Mean-field Dynamics for the Nelson Model with Fermions

The Nelson model (with ultraviolet cutoff) describes a quantum system of non-relativistic particles coupled to a positive or zero mass quantized scalar field. We take the non-relativistic particles to obey Fermi statistics and discuss the time evolution in a mean-field limit of many fermions which is coupled to a semiclassical limit. At time zero, we assume that the bosons of the radiation field are close to a coherent state and that the state of the fermions is close to a Slater determinant with a certain semiclassical structure. We show that the many-body state approximately stays a Slater determinant and retains its semiclassical structure at later times and that its time evolution can be approximated by the fermionic Schroedinger-Klein-Gordon equations. This is proven in terms of reduced density matrices with explicit rates of convergence and for all semiclassical times.


I Introduction
Interacting many-body systems are very difficult to analyze, and analytic or numerical solutions are usually not feasible for many particles. Therefore simpler effective equations are used to analyze these systems throughout the sciences. These approximations often work very well and can be derived with heuristic arguments and good intuition. In mathematical physics the question of a rigorous justification of such effective equations is an active field of research, starting in the 1970's with works like [25,27,12,23,42] (see [41] for an excellent overview). Sparked by the 2001 nobel prize for the experimental realization of a Bose-Einstein condensate there has been great interested in the derivation of effective equations for bosonic systems (we refer to [31,11] for references and an overview of the topic). More recently, there was an increasing interest in the evolution of many fermion systems [32,42,5,6,16,4,21,10,9,34,3,7,37,35]. These works suggest that many aspects of the mean-field regime of weakly correlated bosons and fermions that interact via a pair potential are well understood by now. However, less attention has been paid to systems in which the interaction between the particles is mediated by a second quantized radiation field. Also here effective equations are of great importance because quantized radiation fields are described on Fock space, i.e., Hilbert spaces with infinitely many degrees of freedom. The complexity of such systems is reduced tremendously when the quantized field is approximated by a pair potential or a classical radiation field. The articles [15,26,45] show that the quantized radiation field can sometimes be replaced by a two-particle interaction if the particles are much slower than the bosons of the radiation field. Moreover it is possible to derive classical field equations from second quantized models [22,18,1,19,20,29,30,24,13,14]. While these works focus on bosonic systems or fermionic systems with small particle number the present paper seems to be the first that considers a many particle limit of fermions which interact by means of a quantized radiation field. The scaling, which will be explained in the following, can been seen as a fermionic mean-field limit because it is chosen such that the source term of the radiation field can effectively be replaced by its mean value. Moreover, it can be viewed as the second quantized analogue of the fermionic mean-field model of [10].
We consider N identical fermions that interact by means of a quantized scalar field. The state of the radiation field is represented by elements of the bosonic Fock space F := n≥0 L 2 (R 3 ) ⊗ n s , where the subscript s indicates symmetry under interchange of variables. The Hilbert space of the whole systems is Here the subscript "as" indicates anytisymmetry under exchange of variables. An element where we use the short-hand notation X N = (x 1 , . . . , x N ) and K n = (k 1 , . . . k n ). We define the (pointwise) annihilation and creation operators by (a(k)Ψ N ) (n) (X N , k 1 , . . . , k n ) = (n + 1) 1/2 Ψ wherek j means that k j is left out in the argument of the function. They are operator valued distributions and satisfy the commutation relations Moreover, we introduce the dispersion relation ω(k) = (|k| 2 + m 2 ) 1/2 with mass m ≥ 0 (we set = 1 = c here) and define the form factor of the radiation field bỹ Here, Λ is a momentum cutoff and we assume Λ ≥ 1. The field operator is then given by Φ Λ (x) =ˆd 3 kη(k) e ikx a(k) + e −ikx a * (k) (6) and the free