1 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 \(L^2\!\left( \mathbb {R}^3\right) \otimes \mathcal {F}\left( L^2\!\left( \mathbb {R}^3\right) \right) \), reads

$$\begin{aligned} \mathbb {H}:=-\Delta _x-a\left( w_x\right) -a^\dagger \left( w_x\right) +\mathcal {N}, \end{aligned}$$

where the annihilation and creation operators satisfy the rescaled canonical commutation relations \(\left[ a(f),a^\dagger (g)\right] =\alpha ^{-2}\langle f|g\rangle \) for \(f,g\in L^2\!\left( \mathbb {R}^3\right) \) with \(\alpha >0\) being the coupling strength, the interaction is given by \(w_x(x'):=\pi ^{-\frac{3}{2}}|x'-x|^{-2}\) and \(\mathcal {N}\) is the corresponding (rescaled) particle number operator, i.e. \(\mathcal {N}:=\sum _{n=1}^\infty a^\dagger (\varphi _n)a(\varphi _n)\) where \(\{\varphi _n:n\in \mathbb {N}\}\) is an orthonormal basis of \(L^2\!\left( \mathbb {R}^3\right) \). 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 \(a^\dagger \) in Eq. (1.1) with a (classical) phonon field \(\varphi \in L^2\! \left( \mathbb {R}^3\right) \), i.e. replacing a(f) with \(\langle f|\varphi \rangle \) and \(a^\dagger (f)\) with \(\langle \varphi |f\rangle \), we arrive at the Pekar energy

$$\begin{aligned} \mathcal {E}\left( \psi ,\varphi \right)&:=\big \langle \psi \big |-\Delta _x-\langle w_x|\varphi \rangle -\langle \varphi |w_x\rangle +\Vert \varphi \Vert ^2\, \big |\psi \big \rangle \nonumber \\&=\int \left| \nabla \psi (x)\right| ^2\textrm{d}x-\iint w_x(x')\left( \varphi (x')+\overline{\varphi (x')}\right) \left| \psi (x)\right| ^2\textrm{d}x'\textrm{d}x\nonumber \\&\quad +\int |\varphi (x')|^2\textrm{d}x', \end{aligned}$$

where \(\psi \in L^2\! \left( \mathbb {R}^3\right) \) is the wave-function of the electron. We further define the Pekar functional \(\mathcal {F}^\textrm{Pek}\! \left( \varphi \right) :=\inf _{\Vert \psi \Vert =1}\mathcal {E}\left( \psi ,\varphi \right) \) and the minimal Pekar energy \(e^\textrm{Pek}:=\inf _\varphi \mathcal {F}^\textrm{Pek}\! \left( \varphi \right) \). It is known that the ground state energy \(E_\alpha :=\inf \sigma \left( \mathbb {H}\right) \), as a function of the coupling strength \(\alpha \), is asymptotically given by the minimal Pekar energy \(e^\textrm{Pek}\) in the limit \(\alpha \rightarrow \infty \) [1, 7]. More precisely, one has \(e^\textrm{Pek} \ge E_\alpha =e^\textrm{Pek}+O_{\alpha \rightarrow \infty }\left( \alpha ^{-\frac{1}{5}}\right) \), as shown in [20]. In this work we are going to verify the prediction in the physics literature [2, 3, 30] that the sub-leading term in this energy asymptotics is actually of order \(\alpha ^{-2}\) with a rather explicit pre-factor

$$\begin{aligned} E_\alpha =e^\textrm{Pek}-\frac{1}{2\alpha ^2}\textrm{Tr}\left[ 1-\sqrt{H^\textrm{Pek}}\, \right] +o_{\alpha \rightarrow \infty }\left( \alpha ^{-2}\right) , \end{aligned}$$

where \(\varphi ^\textrm{Pek}\) is a minimizer of \(\mathcal {F}^\textrm{Pek}\) and \(H^\textrm{Pek}\) is the Hessian of \(\mathcal {F}^\textrm{Pek}\) at \(\varphi ^\textrm{Pek}\) restricted to real-valued functions \(\varphi \in L^2_\mathbb {R}\! \left( \mathbb {R}^3\right) \), i.e. \(H^\textrm{Pek}\) is an operator on \(L^2\! \left( \mathbb {R}^3\right) \) defined by

$$\begin{aligned} \langle \varphi |H^\textrm{Pek}|\varphi \rangle =\lim _{\epsilon \rightarrow 0}\frac{1}{\epsilon ^2}\left( \mathcal {F}^\textrm{Pek}\! \left( \varphi ^\textrm{Pek}+\epsilon \varphi \right) -e^\textrm{Pek}\right) \end{aligned}$$

for all \(\varphi \in L^2_\mathbb {R}\! \left( \mathbb {R}^3\right) \). The prediction in Eq. (1.3) has been verified previously for polaron models either confined to a bounded region of \(\mathbb {R}^3\) [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 [5, 15, 16] in the mean-field 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 [9, 11]. Our main result is the proof of Eq. (1.3) in the following Theorem 1.1.

Theorem 1.1

Let \(E_\alpha \) be the ground state energy of \(\mathbb {H}\) in (1.1). For any \(s<\frac{1}{29}\)

$$\begin{aligned} E_\alpha = e^\textrm{Pek}-\frac{1}{2\alpha ^2}\textrm{Tr}\left[ 1-\sqrt{H^\textrm{Pek}}\, \right] +O\left( \alpha ^{-(2+s)}\right) \end{aligned}$$

for all \(\alpha \ge \alpha (s)\), where \(\alpha (s)>0\) 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 \(\varphi ^\textrm{Pek}\) of the Pekar functional \(\mathcal {F}^\textrm{Pek}\). We refer to Theorem 3.13 for a precise statement.

In contrast to the lower bound, the proof of the upper bound on \(E_\alpha \) in Eq. (1.3) is essentially the same as for confined polarons [9, 11] 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 \(E_\alpha (P)\) as a function of the (conserved) total momentum P can be bounded from above by

$$\begin{aligned} E_\alpha (P)\le e^\textrm{Pek}-\frac{1}{2\alpha ^2}\textrm{Tr}\left[ 1-\sqrt{H^\textrm{Pek}}\, \right] +\frac{|P|^2}{2\alpha ^4 m}+C_{\epsilon }\alpha ^{-\frac{5}{2}+\epsilon }, \end{aligned}$$

where \(m:=\frac{2}{3}\Vert \nabla \varphi ^\textrm{Pek}\Vert ^2\) and \(\epsilon >0\), with \(C_{\epsilon }\) a suitable constant. Since \(E_\alpha = E_\alpha (0)\) [8, 13, 23], Eq. (1.6) for the specific case \(P=0\) proves (1.5) as an upper bound, hence to establish Theorem 1.1 it suffices to prove (1.5) as a lower bound. Combining (1.6) with Theorem 1.1, one further obtains an upper bound on the increment \(E_\alpha (P)-E_\alpha \), a quantity related to the effective mass of the polaron [4, 14, 18, 29].

The proof of Eq. (1.3) for confined systems in [9, 11] requires an asymptotically correct local quadratic lower bound on the Pekar functional \(\mathcal {F}^\textrm{Pek}\! \left( \varphi \right) \) 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 \(\Psi \), which is essentially supported close to a minimizer of the Pekar functional \(\mathcal {F}^\textrm{Pek}\), 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 \(E_\alpha \), let us first discuss how test functions providing an asymptotically correct upper bound are expected to look like. In the following let \((\psi ^\textrm{Pek},\varphi ^\textrm{Pek})\) denote a minimizer of the Pekar energy \(\mathcal {E}\) defined in Eq. (1.2). It has been established in [17] that all other minimizers are given by translations \(\varphi ^\textrm{Pek}_x(x'):=\varphi ^\textrm{Pek}(x'-x)\) and \(\psi ^\textrm{Pek}_x(x'):=e^{i\theta }\psi ^\textrm{Pek}(x'-x)\) of \(\varphi ^\textrm{Pek}\) and \(e^{i\theta }\psi ^\textrm{Pek}\), where \(\theta \) is an arbitrary phase. W.l.o.g. let us denote in the following by \((\psi ^\textrm{Pek},\varphi ^\textrm{Pek})\) the unique minimizer of \(\mathcal {E}\) such that \(\varphi ^\textrm{Pek}\) is radial and \(\psi ^\textrm{Pek}\) is non-negative. Then all the product states of the form \(\psi ^\textrm{Pek}_x\otimes \Omega _{\varphi ^\textrm{Pek}_x}\) with \(x\in \mathbb {R}^3\), where \(\Omega _{\varphi ^\textrm{Pek}_x}\) is the coherent state corresponding to \(\varphi ^\textrm{Pek}_x\) (defined by \(a(w) \Omega _\varphi = \langle w|\varphi \rangle \Omega _\varphi \) for all \(w\in L^2\! \left( \mathbb {R}^3\right) \)), have the asymptotically correct leading term in the energy \(\langle \psi ^\textrm{Pek}_x\otimes \Omega _{\varphi ^\textrm{Pek}_x}|\, \mathbb {H}\, |\psi ^\textrm{Pek}_x\otimes \Omega _{\varphi ^\textrm{Pek}_x}\rangle =e^\textrm{Pek}\). By taking convex combinations of these states on the level of density matrices, we can construct a large family of low energy states

$$\begin{aligned} \Gamma _\mu :=\int _{\mathbb {R}^3} \vert \psi ^\textrm{Pek}_x\otimes \Omega _{\varphi ^\textrm{Pek}_x}\rangle \langle \psi ^\textrm{Pek}_x\otimes \Omega _{\varphi ^\textrm{Pek}_x}\vert \textrm{d}\mu (x) \end{aligned}$$

for any given probability measure \(\mu \) on \(\mathbb {R}^3\). Clearly, \(\Gamma _\mu \) exhibits the correct leading energy \(\langle \, \mathbb {H}\, \rangle _{\Gamma _\mu }=e^\textrm{Pek}\). Our proof of the lower bound given in Eq. (1.5) relies on the observation that asymptotically as \(\alpha \rightarrow \infty \), any low energy state \(\Psi \) is of the form \(\Gamma _\mu \) with a suitable probability measure \(\mu \) on \(\mathbb {R}^3\). Since we only need this statement for the phonon part of \(\Psi \), we will verify the weaker statement

$$\begin{aligned} \textrm{Tr}_\textrm{electron}\big [\vert \Psi \rangle \langle \Psi \vert \big ]\approx \int _{\mathbb {R}^3} \vert \Omega _{\varphi ^\textrm{Pek}_x}\rangle \langle \Omega _{\varphi ^\textrm{Pek}_x}\vert \textrm{d}\mu (x) \end{aligned}$$

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 \(\Psi \) where the corresponding measure is a delta measure, \(\mu =\delta _{0}\), i.e. the construction of states where the phonon part is essentially given by a single coherent state \(\Omega _{\varphi ^\textrm{Pek}}\). 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 \(\varphi ^\textrm{Pek}\). Heuristically this means that only field configurations \(\varphi \) close to the minimizer \(\varphi ^\textrm{Pek}\) 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 [9, 11], which starts by introducing an ultraviolet regularization in the interaction \(w_x\) with the aid of a momentum cut-off \(\Lambda \), leading to the study of the truncated Hamiltonian \(\mathbb {H}_\Lambda \). Using a lower bound on the excitation energy \(\mathcal {F}^\textrm{Pek}\! \left( \varphi \right) -e^\textrm{Pek}\) that is, up to a canonical transformation, quadratic in the field variables \(\varphi \) and valid for all \(\varphi \) close to the minimizer \(\varphi ^\textrm{Pek}\), 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 \(\varphi ^\textrm{Pek}\). Finally an explicit diagonalization of this quadratic operator yields the desired lower bound in Eq. (1.5).

The canonical transformation on the phase space \(L^2\! \left( \mathbb {R}^3\right) \), respectively the corresponding unitary transformation on the Hilbert space \(\mathcal {F}\left( L^2\! \left( \mathbb {R}^3\right) \right) \), 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 semi-definite quadratic lower bound on \(\mathcal {F}^\textrm{Pek}\! \left( \varphi \right) - e^\textrm{Pek}\). 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 canonical/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 Sect. 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 \(\varphi ^\textrm{Pek}\) of the Pekar functional \(\mathcal {F}^\textrm{Pek}\). Subsequently we will discuss a large deviation estimate for such condensates in Sect. 4, quantifying the heuristic picture that only configurations close to the point of condensation matter. In Sect. 5 we then discuss properties of the Pekar functional \(\mathcal {F}^\textrm{Pek}\). In particular, we will discuss quadratic approximations around the minimizer \(\varphi ^\textrm{Pek}\) as well as lower bounds that are, up to a coordinate transformation, quadratic in \(\varphi \). Together with the error estimates from Sect. 2 and the large deviation estimate from Sect. 4, applied to the approximate ground state constructed in Sect. 3, this will allow us to verify our main Theorem 1.1 in Sect. 6. The subsequent Sect. 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 \(\varphi ^\textrm{Pek}\) and the projections introduced in Sect. 2, respectively.

Outlook In the second part of this study of the asymptotic properties of the Fröhlich polaron [6], the ground state energy \(E_{\alpha }(P)\) of the operator \(\mathbb {H}\) in the presence of a momentum constraint \(\mathbb {P}=P\) will be investigated, where the total momentum operator is defined as \(\mathbb {P}:=\frac{1}{i}\nabla +\alpha ^2 \int _{\mathbb {R}^3}k a_k^\dagger a_k \textrm{d}k\), and the lower bound

$$\begin{aligned} E_\alpha (P)\ge e^\textrm{Pek}-\frac{1}{2\alpha ^2}\textrm{Tr}\left[ 1-\sqrt{H^\textrm{Pek}}\, \right] +\min \left\{ \frac{|P|^2}{2\alpha ^4 m},\alpha ^{-2}\right\} +o\left( \alpha ^{-2}\right) \end{aligned}$$

will be established. Together with the upper bound in Eq. (1.6) derived in [22], one observes that the energy-momentum relation of a polaron agrees asymptotically with the one of a free particle having an effectively increased mass \(\alpha ^4 m\), where \(\alpha ^4 m\) is the celebrated Landau–Pekar formula for the mass of a polaron in the regime of strong couplings.

2 Models with Cut-off

In this section we will estimate the effect of the introduction of an ultraviolet cut-off \(\Lambda >0\), as well as a discretization in momentum space with box length \(\ell >0\), on the ground state energy, following similar ideas as in [9, 11, 20]. 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 Sect. 7, which applies to low energy states with energy \(e^\textrm{Pek} + o_{\alpha \rightarrow \infty }(1)\), and a fine scale precise enough to yield the correct ground state energy up to errors of order \(o_{\alpha \rightarrow \infty }\left( \alpha ^{-2}\right) \), see the proof of Theorem 1.1 in Sect. 6.

Definition 2.1

Given parameters \(0< \ell <\Lambda \), let us define for \(z\in 2\ell \, \mathbb {Z}^3{\setminus } \{0\}\) the cubes \(C_z:=\left[ z_1-\ell ,z_1+\ell \right) \times \left[ z_2-\ell ,z_2+\ell \right) \times \left[ z_3-\ell ,z_3+\ell \right) \), and let \(z^1,..,z^N\) be an enumeration of the set of all \(z=(z_1,z_2,z_3)\in 2\ell \, \mathbb {Z}^3{\setminus } \{0\}\) such that \(C_z\subset B_{\Lambda }(0)\), where \(B_r(0)\) is the (open) ball of radius r around the origin. Then we define the orthonormal system \(e_n\in L^2\!\left( \mathbb {R}^3\right) \) as

$$\begin{aligned} e_n(x):=\frac{1}{\sqrt{(2\pi )^3 \int _{C_{z^n}}\frac{1}{|k|^2}\,\textrm{d}k}}\int _{C_{z^n}}\frac{e^{i\, k\cdot x}}{|k|}\,\textrm{d}k, \end{aligned}$$

as well as the translated system \(e_{y,n}(x):=e_n(x-y)\) and the orthogonal projection \(\Pi ^y_{\Lambda ,\ell }\) onto the space spanned by \(\{e_{y,1},\dots ,e_{y,N}\}\). Furthermore we denote with \(\Pi _\Lambda \) the projection onto the spectral subspace of momenta \(|k|\le \Lambda \).

Lemma 2.2

Let \(w_x(x'):=\pi ^{-\frac{3}{2}}|x'-x|^{-2}\). Then we obtain for \(0<\ell <\Lambda \) and \(x,y\in \mathbb {R}^3\) the following estimate on the \(L^2\) norm

$$\begin{aligned} \left\| \Pi _\Lambda w_x-\Pi ^y_{\Lambda ,\ell } w_x\right\| \lesssim |x-y|\ell \sqrt{\Lambda }+\sqrt{\ell }. \end{aligned}$$


With \(\widehat{\cdot }\) denoting Fourier transformation, we have

$$\begin{aligned} \sqrt{2\pi ^2}\, \widehat{\Pi ^y_{\Lambda ,\ell } w_x}(k)=\sum _{n=1}^N \frac{1}{\int _{C_{z^n}}\frac{1}{|k'|^2}\textrm{d}k'}\int _{C_{z^n}}\frac{e^{ik'\cdot (y-x)}}{|k'|^2}\textrm{d}k'\, \frac{1}{|k|}\mathbbm {1}_{C_{z^n}}(k), \end{aligned}$$

where we have used that \(\widehat{\Pi _\Lambda w_x}(k)=\frac{1}{\sqrt{2\pi ^2}|k|}\mathbbm {1}_{B_\Lambda (0)}(k)\). Defining the function \(\sigma _n(k,x,y):=\frac{1}{\int _{C_{z^n}}\frac{1}{|k'|^2}\textrm{d}k'}\int _{C_{z^n}}\frac{e^{ik'\cdot (y-x)}-e^{ik\cdot (y-x)}}{|k'|^2}\textrm{d}k'\), we further have

$$\begin{aligned} \sqrt{2\pi ^2}\left( \widehat{\Pi ^y_{\Lambda ,\ell } w_x}(k)-\widehat{\Pi _\Lambda w_x}(k)\right) =\sum _{n=1}^N \sigma _n(k,x,y)\frac{1}{|k|}\mathbbm {1}_{C_{z^n}}(k)-\frac{1}{|k|}\mathbbm {1}_{A}(k) \end{aligned}$$

with \(A:=B_\Lambda (0){\setminus } \left( \bigcup _{n=1}^N C_{z^n}\right) \). Making use of the estimate \(|\sigma _n(k,x,y)|^2\le |y-x|^2\max _{k'\in C_{z^n}}|k'-k|^2\le 12|x-y|^2\ell ^2 \) for \(k\in C_{z^n}\), we therefore obtain

$$\begin{aligned} \sum _{n=1}^N \int _{C_{z^n}} |\sigma _n(k,x,y)|^2\frac{1}{|k|^2}\textrm{d}k\le 12|x-y|^2\ell ^2\int _{|k|\le \Lambda }\frac{1}{|k|^2}\textrm{d}k=48\pi |x-y|^2\ell ^2 \Lambda . \end{aligned}$$

Since \(A\subset B_{2\ell }\cup B_\Lambda \setminus B_{\Lambda -4\ell }\) we consequently have \(\int _A \frac{1}{|k|^2}\textrm{d}k\lesssim \ell \). \(\quad \square \)

Definition 2.3

For \(y\in \mathbb {R}^3\), \(0<\ell <\Lambda \), let us define the cut-off Hamiltonians

$$\begin{aligned} \mathbb {H}^y_{\Lambda ,\ell }&:=-\Delta _x-a\left( \Pi ^y_{\Lambda ,\ell } w_x\right) -a^\dagger \left( \Pi ^y_{\Lambda ,\ell } w_x\right) +\mathcal {N}, \end{aligned}$$
$$\begin{aligned} \mathbb {H}_\Lambda&:=-\Delta _x-a\left( \Pi _{\Lambda } w_x\right) -a^\dagger \left( \Pi _{\Lambda } w_x\right) +\mathcal {N}. \end{aligned}$$

These Hamiltonians can be interpreted as the restriction of \(\mathbb {H}\) (in the quadratic form sense) to states where only the phonon modes in \(\Pi ^y_{\Lambda ,\ell }L^2\! \left( \mathbb {R}^3\right) \), respectively \(\Pi _{\Lambda }L^2\! \left( \mathbb {R}^3\right) \), are occupied. In particular, this implies that \(\inf \sigma (\mathbb {H}^y_{\Lambda ,\ell }) \ge E_\alpha \) as well as \(\inf \sigma (\mathbb {H}_{\Lambda }) \ge E_\alpha \). In the following we shall quantify the energy increase due to the introduction of the cut-offs.

Note that the \(\alpha \)-dependence of the Hamiltonians \(\mathbb {H}\), \(\mathbb {H}^y_{\Lambda ,\ell }\) and \(\mathbb {H}_\Lambda \) only enters through the rescaled canonical commutation relations \(\left[ a(f),a^\dagger (g)\right] =\alpha ^{-2}\langle f|g\rangle \) satisfied by the creation and annihilation operators \(a^\dagger \) and a, and we will usually suppress the \(\alpha \) dependency in our notation for the sake of readability. In the rest of this paper, we will always assume that \(\alpha \) is a parameter satisfying \(\alpha \ge 1\) and, in case it is not stated otherwise, estimates hold uniformly in this parameter for \(\alpha \rightarrow \infty \), i.e. we write \(X\lesssim Y\) in case there exist constants \(C,\alpha _0>0\) such that \(X\le C\, Y\) for all \(\alpha \ge \alpha _0\).

The proof of the subsequent Lemma 2.4 closely follows the arguments in [20, 21], where it was shown that \(\mathbb {H}\) 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 \(L^2\!\left( \mathbb {R}^3\right) \otimes \mathcal {F}\left( L^2\!\left( \mathbb {R}^3\right) \right) \cong L^2\!\left( \mathbb {R}^3,\mathcal {F}\left( L^2\!\left( \mathbb {R}^3\right) \right) \right) \), in order to represent elements \(\Psi \in L^2\!\left( \mathbb {R}^3\right) \otimes \mathcal {F}\left( L^2\!\left( \mathbb {R}^3\right) \right) \) as functions \(x\mapsto \Psi (x)\) with values in \(\mathcal {F}\left( L^2\!\left( \mathbb {R}^3\right) \right) \), allowing us to define the support \(\textrm{supp}\left( \Psi \right) \) as the closure of \(\{x\in \mathbb {R}^3: \Psi (x)\ne 0\}\) and to introduce a space cut-off \(L>0\).

Lemma 2.4

We have for all \(0<\ell <\Lambda \le K\) and \(L>0\), and states \(\Psi \) with \(\textrm{supp}\left( \Psi \right) \subset B_L(y)\) the estimate

$$\begin{aligned} \left| \langle \Psi |\mathbb {H}_K-\mathbb {H}^y_{\Lambda ,\ell }|\Psi \rangle \right| \lesssim \! \left( L\ell \sqrt{\Lambda }+\sqrt{\ell }+\sqrt{\frac{1}{\Lambda }-\frac{1}{K}}\right) \! \langle \Psi |-\Delta _x+\mathcal {N}+1|\Psi \rangle . \end{aligned}$$

Furthermore, there exists a constant \(d>0\) such that

$$\begin{aligned} \mathbb {H}_{K}&\ge -\frac{d}{t^2}-t\, \left( \mathcal {N}+\alpha ^{-2}\right) , \end{aligned}$$
$$\begin{aligned} \mathbb {H}_{K}&\ge -d+\frac{1}{2}\left( -\Delta _x+\mathcal {N}\right) \end{aligned}$$

for all \(t>0\), \(K\ge 0\) and \(\alpha \ge 1\).


Let us define the functions \(u^n_x\) by \(\widehat{u^n_x}(k):=\frac{1}{\sqrt{2\pi ^2}}\mathbbm {1}_{B_K(0){\setminus } B_\Lambda (0)}(k)\frac{k_n e^{ik\cdot x}}{|k|^3}\). We have \(a\left( \partial _{x_n}u^n_x\right) - a^\dagger \left( \partial _{x_n}u^n_x\right) = \left[ \partial _{x_n},a\left( u^n_x\right) \! -\! a^\dagger \left( u^n_x\right) \right] \) and

$$\begin{aligned}&\pm i \left[ \partial _{x_n},a\left( u^n_x\right) \! -\! a^\dagger \left( u^n_x\right) \right] \le - 2\epsilon \partial _{x_n}^2\! \! +\! \frac{1}{\epsilon }\left( a(u^n_x)^\dagger a(u^n_x) \!+\! a(u^n_x)a(u^n_x)^\dagger \right) \\&\quad \le - 2 \epsilon \, \partial _{x_n}^2 + \frac{\Vert u^n_x\Vert ^2}{\epsilon }\left( 2\mathcal {N} + \alpha ^{-2}\right) =2\Vert u^n_x\Vert \left( - \partial _{x_n}^2+\mathcal {N}+\frac{1}{2}\alpha ^{-2}\right) , \end{aligned}$$

where we have applied the Cauchy–Schwarz inequality in the first line and used the specific choice \(\epsilon :=\Vert u^n_x\Vert \) in the last identity. Note that the \(L^2\)-norm \(\Vert u^n_x\Vert \) is independent of x, and furthermore we can express \(\pm \left( \mathbb {H}^y_{\Lambda ,\ell } - \mathbb {H}_K\right) \) as

$$\begin{aligned}&\pm a\left( \Pi _\Lambda w_x\!-\!\Pi ^y_{\Lambda ,\ell } w_x\right) \! \pm \! a^\dagger \left( \Pi _\Lambda w_x\!-\!\Pi ^y_{\Lambda ,\ell } w_x\right) \! \pm i\sum _{n=1}^3 \left( a\left( \partial _{x_n}u^n_x\right) \! - a^\dagger \left( \partial _{x_n}u^n_x\right) \right) \! \\&\quad \le 2\left\| \Pi _\Lambda w_x-\Pi ^y_{\Lambda ,\ell } w_x\right\| \left( 1+\mathcal {N}\right) +2\max _{n\in \{1,2,3\}}\Vert u^n_x\Vert \left( -\Delta _x+3\mathcal {N}+\frac{3}{2}\alpha ^{-2}\right) . \end{aligned}$$

This concludes the proof of Eq. (2.3), since we have \(\left\| \Pi _\Lambda w_x-\Pi ^y_{\Lambda ,\ell } w_x\right\| \lesssim L\ell \sqrt{\Lambda }+\sqrt{\ell }\) for all \(x\in \textrm{supp}\left( \Psi \right) \) by Lemma 2.2 and \(\Vert u^n_x\Vert ^2\lesssim \frac{1}{\Lambda }-\frac{1}{K}\). The other statements in Eqs. (2.4) and (2.5) can be verified similarly, using the decomposition \(\Pi _{K} w_x=\Pi _{K'}w_x+\sum _{n=1}^3 \frac{1}{i}\partial _{x_n} g^n_x\) with \(\widehat{g^n_x}(k):=\frac{1}{\sqrt{2\pi ^2}}\mathbbm {1}_{B_K(0){\setminus } B_{K'}(0)}(k)\frac{k_n e^{ik\cdot x}}{|k|^3}\) where \(K'\le K\) is large enough such that \(\left\| g^n_x\right\| <\frac{1}{12}\). \(\quad \square \)

The subsequent Theorem 2.5 is a direct consequence of the results in [11] and [9, 25], 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 \(\Lambda =\alpha ^{\frac{4}{5}(1+\sigma )}\) with \(\sigma >0\) is only of order \(o_{\alpha \rightarrow \infty }\left( \alpha ^{-2}\right) \). Combined with an application of the IMS localization formula, as was also done in [20], one can furthermore introduce a space cut-off at length scale \(L=\alpha ^{1+\sigma }\) with an energy cost of order \(o_{\alpha \rightarrow \infty }\left( \alpha ^{-2}\right) \) as well.

Theorem 2.5

Given a constant \(0<\sigma \le \frac{1}{4}\), let us introduce the momentum cut-off \(\Lambda :=\alpha ^{\frac{4}{5}(1+\sigma )}\) as well as the space cut-off \(L:=\alpha ^{1+\sigma }\). Then there exists a sequence of states \(\Psi ^\diamond _\alpha \) satisfying \(\langle \Psi ^\diamond _\alpha |\mathbb {H}_{\Lambda }|\Psi ^\diamond _\alpha \rangle -E_\alpha \lesssim \alpha ^{-2(1+\sigma )}\) and \(\textrm{supp}\left( \Psi ^\diamond _\alpha \right) \subset B_{L}(0)\), where \(E_\alpha \) is the ground state energy of \(\mathbb {H}\).


We start by arguing that

$$\begin{aligned} \inf \sigma \left( \mathbb {H}_{\Lambda }\right) -E_\alpha \lesssim \Lambda ^{-\frac{5}{2}}+\alpha ^{-1}\Lambda ^{-\frac{3}{2}}+\alpha ^{-2}\Lambda ^{-\frac{1}{2}} \end{aligned}$$

for large \(\alpha \). An analogous bound was shown in [11, Prop. 7.1] in the confined case, where additional powers of \(\ln \Lambda \) appear due to complications coming from the boundary. In the translation-invariant setting on a torus, (2.6) is shown [9, Prop. 4.5], and that proof applies verbatim also in the unconfined case considered here (as has been worked out also in [25]).

