On the Ultraviolet Limit of the Pauli-Fierz Hamiltonian in the Lieb-Loss Model

Two decades ago, Lieb and Loss proposed to approximate the ground state energy of a free, nonrelativistic electron coupled to the quantized radiation field by the infimum $E_{\alpha, \Lambda}$ of all expectation values $\langle \phi_{el} \otimes \psi_{ph} | H_{\alpha, \Lambda} (\phi_{el} \otimes \psi_{ph}) \rangle$, where $H_{\alpha, \Lambda}$ is the corresponding Hamiltonian with fine structure constant $\alpha>0$ and ultraviolet cutoff $\Lambda<\infty$, and $\phi_{el}$ and $\psi_{ph}$ are normalized electron and photon wave functions, respectively. Lieb and Loss showed that $c \alpha^{1/2} \Lambda^{3/2} \leq E_{\alpha, \Lambda} \leq c^{-1} \alpha^{2/7} \Lambda^{12/7}$ for some constant $c>0$. In the present paper we prove the existence of a constant $C<\infty$, such that \begin{align*} \bigg| \frac{E_{\alpha, \Lambda}}{F[1] \, \alpha^{2/7} \, \Lambda^{12/7}} - 1 \bigg| \ \leq \ C \, \alpha^{4/105} \, \Lambda^{-4/105} \end{align*} holds true, where $F[1]>0$ is an explicit universal number. This result shows that Lieb and Loss' upper bound is actually sharp and gives the asymptotics of $E_{\alpha, \Lambda}$ uniformly in the limit $\alpha \to 0$ and in the ultraviolet limit $\Lambda \to \infty$.


I Introduction and Result
Soon after the discovery of quantum mechanics almost a century ago by Heisenberg and Schrödinger, the quantization of the radiation field was formulated by Born, Heisenberg, and Jordan and by Dirac [11,13], and about seventy years ago quantum electrodynamics (QED) was formulated by Feynman, Schwinger, Tomonaga, and Dyson [14,15,25,28], laying the foundation to answer the question whether light rays consisted of particles or waves that was open for several centuries. Besides being conceptually satisfying, QED is one of the most successful theories with quantitative predictions that match experimental data by more that eight decimals. In spite of its success for applications, however, QED is still lacking essential parts of its mathematical foundation to this very day. Namely, all known formulations require unphysical regularizations at large, ultraviolet, and/or small, infrared, photon energies. While considerable progress has been made in the past three decades on the construction of the infrared limit, i.e., the construction of a theory without regularization at small photon energies [6,7,18,9,5], the construction of the ultraviolet limit is wide open. This difficult problem has been tackled from several angles, e.g., by replacing the fully interacting model by effective mean-field theories of various kinds [20,16,17].
One approach among these is a simplifying variational model proposed by Lieb and Loss in 1999 [23]. Their starting point is the Pauli-Fierz Hamiltonian of a nonrelativistic spinless particle (modelling the electron), minimally coupled to the quantized radiation field. Here 1 i ∇ x is the (particle) momentum operator and A Λ (x) = |k|≤Λ e −ik·x a * (k) + e ik·x a(k) ε(k) dk (2π) 3/2 |k| 1/2 is the magnetic vector potential in Coulomb gauge and cut off for momenta larger than Λ in magnitude. Moreover, H ph = |k| a * (k) a(k) dk is the energy of the radiation field, and α ≈ 1/137 is the (dimensionless) fine structure constant. The Hamiltonian H α,Λ is an unbounded, self-adjoint operator on the domain dom[H 0,0 ] ⊆ H el ⊗ F ph of the noninteracting Hamiltonian H 0,0 = 1 2 (−∆) ⊗ 1 ph + 1 el ⊗ H ph , see [22,21], where H el = L 2 (R 3 ) is the space of square-integrable functions on R 3 , and F ph is the boson Fock space over the space L 2 (R 3 × Z 2 ) of square-integrable, purely transversal vector fields, see Section II for a precise definition.
