The Fröhlich Polaron at Strong Coupling: Part I—The Quantum Correction to the Classical Energy

We study the Fröhlich polaron model in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^3$$\end{document}, and establish the subleading term in the strong coupling asymptotics of its ground state energy, corresponding to the quantum corrections to the classical energy determined by the Pekar approximation.


Introduction and Main Results
This is the first part of a study of the asymptotic properties of the Fröhlich polaron, which is a model describing the interaction between an electron and the optical modes of a polar crystal [12]. In the regime of strong coupling between the electron and the optical modes, also called phonons, it is a well known fact [1,7,20] that the ground state energy of the Fröhlich polaron is asymptotically given by the minimal Pekar energy [26], which can be considered as the ground state energy of an electron interacting with a classical phonon field. This result is motivated by using appropriately scaled units, see e.g. [28], which demonstrates that the strong coupling regime is a semi-classical limit in the phonon field variables. In such units the Fröhlich Hamiltonian, acting on the space L 2 R 3 ⊗ F L 2 R 3 , reads where the annihilation and creation operators satisfy the rescaled canonical commutation relations a(f ), a † (g) = α −2 f |g for f, g ∈ L 2 R 3 with α > 0 being the coupling strength, the interaction is given by w x (x ′ ) := π − 3 2 |x ′ − x| −2 and N is the corresponding (rescaled) particle number operator, i.e. N := ∞ n=1 a † (ϕ n )a(ϕ n ) where {ϕ n : n ∈ N} is an orthonormal basis of L 2 R 3 . The definition of the Fröhlich Hamiltonian in Eq. (1.1) has to be understood in the sense of quadratic forms, see for example [28], due to the ultraviolet singularity in the interaction w x . By substituting the annihilation and creation operators a and a † in Eq. (1.1) with a (classical) phonon field ϕ ∈ L 2 R 3 , i.e. replacing a(f ) with f |ϕ and a † (f ) with ϕ|f , we arrive at the Pekar energy where ψ ∈ L 2 R 3 is the wave-function of the electron. We further define the Pekar functional F Pek (ϕ) := inf ψ =1 E (ψ, ϕ) and the minimal Pekar energy e Pek := inf ϕ F Pek (ϕ). It is known that the ground state energy E α := inf σ (H), as a function of the coupling strength α, is asymptotically given by the minimal Pekar energy e Pek in the limit α → ∞ [1,7]. More precisely, one has e Pek ≥ E α = e Pek + O α→∞ α − 1 5 , as shown in [20]. In this work we are going to verify the prediction in the physics literature [30,2,3] that the sub-leading term in this energy asymptotics is actually of order α −2 with a rather explicit pre-factor where ϕ Pek is a minimizer of F Pek and H Pek is the Hessian of F Pek at ϕ Pek restricted to real-valued functions ϕ ∈ L 2 R R 3 , i.e. H Pek is an operator on L 2 R 3 defined by for all ϕ ∈ L 2 R R 3 . The prediction in Eq. (1.3) has been verified previously for polaron models either confined to a bounded region of R 3 [11] or to a three-dimensional torus [9]. The methods presented there exhibit substantial problems regarding their extension to the unconfined case, however. In this paper we present a new approach, which is partly based on techniques previously developed in the study of Bose-Einstein condensation and the validity of Bogoliubov's approximation for Bose gases [15,16,5] in the meanfield limit. We employ a localization method for the phonon field, which breaks the translation-invariance and effectively reduces the problem to the confined case, allowing for an application of some of the methods developed in [11,9]. Our main result is the following Theorem 1.1 where we verify the lower bound on E α in Eq. (1.3) . for all α ≥ α(s), where α(s) > 0 is a suitable constant.
As an intermediate result, which might be of independent interest, we will establish the existence of a family of approximate ground states, by which we mean states whose energy is given by the right side of (1.3), exhibiting complete Bose-Einstein condensation with respect to a minimizer ϕ Pek of the Pekar functional F Pek . We refer to Theorem 3.13 for a precise statement.
In contrast to the lower bound, the proof of the upper bound on E α in Eq. (1.3) is essentially the same as for confined polarons [11,9] and can be obtained by the same methods. It is also contained as a special case in [22], where it has been verified that the ground state energy E α (P ) as a function of the (conserved) total momentum P can be bounded from above by where m := 2 3 ∇ϕ Pek 2 and ǫ > 0, with C ǫ a suitable constant. Since E α = E α (0) [13,8,23], Theorem 1.1 in combination with Eq. (1.6) for the specific case P = 0 concludes the proof of Eq. (1.3). Combining (1.6) with Theorem 1.1, one further obtains an upper bound on the increment E α (P ) − E α , a quantity related to the effective mass of the polaron [14,18,29,4]. In the second part [6] we will discuss, in analogy to Theorem 1.1, the corresponding lower bound on E α (P ).
The proof of Eq. (1.3) for confined systems in [11,9] requires an asymptotically correct local quadratic lower bound on the Pekar functional F Pek (ϕ) for configurations close to a minimizer, as well as a sufficiently strong quadratic lower bound valid for all configurations. While our proof of Theorem 1.1 makes use of a local quadratic lower bound as well, we believe that in the translation-invariant setting any globally valid quadratic lower bound cannot be sufficiently strong, and therefore new ideas are necessary. As we explain in the following, we circumvent this problem by constructing an approximate ground state Ψ, which is essentially supported close to a minimizer of the Pekar functional F Pek , and consequently we only require a locally valid quadratic lower bound.
