆ d x ˆ d y | x − y | − ( e ψ ϕ s − ψ s )( x ) e ψ ϕ s ( x ) e ψ ϕ s ( y )( e ψ ϕ s − ψ s )( y ) ≤ C (cid:13)(cid:13)(cid:13) e ψ ϕ s ( e ψ ϕ s − ψ s ) (cid:13)(cid:13)(cid:13) / ≤ C (cid:13)(cid:13)(cid:13) e ψ ϕ s − ψ s (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) e ψ ϕ s (cid:13)(cid:13)(cid:13) ≤ C (cid:13)(cid:13)(cid:13) e ψ ϕ s − ψ s (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) e ψ ϕ s (cid:13)(cid:13)(cid:13) H ( R ) ≤ C (cid:13)(cid:13)(cid:13) e ψ ϕ s − ψ s (cid:13)(cid:13)(cid:13) (86)for all | t | ≤ C Λ α by (28). Similarly, ˆ d k | k | − (cid:12)(cid:12)(cid:12)(cid:10)(cid:0) e ψ ϕ s − ψ s (cid:1) , e − ik · ψ s (cid:11)(cid:12)(cid:12)(cid:12) ≤ C k ψ s k H ( R ) (cid:13)(cid:13)(cid:13) ψ s − e ψ ϕ s (cid:13)(cid:13)(cid:13) ≤ C (cid:13)(cid:13)(cid:13) ψ s − e ψ ϕ s (cid:13)(cid:13)(cid:13) (87)by Lemma III.1. Hence, | (82b) | ≤ Cα − ˆ t ds (cid:13)(cid:13)(cid:13) ψ s − e ψ ϕ s (cid:13)(cid:13)(cid:13) (88)for all | t | ≤ C Λ α . Applying Theorem II.1 leads to | (82b) | ≤ Cα − | t | for all | t | ≤ C Λ α . (89)13 he term (82c) In order to continue we note that [18, Theorem X.71], whose assumptions can easily shownto be satisfied by Lemma III.2, guarantees the existence of a two parameter group U h ( s ; τ ) on L ( R ) such that dds U h ( s ; τ ) ψ = − ih ϕ s U h ( s ; τ ) ψ, U h ( τ ; τ ) ψ = ψ for all ψ ∈ H ( R ) . (90)Moreover, we define e U h ( s ; τ ) = e i ´ sτ du e ( ϕ u ) U h ( s ; τ ) . (91)We then have for all s ∈ R dds e U ∗ h ( s ; τ ) = e U ∗ h ( s ; τ ) i (cid:0) h ϕ s − e ( ϕ s ) (cid:1) (92)and e U ∗ h ( s ; τ ) q s f s = − i dds h e U ∗ h ( s ; τ ) R s f s i + i e U ∗ h ( s ; τ ) ˙ R s f s + i e U ∗ h ( s ; τ ) R s ∂ s f s , (93)for f s ∈ L ( R ) . This allows us to express(82c) = 2Im ˆ t ds (cid:10) ξ s − e ψ ϕ s ⊗ Ω , q s Φ − x e ψ ϕ s ⊗ Ω (cid:11) = 2Im ˆ t ds (cid:10) U ∗ h ( s ; 0) (cid:0) ξ s − e ψ ϕ s ⊗ Ω (cid:1) , e U ∗ h ( s ; 0) q s Φ − x ψ ϕ s ⊗ Ω (cid:11) (94)by three integrals which contain a derivative with respect to the the time variable. Note thatwe absorbed the phase factor of e ψ ϕ s in the dynamics e U ∗ h ( s ; 0) . Thus, | (82c) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ds (cid:10) U ∗ h ( s ; 0) (cid:0) ξ s − e ψ ϕ s ⊗ Ω (cid:1) , dds (cid:2) U ∗ h ( s ; 0) R s Φ − x ψ ϕ s ⊗ Ω (cid:3) (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ds (cid:10) U ∗ h ( s ; 0) (cid:0) ξ s − e ψ ϕ s ⊗ Ω (cid:1) , e U ∗ h ( s ; 0) ˙ R s Φ − x ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ds (cid:10) U ∗ h ( s ; 0) (cid:0) ξ s − e ψ ϕ s ⊗ Ω (cid:1) , e U ∗ h ( s ; 0) R s Φ − x ∂ s ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) . (95)In the first term we integrate by parts and we use ξ = e ψ ϕ ⊗ Ω . We find | (82c) | ≤ (cid:12)(cid:12)(cid:12)(cid:10) U ∗ h ( t ; 0) (cid:0) ξ t − e ψ ϕ t ⊗ Ω (cid:1) , U ∗ h ( t ; 0) R t Φ − x e ψ ϕ t ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ds (cid:10) dds h U ∗ h ( s ; 0) (cid:0) ξ s − e ψ ϕ s ⊗ Ω (cid:1)i , U ∗ h ( s ; 0) R s Φ − x e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ds (cid:10)(cid:0) ξ s − e ψ ϕ s ⊗ Ω (cid:1) , ˙ R s Φ − x e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ds (cid:10) ξ s , R s Φ − x ∂ s ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) . (96)In order to compute the time derivative occurring in the second summand, we use (80) andthe notation δH s = Φ x − a ( σ ψ s ) − a ∗ ( σ ψ s ) + N (97)14nd get dds h U ∗ h ( s ; 0) (cid:0) ξ s − e ψ ϕ s ⊗ Ω (cid:1)i = − iU ∗ h ( s ; 0) δH s ξ s − α − U ∗ h ( s ; 0) R s V iϕ s e ψ ϕ s ⊗ Ω (98)as well as U ∗ h ( t ; 0) (cid:0) ξ t − e ψ ϕ t ⊗ Ω (cid:1) = − i ˆ t ds U ∗ h ( s ; 0) δH s ξ s − α − ˆ t ds U ∗ h ( s ; 0) R s V iϕ s e ψ ϕ s ⊗ Ω . (99)Applying (99) to the first term of the r.h.s. of (96) and (98) to the second, we obtain | (82c) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ds (cid:10) δH s ξ s , R s Φ − x e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) (100a) + 2 (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ds (cid:10) δH s ξ s , U ∗ h ( t ; s ) R t Φ − x e ψ ϕ t ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) (100b) + 2 α − (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ds (cid:10) R s V iϕ s e ψ ϕ s ⊗ Ω , R s Φ − x e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) (100c) + 2 α − (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ds (cid:10) R s V iϕ s e ψ ϕ s ⊗ Ω , U ∗ h ( t ; s ) R t Φ − x e ψ ϕ t ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) (100d) + 2 (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ds (cid:10)(cid:0) ξ s − e ψ ϕ s ⊗ Ω (cid:1) , ˙ R s Φ − x e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) (100e) + 2 (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ds (cid:10) ξ s , R s Φ − x ∂ s ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) . (100f) The term (100a) : According to the definition of δH , we decompose (100a) as(100a) ≤ ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) ξ s , N R s Φ − x e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) (101a) + 2 ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) ξ s , (cid:0) a ( σ ψ s ) + a ∗ ( σ ψ s ) (cid:1) R s Φ − x e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) (101b) + 2 ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) ξ s , Φ x R s Φ − x e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) . (101c)We notice that (cid:2) N , R s (cid:3) = 0 and that N Ψ = α − Ψ if Ψ ∈ H is a one-phonon state and writethe first line as (101a) = 2 α − ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) ξ s , R s Φ − x e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) = 2 α − ˆ t ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:10) ξ s , R s ˆ d k | k | − e − ikx a ∗ k e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) . (102)By means of Lemma IV.2 and Lemma A.1 this becomes(101a) ≤ Cα − ˆ t ds k ξ s k (cid:13)(cid:13)(cid:13) R s ( − ∆ + 1) / (cid:13)(cid:13)(cid:13) k ψ ϕ s k ≤ Cα − | t | (103)15or all | t | ≤ C Λ α . In a similar way, we calculate (cid:2) a ( σ ψ s ) , a ∗ k (cid:3) = α − σ ψ s ( k ) for all k ∈ R andestimate (101b) = 2 ˆ t ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:10) ξ s , (cid:0) a ( σ ψ s ) + a ∗ ( σ ψ s ) (cid:1) R s ˆ d k | k | − e − ik · x a ∗ k e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ˆ t ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:10) ξ s , R s ˆ d k | k | − e − ik · x a ∗ ( σ ψ s ) a ∗ k e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) + 2 α − ˆ t ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:10) ξ s , R s ˆ d k | k | − e − ik · x σ ψ s ( k ) e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) . (104)Applying Lemma A.1, Lemma III.2 and Lemma IV.2 to the first line and using the samearguments as in Lemma A.1 for the second line this becomes(101b) ≤ Cα − ˆ t ds k ξ s k (cid:13)(cid:13)(cid:13) R s ( − ∆ + 1) / (cid:13)(cid:13)(cid:13) k σ ψ s k k ψ ϕ s k ≤ Cα − | t | for all | t | ≤ C Λ α . (105)Since Φ x = Φ + x + Φ − x we have(101c) = 2 ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) ξ s , Φ x R s Φ − x e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) ≤ ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) ξ s , Φ + x R s Φ − x e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) (106a) + 2 ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) Φ + x ξ s , R s Φ − x e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) . (106b)Making use of Lemma A.2 the first line can be estimated by(106a) ≤ C ˆ t ds (cid:13)(cid:13)(cid:13) ( − ∆ + 1) / N / R s Φ − x e ψ ϕ s ⊗ Ω (cid:13)(cid:13)(cid:13) = C ˆ t ds (cid:13)(cid:13)(cid:13) ( − ∆ + 1) / R s N / Φ − x e ψ ϕ s ⊗ Ω (cid:13)(cid:13)(cid:13) . (107)Since k ( − ∆+1) / R s ( − ∆+1) / k ≤ C for all | t | ≤ C Λ α by Lemma IV.2 and N / Ψ = α − Ψ if Ψ ∈ H is a one-phonon state, we find(106a) ≤ Cα − ˆ t ds (cid:13)(cid:13)(cid:13) ( − ∆ + 1) − / Φ − x e ψ ϕ s ⊗ Ω (cid:13)(cid:13)(cid:13) (108)for all | t | ≤ C Λ α . With Lemma A.1 we arrive at(106a) ≤ Cα − ˆ t ds k ψ ϕ s k ≤ Cα − | t | (109)16or all | t | < C Λ α . In similar fashion we use Lemma A.2, Lemma A.1 and N R s Φ − x e ψ ϕ s ⊗ Ω = α − R s Φ − x e ψ ϕ s ⊗ Ω to estimate(106b) = 2 ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) Φ + x ξ s , R s Φ − x e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) = 2 ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) ( N + α − ) − / Φ + x ξ s , ( N + α − ) / R s Φ − x e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) ≤ ˆ t ds (cid:13)(cid:13)(cid:13) ( N + α − ) − / Φ + x ξ s (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ( N + α − ) / R s Φ − x e ψ ϕ s ⊗ Ω (cid:13)(cid:13)(cid:13) ≤ Cα − ˆ t ds (cid:13)(cid:13)(cid:13) ( − ∆ + 1) / ξ s (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) R s Φ − x e ψ ϕ s ⊗ Ω (cid:13)(cid:13)(cid:13) ≤ Cα − ˆ t ds (cid:13)(cid:13)(cid:13) ( − ∆ + 1) / ξ s (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) R s ( − ∆ + 