Note that H α,Λ ≥ 0 as a quadratic form. The (nonnegative) ground state of the energy of the system is characterized by the Rayleigh-Ritz variational principle as the infimum of all energy expectation values of the system, E gs (α, Λ) := inf Ψ |H α,Λ Ψ Ψ ∈ H el ⊗ F ph , Ψ = 1 . (I.2) Lieb and Loss proposed [23] to restrict the variation in (I.2) to wave functions of product form Ψ = φ ⊗ ψ, with normalized φ ∈ H el and ψ ∈ F ph , to obtain a new approximation and upper bound E LL (α, Λ) ≥ E gs (α, Λ) to the ground state energy, i.e., E LL (α, Λ) := inf E α,Λ (φ, ψ) φ ∈ H el , ψ ∈ F ph , φ = ψ = 1 , (I. 3) Note that upper bounds on the ground state energy are of particular interest here because the ultraviolet problem is about the understanding of the divergence of E gs (α, Λ) → ∞, as Λ → ∞. We henceforth refer to Eqs. (I. 3)-(I.4) as the Lieb-Loss Model.
In Theorem 1.1 in [23] Lieb and Loss proved the existence of two universal constants C 1 , C 2 ∈ R + such that This is the first of a series of results of Lieb and Loss in [23], extending their model to N ≥ 2 fermions or bosons, taking the electron spin into account by studying the Pauli operator, and replacing the nonrelativistic kinetic energy by a pseudorelativistic one. Note that the Lieb-Loss model does not take the renormalization of the electron mass into account, and the actual value of E LL (α, Λ) is of limited quantitative use in physics. The significance of Eq. (I.5), however, lies in the fact that the formal perturbation expansion of the ground state about the photon vacuum yields E gs (α, Λ) ∼ CαΛ 2 . In contrast, Eq. (I.5) says that this grossly overestimates the ground state energy; it is a warning sign that perturbation theory may not be adequate to construct the ultraviolet limit.
The main result of this paper is to derive the asymptotics of E LL (α, Λ), as Λ → ∞ or α → 0. We obtain an exact characterization of the ground state and ground state energy of the Lieb-Loss Model for any given α > 0 and Λ ≥ 1, in terms of an auxiliary classical functional. To formulate this precisely, we introduce and in [19] the second author shows that the infimum in (I.7) is actually attained and strictly positive, in particular, We briefly sketch the derivation of (I. 10). The intermediate steps yield further insight on the minimizer of the Lieb-Loss model. The latter is described in detail in Section III.3.
(1) For technical reasons we introduce an infrared cutoff σ > 0. The case σ = 0 can be dealt with by a continuity argument in the limit σ → 0 using standard relative bounds on A σ . We do not give details of the argument but refer the reader to [6].
(2) We first analyze the functional E α,Λ . A direct computation yields where ·|· F denotes the scalar product on the photon Fock space F ph and H[ρ, v] is for ρ : R 3 → R + and v : R 3 → R 3 given as (I. 12) In Theorem IV.4 in Section IV.1 we demonstrate that, by a suitably chosen Weyl transformation W φ , the term linear in the fields, i.e., proportional to v = Im{φ ∇φ}, can be eliminated up to an additive constant in the transformed Hamiltonian. The minimization of the energy functional consequently enforces the reality of the wavefunction φ. More precisely, we therefore have that 15) guarantees that we can assume without loss of generality that φ = |φ| ≥ 0, and in this case (I. 16) In Theorem IV.5 in Section IV.2 we give an alternative proof for the observation of Lieb and Loss that , and P C := 1 ( ∇ x ·) = 0 is the projection onto divergence-free vector fields, i.e., vector fields in Coulomb gauge. Inserting this into (I.14)-(I.15), we arrive at with φ(x) ≡ φ(i∇ p ) denoting the corresponding Fourier multiplier (with respect to the momentum representation).
(4) In Section V we introduce the infima of the Lieb-Loss functional E LL (α, Λ)(φ L ) and the auxiliary functional F β (φ L ) under variation only over compactly supported functions φ L ∈ Y L := H 1 (B(0, L)) and compare these infima to E LL (α, Λ) and F [β] by means of the IMS localization formula. More specifically, we prove in Theorem V.1 that for some universal constant C < ∞ and all L > 0. Consequently, the leading orders of E LL (α, Λ) and F [β], respectively, are determined by their behavior on compactly supported functions.