Proof strategy of Theorem 1.1. Even though we want to verify a lower bound on E α , let us first discuss how test functions providing an asymptotically correct upper bound are expected to look like. In the following let (ψ Pek , ϕ Pek ) denote a minimizer of the Pekar energy E defined in Eq. (1.2). It has been established in [17] that all other minimizers are given by translations ϕ Pek x (x ′ ) := ϕ Pek (x ′ − x) and ψ Pek x (x ′ ) := e iθ ψ Pek (x ′ − x) of ϕ Pek and e iθ ψ Pek , where θ is an arbitrary phase. W.l.o.g. let us denote in the following by (ψ Pek , ϕ Pek ) the unique minimizer of E such that ϕ Pek is radial and ψ Pek is non-negative. Then all the product states of the form ψ Pek where Ω ϕ Pek x is the coherent state corresponding to ϕ Pek x (defined by a(w)Ω ϕ = w|ϕ Ω ϕ for all w ∈ L 2 R 3 ), have the asymptotically correct leading term in the energy ψ Pek Pek . By taking convex combinations of these states on the level of density matrices, we can construct a large family of low energy states for any given probability measure µ on R 3 . Clearly, Γ µ exhibits the correct leading energy H Γµ = e Pek . Our proof of the lower bound given in Eq. (1.5) relies on the observation that asymptotically as α → ∞, any low energy state Ψ is of the form Γ µ with a suitable probability measure µ on R 3 . Since we only need this statement for the phonon part of Ψ, we will verify the weaker statement instead, see Theorem 3.2 for a precise formulation. This statement is analogous to a version of the quantum de Finetti theorem used in [15] in order to verify the Hartree approximation for Bose gases in a general setting. The main technical challenge of this paper will be the construction of approximate ground states Ψ where the corresponding measure is a delta measure, µ = δ 0 , i.e. the construction of states where the phonon part is essentially given by a single coherent state Ω ϕ Pek . The method presented here is based on a grand-canonical version of the localization techniques previously developed for translation-invariant Bose gases in [5], and in analogy to the concept of Bose-Einstein condensation we say that such states satisfy (complete) condensation with respect to the Pekar minimizer ϕ Pek . Heuristically this means that only field configurations ϕ close to the minimizer ϕ Pek are relevant, hence the translational degree of freedom has been eliminated and the system is effectively confined. Based on this observation we can adapt the strategy developed for confined polarons in [11,9], which starts by introducing an ultraviolet regularization in the interaction w x with the aid of a momentum cut-off Λ, leading to the study of the truncated Hamiltonian H Λ . Using a lower bound on the excitation energy F Pek (ϕ) − e Pek that is, up to a symplectic transformation, quadratic in the field variables ϕ and valid for all ϕ close to the minimizer ϕ Pek , one can bound the truncated Hamiltonian from below by an operator that is, up to a unitary transformation, quadratic in the creation and annihilation operators. The lower bound is only valid, however, if tested against a state satisfying (complete) condensation in ϕ Pek . Finally an explicit diagonalization of this quadratic operator yields the desired lower bound in Eq. (1.5).
The symplectic transformation on the phase space L 2 R 3 , respectively the corresponding unitary transformation on the Hilbert space F L 2 R 3 , is one of the key novel ingredients in our proof. It turns out to be necessary due to the presence of the translational symmetry, which makes it impossible to find a non-trivial positive semidefinite quadratic lower bound on F Pek (ϕ) − e Pek . This issue has already been addressed in the study of a polaron on the three dimensional torus [9], where a different coordinate transformation is used, however. The symplectic/unitary transformation presented in this paper is an adaptation of the one used in the study of translation-invariant Bose gases in [5].
Outline. The paper is structured as follows. In Section 2 we will introduce an ultraviolet cut-off as well as a discretization in momentum space, and provide estimates on the energy cost associated with such approximations. Section 3 then contains our main technical result Theorem 3.13, in which we verify the existence of approximate ground states satisfying (complete) condensation with respect to a minimizer ϕ Pek of the Pekar functional F Pek . Subsequently we will discuss a large deviation estimate for such condensates in Section 4, quantifying the heuristic picture that only configurations close to the point of condensation matter. In Section 5 we then discuss properties of the Pekar functional F Pek . In particular, we will discuss quadratic approximations around the minimizer ϕ Pek as well as lower bounds that are, up to a coordinate transformation, quadratic in ϕ. Together with the error estimates from Section 2 and the large deviation estimate from Section 4, applied to the approximate ground state constructed in Section 3, this will allow us to verify our main Theorem 1.1 in Section 6. The subsequent Section 7 contains the proof of Theorem 3.2, which can be interpreted as a version of the quantum de Finetti theorem adapted to our setting. Finally, Appendices A and B contain auxiliary results concerning the Pekar minimizer ϕ Pek and the projections introduced in Section 2, respectively.

Models with Cut-off
In this section we will estimate the effect of the introduction of an ultraviolet cut-off, as well as a discretization in momentum space, on the ground state energy, following similar ideas as in [20,11,9]. We will eventually apply these results for two different levels of coarse graining, a rough scale used in the proof of Theorem 3.2 in Section 7, which applies to low energy states with energy e Pek + o α→∞ (1), and a fine scale precise enough to yield the correct ground state energy up to errors of order o α→∞ α −2 , see the proof of Theorem 1.1 in Section 6.
is the (open) ball of radius r around the origin. Then we define the orthonormal system e n ∈ L 2 R 3 as e n (x) := 1 as well as the translated system e y,n (x) := e n (x − y) and the orthogonal projection Π y Λ,ℓ onto the space spanned by {e y,1 , . . . , e y,N }. Furthermore we denote with Π Λ the projection onto the spectral subspace of momenta |k| ≤ Λ.
Then we obtain for 0 < ℓ < Λ and x, y ∈ R 3 the following estimate on the L 2 norm Proof. With · denoting Fourier transformation, we have These Hamiltonians can be interpreted as the restriction of H (in the quadratic form sense) to states where only the phonon modes in Π y Λ,ℓ L 2 R 3 , respectively Π Λ L 2 R 3 , are occupied. In particular, this implies that inf σ(H y Λ,ℓ ) ≥ E α as well as inf σ(H Λ ) ≥ E α . In the following we shall quantify the energy increase due to the introduction of the cut-offs.
Note that the α-dependence of the Hamiltonians H, H y Λ,ℓ and H Λ only enters through the rescaled canonical commutation relations a(f ), a † (g) = α −2 f |g satisfied by the creation and annihilation operators a † and a, and we will usually suppress the α dependency in our notation for the sake of readability. In the rest of this paper, we will always assume that α is a parameter satisfying α ≥ 1 and, in case it is not stated otherwise, estimates hold uniformly in this parameter for α → ∞, i.e. we write X Y in case there exist constants C, α 0 > 0 such that X ≤ C Y for all α ≥ α 0 .
The proof of the subsequent Lemma 2.4 closely follows the arguments in [21,20], where it was shown that H is bounded from below and well approximated by an operator containing only finitely many phonon modes. For the sake of completeness we will illustrate the proof, which is based on the Lieb-Yamazaki commutator method, see [21]. In the following Lemma 2.4, we will use the identification allowing us to define the support supp (Ψ) as the closure of {x ∈ R 3 : Ψ(x) = 0}. Lemma 2.4. We have for all 0 < ℓ < Λ ≤ K and L > 0, and states Ψ with supp (Ψ) ⊂ B L (y) the estimate Furthermore, there exists a constant d > 0 such that for all t > 0, K ≥ 0 and α ≥ 1.
Proof. Let us define the functions u n x by u n x (k) : where we have applied the Cauchy-Schwarz inequality in the first line and used the specific choice ǫ := u n x in the last identity. Note that the L 2 -norm u n x is independent of x, and furthermore we can express ± H y Λ,ℓ − H K as This concludes the proof of Eq.
for all x ∈ supp (Ψ) by Lemma 2.2 and u n x 2 1 Λ − 1 K . The other statements in Eqs. (2.4) and (2.5) can be verified similarly, using the decomposition The subsequent Theorem 2.5 is a direct consequence of the results in [11] and [25,9], where multiple Lieb-Yamazaki bounds as well as a suitable Gross transformation are used in order to verify that the energy cost of introducing an ultraviolet cut-off Λ = α 4 5 (1+σ) with σ > 0 is only of order o α→∞ α −2 . Combined with an application of the IMS localization formula, as was also done in [20], one obtains the following.