By our choice of \(\Lambda =\alpha ^{\frac{4}{5}(1+\sigma )}\), we immediately obtain \(\inf \sigma \left( \mathbb {H}_{\Lambda }\right) -E_\alpha \lesssim \alpha ^{-2(1+\sigma )}\). Hence there exists a state \(\Psi \) satisfying \(\langle \Psi |\mathbb {H}_{\Lambda }|\Psi \rangle -E_\alpha \lesssim \alpha ^{-{2(1+\sigma )}}\). In order to construct a state which is furthermore supported on the ball \(B_{L}(0)\), let \(\chi \) be a non-negative \(H^1\! \left( \mathbb {R}^3\right) \) function with \(\int \chi (y)^2\textrm{d}y=1\) and \(\textrm{supp}\left( \chi \right) \subset B_1(0)\). We define \(\Psi _y(x):=L^{-\frac{3}{2}}\chi \left( L^{-1}(x-y)\right) \Psi (x)\) for \(y\in \mathbb {R}^3\) and compute, using the IMS identity,

$$\begin{aligned} \int \langle \Psi _y|\mathbb {H}_\Lambda |\Psi _y\rangle \, \textrm{d}y&=\langle \Psi |\mathbb {H}_\Lambda |\Psi \rangle +L^{-3}\iint \left| \nabla _x\chi \left( L^{-1}(x-y)\right) \right| ^2 \textrm{d}y\, \Vert \Psi (x)\Vert ^2\textrm{d}x\\&=\langle \Psi |\mathbb {H}_\Lambda |\Psi \rangle +L^{-2}\Vert \nabla \chi \Vert ^2=E_\alpha +O_{\alpha \rightarrow \infty }\left( \alpha ^{-2(1+\sigma )}\right) , \end{aligned}$$

see also [20] where an explicit choice of \(\chi \) is used. Since \(\int \Vert \Psi _y\Vert ^2\textrm{d}y=1\), there clearly exists a \(y\in \mathbb {R}^3\) such that the state \(\Psi ^\diamond _\alpha :=\Vert \Psi _y\Vert ^{-1}\Psi _y\) satisfies \(\langle \Psi ^\diamond _\alpha |\mathbb {H}_\Lambda |\Psi ^\diamond _\alpha \rangle -E_\alpha \lesssim \alpha ^{-2(1+\sigma )}\). By the translation invariance of \(\mathbb {H}_\Lambda \) we can assume that \(y=0\). \(\quad \square \)

3 Construction of a Condensate

The purpose of this section is to construct a sequence of approximate ground states \(\Psi _\alpha \), i.e. states with \(\langle \Psi _\alpha |\mathbb {H}_\Lambda |\Psi _\alpha \rangle =E_\alpha +o_{\alpha \rightarrow \infty }\left( \alpha ^{-2}\right) \) and \(\Lambda \) as in Theorem 2.5, that additionally satisfy complete condensation with respect to a minimizer \(\varphi ^\textrm{Pek}\) of the Pekar functional \(\mathcal {F}^\textrm{Pek}\), i.e. the phonon part of \(\Psi _\alpha \) is in a suitable sense close to a coherent state \(\Omega _{\varphi ^\textrm{Pek}}\) with \(\Omega _{\varphi ^\textrm{Pek}}:=e^{ \alpha ^2 a^\dagger \left( \varphi ^\textrm{Pek}\right) -\alpha ^2 a\left( \varphi ^\textrm{Pek}\right) }\Omega \), where \(\Omega \) is the vacuum in \(\mathcal {F}\left( L^2\!\left( \mathbb {R}^3\right) \right) \), 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 \(\widehat{F}\) defined in the subsequent Definition 3.1.

3.1 Properties of the \(\widehat{F}\) operators

In this subsection we are going to introduce a useful class of operators on \(\mathcal {F}\left( L^2\! \left( \mathbb {R}^3\right) \right) \), which we will refer to as \(\widehat{F}\) operators, and provide an asymptotic formula for their expectation value \(\langle \Psi _\alpha |\widehat{F}|\Psi _\alpha \rangle \) in Theorem 3.2 as well as an estimate on the energy cost of localizing with respect to such an operator in Lemma 3.3.

Definition 3.1

Given a function \(F:\mathcal {M}\left( \mathbb {R}^3\right) \longrightarrow \mathbb {R}\), where \(\mathcal {M}\left( \mathbb {R}^3\right) \) is the set of finite (Borel) measures on \(\mathbb {R}^3\), let us define the operator \(\widehat{F}\) acting on the Fock space \(\mathcal {F}\left( L^2\! \left( \mathbb {R}^3\right) \right) =\bigoplus \limits _{n=0}^\infty L^2_{\textrm{sym}}\! (\mathbb {R}^{3\times n})\) as \(\widehat{F}\bigoplus \limits _{n=0}^\infty \Psi _n:=\bigoplus \limits _{n=0}^\infty F^n\Psi _n\), where

$$\begin{aligned} \left( F^n\Psi _n\right) (x^1,\dots ,x^n):=F\left( \alpha ^{-2}\sum _{k=1}^n\delta _{x^k}\right) \Psi _n(x^1,\dots ,x^n) \end{aligned}$$

and \(F_0\Psi _0=F(0)\Psi _0\), i.e. \(\widehat{F}\) acts component-wise on \(\bigoplus \limits _{n=0}^\infty L^2_{\textrm{sym}}\! (\mathbb {R}^{3\times n})\) by multiplication with the real valued function \((x^1,\dots ,x^n)\mapsto F\left( \alpha ^{-2}\sum _{k=1}^n\delta _{x^k}\right) \).

A particularly important example of an \(\widehat{F}\) operator is the particle number \(\mathcal {N}\), which can be written as \(\mathcal {N}=\widehat{F}\) with \(F(\rho ):=\int \textrm{d}\rho \). More generally we can write, for any bounded and measurable f, \(\int \! f(x)a_x^\dagger a_x \textrm{d}x=\widehat{F_f}\) with \(F_f(\rho ):=\int f \textrm{d}\rho \). Since the assignment \(F\mapsto \widehat{F}\) is linear and multiplicative, we can represent any polynomial in operators of the form \(\int f(x)a_x^\dagger a_x \textrm{d}x\) as an \(\widehat{F}\) operator as well.

Note that in order to keep the notation simple, we will allow \(F:\mathcal {M}\left( \mathbb {R}^3\right) \longrightarrow \mathbb {R}\) to act on non-negative \(L^1\left( \mathbb {R}^3\right) \) functions \(q:\mathbb {R}^3\longrightarrow [0,\infty )\) as well by identifying them with the corresponding measure \(\lambda \in \mathcal {M}\left( \mathbb {R}^3\right) \) defined as \(\frac{\textrm{d}\lambda }{\textrm{d}x}=q(x)\).

Before we discuss the asymptotic formula for the expectation value \(\langle \Psi _\alpha |\widehat{F}|\Psi _\alpha \rangle \), let us introduce a family of cut-off functions \(\chi ^\epsilon \left( a\le f(\rho )\le b\right) \) where \(\epsilon \ge 0\) determines the sharpness of the cut-off. In the following let \(\alpha ,\beta :\mathbb {R}\longrightarrow [0,1]\) be \(C^\infty \) functions such that \(\alpha ^2+\beta ^2=1\), \(\textrm{supp}\left( \alpha \right) \subset (-\infty ,1)\) and \(\textrm{supp}\left( \beta \right) \subset (-1,\infty )\). For a given function \(f:\mathcal {M}\left( \mathbb {R}^3\right) \longrightarrow \mathbb {R}\) and constants \(-\infty \le a<b\le \infty \), let us define the function \(\chi ^\epsilon \left( a\le f\le b\right) :\mathcal {M}\left( \mathbb {R}^3\right) \longrightarrow [0,1]\) as

$$\begin{aligned} \rho \mapsto \chi ^\epsilon \left( a\le f(\rho )\le b\right) := {\left\{ \begin{array}{ll} \alpha \left( \frac{f(\rho )-b}{\epsilon }\right) \beta \left( \frac{f(\rho )-a}{\epsilon }\right) ,\text { for }\epsilon >0\\ \mathbbm {1}_{[a,b]}\left( f(\rho )\right) ,\text { for }\epsilon =0. \end{array}\right. } \end{aligned}$$

Note that \(\sum _{j\in J}\chi ^\epsilon \left( a_j\le f(\rho )\le b_j\right) ^2=1\) in case the intervals \([a_j,b_j)\) are a disjoint partition of \(\mathbb {R}\) with \(-\infty \le a_j<b_j\le \infty \). Usually we will use functions f here that are related to an integral over \(\rho \), e.g. \(f(\rho ):=\int \textrm{d}\rho \).

Similarly, we define the operator \(\chi ^\epsilon \left( a\le T\le b\right) :=\int \chi ^\epsilon \left( a\le t\le b\right) \, \textrm{d}E(t)\), where T is a self-adjoint operator and E is the spectral measure with respect to T. Furthermore we will write \(\chi \left( a\le f\le b\right) \), respectively \(\chi \left( a\le T\le b\right) \), in case \(\epsilon =0\) as well as \(\chi ^{\epsilon }\left( a\le f \right) \), respectively \(\chi ^{\epsilon }\left( a\le T \right) \), and \(\chi ^{\epsilon }\left( f \le b\right) \), respectively \(\chi ^{\epsilon }\left( T \le b\right) \), in case \(b=\infty \) or \(a=-\infty \), respectively.

The proof of the following Theorem 3.2 will be carried out in Sect. 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 \(\{\varphi ^\textrm{Pek}_x\}_{x\in \mathbb {R}^3}\).

Theorem 3.2

Given \(m\in \mathbb {N},C>0\) and \(g\in L^2\! \left( \mathbb {R}^3\right) \), we can find a constant \(T>0\) such that for all \(\alpha \ge 1\) and states \(\Psi \) satisfying \(\chi \left( \mathcal {N}\le C\right) \Psi =\Psi \) and \(\langle \Psi |\mathbb {H}_K|\Psi \rangle \le e^\textrm{Pek}+\delta e\) with \(\delta e\ge 0\) and \(K\ge \alpha ^{\frac{8}{29}}\), there exists a probability measure \(\mu \) on \(\mathbb {R}^3\), with the property

$$\begin{aligned} \left| \big \langle \Psi \big |\widehat{F}\, \big |\Psi \big \rangle -\int _{\mathbb {R}^3} F\left( |\varphi _x^\textrm{Pek}|^2\right) \, \textrm{d}\mu (x)\right| \le T\Vert f\Vert _\infty \max \Big \{\sqrt{\delta e},\alpha ^{-\frac{2}{29}}\Big \} \end{aligned}$$

for all \(F:\mathcal {M}\left( \mathbb {R}^3\right) \longrightarrow \mathbb {R}\) of the form \(F\left( \rho \right) =\int \dots \int f(x_1,\dots , x_m)\, \textrm{d}\rho (x_1)\dots \textrm{d}\rho (x_m)\) with bounded \(f:\mathbb {R}^{3\times m}\longrightarrow \mathbb {R}\), and furthermore

$$\begin{aligned}&\left| \Big \langle \Psi \Big |W_g^{-1}\mathcal {N}W_g\Big |\Psi \Big \rangle -\int _{\mathbb {R}^3} \left\| \varphi _x^\textrm{Pek}-g\right\| ^2\, \textrm{d}\mu \left( x\right) \right| \le T\max \Big \{\sqrt{\delta e},\alpha ^{-\frac{2}{29}}\Big \}, \end{aligned}$$

where \(W_g\) is the Weyl operator characterized by \(W_g^{-1}a(h)W_g=a(h)-\langle h|g\rangle \).

In the subsequent Lemma 3.3 we introduce a generalized IMS-type estimate quantifying the energy cost of localizing with respect to an \(\widehat{F}\)-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 \(\Omega \subset \mathcal {M}\left( \mathbb {R}^3\right) \) and a (quadratic) partition of unity \(\mathcal {P}=\{F_j:\mathcal {M}\left( \mathbb {R}^3\right) \longrightarrow \mathbb {R}:j\in J\}\), i.e. \(0\le F_j\le 1\) and \(\sum _{j\in J}F_j^2=1\), the variation of this partition on \(\Omega \) as

$$\begin{aligned} V_\Omega \left( \mathcal {P}\right) :=\alpha ^{4}\underset{\rho \in \Omega ,y\in \mathbb {R}^3}{\sup }\sum _{j\in J}\left| F_j\left( \rho +\alpha ^{-2}\delta _y\right) -F_j\left( \rho \right) \right| ^2. \end{aligned}$$

Lemma 3.3

There exists a constant \(c>0\), such that for any partition of unity \(\mathcal {P}=\{F_j:\mathcal {M}\left( \mathbb {R}^3\right) \longrightarrow \mathbb {R}:j\in J\}\), \(\Omega \subset \mathcal {M}\left( \mathbb {R}^3\right) \), \(K>0\), \(\alpha \ge 1\) and state \(\Psi \) with \(\widehat{\mathbbm {1}_{\Omega }}\Psi =\Psi \)

$$\begin{aligned} \left| \sum _{j\in J }\big \langle \widehat{F_j}\Psi \big |\mathbb {H}_K\big |\widehat{F_j}\Psi \big \rangle -\langle \Psi |\mathbb {H}_K|\Psi \rangle \right| \le c\sqrt{K} \alpha ^{-4}V_\Omega \left( \mathcal {P}\right) \big \langle \Psi \big |\sqrt{\mathcal {N}+\alpha ^{-2}}\, \big |\Psi \big \rangle . \end{aligned}$$

Furthermore given \(M>0\), there exists a constant \(c'>0\) such that we have for any \(\varphi \in L^2\! \left( \mathbb {R}^3\right) \) satisfying \(\Vert \varphi \Vert \le M\), partition of unity \(\{f_j:\mathbb {R}\longrightarrow \mathbb {R}:j\in J\}\), \(K\ge 1\), \(\alpha \ge 1\) and state \(\Psi \)

$$\begin{aligned} \left| \sum _{j\in J }\langle \Psi _j|\mathbb {H}_K|\Psi _j\rangle -\langle \Psi |\mathbb {H}_K|\Psi \rangle \right| \le c'\sqrt{K} \alpha ^{-4} V_{\mathcal {M}\left( \mathbb {R}^3\right) }\left( \mathcal {P}'\right) \Big \langle \Psi \Big | \sqrt{\mathcal {N}+1}\Big | \Psi \Big \rangle , \end{aligned}$$

where we define \(\Psi _j:=f_j\left( W_\varphi ^{-1}\mathcal {N}W_\varphi \right) \Psi \) with \(W_\varphi \) being the corresponding Weyl operator and \(\mathcal {P}':=\{F'_j:\mathcal {M}\left( \mathbb {R}^3\right) \longrightarrow \mathbb {R}:j\in J\}\) with \(F'_j(\rho ):=f_j(\int \textrm{d}\rho )\).


By applying the IMS identity, we obtain

$$\begin{aligned} \sum _{j\in J }\widehat{F_j}\mathbb {H}_K\widehat{F_j}-\mathbb {H}_K=\frac{1}{2}\sum _{j\in J } \left[ \left[ \widehat{F_j},\mathbb {H}_K\right] ,\widehat{F_j}\right] =-\sum _{j\in J }\mathfrak {Re}\, \left[ \left[ \widehat{F_j},a\left( \Pi _K w_x\right) \right] ,\widehat{F_j}\right] , \end{aligned}$$

where we have used the fact that \(F_j\) commutes with \(-\Delta _x\) and \(\mathcal {N}\) in the last identity. Since a state \(\Psi \) is a function with values in \(\mathcal {F}\left( L^2\! \left( \mathbb {R}^3\right) \right) =\bigoplus \limits _{n=0}^\infty L^2_{\textrm{sym}}\! (\mathbb {R}^{3\times n})\), we can represent it as \(\Psi =\bigoplus _{n=0}^\infty \Psi _n\) where \(\Psi _n(y,x^1,\dots ,x^{n})\) is a function of the electron variable y and the n phonon coordinates \(x^k\in \mathbb {R}^3\). In order to simplify the notation, we will suppress the dependence on the electron variable y in our notation. By an explicit computation, we obtain \(\left[ \left[ \widehat{F},a\left( v\right) \right] ,\widehat{F}\right] \bigoplus _{n=0}^\infty \Psi _n=-\bigoplus _{n=0}^\infty \sqrt{\frac{n+1}{\alpha ^2}}\Psi '_n\) with

$$\begin{aligned} \Psi '_n(x^1, \dots ,x^n)\! =\! \! \! \int \! \! \Bigg [\! F\bigg (\alpha ^{-2}\sum _{k=1}^{n+1}\delta _{x^k}\bigg )\!-\!F\bigg (\alpha ^{-2}\sum _{k=1}^{n}\delta _{x^k}\bigg )\!\Bigg ]^2 \! \! \! v(x^{n\!+\!1})\Psi _{n\!+\!1}(x^1,\dots ,x^{n\!+\!1})\textrm{d}x^{n\!+\!1}, \end{aligned}$$

for \(v\in L^2\! \left( \mathbb {R}^3\right) \) and \(F:\mathcal {M}\left( \mathbb {R}^3\right) \longrightarrow \mathbb {R}\). By the definition of \(V_\Omega \left( \mathcal {P}\right) \) we obtain that

$$\begin{aligned} \sigma (x^1,\dots ,x^{n+1}):=\sum _{j\in J}\bigg [ F_j\Big (\alpha ^{-2}\sum _{k=1}^{n+1}\delta _{x^k}\Big )-F_j\Big (\alpha ^{-2}\sum _{k=1}^{n}\delta _{x^k}\Big )\bigg ]^2\le \alpha ^{-4}V_\Omega \left( \mathcal {P}\right) \end{aligned}$$

for all \(x^{n+1}\in \mathbb {R}^3\) and every \((x^1,\dots ,x^n)\in \mathbb {R}^{3n}\) with \(\alpha ^{-2}\sum _{k=1}^n \delta _{x^k}\in \Omega \). Hence we can estimate \(\left| \Big \langle \Psi \Big | \sum _{j\in J}\mathfrak {Re}\left[ \left[ \widehat{F}_j,a\left( v\right) \right] ,\widehat{F}_j\right] \Big | \Psi \Big \rangle \right| \), using the notation \(X=(x^1,\dots ,x^n)\), by

$$\begin{aligned}&\sum _{n=0}^\infty \sqrt{\frac{n\! +\! 1}{\alpha ^2}} \int |\Psi _n(X)| \int \sigma (X,x^{n\!+\! 1})|v(x_{n+1})\Psi _{n+1}(X,x_{n\!+\! 1})|\textrm{d}x_{n + 1}\textrm{d}X\\&\quad \le \alpha ^{-5} V_\Omega \left( \mathcal {P}\right) \sum _{n=0}^\infty \sqrt{n+1}\int \!|\Psi _n(X)|\int |v(x_{n+1})\Psi _{n+1}(X,x_{n+1})|\textrm{d}x^{n+1}\textrm{d}X\\&\quad \le \alpha ^{-5} V_\Omega \left( \mathcal {P}\right) \Vert v\Vert \sum _{n=0}^\infty \sqrt{n+1}\big \Vert \Psi _n\big \Vert \, \big \Vert \Psi _{n+1}\big \Vert \le \alpha ^{-4}V_\Omega \left( \mathcal {P}\right) \Vert v\Vert \Big \langle \Psi \Big | \sqrt{\mathcal {N}+\alpha ^{-2}}\,\Big |\Psi \Big \rangle . \end{aligned}$$

This concludes the proof of Eq. (3.4), using the concrete choice \(v:=\Pi _K w_x\), since \(\left\| \Pi _K w_x\right\| ^2=\frac{1}{2\pi ^2} \int _{|k|\le K}\frac{1}{|k|^2}=\frac{2}{\pi } K\).

In order to verify the second statement we apply the unitary transformation \(W_\varphi \) to the operator \(\mathbb {X}:=\sum _{j\in J }f_j\left( W_\varphi ^{-1}\mathcal {N}W_\varphi \right) \mathbb {H}_Kf_j\left( W_\varphi ^{-1}\mathcal {N}W_\varphi \right) -\mathbb {H}_K\) and compute

$$\begin{aligned} W_\varphi \mathbb {X}\, W_\varphi ^{-1}&=\frac{1}{2}\sum _{j\in J } \Big [\Big [f_j\left( \mathcal {N}\right) ,W_\varphi \mathbb {H}_K W_\varphi ^{-1}\Big ],f_j\left( \mathcal {N}\right) \Big ]\\&=\sum _{j\in J } \mathfrak {Re}\Big [\Big [f_j\left( \mathcal {N}\right) ,a\left( \varphi -\Pi _K w_x\right) \Big ],f_j\left( \mathcal {N}\right) \Big ]=\sum _{j\in J } \mathfrak {Re}\Big [\Big [\widehat{F}_j',a\left( v\right) \Big ],\widehat{F}_j'\Big ], \end{aligned}$$

where we defined \(v:=\varphi -\Pi _K w_x\) and applied the definition \(F_j'(\rho )=f_j\left( \int \textrm{d}\rho \right) \). We know from the previous estimates that

$$\begin{aligned} \pm \sum _{j\in J } \mathfrak {Re}\Big [\Big [f_j\left( \mathcal {N}\right) ,a\left( v\right) \Big ],f_j\left( \mathcal {N}\right) \Big ]\le \alpha ^{-4}V_{\mathcal {M}\left( \mathbb {R}^3\right) }\left( \mathcal {P}'\right) \Vert v\Vert \sqrt{\mathcal {N}+\alpha ^{-2}}. \end{aligned}$$

Clearly \(\Vert v\Vert \le \Vert \varphi \Vert +\Vert \Pi _K w_x\Vert \lesssim \sqrt{K}\) for \(K\ge 1\), and consequently

$$\begin{aligned}&\left| \sum _{j\in J }\langle \Psi _j|\mathbb {H}_K|\Psi _j\rangle -\langle \Psi |\mathbb {H}_K|\Psi \rangle \right| \lesssim \sqrt{K}\alpha ^{-4}V_{\mathcal {M}\left( \mathbb {R}^3\right) }\left( \mathcal {P}'\right) \Big \langle \Psi \Big | \sqrt{W_\varphi ^{-1}\mathcal {N}W_\varphi +\alpha ^{-2}}\,\Big |\Psi \Big \rangle \\&\quad \lesssim \sqrt{K}\alpha ^{-4}V_{\mathcal {M}\left( \mathbb {R}^3\right) }\left( \mathcal {P}'\right) \Big \langle \Psi \Big | \sqrt{\mathcal {N}+1}\,\Big |\Psi \Big \rangle , \end{aligned}$$

where we have used that \(W_\varphi ^{-1}\mathcal {N}W_\varphi \le 2\left( \mathcal {N}+\Vert \varphi \Vert ^2\right) \) and the operator-monotonicity of the square root. \(\quad \square \)

3.2 Auxiliary localization procedures

In the following Eqs. (3.6) and (3.10), we will apply localizations procedures to a given sequence \(\Psi _\alpha ^\bullet \) 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 \(\langle \Psi _\alpha |\mathbb {H}_\Lambda |\Psi _\alpha \rangle -\widetilde{E}_\alpha \lesssim \alpha ^{-3}\) in the Lemmas 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].

In the following let \(L:=\alpha ^{1+\sigma }\) and \(\Lambda :=\alpha ^{\frac{4}{5}(1+\sigma )}\) with \(0<\sigma \le \frac{1}{4}\), and let \(\Psi _\alpha ^\bullet \) be a sequence of states satisfying \(\textrm{supp}\left( \Psi _\alpha ^\bullet \right) \subset B_L(0)\) and \(\widetilde{E}_\alpha - E_\alpha \lesssim \alpha ^{-\frac{4}{29}}\), where

$$\begin{aligned} \widetilde{E}_\alpha :=\langle \Psi _\alpha ^\bullet |\mathbb {H}_\Lambda |\Psi _\alpha ^\bullet \rangle . \end{aligned}$$

The exponent \(\frac{4}{29}\) is chosen for convenience, as it allows to simplify the right hand side of Eq. (3.2) to \(\Vert f\Vert _\infty \alpha ^{-\frac{2}{29}}\) (using that \(E_\alpha \le e^\textrm{Pek}\)). For the proof of Theorem 1.1 we shall use the specific choice \(\Psi _\alpha ^\diamond \) from Theorem 2.5 for \(\Psi _\alpha ^\bullet \), 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 \(\Psi _\alpha ^\bullet \).

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 \(\Psi '_\alpha \) having a (rescaled) particle number \(\mathcal {N}\) between some fixed constants \(c_-\) and \(c_+\). To be precise, for given \(c_-,c_+\) and \(\epsilon '\) we use the function \(F_*(\rho ):=\chi ^{\epsilon '}\left( c_-+\epsilon '\le \int \textrm{d}\rho \le c_+-\epsilon '\right) \) in order to define the states

$$\begin{aligned} \Psi _\alpha ':=Z_{\alpha }^{-1}\widehat{F}_*\Psi ^\bullet _\alpha , \end{aligned}$$

with the corresponding normalization constants \(Z_{\alpha }:=\Vert \widehat{F}_*\Psi ^\bullet _\alpha \Vert \). By construction we have \(\chi \left( c_-\le \mathcal {N}\le c_+\right) \Psi '_\alpha =\Psi '_\alpha \) as well as \(\textrm{supp}\left( \Psi '_\alpha \right) \subset B_{L}(0)\). In the following Lemma 3.4 we derive an upper bound on the energy of \(\Psi _\alpha '\), and in addition we will investigate the large \(\alpha \) behavior of \(Z_\alpha \), which will be useful in the second part.

Lemma 3.4

Let \(\Psi _\alpha ^\bullet \) be the sequence introduced above Eq. (3.5). Then there exist \(\alpha \)-independent constants \(c_-,c_+,\epsilon '>0\) such that the corresponding states \(\Psi _\alpha '\) defined in Eq. (3.6) satisfy \(\langle \Psi '_\alpha |\mathbb {H}_\Lambda |\Psi '_\alpha \rangle -\widetilde{E}_\alpha \lesssim \alpha ^{-\frac{7}{2}}\). Furthermore, \(Z_\alpha \underset{\alpha \rightarrow \infty }{\longrightarrow }\ 1\).


In the following let \(F_*\) be the function defined above Eq. (3.6) and let us complete it to a quadratic partition of unity \(\mathcal {P}:=\{F_-,F_*,F_+\}\) with the aid of the functions \(F_-(\rho ):=\chi ^{\epsilon '}\left( \int \textrm{d}\rho \le c_-+\epsilon '\right) \) and \(F_+(\rho ):=\chi ^{\epsilon '}\left( c_+-\epsilon '\le \int \textrm{d}\rho \right) \). Making use of Lemma 3.3 and \(\Lambda =\alpha ^{\frac{4}{5}(1+\sigma )}\le \alpha \), we then obtain

$$\begin{aligned}&Z_{\alpha ,-}^2 \langle \Psi _{\alpha ,-}|\mathbb {H}_\Lambda |\Psi _{\alpha ,-}\rangle + Z_{\alpha }^2 \langle \Psi _{\alpha }'|\mathbb {H}_\Lambda |\Psi _{\alpha }'\rangle + Z_{\alpha ,+}^2 \langle \Psi _{\alpha ,+}|\mathbb {H}_\Lambda |\Psi _{\alpha ,+}\rangle \nonumber \\&\quad \le \langle \Psi ^\bullet _\alpha |\mathbb {H}_\Lambda |\Psi ^\bullet _\alpha \rangle + c\, \alpha ^{-\frac{7}{2}} V_{\mathcal {M}\left( \mathbb {R}^3\right) } \left( \mathcal {P}\right) \big \langle \Psi ^\bullet _\alpha \big |\sqrt{\mathcal {N} + \alpha ^{-2} }\big |\Psi ^\bullet _\alpha \big \rangle , \end{aligned}$$

where \(\Psi _{\alpha ,\pm }:=Z_{\alpha ,\pm }^{-1}\widehat{F}_{(\pm )}\Psi ^\bullet _\alpha \), with corresponding normalization factors \(Z_{\alpha ,\pm }:=\Vert \widehat{F}_{(\pm )}\Psi ^\bullet _\alpha \Vert \). By Eq. (2.5) there exists a constant d s.t. \(\big \langle \Psi ^\bullet _\alpha \big |\mathcal {N} \big |\Psi ^\bullet _\alpha \big \rangle \le \big \langle \Psi ^\bullet _\alpha \big |2\mathbb {H}_\Lambda +d \big |\Psi ^\bullet _\alpha \big \rangle \lesssim d+\alpha ^{-\frac{4}{29}}\), where we have used the assumption \(\big \langle \Psi ^\bullet _\alpha \big |\mathbb {H}_\Lambda \big |\Psi ^\bullet _\alpha \big \rangle =\widetilde{E}_\alpha \le \widetilde{E}_\alpha -E_\alpha \lesssim \alpha ^{-\frac{4}{29}}\). The first derivative of the functions \(\chi ^{\epsilon '}(\cdot \le c_-+{\epsilon '}),\chi ^{\epsilon '}(c_-+{\epsilon '}\le \cdot \le c_+-{\epsilon '})\) and \(\chi ^{\epsilon '}(\cdot \le c_+-{\epsilon '})\) is uniformly bounded by some \(\epsilon '\)-dependent constant D, and consequently we have for all finite measures \(\rho \) and \(\rho ':=\rho +\alpha ^{-2}\delta _y\) with \(y\in \mathbb {R}^3\), and \(\diamond \in \{-,*,+\}\),

$$\begin{aligned} \left| F_\diamond \left( \rho '\right) -F_\diamond \left( \rho \right) \right| \le D \left| \int \textrm{d}\rho '-\int \textrm{d}\rho \right| =D\alpha ^{-2}. \end{aligned}$$

This implies that \(V_{\mathcal {M}\left( \mathbb {R}^3\right) }\left( \mathcal {P}\right) \lesssim 1\), and therefore the right hand side of Eq. (3.7) is bounded by \(\langle \Psi ^\bullet _\alpha |\mathbb {H}_\Lambda |\Psi ^\bullet _\alpha \rangle +C\alpha ^{-\frac{7}{2}}\) for a suitable \(C>0\). Since \(Z_{\alpha ,-}^2+Z_{\alpha }^2+Z_{\alpha ,+}^2=1\), this means that at least one of the terms \(\langle \Psi _{\alpha ,-}|\mathbb {H}_\Lambda |\Psi _{\alpha ,-}\rangle ,\langle \Psi _{\alpha }'|\mathbb {H}_\Lambda |\Psi _{\alpha }'\rangle \) or \(\langle \Psi _{\alpha ,+}|\mathbb {H}_\Lambda |\Psi _{\alpha ,+}\rangle \) is bounded from above by \(\langle \Psi ^\bullet _\alpha |\mathbb {H}_\Lambda |\Psi ^\bullet _\alpha \rangle +C\alpha ^{-\frac{7}{2}}=\widetilde{E}_\alpha +C\alpha ^{-\frac{7}{2}}\). We can however rule out that \(\langle \Psi _{\alpha ,-}|\mathbb {H}_\Lambda |\Psi _{\alpha ,-}\rangle \), respectively \(\langle \Psi _{\alpha ,+}|\mathbb {H}_\Lambda |\Psi _{\alpha ,+}\rangle \), satisfy this upper bound for all small \(c_-,{\epsilon '}\) and large \(\alpha ,c_+\), since \(\widetilde{E}_\alpha \le E_\alpha +C'\alpha ^{-\frac{4}{29}}\le e^\textrm{Pek}+C'\alpha ^{-\frac{4}{29}}< \frac{e^\textrm{Pek}}{2}<0\) for \(\alpha \) large enough and a suitable \(C'\), and since we have by Eqs. (2.4) and (2.5) for all \(t>0\)

$$\begin{aligned} \langle \Psi _{\alpha ,-}|\mathbb {H}_\Lambda |\Psi _{\alpha ,-}\rangle&\ge \langle \Psi _{\alpha ,-}| - \frac{d}{t^2} - t \left( \mathcal {N} + \alpha ^{-2}\right) |\Psi _{\alpha ,-}\rangle \ge -\frac{d}{t^2} - t \left( c_- + 2{\epsilon '} + \alpha ^{-2}\right) \ge -\frac{e^\textrm{Pek}}{2}, \end{aligned}$$
$$\begin{aligned} \langle \Psi _{\alpha ,+}|\mathbb {H}_\Lambda |\Psi _{\alpha ,+}\rangle&\ge \langle \Psi _{\alpha ,+}|-d+\frac{1}{2}\mathcal {N}|\Psi _{\alpha ,+}\rangle \ge -d+\frac{1}{2}(c_+-2{\epsilon '})\ge 0, \end{aligned}$$

where the last inequality in Eq. (3.8), respectively Eq. (3.9), holds for small \(c_-,{\epsilon '}\) and large \(\alpha ,c_+\) with the concrete choice \(t:=\left( \frac{d}{c_-+2{\epsilon '}+\alpha ^{-2}}\right) ^{\frac{1}{3}}\). Using again that the right hand side of Eq. (3.7) is bounded by \(\langle \Psi ^\bullet _\alpha |\mathbb {H}_\Lambda |\Psi ^\bullet _\alpha \rangle +C\alpha ^{-\frac{7}{2}}\) together with Eqs. (3.8) and (3.9), and the fact that \(\mathbb {H}_\Lambda \ge E_\alpha \) and \(E_\alpha \le e^\textrm{Pek}\), yields furthermore

$$\begin{aligned} (1-Z_\alpha ^2)\left( E_\alpha -\frac{e^\textrm{Pek}}{2}\right) +Z_\alpha ^2 E_\alpha \le (1-Z_\alpha ^2)\frac{e^\textrm{Pek}}{2}+Z_\alpha ^2 E_\alpha \le \widetilde{E}_\alpha +C\alpha ^{-\frac{7}{2}}, \end{aligned}$$

and therefore \(-(1-Z_\alpha ^2)\frac{e^\textrm{Pek}}{2}\le \widetilde{E}_\alpha -E_\alpha +C\alpha ^{-\frac{7}{2}}\underset{\alpha \rightarrow \infty }{\longrightarrow }\ 0\). Since \(e^\textrm{Pek}<0\), this immediately implies \(Z_\alpha \underset{\alpha \rightarrow \infty }{\longrightarrow }\ 1\). \(\quad \square \)

Regarding the next localization step in Lemma 3.5, let us introduce for given R and \(\epsilon >0\) satisfying \(R>2\epsilon \) the function \(K_R\left( \rho \right) :=\iint \chi ^{\epsilon }\left( R-\epsilon \le |x-y|\right) \textrm{d}\rho (x)\textrm{d}\rho (y)\), which measures how sharply the mass of the measure \(\rho \) is concentrated. It will be convenient in the second part to have \(K_R\) defined for arbitrary \(\epsilon \ge 0\) even though we only need it for \(\epsilon =0\) in the following. We also define the function \(F_R\left( \rho \right) :=\chi ^{\frac{\delta }{3}}\Big (K_R\left( \rho \right) \le \frac{2\delta }{3}\Big )\) for \(R,\delta >0\), as well as the states

$$\begin{aligned} \Psi ''_\alpha :=Z_{R,\alpha }^{-1}\widehat{F}_R\Psi '_\alpha , \end{aligned}$$

where \(\Psi '_\alpha \) is as in Lemma 3.4 and \(Z_{R,\alpha }:=\Vert \widehat{F}_R\Psi '_\alpha \Vert \). Since \(\Psi '_\alpha \) satisfies \(\textrm{supp}\left( \Psi '_\alpha \right) \subset B_L(0)\), we have \(\textrm{supp}\left( \Psi ''_\alpha \right) \subset B_L(0)\) as well. Furthermore \(\chi \left( \widehat{K}_R\le \delta \right) \Psi ''_\alpha =\Psi ''_\alpha \). Heuristically this means that we can restrict our attention to phonon configurations that concentrate in a ball of fixed radius R.