(5) The fourth step carried out in Sections VI and VII is to find upper and lower bounds for all compactly supported φ = |φ| ∈ Y L := H 1 B(0, L) on X(Θ φ L ,α ). In Theorem VI.1 we prove the existence of a universal constant C < ∞ such that, for all 0 < ε ≤ 1, L ≥ 1/Λ, and φ ∈ Y L , This is complemented by the lower bound in Theorem VII.2 which asserts that, there exists a universal constant C < ∞ such that, for all L ≥ 1/Λ and φ ∈ Y L , 9π Λ 3 which is done in Section VIII. Indeed, this leads us to introduce the family of auxiliary functionals and their infima This family of functionals is analyzed by direct methods of the calculus of variations in detail by the second author in a separate paper [19], and here we describe its properties only briefly.
-For fixed β > 0, the functional F β possesses a minimizer, which is unique up to translations, nonnegative, spherically symmetric and decreasing. In particular, its infimum F [β] is attained and hence a strictly positive minimum. In Section VIII we use the information on the auxiliary functional and especially the scaling relation F [β] = β 4/7 F 1 to finally derive (I.10), formulated again as (VIII.3) in Theorem VIII.1. In order to simultaneously control the errors on the right side of (I. 25)  Note that both estimates suggest that the length scale ℓ(Λ) of the particle in the ground state of the Lieb-Loss model is of order ℓ(Λ) ≈ α τ −1 Λ τ , with τ = 6 7 ≈ 0.86.

II The Lieb-Loss Model
The Lieb-Loss model is a variational model for the study of the ground state energy of a system containing a single nonrelativistic spinless particle which is minimally coupled to the quantized radiation field. The dynamics of such a quantum system is generated by the Pauli-Fierz Hamiltonian which we define here as a quadratic form on is the Sobolev space of square-integrable functions whose gradient is square-integrable, as well, and D(N The one-photon Hilbert space h pol := L 2 (S σ,Λ × Z 2 ) is the space of squareintegrable, divergence-free vector fields k → ǫ( k, +)f (k, +) + ǫ( k, −)f (k, −) supported in the momentum shell S σ,Λ := { k ∈ R 3 : σ ≤ | k| < Λ} ⊆ R 3 which excludes momenta of magnitude below the infrared cutoff σ ≥ 0 and above the ultraviolet cutoff 1 ≤ Λ < ∞. The two transversal polarizations are parametrized by the polarization vectors ǫ(k, ±) ⊥ k that are chosen so as to form an orthonormal frame k/| k|, ǫ( k, +), ǫ( k, −) in C ⊗ R 3 , for all k ∈ S σ,Λ \ { 0}. Of course, the map k → ǫ(k) is assumed to be measurable and, for convenience, chosen to be real, ǫ(k, ±) ∈ R 3 , almost everywhere in R 3 × Z 2 .
The Lieb-Loss model is defined by the Lieb-Loss (energy) functional E α,σ,Λ : ph ) → R which results from varying only over products φ ⊗ ψ of normalized wave functions of the particle φ ∈ L 2 (R 3 ) and the photon state ψ ∈ F ph in the Rayleigh-Ritz principle, i.e., (II.6) Note that, given a fixed φ ∈ H 1 (R 3 ) and varying only over ψ ∈ D(N 1/2 ph ), the Lieb-Loss functional ψ → E LL (φ, ψ) becomes the expectation value in ψ of a Hamiltonian that is quadratic in the boson fields. More specifically, a simple computation shows that where ·|· F denotes the scalar product on the photon Fock space F ph and, for fixed ρ : R 3 → R + and v : R 3 → R 3 , the quadratic Hamiltonian H[ρ, v] is given as As we show below it turns out that the minimal values of the Lieb-Loss functional is attained for positive wave functions. To exhibit this we define r := |φ| ∈ H 1 (R 3 ; R + 0 ) and choose γ ∈ H 1 (R 3 ; R), for a given φ ∈ H 1 (R 3 ; C), so that φ = r e iγ , |φ| 2 = r 2 , Im{φ ∇φ} = r 2 ∇γ , (II.9) ∇φ 2 2 = ∇r 2 2 + r ∇γ 2 2 , (II. 10) and thus E α,σ,Λ (r e iγ , ψ) = 1 2 Although convenient, the explicit parametrization of Couloumb gauge by polarization vectors ǫ( k, ±) tends to obscure the picture by introducing a seeming dependence of the model on the choice of