Theorem 2.5. Given a constant 0 < σ ≤ 1 4 , let us introduce the momentum cut-off Λ := α 4 5 (1+σ) as well as the space cut-off L := α 1+σ . Then there exists a sequence of , where E α is the ground state energy of H.

Construction of a Condensate
The purpose of this section is to construct a sequence of approximate ground states Ψ α , i.e. states with Ψ α |H Λ |Ψ α = E α + o α→∞ α −2 and Λ as in Theorem 2.5, that additionally satisfy complete condensation with respect to a minimizer ϕ Pek of the Pekar functional F Pek , i.e. the phonon part of Ψ α is in a suitable sense close to a coherent state Ω ϕ Pek with Ω ϕ Pek := e α 2 a † (ϕ Pek )−α 2 a(ϕ Pek ) Ω, where Ω is the vacuum in F L 2 R 3 , see Lemma 3.12 and Theorem 3.13. The construction will be based on various localization procedures of the phonon field with respect to operators of the form F defined in the subsequent Definition 3.1. Before we start with the localization procedures, we will discuss an asymptotic formula for the expectation value Ψ α | F |Ψ α in Theorem 3.2 as well as the energy cost of localizing with respect to such an operator F in Lemma 3.3.
L 2 sym (R 3×n ) by multiplication with the real valued function (x 1 , . . . , x n ) → F α −2 n k=1 δ x k . In order to keep the notation simple, we will allow F : M R 3 −→ R to act on non-negative L 1 R 3 functions q : R 3 −→ [0, ∞) as well by identifying them with the corresponding measure λ ∈ M R 3 defined as dλ dx = q(x). Before we discuss the asymptotic formula for the expectation value Ψ α | F |Ψ α , let us introduce a family of cut-off functions χ ǫ (a ≤ f (x) ≤ b) where ǫ ≥ 0 determines the sharpness of the cut-off. In the following let α, β : R −→ [0, 1] be C ∞ functions such that α 2 + β 2 = 1, supp (α) ⊂ (−∞, 1) and supp (β) ⊂ (−1, ∞). For a given function f : X −→ R and constants −∞ ≤ a < b ≤ ∞, let us define the function Similarly, we define the operator where T is a self-adjoint operator and E is the spectral measure with respect to T . Furthermore we will write χ (a ≤ f ≤ b), respectively χ (a ≤ T ≤ b), in case ǫ = 0 as well as χ ǫ (a ≤ ·) and χ ǫ (· ≤ b) in case b = ∞ or a = −∞, respectively.
The proof of the following Theorem 3.2 will be carried out in Section 7. It is reminiscent of the quantum de-Finetti Theorem, and establishes in addition that for low energy states phonon field configurations are necessarily close to the set of Pekar minimizers given by {ϕ Pek x } x∈R 3 . Theorem 3.2. Given m ∈ N, C > 0 and g ∈ L 2 R 3 , we can find a constant T > 0 such that for all α ≥ 1 and states Ψ satisfying χ (N ≤ C) Ψ = Ψ and Ψ|H K |Ψ ≤ e Pek + δe with δe ≥ 0 and K ≥ α 8 29 , there exists a probability measure µ on R 3 , with the property where W g is the Weyl operator characterized by W −1 g a(h)W g = a(h) − h|g .
In the subsequent Lemma 3.3 we introduce a generalized IMS-type estimate quantifying the energy cost of localizing with respect to an F -operator, similar to the generalized IMS results in [19,Theorem A.1] and [16,Proposition 6.1]. In order to formulate the result, let us define for a given subset Ω ⊂ M R 3 and a (quadratic) partition of unity P = {F j : M R 3 −→ R : j ∈ J}, i.e. 0 ≤ F j ≤ 1 and j∈J F 2 j = 1, the variation of this partition on Ω as Lemma 3.3. There exists a constant c > 0, such that for any partition of unity P = Furthermore given M > 0, there exists a constant c ′ > 0 such that we have for any where we define Ψ j := f j W −1 ϕ N W ϕ Ψ with W ϕ being the corresponding Weyl operator and P ′ : where we have used the fact that F j commutes with −∆ x and N in the last identity. Since a state Ψ is a function with values in F L 2 R 3 = ∞ n=0 L 2 sym (R 3×n ), we can represent it as Ψ = ∞ n=0 Ψ n where Ψ n (y, x 1 , . . . , x n ) is a function of the electron variable y and the n phonon coordinates x k ∈ R 3 . In order to simplify the notation, we will suppress the dependence on the electron variable y in our notation. By an explicit computation, for v ∈ L 2 R 3 and F : M R 3 −→ R. By the definition of V Ω (P) we obtain that for all x n+1 ∈ R 3 and every (x 1 , . . . , x n ) ∈ R 3n with α −2 n k=1 δ x k ∈ Ω. Hence we can estimate Ψ j∈J Re F j , a (v) , F j Ψ , using the notation X = (x 1 , . . . , x n ), by This concludes the proof of Eq. (3.4), using the concrete choice v : In order to verify the second statement we apply the unitary transformation W ϕ to the operator X : where we defined v := ϕ − Π K w x and applied the definition F ′ j (ρ) = f j dρ . We know from the previous estimates that Clearly v ≤ ϕ + Π K w x √ K for K ≥ 1, and consequently where we have used that W −1 ϕ N W ϕ ≤ 2 N + ϕ 2 and the operator-monotonicity of the square root.
In the following let L := α 1+σ and Λ := α 4 5 (1+σ) with 0 < σ ≤ 1 4 , and let Ψ • α be a sequence of states satisfying supp ( The exponent 4 29 is chosen for convenience, as it allows to simplify the right hand side of Eq. . For the proof of Theorem 1.1 we shall use the specific choice Ψ ⋄ α from Theorem 2.5 for Ψ • α , but it will be useful in the second part to have the first two localization procedures in Lemma 3.4 and 3.5 formulated for a more general sequence Ψ • α . In the following Eq. (3.6) and Eq. (3.10), we will apply localizations procedures to the given sequence Ψ • α in order to construct states having additional useful properties, which we will use in Lemma 3.12 in order to construct a sequence of approximate ground states satisfying complete condensation. Furthermore we will quantify the energy cost of these localizations by Ψ α |H Λ |Ψ α − E α α −3 in the Lemmata 3.4 and 3.5. In Theorem 3.13 we will then apply a final localization procedure, in order to lift the (weak) condensation from Lemma 3.12 to a strong one, following the argument in [16].