Lemma 3.5

Let \(\Psi '_\alpha \) be the sequence from Lemma 3.4, and let \(\epsilon \ge 0\) and \(\delta >0\) be given constants. Then there exists a \(\alpha \) independent \(R>0\), such that the states \(\Psi ''_\alpha \) defined in Eq. (3.10) satisfy \(\langle \Psi ''_\alpha |\mathbb {H}_\Lambda |\Psi ''_\alpha \rangle -\widetilde{E}_\alpha \lesssim \alpha ^{-\frac{7}{2}}\), where \(\widetilde{E}_\alpha \) is defined in Eq. (3.5). Furthermore, \(Z_{R,\alpha }\underset{\alpha \rightarrow \infty }{\longrightarrow }1\).


Since \(\mathcal {P}:=\{F_R,G_R\}\) with \(G_R:=\sqrt{1-F_R^2}=\chi ^{\frac{\delta }{3}}\left( \frac{2\delta }{3}\le K_R\left( \rho \right) \right) \) is a partition of unity, we obtain by Lemma 3.3

$$\begin{aligned}&\langle \widehat{F}_R\Psi '_\alpha |\mathbb {H}_\Lambda |\widehat{F}_R\Psi '_\alpha \rangle + \langle \widehat{G}_R\Psi '_\alpha |\mathbb {H}_\Lambda |\widehat{G}_R\Psi '_\alpha \rangle \nonumber \\&\le \langle \Psi '_\alpha |\mathbb {H}_\Lambda |\Psi '_\alpha \rangle + c\, \alpha ^{-\frac{7}{2}} V_\Omega \left( \mathcal {P}\right) \big \langle \Psi '_\alpha \big |\sqrt{c_+ + \alpha ^{-2} }\,\big |\Psi '_\alpha \big \rangle \end{aligned}$$

with \(\Omega :=\{\rho :\int \textrm{d}\rho \le c_+\}\), where we have used \(\chi \left( \mathcal {N}\le c_+\right) \Psi '_\alpha =\Psi '_\alpha \) and \(\Lambda \le \alpha \). Since \(\frac{\textrm{d}}{\textrm{d}x}\chi ^{\frac{\delta }{3}}\left( \frac{2\delta }{3}\le x \right) \) and \(\frac{\textrm{d}}{\textrm{d}x}\chi ^{\frac{\delta }{3}}\left( x \le \frac{2\delta }{3}\right) \) are bounded by some \(\delta \)-dependent constant D, we have for all \(\rho \in \Omega \) and \(\rho ':=\rho +\alpha ^{-2}\delta _z\) with \(z\in \mathbb {R}^3\), and \(R>2\epsilon \), the estimate

$$\begin{aligned} \left| F_R\left( \rho '\right) -F_R\left( \rho \right) \right|&\le D\left| K_R(\rho ')-K_R(\rho )\right| =2D\alpha ^{-2}\int \chi ^{\epsilon }\left( R-\epsilon \le |y-z|\right) \textrm{d}\rho (y)\\&\le 2D\alpha ^{-2} c_+, \end{aligned}$$

and the same result holds for \(G_R\). Therefore we have by Eq. (3.11) and Lemma 3.4

$$\begin{aligned} \langle \widehat{F}_R\Psi '_\alpha |\mathbb {H}_\Lambda |\widehat{F}_R\Psi '_\alpha \rangle +\langle \widehat{G}_R\Psi '_\alpha |\mathbb {H}_\Lambda |\widehat{G}_R\Psi '_\alpha \rangle \le \langle \Psi '_\alpha |\mathbb {H}_\Lambda |\Psi '_\alpha \rangle +C_1\alpha ^{-\frac{7}{2}}\le \widetilde{E}_\alpha +C_2 \alpha ^{-\frac{7}{2}} \end{aligned}$$

for suitable constants \(C_1,C_2>0\). Since \(\Vert \widehat{F}_R\Psi '_\alpha \Vert ^2+\Vert \widehat{G}_R\Psi '_\alpha \Vert ^2=1\), this means that we either have \(\langle \Psi ''_\alpha |\mathbb {H}_\Lambda |\Psi ''_\alpha \rangle \le \widetilde{E}_\alpha +C_2\alpha ^{-\frac{7}{2}}\) or \(\langle \widetilde{\Psi }_\alpha |\mathbb {H}_\Lambda |\widetilde{\Psi }_\alpha \rangle \le \widetilde{E}_\alpha +C_2\alpha ^{-\frac{7}{2}}\), where \(\widetilde{\Psi }_\alpha :=\Vert \widehat{G}_R\Psi '_\alpha \Vert ^{-1}\widehat{G}_R\Psi '_\alpha \). In the following we are going to rule out the second case for R and \(\alpha \) large enough, to be precise we are going to verify \(\langle \widetilde{\Psi }_\alpha |\mathbb {H}_\Lambda |\widetilde{\Psi }_\alpha \rangle > \widetilde{E}_\alpha +d\alpha ^{-\frac{4}{29}}\) for any \(d>0\) and large enough R and \(\alpha \) by contradiction. In order to do this, let us assume \(\langle \widetilde{\Psi }_\alpha |\mathbb {H}_\Lambda |\widetilde{\Psi }_\alpha \rangle \le \widetilde{E}_\alpha +d\alpha ^{-\frac{4}{29}}\). Since \(\widetilde{E}_\alpha \le E_\alpha +C\alpha ^{-\frac{4}{29}}\le e^\textrm{Pek}+C\alpha ^{-\frac{4}{29}}\) by assumption for a suitable constant C, \(\widetilde{\Psi }_\alpha \) satisfies the assumptions of Theorem 3.2 with \(\delta e:=(d+C)\alpha ^{-\frac{4}{29}}\). Hence there exists a measure \(\mu \) such that Eq. (3.2) holds. By the support properties of \(G_R\) we obtain

$$\begin{aligned} \frac{\delta }{3}\le \big \langle \widetilde{\Psi }_\alpha \big |\widehat{K}_R\big |\widetilde{\Psi }_\alpha \big \rangle&=\int K_R\left( \left| \varphi _x^\textrm{Pek}\right| ^2\right) \, \textrm{d}\mu + O_{\alpha \rightarrow \infty }\left( \alpha ^{-\frac{2}{29}}\right) \nonumber \\&= K_R\left( \left| \varphi ^\textrm{Pek}\right| ^2\right) + O_{\alpha \rightarrow \infty }\left( \alpha ^{-\frac{2}{29}}\right) . \end{aligned}$$

Since \(\lim _{R\rightarrow \infty }K_R\left( \left| \varphi ^\textrm{Pek}\right| ^2\right) =0\), Eq. (3.13) is a contradiction for large R and \(\alpha \), and consequently we have \(\langle \widetilde{\Psi }_\alpha |\mathbb {H}_\Lambda |\widetilde{\Psi }_\alpha \rangle > \widetilde{E}_\alpha +d\alpha ^{-\frac{4}{29}}\) for such R and \(\alpha \). In combination with Eq. (3.12) this furthermore yields

$$\begin{aligned} Z_{R,\alpha }^2 E_\alpha \! +\! (1\! -\! Z_{R,\alpha }^2)\! \left( E_\alpha \! +\! d\alpha ^{-\frac{4}{29}}\right) \le Z_{R,\alpha }^2 E_\alpha \! +\! (1\! -\! Z_{R,\alpha }^2)\! \left( \widetilde{E}_\alpha \! +\! d\alpha ^{-\frac{4}{29}}\right) \le \widetilde{E}_\alpha \! +\! C_2 \alpha ^{-\frac{7}{2}}, \end{aligned}$$

and therefore \(1-Z_{R,\alpha }^2\le \frac{\alpha ^{\frac{4}{29}}}{d}\left( \widetilde{E}_\alpha -E_\alpha +C_2\alpha ^{-\frac{7}{2}}\right) \le \frac{1}{d}+\frac{C_2}{d}\alpha ^{\frac{4}{29}-\frac{7}{2}}\). Since this holds for any \(d>0\) and \(\alpha \) large enough, we conclude that \(Z_{R,\alpha }\underset{\alpha \rightarrow \infty }{\longrightarrow }1\). \(\quad \square \)

3.3 Localization of the median

The previous localizations in the Lemmas 3.4 and 3.5 will allow us to control the energy error in the proof of Lemma 3.12, where we carry out the main localization procedure with respect to the (regularized) median \(m_q\) defined in Definition 3.8. Before we come to the proof of Lemma 3.12, we are going to derive Lemma 3.10, which provides an upper bound on the variation \(V_\Omega \left( \mathcal {P}\right) \) for partitions \(\mathcal {P}=\{F_j:j\in J\}\) of the form \(F_j(\rho )=f_j\left( m_q(\rho )\right) \). 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 \(\Omega _\textrm{reg}\) as the set of all \(\rho \in \mathcal {M}\left( \mathbb {R}^3\right) \) satisfying

$$\begin{aligned} \rho _i(\{t\})\le \alpha ^{-2} \end{aligned}$$

for all \(t\in \mathbb {R}\) and \(i\in \{1,2,3\}\), where \(\rho _1,\rho _2\) and \(\rho _3\) are the marginal measures of \(\rho \) defined by \(\rho _i\left( A\right) :=\rho \left( \left[ x_i\in A\right] \right) \). Then \(\widehat{\mathbbm {1}_{\Omega _\textrm{reg}}}\Psi =\Psi \) for all \(\Psi \in \mathcal {F}\left( L^2\! \left( \mathbb {R}^3\right) \right) \).


For given \(x=(x^1,\dots ,x^n)\in \mathbb {R}^{3\times n}\), define the measure \(\rho _x:=\alpha ^{-2}\sum _{k=1}^n \delta _{x^k}\). Note that \(\rho _x\notin \Omega _\textrm{reg}\) if and only if there exists an \(i\in \{1,2,3\}\) such that \(x_i^k=x_i^{k'}\) for indices \(k\ne k'\). Clearly the set of all such \(x\in \mathbb {R}^{3\times n}\) has Lebesgue measure zero. Hence the multiplication operator by the function \((x^1,\dots ,x^n)\mapsto \mathbbm {1}_{\Omega _\textrm{reg}}\left( \rho _x\right) \) is equal to the identity on \(L^2_{\textrm{sym}}\! \left( \mathbb {R}^{3\times n}\right) \), which concludes the proof according to Definition 3.1. \(\square \)

Lemma 3.7

Let \(\nu ,\nu '\) be finite measures on \(\mathbb {R}\) such that \(\nu \left( \{t\}\right) \le \epsilon \) and \(\nu '\left( \{t\}\right) \le \epsilon \) for all \(t\in \mathbb {R}\), and let \(x^\kappa (\nu )\) be the \(\kappa \)-quantile of the measure \(\nu \) with \(0\le \kappa \le 1\), to be precise \(x^{\kappa }(\nu )\) is the supremum over all numbers \(t\in \mathbb {R}\) satisfying \(\int _{-\infty }^t\textrm{d}\nu \le \kappa \int \textrm{d}\nu \), where we use the convention that the boundaries are included in the domain of integration \(\int _{a}^b f\textrm{d}\nu :=\int _{[a,b]}f\textrm{d}\nu \). Then

$$\begin{aligned} \Big |\int _{-\infty }^{x^\kappa \left( \nu '\right) } \textrm{d}\nu -\int _{-\infty }^{x^\kappa \left( \nu \right) } \textrm{d}\nu \Big |\le 2\Vert \nu '-\nu \Vert _{\textrm{TV}}+\epsilon , \end{aligned}$$

where \(\Vert \nu '-\nu \Vert _{\textrm{TV}}:=\underset{\Vert f\Vert _\infty =1}{\sup }\left| \int f\, \textrm{d}\nu '-\int f\, \textrm{d}\nu \right| \).


We estimate

$$\begin{aligned}&\int _{-\infty }^{x^\kappa \left( \nu '\right) } \textrm{d}\nu -\int _{-\infty }^{x^\kappa \left( \nu \right) } \textrm{d}\nu \le \int _{-\infty }^{x^\kappa \left( \nu '\right) }\textrm{d}\nu -\kappa \int \textrm{d}\nu \\&\quad \le \int _{-\infty }^{x^\kappa \left( \nu '\right) }\textrm{d}\nu '+\Vert \nu '-\nu \Vert _{\textrm{TV}}-\kappa \int \textrm{d}\nu \\&\quad \le \kappa \int \textrm{d}\nu '+\epsilon +\Vert \nu '-\nu \Vert _{\textrm{TV}}-\kappa \int \textrm{d}\nu \le 2\Vert \nu '-\nu \Vert _{\textrm{TV}}+\epsilon , \end{aligned}$$

where we have used \(\int _{-\infty }^{x^\kappa \left( \nu \right) } \textrm{d}\nu \ge \kappa \int \textrm{d}\nu \) and \(\int _{-\infty }^{x^\kappa \left( \nu '\right) }\textrm{d}\nu '\le \kappa \int \textrm{d}\nu '+\epsilon \). The bound from below can be obtained by interchanging the role of \(\nu \) and \(\nu '\). \(\quad \square \)

Definition 3.8

Let \(x^\kappa \! \left( \nu \right) \) be the \(\kappa \)-quantile of a measure \(\nu \) on \(\mathbb {R}\) defined in Lemma 3.7 and let us denote \(K_q(\nu ):=[x^{\frac{1}{2}-q}(\nu ),x^{\frac{1}{2}+q}(\nu )]\) for \(0< q< \frac{1}{2}\). Then we define

$$\begin{aligned} m_q(\nu ):=\frac{1}{\int _{K_q(\nu )} \textrm{d}\nu }\int _{K_q(\nu )}t\, \textrm{d}\nu (t)\in \mathbb {R} \end{aligned}$$

for \(\nu \ne 0\) and \(m_q(0):=0\). Furthermore we define for a measure \(\rho \) on \(\mathbb {R}^3\) the regularized median as \(m_q(\rho ):=\Big (m_q(\rho _1),m_q(\rho _2),m_q(\rho _3)\Big )\in \mathbb {R}^3\), where \(\rho _1,\rho _2\) and \(\rho _3\) are the marginal measures of \(\rho \).

Note that \(x^\kappa (\nu )\) is the largest value, such that both \(\int _{-\infty }^{x^\kappa (\nu )}\textrm{d}\nu \ge \kappa \int \textrm{d}\nu \) and \(\int _{x^\kappa (\nu )}^\infty \textrm{d}\nu \ge (1-\kappa )\int \textrm{d}\nu \) hold. As an immediate consequence, we obtain that the expression in Eq. (3.14) is well-defined for \(\nu \ne 0\) and \(0<q<\frac{1}{2}\), since

$$\begin{aligned} \int _{K_q(\nu )} \textrm{d}\nu =\int _{-\infty }^{x^{\frac{1}{2}+q}(\nu )}\textrm{d}\nu +\int _{x^{\frac{1}{2}-q}(\nu )}^{\infty }\textrm{d}\nu -\int \textrm{d}\nu \ge 2q\int \textrm{d}\nu >0. \end{aligned}$$

In the following Lemma 3.9 we are going to show that the quantiles \(x^{\frac{1}{2}\pm q}\) are positioned in a ball of radius R around the median \(x^{\frac{1}{2}}\) for all measures \(\rho \) that concentrate in a ball of radius R around the median, in the sense that \(\underset{|x-y|\ge R}{\int \int } \textrm{d}\rho (x)\textrm{d}\rho (y)\le \delta \), see also the definition of \(F_R\) above Eq. (3.10), where \(\delta \) is a small enough constant depending on q and the total mass \(\int \textrm{d}\rho \).

Lemma 3.9

Given constants \(R,c>0\) and \(0<\delta <\frac{c^2}{2}\), let \(\rho \) satisfy \(c\le \int \textrm{d}\rho \) and \(\underset{|x-y|\ge R}{\int \int }\ \textrm{d}\rho (x)\textrm{d}\rho (y)\le \delta \) and let q be a constant satisfying \(0<q\le \frac{1}{2}-\frac{\delta }{c^2}\). Then we have for all \(i\in \{1,2,3\}\) that \(x^{\frac{1}{2}}(\rho _i)-R\le x^{\frac{1}{2}-q}(\rho _i)\le x^{\frac{1}{2}+q}(\rho _i)\le x^{\frac{1}{2}}(\rho _i)+R\).


Since \(x^\kappa \) is translation covariant, i.e. \(x^\kappa \left( \nu (\cdot -t)\right) =x^\kappa (\nu )+t\), we can assume w.l.o.g. that \(x^{\frac{1}{2}}(\rho _i)=0\) for \(i\in \{1,2,3\}\). Then

$$\begin{aligned}&\delta \ge \! \! \underset{|x-y|\ge R}{\int \int }\! \! \textrm{d}\rho (x)\textrm{d}\rho (y)\ge 2\int _{x_i\ge 0}\! \! \textrm{d}\rho (x)\int _{y_i\le -R}\! \! \textrm{d}\rho (y)\ge \int \! \textrm{d}\rho \int _{y_i\le -R}\! \! \textrm{d}\rho (y)\\&\quad \ge c\int _{y_i\le -R}\! \! \textrm{d}\rho (y), \end{aligned}$$

where we have used that \(x^{\frac{1}{2}}(\rho _i)=0\) and \(\int \textrm{d}\rho \ge c\) in the last two inequalities. Hence

$$\begin{aligned} \int _{y_i\le -R}\textrm{d}\rho (y)\le \frac{\delta }{c}\le \frac{\delta }{c^2}\int \textrm{d}\rho \le \kappa \int \textrm{d}\rho \end{aligned}$$

for all \(\kappa \ge \frac{\delta }{c^2}\) and consequently we have \(-R\le x^\kappa (\rho _i)\) for all such \(\kappa \) by the definition of \(x^\kappa (\rho _i)\). Similarly we obtain \(x^\kappa (\rho _i)\le R\) for all \(\kappa \) satisfying \(\kappa \le 1-\frac{\delta }{c^2}\). Therefore \(|x^{\frac{1}{2}\pm q}(\rho _i)|\le R\) for \(q\le \frac{1}{2}-\frac{\delta }{c^2}\). \(\quad \square \)

Lemma 3.10

Given constants \(R,c>0\) and \(0<\delta <\frac{c^2}{2}\), let \(\Omega \) be the set of \(\rho \in \Omega _\textrm{reg}\) satisfying \(c\le \int \textrm{d}\rho \) and \(\underset{|x-y|\ge R}{\int \int } \textrm{d}\rho (x)\textrm{d}\rho (y)\le \delta \). Then

$$\begin{aligned} \left| m_q\left( \rho +\alpha ^{-2}\delta _x\right) -m_q\left( \rho \right) \right| \lesssim \frac{R}{c\alpha ^2 q} \end{aligned}$$

for all \(\rho \in \Omega \), \(x\in \mathbb {R}^3\) and \(0<q<\frac{1}{2}-\frac{\delta }{c^2}\), where \(m_q\) is defined in Definition 3.8.


Since \(m_q\) acts translation covariant on any \(\rho \ne 0\), i.e. \(m_q\left( \rho (\cdot -y)\right) =m_q(\rho )+y\), we can assume w.l.o.g. that \(x^{\frac{1}{2}}(\rho _i)=0\) for \(i\in \{1,2,3\}\). By Lemma 3.9 we therefore obtain \(|x^{\frac{1}{2}\pm q}(\rho _i)|\le R\) for \(\rho \in \Omega \) and \(0<q\le \frac{1}{2}-\frac{\delta }{c^2}\). Note that the marginal measures \(\rho _i\) and \(\rho '_i\), where \(\rho ':=\rho +\alpha ^{-2}\delta _x\), satisfy \(\rho _i\left( \{y\}\right) \le \alpha ^{-2}\) and \(\rho '_i\left( \{y\}\right) \le 2\alpha ^{-2}\) by our assumption \(\rho \in \Omega _\textrm{reg}\). Therefore \(x^{\kappa _*}(\rho _i)\le x^\kappa (\rho '_i)\le x^{\kappa ^*}(\rho _i)\) for \(\rho \in \Omega \) and \(\kappa >0\), with \(\kappa _*:=\kappa -2\frac{1}{c}\alpha ^{-2}\) and \(\kappa ^*:=\kappa +3\frac{1}{c}\alpha ^{-2}\). In particular, this implies \(|x^{\frac{1}{2}\pm q}(\rho '_i)|\le R\) for \(0<q<1/2-\delta /c^2\) and \(\alpha \) large enough. In the following it will be convenient to write the difference \(m_q\left( \rho '_i\right) \!-\!m_q(\rho _i)\) as

$$\begin{aligned}&\left( \frac{1}{\int _{K_q\left( \rho '_i\right) }\textrm{d}\rho '_i}\! -\! \frac{1}{\int _{K_q\left( \rho _i\right) }\textrm{d}\rho _i}\right) \int _{K_q(\rho '_i)} t\, \textrm{d}\rho '_i(t)\nonumber \\&\quad +\frac{1}{\int _{K_q\left( \rho _i\right) }\textrm{d}\rho _i}\left( \int _{K_q(\rho '_i)}t\, \textrm{d}\rho '_i(t)\!- \! \!\int _{K_q(\rho _i)} t\, \textrm{d}\rho _i(t)\! \right) . \end{aligned}$$

Making use of \(\int _{K_q\left( \rho _i\right) }\textrm{d}\rho _i\ge 2qc\), see Eq. (3.15), and \(K_q(\rho '_i)\subset [-R,R]\) for all \(\rho \in \Omega \), we can estimate the individual terms in Eq. (3.16) by

$$\begin{aligned}&\left| \left( \frac{1}{\int _{K_q\left( \rho '_i\right) }\textrm{d}\rho '_i}\! -\! \frac{1}{\int _{K_q\left( \rho _i\right) }\textrm{d}\rho _i}\right) \int _{K_q(\rho '_i)} t\, \textrm{d}\rho '_i(t)\right| \le R\frac{\left| \int _{K_q\left( \rho '_i\right) }\textrm{d}\rho '_i-\int _{K_q\left( \rho _i\right) }\textrm{d}\rho _i\right| }{2qc},\\&\left| \frac{1}{\int _{K_q\left( \rho _i\right) }\textrm{d}\rho _i}\left( \int _{K_q(\rho '_i)}t\, \textrm{d}\rho '_i(t)\!- \! \!\int _{K_q(\rho _i)} t\, \textrm{d}\rho _i(t)\! \right) \right| \le \frac{\left| \int _{K_q(\rho '_i)}t\, \textrm{d}\rho '_i(t)\!- \! \!\int _{K_q(\rho _i)} t\, \textrm{d}\rho _i(t)\right| }{2qc}. \end{aligned}$$

Note that \(K_q(\rho _i)\) is contained in \([-R,R]\) as well and consequently t is bounded by R on the subset \(K_q(\rho _i)\cup K_q(\rho '_i)\). In order to verify the statement of the Lemma, it is therefore sufficient to prove that \(\left| \int _{K_q(\rho '_i)}f(t)\, \textrm{d}\rho '_i(t)\!- \! \!\int _{K_q(\rho _i)} f(t)\, \textrm{d}\rho _i(t)\right| \lesssim \alpha ^{-2}\Vert f\Vert _\infty \) for an arbitrary measurable and bounded \(f:\mathbb {R}\rightarrow \mathbb {R}\). We estimate

$$\begin{aligned} \left| \int _{K_q(\rho '_i)}f(t)\, \textrm{d}\rho '_i(t)\!- \! \!\int _{K_q(\rho _i)} f(t)\, \textrm{d}\rho _i(t)\right|&\le \left| \int _{K_q(\rho '_i)}f(t)\, \textrm{d}\rho '_i(t)\!- \! \!\int _{K_q(\rho '_i)} f(t)\, \textrm{d}\rho _i(t)\right| \\&\quad +\left| \int _{K_q(\rho '_i)}f(t)\, \textrm{d}\rho _i(t)\!- \! \!\int _{K_q(\rho _i)} f(t)\, \textrm{d}\rho _i(t)\right| \\&\quad \le \Vert f\Vert _\infty \left( \Vert \rho '_i-\rho _i\Vert _{\textrm{TV}}+\int _{K_q(\rho '_i)\Delta K_q(\rho _i)}\textrm{d}\rho _i\right) , \end{aligned}$$

where \(A\Delta B:=\left( A\cup B\right) {\setminus } \left( A\cap B\right) \) is the symmetric difference. Note that \(\Vert \rho '_i-\rho _i\Vert _{\textrm{TV}}=\alpha ^{-2}\). Furthermore we can estimate the expression \(\int _{K_q(\rho '_i)\Delta K_q(\rho _i)} \textrm{d}\rho _i\) by

$$\begin{aligned} \left| \int _{-\infty }^{x^{\frac{1}{2}-q}(\rho '_i)}\textrm{d}\rho _i-\int _{-\infty }^{x^{\frac{1}{2}-q}(\rho _i)}\textrm{d}\rho _i\right| +\left| \int _{-\infty }^{x^{\frac{1}{2}+q}(\rho '_i)}\textrm{d}\rho _i-\int _{-\infty }^{x^{\frac{1}{2}+q}(\rho _i)}\textrm{d}\rho _i\right| . \end{aligned}$$

Since the distributions \(\rho _i\) and \(\rho _i'\) satisfy the assumptions of Lemma 3.7 with \(\epsilon :=2\alpha ^{-2}\), we conclude that every term in the sum above is bounded by \(2\Vert \rho '-\rho \Vert _{\textrm{TV}}+\epsilon =4\alpha ^{-2}\).

\(\square \)

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.

Lemma 3.11

Given a constant \(C>0\), there exists a constant \(T>0\), such that

$$\begin{aligned} \Big \langle \Psi \Big |W_{\varphi ^\textrm{Pek}}^{-1}\, \mathcal {N}W_{\varphi ^\textrm{Pek}}\Big | \Psi \Big \rangle \le T\left( \alpha ^{-\frac{2}{29}}+q+\epsilon \right) \end{aligned}$$

for all states \(\Psi \) satisfying \(\langle \Psi |\mathbb {H}_K|\Psi \rangle \le e^\textrm{Pek}+\alpha ^{-\frac{4}{29}}\) with \(K\ge \alpha ^{\frac{8}{29}}\) and \(\widehat{\mathbbm {1}_{\Omega ^*}}\Psi =\Psi \), where \(\Omega ^*\) is the set of all \(\rho \) satisfying \(\int \textrm{d}\rho \le C\) and \(|m_q(\rho )|\le \epsilon \) with \(q,\epsilon >0\).