Hamiltonian of the scalar field is the self-adjoint operator H f =ˆd 3 k ω(k)a * (k)a(k) with The full system is described by the Nelson Hamiltonian The factor δ N is an arbitrary particle number dependent scaling parameter that allows to scale the velocity of the bosons. The Nelson Hamiltonian is self-adjoint on the domain D (H N ) = D N j=1 −∆ j + H f , which can be shown by applying Kato's theorem as in [33,43]. The time evolution of the wave function Ψ N,t is governed by the Schrödinger equation The appearance of N −1/3 in (10) stems from the fact that we are interested in initial conditions which are localized in a volume of order one. Then, due to the Fermi statistics, the average kinetic energy per fermion is of order N 2/3 , and the average momentum per fermion of order N 1/3 . Therefore, we rescale time so we track the particles only while they move in the volume of order one, i.e., we go to time scales N −1/3 . This gives rise to a factor N 1/3 in front of the time derivative. If we use the Schrödinger equation (10) to compute the Ehrenfest equation for the field operator, we obtain Note that the integral on the right-hand side is proportional to N −1 times the smeared out electron density (i.e., for Λ → ∞ the electron density), which for our initial conditions is a function of order one in a volume of order one. Equation (11) also shows that not only the coupling constant in front of the radiation field (which we set equal to one) but also δ N determines the variation of the mean of the field operator. While our main result Theorem II.3 holds for arbitrary δ N , we believe that two choices are of particular interest.
1. For δ N = N 1/3 the velocities of the electrons and the bosons scale equally. Moreover, it ensures that the right hand side of (11) and hence the variation of the mean of the field operator is of order N 2/3 . This gives rise to interesting evolution equations (16) which capture the effect of the interaction.
2. If we set δ N = 1 our model corresponds to an unscaled system whose dynamics is studied for time scales of order N −1/3 . This is interesting because usually mean-field results for systems with two-particle interaction require a scaling of the coupling constant. It should be noted that most of the electrons travel on a distance of order one and hence could interact with the other electrons. However, a look at (11) shows that the group velocity of the bosons is too slow to mediate an interaction between the electrons. This suggests that the electrons effectively evolve like free electrons in an external potential.
Further insight concerning the scaling can be gained if we set N = N −1/3 and multiply (10) by N . This gives and shows that for δ N = N 1/3 our limit can be viewed as a combined weak coupling and semiclassical limit. Moreover, it displays a connection to the fermionic mean-field scaling considered in [10], i.e., to the model with χ N,t ∈ L 2 as (R 3N ). Like in [10] it will be crucial for us to consider initial data with a semiclassical structure, meaning that the kernel of the one-particle reduced density matrix is concentrated along its diagonal (see Remark II.6 for more details). Moreover, we assume the initial states to be approximately of product form Here, α 0 ∈ L 2 (R 3 ), N j=1 ϕ 0 j denotes the antisymmetrized product (wedge product) of or- , Ω denotes the vacuum in F and W is the Weyl operator for all f ∈ L 2 (R 3 ) (f (k) denotes the complex conjugate of f (k)). In such a state the only correlations are due to the antisymmetry of the electrons. During the time evolution correlations emerge but the product structure (as will be shown) is preserved in the limit N → ∞. This suggests to approximate the action of the scaled field operator N −2/3 Φ Λ on Ψ N,t by a classical radiation field and replace the right hand side of (11) by a coupling to the mean electron density. In fact, Theorem II.3 says that Ψ N,t can be approximated by N . This system of equations is formally equivalent to Its solutions have nice regularity properties because of the ultraviolet cutoff in the radiation field. Let m ∈ N, H m (R 3 ) denote the Sobolev space of order m and L 2 m (R 3 ) a weighted L 2 -space with norm ||α|| L 2 m (R 3 ) = (1 + |·| 2 ) m/2 α L 2 (R 3 ) . Throughout this paper, we use (16). Moreover, if ϕ 0 1 , . . . , ϕ 0 N are orthonormal, then so are ϕ t 1 , . . . , ϕ t N for all t ∈ [0, ∞). Proof. The proof follows by a standard fixed point argument, which is given in the Appendix A.