Having Lemma 3.3 at hand, we can verify our first localization result in Lemma 3.4, which allows us to restrict our attention to states Ψ ′ α having a (rescaled) particle number N between some fixed constants c − and c + . To be precise, for given c − , c + and ǫ ′ we use the function with the corresponding normalization constants . In the following Lemma 3.4 we derive an upper bound on the energy of Ψ ′ α , and in addition we will investigate the large α behavior of Z α , which will be useful in the second part.
Proof. In the following let F * be the function defined above Eq. (3.6) and let us complete it to a quadratic partition of unity P := {F − , F * , F + } with the aid of the functions Making use of Lemma 3.3 and Λ = α 4 5 (1+σ) ≤ α, we then obtain is uniformly bounded by some ǫ ′ -dependent constant D, and consequently we have for all finite measures ρ and ρ ′ := ρ + α −2 δ y with y ∈ R 3 , and ⋄ ∈ {−, * , +}, This implies that V M(R 3 ) (P) 1, and therefore the right hand side of Eq.
We can however rule out that Ψ α,− |H Λ |Ψ α,− , respectively Ψ α,+ |H Λ |Ψ α,+ , satisfy this upper bound for all small c − , ǫ ′ and large α, Pek 2 < 0 for α large enough and a suitable C ′ , and since we have by Eqs. (2.4) and (2.5) for all t > 0 where the last inequality in Eq. (3.8), respectively Eq. (3.9), holds for small c − , ǫ ′ and 3 . Using again that the right hand .8) and (3.9), and the fact that H Λ ≥ E α and E α ≤ e Pek , yields furthermore Regarding the next localization step in Lemma 3.5, let us introduce for given R and ǫ > 0 satisfying R > 2ǫ the function K R (ρ) := χ ǫ (R − ǫ ≤ |x − y|) dρ(x)dρ(y), which measures how sharply the mass of the measure ρ is concentrated. It will be convenient in the second part to have K R defined for arbitrary ǫ ≥ 0 even though we only need it for ǫ = 0 in the following. We also define the function F R (ρ) := χ δ 3 K R (ρ) ≤ 2δ 3 for R, δ > 0, as well as the states where Ψ ′ α is as in Lemma 3.4 and Z R,α : Heuristically this means that we can restrict our attention to phonon configurations that concentrate in a ball of fixed radius R.
α be the sequence from Lemma 3.4, and let ǫ ≥ 0 and δ > 0 be given constants. Then there exists a α independent R > 0, such that the states Ψ ′′ is a partition of unity, we obtain by Lemma 3.3 with Ω : are bounded by some δ-dependent constant D, we have for all ρ ∈ Ω and ρ ′ := ρ + α −2 δ z with z ∈ R 3 , and R > 2ǫ, the estimate and the same result holds for G R . Therefore we have by Eq. (3.11) and Lemma 3.4 In the following we are going to rule out the second case for R and α large enough, to be precise we are going to verify for any d > 0 and large enough R and α by contradiction. In order to do this, let by assumption for a suitable constant C, Ψ α satisfies the assumptions of Theorem 3.2 with δe := (d + C)α − 4 29 . Hence there exists a measure µ such that Eq. (3.2) holds. By the support properties of G R we obtain . (3.14) Since lim R→∞ K R ϕ Pek 2 = 0, Eq. (3.13) is a contradiction for large R and α, and for such R and α. In combination with Eq. (3.12) this furthermore yields 2 . Since this holds for any d > 0 and α large enough, we conclude that Z R,α −→ α→∞ 1.
The previous localizations in the Lemmas 3.4 and 3.5 will allow us to control the energy cost of the main localization in the proof of Lemma 3.12. Before we come to Lemma 3.12 we need to define the regularized median m q in Definition 3.8 and verify Lemma 3.10, which provides an upper bound on the variation V Ω (P) for partitions P = {F j : j ∈ J} of the form F j (ρ) = f j (m q (ρ)). The following auxiliary Lemmas 3.6 , 3.7 and 3.9 will be useful in proving Lemma 3.10.
Lemma 3.6. Let us define the set Ω reg as the set of all ρ ∈ M R 3 satisfying Clearly the set of all such x ∈ R 3×n has Lebesgue measure zero. Hence the multiplication operator by the function (x 1 , . . . , x n ) → 1 Ωreg (ρ x ) is equal to the identity on L 2 sym R 3×n , which concludes the proof according to Definition 3.1.
where we have used The bound from below can be obtained by interchanging the role of ν and ν ′ .
for ν = 0 and m q (0) := 0. Furthermore we define for a measure ρ on R 3 the regularized Note that x κ (ν) is the largest value, such that both As an immediate consequence, we obtain that the expression in Eq. (3.15) is well-defined for ν = 0 and 0 < q < 1 2 , since Lemma 3.9. Given constants R, c > 0 and 0 < δ < c 2 2 , let ρ satisfy c ≤ dρ and where we have used that x 1 2 (ρ i ) = 0 and dρ ≥ c in the last two inequalities. Hence Lemma 3.10. Given constants R, c > 0 and 0 < δ < c 2 2 , let Ω be the set of ρ ∈ Ω reg satisfying c ≤ dρ and where m q is defined in Definition 3.8. Proof. Since m q acts translation covariant on any ρ = 0, i.e. m q (ρ(· − y)) = m q (ρ) + y, we can assume w.l.o.g. that x 1 2 (ρ i ) = 0 for i ∈ {1, 2, 3}. By Lemma 3.9 we therefore obtain |x In particular, this implies |x 1 2 ±q (ρ ′ i )| ≤ R for 0 < q < 1/2 − δ/c 2 and α large enough. In the following it will be convenient to write (3.17) In order to verify the statement of the Lemma, it is therefore sufficient to prove that Kq Since the distributions ρ i and ρ ′ i satisfy the assumptions of Lemma 3.7 with ǫ := 2α −2 , we conclude that every term in the sum above is bounded by 2 ρ ′ −ρ TV +ǫ = 4α −2 .
Before we state the central Lemma 3.12, let us verify in the subsequent Lemma 3.11 that low energy states with a localized median necessarily satisfy (complete) condensation with respect to a minimizer of the Pekar functional.
Proof. Let us begin by defining the functions q for all ρ ∈ Ω * , and consequently the measure µ from Theorem 3.2 corresponding to the state Ψ satisfies P ǫ i ϕ Pek Lemma 3.12. Given 0 < σ ≤ 1 4 , let Λ and L be as in Theorem 2.5. Then there exist where W ϕ Pek is the Weyl operator corresponding to the Pekar minimizer ϕ Pek .