ǫ( k, ±), which, however, should be physically meaningless. For this reason we choose the one-photon space to be the Hilbert space is the orthogonal projection acting as P C f ( k) := P ⊥ k f ( k), with P k : R 3 → R 3 being the projection in R 3 onto the unit vector k/ k| ∈ S 2 . Note that for any arbitrary, but fixed, choice of polarization vectors basis { ǫ( k, +), ǫ( k, −)} k∈S σ,Λ described above, the map , and allows us to switch between the photon representations, if necessary. Accordingly, the photon Fock space we use is F ph := F b [h] the bosonic Fock space over divergence-free vector fields. On F ph we have a Fock representation of the CCR of the form a( k 1 , ν 1 ), a( k 2 , ν 2 ) = a * ( k 1 , ν 1 ), a * ( k 2 , ν 2 ) = 0 , (II.14) for all k 1 , k 2 ∈ S σ,Λ and ν 1 , ν 2 ∈ Z 3 , as operator-valued distributions, or In this representation the operator A σ,Λ (x) of the magnetic vector potential becomes 20) and the Hamiltonian H r 2 , r 2 ∇γ in (II.11) turns into Note that the dependence of A(x) on the cutoff parameters 0 < σ ≤ 1 and 1 ≤ Λ < ∞ is not displayed anymore.

III Bogolubov Transformations
Next, we analyze the infimum of ψ → ψ H r 2 , r 2 ∇γ ψ , as ψ ∈ D(N 1/2 ph ) varies over normalized states, by means of Bogolubov transformations. For a suitable definition of these in the present context, the choice of the antilinear involution J : h → h defined by plays a key role. Before using J, we recall a few facts about antiunitary maps and generalized creation and annihilation operators.

III.1 Antiunitary Maps and Generalized Field Operators
Conversely, any antiunitary from h to h * is of this form.
In the present paper we prefer to work with an antiunitary J which additionally constitutes an antilinear involution or real structure. Given a general complex Hilbert space h these are antiunitary bijections J : h → h, which obey Given an antiunitary involution J : h → h we can define the maximal J-invariant subspace This gives us freedom to make a suitable choice for the problem to solve, namely, (III.1) in the present case.
To define Bogolubov transformations it is convenient to use generalized creation and annihilation operators which were first introduced by Araki and Shiraishi in [2,3] to describe the second quantization of one-body Hamiltonians. Bogolubov transformations are also discussed in detail in [27,8]. Given an antiunitary involution J : h → h, the generalized creation and annihilation (field) being an antiunitary involution on h ⊕ h. The vectors in h ⊕ h which are invariant under J are of the form y ⊕ Jy, with y ∈ h. They form a real subspace One advantage of the generalized formalism consists in encoding all orderings in the second quantization of operators, so that we need not worry about imposing normal-ordering. The price for this is the slightly modified form of the canonical commutation relations (CCR), the generalized field operators obey, namely, where S is a natural symplectic form on h ⊕ h given by

III.2 Second Quantization and Bogolubov Transformations
Next, we introduce the second quantization of one-photon operators. Let J : h → h be an antiunitary involution and Note that the definition (III.10) of dΓ J [T, y] is independent of the choice of the orthonormal basis , y , and we can and will henceforth always assume that JbJ d . (III.11) A second advantage of the generalized creation and annihilation operators is that their use eases the definition of Bogolubov transformations. We recall that Bogolubov transformations are unitary transformations U on Fock space F ph which preserve (III.7) and are linear in the field operators, i.e., they act as for all f ∈ h, where U and V are linear operators on h and η ∈ h. The Bogolubov transformations form a group which is the semidirect product of the group of homogenous Bogolubov transformations and the group of Weyl transformations.