Let us begin by defining the functions

$$\begin{aligned} P_i^\epsilon (\rho ):=\left( \frac{1}{2}\int \textrm{d}\rho \right) ^2-\int _{x_i\le \epsilon }\textrm{d}\rho (x)\int _{y_i\ge -\epsilon }\textrm{d}\rho (y). \end{aligned}$$

Observe that \(|m_{q}(\rho )|\le \epsilon \) implies \(-\epsilon \le x^{\frac{1}{2}+q}(\rho _i)\) and \(x^{\frac{1}{2}-q}(\rho _i)\le \epsilon \) for all such \(\rho \) which additionally satisfy \(\rho \ne 0\), see Definition 3.8. Therefore \(P^\epsilon _i(\rho )\le \left( \int \textrm{d}\rho \right) ^2 \big (\frac{1}{4}-\left( \frac{1}{2}-q\right) ^2\big )\lesssim q\) for all \(\rho \in {\Omega ^*}\), and consequently the measure \(\mu \) from Theorem 3.2 corresponding to the state \(\Psi \) satisfies \(\int P^\epsilon _i\left( \left| \varphi ^\textrm{Pek}_x\right| ^2\right) \, \textrm{d}\mu (x)\le \big \langle \Psi \big |\widehat{P}^\epsilon _i\, \big |\Psi \big \rangle +D \alpha ^{-\frac{2}{29}}\lesssim q+\alpha ^{-\frac{2}{29}}\) for a suitable \(D>0\), where we have used Eq. (3.2) in the first inequality. Furthermore we know that \(\Vert \varphi ^\textrm{Pek}_x-\varphi ^\textrm{Pek}\Vert ^2\lesssim \sum _{i=1}^3 P^\epsilon _i\left( \left| \varphi ^\textrm{Pek}_x\right| ^2\right) +\epsilon \) by Lemma A.3, hence

$$\begin{aligned} \int \Vert \varphi ^\textrm{Pek}_x\! -\! \varphi ^\textrm{Pek}\Vert ^2\, \textrm{d}\mu (x)\lesssim \sum _{i=1}^3\int P^\epsilon _i\left( \left| \varphi ^\textrm{Pek}_x\right| ^2\right) \, \textrm{d}\mu (x)\! +\! \epsilon \lesssim q\! + \! \alpha ^{-\frac{2}{29}}\! +\! \epsilon . \end{aligned}$$

Therefore Eq. (3.3) immediately concludes the proof of Eq. (3.18). \(\quad \square \)

Lemma 3.12

Given \(0<\sigma \le \frac{1}{4}\), let \(\Lambda \) and L be as in Theorem 2.5. Then there exist states \(\Psi _\alpha '''\) satisfying \(\langle \Psi '''_\alpha |\mathbb {H}_\Lambda |\Psi '''_\alpha \rangle -E_\alpha \lesssim \alpha ^{-2(1+\sigma )}\), \(\textrm{supp}\left( \Psi _\alpha '''\right) \subset B_{4L}(0)\) and

$$\begin{aligned} \Big \langle \Psi '''_\alpha \Big |W_{\varphi ^\textrm{Pek}}^{-1}\, \mathcal {N}W_{\varphi ^\textrm{Pek}}\Big | \Psi '''_\alpha \Big \rangle \lesssim \alpha ^{-\frac{2}{29}}, \end{aligned}$$

where \(W_{\varphi ^\textrm{Pek}}\) is the Weyl operator corresponding to the Pekar minimizer \(\varphi ^\textrm{Pek}\).


It is clearly sufficient to consider only the case \(\alpha \ge \alpha _0\) for a suitable (large) \(\alpha _0\), since we can always re-define \(\Psi '''_\alpha :=\Psi \) for \(\alpha <\alpha _0\) where \(\Psi \) is an arbitrary state satisfying \(\textrm{supp}\left( \Psi \right) \subset B_{4L}(0)\). In the following let us use the concrete choice \(\Psi _\alpha ^\bullet :=\Psi _\alpha ^\diamond \) for the sequence in Eq. (3.5), where \(\Psi _\alpha ^\diamond \) is defined in in Theorem 2.5, which is a valid choice since it satisfies the assumptions \(\textrm{supp}\left( \Psi _\alpha ^\diamond \right) \subset B_L(0)\) and \(\widetilde{E}_\alpha -E_\alpha \lesssim \alpha ^{-2(1+\sigma )}\le \alpha ^{-\frac{4}{29}}\). Furthermore let \(\{\chi _z:z\in \mathbb {Z}^3\}\) be a smooth (quadratic) partition of unity on \(\mathbb {R}^3\), i.e. \(0\le \chi _z\le 1\) and \(\sum _{z\in \mathbb {Z}^3}\chi _z^2=1\), with \(\chi _z(x)=\chi _0(x-z)\) and \(\textrm{supp} \left( \chi _0\right) \subset B_1(0)\). Then we define for \(z\in \mathbb {Z}^3\) and \(u,v\ge \frac{2}{29}\) with \(u+v\le \frac{1}{4}\) the function \(F_z(\rho ):=\chi _z\big (\alpha ^{u}\, m_{\alpha ^{-{v}}}(\rho )\big )\), as well as the states

$$\begin{aligned} \Psi _{\alpha ,z}:=Z_{\alpha ,z}^{-1}\widehat{F}_z \Psi ''_\alpha \end{aligned}$$

with \(Z_{\alpha ,z}:=\Vert \widehat{F}_z \Psi ''_\alpha \Vert \) and \(\Psi ''_\alpha \) as in Lemma 3.5 for \(\epsilon =0\) and \(0<\delta <\frac{c^2}{2}\), where \(c:=c_-\) is as in Lemma 3.4. Applying Lemma 3.3 with respect to \(\mathcal {P}:=\{F_z:z\in \mathbb {Z}^3\}\), where the functions \(F_z\) are defined above Eq. (3.19) and \(\Omega \) is defined as the set of all \(\rho \in \Omega _\textrm{reg}\) satisfying \(c_-\le \int \textrm{d}\rho \le c_+\) and \(\underset{|x-y|\ge R}{\int \int } \textrm{d}\rho (x)\textrm{d}\rho (y)\le \delta \), yields

$$\begin{aligned} \sum _{z\in \mathbb {Z}^3}Z^2_{\alpha ,z}\langle \Psi _{\alpha ,z}|\mathbb {H}_\Lambda |\Psi _{\alpha ,z}\rangle \le \langle \Psi ''_\alpha |\mathbb {H}_\Lambda |\Psi ''_\alpha \rangle +c\alpha ^{-\frac{7}{2}}V_\Omega \left( \mathcal {P}\right) \sqrt{c_+ +\alpha ^{-2}}, \end{aligned}$$

where we used Lemma 3.6, \(\Lambda \le \alpha \) and \(\widehat{\mathbbm {1}_\Omega }\Psi ''_\alpha =\Psi ''_\alpha \) by the definition of \(\Psi ''_\alpha \) in Eq. (3.10). Since the support of \(\chi _z\) only overlaps with the support of finitely many other \(\chi _{z'}\), we obtain for \(v>0\) and \(\alpha \) large enough

$$\begin{aligned} V_\Omega \left( \mathcal {P}\right)&\lesssim \alpha ^{4} \textrm{sup}_{\rho \in \Omega ,y\in \mathbb {R}^3}\, \textrm{sup}_{z\in \mathbb {Z}^3}\left| \chi _z\big (\alpha ^{u} m_{\alpha ^{-{v}}}(\rho +\alpha ^{-2}\delta _y)\big )-\chi _z\big (\alpha ^{u} m_{\alpha ^{-{v}}}(\rho )\big )\right| ^2\\&\lesssim \alpha ^{2u+4}\textrm{sup}_{\rho \in \Omega ,y\in \mathbb {R}^3}\left| m_{\alpha ^{-{v}}}(\rho +\alpha ^{-2}\delta _y) -m_{\alpha ^{-{v}}}(\rho )\right| ^2\lesssim \alpha ^{2(u+v)}, \end{aligned}$$

where we have used \(\textrm{sup}_{z\in \mathbb {Z}^3}\left| \chi _z\left( y\right) -\chi _z\left( x\right) \right| \le \big \Vert \nabla \chi _0\big \Vert _\infty |y-x|\) in the first inequality and Lemma 3.10 in the second one. Combining this with Eq. (3.20) and the fact that \(u+v\le \frac{1}{4 }\) yields

$$\begin{aligned} \sum _{z\in \mathbb {Z}^3}Z^2_{\alpha ,z}\langle \Psi _{\alpha ,z}|\mathbb {H}_\Lambda |\Psi _{\alpha ,z}\rangle -\langle \Psi ''_\alpha |\mathbb {H}_\Lambda |\Psi ''_\alpha \rangle \lesssim \alpha ^{-3}. \end{aligned}$$

Since \(\sum _{z\in \mathbb {Z}^3}Z_{\alpha ,z}^2=1\), this in particular means that there exists a \(z_\alpha \in \mathbb {Z}^3\) such that \(\langle \Psi _{\alpha ,z_\alpha }|\mathbb {H}_\Lambda |\Psi _{\alpha ,z_\alpha }\rangle - E_\alpha \lesssim \alpha ^{-2(1+\sigma )}\), and by the translation invariance of \(\mathbb {H}_\Lambda \) we obtain \(\langle \Psi '''_{\alpha }|\mathbb {H}_\Lambda |\Psi '''_{\alpha }\rangle - E_\alpha \lesssim \alpha ^{-2(1+\sigma )}\) where \(\Psi _\alpha '''=\mathcal {T}_{-\alpha ^{-u}z_\alpha }\Psi _{\alpha ,z_\alpha }\). Using the fact that \(\mathbbm {1}_{\Omega ^*}\Psi _\alpha '''=\Psi _\alpha '''\), where \({\Omega ^*}\) is the set of all \(\rho \) satisfying \(\int \textrm{d}\rho \le c_+\) and \(|m_{\alpha ^{-v}}(\rho )|\le \alpha ^{-u}\), together with Lemma 3.11, immediately concludes the proof of Eq. (3.18).

Finally let us verify that \(\textrm{supp}\left( \Psi '''_\alpha \right) \subset B_{4L}(0)\). By definition of \(\Psi _\alpha '''=\mathcal {T}_{-\alpha ^{-u}z_\alpha }\Psi _{\alpha ,z_\alpha }\), and the fact that \(\textrm{supp}\left( \Psi _{\alpha ,z_\alpha }\right) \subset B_{L}(0)\), it is clear that \(\textrm{supp}\left( \Psi '''_\alpha \right) \subset B_{L}(-w_\alpha )\) with \(w_\alpha :=\alpha ^{-u}z_\alpha \). In the following we show that \(|w_\alpha |\le 3L\) by contradiction for \(\alpha \) large enough, and therefore \(\textrm{supp}\left( \Psi '''_\alpha \right) \subset B_{L+|w_\alpha |}(0)\subset B_{4\,L}(0)\). Assuming \(|w_\alpha |>3\,L\), we obtain \(\textrm{supp}\left( \Psi '''_\alpha \right) \subset \mathbb {R}^3{\setminus } B_{2\,L}(0)\) and Corollary B.7 consequently yields \(\langle \Psi _\alpha '''|\mathbb {H}_\Lambda |\Psi _\alpha '''\rangle \ge E_\alpha +\langle \Psi _\alpha '''|\mathcal {N}_{B_L(0)}|\Psi _\alpha '''\rangle -\sqrt{\frac{D}{L}}\), where \(\mathcal {N}_{B_L(0)}\) denotes the number operator in the ball \(B_L(0)\) (as defined in Cor. B.7). Defining \(\varphi _L(x):=\chi \left( |x|\le L\right) \varphi ^\textrm{Pek}(x)\), we further have

$$\begin{aligned} \langle \Psi '''_\alpha |\mathcal {N}_{B_L(0)}|\Psi _\alpha '''\rangle&=\Big \langle \Psi _\alpha '''\Big |W_{\varphi ^\textrm{Pek}}^{-1}\left( \mathcal {N}_{B_L(0)}+a(\varphi _L)+a^\dagger (\varphi _L)+\Vert \varphi _L\Vert ^2\right) W_{\varphi ^\textrm{Pek}}\Big | \Psi '''_\alpha \Big \rangle \\&\ge -\Big \langle \Psi '''_\alpha \Big |W_{\varphi ^\textrm{Pek}}^{-1}\mathcal {N} W_{\varphi ^\textrm{Pek}}\Big | \Psi '''_\alpha \Big \rangle +\frac{1}{2}\Vert \varphi _L\Vert ^2\ge -D'\alpha ^{-\frac{2}{29}}+\frac{1}{2}\Vert \varphi _L\Vert ^2 \end{aligned}$$

for a suitable constant \(D'\), where we have used the operator inequality \(\mathcal {N}_{B_L(0)}+a(\varphi _L)+a^\dagger (\varphi _L)+\Vert \varphi _L\Vert ^2\ge -\mathcal {N}+\frac{1}{2}\Vert \varphi _L\Vert ^2\) as well as Eq. (3.18). Therefore we obtain

$$\begin{aligned} \langle \Psi _\alpha '''|\mathbb {H}_\Lambda |\Psi _\alpha '''\rangle -E_\alpha \ge \frac{1}{2}\Vert \varphi _L\Vert ^2-D'\alpha ^{-\frac{2}{29}}-\sqrt{\frac{D}{L}}\underset{\alpha \rightarrow \infty }{\longrightarrow }\frac{1}{2}\Vert \varphi ^\textrm{Pek}\Vert ^2>0, \end{aligned}$$

where we have used that \(L=\alpha ^{1+\sigma }\underset{\alpha \rightarrow \infty }{\longrightarrow }\infty \). This, however, is a contradiction to \(\langle \Psi '''_\alpha |\mathbb {H}_\Lambda |\Psi '''_\alpha \rangle -E_\alpha \lesssim \alpha ^{-2(1+\sigma )}\). \(\quad \square \)

Following the method in [16], we are going to lift the weak condensation derived in Lemma 3.12 to a strong one in the subsequent Theorem 3.13, which represents the main result of this section.

Theorem 3.13

Given \(0<\sigma \le \frac{1}{4}\) and \(h<\frac{2}{29}\), let \(\Lambda \) and L be as in Theorem 2.5. Then there exist states \(\Psi _\alpha \) with \(\langle \Psi _\alpha |\mathbb {H}_\Lambda |\Psi _\alpha \rangle - E_\alpha \lesssim \alpha ^{-2(1+\sigma )}\) and \(\textrm{supp}\left( \Psi _\alpha \right) \subset B_{4L}(0)\), satisfying

$$\begin{aligned} \chi \left( W_{\varphi ^\textrm{Pek}}^{-1}\, \mathcal {N}W_{\varphi ^\textrm{Pek}}\le \alpha ^{-h}\right) \Psi _\alpha =\Psi _\alpha \end{aligned}$$

for large enough \(\alpha \).


Using the states \(\Psi _\alpha '''\) from Lemma 3.12, we define for \(0<\epsilon <\frac{1}{2}\)

$$\begin{aligned} \Psi _\alpha :=Z_\alpha ^{-1}\chi ^{\epsilon }\left( \alpha ^{h}W_{\varphi ^\textrm{Pek}}^{-1}\mathcal {N}W_{\varphi ^\textrm{Pek}} \le \frac{1}{2}\right) \Psi '''_\alpha \end{aligned}$$

where \(Z_\alpha \) is a normalizing constant. Clearly the states \(\Psi _\alpha \) satisfy the strong condensation property \(\chi \left( W_{\varphi ^\textrm{Pek}}^{-1}\mathcal {N}W_{\varphi ^\textrm{Pek}} \le \alpha ^{-h}\right) \Psi _\alpha =\Psi _\alpha \). In order to control the energy cost of the localization with respect to the operator \(W_{\varphi ^\textrm{Pek}}^{-1}\mathcal {N}W_{\varphi ^\textrm{Pek}}\), note that the partition \(\mathcal {P}':=\{F',G'\}\) with \(F'(\rho ):=\chi ^{\epsilon }\left( \alpha ^{h}\int \textrm{d}\rho \le \frac{1}{2}\right) \) and \(G'(\rho ):=\chi ^{\epsilon }\left( \frac{1}{2}\le \alpha ^{h}\int \textrm{d}\rho \right) \) satisfies

$$\begin{aligned} \kappa :=V_{\mathcal {M}\left( \mathbb {R}^3\right) }\left( \mathcal {P}'\right) \lesssim \alpha ^{4} \textrm{sup}_{\rho ,x\in \mathbb {R}^3}\left| \alpha ^h\int \textrm{d}\! \left( \rho +\alpha ^{-2}\delta _x\right) -\alpha ^h\int \textrm{d}\rho \right| ^2=\alpha ^{2h}, \end{aligned}$$

where we used \(\left| \chi ^{\epsilon }\left( y \le \frac{1}{2}\right) -\chi ^{\epsilon }\left( x \le \frac{1}{2}\right) \right| \le \big \Vert \frac{\textrm{d}}{\textrm{d}x} \chi ^{\epsilon }\Big (\cdot \le \frac{1}{2}\Big )\big \Vert _\infty |y-x|\) and the corresponding estimate for \(\chi ^{\epsilon }\left( \frac{1}{2}\le \cdot \right) \). Therefore we obtain by Lemma 3.3, using \(\Lambda \le \alpha \),

$$\begin{aligned}&Z_\alpha ^{2}\! \langle \Psi _\alpha |\mathbb {H}_\Lambda |\Psi _\alpha \rangle \! +\! (1\! -\! Z_\alpha ^2)\! \langle \widetilde{\Psi }_\alpha |\mathbb {H}_\Lambda |\widetilde{\Psi }_\alpha \rangle \! \le \! \langle \Psi '''_\alpha |\mathbb {H}_\Lambda |\Psi '''_\alpha \rangle \! +\! c'\, \alpha ^{-\frac{7}{2}} \kappa \Big \langle \Psi ''' \Big | \sqrt{\mathcal {N}\! +\! 1}\Big | \Psi ''' \Big \rangle \nonumber \\&\quad \le E_\alpha +O_{\alpha \rightarrow \infty }\left( \alpha ^{-2(1+\sigma )}\right) +O_{\alpha \rightarrow \infty }\left( \alpha ^{2h-\frac{7}{2}}\right) =E_\alpha +O_{\alpha \rightarrow \infty }\left( \alpha ^{-2(1+\sigma )}\right) , \end{aligned}$$

with \(\widetilde{\Psi }_\alpha :=\sqrt{1-Z_\alpha ^2}^{-1}\chi ^{\epsilon }\left( \frac{1}{2}\le \alpha ^{h}W_{\varphi ^\textrm{Pek}}^{-1}\mathcal {N}W_{\varphi ^\textrm{Pek}} \right) \Psi '''_\alpha \). Making use of the trivial lower bound \(E_\alpha \le \langle \widetilde{\Psi }_\alpha |\mathbb {H}_\Lambda |\widetilde{\Psi }_\alpha \rangle \), Eq. (3.23) implies \(\langle \Psi _\alpha |\mathbb {H}_\Lambda |\Psi _\alpha \rangle \le E_\alpha +Z_\alpha ^{-2}O_{\alpha \rightarrow \infty }\left( \alpha ^{-2(1+\sigma )}\right) \), which concludes the proof since

$$\begin{aligned} 1-Z_\alpha ^2&=\Big \langle \Psi '''_\alpha \Big |\chi ^{\epsilon }\left( \frac{1}{2}\le \alpha ^{h}W_{\varphi ^\textrm{Pek}}^{-1}\mathcal {N}W_{\varphi ^\textrm{Pek}} \right) ^2\Big | \Psi '''_\alpha \Big \rangle \\&\le \frac{1}{\frac{1}{2}-\epsilon }\alpha ^{h}\Big \langle \Psi '''_\alpha \Big | W_{\varphi ^\textrm{Pek}}^{-1}\mathcal {N}W_{\varphi ^\textrm{Pek}} \Big |\Psi '''_\alpha \Big \rangle \lesssim \frac{1}{\frac{1}{2}-\epsilon }\alpha ^{h}\alpha ^{-\frac{2}{29}}\underset{\alpha \rightarrow \infty }{\longrightarrow }0. \end{aligned}$$

\(\square \)

4 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 \(\alpha ^2\)), 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 \(\alpha ^2\), contained in the spectral subspace \(\left| a(f)+a^\dagger (f)\right| \le \epsilon \), 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 \(0<\sigma <\frac{1}{4}\) let us define the momentum cut-off \(\Lambda :=\alpha ^{\frac{4}{5}(1+\sigma )}\) and the discretization parameter of the momentum space \(\ell :=\alpha ^{-4(1+\sigma )}\), and the associated projection

$$\begin{aligned} \Pi :=\Pi ^0_{\Lambda ,\ell }, \end{aligned}$$

see Definition 2.1, and let us identify \(\mathcal {F}\left( \Pi L^2\! \left( \mathbb {R}^3\right) \right) \) with \(L^2\! \left( \mathbb {R}^{N}\right) \) using the representation of real functions \(\varphi =\sum _{n=1}^{N} \lambda _n \varphi _n \in \Pi L^2\! \left( \mathbb {R}^3\right) \) by points \(\lambda =(\lambda _1,\dots ,\lambda _{{N}})\in \mathbb {R}^{N}\), where \(N:=\textrm{dim}\Pi L^2\! \left( \mathbb {R}^3\right) \) and \(\{\varphi _1,\dots ,\varphi _{N}\}\) is a real orthonormal basis of \(\Pi L^2\! \left( \mathbb {R}^3\right) \). We choose this identification such that the annihilation operators \(a_n:=a\left( \varphi _n\right) \) read

$$\begin{aligned} a_n=\lambda _n+\frac{1}{2\alpha ^2}\partial _{\lambda _n}, \end{aligned}$$

where \(\lambda _n\) is the multiplication operator by the function \(\lambda \mapsto \lambda _n\) on \(L^2\! \left( \mathbb {R}^{N}\right) \). From the construction one readily checks that \(N \lesssim (\Lambda / \ell )^3 \le \alpha ^p\) for suitable \(p>0\).

In the following we will verify a large deviation principle for the density function \(\rho (\lambda ):=\gamma (\lambda ,\lambda )\) corresponding to a density matrix \(\gamma \) on \(\mathcal {F}\left( \Pi L^2\! \left( \mathbb {R}^3\right) \right) \) that satisfies the strong condensation condition

$$\begin{aligned} \chi \left( \sum _{n=1}^{N} a_n^\dagger a_n\le \alpha ^{-h}\right) \gamma =\gamma \end{aligned}$$

for some condensation rate \(h>0\). This result is comparable to [5, Lemma C.2]. For this purpose, we define a convenient norm \(|\cdot |_\diamond \) on \(\mathbb {R}^{N}\) in the subsequent Definition.

Definition 4.1

Let \(|\lambda |:=\sqrt{\sum _{n=1}^{{N}} \lambda _n^2}\) denote the standard norm on \(\mathbb {R}^{N}\) and let us define the norm \(|\cdot |_\diamond \) on \(\mathbb {R}^{N}\), using the identification \(\varphi =\sum _{n=1}^{{N}} \lambda _n \varphi _n\), as

$$\begin{aligned} |\lambda |_\diamond :=2\sup _{x\in \mathbb {R}^3}\sqrt{\int _{B_1(x)}\left| \left( \left( -\Delta \right) ^{-\frac{1}{2}}\varphi \right) (y)\right| ^2\textrm{d}y}. \end{aligned}$$

The norm \(|\cdot |_\diamond \) will again appear naturally in Sect. 5 where we investigate properties of the Pekar functional \(\mathcal {F}^\textrm{Pek}\) (see Eq. (5.2) and the subsequent comment). In the following Proposition 4.2 we establish a large deviation like estimate on the \(\epsilon \)-tail of the probability distribution \(\rho \) in the limit of \(\alpha \rightarrow \infty \), where \(\epsilon \) is even allowed to go to zero simultaneously as \(\alpha \) goes to infinity, as long as \(\epsilon \ge D\alpha ^{-s}\) for a h and \(\sigma \) dependent constant s.

Proposition 4.2

Let \(0<s<\min \big \{\frac{h}{2},\frac{1}{5}(1-4\sigma )\big \}\) and \(D>0\). Then there exist constants \(\beta ,\alpha _0>0\), such that we have for all \(\alpha \ge \alpha _0\), \(\epsilon \ge D\alpha ^{-s}\) and \(\gamma \) satisfying Eq. (4.3)

$$\begin{aligned} \int _{|\lambda |_\diamond \ge \epsilon }\left( 1+|\lambda |^2\right) \rho (\lambda )\textrm{d}\lambda \le e^{-\beta \epsilon ^2\alpha ^2}, \end{aligned}$$

where \(\rho (\lambda ):=\gamma (\lambda ,\lambda )\) is the density function corresponding to the state \(\gamma \). Furthermore for all \(\zeta \in \mathbb {R}^{N}\) and \(\beta <\frac{1}{|\zeta |^2}\), there exists a constant \(\alpha (\beta ,|\zeta |)\) such that

$$\begin{aligned} \int _{|\langle \zeta |\lambda \rangle |\ge \epsilon }\left( 1+|\lambda |^2\right) \rho (\lambda )\textrm{d}\lambda \le e^{-\beta \epsilon ^2\alpha ^2} \end{aligned}$$

for all \(\alpha \ge \alpha (\beta ,|\zeta |)\) and \(\epsilon \ge D\alpha ^{-s}\).

The restriction to the finite dimensional space \(\Pi L^2\! \left( \mathbb {R}^3\right) \) will be essential in the proof of Proposition 4.2, to be precise we will make use of the fact that \(N\lesssim \alpha ^p\) for a suitable \(p>0\), which in particular implies that \(N\lesssim e^{\alpha ^t}\), uniformly in \(\alpha \), for any \(t>0\). Before we prove Proposition 4.2, we first need auxiliary results concerning the \(|\cdot |_\diamond \) norm.

Definition 4.3

For \(x\in \mathbb {R}^3\) and \(r>0\), let us define \(T_x\lambda :=-2\chi \left( |\cdot -x|\le 1\right) \left( -\Delta \right) ^{-\frac{1}{2}}\varphi \) and \(T_{\ge r} \lambda :=-2\chi \left( |\cdot |\ge r\right) \left( -\Delta \right) ^{-\frac{1}{2}}\varphi \) with the above identification \(\varphi =\sum _{n=1}^{{N}} \lambda _n \varphi _n\). Furthermore let us define the operators \(A_x:=\sqrt{T_x^\dagger T_x}\) and \(A_{\ge r}:=\sqrt{T_{\ge r}^\dagger T_{\ge r}}\), as well as the constant \(\beta _0:=\inf _{x\in \mathbb {R}^3}\Vert A_x\Vert ^{-2}\).

Using the operators \(A_x\) we can write \(|\lambda |_{\diamond }=\sup _{x\in \mathbb {R}^3}|A_x \lambda |\), which is bounded by

$$\begin{aligned} |\lambda |_{\diamond }\le 65\max \bigg \{\sup _{z\in \mathbb {Z}^3:|z|\le r+1}|A_z \lambda |,\, |A_{\ge r} \lambda |\bigg \} \end{aligned}$$

for any \(r>0\). In order to see this, note that for any \(y\in \mathbb {R}^3\) there exists a \(z\in \mathbb {Z}^3\) with \(|y-z|< 1\). In case \(y\in B_r(0)\cap B_1(x)\), where \(x\in \mathbb {R}^3\), we see that z satisfies \(|z|\le r+1\) and \(|x-z|<2\). Denoting the set of such z by \(M(x,r)\subset \mathbb {Z}^3\), we obtain \(B_1(x)\subset \bigcup _{z\in M(x,r)}B_1(z)\cup \left( \mathbb {R}^3{\setminus } B_r(0)\right) \). Consequently

$$\begin{aligned} |\lambda |_{\diamond }&\le \sup _x \sum _{z\in M(x,r)}|A_z \lambda |+|A_{\ge r} \lambda |\\&\le \sup _x\Big ( |M(x,r)|+1\Big )\max _x\Big \{\sup _{z\in M(x,r)}|A_z \lambda |,\, |A_{\ge r} \lambda |\Big \}. \end{aligned}$$

This concludes the proof of Eq. (4.7), since there are at most 64 elements \(z\in \mathbb {Z}^3\) satisfying \(|x-z|<2\).

Lemma 4.4

The constant \(\beta _0\) from Definition 4.3 is positive, uniformly in \(\alpha \), and \(\Vert A_x\Vert _{\textrm{HS}}\lesssim \Lambda \) uniformly in \(x\in \mathbb {R}^3\), where \(\Lambda \) is defined above Eq. (4.1). Furthermore there exists a constant \(v>0\) such that \(\Vert A_{\ge r}\Vert _{\textrm{HS}}\lesssim \frac{\alpha ^v}{\sqrt{r}}\) for all \(\alpha \ge 1\) and \(r>0\).