Proof. It is clearly sufficient to consider only the case α ≥ α 0 for a suitable (large) α 0 , since we can always re-define Ψ ′′′ α := Ψ for α < α 0 where Ψ is an arbitrary state satisfying supp (Ψ) ⊂ B 4L (0). In the following let us use the concrete choice Ψ • α := Ψ ⋄ α for the sequence in Eq. (3.5), where Ψ ⋄ α is defined in in Theorem 2.5, which is a valid choice since it satisfies the assumptions supp ( with Z α,z := F z Ψ ′′ α and Ψ ′′ α as in Lemma 3.5 for ǫ = 0 and 0 < δ < c 2 2 , where c := c − is as in Lemma 3.4. Applying Lemma 3.3 with respect to P := {F z : z ∈ Z 3 }, where the functions F z are defined above Eq. (3.20) and Ω is defined as the set of all ρ ∈ Ω reg satisfying c − ≤ dρ ≤ c + and where we used Lemma 3.6, Λ ≤ α and 1 Ω Ψ ′′ α = Ψ ′′ α by the definition of Ψ ′′ α in Eq. (3.10). Since the support of χ z only overlaps with the support of finitely many other χ z ′ , we obtain for v > 0 and α large enough where we have used sup z∈Z 3 |χ z (y) − χ z (x)| ≤ ∇χ 0 ∞ |y − x| in the first inequality and Lemma 3.10 in the second one. Combining this with Eq. (3.21) and the fact that Since z∈Z 3 Z 2 α,z = 1, this in particular means that there exists a z α ∈ Z 3 such that Ψ α,zα |H Λ |Ψ α,zα − E α α −2(1+σ) , and by the translation invariance of H Λ we obtain where Ω * is the set of all ρ satisfying dρ ≤ c + and |m α −v (ρ)| ≤ α −u , together with Lemma 3.11, immediately concludes the proof of Eq. (3.19).
Proof. Using the states Ψ ′′′ α from Lemma 3.12, we define for 0 < ǫ < 1 where Z α is a normalizing constant. Clearly the states Ψ α satisfy the strong condensation In order to control the energy cost of the localization with respect to the operator W −1 ϕ Pek N W ϕ Pek , note that the partition ∞ |y − x| and the corresponding estimate for χ ǫ 1 2 ≤ · . Therefore we obtain by Lemma 3.3, using Λ ≤ α,

Large Deviation Estimates for Strong Condensates
In this Section we will derive a large deviation principle for states with suitably small particle number (compared to α 2 ), which can be interpreted as complete condensation with respect to the vacuum. We will show that such states are, up to an error which is exponentially small in α 2 , contained in the spectral subspace a(f ) + a † (f ) ≤ ǫ, see Eq. (4.6). Note that taking the point of condensation to be the vacuum is not a real restriction, since this is the case after applying a suitable Weyl transformation. Before we can formulate the main result of this section in Proposition 4.2, we need to introduce some notation.
For 0 < σ < 1 4 let us define Λ := α see Definition 2.1, and let us identify F ΠL 2 R 3 with L 2 R N using the representation of real functions ϕ = N n=1 λ n ϕ n ∈ ΠL 2 R 3 by points λ = (λ 1 , . . . , λ N ) ∈ R N , where N := dimΠL 2 R 3 and {ϕ 1 , . . . , ϕ N } is a real orthonormal basis of ΠL 2 R 3 . We choose this identification such that the annihilation operators a n := a (ϕ n ) read a n = λ n + 1 where λ n is the multiplication operator by the function λ → λ n on L 2 R N . From the construction one readily checks that N (Λ/ℓ) 3 ≤ α p for suitable p > 0. In the following we will verify a large deviation principle for the density function ρ(λ) := γ(λ, λ) corresponding to a density matrix γ on F ΠL 2 R 3 that satisfies the strong condensation condition for some h > 0. This result is comparable to [5,Lemma C.2]. For this purpose, we define a convenient norm | · | ⋄ on R N in the subsequent Definition.
The restriction to the finite dimensional space ΠL 2 R 3 will be essential in the proof of Proposition 4.2, to be precise we will make use of the fact that N α p for a suitable p > 0, which in particular implies that N e α t , uniformly in α, for any t > 0. Before we prove Proposition 4.2, we first need auxiliary results concerning the | · | ⋄ norm.
Using the operators A x we can write |λ| ⋄ = sup x∈R 3 |A x λ|, which is bounded by |A z λ|, |A ≥r λ| (4.7) for any r > 0. In order to see this, note that for any y ∈ R 3 there exists a z ∈ Z 3 with |y − z| < 1. In case y ∈ B r (0) ∩ B 1 (x), where x ∈ R 3 , we see that z satisfies |z| ≤ r + 1 and |x − z| < 2. Denoting the set of such z by M (x, r) ⊂ Z 3 , we obtain This concludes the proof of Eq. (4.7), since there are at most 64 elements z ∈ Z 3 satisfying |x − z| < 2. Proof. Note that the space ΠL 2 R 3 is contained in the spectral subspace −∆ ≤ Λ 2 , hence Π ≤ 1 + Λ 2 (1 − ∆) −1 , and therefore In order to prove the uniform lower bound β 0 > 0, it is enough to verify the boundedness of t . An explicit computation in Fourier space yields for ϕ ∈ L 2 R 3 Finally we are going to verify A ≥r HS α v √ r , using that This concludes the proof, since N α p for some p > 0. where the sum runs over z ∈ Z 3 with |z| ≤ r + 1. In the following we are going to verify that every contribution of the form |Axλ|≥ǫ * 1+|λ| 2 ρ(λ)dλ is exponentially small uniformly in x ∈ R 3 . As a consequence of Eq. (4.3), we have for t ≥ 0 the estimate By our assumption on s, there exists a h ′ such that 2s < h ′ < h. Consequently we obtain for t := α 2+(h−h ′ ) , using Mehler's kernel, is bounded uniformly in α. Since w α ≥ 0 is strictly increasing in α, we can choose 0 < β ′ < β 0 inf α≥1 w α , where β 0 is the constant from Definition 4.3. Consequently β ′ wα |A x | 2 < 1 uniformly in x ∈ R 3 and α ≥ 1, and in particular 1 − β ′ wα |A x | 2 −1 is a bounded operator. Hence we obtain for x ∈ R 3 Furthermore, for a suitable, x-independent, constant µ where we have used the rough estimate w α + α −2 Tr 1 − β ′ wα |A x | 2 −1 1 + α −2 N α p for a suitable exponent p > 0 in the first inequality and Lemma 4.4 in the last inequality.