That is, every Bogolubov transformation U can be written as a composition of a homogeneous Bogolubov transformation U B and a Weyl transformation W η or a composition of a Weyl transformation W µ and U B , but with µ = η, in general. Homogeneous Bogolubov transformations U B are the special case η = 0 of (III.12). In terms of the generalized field operators they assume the form where the form of B is determined by (III.5), i.e., J B = BJ , and (III.12). Note that this makes explicit use of the antiunitary involution J : h → h. The homogeneous Bogolubov transformation U B is unitary iff it leaves the CCR invariant and preserves the norm of the vacuum vector Ω ∈ F ph , which is equivalent to The second identity in (III.15) is actually a consequence of the first, as the latter implies the invertibility of B, and then the second identity follows from the uniqueness of the inverse. The requirement that V be a Hilbert-Schmidt operator is known as the Shale-Stinespring condition. A simple computation shows that the second quantization dΓ J [T, y] of T and y transforms under a homogeneous Bogolubov transformation Weyl transformations W η are the special case U = 1 h and V = 0 of (III.12). They act on the generalized field operators as The unitarity of W η is equivalent to the requirement η ∈ h. Another simple computation shows that the second quantization dΓ J [T, y] of T and y transforms under a Weyl transformation W η as

III.3 The Lieb-Loss Model in Terms of Second Quantization
We We identify H(r 2 , r 2 ∇γ) with dΓ J [T r,α , y r,γ,α ], for suitably chosen T r,α and y r,γ,α . We state the result in form of Lemma III.1 below.
We obtain the assertion from here by (III.18).
As a corollary of Lemma IV.2, we now find the following lower bound on the Lieb-Loss functional defined in (I.4).
Corollary IV.3. Let φ ∈ H 1 (R 3 ) and ψ ∈ F ph be normalized wave functions. Then there exists a unitary Weyl transformation W φ such that (IV.14) As a consequence it follows that the partial minimization of the Lieb-Loss functional where σ(A) ⊆ R denotes the spectrum of a self-adjoint operator A and T |φ| is as defined in (III.21)-(III.23).

IV.2 The Ground State Energy of T |φ|,α
In this section we show that the infimum of the spectrum of 1 2 dΓ J [T |φ| , 0] equals X(Θ |φ|,α ), as defined in (I. 19) and (III.21)-(III.23). This fact had already been observed in [23], and we give an alternative and detailed proof here. More specifically, we prove the following theorem in this section.
Theorem IV.5. Let J : h → h be defined by (III. 19), suppose that φ = |φ| ∈ H 1 (R 3 ), and let T φ,α and Θ φ,α be given as in (III.21)-(III.24). Then where qfDM denotes the set of quasifree density matrices. In other words, for the computation of the Bogolubov-Hartree-Fock energy of the system, one may restrict the variation over all quasifree states to pure states. This statement may be viewed as a generalization of Lieb's variational principle [24]. In Lemma IV.7 below, the observation from [4] is applied to the Hamiltonian dΓ J [T φ,α , 0] and yields the statement, that its ground state energy is the lowest vacuum expectation value of all homogeneous Bogolubov transforms of dΓ Tr 2|k| 1/2 v 2 |k| 1/2 + Θ The infimum on the right side of the lower bound (IV.21) is explicitly computed in Lemma IV.9 below, using σ · 1 ≤ a := 2|k| ≤ Λ · 1, b := |k| −1/2 Θ φ,α |k| −1/2 ≥ 0, and d := 0 again. Consequently, We finally define . Moreover, as |k| and Θ φ,α are J-invariant, so are y * and hence also v * and 1 + v 2 * . It follows that B * ∈ Bog J [h] is a homogeneous Bogolubov transformation. Finally, The first step in our derivation rests on an observation made in [4] which may be viewed as a generalization of Lieb's variational principle [24]. where DM := ρ ∈ B[F ph ] 0 ≤ ρ ≤ Tr{ρ} = 1 denotes the set of density matrices on F ph . In [4] it is shown that the Bogolubov-Hartree-Fock energy is already obtained by taking the infimum over all pure quasifree states, E BHF (H) = inf Tr{ρ 1/2 H ρ 1/2 } ρ ∈ DM , ρ is quasifree and pure .