II Main result
As mentioned above, our goal is to show that Ψ N,t ≈ N j=1 ϕ t j ⊗ W (N 2/3 α t )Ω holds during the time evolution. In the following, this will be proven in the trace-norm distance of reduced density matrices. Let us introduce the number operator with domain Moreover, we choose ||Ψ N,0 || = 1 and Ψ N,0 ∈ H (N ) ∩ D(N ) ∩ D(N H N ) (note that for the definition of the reduced density matrix below we only need Ψ N,0 ∈ H (N ) ∩ D(N )). By unitarity also ||Ψ N,t || = 1 and the following lemma holds.
For k ∈ N, we define the k-particle reduced density matrices of the fermions (as operators on L 2 (R 3k )) by γ (k,0) where Tr k+1,...,N denotes the partial trace over the coordinates x k+1 , . . . , x N and Tr F the trace over Fock space. Additionally, we consider on L 2 (R 3 ) the one-particle reduced density matrix of the bosons with kernel The operator γ (0,1) N,t is trace class with Tr γ (0,1) trace one) by a constant. In our choice we only measure deviations from the classical mode function that are at least of order N 4/3 . This is important if one starts initially with no bosons and examines the one-particle reduced density matrix after short times when only a few bosons have been created. Then, the state of the bosons might not be coherent and the usual definition of the one-particle reduced density matrix may not converge to the classical mode function. However, such mismatches are not important for the dynamics (and hence neglected in our definition) because the field operator is rescaled by a factor of N −2/3 , see (12). Let us now state the main result of this article. We summarize the conditions on our initial data in the following assumption. We denote the trace norm of an operator A by ||A|| Tr := Tr |A|.
Theorem II.3. Let Assumption II.2 hold and let Ψ N,t be the solution to (10) with initial condition Ψ N,0 , and ϕ t 1 , . . . , ϕ t N , α t the solution to (16) with initial condition ϕ 0 1 , . . . , ϕ 0 N , α 0 . We define Then there exists C > 0 (independent of N , Λ and t) such that for any t ≥ 0, and for The theorem is proved in Section VI.
Remark II.4. In [16] and [10] a similar limit was considered for fermions that interact by means of a pair potential. From these works we learned the importance of the semiclassical structure. The most related works from a technical point of view are [34] and [18,29,30].
Remark II.5. There are different ways of stating the main result. For example, one could get rid of the b N by simply noting that b N ≤ 1 2 N 1/3 a N (which follows directly from (118)). This leads to stronger conditions on the initial state. Also, a slightly more general statement which implies Theorem II.3 can be found in Lemma V.7. There, the inequalities are expressed in terms of the number of particles outside the state N j=1 ϕ t j ⊗ W (N 2/3 α t )Ω rather than in terms of trace norms.
The first inequality in (22) means that the kernel p 0 (x, y) is localized around a distance smaller than of order N −1/3 around the diagonal x − y. The second inequality means that the density varies on scales of order one. In fact, these conditions should imply that the time evolution of p 0 (or, say, its Wigner transform) is close to a classical evolution equation, which here is the Vlasov equation. This has indeed been shown in the two-body interaction case, let us refer to [8] and references therein. Note also that for simple cases like plane waves in a box of volume of order one, (22) indeed holds, see [37]. For a more thorough discussion of these conditions we refer to [10,37].
Remark II.7. Let us give a bit more intuition about c N . We first note that the Weyl operator One of its well-known properties (see, e.g., [39] for a nice exposition) is With that at hand we can rewrite c N and find from which it might become more clear that c N measures the initial deviations around the classical radiation field α 0 .
Remark II.8. An extension to semirelativistic or Dirac-type dispersion relations for the fermions similar to [9] seems possible. Also regular enough external potentials could be included.