Properties of the Pekar Functional
In this section we are going to discuss essential properties of the Pekar functional F Pek , and we are going to verify an asymptomatically sharp quadratic approximation for F Pek (ϕ), which is valid for all field configurations ϕ close to a minimizer ϕ Pek . It has been proven in [11] that a suitable quadratic approximation of F Pek holds for all configurations ϕ satisfying V ϕ−ϕ Pek ≪ 1, where In the following we are showing that this result is still valid, in case we substitute the L 2 -norm with the weaker · ⋄ norm, which is a hybrid between the L 2 and the L ∞ norm defined as where B 1 (x) is the unit ball centered at x ∈ R 3 . This will be the content of Lemma 5.2 and Theorem 5.4, respectively. We have V ϕ ⋄ = |λ| ⋄ for ϕ = N n=1 λ n ϕ n , where | · | ⋄ is the norm defined in Eq. (4.4). Before we come to the proof of Lemma 5.2, we first need the subsequent auxiliary Lemma 5.1.
Lemma 5.1. There exists a constant C > 0 such that the operator inequality holds for all (measurable) V : R 3 −→ R, where V 2 is interpreted as a multiplication operator.
Proof. As a first step, we are going to verify that Eq. (5.3) holds in case we use the L 2 norm V instead of V ⋄ . This follows from where · HS is the Hilbert-Schmidt norm, and with K(y − x) being the kernel of the operator (1 − ∆) −1 . Note that C ′ := K(y) 2 dy is finite, which concludes the first step. In order to obtain the analogue statement for V ⋄ , let χ be a smooth, non-negative, function with supp (χ) ⊂ B 1 (0) and R 3 χ(y) 2 dy = 1.
In the following we are going to use that we can write the Pekar energy as where V ϕ is defined in Eq. (5.1). As an immediate consequence of Eq. (5.3) we have ±V ≤ √ C V ⋄ (1 − ∆) and consequently there exists a δ 0 > 0 and a contour C ⊂ C, such that C separates the ground state energy inf σ (−∆ + V ) from the excitation spectrum of H V := −∆ + V for all V with V − V ϕ Pek ⋄ < δ 0 , see also [11]. This allows us to further identify F Pek (ϕ) as for all ϕ satisfying V ϕ−ϕ Pek ⋄ < δ 0 . Following the strategy in [11], we will use Eq. (5.5) to compare F Pek (ϕ) with e Pek = F Pek ϕ Pek . Before we do this let us introduce the operators Proof. By taking δ 0 small enough, we can assume for all V with V ϕ−ϕ Pek ⋄ < δ 0 that where · op denotes the operator norm. This immediately follows from where we used Eq. (5.3) and the fact that the spectrum of H V Pek has a positive distance to the contour C, allowing us to bound the operator norm of (1 − ∆) 1 H V Pek −z uniformly in z ∈ C. Given Eq. (5.9), it has been verified in the proof of [11, Proposition 3.3] that

3). Similarly
Lemma 5.2 gives a lower bound on F Pek ϕ Pek + ξ − e Pek in terms of a quadratic function ξ → ξ|1 − K Pek + ǫL Pek |ξ for ξ satisfying V ξ ⋄ < min{ ǫ c , δ 0 }. Due to the translation invariance of F Pek , this lower bound is however insufficient, since we have for all ξ ∈ span{∂ y 1 ϕ Pek , ∂ y 2 ϕ Pek , ∂ y 3 ϕ Pek } \ {0} (5.10) i.e. the quadratic lower bound is not even non-negative. In order to improve this lower bound, we will introduce a suitable coordinate transformation τ in Definition 5.3. Before we can formulate Definition 5.3 we need some auxiliary preparations.
In the following let Π be the projection defined in Eq. (4.1) and let us define the real orthonormal system for n ∈ {1, 2, 3}, which we complete to a real orthonormal basis {ϕ 1 , .., ϕ N } of ΠL 2 R 3 . Furthermore let us write ϕ Pek x (y) := ϕ Pek (y − x) for the translations of ϕ Pek and let us define the map ω : and since the operator Π commutes with the reflections y i → −y i and permutations y i ↔ y j , it is clear that ω(0) = 0. By the same argument we see that D| 0 ω has full rank and therefore there exists a local inverse t → x t for |t| < δ * and a suitable constant δ * > 0.
Definition 5.3. We define the map τ : where t ϕ := ϕ 1 |ϕ , ϕ 2 |ϕ , ϕ 3 |ϕ ∈ R 3 and f (t) is defined as The map τ is constructed in a way such that it "flattens" the manifold of Pekar minimizers {ϕ Pek x : x ∈ R 3 }. More precisely, we have that τ Πϕ Pek x is for all small enough x ∈ R 3 an element of the linear space spanned by {ϕ 1 , ϕ 2 , ϕ 3 }. A similar construction appears in [5] and, in a somewhat different way, in [9].
Recall the operators K Pek and L Pek from Eqs. (5.6) and (5.7), and let T x be the translation operator defined by (T x ϕ)(y) := ϕ(y − x). Then we define the operators K Pek x := T x K Pek T −x and L Pek x := T x L Pek T −x , as well as for |t| < ǫ with ǫ < δ * J t,ǫ := π 1 − (1 + ǫ) K Pek xt + ǫL Pek xt π, (5.13) where π : L 2 R 3 −→ L 2 R 3 is the orthogonal projection onto the subspace spanned by {ϕ 4 , . . . , ϕ N }. Furthermore we define J t,ǫ := π for |t| ≥ ǫ. In contrast to the operator 1 − K Pek + ǫL Pek from Eq. (5.10), the operator J t,ǫ is non-negative for ǫ small enough, as will be shown in Lemma B.5. With the operator J t,ǫ and the transformation τ at hand we can formulate a strong lower bound for F Pek (ϕ) − e Pek in the subsequent Theorem 5.4, where we use the shorthand notation J t,ǫ ϕ := ϕ|J t,ǫ |ϕ .
Proof. In the following we use the abbreviation t := t ϕ . Since V ϕ Pek −ϕ Pek x ⋄ |x| and |x t | |t| for |t| ≤ δ * 2 , we have for all ϕ satisfying V ϕ−ϕ Pek ⋄ < Dǫ and |t| < min{Dǫ, δ * 2 } By taking D small enough we obtain V T −x t ϕ−ϕ Pek ⋄ ≤ ǫ c where c is the constant from Lemma 5.2. Let us define ǫ 0 := min cδ 0 , δ * 2D , δ * . Using the translation-invariance of F Pek and applying Lemma 5.2 yields where we have used the positivity of K Pek x and L Pek x , and the Cauchy-Schwarz inequality in the last estimate. Note that by construction of x t as the local inverse of the function ω from Eq. (5.12), we have ϕ n |ϕ − Πϕ Pek xt = 0 for n ∈ {1, 2, 3} and therefore with π being defined below Eq. (5.13), where we used |t| < δ * . This concludes the proof with C := (1 + ǫ 0 ) ( K op + ǫ 0 L op ).