(IV.30)
Since dΓ J [T |φ| , 0] is quadratic in the field operators, its ground state energy agrees with its Bogolubov-Hartree-Fock energy, On the other hand, the pure quasifree density matrices ρ pure ∈ DM are precisely the rank-one orthogonal projections ρ pure = |U * B W * η Ω U * B W * η Ω| onto Bogolubov and Weyl transforms U * B W * η Ω of the vacuum vector Ω, using that, U * B = U SB * S is a homogeneous Bogolubov transformation, for B ∈ Bog J [h], and W * η = W −η is a Weyl transformation, for η ∈ h. Thus we obtain it follows that the infimum on the right side of (IV.32) is attained for η = 0. and Proof. First, we note that then a simple computation using that j commutes with a, b, and d, shows that Using (IV.37), this yields From the Cauchy-Schwarz inequality for traces we obtain for any bounded and invertible positive operator x ≥ µ · 1 > 0.
Next we remark that, due to (III.15), we have For any r > 0, this implies that which, in turn, gives Writing the square root as an integral over resolvents according to for all r > 0. For small 0 < ε < 1, we define x ε := (1 + vv * ) −1/2 (ε + vv * ) 1/2 (IV.46) and observe that, due to (IV.45) and (IV.42), we have Inserting (IV.47) and (IV.48) into (IV.41) and taking the limit ε → 0, we obtain where y ≥ 1 is a positive operator defined by (IV.52) through functional calculus. Note in passing that y is uniquely determined by v up to ker(y − 1) and that y − 1 ∈ L 2 [h], due to Lemma IV.10 (i). Then (IV.53) Hence we have that Inserting the parametrization (IV.52) into the trace in (IV.51), we obtain Obviously, y → G(y) is convex. We define y * ≥ 1 by and observe that y * −1 ∈ L 2 [h], by Lemma IV.10 (ii), and that y * m 2 y * = m 2 +4b which is equivalent to The latter is the formal condition for stationarity of y → G(y). We refrain from turning this formal into a mathematically rigorous condition by establishing differentiability of G in a suitable sense. Instead, we simply check by computation that y * is the minimizer of G. Namely, we have that G(y) − G(y * ) := Tr m 2 (y − y * ) + (m 2 + 4b) y −1 − y −1 * , (IV.59) and the second resolvent equation gives Thus from (IV.58) derives is Hilbert-Schmidt, y ≥ 1, and m ≥ σ 1/2 · 1, for some σ > 0. Then the following assertions hold true.

V Localization Estimates
In this section we turn to the analysis of the effective energy functional Ultimately, we compare E α,Λ and its infimum E LL (α, Λ) to F β(α,Λ) and its infimum F [β(α, Λ)], respectively, where In the present section we demonstrate that the minimization in (V.3) may be restricted to functions supported in the ball B(0, L) = {x ∈ R 3 : |x| < L} of radius L < ∞, provided L ≫ 1 is sufficiently large. That is, we prove in Theorem V.1 below that approximates E LL (α, Λ), as L → ∞, by showing that the error made by this restriction is of order L −2 , as suggested by the IMS localization formula. Here, and we correspondingly approximate F [β] by Theorem V.1. There exists a universal constant C < ∞ such that, for all α, β, L > 0, σ ≥ 0, and Λ ≥ 1, with E LL (α, Λ), F (α, Λ), E For the derivation of the lower bound (V.10) on E LL (α, Λ) we pick a smooth and compactly supported function η ∈ C ∞ 0 (R 3 ; R + 0 ), chosen such that supp(η) ⊆ B(0, 1) and η 2 = 1. Then we define for all L > 0, and we observe that η L,z 2 = 1 and η 2 L,z (x) d 3 z = 1. We further set 13) and observe that ρ L is a probability density on R 3 . A variant of the IMS localization formula [12] now yields Note that and since A → X(A) is concave according to Lemma V.3 (ii), we obtain Taking the infimum over φ ∈ H 1 (R 3 ) concludes the proof of the first inequality in (V.10). The proof of the first inequality in (V.11) is similar.
For the proof of Theorem V.1, we supply various properties of X(A) in the following two lemmata. To formulate these it is convenient to denote Since on h, the multiplication operator σ · 1 ≤ |k| ≤ Λ · 1 is bounded and bounded invertible, we observe that Lemma V.2. Let A = A * ≥ 0 be a bounded self-adjoint operator on h such that (k 2 + A) 1 2 − |k| is trace class. Then Proof. Using (V.18)-(V.20), we have that where the finiteness of the left side of (V.22) implies finiteness of all following lines.