1) / (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ( − ∆ + 1) − / Φ − x e ψ ϕ s ⊗ Ω (cid:13)(cid:13)(cid:13) ≤ Cα − ˆ t ds (cid:13)(cid:13)(cid:13) ( − ∆ + 1) / ξ s (cid:13)(cid:13)(cid:13) k ψ ϕ s k = Cα − ˆ t ds (cid:13)(cid:13)(cid:13) ( − ∆ + 1) / e − iH α s ψ ϕ ⊗ W ( α ϕ )Ω (cid:13)(cid:13)(cid:13) (110)for all | t | < C Λ α . Thus, if we now use − ∆ + 1 ≤ C ( H α + 1) (see Lemma A.3) this becomesusing the properties (7) of the Weyl operators(106b) ≤ Cα − ˆ t ds (cid:13)(cid:13)(cid:13) ( H α + 1) / e − iH α s ψ ϕ ⊗ W ( α ϕ )Ω (cid:13)(cid:13)(cid:13) = Cα − ˆ t ds (cid:13)(cid:13)(cid:13) ( H α + 1) / ψ ϕ ⊗ W ( α ϕ )Ω (cid:13)(cid:13)(cid:13) = Cα − ˆ t ds (cid:0) e ( ϕ ) + k ϕ k + 1 (cid:1) / = Cα − | t | (111)for all | t | < C Λ α . In total, we obtain (101c) ≤ Cα − | t | and hence (100a) ≤ Cα − | t | for all | t | < C Λ α . The term (100b) : For the next estimate, we recall the notation (97) to write (100b) as(100b) ≤ ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) ξ s , N U ∗ h ( t ; s ) R t Φ − x e ψ ϕ t ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) + 2 ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) ξ s , (cid:0) a ( σ ψ s ) + a ∗ ( σ ψ s (cid:1) U ∗ h ( t ; s ) R t Φ − x e ψ ϕ t ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) + 2 ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) ξ s , Φ x U ∗ h ( t ; s ) R t Φ − x e ψ ϕ t ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) . (112)Using [ N , U ∗ h ( t ; s )] = [ a ( σ ψ s ) , U ∗ h ( t ; s )] = [ a ∗ ( σ ψ s ) , U ∗ h ( t ; s )] = 0 allows us to estimate the firsttwo lines in exactly the same way as (101a) and (101b) and leaves us with(100b) ≤ Cα − | t | + 2 ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) ξ s , Φ x U ∗ h ( t ; s )) R t Φ − x e ψ ϕ t ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) (113)17or all | t | < C Λ α . The difficulty of this term is the fact that the operators Φ x and U ∗ h ( t ; s ) do not commute. Nevertheless, we can use Φ x = Φ + x + Φ − x to get(100b) ≤ Cα − | t | + 2 ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) Φ + x ξ s , U ∗ h ( t ; s ) R t Φ − x e ψ ϕ t ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) (114a) + 2 ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) ξ s , Φ + x U ∗ h ( t ; s ) R t Φ − x e ψ ϕ t ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) . (114b)Using the same estimates as in (110) and (111) we bound the first integral by ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) Φ + x ξ s , U ∗ h ( t ; s ) R t Φ − x ˜ ψ ϕ t ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) = 2 ˆ t ds (cid:12)(cid:12)(cid:12)(cid:10) ( N + α − ) − / Φ + x ξ s , U ∗ h ( t ; s )( N + α − ) / R t Φ − x ˜ ψ ϕ t ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12) ≤ ˆ t ds (cid:13)(cid:13)(cid:13) ( N + α − ) − / Φ + x ξ s (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ( N + α − ) / R t Φ − x ˜ ψ ϕ t ⊗ Ω (cid:13)(cid:13)(cid:13) ≤ Cα − | t | for all | t | ≤ C Λ α . (115)For the second term Lemma A.2 and U ∗ h ( t ; s ) = U h ( s ; t ) imply(114b) ≤ C ˆ t ds (cid:13)(cid:13)(cid:13) ( − ∆ + 1) / N / U h ( s ; t ) R t Φ − x ˜ ψ ϕ t ⊗ Ω (cid:13)(cid:13)(cid:13) . (116)It follows from Lemma III.2 and (19) that for ξ ∈ L ( R ) ⊗ Fh ξ, U ∗ h ( s ; τ )( − ∆ + 1) U h ( s ; τ ) ξ i ≤ C h ξ, U ∗ h ( s ; τ )( h ϕ s + 1) U h ( s ; τ ) ξ i = C h ξ, ( h ϕ τ + 1) ξ i − α − ˆ sτ dτ ′ h ξ, U ∗ h ( τ ′ ; τ ) V iϕ τ ′ U h ( τ ′ ; τ ) ξ i≤ C h ξ, ( − ∆ + 1) ξ i + α − ˆ sτ dτ ′ h ξ, U ∗ h ( τ ′ ; τ )( − ∆ + 1) U h ( τ ′ ; τ ) ξ i . (117)The Gronwall inequality yields k ( − ∆ + 1) / U h ( s ; τ ) ξ k ≤ Ce α − | s − τ | k ( − ∆ + 1) / ξ k ≤ C k ( − ∆ + 1) / ξ k (118)for all | s − τ | ≤ C Λ α . Thus(114b) ≤ C ˆ t ds (cid:13)(cid:13)(cid:13) ( − ∆ + 1) / N / R t Φ − x e ψ ϕ t ⊗ Ω (cid:13)(cid:13)(cid:13) ≤ Cα − | t | (119)for all | t | ≤ C Λ α − , where we concluded by Lemma A.1 and Lemma IV.2 as for the term(106a). The terms (100c) and (100d) : With the help of Lemma III.2, Lemma A.1, Lemma IV.2and (28) one obtains(100c) = 2 α − (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ds (cid:10) R s V iϕ s e ψ ϕ s ⊗ Ω , R s ˆ d k | k | − e − ik · x a ∗ k e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α − ˆ t ds k R s k k ψ ϕ s k H ( R ) (cid:13)(cid:13)(cid:13) R s ( − ∆ + 1) / (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13) ( − ∆ + 1) − / ˆ d k | k | − e − ikx a ∗ k e ψ ϕ s ⊗ Ω (cid:13)(cid:13)(cid:13)(cid:13) ≤ Cα − | t | (120)and (100d) ≤ Cα − | t | for all | t | < C Λ α . 18 he term (100e) : Applying Lemma A.1 once more we estimate(100e) = 2 (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ds (cid:10)(cid:0) ξ s − e ψ ϕ s ⊗ Ω (cid:1) , ˙ R s ˆ d k | k | − e − ik · x a ∗ k e ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ˆ t ds (cid:13)(cid:13)(cid:13) ˙ R s (cid:0) − ∆ + 1 (cid:1) / (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13) ( − ∆ + 1) − / ˆ d k | k | − e − ik · x a ∗ k e ψ ϕ s ⊗ Ω (cid:13)(cid:13)(cid:13)(cid:13) . (121)From (28), (55) and Lemma IV.2 we get(100e) ≤ Cα − | t | for all | t | < C Λ α . (122) The term (100f) : With the help of Lemma IV.1 and Lemma III.2 we get(100f) = 2 α − (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ds (cid:10) ξ s , R s Φ − x R s V iϕ s ψ ϕ s ⊗ Ω (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α − ˆ t ds (cid:13)(cid:13)(cid:13) R s ( − ∆ + 1) / (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ( − ∆ + 1) − / Φ − x R s V iϕ s ψ ϕ s ⊗ Ω (cid:13)(cid:13)(cid:13) ≤ Cα − ˆ t ds (cid:13)(cid:13)(cid:13) R s ( − ∆ + 1) / (cid:13)(cid:13)(cid:13) k R s V iϕ s k k ψ ϕ s k ≤ Cα − | t | . (123)Here we used again Lemma A.1 and Lemma IV.2. In total, we obtain | (82c) | ≤ Cα − | t | for all | t | < C Λ α . (124)Summing up, we have shown that (cid:13)(cid:13)(cid:13) e − iH α t ψ ϕ ⊗ W ( α ϕ )Ω − e − i ´ t du ω ( u ) e ψ ϕ t ⊗ W ( α ϕ t )Ω (cid:13)(cid:13)(cid:13) ≤ Cα − | t | , (125)for all | t | ≤ C Λ α . A Auxiliary estimates
Lemma A.1.