Proof of Theorem 1.1
In the following we will combine the results of the previous sections in order to prove the lower bound on the ground state energy E α in Theorem 1.1. We start by verifying the subsequent Lemma 6.1, which provides a lower bound on E α in terms of an operator that is, up to a coordinate transformation τ and a non-negative term, a harmonic oscillator. Let us again use the identification F ΠL 2 R 3 ∼ = L 2 R N utilizing the representation of real functions ϕ = N n=1 λ n ϕ n ∈ ΠL 2 R 3 by points λ = (λ 1 , . . . , λ N ) ∈ R N , such that the annihilation operators a n := a (ϕ n ) are given by a n = λ n + 1 2α 2 ∂ λn , where λ n is the multiplication operator by the function λ → λ n on L 2 R N , see also Eq. (4.2), where Π is the projection from Eq. (4.1) and {ϕ 1 , . . . , ϕ N } is the orthonormal basis of ΠL 2 R 3 constructed around Eq. (5.11). Let us also use for functions ϕ → g(ϕ) depending on elements ϕ ∈ ΠL 2 R 3 the convenient notation g(λ) := g N n=1 λ n ϕ n , where λ ∈ R N . Lemma 6.1. Let C > 0 and 0 < σ ≤ 1 4 , and assume s, h and σ satisfy 2s < h and σ < 1−5s 4 . Furthermore let us define Λ := α 4 5 (1+σ) and L := α 1+σ . Then we obtain for any state Ψ satisfying Ψ|H Λ |Ψ ≤ C, supp (Ψ) ⊂ B 4L (0) and where t ϕ and τ (ϕ) are defined in Lemma 5.3 and J t,ǫ is defined in Eq. (5.13). Furthermore, there exists a β > 0, such that Ψ|1 − B|Ψ ≤ e −βα 2(1−s) , where B is the multiplication operator by the function λ → χ |t λ | < α −s .
Proof. Applying Eq. (2.3) with Λ and ℓ as in the definition of Π, see Eq. (4.1), and K := Λ, and utilizing Eq. (2.5), we obtain for a suitable C ′ Making use of N n=1 a † n a n = N n=1 − 1 4α 4 ∂ 2 λn + λ 2 n − N 2α 2 and a n + a † n = 2λ n , we further have the identity a † n a n , n , which yields the inequality H 0 Λ,ℓ ≥ K + N − N n=1 a † n a n with Combining Eqs. (6.3) and (6.4), we obtain where γ is the reduced density matrix on the Hilbert space F ΠL 2 R 3 ∼ = L 2 R N corresponding to the state Ψ, i.e. we trace out the electron component as well as all the modes in the orthogonal complement of ΠL 2 R 3 , Note that we have the inequality W −1 The operators on the left and right hand side commute, and consequently (6.1) implies that χ W −1 Πϕ Pek N n=1 a † n a n W Πϕ Pek ≤ α −h Ψ = Ψ. This in particular means that the transformed reduced density matrix γ : Using the identification ϕ = N n=1 λ n ϕ n as before, the Weyl operator W Πϕ Pek acts as W Πϕ Pek Ψ (λ) = Ψ λ + λ Pek with λ Pek := ϕ 1 |ϕ Pek , . . . , ϕ N |ϕ Pek . Due to Eq. (6.6), and the fact that 2s < h and σ < 1−5s 4 , the assumptions of Proposition 4.2 are satisfied, and therefore we obtain for any D > 0 the existence of a constant β > 0 such that for α large enough where ρ and ρ are the density functions corresponding to γ and γ, respectively, and where we have used t λ = (λ 1 , λ 2 , λ 3 ) ∈ R 3 . For the concrete choice D := 1, Eq. (6.8) immediately yields the claim Ψ|1 − B|Ψ = |t λ |≥α −s ρ(λ)dλ ≤ e −βα 2(1−s) .

Approximation by Coherent States
This section is devoted to the proof of Theorem 3.2, which states that asymptotically the phonon part of any low energy state is a convex combination of the coherent states Ω ϕ Pek x with x ∈ R 3 , where the convex combination is taken on the level of density matrices. As a central tool we will verify in Lemma 7.2 an asymptotic formula for the expectation value Ψ F Ψ in terms of the lower symbol P y corresponding to the state Ψ, see Eq. (7.6). Furthermore we will make use of the inequality derived in [10, Lemma 7], which implies that the only coherent states Ω ϕ with a low energy have their point of condensation ϕ close to the manifold of Pekar minimizers {ϕ Pek x : x ∈ R 3 }. We start with the subsequent Lemma 7.1, which provides an asymptotic formula for F operators in terms of creation and annihilation operators.
Lemma 7.1. Let m ∈ N and C > 0 be given constants, {g n : n ∈ N} an orthonormal basis of L 2 R 3 and let us denote a n := a(g n ). Then there exists a constant T > 0 such that for all functions F of the form F (ρ) = · · · f (x 1 , . . . , x m ) dρ(x 1 ) . . . dρ(x m ), (7.2) with f : R 3×m −→ R bounded, and states Ψ satisfying χ (N ≤ C) Ψ = Ψ, we can approximate the operator F from Definition 3.1 by where we interpret f as a multiplication operator on L 2 R 3 ⊗ m ∼ = L 2 R 3×m and denote the matrix elements f I,J := g I 1 ⊗ · · · ⊗ g Im |f |g J 1 ⊗ · · · ⊗ g Jm .
Proof. By the assumption χ (N ≤ C) Ψ = Ψ, we can represent the state Ψ as Ψ = n≤Cα 2 Ψ n where Ψ n (y, x 1 , . . . , x n ) is a function of the electron variable y and the n phonon coordinates x k ∈ R 3 . As in the proof of Lemma 3.3, we will suppress the dependence on the electron variable y in our notation. Using the definition of F in Definition 3.1, as well as the notation X = (x 1 , . . . , x n ), we can write Defining K as the set of all k ∈ {1, . . . , n} m satisfying k i = k j for all i = j, we can further express the operator I,J∈N m f I,J a † I 1 . . . a † Im a J 1 . . . a Jm as Consequently we can identify the left hand side of Eq. (7.3) as Cα 2 and since n≤Cα 2 R 3n |Ψ n (X)| 2 dX = Ψ 2 = 1, this concludes the proof.