III Structure of the proof
To prove Theorem II.3 it is important to define and control the right macroscopic variables. For that, we adapt techniques that are based on the method from [36] and that were further developed in [34,29,30]. In addition, it is crucial to find the right measure for the correlations between the electrons and to consider only initial states with semiclassical structure. The key idea of the proof is to define a suitable functional β(Ψ N , ϕ 1 , . . . , ϕ N , α) which measures if the fermions are close to an antisymmetrized product state N j=1 ϕ j with ϕ 1 , . . . , ϕ N orthonormal and if the state of the radiation field is approximately coherent. To this end, we introduce the following operators.
Moreover, we define the projectors p ϕ 1 ,...,ϕ N m : The correlations between the electrons are controlled by means of two functionals.
We note that β a,1 (Ψ N , ϕ 1 , . . . , ϕ N ) corresponds to the expectation value of the relative number of fermions outside the antisymmetric product N j=1 ϕ j (i.e., the relative number of excitations around the state N j=1 ϕ j ). The functional N −1/3 β a,2 (Ψ N , ϕ 1 , . . . , ϕ N ) corresponds (up to a small error) to the expectation value of the square of this number. More details about the technical relevance of β a,2 are given at the beginning of Section V.
In order to investigate whether the state of the radiation field is coherent, we define β b , which measures the fluctuations of the field modes around the complex function α.
Note that β b (Ψ N,0 , α 0 ) = c N as we showed in (32). Let us also remark that when Ψ N,t is a solution to (10) and ϕ t 1 , . . . , ϕ t N , α t a solution to (16), then the functional β b Ψ N,t , α t coincides (up to scaling) with the one used in the coherent states approach, see, e.g., [11,Chapter 3]. Finally, the functional β is defined by In the following, we are interested in the value of β Ψ N,t , ϕ t 1 , . . . , ϕ t N , α t , where (ϕ t 1 , . . . , ϕ t N , α t ) is a solution of the Schrödinger-Klein-Gordon equations (16) and Ψ N,t evolves according to the Schrödinger equation (10). In this case, we apply the shorthand notations β(t), β a,1 (t), β a,2 (t) and β b (t). Moreover, we use the abbreviations p For the proof of Theorem II.3 we pursue the following strategy.
A) We choose initial data (ϕ 0 1 , . . . , ϕ 0 N , α 0 ) of the Schrödinger-Klein-Gordon system (16) and a many-body wave function Ψ N,0 that satisfy our Assumption II.2. Theorem I.1 and Lemma II.1 make sure that the solutions at any time t ≥ 0 are regular enough, and in Section IV we show that the solutions still have the semiclassical structure. B) After that, we control the change of β(t) in time. For this, we use the semiclassical structure to estimate d dt β(t) ≤ e Ct β(t) + N −1 for some C > 0 at each time t ≥ 0. Gronwall's lemma then yields β(t) ≤ e e Ct β(0) + N −1 .
C) Finally, we relate the initial states of Theorem II.3 and the trace norm convergence of the reduced density matrices to the counting functional.
Notation III.5. In the rest of this article the letter C denotes a generic positive constant and its value might change from line to line for notational convenience.

IV Semiclassical structure
We first prove that the semiclassical structure from Equation (22) can be propagated in time.
The Hilbert-Schmidt norm of an operator A is denoted by ||A|| HS := √ Tr A * A. (16). We assume that for all k ∈ R 3 and for someC > 0. Then there exists some C > 0 (independent of N , Λ and t) such that for all k ∈ R 3 and for all t ∈ R.
Remark IV.2. We could formulate Lemma IV.1 likewise in terms of p t , e ikx Tr and p t , ∇ Tr as was done in [10], because These inequalities hold since p t q t = 0, ||AB|| Tr ≤ ||A|| ||B|| Tr and ||BA|| Tr ≤ ||A|| ||B|| Tr for A bounded and B trace class, q t = 1, and ||B|| Tr = ||B * || Tr for B trace class (which follows from the fact that B and B * have the same singular values, and thus the positive operators |B| and |B * | have the same eigenvalues).