Note that the space \(\Pi L^2\! \left( \mathbb {R}^3\right) \) is contained in the spectral subspace \(-\Delta \le \Lambda ^2\), hence \(\Pi \le \left( 1+ \Lambda ^2\right) \left( 1-\Delta \right) ^{-1}\), and therefore

$$\begin{aligned} \Vert A_x\Vert ^2_{\textrm{HS}}&\! =\! 4\! \left\| \chi \left( |\cdot \! -x|\le 1\right) \left( \! -\! \Delta \right) ^{-\frac{1}{2}}\Pi \right\| ^2_{\textrm{HS}}\! \\&\le \! 4\! \left( \! 1\! +\! \Lambda ^2\right) \! \left\| \chi \left( |\cdot \! -x|\le 1\right) \left( -\Delta \right) ^{-\frac{1}{2}}\left( 1-\Delta \right) ^{-\frac{1}{2}}\right\| ^2_{\textrm{HS}}\\&=4\left( 1+ \Lambda ^2\right) \left\| \chi \left( |\cdot |\le 1\right) \left( -\Delta \right) ^{-\frac{1}{2}}\left( 1-\Delta \right) ^{-\frac{1}{2}}\right\| ^2_{\textrm{HS}}. \end{aligned}$$

Applying Eq. (B.5) with \(\psi =\chi \left( |.|\le 1\right) \) yields that \(\chi \left( |\cdot |\le 1\right) \left( -\Delta \right) ^{-\frac{1}{2}}\left( 1-\Delta \right) ^{-\frac{1}{2}}\) is Hilbert-Schmidt, hence \(\Vert A_x\Vert _{\textrm{HS}}\lesssim \Lambda \). In order to prove the uniform lower bound \(\beta _0>0\), it is enough to verify the boundedness of \(\chi \left( |.|\le 1\right) f(-\Delta )\), where \(f(t):=\frac{\chi (|t|\le 1)}{\sqrt{t}}\). An explicit computation in Fourier space yields for \(\varphi \in L^2 \! \left( \mathbb {R}^3\right) \)

$$\begin{aligned}&\langle \varphi |f(-\Delta )\chi \left( |.|\le 1\right) f(-\Delta )|\varphi \rangle =\int _{|k|\le 1}\int _{|k|\le 1}\frac{\widehat{\chi \left( |\cdot |\le 1\right) }(k-k')\overline{\widehat{\varphi }(k)}\widehat{\varphi }(k')}{|k|\, |k'|}\, \textrm{dkdk}\\&\quad \le \left\| \widehat{\chi \left( |\cdot |\le 1\right) }\right\| _\infty \left| \int _{|k|\le 1}\frac{|\widehat{\varphi }(k)|}{|k|}\, \textrm{dk}\right| ^2\lesssim \Vert \varphi \Vert ^2. \end{aligned}$$

Finally we are going to verify \(\Vert A_{\ge r}\Vert _{\textrm{HS}}\lesssim \frac{\alpha ^v}{\sqrt{r}}\), using that

$$\begin{aligned} \Vert A_{\ge r}\Vert _{\textrm{HS}}= 2 \sqrt{\sum _{n=1}^{{N}} \left\| \chi \left( |\cdot |\ge r\right) \left( -\Delta \right) ^{-\frac{1}{2}}\varphi _n\right\| ^2}\lesssim \sqrt{{N}}\frac{\alpha ^v}{\sqrt{r}} \end{aligned}$$

for a suitable constant \(v>0\) by Corollary B.2, where N is the dimension of \(\Pi L^2\! \left( \mathbb {R}^3\right) \). This concludes the proof, since \({N}\lesssim \alpha ^p\) for some \(p>0\). \(\quad \square \)

Proof of Proposition 4.2

Making use of Eq. (4.7) and defining \(\epsilon _*:=\frac{\epsilon }{65}\), we obtain

$$\begin{aligned} \int _{|\lambda |_\diamond \ge \epsilon }\! \! \left( 1\! +\! |\lambda |^2\right) \! \rho (\lambda )\textrm{d}\lambda \! \le \! \!\! \! \sum _{|z|\le r+1}\! \int _{|A_z\lambda |\ge \epsilon _*}\! \! \! \left( 1\! +\! |\lambda |^2\right) \! \rho (\lambda )\textrm{d}\lambda \,+\!\int _{|A_{\ge r}\lambda |\ge \epsilon _*}\! \! \! \! \left( 1\! +\! |\lambda |^2\right) \! \rho (\lambda )\textrm{d}\lambda , \end{aligned}$$

where the sum runs over \(z\in \mathbb {Z}^3\) with \(|z|\le r+1\). In the following we are going to verify that every contribution of the form \(\int _{|A_x\lambda |\ge \epsilon _*}\! \! \! \left( 1\! +\! |\lambda |^2\right) \! \rho (\lambda )\textrm{d}\lambda \) is exponentially small uniformly in \(x\in \mathbb {R}^3\). As a consequence of Eq. (4.3), we have for \(t\ge 0\) the estimate

$$\begin{aligned} \gamma \le \chi \left( \sum _{n=1}^{{N}} a_n^\dagger a_n\le \alpha ^{-h}\right) \le e^{t\left( \alpha ^{-h}-\sum _{n=1}^{{N}} a_n^\dagger a_n\right) }. \end{aligned}$$

By our assumption on s, there exists a \(h'\) such that \(2\,s<h'<h\). Consequently we obtain for \(t:=\alpha ^{2+(h-h')}\), using Mehler’s kernel,

$$\begin{aligned} \rho (\lambda )\! =\! \gamma (\lambda ,\lambda )\! \le \! e^{\alpha ^{2-h'}}e^{-t\sum _{n=1}^{{N}} a_n^\dagger a_n}(\lambda ,\lambda )\! = \! e^{\alpha ^{2-h'}}\! \left( \frac{1}{1-e^{-\alpha ^{h-h'}}}\right) ^{N}\! \! \left( \frac{ \alpha ^2 w_\alpha }{\pi }\right) ^{\frac{{N}}{2}}\! \! \! \! e^{-\alpha ^2 w_\alpha |\lambda |^2}, \end{aligned}$$

with \(w_\alpha :=\textrm{coth}\left( \alpha ^{h-h'}\right) -\textrm{cosech}\left( \alpha ^{h-h'}\right) \). Since \(N e^{-\alpha ^{h-h'}}\underset{\alpha \rightarrow \infty }{\longrightarrow }0\), it is clear that \(\left( \frac{1}{1-e^{-\alpha ^{h-h'}}}\right) ^{N}\) is bounded uniformly in \(\alpha \). Since \(w_\alpha \ge 0\) is strictly increasing in \(\alpha \), we can choose \(0<\beta '<\beta _0\inf _{\alpha \ge 1}w_\alpha \), where \(\beta _0\) is the constant from Definition 4.3. Consequently \(\Vert \frac{\beta '}{w_\alpha }|A_x|^2\Vert <1\) uniformly in \(x\in \mathbb {R}^3\) and \(\alpha \ge 1\), and in particular \(\left( 1-\frac{\beta '}{w_\alpha }|A_x|^2\right) ^{-1}\) is a bounded operator. Hence we obtain for \(x\in \mathbb {R}^3\)

$$\begin{aligned}&\int _{|A_x\lambda |\ge \epsilon _*}\left( 1+|\lambda |^2\right) \rho (\lambda )\textrm{d}\lambda \lesssim e^{\alpha ^{2-h'}}\left( \frac{ \alpha ^2 w_\alpha }{\pi }\right) ^{\frac{{N}}{2}}\int _{|A_x\lambda |\ge \epsilon _*}\left( 1+|\lambda |^2\right) e^{-\alpha ^2 w_\alpha |\lambda |^2}\textrm{d}\lambda \\&\quad \le e^{\alpha ^{2-h'}}\left( \frac{ \alpha ^2 w_\alpha }{\pi }\right) ^{\frac{{N}}{2}}\int _{\mathbb {R}^{N}} \left( 1+|\lambda |^2\right) e^{-\alpha ^2 \left( w_\alpha |\lambda |^2+\beta '\epsilon _*^2 -\beta ' |A_x \lambda |^2\right) }\textrm{d}\lambda \\&\quad =e^{\alpha ^{2-h'}}\frac{w_\alpha +\alpha ^{-2}\textrm{Tr}\left( 1-\frac{\beta '}{w_\alpha }|A_x|^2\right) ^{-1}}{w_\alpha \det \sqrt{1-\frac{\beta '}{w_\alpha }|A_x|^2}}\ e^{-\beta '\epsilon _*^2 \alpha ^2}. \end{aligned}$$

Furthermore, for a suitable, x-independent, constant \(\mu \)

$$\begin{aligned}&e^{\alpha ^{2-h'}}\frac{w_\alpha \! +\! \alpha ^{-2}\textrm{Tr}\left( 1\! -\! \frac{\beta '}{w_\alpha }|A_x|^2\right) ^{-1}}{w_\alpha \det \sqrt{1\! -\! \frac{\beta '}{w_\alpha }|A_x|^2}}\! \lesssim \! e^{\alpha ^{2-h'}}\! \! \frac{\alpha ^p}{\det \! \sqrt{1\! -\! \frac{\beta '}{w_\alpha }|A_x|^2}}\! \nonumber \\&\quad = e^{\alpha ^{2-h'}+p\ln {\alpha }-\frac{1}{2}\textrm{Tr}\ln \left( 1-\frac{\beta '}{w_\alpha }|A_x|^2\right) }\le e^{\alpha ^{2-h'}+p\ln {\alpha }+\mu \Vert A_x\Vert _{\textrm{HS}}^2}\le e^{\alpha ^{2-h'}+p\ln \alpha +\mu C\Lambda ^2}, \end{aligned}$$

where we have used the rough estimate \(w_\alpha + \alpha ^{-2}\textrm{Tr}\left( 1 - \frac{\beta '}{w_\alpha }|A_x|^2\right) ^{-1}\lesssim 1+\alpha ^{-2}N\lesssim \alpha ^p\) for a suitable exponent \(p>0\) in the first inequality and Lemma 4.4 in the last inequality. Note that the exponent in Eq. (4.9) is of order \(\alpha ^{\textrm{max}\big \{2-h',\frac{8}{5}(1+\sigma )\big \}}\ll \epsilon _*^2 \alpha ^2\) since \(\Lambda ^2=\alpha ^{\frac{8}{5}(1+\sigma )}\) and \(\epsilon \ge D\alpha ^{-s}\) with \(s<\textrm{min}\big \{\frac{h'}{2},\frac{1}{5}(1-4\sigma )\big \}\).

Defining \(r:=\alpha ^{2q}\) with \(q>v\), where v is the constant from Lemma 4.4 and making use of the fact that the number of \(z\in \mathbb {Z}^3\) with \(|z|\le r+1\) is of order \(r^3=\alpha ^{6q}\), we obtain

$$\begin{aligned} \sum _{|z|\le r+1}\int _{|A_{z}\lambda |\ge \epsilon _*}\left( 1+|\lambda |^2\right) \rho (\lambda )\textrm{d}\lambda \, \textrm{d}x\lesssim \alpha ^{6q}e^{\alpha ^{2-h'}+p\ln \alpha +\mu C\Lambda ^2-\beta '\epsilon _*^2\alpha ^2}\le e^{-\beta \epsilon _*^2\alpha ^2} \end{aligned}$$

for \(\beta <\beta '\) and \(\alpha \) large enough. We have \(\Vert A_{\ge r}\Vert _\textrm{HS} \underset{\alpha \rightarrow \infty }{\longrightarrow }0\) by Lemma 4.4 and our choice \(r=\alpha ^{2q}\) with \(q>v\). Using Eq. (4.8), and an argument similar to the one in Eq. (4.9), we can therefore estimate \(\int _{|A_{\ge r}\lambda |\ge \epsilon _*}\left( 1+|\lambda |^2\right) \rho (\lambda )\textrm{d}\lambda \) by

$$\begin{aligned}&\int _{|A_{\ge r}\lambda |\ge \epsilon _*}\left( 1+|\lambda |^2\right) \rho (\lambda )\textrm{d}\lambda \lesssim e^{\alpha ^{2-h'}}\left( \frac{ \alpha ^2 w_\alpha }{\pi }\right) ^{\frac{{N}}{2}}\int _{|A_{\ge r}\lambda |\ge \epsilon _*}\left( 1+|\lambda |^2\right) e^{-\alpha ^2 w_\alpha |\lambda |^2}\textrm{d}\lambda \\&\quad \lesssim e^{\alpha ^{2-h'}}\frac{\alpha ^p}{\det \sqrt{1-\frac{\beta '}{w_\alpha }|A_{\ge r}|^2}}\ e^{-\beta '\epsilon _*^2 \alpha ^2}\lesssim e^{\alpha ^{2-h'}+p\ln \alpha +\mu \Vert A_{\ge r}\Vert _{\textrm{HS}}^2-\beta '\epsilon _*^2 \alpha ^2}. \end{aligned}$$

Again we observe that the exponent \(\alpha ^{2-h'}+p\ln \alpha +\mu \Vert A_{\ge r}\Vert _{\textrm{HS}}^2\) is small compared to \(\epsilon _*^2 \alpha ^2\), which concludes the proof of Eq. (4.5).

The proof of Eq. (4.6) can be carried out analogously with the help of the operator \(A_\zeta \lambda :=\langle \zeta |\lambda \rangle \frac{\zeta }{\Vert \zeta \Vert }\) using the fact that \(\Vert A_\zeta \Vert _{\textrm{HS}}=\Vert A_\zeta \Vert =|\zeta |\) and the assumption \(\beta <\frac{1}{|\zeta |^2}\). More precisely we obtain for \(\beta<\beta '<\frac{1}{|\zeta |^2}\)

$$\begin{aligned}&\int _{|\langle \zeta |\lambda \rangle |\ge \epsilon }\left( 1+|\lambda |^2\right) \rho (\lambda )\textrm{d}\lambda \lesssim e^{\alpha ^{2-h'}}\left( \frac{ \alpha ^2 w_\alpha }{\pi }\right) ^{\frac{{N}}{2}}\int _{|A_\zeta \lambda |\ge \epsilon }\left( 1+|\lambda |^2\right) e^{-\alpha ^2 w_\alpha |\lambda |^2}\textrm{d}\lambda \\&\quad \lesssim e^{\alpha ^{2-h'}}\frac{\alpha ^p}{\det \sqrt{1-\frac{\beta '}{w_\alpha }|A_\zeta |^2}}\ e^{-\beta '\epsilon ^2 \alpha ^2}\lesssim e^{\alpha ^{2-h'}+p\ln \alpha +\mu \Vert A_\zeta \Vert _{\textrm{HS}}^2-\beta '\epsilon ^2 \alpha ^2}\le e^{-\beta \epsilon ^2\alpha ^2}. \end{aligned}$$

\(\square \)

5 Properties of the Pekar Functional

In this section we are going to discuss essential properties of the Pekar functional \(\mathcal {F}^\textrm{Pek}\), and we are going to verify an asymptomatically sharp quadratic approximation for \(\mathcal {F}^\textrm{Pek}\! \left( \varphi \right) \), which is valid for all field configurations \(\varphi \) close to a minimizer \(\varphi ^\textrm{Pek}\). It has been proven in [11] that a suitable quadratic approximation of \(\mathcal {F}^\textrm{Pek}\) holds for all configurations \(\varphi \) satisfying \(\Vert V_{\varphi -\varphi ^\textrm{Pek}}\Vert \ll 1\), where

$$\begin{aligned} V_\varphi :=-2\left( -\Delta \right) ^{-\frac{1}{2}}\mathfrak {Re}\, \varphi . \end{aligned}$$

In the following we are showing that this result is still valid, in case we substitute the \(L^2\)-norm with the weaker \(\Vert \cdot \Vert _\diamond \) norm, which is a hybrid between the \(L^2\) and the \(L^\infty \) norm defined as

$$\begin{aligned} \Vert V \Vert _\diamond :=\sup _{x\in \mathbb {R}^3}\sqrt{\int _{B_{1}(x)}|V(y)|^2\ \textrm{d}y}, \end{aligned}$$

where \(B_{1}(x)\) is the unit ball centered at \(x\in \mathbb {R}^3\). This will be the content of Lemma 5.2 and Theorem 5.4, respectively. We have \(\Vert V_\varphi \Vert _\diamond =|\lambda |_\diamond \) for \(\varphi =\sum _{n=1}^N \lambda _n \varphi _n\), where \(|\cdot |_\diamond \) 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

$$\begin{aligned} V^2\le C\Vert V\Vert _\diamond ^2 \left( 1-\Delta \right) ^2 \end{aligned}$$

holds for all (measurable) \(V:\mathbb {R}^3\longrightarrow \mathbb {R}\), where \(V^2\) is interpreted as a multiplication operator.


As a first step, we are going to verify that Eq. (5.3) holds in case we use the \(L^2\) norm \(\Vert V\Vert \) instead of \(\Vert V\Vert _\diamond \). This follows from \(V^2\le \left\| V(1-\Delta )^{-1}\right\| _{\textrm{HS}}^2 \left( 1-\Delta \right) ^2\), where \(\Vert \cdot \Vert _{\textrm{HS}}\) is the Hilbert-Schmidt norm, and

$$\begin{aligned} \left\| V(1-\Delta )^{-1}\right\| _{\textrm{HS}}^2=\int \int V(x)^2 K(y-x)^2\, \textrm{d}x\textrm{d}y=\int K(y)^2\textrm{d}y\, \Vert V\Vert ^2 \end{aligned}$$

with \(K(y-x)\) being the kernel of the operator \((1-\Delta )^{-1}\). Note that \(C':=\int K(y)^2\textrm{d}y\) is finite, which concludes the first step. In order to obtain the analogue statement for \(\Vert V\Vert _\diamond \), let \(\chi \) be a smooth, non-negative, function with \(\textrm{supp} \left( \chi \right) \subset B_1(0)\) and \(\int _{\mathbb {R}^3} \chi (y)^2\, \textrm{d}y=1\). Defining \(\chi _y(x):=\chi (x-y)\) for \(y\in \mathbb {R}^3\) and using the previously derived inequality \(V^2\le C'\Vert V\Vert ^2 \left( 1-\Delta \right) ^2\), which holds for any \(V\in L^2\! \left( \mathbb {R}^3\right) \), we obtain

$$\begin{aligned} V^2&=\! \int \! \chi _y V^2\chi _y\, \textrm{d}y=\! \int \! \chi _y \left( \mathbbm {1}_{B_1(y)}V\right) ^2\chi _y\, \textrm{d}y\le C'\!\int \! \Vert \mathbbm {1}_{B_1(y)}V\Vert ^2\, \chi _y \left( 1-\Delta \right) ^2\chi _y\, \textrm{d}y\\&\le C'\Vert V\Vert ^2_\diamond \int \chi _y \left( 1-\Delta \right) ^2\chi _y\, \textrm{d}y=C'\Vert V\Vert ^2_\diamond \int \big |(1-\Delta )\chi _y\big |^2\, \textrm{d}y, \end{aligned}$$

where \(\left| A\right| ^2=A^\dagger A\). Furthermore \((1-\Delta )\chi _y=\chi _y(1-\Delta )-2\left( \nabla \chi _y\right) \nabla -\left( \Delta \chi _y\right) \), which yields together with a Cauchy–Schwarz inequality the estimate

$$\begin{aligned}&\int \big |(1-\Delta )\chi _y\big |^2 \textrm{d}y\le 3\int \Big ((1-\Delta )\chi _y^2(1-\Delta )-4\nabla \left| \nabla \chi _y\right| ^2 \nabla +|\Delta \chi _y|^2\Big )\textrm{d}y\\&\quad = 3(1-\Delta )^2-12\, \nabla \! \left( \int \left| \nabla \chi _y^2\right| \textrm{d}y\right) \! \nabla +3\int |\Delta \chi _y|^2\textrm{d}y\lesssim \left( 1-\Delta \right) ^2, \end{aligned}$$

where we have used that \( \int \left| \nabla \chi (y)\right| ^2\, \textrm{d}y\) and \(\int \left| \Delta \chi (y)\right| ^2\, \textrm{d}y\) are finite. \(\quad \square \)

In the following we are going to use that we can write the Pekar energy as

$$\begin{aligned} \mathcal {F}^\textrm{Pek}\! \left( \varphi \right) =\Vert \varphi \Vert ^2+\inf \sigma \left( -\Delta +V_\varphi \right) , \end{aligned}$$

where \(V_\varphi \) is defined in Eq. (5.1). As an immediate consequence of Eq. (5.3) we have \(\pm V\le \sqrt{C}\Vert V\Vert _\diamond \left( 1-\Delta \right) \) and consequently there exists a \(\delta _0>0\) and a contour \(\mathcal {C}\subset \mathbb {C}\), such that \(\mathcal {C}\) separates the ground state energy \(\inf \sigma \left( -\Delta +V\right) \) from the excitation spectrum of \(H_V:=-\Delta +V\) for all V with \(\Vert V-V_{\varphi ^\textrm{Pek}}\Vert _\diamond <\delta _0\), see also the proof of Proposition 3.1 in [11]. This allows us to further identify \(\mathcal {F}^\textrm{Pek}\! \left( \varphi \right) \) as

$$\begin{aligned} \mathcal {F}^\textrm{Pek}\! \left( \varphi \right) =\Vert \varphi \Vert ^2+\textrm{Tr}\int _\mathcal {C} \frac{z}{z-H_{V_\varphi }}\frac{\textrm{d}z}{2\pi i} \end{aligned}$$

for all \(\varphi \) satisfying \(\Vert V_{\varphi -\varphi ^\textrm{Pek}}\Vert _\diamond <\delta _0\). Following the strategy in [11], we will use Eq. (5.5) to compare \(\mathcal {F}^\textrm{Pek}\! \left( \varphi \right) \) with \(e^\textrm{Pek}=\mathcal {F}^\textrm{Pek}\! \left( \varphi ^\textrm{Pek}\right) \). Before we do this let us introduce the operators

$$\begin{aligned} K^\textrm{Pek}&:=1-H^\textrm{Pek}=4\left( -\Delta \right) ^{-\frac{1}{2}}\psi ^\textrm{Pek}\frac{1-\vert \psi ^\textrm{Pek}\rangle \langle \psi ^\textrm{Pek}\vert }{H_{V^\textrm{Pek}}-\mu ^\textrm{Pek}}\psi ^\textrm{Pek}\left( -\Delta \right) ^{-\frac{1}{2}}, \end{aligned}$$
$$\begin{aligned} L^\textrm{Pek}&:=4\left( -\Delta \right) ^{-\frac{1}{2}}\psi ^\textrm{Pek}\left( 1-\Delta \right) ^{-1}\psi ^\textrm{Pek}\left( -\Delta \right) ^{-\frac{1}{2}}, \end{aligned}$$

where \(H^\textrm{Pek}\) is defined in Eq. (1.4), \(\mu ^\textrm{Pek}:=e^\textrm{Pek}-\Vert \varphi ^\textrm{Pek}\Vert ^2\) and \(\psi ^\textrm{Pek}\) is the, non-negative, ground state of the operator \(H_{V^\textrm{Pek}}\) with \(V^\textrm{Pek}:=V_{\varphi ^\textrm{Pek}}\), which we interpret as a multiplication operator in Eqs. (5.6) and  (5.7). The following Lemma 5.2 can be proved in the same way as [11, Proposition 3.3], using Lemma 5.1.

Lemma 5.2

There exist constants \(c,\delta _0>0\) such that for all \(\varphi \) with \(\Vert V_{\varphi -\varphi ^\textrm{Pek}}\Vert _\diamond <\delta _0\)

$$\begin{aligned}&\left| \mathcal {F}^\textrm{Pek} \left( \varphi \right) - e^\textrm{Pek}-\langle \varphi - \varphi ^\textrm{Pek}|1 - K^\textrm{Pek}|\varphi -\varphi ^\textrm{Pek}\rangle \right| \nonumber \\ {}&\quad \le c\Vert V_{\varphi -\varphi ^\textrm{Pek}}\Vert _\diamond \langle \varphi - \varphi ^\textrm{Pek}|L^\textrm{Pek}|\varphi - \varphi ^\textrm{Pek}\rangle . \end{aligned}$$


By taking \(\delta _0\) small enough, we can assume for all V with \(\Vert V_{\varphi -\varphi ^\textrm{Pek}}\Vert _\diamond <\delta _0\) that

$$\begin{aligned} \sup _{z\in \mathcal {C}}\left\| V_{\varphi -\varphi ^\textrm{Pek}}\frac{1}{z-H_{V^\textrm{Pek}}}\right\| _{\textrm{op}}<1, \end{aligned}$$

where \(\Vert \cdot \Vert _\textrm{op}\) denotes the operator norm. This immediately follows from

$$\begin{aligned} \left\| V_{\varphi -\varphi ^\textrm{Pek}}\frac{1}{H_{V^\textrm{Pek}}-z}\right\| _\textrm{op}^2\lesssim \left\| \left( V_{\varphi }-V^\textrm{Pek}\right) \left( 1-\Delta \right) ^{-1}\right\| _\textrm{op}^2\le C\Vert V_{\varphi -\varphi ^\textrm{Pek}}\Vert ^2_\diamond , \end{aligned}$$

where we used Eq. (5.3) and the fact that the spectrum of \(H_{V^\textrm{Pek}}\) has a positive distance to the contour \(\mathcal {C}\), allowing us to bound the operator norm of \(\left( 1-\Delta \right) \frac{1}{H_{V^\textrm{Pek}}-z}\) uniformly in \(z\in \mathcal {C}\). Given Eq. (5.9), it has been verified in the proof of [11, Proposition 3.3] that

$$\begin{aligned}&\left| \Vert \varphi \Vert ^2+\textrm{Tr}\int _\mathcal {C} \frac{z}{z-H_{V_\varphi }}\frac{\textrm{d}z}{2\pi i}-e^\textrm{Pek}-\langle \varphi -\varphi ^\textrm{Pek}|1-K^\textrm{Pek}|\varphi -\varphi ^\textrm{Pek}\rangle \right| \\&\quad \lesssim \epsilon \langle \varphi -\varphi ^\textrm{Pek}|L^\textrm{Pek}|\varphi -\varphi ^\textrm{Pek}\rangle \end{aligned}$$

for \(\epsilon :=\sup _{z\in \mathcal {C}}\Big \{\left\| \frac{A}{1-A}\right\| _\textrm{op}+\left\| \frac{B}{1-B}\right\| _\textrm{op}+\left\| \left( 1-\Delta \right) ^{\frac{1}{2}}\frac{1}{z-H_{V^\textrm{Pek}}}\frac{A}{1-A}\left( 1-\Delta \right) ^{\frac{1}{2}}\right\| _\textrm{op}\Big \}\), where we denote \(A:=\left( V_{\varphi -\varphi ^\textrm{Pek}}\right) \frac{1}{z-H_{V^\textrm{Pek}}}\) and \(B:=\left( 1-\vert \psi ^\textrm{Pek}\rangle \langle \psi ^\textrm{Pek}\vert \right) A^\dagger \). In the following we want to verify that \(\epsilon \lesssim \Vert V_{\varphi -\varphi ^\textrm{Pek}}\Vert _\diamond \), which concludes the proof by Eq. (5.5). Since \((1-\Delta )\frac{1}{z-H_{V^\textrm{Pek}}}\) is uniformly bounded in z, \(\left\| \frac{A}{1-A}\right\| _\textrm{op}\!\le \! \frac{\Vert A\Vert _\textrm{op}}{1-\Vert A\Vert _\textrm{op}}\!\lesssim \! \Vert \left( V_{\varphi -\varphi ^\textrm{Pek}}\right) \left( 1\!-\!\Delta \right) ^{-1}\Vert _\textrm{op}\!\lesssim \!\)\( \Vert V_{\varphi -\varphi ^\textrm{Pek}}\Vert _\diamond \) by Eq. (5.3). Similarly \(\left\| \frac{B}{1-B}\right\| _\textrm{op}\lesssim \Vert V_{\varphi -\varphi ^\textrm{Pek}}\Vert _\diamond \). Regarding the final term in the definition of \(\epsilon \), note that \((1-\Delta )^{\frac{1}{2}}\frac{1}{z-H_{V^\textrm{Pek}}}(1-\Delta )^{\frac{1}{2}}\) is uniformly bounded in z, and therefore

$$\begin{aligned}&\left\| \left( 1\! -\! \Delta \right) ^{\frac{1}{2}}\frac{1}{z\! -\! H_{V^\textrm{Pek}}}\frac{A}{1\! -\! A}\left( 1\! -\! \Delta \right) ^{\frac{1}{2}}\right\| _\textrm{op}\! \lesssim \! \left\| \left( 1\! -\! \Delta \right) ^{-\frac{1}{2}}\frac{A}{1\! -\! A}\left( 1\! -\! \Delta \right) ^{\frac{1}{2}}\right\| _\textrm{op}\! =\! \left\| \frac{A'}{1\! -\! A'}\right\| _\textrm{op}\! , \end{aligned}$$

with \(A'\!:=\! \left( 1-\Delta \right) ^{-\frac{1}{2}}A\left( 1-\Delta \right) ^{\frac{1}{2}}\). Furthermore \(\left\| \frac{A'}{1\! -\! A'}\right\| _\textrm{op}\! \! \! \le \! \frac{\Vert A'\Vert _\textrm{op}}{1\! -\! \Vert A'\Vert _\textrm{op}}\) and

$$\begin{aligned} \Vert A'\Vert _\textrm{op}&\lesssim \left\| \left( 1-\Delta \right) ^{-\frac{1}{2}}\left( V_{\varphi -\varphi ^\textrm{Pek}}\right) \left( 1-\Delta \right) ^{-\frac{1}{2}}\right\| _\textrm{op}\le \Vert \left( V_{\varphi -\varphi ^\textrm{Pek}}\right) \left( 1-\Delta \right) ^{-1}\Vert _\textrm{op}\\&\lesssim \Vert V_{\varphi -\varphi ^\textrm{Pek}}\Vert _\diamond . \end{aligned}$$

\(\square \)

Lemma 5.2 gives a lower bound on \(\mathcal {F}^\textrm{Pek}\! \left( \varphi ^\textrm{Pek}+\xi \right) -e^\textrm{Pek}\) in terms of a quadratic function \(\xi \mapsto \langle \xi |1-\left( K^\textrm{Pek}+\epsilon L^\textrm{Pek}\right) |\xi \rangle \) for \(\xi \) satisfying \(\Vert V_\xi \Vert _\diamond < \min \{\frac{\epsilon }{c},\delta _0\}\). Due to the translation invariance of \(\mathcal {F}^\textrm{Pek}\), this lower bound is however insufficient, since we have for all \(\xi \in \textrm{span}\{\partial _{y_1}\varphi ^\textrm{Pek},\partial _{y_2}\varphi ^\textrm{Pek},\partial _{y_3}\varphi ^\textrm{Pek}\}{\setminus } \{0\}\)

$$\begin{aligned} \langle \xi |1-\left( K^\textrm{Pek}+\epsilon L^\textrm{Pek}\right) |\xi \rangle =\textrm{Hess}|_{\varphi ^\textrm{Pek}}\mathcal {F}^\textrm{Pek}[\xi ]-\epsilon \langle \xi |L^\textrm{Pek}|\xi \rangle =-\epsilon \langle \xi |L^\textrm{Pek}|\xi \rangle <0, \end{aligned}$$

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 \(\tau \) in Definition 5.3. Before we can formulate Definition 5.3 we need some auxiliary preparations.