There exists a constant
C > such that for all u ∈ L ( R ) and f ∈ L ( R ) (cid:13)(cid:13)(cid:13)(cid:13) ( − ∆ + 1) − / ˆ d k | k | − e − ik · x a ∗ k u ⊗ Ω (cid:13)(cid:13)(cid:13)(cid:13) ≤ Cα − k u k , (126) (cid:13)(cid:13)(cid:13)(cid:13) ( − ∆ + 1) − / ˆ d k | k | − e − ik · x a ∗ ( f ) a ∗ k u ⊗ Ω (cid:13)(cid:13)(cid:13)(cid:13) ≤ Cα − k u k k f k . (127) Proof.
The commutation relations imply k ( − ∆ + 1) − / ˆ d k | k | − e − ik · x a ∗ k u ⊗ Ω k = ˆ d k ˆ d k ′ | k | − | k ′ | − h e − ik · x a ∗ k u ⊗ Ω , ( − ∆ + 1) − e − ik ′ · x a ∗ k ′ u ⊗ Ω i = α − ˆ d k | k | − h e − ik · x u, ( − ∆ + 1) − e − ik · x u i = α − ˆ d k | k | − h u, (( − i ∇ − k ) + 1) − u i = α − ˆ d p | ˆ u ( p ) | ˆ d k p − k ) + 1) | k | . (128)19ince | · | − and ( | · | + 1) − are radial symmetric and decreasing functions we have sup p ∈ R ˆ d k p − k ) + 1) | k | = ˆ d k k + 1) | k | < ∞ (129)by the rearrangement inequality. Hence, k ( − ∆ + 1) − / ˆ d | k | − e − ik · x a ∗ k u ⊗ Ω k ≤ Cα − ˆ d p | ˆ u ( p ) | = Cα − k u k . (130)For the second bound of the Lemma we use again the commutation relations. We find k ( − ∆ + 1) − / ˆ d k | k | − e − ik · x a ∗ ( f ) a ∗ k u ⊗ Ω k = ˆ d k ˆ d k ′ | k | − | k ′ | − h e − ik · x a ∗ ( f ) a ∗ k u ⊗ Ω , ( − ∆ + 1) − e − ik ′ · x a ∗ ( f ) a ∗ k ′ u ⊗ Ω i = α − k f k ˆ d k ˆ d k ′ | k | − | k ′ | − h e − ik · x a ∗ k u ⊗ Ω , ( − ∆ + 1) − e − ik ′ · x a ∗ k ′ u ⊗ Ω i + α − ˆ d kd k ′ | k | − | k ′ | − f ( k ) f ( k ′ ) h e − ik · x u, ( − ∆ + 1) − e − ik ′ · x u i . (131)Using the Cauchy-Schwarz inequality twice implies k ( − ∆ + 1) − / ˆ d k | k | − e − ik · x a ∗ ( f ) a ∗ k u ⊗ Ω k ≤ α − k f k ˆ d k ˆ d k ′ | k | − | k ′ | − h e − ik · x a ∗ k u ⊗ Ω , ( − ∆ + 1) − e − ik ′ · x a ∗ k ′ u ⊗ Ω i + α − k f k ˆ d k | k | − h e − ik · x u, ( − ∆ + 1) − e − ik · x u i≤ Cα − k f k k u k , (132)where we concluded using the first estimate of the Lemma resp. the arguments used for itsproof. Lemma A.2 (Lemma 4, Lemma 10 in [5]) . Let Φ + x = ´ d k | k | − e ik · x a k and N = ´ d k a ∗ k a k .Then (cid:13)(cid:13) Φ + x Ψ (cid:13)(cid:13) ≤ C (cid:13)(cid:13)(cid:13) ( − ∆ + 1) / N / Ψ (cid:13)(cid:13)(cid:13) and (cid:13)(cid:13)(cid:13) ( N + α − ) − / Φ + x Ψ (cid:13)(cid:13)(cid:13) ≤ C (cid:13)(cid:13)(cid:13) ( − ∆ + 1) / Ψ (cid:13)(cid:13)(cid:13) . (133) Proof.
We split the operator Φ + x = Φ + ,>x + Φ + ,
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 (grantagreement 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 thankAndreas Deuchert and David Mitrouskas for interesting discussions. B. S. and R. S. wouldlike to thank Rupert Frank for stimulating discussions about the time-evolution of a polaron.
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