Proof of Lemma IV.1. The propagation of the semiclassical structure is shown in a similar way as in [10,Section 5]. Recall that due to Theorem I.1 the solution From and using p t + q t = 1, we conclude Next, we define the time dependent self-adjoint operators and their respective unitary propagators U +k (t; s) and U −k (t; s). These are indeed welldefined, which follows from [ which leads to and thus q t e ikx p t = U +k (t; 0)q 0 e ikx p 0 U * −k (t; 0) For the trace norm, we then obtain the estimate q t e ikx p t Tr ≤ q 0 e ikx p 0 and ||AB|| Tr ≤ ||A|| ||B|| Tr and ||BA|| Tr ≤ ||A|| ||B|| Tr for A bounded and B trace class. Thus, In order to control the latter term, we calculate the time derivative of q t ∇p t . We find In analogy to the previous calculation, we define the two-parameter group U h (t; s) satisfying for all ϕ ∈ H 2 (R 3 ) and U h (s; s) = 1. Then, we calculate which implies Using the same inequalities as for (52), this leads to By Lemma A.3, which says that α t 2 ≤ α 0 2 + ||η|| 2 |t|, we can estimate (1 + |k|) −1 q t e ikx p t where C(Λ, s, α 0 2 ) := 4 + 2 (1 + |·|) 2η By means of Gronwall's lemma and the chosen initial conditions, we obtain Finally, note that Tr . (65)

V Estimates on the time derivative
In this section we control the change of β(t) in time by separately estimating the time derivatives of β a,1 (t), β a,2 (t) and β b (t). Note that the time derivative of β a,1 (t) can be controlled in terms of β a,1 (t) itself, β b (t), and an error of order N −1 . The time derivative of β b (t), however, is controlled in terms of β a,1 (t), β a,2 (t), β b (t) itself, and an error of order N −1 . This is why we also introduced β a,2 (t). It allows us to close the Gronwall argument, since its time derivative can be bounded in terms of β a,1 (t), β a,2 (t) itself, β b (t), and an error of order N −5/3 . We first derive simple expressions for the corresponding time derivatives by direct computation. Then, in the following subsections, we bound these expressions as explained above.
Let Ψ N,t be the solution to (10) with initial condition Ψ N,0 , and ϕ t 1 , . . . , ϕ t N , α t the solution to (16) with initial condition ϕ 0 1 , . . . , ϕ 0 N , α 0 . Then Proof. The functional β a,1 (t) is time-dependent, because Ψ N,t and (ϕ t 1 , . . . , ϕ t N , α t ) evolve according to (10) and (16) is the effective Hamiltonian acting on the m-th variable. This leads to d dt Similarly, one derives The time derivative of β b (t) is obtained by the following calculations. Note that the expressions in the calculations are all indeed well-defined, since the domain D (N ) ∩ D (N H N ) is invariant under the time evolution, see Lemma II.1. Using (16) we find For the commutator we find It follows that since the scalar product in the first line is real.
Before we prove appropriate estimates for the time derivative of β(t), let us state a technical lemma which was already proven, e.g., in [34,3]; we give a proof here for convenience. Note that this is an important point where the antisymmetry of the wave function is used.
Proof. Let us omit the subscripts l 1 , . . . , l j in the following for better readability. In order to prove the inequality, it is convenient to use the singular value decomposition A = Cauchy-Schwarz, this allows us to estimate Note that k∈K |χ i χ i | k is for all i ∈ N and K ⊂ {1, . . . , N } a projector on functions antisymmetric in all K-variables, since where the last step is true because the non-diagonal terms vanish due to the antisymmetry. It follows that V.1 Estimate on the time derivative of β a,1 (t) Lemma V.3. Let Assumption II.2 hold and let Ψ N,t be the solution to (10) with initial condition Ψ N,0 , and ϕ t 1 , . . . , ϕ t N , α t the solution to (16) with initial condition ϕ 0 1 , . . . , ϕ 0 N , α 0 . Then there is a C > 0 (independent of N , Λ, t) such that for all t > 0, Proof. Using the Fourier expansion of the radiation field we write We need to estimate this expression in terms of β(t). Let us consider (81) first. By using Cauchy-Schwarz we can obtain β b (t)β a,1 (t) if we estimate p t 1 e −ikx 1 ≤ 1. But then there is an extra factor N 1/6 too much (recall that there is one factor N 1/3 in the definition of β b (t)). This difficulty is solved by using the estimates from Section IV, i.e., the semiclassical structure of the solution. In (80) there is an additional difficulty, namely that the q t 1 projector is not next to the right Ψ N,t in order to gain β a,1 (t) by Cauchy-Schwarz. This problem can be solved by symmetrization. Using the antisymmetry of Ψ N,t (in the x variables), we find for the first summand We now use that by Lemma V.2, HS ||Ψ N || 2 for all antisymmetric Ψ N , Hilbert-Schmidt operators A and bounded operators B. This is a type of estimate where we crucially use the antisymmetry of Ψ N . In the end we use the semiclassical structure, i.e., Lemma IV.1, and find Here, C(Λ, α 0 , t) = CΛ 4 (1 + α 0 )(1 + t 2 ). Thus, For the second summand we can directly use Cauchy-Schwarz without symmetrization. We use again ||A 1 Ψ N || 2 ≤ N −1 ||A|| 2 HS ||Ψ N || 2 and Lemma IV.1 in the end and find To summarize, we have Since (1 + |·|) 1/2η 2 ≤ CΛ 3/2 and using for ease of notation |x| ≤ exp(|x|), this gives V.2 Estimate on the time derivative of β a,2 (t) Lemma V.4. Let Assumption II.2 hold and let Ψ N,t be the solution to (10) with initial condition Ψ N,0 , and ϕ t 1 , . . . , ϕ t N , α t the solution to (16) with initial condition ϕ 0 1 , . . . , ϕ 0 N , α 0 . Then there is a C > 0 (independent of N , Λ, t) such that for all t > 0, Proof. We write the time derivative of β a,2 (t) as d dt β a,2 (t) Here, we have symmetrized the q t 2 so that we can bound the time derivative appropriately in terms of β a,2 (t). Note that We can then use Lemma V.2, together with Lemma IV.1 and find Since ||(1 + |·|)η|| 2 ≤ CΛ 2 , we arrive at

V.3 Estimate on the time derivative of β b (t)
The crucial terms in the time derivative of β b (t) can be estimated with a diagonalization trick similar to the one used in [34]. For the following estimates it is helpful to introduce the operators where ϕ ∈ L 2 (R 3 ). They have the following properties.