In the following let \(\Pi \) be the projection defined in Eq. (4.1) and let us define the real orthonormal system

$$\begin{aligned} \varphi _n:=\frac{\Pi \partial _{y_n}\varphi ^\textrm{Pek}}{\Vert \Pi \partial _{y_n}\varphi ^\textrm{Pek}\Vert } \end{aligned}$$

for \(n\in \{1,2,3\}\), which we complete to a real orthonormal basis \(\{\varphi _1,..,\varphi _{N}\}\) of \(\Pi L^2\! \left( \mathbb {R}^3\right) \). Furthermore let us write \(\varphi ^\textrm{Pek}_{x}(y):=\varphi ^\textrm{Pek}(y-x)\) for the translations of \(\varphi ^\textrm{Pek}\) and let us define the map \(\omega :\mathbb {R}^3\longrightarrow \mathbb {R}^3\) as

$$\begin{aligned} \omega \left( x\right) :=\Big (\langle \varphi _n|\varphi ^\textrm{Pek}_{x}\rangle \Big )_{n=1}^3\in \mathbb {R}^3. \end{aligned}$$

Since \(\varphi ^\textrm{Pek} \in H^1\! \left( \mathbb {R}^3\right) \), \(\omega \) is differentiable. Moreover, since \(\varphi ^\textrm{Pek}\) is invariant under the action of \(O\left( 3\right) \) and since the operator \(\Pi \) commutes with the reflections \(y_i\rightarrow -y_i\) and permutations \(y_i\leftrightarrow y_j\), it is clear that \(\omega (0)=0\). By the same argument we see that \(D|_0 \omega \) has full rank and therefore there exists a local inverse \(t\mapsto x_t\) for \(|t|< \delta _*\) and a suitable constant \(\delta _*>0\).

Definition 5.3

We define the map \(\tau :\Pi L^2\! \left( \mathbb {R}^3\right) \longrightarrow \Pi L^2\! \left( \mathbb {R}^3\right) \) as

$$\begin{aligned} \tau \left( \varphi \right) :=\varphi -f(t^\varphi ), \end{aligned}$$

where \(t^\varphi :=\big (\langle \varphi _1|\varphi \rangle ,\langle \varphi _2|\varphi \rangle ,\langle \varphi _3|\varphi \rangle \big )\in \mathbb {R}^3\) and f(t) is defined as

$$\begin{aligned} f(t):=\chi \big (|t|< \delta _*\big )\left( \Pi \varphi ^\textrm{Pek}_{x_t}-\sum _{n=1}^3 t_n \varphi _n\right) . \end{aligned}$$

The map \(\tau \) is constructed in a way such that it “flattens” the manifold of Pekar minimizers \(\{\varphi ^\textrm{Pek}_x:x\in \mathbb {R}^3\}\). More precisely, we have that \(\tau \left( \Pi \varphi ^\textrm{Pek}_x\right) \) is for all small enough \(x\in \mathbb {R}^3\) an element of the linear space spanned by \(\{\varphi _1,\varphi _2,\varphi _3\}\). A similar construction appears in [5] and, in a somewhat different way, in [9].

Recall the operators \(K^\textrm{Pek}\) and \(L^\textrm{Pek}\) from Eqs. (5.6) and (5.7), and let \(T_x\) be the translation operator defined by \((T_x \varphi )(y):=\varphi (y-x)\). Then we define the operators \(K^\textrm{Pek}_x:=T_x\, K^\textrm{Pek}\, T_{-x}\) and \(L^\textrm{Pek}_x:=T_x\, L^\textrm{Pek}\, T_{-x}\), as well as for \(|t|<\epsilon \) with \(\epsilon <\delta _*\)

$$\begin{aligned} J_{t,\epsilon }:=\pi \Big (1-(1+\epsilon )\left( K^\textrm{Pek}_{x_t}+\epsilon L^\textrm{Pek}_{x_t}\right) \Big )\pi , \end{aligned}$$

where \(\pi :L^2\! \left( \mathbb {R}^3\right) \longrightarrow L^2\! \left( \mathbb {R}^3\right) \) is the orthogonal projection onto the subspace spanned by \(\{\varphi _4,\dots , \varphi _N\}\). Furthermore we define \(J_{t,\epsilon }:=\pi \) for \(|t|\ge \epsilon \). In contrast to the operator \(1-\left( K^\textrm{Pek}+\epsilon L^\textrm{Pek}\right) \) from Eq. (5.10), the operator \(J_{t,\epsilon }\) is non-negative for \(\epsilon \) small enough, as will be shown in Lemma B.5. With the operator \(J_{t,\epsilon }\) and the transformation \(\tau \) at hand we can formulate a strong lower bound for \(\mathcal {F}^\textrm{Pek}\! \left( \varphi \right) -e^\textrm{Pek}\) in the subsequent Theorem 5.4, where we use the shorthand notation \(J_{t,\epsilon } \big [ \varphi \big ]:=\langle \varphi |J_{t,\epsilon }|\varphi \rangle \).

Theorem 5.4

There exist constants \(C>0\), \(0<\epsilon _0\le \delta _*\) and \(0<D\le 1\) such that

$$\begin{aligned} \mathcal {F}^\textrm{Pek}\! \left( \varphi \right) \ge e^\textrm{Pek}+J_{t^\varphi ,\epsilon } \big [ \tau \! \left( \varphi \right) \big ]-\frac{C}{\epsilon }\left\| \left( 1-\Pi \right) \varphi ^\textrm{Pek}_{x_{t^\varphi }}\right\| ^2 \end{aligned}$$

for all \(0<\epsilon < \epsilon _0\) and \(\varphi \in \Pi L^2\! \left( \mathbb {R}^3\right) \) satisfying \(\left\| V_{\varphi -\varphi ^\textrm{Pek}}\right\| _\diamond <\epsilon D\) and \(|t^\varphi |<\epsilon D\), where \(J_{t,\epsilon }\) is defined in Eq. (5.13).


In the following we use the abbreviation \(t:=t^\varphi \). Since \(\big \Vert V_{\varphi ^\textrm{Pek}-\varphi ^\textrm{Pek}_{x}}\big \Vert _\diamond \lesssim |x|\) and \(|x_t|\lesssim |t|\) for \(|t|\le \frac{\delta _*}{2}\), we have for all \(\varphi \) satisfying \(\left\| V_{\varphi -\varphi ^\textrm{Pek}}\right\| _\diamond <D\epsilon \) and \(|t|<\min \{D\epsilon ,\frac{\delta _*}{2}\}\)

$$\begin{aligned} \big \Vert V_{T_{-x_t}\varphi -\varphi ^\textrm{Pek}}\big \Vert _\diamond&= \big \Vert V_{\varphi -\varphi ^\textrm{Pek}_{x_t}}\big \Vert _\diamond \le \big \Vert V_{\varphi -\varphi ^\textrm{Pek}}\big \Vert _\diamond +\big \Vert V_{\varphi ^\textrm{Pek}-\varphi ^\textrm{Pek}_{x_t}}\big \Vert _\diamond \\ {}&\lesssim \big \Vert V_{\varphi -\varphi ^\textrm{Pek}}\big \Vert _\diamond +|t|\lesssim D\epsilon . \end{aligned}$$

By taking D small enough we obtain \(\big \Vert V_{T_{-x_t}\varphi -\varphi ^\textrm{Pek}}\big \Vert _\diamond \le \frac{\epsilon }{c}\) where c is the constant from Lemma 5.2. Let us define \(\epsilon _0:=\min \left\{ c \delta _0,\frac{\delta _*}{2D},\delta _*\right\} \). Using the translation-invariance of \(\mathcal {F}^\textrm{Pek}\) and applying Lemma 5.2 yields

$$\begin{aligned} \mathcal {F}^\textrm{Pek}\! \left( \varphi \right) \! -\! e^\textrm{Pek}\!&=\! \mathcal {F}^\textrm{Pek}\! \left( T_{-x_t}\varphi \right) \! -\! e^\textrm{Pek}\! \ge \! \langle T_{-x_t}\varphi \! -\! \varphi ^\textrm{Pek}|1\! -\! \left( \! K^\textrm{Pek}\! +\! \epsilon L^\textrm{Pek}\! \right) \! |T_{-x_t}\varphi \! -\! \varphi ^\textrm{Pek}\rangle \nonumber \\&=\langle \varphi -\varphi ^\textrm{Pek}_{x_t}|1 - \left( K^\textrm{Pek}_{x_t}+\epsilon L^\textrm{Pek}_{x_t}\right) |\varphi -\varphi ^\textrm{Pek}_{x_t}\rangle \nonumber \\&\ge \Vert \varphi -\Pi \varphi ^\textrm{Pek}_{x_t}\Vert ^2 - \langle \varphi -\varphi ^\textrm{Pek}_{x_t}|K^\textrm{Pek}_{x_t} + \epsilon L^\textrm{Pek}_{x_t}|\varphi -\varphi ^\textrm{Pek}_{x_t}\rangle \nonumber \\&\ge \Vert \varphi -\Pi \varphi ^\textrm{Pek}_{x_t}\Vert ^2-\left( 1+\epsilon \right) \langle \varphi -\Pi \varphi ^\textrm{Pek}_{x_t}|K^\textrm{Pek}_{x_t}+\epsilon L^\textrm{Pek}_{x_t}|\varphi -\Pi \varphi ^\textrm{Pek}_{x_t}\rangle \nonumber \\&\quad -\left( 1+\epsilon ^{-1}\right) \langle \left( 1-\Pi \right) \varphi ^\textrm{Pek}_{x_t}|K^\textrm{Pek}_{x_t}+\epsilon L^\textrm{Pek}_{x_t}|\left( 1-\Pi \right) \varphi ^\textrm{Pek}_{x_t}\rangle , \end{aligned}$$

where we have used the positivity of \(K^\textrm{Pek}_x\) and \(L^\textrm{Pek}_x\), and the Cauchy–Schwarz inequality in the last estimate. Note that by construction of \(x_t\) as the local inverse of the function \(\omega \) from Eq. (5.12), we have \(\langle \varphi _n|\varphi -\Pi \varphi ^\textrm{Pek}_{x_t}\rangle =0\) for \(n\in \{1,2,3\}\) and therefore

$$\begin{aligned} \varphi -\Pi \varphi ^\textrm{Pek}_{x_t}=\pi \left( \varphi -\Pi \varphi ^\textrm{Pek}_{x_t}\right) =\pi \left( \varphi -f(t)\right) =\pi \left( \tau \left( \varphi \right) \right) \end{aligned}$$

with \(\pi \) being defined below Eq. (5.13), where we used \(|t|<\delta _*\). This concludes the proof with \(C:=\left( 1+\epsilon _0\right) \left( \Vert K\Vert _\textrm{op}+\epsilon _0 \Vert L\Vert _\textrm{op}\right) \). \(\quad \square \)

6 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 \(E_\alpha \) in Theorem 1.1. We start by verifying the subsequent Lemma 6.1, which provides a lower bound on \(E_\alpha \) in terms of an operator that is, up to a coordinate transformation \(\tau \) and a non-negative term, a harmonic oscillator.

Let us again use the identification \(\mathcal {F}\left( \Pi L^2\! \left( \mathbb {R}^3\right) \right) \cong L^2\! \left( \mathbb {R}^{N}\right) \) utilizing the representation of real functions \(\varphi =\sum _{n=1}^{N} \lambda _n \varphi _n\in \Pi L^2\! \left( \mathbb {R}^3\right) \) by points \(\lambda =(\lambda _1,\dots ,\lambda _{{N}})\in \mathbb {R}^{N}\), such that the annihilation operators \(a_n:=a\left( \varphi _n\right) \) are given by \(a_n=\lambda _n+\frac{1}{2\alpha ^2}\partial _{\lambda _n}\), where \(\lambda _n\) is the multiplication operator by the function \(\lambda \mapsto \lambda _n\) on \(L^2\! \left( \mathbb {R}^{N}\right) \), see also Eq. (4.2), where \(\Pi \) is the projection from Eq. (4.1) and \(\{\varphi _1,\dots ,\varphi _{N}\}\) is the orthonormal basis of \(\Pi L^2\! \left( \mathbb {R}^3\right) \) constructed around Eq. (5.11). Let us also use for functions \(\varphi \mapsto g(\varphi )\) depending on elements \(\varphi \in \Pi L^2\! \left( \mathbb {R}^3\right) \) the convenient notation \(g(\lambda ):=g\left( \sum _{n=1}^{N} \lambda _n \varphi _n\right) \), where \(\lambda \in \mathbb {R}^N\).

Lemma 6.1

Let \(C>0\) and \(0<\sigma \le \frac{1}{4}\), and assume sh and \(\sigma \) satisfy \(2\,s<h\) and \(\sigma <\frac{1-5\,s}{4}\). Furthermore let us define \(\Lambda :=\alpha ^{\frac{4}{5}(1+\sigma )}\) and \(L:=\alpha ^{1+\sigma }\). Then we obtain for any state \(\Psi \) satisfying \(\langle \Psi |\mathbb {H}_\Lambda |\Psi \rangle \le C\), \(\textrm{supp}\left( \Psi \right) \subset B_{4L}(0)\) and

$$\begin{aligned} \chi \left( W^{-1}_{\varphi ^\textrm{Pek}}\mathcal {N}W_{\varphi ^\textrm{Pek}}\le \alpha ^{-h}\right) \Psi =\Psi , \end{aligned}$$


$$\begin{aligned}&\langle \Psi |\mathbb {H}_\Lambda |\Psi \rangle \ge e^\textrm{Pek}+\Big \langle \Psi \Big | -\frac{1}{4\alpha ^4}\sum _{n=1}^{N}\partial _{\lambda _n}^2+J_{t^\lambda ,\alpha ^{-s}}\! \big [ \tau \! \left( \lambda \right) \big ]+\mathcal {N}-\sum _{n=1}^N a_n^\dagger a_n\Big |\Psi \Big \rangle \! -\frac{{N}}{2\alpha ^2}\nonumber \\&\quad +O\left( \alpha ^{s-\frac{12}{5}(1+\sigma )}+\alpha ^{-2(1+\sigma )}\right) , \end{aligned}$$

where \(t^\varphi \) and \(\tau (\varphi )\) are defined in Lemma 5.3 and \(J_{t,\epsilon }\) is defined in Eq. (5.13). Furthermore, there exists a \(\beta >0\), such that \(\langle \Psi |1-\mathbb {B}|\Psi \rangle \le e^{-\beta \alpha ^{2(1-s)}}\), where \(\mathbb {B}\) is the multiplication operator by the function \(\lambda \mapsto \chi \left( |t^\lambda |<\alpha ^{-s}\right) \).


Applying Eq. (2.3) with \(\Lambda \) and \(\ell \) as in the definition of \(\Pi \), see Eq. (4.1), and \(K:=\Lambda \), and utilizing Eq. (2.5), we obtain for a suitable \(C'\)

$$\begin{aligned} \langle \Psi |\mathbb {H}_\Lambda |\Psi \rangle \ge \langle \Psi |\mathbb {H}^0_{\Lambda ,\ell }|\Psi \rangle -C' \alpha ^{-2(1+\sigma )}. \end{aligned}$$

Making use of \(\sum _{n=1}^{N} a_n^\dagger a_n=\sum _{n=1}^{N}\left( -\frac{1}{4\alpha ^4}\partial _{\lambda _n}^2+\lambda _n^2\right) -\frac{{N}}{2\alpha ^2}\) and \(a_n+a_n^\dagger =2\lambda _n\), we further have the identity

$$\begin{aligned} \mathbb {H}^0_{\Lambda ,\ell }&=-\Delta _x-2\sum _{n=1}^N \langle \varphi _n|w_x\rangle \lambda _n+\sum _{n=1}^{N}\left( -\frac{1}{4\alpha ^4}\partial _{\lambda _n}^2+\lambda _n^2\right) -\frac{{N}}{2\alpha ^2}+\mathcal {N}-\sum _{n=1}^N a_n^\dagger a_n\\&= -\Delta _x+V_\lambda (x)+\sum _{n=1}^{N}\left( -\frac{1}{4\alpha ^4}\partial _{\lambda _n}^2+\lambda _n^2\right) -\frac{{N}}{2\alpha ^2}+\mathcal {N}-\sum _{n=1}^N a_n^\dagger a_n, \end{aligned}$$

with \(V_\varphi \) defined in Eq. (5.1). Clearly \(-\Delta _x+V_\lambda \ge \inf \sigma \left( -\Delta _x+V_\lambda \right) =\mathcal {F}^\textrm{Pek}\! \left( \lambda \right) -\sum _{n=1}^{N} \lambda _n^2\), which yields the inequality \(\mathbb {H}^0_{\Lambda ,\ell }\ge \mathbb {K}+\mathcal {N}-\sum _{n=1}^N a_n^\dagger a_n\) with

$$\begin{aligned} \mathbb {K}:=-\frac{1}{4\alpha ^4}\sum _{n=1}^{N}\partial _{\lambda _n}^2+\mathcal {F}^\textrm{Pek}\! \left( \lambda \right) -\frac{{N}}{2\alpha ^2}. \end{aligned}$$

Combining Eqs. (6.3) and (6.4), we obtain

$$\begin{aligned} \Big \langle \Psi \Big |\mathbb {H}_\Lambda -\mathcal {N}+\sum _{n=1}^N a_n^\dagger a_n\Big |\Psi \Big \rangle +C'\alpha ^{-2(1+\sigma )}\ge \langle \Psi |\mathbb {K}|\Psi \rangle =\langle \mathbb {K}\rangle _{\gamma }, \end{aligned}$$

where \(\gamma \) is the reduced density matrix on the Hilbert space \(\mathcal {F}\left( \Pi L^2\! \left( \mathbb {R}^3\right) \right) \cong L^2\! \left( \mathbb {R}^{N}\right) \) corresponding to the state \(\Psi \), i.e. we trace out the electron component as well as all the modes in the orthogonal complement of \(\Pi L^2\! \left( \mathbb {R}^3\right) \),

$$\begin{aligned} \gamma :=\textrm{Tr}_{L^2\! \left( \mathbb {R}^3\right) \otimes \mathcal {F}\left( L^2\! \left( \mathbb {R}^3\right) \right) \rightarrow \mathcal {F}\left( \Pi L^2\! \left( \mathbb {R}^3\right) \right) }\left[ \, \vert \Psi \rangle \langle \Psi \vert \, \right] . \end{aligned}$$

Note that we have the inequality \(W_{\Pi \varphi ^\textrm{Pek}}^{-1}\left( \sum _{n=1}^{N} a_n^\dagger a_n\right) W_{\Pi \varphi ^\textrm{Pek}}\le W_{\varphi ^\textrm{Pek}}^{-1}\, \mathcal {N}W_{\varphi ^\textrm{Pek}}\). The operators on the left and right hand side commute, and consequently (6.1) implies that \(\chi \left( W_{\Pi \varphi ^\textrm{Pek}}^{-1}\left( \sum _{n=1}^{N} a_n^\dagger a_n\right) W_{\Pi \varphi ^\textrm{Pek}}\le \alpha ^{-h}\right) \Psi =\Psi \). This in particular means that the transformed reduced density matrix \(\widetilde{\gamma }:=W_{\Pi \varphi ^\textrm{Pek}} \gamma W_{\Pi \varphi ^\textrm{Pek}}^{-1}\) satisfies

$$\begin{aligned} \chi \left( \sum _{n=1}^{N} a_n^\dagger a_n\le \alpha ^{-h}\right) \widetilde{\gamma }=\widetilde{\gamma }. \end{aligned}$$

Using the identification \(\varphi =\sum _{n=1}^{N} \lambda _n \varphi _n\) as before, the Weyl operator \(W_{\Pi \varphi ^\textrm{Pek}}\) acts as \(\left( W_{\Pi \varphi ^\textrm{Pek}}\Psi \right) \left( \lambda \right) =\Psi \left( \lambda +\lambda ^\textrm{Pek}\right) \) with \(\lambda ^\textrm{Pek}:=\big (\langle \varphi _1|\varphi ^\textrm{Pek}\rangle ,\dots ,\langle \varphi _{N}|\varphi ^\textrm{Pek}\rangle \big )\). Due to Eq. (6.6), and the fact that \(2\,s<h\) and \(\sigma < \frac{1-5\,s}{4}\), the assumptions of Proposition 4.2 are satisfied, and therefore we obtain for any \(D>0\) the existence of a constant \(\beta >0\) such that for \(\alpha \) large enough

$$\begin{aligned} \int _{|\lambda -\lambda ^\text {Pek}|_\diamond \ge \alpha ^{-s}D}\left( 1+|\lambda -\lambda ^\text {Pek}|^2\right) \rho (\lambda )\text {d}\lambda&=\int _{|\lambda |_\diamond \ge \alpha ^{-s}D}\left( 1+|\lambda |^2\right) \widetilde{\rho }(\lambda )\text {d}\lambda \le e^{-\beta \alpha ^{2(1-s)}}, \end{aligned}$$
$$\begin{aligned} \int _{\left| t^\lambda \right| \ge \alpha ^{-s}D}\left( 1+|\lambda -\lambda ^\textrm{Pek}|^2\right) \rho (\lambda )\textrm{d}\lambda&\le \sum _{n=1}^3\int _{|\lambda _n|\ge \frac{\alpha ^{-s}}{\sqrt{3}}D}\left( 1+|\lambda -\lambda ^\textrm{Pek}|^2\right) \rho (\lambda )\textrm{d}\lambda \nonumber \\&= \sum _{n=1}^3\int _{|\lambda _n|\ge \frac{\alpha ^{-s}}{\sqrt{3}}D}\left( 1+|\lambda |^2\right) \widetilde{\rho }(\lambda )\textrm{d}\lambda \le e^{-\beta \alpha ^{2(1-s)}}, \end{aligned}$$

where \(\rho \) and \(\widetilde{\rho }\) are the density functions corresponding to \(\gamma \) and \(\widetilde{\gamma }\), respectively, and where we have used \(t^\lambda =(\lambda _1,\lambda _2,\lambda _3)\in \mathbb {R}^3\). For the concrete choice \(D:=1\), Eq. (6.8) immediately yields the claim \(\langle \Psi |1-\mathbb {B}|\Psi \rangle =\int _{\left| t^\lambda \right| \ge \alpha ^{-s}} \rho (\lambda )\textrm{d}\lambda \le e^{-\beta \alpha ^{2(1-s)}}\).

In order to verify Eq. (6.2), we need to find a sufficient lower bound for the expectation value \(\langle \mathbb {K}\rangle _{\gamma }\), where \(\mathbb {K}\) is the operator from Eq. (6.4). Recall the definition of the transformation \(\tau :\Pi L^2\! \left( \mathbb {R}^3\right) \longrightarrow \Pi L^2\! \left( \mathbb {R}^3\right) \) from Definition 5.3 and the operator \(J_{t,\epsilon }\) from Eq. (5.13). As a first step we will provide a lower bound on \(\langle \mathcal {F}^\textrm{Pek}\!\left( \lambda \right) \rangle _{\gamma }\), using Eq. (5.14) and the fact that \(\sup _{|t|\le t_0}\left\| \left( 1-\Pi \right) \varphi ^\textrm{Pek}_{x_{t}}\right\| ^2\lesssim \alpha ^{-\frac{12}{5}(1+\sigma )}\) for \(t_0\) small enough, which follows from Lemma A.1 together with \(x_t\underset{t\rightarrow 0}{\longrightarrow }\ 0\). We define the operator \(\mathbb {A}:=\chi \left( |\lambda -\lambda ^\textrm{Pek}|_\diamond<\alpha ^{-s}D\right) \chi \left( \left| t^\lambda \right| <\alpha ^{-s}D\right) \), where D is as in Theorem 5.4, and estimate

$$\begin{aligned}&\langle \mathcal {F}^\textrm{Pek}\! \left( \lambda \right) \rangle _{\gamma }=\langle \mathcal {F}^\textrm{Pek}\! \left( \lambda \right) \mathbb {A}\rangle _{\gamma }+\langle \mathcal {F}^\textrm{Pek}\! \left( \lambda \right) \left( 1-\mathbb {A}\right) \rangle _{\gamma }\nonumber \\&\quad \ge \Big \langle \big (e^\textrm{Pek}+J_{t^\lambda ,\alpha ^{-s}}\! \big [ \tau \! \left( \lambda \right) \big ]\big )\mathbb {A}\Big \rangle _{\gamma }+\langle \mathcal {F}^\textrm{Pek}\! \left( \lambda \right) \left( 1-\mathbb {A}\right) \rangle _{\gamma }+O\left( \alpha ^{s-\frac{12}{5}(1+\sigma )}\right) \nonumber \\&\quad =\Big \langle e^\textrm{Pek}\!+\!J_{t^\lambda ,\alpha ^{-s}}\! \big [ \tau \! \left( \lambda \right) \big ]\Big \rangle _{\gamma }\! \! \!+\!\Big \langle X \Big \rangle _{\gamma }+\!O\left( \! \alpha ^{s-\frac{12}{5}(1+\sigma )}\! \right) \end{aligned}$$

with \(X:=\left( \mathcal {F}^\textrm{Pek}\! \left( \lambda \right) -e^\textrm{Pek}-J_{t^\lambda ,\alpha ^{-s}}\! \big [ \tau \! \left( \lambda \right) \big ]\right) \left( 1-\mathbb {A}\right) \). Using Eqs. (6.7) and (6.8) as well as the fact that \(1-\mathbb {A}\le \chi \left( |\lambda -\lambda ^\textrm{Pek}|_\diamond \ge D\alpha ^{-s}\right) +\chi \left( \left| t^\lambda \right| \ge D\alpha ^{-s}\right) \), we obtain \(\Big \langle X \Big \rangle _{\gamma }\lesssim e^{-\beta \alpha ^{2(1-s)}}\), where we have used that \(\mathcal {F}^\textrm{Pek}\! \left( \lambda \right) \) and \(J_{t^\lambda ,\alpha ^{-s}}\! \big [ \tau \! \left( \lambda \right) \big ]\) are bounded by \(C(1+|\lambda |^2)\) for suitable \(C>0\). By Eq. (6.9) we therefore have the estimate \(\langle \mathcal {F}^\textrm{Pek}\! \left( \lambda \right) \rangle _{\gamma }\ge \Big \langle e^\textrm{Pek}\!+\!J_{t^\lambda ,\alpha ^{-s}}\! \big [ \tau \! \left( \lambda \right) \big ]\Big \rangle _{\gamma }\! \! \!+\!O\left( \! \alpha ^{s-\frac{12}{5}(1+\sigma )}\! \right) \), and consequently

$$\begin{aligned} \langle \mathbb {K}\rangle _{\gamma }\ge e^\textrm{Pek}\! \! +\Big \langle \! -\frac{1}{4\alpha ^4}\sum _{n=1}^{N}\partial _{\lambda _n}^2+J_{t^\lambda ,\alpha ^{-s}}\! \big [ \tau \! \left( \lambda \right) \big ]\Big \rangle _{\gamma }\! \! \! \! -\frac{{N}}{2\alpha ^2}+O\left( \alpha ^{s-\frac{12}{5}(1+\sigma )}\right) . \end{aligned}$$

Since \(\Big \langle \! -\frac{1}{4\alpha ^4}\sum _{n=4}^{N}\partial _{\lambda _n}^2+J_{t^\lambda ,\alpha ^{-s}}\! \big [ \tau \! \left( \lambda \right) \big ]\Big \rangle _{\gamma }=\Big \langle \Psi \Big |\! -\frac{1}{4\alpha ^4}\sum _{n=4}^{N}\partial _{\lambda _n}^2+J_{t^\lambda ,\alpha ^{-s}}\! \big [ \tau \! \left( \lambda \right) \big ]\Big |\Psi \Big \rangle \), this concludes the proof together with Eq. (6.5). \(\quad \square \)