Lemma V.3. Let A = A * , B = B * ≥ 0 be two bounded self-adjoint operators on h such that K A − K 0 and K B − K 0 are trace class. Then Proof. Since A, B ≥ 0 we may use the operator monotonicity of the square root and the inverse to infer that A → K A and A → K −1 A are monotone and thus and additionally using Lemma V.2. This yields (V.23).
As for (V.24), we note that which yields midpoint concavity of A → X(A), i.e., Finally, midpoint concavity and continuity of A → X(A) implies general concavity and hence (V.24).
Proof. We first apply Lemma V.2 and the operator monotonicity of A → √ A and A → A −1 and observe that Secondly, we note that (φ * ) * (φ * ) = F φ 2 F * , where F is (componentwise) Fourier transformation. As is customary, we denote by φ(x) := F φF * ≥ 0 the corresponding nonnegative multiplication operator, indicating the change from momentum to position space by explicitly keeping the argument "x" for the spatial variable. Using (VI.4), the decomposition 1 = χ 0,σ + χ σ,Λ + χ Λ,(1+ε)Λ + χ (1+ε)Λ , where χ r := 1 − χ r , and the triangle inequality for the trace norm, we obtain is the main term. Note that the factor 2 takes into account that P C is an orthogonal projection of rank 2 on C ⊗ R 3 . Moreover, we denote by Vol[M] := 1 M (k) d 3 the three-dimensional Lebesgue measure of a measurable set M ⊆ R 3 in (VI.6) and henceforth. Furthermore, are error terms we proceed to estimate next. Before we remark that the Hilbert space in (VI.7)-(VI.9) is the space h 0 := L 2 (R 3 ) of complex-valued (scalar) square-integrable functions, as opposed to the one-photon Hilbert space h of square-integrable divergence-free vector fields used before. The factors 3 on the right side of (VI.5) account for the three components of the latter. Using the trace inequality (VI.10) 11) similarly to (VI.6).

VII Lower Bound on X(Θ φ,α )
In order to complement the upper bound on X(Θ φ,α ) from Section VI by a corresponding lower bound we first derive a general inequality on X(A) of the form X(A) ≥ Tr A 1/2 − 2Λ 1−p Tr A p/2 , where p is any exponent between 1 2 and 1. By another application of the Birman-Solomyak inequality (VI.1) we then estimate the emerging error term by a multiple of φ p 1 . We begin by deriving a general lower bound on X(A) only using that |k| ≤ Λ · 1 on h.
Lemma VII.1. Let A ≥ 0 be a nonnegative self-adjoint operator on h such that A 1/2 ∈ L[h] is trace-class and assume that 0 < p < 1. Then Proof. We recall from Lemma V.2 that From the second resolvent identity we derive which implies and K A ≥ A 1/2 , we have that By operator monotonicity we further have Inserting (VII.6) and (VII.7) into (VII.4), we arrive at the claim.
As described above, we now use Lemma VII.1 to derive a lower bound on X(2Θ φ,α ).

VIII Asymptotics of the Lieb-Loss Energy
We turn to the proof of the main result of this paper, Theorem I.1, stated below again for the reader's convenience. In our proof a key role is played by the scaling relation the effective energy F [β] obeys. F [β] is defined in (V.5) as the infimum of the functional F β > 0 over L 2 -normalized functions in H 1 (R 3 ) ∩ L 1 (R 3 ).
In [19] one of us showed that this infimum is attained for some φ β and hence is actually a minimum. A major issue in this regard is non-reflexivity of the L 1 -space precluding a naive application of the direct method of the calculus of variations. This was remedied by using the theory of uniform convex spaces and the Milman-Pettis theorem. Subsequently an explicit characterization of the minimizer (up to spherical rearrangement) can be given in terms of a Bessel function. In particular, Proof. We first take the infrared limit σ → 0. Note that E LL , E LL , F , F (L) , X(2Θ φ,α ), and all error terms are continuous at σ = 0, and we can simply set σ := 0 everywhere. Then Theorems VI.1 and VII.2 with p = 1/2 yield some constant C 1 < ∞ and any φ = |φ| ∈ Y L with φ 2 = 1, provided that L ≥ Λ −1 .