Lemma V.5. The operators p ϕ and q ϕ as defined in (94) are projectors on H (N ) for all ϕ ∈ L 2 (R 3 ). Moreover, let χ 1 , . . . , χ N ∈ L 2 (R 3 ) and ϕ 1 , . . . , ϕ N ∈ L 2 (R 3 ) each be orthonormal, and such that span{χ 1 , . . . , χ N } = span{ϕ 1 , . . . , ϕ N }. Then q χ j , q χ k = 0 ∀j, k = 1, . . . , N and Lemma V.6. Let Assumption II.2 hold and let Ψ N,t be the solution to (10) with initial condition Ψ N,0 , and ϕ t 1 , . . . , ϕ t N , α t the solution to (16) with initial condition ϕ 0 1 , . . . , ϕ 0 N , α 0 . Then there is a C > 0 (independent of N , Λ, t) such that for all t > 0, Proof. If we insert the identity p t 1 + q t 1 = 1 twice, (68) can be written as Note that for estimating the pp−Term it is crucial to use that parts of the first line of (100) cancel with parts of the second line of (100) which comes from the source term in the effective equations. To estimate the pp−Term, we split e −ikx 1 = cos(kx 1 ) − i sin(kx 1 ) into its real and imaginary part. This allows us to use the selfadjointness of the operators p t 1 cos(kx 1 )p t 1 and p t 1 sin(kx 1 )p t 1 . Subsequently we only estimate the cos-terms pp−Term cos ; the sin-terms are estimated in exactly the same manner. Note that for each t > 0 we can find N j=1 λ t j (k) = Tr p t cos(kx)p t =ˆd 3 y cos(ky)ρ t (y), and (105) The cos-part of the pp−Term can then be written as and be estimated by If one makes use of λ t j (k) ≤ 1 and Lemma V.5 one finds and obtains |pp−Term cos | ≤ 2 ||η|| 2 β b (t) β a,1 (t) + β a,2 (t) ≤ CΛ β a,1 (t) + β a,2 (t) + β b (t) . (110) In exactly the same manner one estimates pp−Term sin and obtains |pp−Term| ≤ CΛβ(t).
Summing all terms up then shows Lemma V.6.

VI Proof of Theorem II.3
In order to state our main result in terms of the trace norm difference of reduced density matrices let us add the following lemma.
Proof. This is a standard result. For example, a proof of (118) can be found in [34, Section 3.1] and a proof of (119) in [30, Section VII].
Let us now summarize all estimates and put them together for a proof of our main result.
In the theorem we also provide bounds for the specific initial state N j=1 ϕ 0 j ⊗W (N 2/3 α 0 )Ω. Since for this state γ (1,0) which can be checked by direct calculation as in [29, Section IX].

A Appendix: The fermionic Schrödinger-Klein-Gordon equations
Subsequently, we prove the existence and uniqueness of solutions of the effective equations (16). We start with the notation Then, we define the operator A : D(A) → H as the orthogonal sum Moreover, we define J : The fermionic Schrödinger-Klein-Gordon equations (16) can then be written as d dt The goal of this section is to show where all constants are monotone increasing (everywhere finite) functions of all its variables. Moreover, let ( ϕ 0 , α 0 ) ∈ D(A) and assume there is a T > 0 so that (130) has a unique continuously differentiable solution for t ∈ [0, T ). Then, ( ϕ t , α t ) is bounded from above for all t ∈ [0, T ).
We give the proof of the Lemma below. In order to prove Theorem I.1 we use Theorem A.2 (Theorem X.74 in [38] with n = 1). Let A be a self-adjoint operator on a Hilbert space H . Let J be a mapping which takes D(A) into D(A) and which satisfies for j ∈ {0, 1} for all ϕ, ψ ∈ D(A) where each constant C is a monotone increasing (everywhere finite) function of all its variables. Let ϕ 0 ∈ D(A) and suppose that on any finite interval of existence the solution ϕ(t) guaranteed by part (a) of Theorem X.73 in [38] has the property that ||ϕ(t)|| is bounded from above. Then there is a strongly differentiable D Proof. We define U ω (t) = e −iN −1/3 δ N ω(k)t . Then the Duhamel expansion of Equation (16) for α t can be written as Then, since FT [ρ ϕ s ] ∞ ≤ (2π) −3/2 ρ ϕ s 1 = (2π) −3/2 N for all s ∈ R, we have Proof of Lemma A.1.

Part a)
The operators are self-adjoint. Moreover, due to and we have {α ∈ L 2 (R 3 )|A N +1 α ∈ L 2 (R 3 )} = L 2 1 (R 3 ) and thus as a multiplication operator with dense domain is self-adjoint. Since direct sums of selfadjoint operators are self-adjoint (see, e.g., [44,Theorem 2.24] is a self-adjoint operator.

Part c)
To show part c) of Lemma A.1, we note that the classical radiation field Φ α Λ is linear in α, i.e.