In the following, let \(\Psi _\alpha \) be the sequence of states constructed in Theorem 3.13, satisfying \(\langle \Psi _\alpha |\mathbb {H}_\Lambda |\Psi _\alpha \rangle -E_\alpha \lesssim \alpha ^{-2(1+\sigma )}\), \(\textrm{supp}\left( \Psi _\alpha \right) \subset B_{4L}(0)\) with \(L=\alpha ^{1+\sigma }\) and strong condensation with respect to \(\varphi ^\textrm{Pek}\), i.e. \(\chi \left( W_{\varphi ^\textrm{Pek}}^{-1}\, \mathcal {N}W_{\varphi ^\textrm{Pek}}\le \alpha ^{-h}\right) \Psi _\alpha =\Psi _\alpha \), and furthermore let \(s<\frac{1}{29}\) be a given constant and let us choose \(\sigma \) and h such that \(2\,s<h<\frac{2}{29}\) and \(\frac{s}{2}\le \sigma <\frac{1-5\,s}{4}\). Note that \(h<\frac{2}{29}\) makes sure that the assumption of Theorem 3.13 is satisfied, while \(2\,s<h\) and \(\sigma <\frac{1-5\,s}{4}\) are necessary in order to apply Lemma 6.1. The final assumption \(\frac{s}{2}\le \sigma \) will be useful later in Eq. (6.15) in order to make sure that \(\alpha ^{-2(1+\sigma )}\le \alpha ^{-(2+s)}\). Making use of \(-\frac{1}{4\alpha ^4}\sum _{n=1}^3 \partial _{\lambda _n}^2\ge 0\) and \(\mathcal {N}\ge \sum _{n=1}^N a_n^\dagger a_n\), we obtain by Lemma 6.1 that

$$\begin{aligned} E_\alpha \ge e^\textrm{Pek} \!+ \!\Big \langle \Psi _\alpha \Big | \! -\frac{1}{4\alpha ^4}\sum _{n=4}^{N}\partial _{\lambda _n}^2 \!+ \!J_{t^\lambda ,\alpha ^{-s}}\! \big [ \tau \! \left( \lambda \right) \big ]\Big |\Psi _\alpha \Big \rangle \! - \!\frac{{N}}{2\alpha ^2} \!+ \!O\left( \alpha ^{-2(1+\sigma )}\right) \end{aligned}$$

for a suitable \(C'\), where we have used \(\alpha ^{s-\frac{12}{5}(1+\sigma )}\le \alpha ^{-2(1+\sigma )}\) and \(E_\alpha -\langle \Psi _\alpha |\mathbb {H}_\Lambda |\Psi _\alpha \rangle \gtrsim -\alpha ^{-2(1+\sigma )}\). In order to further estimate the expectation value in Eq. (6.11), let us define the unitary transformation \(\left( \mathcal {U}\Psi \right) (\lambda ):=\Psi \left( \tau '\left( \lambda \right) \right) \) with \(\tau '\left( \lambda \right) :=\Big (\big \langle \varphi _n|\tau \left( \lambda \right) \big \rangle \Big )_{n=1}^{N}\in \mathbb {R}^N\). Since \(\tau '\) acts as a shift operator on each of the planes \(X_t:=\{\lambda :(\lambda _1,\lambda _2,\lambda _3)=t\}\) for \(t\in \mathbb {R}^3\), it is clear that \(\textrm{det}\, \textrm{D}|_\lambda \tau '=1\), which in particular means that the operator \(\mathcal {U}\) is indeed unitary, and we have \(\partial _{\lambda _n}=\mathcal {U}^{-1}\partial _{\lambda _n}\mathcal {U}\) for \(n\ge 4\). Furthermore we define the operator

$$\begin{aligned} \mathbb {Q}_{t,\epsilon }:= -\frac{1}{4\alpha ^4}\sum _{n=4}^{N}\partial _{\lambda _n}^2+\sum _{n,m=1}^{N}\left( J_{t,\epsilon }\right) _{n,m}\, \lambda _n\lambda _m \end{aligned}$$

with \(\left( J_{t,\epsilon }\right) _{n,m}:=\langle \varphi _n|J_{t,\epsilon }|\varphi _m\rangle \). Note that \(\left( J_{t,\epsilon }\right) _{n,m}=\left( J_{t,\epsilon }\right) _{m,n}=0\) in case \(n\in \{1,2,3\}\), i.e. the operator \(\mathbb {Q}_{t,\epsilon }\) depends only on the variables \(\lambda _n\) for \(n\ge 4\) and not on \(t^\lambda =(\lambda _1,\lambda _2,\lambda _3)\), hence it acts on the Fock space \(\mathcal {F}\big (\textrm{span}\{\varphi _4,\dots ,\varphi _{N}\}\big )\cong L^2\! \left( \mathbb {R}^{N-3}\right) \) only. Utilizing the fact that \(\mathcal {U}^{-1}J_{t^\lambda ,\alpha ^{-s}}\! \big [ \tau \! \left( \lambda \right) \big ]\, \mathcal {U}=J_{t^\lambda ,\alpha ^{-s}}\! \big [ \lambda \big ]=\sum _{n,m=1}^{N}\left( J_{t^\lambda ,\alpha ^{-s}}\right) _{n,m}\, \lambda _n\lambda _m\), where we used that \(\mathcal {U}^{-1}t^\lambda \mathcal {U}=t^\lambda \), we obtain

$$\begin{aligned}&\mathcal {U}^{-1}\! \! \left( \! -\frac{1}{4\alpha ^4}\sum _{n=4}^{N}\partial _{\lambda _n}^2\!\! +\!J_{t^\lambda ,\alpha ^{-s}}\! \big [ \tau \! \left( \lambda \right) \big ] \right) \! \mathcal {U}=\mathbb {Q}_{t^\lambda ,\alpha ^{-s}}\ge \mathbb {Q}_{t^\lambda ,\alpha ^{-s}}\mathbb {B}\ge \inf _{|t|<\alpha ^{-s}}\inf \sigma \left( \mathbb {Q}_{t,\alpha ^{-s}}\right) \mathbb {B}, \end{aligned}$$

where \(\mathbb {B}\) is as in Lemma 6.1. Here we have used \(\mathbb {Q}_{t^\lambda ,\alpha ^{-s}}\ge 0\), which follows from Lemma B.5, as well as the fact that \(1-\mathbb {B}\) is non-negative and commutes with \(\mathbb {Q}_{t^\lambda ,\alpha ^{-s}}\). Applying this inequality with respect to the state \(\widetilde{\Psi }_\alpha :=\mathcal {U}^{-1}\Psi _\alpha \) yields

$$\begin{aligned}&\Big \langle \Psi _\alpha \Big | -\frac{1}{4\alpha ^4}\sum _{n=4}^{N}\partial _{\lambda _n}^2\! +\! J_{t^\lambda ,\alpha ^{-s}}\! \big [ \tau \! \left( \lambda \right) \big ]\Big |\Psi _\alpha \Big \rangle \ge \inf _{|t|<\alpha ^{-s}}\inf \sigma \left( \mathbb {Q}_{t,\alpha ^{-s}}\right) \big \langle \widetilde{\Psi }_\alpha \big | \mathbb {B} \big | \widetilde{\Psi }_\alpha \big \rangle \nonumber \\&\quad \ge \inf _{\left| t\right| <\alpha ^{-s}}\! \! \! \inf \! \sigma \left( \mathbb {Q}_{t,\alpha ^{-s}}\right) \! -\! \frac{{N}}{2\alpha ^2}\big \langle \widetilde{\Psi }_\alpha \big | 1-\mathbb {B}\big | \widetilde{\Psi }_\alpha \big \rangle \end{aligned}$$

where we have used \(J_{t,\epsilon }\le 1\), and therefore \(\inf \sigma \left( \mathbb {Q}_{t,\epsilon }\right) \le \frac{{N}}{2\alpha ^2}\). By Lemma 6.1, we know that \(\big \langle \widetilde{\Psi }_\alpha \big | 1-\mathbb {B}\big | \widetilde{\Psi }_\alpha \big \rangle =\big \langle \Psi _\alpha \big | 1-\mathbb {B}\big | \Psi _\alpha \big \rangle \le e^{-\beta \alpha ^{2-2\,s}}\). Combining Eqs. (6.11) and (6.12), and making use of the fact that \({N}\lesssim \alpha ^p\) for some \(p>0\), yields

$$\begin{aligned} E_\alpha \ge e^\textrm{Pek}+\inf _{\left| t\right| <\alpha ^{-s}}\inf \sigma \left( \mathbb {Q}_{t,\alpha ^{-s}}\right) -\frac{{N}}{2\alpha ^2}+O\left( \alpha ^{-2(1+\sigma )}\right) . \end{aligned}$$

Since the operator \(\mathbb {Q}_{t,\alpha ^{-s}}\) is quadratic in \(\partial _{\lambda _n}\) and \(\lambda _n\), we have an explicit formula for its ground state energy, given by

$$\begin{aligned} \inf \sigma \left( \mathbb {Q}_{t,\alpha ^{-s}}\right) -\frac{N}{2\alpha ^2}=-\frac{\textrm{Tr}_{\Pi L^2\! \left( \mathbb {R}^3\right) }\! \! \left[ 1 -\sqrt{ J_{t,\alpha ^{-s}} }\right] }{2\alpha ^2}, \end{aligned}$$

where we used the fact that \(J_{t,\alpha ^{-s}}\ge 0\) for \(\alpha \) large enough, as shown in Lemma B.5. Using Eq. (B.7), we can approximate this quantity by

$$\begin{aligned} \sup _{|t|<\alpha ^{-s}}\left| \textrm{Tr}_{\Pi L^2\! \left( \mathbb {R}^3\right) }\! \!\left[ 1 -\sqrt{ J_{t,\alpha ^{-s}} }\, \right] -\textrm{Tr}\left[ 1-\sqrt{H^\textrm{Pek}}\, \right] \right| \lesssim \alpha ^{-s}+\alpha ^{-\frac{1}{5}}, \end{aligned}$$

where \(H^\textrm{Pek}\) is defined in Eq. (1.4). Consequently Eq. (6.13) yields

$$\begin{aligned} E_\alpha \! -\! e^\textrm{Pek}\! +\! \frac{1}{2\alpha ^2}\textrm{Tr}\left[ 1\! -\! \sqrt{H^\textrm{Pek}}\, \right] \gtrsim \! - \alpha ^{-2(1+\sigma )}\! -\! \alpha ^{-(2+s)}\! -\! \alpha ^{-\left( 2+\frac{1}{5}\right) }, \end{aligned}$$

which concludes the proof, since all the terms on the right side are of order \(\alpha ^{-(2+s)}\).

7 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 \(\Omega _{\varphi ^\textrm{Pek}_x}\) with \(x\in \mathbb {R}^3\), 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 \(\big \langle \Psi \big |\widehat{F}\big |\Psi \big \rangle \) in terms of the lower symbol \(\mathbb {P}_y\) corresponding to the state \(\Psi \), see Eq. (7.6). Furthermore we will make use of the inequality

$$\begin{aligned} \inf _{x\in \mathbb {R}^3}\Vert \varphi -\varphi ^\textrm{Pek}_x\Vert ^2\lesssim \mathcal {F}^\textrm{Pek}\! \left( \varphi \right) - e^\textrm{Pek} \end{aligned}$$

derived in [10, Lemma 7], which implies that the only coherent states \(\Omega _\varphi \) with a low energy have their point of condensation \(\varphi \) close to the manifold of Pekar minimizers \(\{\varphi ^\textrm{Pek}_x:x\in \mathbb {R}^3\}\). We start with the subsequent Lemma 7.1, which provides an asymptotic formula for \(\widehat{F}\) operators in terms of creation and annihilation operators.

Lemma 7.1

Let \(m\in \mathbb {N}\) and \(C>0\) be given constants, \(\{g_n:n\in \mathbb {N}\}\) an orthonormal basis of \(L^2\!\left( \mathbb {R}^3\right) \) 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

$$\begin{aligned} F\left( \rho \right) =\int \dots \int f(x_1,\dots , x_m)\, \textrm{d}\rho (x_1)\dots \textrm{d}\rho (x_m), \end{aligned}$$

with \(f:\mathbb {R}^{3\times m}\longrightarrow \mathbb {R}\) bounded, and states \(\Psi \) satisfying \(\chi \left( \mathcal {N}\le C\right) \Psi =\Psi \), we can approximate the operator \(\widehat{F}\) from Definition 3.1 by

$$\begin{aligned} \left| \big \langle \Psi \big |\widehat{F}\big |\Psi \big \rangle -\sum _{I,J\in \mathbb {N}^m} f_{I,J}\big \langle \Psi \big |a_{I_1}^\dagger \dots a_{I_m}^\dagger a_{J_1}\dots a_{J_m}\big |\Psi \big \rangle \right| \le T\Vert f\Vert _\infty \alpha ^{-2}, \end{aligned}$$

where we interpret f as a multiplication operator on \(L^2\! \left( \mathbb {R}^3\right) ^{\otimes ^m}\cong L^2\! \left( \mathbb {R}^{3\times m}\right) \) and denote the matrix elements \(f_{I,J}:=\langle g_{I_1}\otimes \dots \otimes g_{I_m}|f|g_{J_1}\otimes \dots \otimes g_{J_m}\rangle \).


By the assumption \(\chi \left( \mathcal {N}\le C\right) \Psi =\Psi \), we can represent the state \(\Psi \) as \(\Psi =\bigoplus _{n\le C\alpha ^2} \Psi _n\) where \(\Psi _n(y,x^1,\dots ,x^{n})\) is a function of the electron variable y and the n phonon coordinates \(x^k\in \mathbb {R}^3\). 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 \(\widehat{F}\) in Definition 3.1, as well as the notation \(X=(x^1,\dots ,x^n)\), we can write

$$\begin{aligned} \big \langle \Psi \big |\widehat{F}\big |\Psi \big \rangle&=\sum _{n\le C\alpha ^2}\int _{\mathbb {R}^{3n}}F\left( \alpha ^{-2}\sum _{k=1}^n\delta _{x^k}\right) |\Psi _n(X)|^2\textrm{d}X\\&=\alpha ^{-2m}\sum _{n\le C\alpha ^2}\sum _{k\in \{1,\dots ,n\}^m}\int _{\mathbb {R}^{3n}}f(x^{k_1},\dots ,x^{k_m})|\Psi _n(X)|^2\textrm{d}X. \end{aligned}$$

Defining \(\mathcal {K}\) as the set of all \(k\in \{1,\dots ,n\}^m\) satisfying \(k_i\ne k_j\) for all \(i\ne j\), we can further express the operator \(\sum _{I,J\in \mathbb {N}^m} f_{I,J}\, a_{I_1}^\dagger \dots a_{I_m}^\dagger a_{J_1}\dots a_{J_m}\) as

$$\begin{aligned} \sum _{I,J\in \mathbb {N}^m}\! \! \! f_{I,J}\big \langle \Psi \big |a_{I_1}^\dagger \dots a_{I_m}^\dagger a_{J_1}\dots a_{J_m}\big |\Psi \big \rangle = \alpha ^{-2m}\! \! \sum _{n\le C\alpha ^2}\sum _{k\in \mathcal {K}}\int _{\mathbb {R}^{3n}}\! \! f(x^{k_1},\dots ,x^{k_m})|\Psi _n(X)|^2\textrm{d}X. \end{aligned}$$

Consequently we can identify the left hand side of Eq. (7.3) as

$$\begin{aligned}&\left| \alpha ^{-2m}\sum _{n\le C\alpha ^2}\sum _{k\in \{1,\dots ,n\}^m\setminus \mathcal {K}}\int _{\mathbb {R}^{3n}}f(x^{k_1},\dots ,x^{k_m})|\Psi _n(X)|^2\textrm{d}X\right| \\&\quad \le \Vert f\Vert _\infty \sum _{n\le C\alpha ^2}\left( \sum _{k\in \{1,\dots ,n\}^m\setminus \mathcal {K}}\alpha ^{-2m}\right) \int _{\mathbb {R}^{3n}}|\Psi _n(X)|^2\textrm{d}X. \end{aligned}$$

Since \(\sum _{k\in \{1,\dots ,n\}^m{\setminus } \mathcal {K}}\alpha ^{-2\,m}=\left( n^m-\frac{n!}{(n-m)!}\right) \alpha ^{-2\,m}\le m 2^m n^{m-1}\alpha ^{-2\,m}\lesssim \alpha ^{-2}\) for \(n\le C\alpha ^2\) and since \(\sum _{n\le C\alpha ^2}\int _{\mathbb {R}^{3n}}|\Psi _n(X)|^2\textrm{d}X=\Vert \Psi \Vert ^2=1\), this concludes the proof. \(\square \)

In the following we are going to define the lower symbol \(\mathbb {P}_y\) corresponding to a state \(\Psi \in L^2\Big (\mathbb {R}^3,\mathcal {F}\big (L^2\! \left( \mathbb {R}^3\right) \big )\Big )\). Since we consider the Fock space over the infinite dimensional Hilbert space \(L^2\! \left( \mathbb {R}^3\right) \), we need to modify the usual definition of the lower symbol by introducing suitable localizations. For \(0<s\le \frac{4}{27}\) and \(y\in \mathbb {R}^3\), let us define \(\ell _*:=\alpha ^{-\frac{5}{2}s}\) and \(\Lambda _*:=\alpha ^{2\,s}\), and the projection

$$\begin{aligned} \Pi _{y}:=\Pi ^y_{\Lambda _*,\ell _*}, \end{aligned}$$

see Definition 2.1. We have \(N_*:=\textrm{dim}\,\Pi _{y}L^2\! \left( \mathbb {R}^3\right) \lesssim \left( {\Lambda _*}/{\ell _*}\right) ^3\le \alpha ^2\) by our assumption \(s\le \frac{4}{27}\). Using the notation \(\{e_{y,1},\dots ,e_{y,{N_*}}\}\) for the orthonormal basis of \(\Pi _{y}L^2\! \left( \mathbb {R}^3\right) \) from Definition 2.1, we introduce for \(\xi \in \mathbb {C}^{N_*}\) the coherent states \(\Omega _{y,\xi }:=e^{\alpha ^2 a^\dagger \left( \varphi _{y,\xi }\right) - \alpha ^2 a\left( \varphi _{y,\xi }\right) }\Omega \), where \(\Omega \) is the vacuum in \(\mathcal {F}\left( \Pi _y L^2\! \left( \mathbb {R}^3\right) \right) \) and \(\varphi _{y,\xi }:=\)\(\sum _{n=1}^{N_*} \xi _n e_{y,n}\in \Pi _{y}L^2\! \left( \mathbb {R}^3\right) \). Furthermore we define wave-functions \(\Psi _y\) localized in the electron coordinates x as

$$\begin{aligned} \Psi _{y}(x):=L_*^{-\frac{3}{2}}\chi \left( \frac{x-y}{L_*}\right) \Psi (x), \end{aligned}$$

where \(y\in \mathbb {R}^3\) and \(L_*:=\alpha ^{\frac{s}{2}}\), and \(\chi \) is a smooth non-negative function with \(\textrm{supp} \left( \chi \right) \subset B_1(0)\) and \(\int \chi (y)^2\, \textrm{d}y=1\). For the following construction, note that we can identify \(L^2\Big (\mathbb {R}^3,\mathcal {F}\big (L^2\! \left( \mathbb {R}^3\right) \big )\Big )\cong \mathcal {F}\left( \Pi _y L^2\! \left( \mathbb {R}^3\right) \right) \otimes L^2\! \Big (\mathbb {R}^3,\mathcal {F}\Big (\Pi _y L^2\! \left( \mathbb {R}^3\right) ^\perp \Big )\Big )\). Let us now define measures \(\mathbb {P}_y\) on \(\mathbb {C}^{N_*}\cong \mathbb {R}^{2{N_*}}\) corresponding to the state \(\Psi _y\) as

$$\begin{aligned} \frac{\textrm{d}\mathbb {P}_y}{\textrm{d}\xi }:=\frac{1}{\pi ^{N_*}}\left\| \Theta _{y,\xi }\Psi _{y}\right\| ^2, \end{aligned}$$

where \(\Theta _{y,\xi }\) is the orthogonal projection onto the set spanned by elements of the form \(\Omega _{y,\xi }\otimes \widetilde{\Psi }\) with \(\widetilde{\Psi }\in L^2\Big (\mathbb {R}^3,\mathcal {F}\Big (\Pi _y L^2\! \left( \mathbb {R}^3\right) ^\perp \Big )\Big )\). Note that the coherent states \(\Omega _{y,\xi }\) provide a resolution of the identity \(\frac{1}{\pi ^{N_*}}\int _{\mathbb {C}^{N_*}}\vert \Omega _{y,\xi }\rangle \langle \Omega _{y,\xi }\vert \textrm{d}\xi =1_{\mathcal {F}\left( \Pi _y L^2\! \left( \mathbb {R}^3\right) \right) }\), see for example [20], and consequently the projections \(\Theta _{y,\xi }\) satisfy \(\frac{1}{\pi ^{N_*}}\int _{\mathbb {C}^{N_*}}\Theta _{y,\xi }\, \textrm{d}\xi =1\). In particular we see that the total mass of the measure \(\mathbb {P}_y\) is \(\int \textrm{d}\mathbb {P}_y=\Vert \Psi _y\Vert ^2\) and therefore

$$\begin{aligned} \iint \textrm{d}\mathbb {P}_y\textrm{d}y=\int \Vert \Psi _y\Vert ^2\textrm{d}y=\Vert \Psi \Vert ^2=1. \end{aligned}$$

In the following Lemma 7.2 and Corollary 7.3 we will provide an asymptotic formula for the expectation value \(\big \langle \Psi _y\big |\widehat{F}\big |\Psi _y\big \rangle \), respectively \(\big \langle \Psi \big |\widehat{F}\big |\Psi \big \rangle \), in terms of the measures \(\mathbb {P}_y\).

Lemma 7.2

Given \(m\in \mathbb {N}\), \(C>0\) and \(g\in L^2\! \left( \mathbb {R}^3\right) \), there exists a \(T>0\) such that for all F of the form (7.2), \(y\in \mathbb {R}^3\) and \(\epsilon >0\), and states \(\Psi \) satisfying \(\chi \left( \mathcal {N}\le C\right) \Psi =\Psi \)

$$\begin{aligned} \frac{1}{T\Vert f\Vert _{\infty }}\! \left| \big \langle \Psi _y\big |\widehat{F}\,\big |\Psi _y \big \rangle \! -\! \int F\left( |\varphi _{y,\xi }|^2\right) \textrm{d}\mathbb {P}_y\! \left( \xi \right) \right| \! \le \! \left( \! \frac{{N_*}}{\alpha ^2}+\epsilon \! \right) \! \Vert \Psi _y\Vert ^2\! +\! \epsilon ^{-1}\big \langle \Psi _y\big |\mathcal {N}^y_{>{N_*}}\big |\Psi _y\big \rangle , \end{aligned}$$

with \(\mathcal {N}^y_{>{N_*}}:=\mathcal {N}-\sum _{n=1}^{N_*} a_{y,n}^\dagger a_{y,n}\) and \(a_{y,n}:=a\left( e_{y,n}\right) \), and furthermore

$$\begin{aligned} \frac{1}{T}\left| \big \langle \Psi _y\big |W_{g}^{-1} \mathcal {N}W_{g}\big |\Psi _y \big \rangle \! - \! \! \int \! \! \Vert \varphi _{y,\xi }\! -\! g\Vert ^2 \textrm{d}\mathbb {P}_y\! \left( \xi \right) \right| \! \le \! \left( \! \frac{{N_*}}{\alpha ^2}\! +\! \epsilon \! \right) \! \Vert \Psi _y\Vert ^2\! +\! \epsilon ^{-1}\big \langle \Psi _y\big |\mathcal {N}^y_{>{N_*}}\big |\Psi _y\big \rangle , \end{aligned}$$

where \(W_{g}\) is the corresponding Weyl transformation.


Let \(\{g_n:n\in \mathbb {N}\}\) be a completion of \(\{e_{y,1},\dots ,e_{y,{N_*}}\}\) to an orthonormal basis of \(L^2\! \left( \mathbb {R}^3\right) \) and let us define \(a_n:=a\left( g_n\right) \). We further introduce an operator \(\widetilde{F}\) as

$$\begin{aligned} \widetilde{F}:=\! \! \! \! \! \! \! \sum _{I,J\in \{1,\dots ,{N_*}\}^m}\! \! \! \! \! \! \! \! f_{I,J}\, a_{I_1}^\dagger \! \dots a_{I_m}^\dagger a_{J_1}\! \dots a_{J_m}\! =\! \! \! \! \! \sum _{I,J\in \mathbb {{N}}^m}\! \! \! \! \! \left( \Pi _y^{\otimes ^m}f\Pi _y^{\otimes ^m}\right) _{I,J} a_{I_1}^\dagger \! \dots a_{I_m}^\dagger a_{J_1}\! \dots a_{J_m}. \end{aligned}$$

In the following we want to verify that both \(\Vert f\Vert _\infty ^{-1} \left| \big \langle \Psi _y\big |\widehat{F}\,\big |\Psi _y \big \rangle -\big \langle \Psi _y\big |\widetilde{F}\,\big |\Psi _y \big \rangle \right| \) and \(\Vert f\Vert _\infty ^{-1} \left| \big \langle \Psi _y\big |\widetilde{F}\,\big |\Psi _y \big \rangle -\int F\left( |\varphi _{y,\xi }|^2\right) \, \textrm{d}\mathbb {P}_y\left( \xi \right) \right| \) 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 \(\epsilon >0\)

$$\begin{aligned} \pm \! \left( f\! -\!\Pi _y^{\otimes ^m}f\, \Pi _y^{\otimes ^m}\right)&=\pm f\left( 1\! -\! \Pi _y^{\otimes ^m}\right) \! \pm \! \left( 1\! -\! \Pi _y^{\otimes ^m}\right) f\, \Pi _y^{\otimes ^m}\\&\le \epsilon \Vert f\Vert _{\infty }\!+\!\epsilon ^{-1}\Vert f\Vert _{\infty }\left( 1\!-\!\Pi _y^{\otimes ^m}\right) \\&\le \epsilon \Vert f\Vert _{\infty }\!+\!\epsilon ^{-1}\Vert f\Vert _{\infty }\big ((1\!-\!\Pi _y)_1\!+\!\dots \!+\!(1\!-\!\Pi _y)_m\big ), \end{aligned}$$

where \((1-\Pi _y)_j\) means that the operator \(1-\Pi _y\) acts on the j-th factor in the tensor product. Consequently we have the operator inequality

$$\begin{aligned} \pm \left( \sum _{I,J\in \mathbb {N}^m} f_{I,J}\, a_{I_1}^\dagger \dots a_{I_m}^\dagger a_{J_1}\dots a_{J_m}-\widetilde{F}\right) \le \epsilon \Vert f\Vert _{\infty }\mathcal {N}^m+\epsilon ^{-1}\Vert f\Vert _{\infty }m\, \mathcal {N}^y_{>{N_*}}\mathcal {N}^{m-1}. \end{aligned}$$

Making use of Eq. (7.3) and the fact that \(\chi \left( \mathcal {N}\le C\right) \Psi _y=\Psi _y\) further yields

$$\begin{aligned} \left| \langle \Psi _y|\widehat{F}|\Psi _y\rangle -\sum _{I,J\in \mathbb {N}^m} f_{I,J}\,\langle \Psi _y| a_{I_1}^\dagger \dots a_{I_m}^\dagger a_{J_1}\dots a_{J_m}|\Psi _y\rangle \right| \le d\alpha ^{-2}\Vert f\Vert _\infty \Vert \Psi _y\Vert ^2 \end{aligned}$$

for a suitable constant \(d>0\). We have thus shown the bound

$$\begin{aligned} \frac{1}{\Vert f\Vert _{\infty }}\left| \! \langle \Psi _y|\widehat{F}\, |\Psi _y\rangle \! -\! \langle \Psi _y|\widetilde{F}\, |\Psi _y \rangle \! \right| \! \le \! \left( d\alpha ^{-2}\! +\! \epsilon C^m\right) \! \Vert \Psi _y\Vert ^2\! +\! \epsilon ^{-1}m C^{m-1}\! \big \langle \Psi _y\big |\mathcal {N}^y_{>{N_*}}\big |\Psi _y\big \rangle \end{aligned}$$

which is of the desired form.

In order to verify that \(\frac{1}{\Vert f\Vert _{\infty }}\left| \big \langle \Psi _y\big |\widetilde{F}\,\big |\Psi _y \big \rangle -\int F\left( |\varphi _{y,\xi }|^2\right) \, \textrm{d}\mathbb {P}_y\left( \xi \right) \right| \) is of the same order as the right hand side of Eq. (7.7) as well, we will first compute \(\widetilde{F}\) with reversed operator ordering, i.e. we compute

$$\begin{aligned}&\sum _{I,J\in \{1,\dots ,{N_*}\}^m} f_{I,J}\, a_{J_1}\dots a_{J_m} a_{I_1}^\dagger \dots a_{I_m}^\dagger = \sum _{I,J\in \{1,\dots ,{N_*}\}^m}\! \! \! \! \! \! \! \! f_{I,J}\, a_{I_1}^\dagger \! \dots a_{I_m}^\dagger a_{J_1}\! \dots a_{J_m}\nonumber \\&\quad +\sum _{n=1}^m \frac{1}{\alpha ^{2n}n!}\sum _{\sigma ,\tau \in \mathcal {S}^{m,n}} \! \! \left( \sum _{I',J'} f^{\sigma ,\tau }_{I',J'}\! \prod _{k\notin \{\sigma _1,\dots ,\sigma _n\}}a_{I'_k}^\dagger \prod _{\ell \notin \{\tau _1,\dots ,\tau _n\}}a_{J'_\ell }\right) \end{aligned}$$

where \(\mathcal {S}^{m,n}\) is the set of all sequences \(\sigma =(\sigma _1,\dots ,\sigma _n)\) without repetitions having values \(\sigma _k\in \{1,\dots ,m\}\) and the coordinate matrices \(f^{\sigma ,\tau }\) are defined as

$$\begin{aligned} f^{\sigma ,\tau }_{I',J'}:=\sum _{I,J\in \{1,\dots ,N_*\}^m} f_{I,J}\, \delta _{I_{\sigma _1},J_{\tau _1}}\dots \delta _{I_{\sigma _n},J_{\tau _n}}\prod _{k\notin \{\sigma _1,\dots ,\sigma _n\}}\delta _{I_k,I'_k}\prod _{\ell \notin \{\tau _1,\dots ,\tau _n\}}\delta _{J_\ell ,J'_\ell } \end{aligned}$$

for \(I'\in \{1,\dots ,N_*\}^{\{1,\dots ,m\}{\setminus } \{\sigma _1,\dots ,\sigma _n\}}\) and \(J'\in \{1,\dots ,N_*\}^{\{1,\dots ,m\}{\setminus }\{\tau _1,\dots ,\tau _n\}}\). One can verify Eq. (7.11) either by iteratively applying the (rescaled) canonical commutation relations \([a_i,a_j^\dagger ]=\alpha ^{-2}\delta _{i,j}\), or by using the fact that the operator \(e^{\alpha ^{-2}\nabla _{\bar{\xi }}\nabla _\xi }\), which is well defined on polynomials in \(\xi \) and \(\bar{\xi }\), transforms the upper symbol into the lower symbol (see e.g. [27]), and computing its action on \(P(\xi ):=\sum _{I,J\in \{1,\dots ,{N_*}\}^{m}} f_{I,J} \overline{\xi _{I_1}\dots \xi _{I_m}}\xi _{J_1}\dots \xi _{J_m} \) as

$$\begin{aligned} e^{\alpha ^{-2}\nabla _{\bar{\xi }}\nabla _\xi }\left( P\right) (\xi )= P(\xi ) +\sum _{n=1}^m \frac{1}{\alpha ^{2n}n!}\sum _{\sigma ,\tau \in \mathcal {S}^{m,n}} \! \! \left( \sum _{I',J'} f^{\sigma ,\tau }_{I',J'}\! \prod _{k\notin \{\sigma _1,\dots ,\sigma _n\}}\overline{\xi _{I'_k}} \prod _{\ell \notin \{\tau _1,\dots ,\tau _n\}}\xi _{J'_\ell }\right) . \end{aligned}$$

In order to identify the left hand side of Eq. (7.11), we will make use of the resolution of identity \(\frac{1}{\pi ^{N_*}}\int _{\mathbb {C}^{N_*}}\Theta _{y,\xi }\, \textrm{d}\xi =1\), where \(\Theta _{y,\xi }\) is defined below Eq. (7.6), which allows us to rewrite the anti-wick ordered term \(a_{J_1}\dots a_{J_m}a_{I_1}^\dagger \dots a_{I_m}^\dagger \) as

$$\begin{aligned} \frac{1}{\pi ^{N_*}}\int _{\mathbb {C}^{N_*}}a_{J_1}\dots a_{J_m}\Theta _{y,\xi }a_{I_1}^\dagger \dots a_{I_m}^\dagger \, \textrm{d}\xi =\frac{1}{\pi ^{N_*}}\int _{\mathbb {C}^{N_*}}\xi _{J_1}\dots \xi _{J_m} \overline{\xi _{I_1}\dots \xi _{I_m}}\Theta _{y,\xi }\, \textrm{d}\xi . \end{aligned}$$

Here we have used that \(a_i\Theta _{y,\xi }=\xi _i\Theta _{y,\xi }\) for all \(i\in \{1,\dots ,N_*\}\). By the definition of \(\mathbb {P}_y\) in Eq. (7.6) we can therefore rewrite the expectation value of the first term on the left hand side of Eq. (7.11) with respect to the state \(\Psi _y\) as

$$\begin{aligned}&\sum _{I,J\in \{1,\dots ,{N_*}\}^{m}}\! \! \! \! \! \! \! \! \!\! \! \! \! f_{I,J}\, \big \langle \Psi _y\big | a_{J_1}\dots a_{J_m} a_{I_1}^\dagger \dots a_{I_m}^\dagger \big | \Psi _y\big \rangle \! = \! \! \! \! \! \! \! \sum _{I,J\in \{1,\dots ,{N_*}\}^{m}} \! \! \! \! \! \! \! \! \! \! \! f_{I,J}\int \xi _{J_1}\dots \xi _{J_m} \overline{\xi _{I_1}\dots \xi _{I_m}}\textrm{d}\mathbb {P}_y(\xi )\nonumber \\&\quad =\int \big \langle \varphi _{y,\xi }^{\otimes ^m}\big |f\big |\varphi _{y,\xi }^{\otimes ^m}\big \rangle \, \textrm{d}\mathbb {P}_y(\xi )=\int F\left( \left| \varphi _{y,\xi }\right| ^2\right) \, \textrm{d}\mathbb {P}_y\left( \xi \right) . \end{aligned}$$

In order to control the terms in the second line of Eq. (7.11), we can estimate the norm \(\left\| f^{\sigma ,\tau }\right\| _{\textrm{op}}\le \Vert f\Vert _\infty N_*^{n}\) for all \(\sigma ,\tau \in \mathcal {S}^{m,n}\), which follow from

$$\begin{aligned} \langle v|f^{\sigma ,\tau }|w\rangle&=\sum _{I,J\in \{1,\dots ,N_*\}^m}f_{I,J}\delta _{I_{\sigma _1},J_{\tau _1}}\dots \delta _{I_{\sigma _n},J_{\tau _n}}\overline{v_{I'}}w_{J'}=\sum _{k\in \{1,\dots ,N_*\}^n}\big \langle v^k\big |f\big |w^k\big \rangle \\&\le \Vert f\Vert _\infty \sum _{k\in \{1,\dots ,N_*\}^n}\Vert v^k\Vert \Vert w^k\Vert \le \Vert f\Vert _\infty N_*^n \Vert v\Vert \Vert w\Vert , \end{aligned}$$

where \(I'\) denotes the restriction of I to \(\{1,\dots ,m\}{\setminus } \{\sigma _1,\dots ,\sigma _n\}\) and \(v^k\) is defined as \(\left( v^k\right) _I:=\delta _{I_{\sigma _1},k_1}\dots \delta _{I_{\sigma _n},k_n}v_{I'}\), and \(J'\) and \(w^k\) are defined analogue. Hence we obtain

$$\begin{aligned} \left| \frac{1}{\alpha ^{2n}}\sum _{I',J'}f^{\sigma ,\tau }_{I',J'}\big \langle \Psi _y\big | \prod _{k\notin \{\sigma _1,\dots ,\sigma _n\}}a_{I'_k}^\dagger \prod _{\ell \notin \{\tau _1,\dots ,\tau _n\}}a_{J'_\ell }\big | \Psi _y\big \rangle \right| \le \Vert f\Vert _\infty \left( \frac{N_*}{\alpha ^2}\right) ^n\langle \Psi _y|\mathcal {{N}}^{m-n}|\Psi _y\rangle \end{aligned}$$

for \(n\ge 1\). Since \(\chi \left( \mathcal {N}\le C\right) \Psi _y=\Psi _y\) and \(N_*\lesssim \alpha ^2\), see the comment below Eq. (7.4), this is a quantity of order \(\Vert f\Vert _{\infty }\frac{N_*}{\alpha ^2}\Vert \Psi _y\Vert ^2\). Combing this estimate with Eq. (7.11) and Eq. (7.12) yields that \(\frac{1}{\Vert f\Vert _{\infty }}\left| \big \langle \Psi _y\big |\widetilde{F}\,\big |\Psi _y \big \rangle -\int F\left( |\varphi _{y,\xi }|^2\right) \, \textrm{d}\mathbb {P}_y\left( \xi \right) \right| \) is, up to a multiplicative factor, bounded by the right hand side of Eq. (7.7). Together with Eq. (7.10), this concludes the proof of Eq. (7.7).