In the following we are going to define the lower symbol P y corresponding to a state Ψ ∈ L 2 R 3 , F L 2 R 3 . Since we consider the Fock space over the infinite dimensional Hilbert space L 2 R 3 , we need to modify the usual definition of the lower symbol by introducing suitable localizations. For 0 < s ≤ 4 27 and y ∈ R 3 , let us define ℓ * := α − 5 2 s and Λ * := α 2s , and the projection Π y := Π y Λ * ,ℓ * , (7.4) see Definition 2.1. We have N * := dim Π y L 2 R 3 (Λ * /ℓ * ) 3 ≤ α 2 by our assumption s ≤ 4 27 . Using the notation {e y,1 , . . . , e y,N * } for the orthonormal basis of Π y L 2 R 3 from Definition 2.1, we introduce for ξ ∈ C N * the coherent states Ω y,ξ := e α 2 a † (ϕy,ξ)−α 2 a(ϕ y,ξ) Ω, where Ω is the vacuum in F Π y L 2 R 3 and ϕ y,ξ := N * n=1 ξ n e y,n ∈ Π y L 2 R 3 . Furthermore we define wave-functions Ψ y localized in the electron coordinates x as where y ∈ R 3 and L * := α s 2 , and χ is a smooth non-negative function with supp (χ) ⊂ B 1 (0) and χ(y) 2 dy = 1. For the following construction, note that we can identify Let us now define measures P y on C N * ∼ = R 2N * corresponding to the state Ψ y as dP y dξ := 1 π N * Θ y,ξ Ψ y 2 , (7.6) where Θ y,ξ is the orthogonal projection onto the set spanned by elements of the form Note that the coherent states Ω y,ξ provide a resolution of the identity 1 π N * C N * |Ω y,ξ Ω y,ξ | dξ = 1 F (ΠyL 2 (R 3 )) , see for example [20], and consequently the projections Θ y,ξ satisfy 1 π N * C N * Θ y,ξ dξ = 1. In particular we see that the total mass of the measure P y is dP y = Ψ y 2 and therefore dP y dy = Ψ y 2 dy = Ψ 2 = 1.
In the following Lemma 7.2 and Corollary 7.3 we will provide an asymptotic formula for the expectation value Ψ y F Ψ y , respectively Ψ F Ψ , in terms of the measures P y .
Lemma 7.2. Given m ∈ N, C > 0 and g ∈ L 2 R 3 , there exists a T > 0 such that for all F of the form (7.2), y ∈ R 3 and ǫ > 0, and states Ψ satisfying with N y >N * := N − N * n=1 a † y,n a y,n and a y,n := a (e y,n ), and furthermore

8)
where W g is the corresponding Weyl transformation.
Proof. Let {g n : n ∈ N} be a completion of {e y,1 , . . . , e y,N * } to an orthonormal basis of L 2 R 3 and let us define a n := a (g n ). We further introduce an operator F as In the following we want to verify that both f −1 are, up to a multiplicative constant, bounded by the right hand side of Eq. (7.7). Applying the Cauchy-Schwarz inequality, we obtain for all ǫ > 0 where (1 − Π y ) j means that the operator 1 − Π y acts on the j-th factor in the tensor product. Consequently we have the operator inequality Making use of Eq. (7.3) and the fact that χ (N ≤ C) Ψ y = Ψ y further yields for a suitable constant d > 0. We have thus shown the bound which is of the desired form.
In order to verify that 1 f ∞ Ψ y F Ψ y − F |ϕ y,ξ | 2 dP y (ξ) is of the same order as the right hand side of Eq. (7.7) as well, we will first compute F with reversed operator ordering, i.e. we compute In order to identify the left hand side of Eq. (7.11), we will make use of the resolution of identity 1 π N * C N * Θ y,ξ dξ = 1, where Θ y,ξ is defined below Eq. (7.6), which allows us to rewrite the anti-wick ordered term a J 1 . . . a Jm a † I 1 . . . a † Im as 1 Here we have used that a i Θ y,ξ = ξ i Θ y,ξ for all i ∈ {1, . . . , N * }. By the definition of P y in Eq. (7.6) we can therefore rewrite the expectation value of the first term on the left hand side of Eq. = ϕ ⊗ m y,ξ f ϕ ⊗ m y,ξ dP y (ξ) = F |ϕ y,ξ | 2 dP y (ξ) .
The bound in Eq. (7.19) suggests that ϕ y,ξ is close to ϕ Pek x y,ξ with a high probability, where x y,ξ is the minimizer of x → ϕ y,ξ −ϕ Pek x . Motivated by this observation we expect F |ϕ y,ξ | 2 dP y dy ≈ F ϕ Pek x y,ξ 2 dP y dy for measures P y for low energy states Ψ, and therefore it seems natural to define the measure µ in Theorem 3.2 as f dµ := f x y,ξ dP y dy, allowing us to identify F ϕ Pek This expression is however ill-defined, since the infimum inf x∈R 3 ϕ y,ξ − ϕ Pek x is not necessarily attained and it is not necessarily unique. In order to avoid these difficulties, we will slightly modify the definition of the measure µ in the proof of Lemma 7.5.

A. Properties of the Pekar Minimizer
In the following section we derive certain useful properties concerning the minimizer ϕ Pek of the Pekar functional F Pek in (5.4). We start with Lemma A.1, where we quantify the error of applying the cut-off Π to a minimizer, where Π is the projection defined in Eq. (4.1) for a given parameter 0 < σ < 1 4 . The subsequent Lemmas A.2 and A.3 then concern the concentration of the density ϕ Pek 2 around the origin.
Proof. By the reflection symmetry of the Pekar minimizer, it is enough to prove the statement for i = 1. For this purpose, let us define the function D : R → R as In order to prove the Lemma, we are going to show that D is a bounded function. Since D(t) −→ t→±∞ 0, we have D ∞ ≤ |D ′ (t)|dt and furthermore where we have used that ϕ Pek ∈ H 1 R 3 .

B. Properties of the Projection Π
In the following section we discuss properties of the Projections Π defined in Eq. (4.1) and Π K defined in Definition 2.1. The first two results in Lemma B.1 and Corollary B.2 concern the space confinement of elements in the range of Π, to be precise we show that the associated potentials V ϕ defined in Eq. (5.1) are concentrated in a ball of radius α q for a suitable q > 0. While Lemma B.3 is an auxiliary result, we will show in the subsequent Lemmas B.4 and B.5 that the operator J t,ǫ is an approximation of the Hessian Hess| ϕ Pek F Pek , where J t,ǫ is the operator defined in Eq. (5.14). Finally, we will show in Lemma B.6 that the functions Π K w x , which appear in the definition of H K in Eq. (2.2), are confined in space around the origin. We will then use this result in order to quantify the energy cost of having the electron and the phonon field localized in different regions of space, see Corollary B.7. The proof of the following auxiliary Lemma B.1 is an easy analysis exercise and is left to the reader.
Corollary B.2. There exists a constant v > 0, such that for all r > 0 and ϕ ∈ ΠL 2 R 3 where Π is defined in Eq. (4.1) and V ϕ is defined in Eq. (5.1).
Lemma B.5. Recall the operator H Pek from Eq. (1.4). Then there exists a constant c > 0 such that J t,ǫ ≥ c π for ǫ small enough and α large enough. Furthermore for |t| < ǫ, ǫ small enough and α large enough.