In order to verify Eq. (7.8), let us define \(G(\rho ):=\int \textrm{d}\rho \). Note that \(W_{g}^{-1} \mathcal {N}W_{g}=\mathcal {N}-a(g)-a^\dagger (g)+\Vert g\Vert ^2=\widehat{G\, }-a(g)-a^\dagger (g)+\Vert g\Vert ^2\). Furthermore we have \(\big \langle \Psi _{y}\big |a(\Pi _{y}g)+a^\dagger (\Pi _{y}g)\,\big |\Psi _{y} \big \rangle =\int \left( \langle g|\varphi _{y,\xi }\rangle +\langle \varphi _{y,\xi }|g\rangle \right) \textrm{d}\mathbb {P}_{y}\left( \xi \right) \), where we used that \(a(g)+a^\dagger (g)\) is anti-Wick ordered, and

$$\begin{aligned} \left| \Big \langle \Psi _{y}\Big | a(g)+a^\dagger (g)- a(\Pi _{y} g)-a^\dagger (\Pi _{y} g)\,\Big |\Psi _{y} \Big \rangle \right| \le \epsilon ^{-1}\langle \Psi _{y}|\mathcal {N}^y_{>{N_*}}|\Psi _{y}\rangle +\epsilon \Vert g\Vert ^2 \Vert \Psi _y\Vert ^2. \end{aligned}$$

Hence, applying Eq. (7.7) with respect to the function G and using that \(\int \! \! \Vert \varphi _{y,\xi } - g\Vert ^2 \textrm{d}\mathbb {P}_y=\int \left( G\left( |\varphi _{y,\xi }|^2\right) +\langle g|\varphi _{y,\xi }\rangle +\langle \varphi _{y,\xi }|g\rangle \right) \textrm{d}\mathbb {P}_{y}\left( \xi \right) +\Vert g\Vert ^2\Vert \Psi _y\Vert ^2\) concludes the proof of Eq. (7.8). \(\quad \square \)

Corollary 7.3

Given constants \(m\in \mathbb {N},C>0\) and \(g\in L^2\! \left( \mathbb {R}^3\right) \), there exists a constant \(T>0\) such that for all F of the form (7.2) and states \(\Psi \) satisfying \(\chi \left( \mathcal {N}\le C\right) \Psi =\Psi \) and \(\langle \Psi |\mathbb {H}_K|\Psi \rangle \le e^\textrm{Pek}+\delta e\), with \(\delta e\ge 0\) and \(K\ge \Lambda _*=\alpha ^{2\,s}\),

$$\begin{aligned} \frac{1}{T\Vert f\Vert _{\infty }}\left| \big \langle \Psi \big |\widehat{F}\,\big |\Psi \big \rangle -\iint F\left( \left| \varphi _{y,\xi }\right| ^2\right) \textrm{d}\mathbb {P}_{y}\left( \xi \right) \textrm{d}y\right| \le \sqrt{\delta e}+ \alpha ^{-\frac{s}{2}}+ \alpha ^{\frac{27}{2}s-2}, \end{aligned}$$

and furthermore

$$\begin{aligned} \frac{1}{T}\left| \Big \langle \Psi \Big |W_g^{-1}\mathcal {N}W_g\Big |\Psi \Big \rangle -\iint \Vert \varphi _{y,\xi }-g\Vert ^2\textrm{d}\mathbb {P}_{y}\left( \xi \right) \textrm{d}y\right| \le \sqrt{\delta e}+ \alpha ^{-\frac{s}{2}}+ \alpha ^{\frac{27}{2}s-2}. \end{aligned}$$


Using the fact that we have \(\big \langle \Psi \big |\widehat{F}\,\big |\Psi \big \rangle =\int \big \langle \Psi _y\big |\widehat{F}\,\big |\Psi _y \big \rangle \textrm{d}y\) and \(\big \langle \Psi \big |W_g^{-1}\mathcal {N}W_g\big |\Psi \big \rangle =\int \big \langle \Psi _y\big |W_g^{-1}\mathcal {N}W_g\big |\Psi _y \big \rangle \textrm{d}y\), and applying Eq. (7.7), respectively Eq. (7.8), immediately yields that the left hand sides of Eqs. (7.13) and (7.14) are bounded by

$$\begin{aligned} \! \frac{{N_*}}{\alpha ^2}+\epsilon +\epsilon ^{-1}\! \!\int \! \langle \Psi _{y}|\mathcal {N}^y_{>{N_*}}|\Psi _{y}\rangle \textrm{d}y \end{aligned}$$

for any \(\epsilon >0\). In order to bound \(\int \! \langle \Psi _{y}|\mathcal {N}^y_{>{N_*}}|\Psi _{y}\rangle \textrm{d}y\) from above, let us first apply Eq. (2.3) together with Eq. (2.5), which provides the auxiliary estimate

$$\begin{aligned}&\int |\langle \Psi _{y}|\mathbb {H}^{y}_{\Lambda _*,\ell _*}|\Psi _{y}\rangle -\langle \Psi _{y}|\mathbb {H}_K|\Psi _{y}\rangle |\textrm{d}y\lesssim \alpha ^{-s}\int \langle \Psi _{y}|-\Delta _x+\mathcal {N}+1|\Psi _{y}\rangle \textrm{d}y\\&\quad \le \alpha ^{-s}\int \langle \Psi _{y}|2\mathbb {H}_K+d+1|\Psi _{y}\rangle \textrm{d}y. \end{aligned}$$

Note that the assumptions of Eq. (2.3) are indeed satisfied, since \(K\ge \Lambda _*\) and \(\textrm{supp}\left( \Psi _{y}\right) \subset B_{L_*}(y)\). In combination with the IMS identity \(\int \langle \Psi _{y}|\mathbb {H}_K|\Psi _{y}\rangle \textrm{d}y=\langle \Psi |\mathbb {H}_K|\Psi \rangle +L_*^{-2}\Vert \nabla \chi \Vert ^2\), where \(\chi \) is the function from Eq. (7.5), this furthermore yields

$$\begin{aligned}&\left| \int \langle \Psi _{y}|\mathbb {H}^{y}_{\Lambda _*,\ell _*}|\Psi _{y}\rangle \textrm{d}y-\langle \Psi |\mathbb {H}_K|\Psi \rangle \right| \lesssim \alpha ^{-s}\left( \langle \Psi |\mathbb {H}_K|\Psi \rangle +d+1\right) , \end{aligned}$$

where we have used \(L_*^{-2}=\alpha ^{-s}\). Furthermore \(\langle \Psi |\mathbb {H}_K|\Psi \rangle \le e^\textrm{Pek}+\delta e\) by assumption, and consequently \(|\int \langle \Psi _{y}|\mathbb {H}^{y}_{\Lambda _*,\ell _*}|\Psi _{y}\rangle \textrm{d}y-\langle \Psi |\mathbb {H}_K|\Psi \rangle |\le D \alpha ^{-s}(\delta e+1)\) for a suitable D. Consequently

$$\begin{aligned} \langle \Psi |\mathbb {H}_K|\Psi \rangle&\ge \int \langle \Psi _{y}|\mathbb {H}^{y}_{\Lambda _*,\ell _*}|\Psi _{y}\rangle \textrm{d}y-D \alpha ^{-s}(\delta e+1)\nonumber \\&\ge E_\alpha +\int \langle \Psi _{y}|\mathcal {N}^y_{>{N_*}}|\Psi _{y}\rangle \textrm{d}y-D \alpha ^{-s}(\delta e+1). \end{aligned}$$

where we have used that \(\mathbb {H}^{y}_{\Lambda _*,\ell _*}\ge E_\alpha +\mathcal {N}^y_{>{N_*}}\) in the second inequality. Using Eq. (7.17) as well as the fact that \(E_\alpha -e^\textrm{Pek}\gtrsim -\alpha ^{-\frac{1}{5}}\ge -\alpha ^{-s}\), see [20], we obtain the upper bound

$$\begin{aligned} \int \langle \Psi _{y}|\mathcal {N}^y_{>{N_*}}|\Psi _{y}\rangle \textrm{d}y\lesssim \langle \Psi |\mathbb {H}_K|\Psi \rangle - e^\textrm{Pek}+\alpha ^{-s}(\delta e+1)\lesssim \delta e+\alpha ^{-s}. \end{aligned}$$

Choosing \(\epsilon :=\sqrt{\delta e+\alpha ^{-s}}\) in Eq. (7.15) therefore concludes the proof together with the observation that \(\frac{N_*}{ \alpha ^2}\lesssim \alpha ^{\frac{27}{2}s-2}\). \(\quad \square \)

In the following Lemma 7.4 we are investigating the support properties of the lower symbol \(\mathbb {P}_y\). In particular we derive bounds on the associated moments and verify that \(\varphi _{y,\xi }\) is typically close to the manifold of minimizers \(\{\varphi _x^\textrm{Pek}:x\in \mathbb {R}^3\}\).

Lemma 7.4

Given constants \(m\in \mathbb {N}\) and \(C>0\), there exists a \(T>0\), such that \(\iint |\xi |^{2\,m}\textrm{d}\mathbb {P}_{y}\left( \xi \right) \textrm{d}y\le T\) for all \(\Psi \) satisfying \(\chi \left( \mathcal {N}\le C\right) \Psi =\Psi \), and furthermore we have for all \(K\ge \Lambda _*\), where \(\Lambda _*\) is as in the definition of \(\Pi ^{y}\) in Eq. (7.4),

$$\begin{aligned}&\frac{1}{T}\iint \inf _{x\in \mathbb {R}^3}\Vert \varphi _{y,\xi }-\varphi ^\textrm{Pek}_x\Vert ^2\textrm{d}\mathbb {P}_{y}\left( \xi \right) \textrm{d}y\le \langle \Psi |\mathbb {H}_K|\Psi \rangle -e^\textrm{Pek}+\alpha ^{-s}+\alpha ^{\frac{27}{2}s-2}. \end{aligned}$$


For \(m\in \mathbb {N}\), let us define the function \(G(\rho ):=\left( \int \textrm{d}\rho (x)\right) ^m=\int \dots \int \textrm{d}\rho (x_1)\dots \textrm{d}\rho (x_m)\), which is clearly of the form given in Eq. (7.2). Consequently by Lemma 7.2

$$\begin{aligned}&\int \! |\xi |^{2m}\textrm{d}\mathbb {P}_{y}\left( \xi \right) =\! \int \! G\left( |\varphi _{y,\xi }|^{2}\right) \textrm{d}\mathbb {P}_{y}\left( \xi \right) \\&\quad \lesssim \big \langle \Psi _{y}\big |\widehat{G}\,\big |\Psi _{y} \big \rangle +\left( \frac{{N_*}}{\alpha ^2}+1\right) \Vert \Psi _y\Vert ^2+\langle \Psi _{y}|\mathcal {N}^y_{>{N_*}}|\Psi _{y}\rangle \\&\quad =\big \langle \Psi _{y}\big | \mathcal {N}^{\, 2m}\big |\Psi _{y}\big \rangle +\left( \frac{{N_*}}{\alpha ^2}+1\right) \Vert \Psi _y\Vert ^2+\langle \Psi _{y}|\mathcal {N}^y_{>{N_*}}|\Psi _{y}\rangle \\&\quad \le \left( C^{2m}+\frac{{N_*}}{\alpha ^2}+1+C\right) \Vert \Psi _y\Vert ^2, \end{aligned}$$

which concludes the proof of the first part, since \(N_*\lesssim \alpha ^2\) and \(\int \Vert \Psi _y\Vert ^2\textrm{d}y=\Vert \Psi \Vert ^2=1\).

Regarding the proof of Eq. (7.19), we have the simple bound

$$\begin{aligned} \mathbb {H}^{y}_{\Lambda _*,\ell _*}&\!=\! -\!\Delta _x\!-\!a\left( \Pi _{y} w_x\right) \!-\!a^\dagger \left( \Pi _{y} w_x\right) \!+\!\mathcal {N}\nonumber \\&\ge \! -\!\Delta _x\!-\!a\left( \Pi _{y} w_x\right) -\!a^\dagger \left( \Pi _{y} w_x\right) \!+\!\sum _{n=1}^{N_*} a_{y,n}^\dagger a_{y,n}\nonumber \\&=-\Delta _x-a\left( \Pi _{y} w_{x}\right) -a^\dagger \left( \Pi _{y} w_{x}\right) +\sum _{n=1}^{N_*} a_{y,n} a_{y,n}^\dagger -\frac{{N_*}}{\alpha ^2}. \end{aligned}$$

Since all terms in Eq. (7.20) are represented in anti-Wick ordering, we can follow [20] and express, similar as in the proof of Lemma 7.2, their expectation value as

$$\begin{aligned}&\Big \langle \! \Psi _{y}\! \Big |\!\! -\! \!\Delta _x\! -\! a\left( \Pi _{y} w_{x}\right) \! -\! a^\dagger \left( \Pi _{y} w_{x}\right) \! +\! \! \sum _{n=1}^{N_*} a_{y,n} a_{y,n}^\dagger \Big |\! \Psi _{y}\! \Big \rangle \! \!\nonumber \\&\quad =\! \! \! \int \! \! \! \left( \! \langle \psi ^\xi _{y}|\! -\! \! \Delta _x\! +\! V_{\varphi _{y,\xi }}|\psi ^\xi _{y}\rangle \! +\! \Vert \varphi _{y,\xi }\Vert ^2\! \right) \! \textrm{d}\mathbb {P}_{y}\! \left( \xi \right) \nonumber \\&\quad \ge \int \left( \inf \sigma \left( -\Delta _x+V_{\varphi _{y,\xi }}\right) +\Vert \varphi _{y,\xi }\Vert ^2\right) \textrm{d}\mathbb {P}_{y}\left( \xi \right) =\int \mathcal {F}^\textrm{Pek}\! \left( \varphi _{y,\xi }\right) \textrm{d}\mathbb {P}_{y}\left( \xi \right) , \end{aligned}$$

with \(\psi ^\xi _{y}:=\frac{\Theta _{y,\xi } \Psi _{y}}{\Vert \Theta _{y,\xi } \Psi _{y}\Vert }\) where \(\Theta _{y,\xi }\) is defined below Eq. (7.6), \(\mathcal {F}^\textrm{Pek}\) is the Pekar functional and \(V_{\varphi }\) is defined in Eq. (5.1). Making use of Eq. (7.1) we obtain together with Eqs. (7.16), (7.20) and (7.21)

$$\begin{aligned}&\int \int \inf _{x\in \mathbb {R}^3}\Vert \varphi _{y,\xi }-\varphi ^\textrm{Pek}_x\Vert ^2\textrm{d}\mathbb {P}_{y}\left( \xi \right) \textrm{d}y\lesssim \int \langle \Psi _{y}|\mathbb {H}^{y}_{\Lambda _*,\ell _*}| \Psi _{y}\rangle \textrm{d}y-e^\textrm{Pek}+\frac{{N_*}}{\alpha ^2}\\&\quad \lesssim \langle \Psi |\mathbb {H}_K| \Psi \rangle -e^\textrm{Pek}+\frac{{N_*}}{\alpha ^2}+D\alpha ^{-s}\left( \langle \Psi |\mathbb {H}_K|\Psi \rangle +d+1\right) , \end{aligned}$$

for a suitable \(D>0\). This concludes the proof, since we have \({N_*}\lesssim \alpha ^{\frac{27}{2}s}\). \(\quad \square \)

The bound in Eq. (7.19) suggests that \(\varphi _{y,\xi }\) is close to \(\varphi ^\textrm{Pek}_{x^{y,\xi }}\) with a high probability, where \(x^{y,\xi }\) is the minimizer of \(x\mapsto \Vert \varphi _{y,\xi }-\varphi ^\textrm{Pek}_{x}\Vert \). Motivated by this observation we expect \(\iint F\left( \left| \varphi _{y,\xi }\right| ^2\right) \textrm{d}\mathbb {P}_y\textrm{d}y\approx \iint F\left( \left| \varphi ^\textrm{Pek}_{x^{y,\xi }}\right| ^2\right) \textrm{d}\mathbb {P}_y\textrm{d}y\) for measures \(\mathbb {P}_y\) for low energy states \(\Psi \), and therefore it seems natural to define the measure \(\mu \) in Theorem 3.2 as \(\int f\textrm{d}\mu :=\iint f\left( x^{y,\xi }\right) \textrm{d}\mathbb {P}_y\textrm{d}y\), allowing us to identify \(\iint F\left( \left| \varphi ^\textrm{Pek}_{x^{y,\xi }}\right| ^2\right) \textrm{d}\mathbb {P}_y\textrm{d}y=\int F\left( \left| \varphi ^\textrm{Pek}_x\right| ^2\right) \textrm{d}\mu \). This expression is however ill-defined, since the infimum \(\inf _{x\in \mathbb {R}^3}\Vert \varphi _{y,\xi }-\varphi ^\textrm{Pek}_{x}\Vert \) 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 \(\mu \) in the proof of Lemma 7.5.

Lemma 7.5

Given \(m\in \mathbb {N},C>0\) and \(g \in L^2\! \left( \mathbb {R}^3\right) \) we can find a constant \(T>0\), such that for all states \(\Psi \) satisfying \(\chi \left( \mathcal {N}\le C\right) \Psi =\Psi \) and \(\langle \Psi |\mathbb {H}_K|\Psi \rangle \le e^\textrm{Pek}+\delta e\), with \(\delta e\ge 0\) and \(K\ge \Lambda _*\), there exists a probability measure \(\mu \) on \(\mathbb {R}^3\) with the property

$$\begin{aligned} \frac{1}{T\Vert f\Vert _{\infty }}\left| \iint \! F\left( \left| \varphi _{y,\xi }\right| ^2\right) \! \textrm{d}\mathbb {P}_{y}\! \left( \xi \right) \textrm{d}y-\int \! \! F\! \left( \left| \varphi ^\textrm{Pek}_{x}\right| ^2\right) \! \textrm{d}\mu \! \left( x\right) \right| \le \sqrt{\delta e}+ \alpha ^{-\frac{s}{2}}+ \alpha ^{\frac{27}{4}s-1}, \end{aligned}$$

for all F of the form (7.2), and furthermore

$$\begin{aligned} \frac{1}{T}\left| \iint \Vert \varphi _{y,\xi }-g\Vert ^2 \textrm{d}\mathbb {P}_{y}\! \left( \xi \right) \textrm{d}y-\int \! \! \Vert \varphi ^\textrm{Pek}_x-g\Vert ^2 \textrm{d}\mu \! \left( x\right) \right| \le \sqrt{\delta e}+ \alpha ^{-\frac{s}{2}}+ \alpha ^{\frac{27}{4}s-1}. \end{aligned}$$


For \(\epsilon >0\), let \(\bigcup _{n=1}^\infty A_{\epsilon ,n}=\mathbb {C}^{N_*}\) be a partition of \(\mathbb {C}^{N_*}\) consisting of non-empty measurable sets \(A_{\epsilon ,n}\) having a diameter bounded by \(d(A_{\epsilon ,n})\le \epsilon \). Furthermore choose \(\xi _{\epsilon ,n}\in A_{\epsilon ,n}\) and \(x_{\epsilon ,n}\in \mathbb {R}^3\) satisfying \(\Vert \varphi _{0,\xi _{\epsilon ,n}}-\varphi ^\textrm{Pek}_{x_{\epsilon ,n}}\Vert \le \inf _{x\in \mathbb {R}^3}\Vert \varphi _{0,\xi _{\epsilon ,n}}-\varphi ^\textrm{Pek}_{x}\Vert +\epsilon \). Then

$$\begin{aligned}&\Vert \varphi _{y,\xi }\! -\! \varphi ^\textrm{Pek}_{y+x_{\epsilon ,n}}\Vert = \Vert \varphi _{0,\xi }\! -\! \varphi ^\textrm{Pek}_{x_{\epsilon ,n}}\Vert \le \Vert \varphi _{0,\xi _{\epsilon ,n}}\! -\! \varphi ^\textrm{Pek}_{x_{\epsilon ,n}}\Vert \! +\! \Vert \varphi _{0,\xi }\! -\! \varphi _{0,\xi _{\epsilon ,n}}\Vert \nonumber \\&\quad \le \Vert \varphi _{0,\xi _{\epsilon ,n}}\! -\! \varphi ^\textrm{Pek}_{x_{\epsilon ,n}}\Vert \! +\! \epsilon \nonumber \\&\quad \le \inf _{x\in \mathbb {R}^3}\! \Vert \varphi _{0,\xi _{\epsilon ,n}}\! -\! \varphi ^\textrm{Pek}_{x}\Vert \! +\! 2\epsilon \le \inf _{x\in \mathbb {R}^3}\! \Vert \varphi _{0,\xi }\! -\! \varphi ^\textrm{Pek}_{x}\Vert \! +\! 3\epsilon =\inf _{x\in \mathbb {R}^3}\! \Vert \varphi _{y,\xi }\! -\! \varphi ^\textrm{Pek}_{x}\Vert \! +\! 3\epsilon . \end{aligned}$$

Let us now define the probability measure \(\mu \) on \(\mathbb {R}^3\) by specifying its action on functions \(f\in C\left( \mathbb {R}^3\right) \) as

$$\begin{aligned} \int f\textrm{d}\mu :=\sum _{n=1}^\infty \int \! f\left( y+x_{\epsilon ,n}\right) \! \mathbb {P}_{y}\! \left( A_{\epsilon ,n}\right) \textrm{d}y=\sum _{n=1}^\infty \int \! \int _{A_{\epsilon ,n}}\! f\left( y+x_{\epsilon ,n}\right) \textrm{d}\mathbb {P}_{y} \textrm{d}y. \end{aligned}$$

Since \(\int \! F\left( \left| \varphi _{y,\xi }\right| ^2\right) \! \textrm{d}\mathbb {P}_{y}\! \left( \xi \right) =\sum _{n=1}^\infty \int _{A_{\epsilon ,n}} \! F\left( \left| \varphi _{y,\xi }\right| ^2\right) \! \textrm{d}\mathbb {P}_{y}\! \left( \xi \right) \), we can estimate the left hand side of Eq. (7.22) with the aid of the triangle inequality by

$$\begin{aligned} \sum _{n=1}^\infty \int \! \int _{A_{\epsilon ,n}}\! \left| F\left( \left| \varphi _{y,\xi }\right| ^2\right) -F\left( \left| \varphi ^\textrm{Pek}_{y+x_{\epsilon ,n}}\right| ^2\right) \right| \textrm{d}\mathbb {P}_{y}\! \left( \xi \right) \textrm{d}y. \end{aligned}$$

From the concrete form of the function F given in Eq. (7.2), as well as the facts that \(\Vert \varphi ^\textrm{Pek}_{y+x_{\epsilon ,n}}\Vert =\Vert \varphi ^\textrm{Pek}_{0}\Vert \) is finite and \(\Vert \varphi _{y,\xi }\Vert =|\xi |\), one readily concludes that

$$\begin{aligned} \left| F\left( \left| \varphi _{y,\xi }\right| ^2\right) -F\left( \left| \varphi ^\textrm{Pek}_{y+x_{\epsilon ,n}}\right| ^2\right) \right| \lesssim \Vert f\Vert _\infty \left\| \varphi _{y,\xi }-\varphi ^\textrm{Pek}_{y+x_{\epsilon ,n}}\right\| \left( 1+|\xi |\right) ^{2m-1}. \end{aligned}$$

Using Eq. (7.24) we further obtain for any \(\kappa >0\) and \(\xi \in A_{\epsilon ,n}\)

$$\begin{aligned}&\left\| \varphi _{y,\xi }-\varphi ^\textrm{Pek}_{y+x_{\epsilon ,n}}\right\| \left( 1+|\xi |\right) ^{2m-1}\le \left( \inf _{x\in \mathbb {R}^3}\! \Vert \varphi _{y,\xi }\! -\! \varphi ^\textrm{Pek}_{x}\Vert +3\epsilon \right) \left( 1+|\xi |\right) ^{2m-1}\\&\quad \le \kappa ^{-1}\inf _{x\in \mathbb {R}^3}\! \Vert \varphi _{y,\xi }\! -\! \varphi ^\textrm{Pek}_{x}\Vert ^2+\frac{\kappa }{4}\left( 1+|\xi |\right) ^{4m-2}+3\epsilon \left( 1+|\xi |\right) ^{2m-1}, \end{aligned}$$

and therefore the expression in Eq. (7.25) can be bounded from above by

$$\begin{aligned}&\Vert f\Vert _\infty \bigg (\kappa ^{-1}\iint \inf _{x\in \mathbb {R}^3}\! \Vert \varphi _{y,\xi }\! -\! \varphi ^\textrm{Pek}_{x}\Vert ^2 \textrm{d}\mathbb {P}_{y}\! \left( \xi \right) \textrm{d}y + \frac{\kappa }{4}\! \iint \left( 1+|\xi |\right) ^{4m-2}\textrm{d}\mathbb {P}_{y}\! \left( \xi \right) \textrm{d}y\\&\quad + 3\epsilon \! \iint \left( 1+|\xi |\right) ^{2m-1}\textrm{d}\mathbb {P}_{y}\! \left( \xi \right) \textrm{d}y\bigg ). \end{aligned}$$

By Lemma 7.4 this concludes the proof of (7.22) with \(\epsilon :=\kappa :=\sqrt{\delta e+\alpha ^{-s}+\alpha ^{\frac{27}{2}s-2}}\). Equation (7.23) can be proven analogously, using the estimate

$$\begin{aligned} \left| \Vert \varphi _{y,\xi }-g\Vert ^2-\Vert \varphi ^\textrm{Pek}_{y+x_{\epsilon ,n}}-g\Vert ^2\right| \lesssim \left\| \varphi _{y,\xi }-\varphi ^\textrm{Pek}_{y+x_{\epsilon ,n}}\right\| \left( 1+|\xi |\right) \end{aligned}$$

for \(\xi \in A_{\epsilon ,n}\). \(\quad \square \)

Combining Eq. (7.13), respectively Eq. (7.14), with Eq. (7.22), respectively Eq. (7.23), immediately yields that the left hand side of Eq. (3.2), respectively Eq. (3.3), is of the order \(\sqrt{\delta e}+ \alpha ^{-\frac{s}{2}}+ \alpha ^{\frac{27}{4}s-1}\). Optimizing in the parameter \(0<s\le \frac{4}{27}\) concludes the proof of Theorem 3.2 with the concrete choice \(s:=\frac{4}{29}\).