On the Ground State Energy

of the Translation Invariant Pauli-Fierz Model. II.

Jean-Marie Barbaroux and Semjon A. Vugalter

Received: February 17, 2012

Communicated by Heinz Siedentop

Abstract.

We determine the ground state energy of the translation invariant Pauli-Fierz model for an electron with spin, to subleading order O(α^{2}) with respect to powers of the finestructure constant α and prove rigorous error bounds of order O(α^{3}). A main objective of our argument is its brevity.

2000 Mathematics Subject Classification: 81Q10, 35P15, 46N50, 47N50 Keywords and Phrases: Translation invariant Pauli-Fierz Hamiltonian, spectral theory, ground state energy.

1. Introduction

We continue the study of the translation invariant Pauli-Fierz model [2], describing a nonrelativistic free electron interacting with the quantized electromagnetic field. In contrast with [2], we study now electron with spin. We are interested in quantitative properties of the ground state energy (Theorem 2.1) and its associated eigenfunctions (Theorem 2.2). In particular, we determine the subleading terms of the ground state energy up to order α^{2}, where α denotes the finestructure constant, and rigorously bound the error by a term of order α^{3}. In comparison with [2], the ground state energy is an order of magnitude larger in powers of α, due to the presence of electron spin. Following the technique developed in [2] (see also [4]), our method is based on perturbations around the true ground state of the translation invariant operator, together with a bound on the expected photon number for this ground state, obtained by Chen and Fröhlich [8]. In particular, an important ingredient of the proof is the improvement of photon number estimates for different parts of the ground state. A well-known difficulty connected to this problem arises from the fact that the ground state energy is not an isolated eigenvalue of the Hamiltonian, and that the form factor in the interaction term of the Hamiltonian contains a critical frequency space singularity (the infrared problem of Quantum Electrodynamics (QED)). Estimates on the ground state energy play an important role, for instance, in binding problems, e.g., the determination of the Hydrogen binding energy [3]. The systematic study of Pauli-Fierz Hamiltonian was initiated in [1]. The first estimate for the translation invariant operator for spinless electron was obtained by [12]. Later on in [6], the model for electron with spin was considered, and the bound was obtained up to the order α^{2} with an error term of the order α^{\frac{5}{2}} log α. Such estimates are not sufficient to compute the correction to the binding energy due to the interaction with the radiation field. In [2] a new effective method was developed to obtain the self energy in the spinless case up to the order α^{3} with an error O(α^{4}). This result was later improved in [5] with computing the term O(α^{4}) with error term O(α^{5}). These last two results [2, 5] were crucial for proving that the binding energy in the case of the Hydrogen atom with spinless electron contained an α^{5} log α term and that this term comes from the ground state energy of the Hydrogen atom and does not exist in the translation invariant case [3]. In the work at hand, we are starting to implement the same program for the model of a Hydrogen atom with spin 1/2 electron interacting with the quantized radiation field. The first step of this program is, as in [2], computing the selfenergy, for the electron with spin, up to the order O(α^{3}). The Pauli-Fierz Hamiltonian H for a free electron coupled to the quantized electromagnetic field is defined by

\begin{aligned}& H = : (i∇_{x} ⊗ I_{f} − \sqrt{α}A(x))^{2} : +\sqrt{α}σ · B(x) + I_{el} ⊗ H_{f}. \\ (1) \end{aligned}

where : ⋯: denotes normal ordering, corresponding to the subtraction of a normal ordering constant proportional to α. The operator H acts on the Hilbert space H := H_{el} ⊗ F, where H_{el} = L^{2}(ℝ^{3}, ℂ^{2}), is the Hilbert space of one non-relativistic electron, ℝ^{3} is the configuration space of the electron, and ℂ^{2} accomodates its spin. We describe the quantized electromagnetic field by use of the Coulomb gauge condition. Accordingly, the one-photon Hilbert space is given by L^{2}(ℝ^{3}) ⊗ ℂ^{2}, where ℝ^{3} denotes the photon momentum and ℂ^{2} accounts for the two independent transversal polarizations of the photon. The photon Fock space is then defined by

F = ⨁_{n∈ℕ} F_{s}^{(n)},

where the n-photons space F_{s}^{(n)} = ⨂_{s}^{n} (L^{2}(ℝ^{3}) ⊗ ℂ^{2}) is the symmetric tensor product of n copies of L^{2}(ℝ^{3}) ⊗ ℂ^{2}. We use units such that ℏ = c = 1, and where the mass of the electron equals m = 1/2. The electron charge is then given by e = \sqrt{α}, with α ≈ 1/137 denoting the fine structure constant. As usual, we will consider α as a parameter.

The operator that couples an electron to the quantized vector potential is given by

A(x) = ∑_{λ=1,2} ∫_{ℝ^{3}} \frac{ζ(|k|)}{2π|k|^{1/2}} ε_{λ}(k)[e^{ikx} ⊗ a_{λ}(k) + e^{−ikx} ⊗ a_{λ}^{∗}(k)]dk ,

where by the Coulomb gauge condition, divA = 0. The operators a_{λ}, a_{λ}^{∗} satisfy the usual commutation relations

[a_{ν}(k), a_{λ}^{∗}(k′)] = δ(k − k′)δ_{λ,ν}, [a_{ν}(k), a_{λ}(k′)] = 0,

and there exists a unique unit ray Ω_{f} ∈ F, the Fock vacuum, which satisfies a_{λ}(k)Ω_{f} = 0 for all k ∈ ℝ^{3} and λ ∈ {1, 2}. The vectors ε_{λ}(k) ∈ ℝ^{3} are the following two orthonormal polarization vectors perpendicular to k,

ε_{1}(k) = \frac{(k_{2}, −k_{1}, 0)}{\sqrt{k_{1}^{2} + k_{2}^{2}}}  and  ε_{2}(k) = \frac{k}{|k|} ∧ ε_{1}(k).

The function ζ(|k|) describes the ultraviolet cutoff on the wavenumbers k. We assume ζ to be of class C^{1}, with compact support. The operator that couples the electron to the magnetic field is

B(x) = ∑_{λ=1,2} ∫_{ℝ^{3}} \frac{ζ(|k|)}{2π|k|^{1/2}} k × iε_{λ}(k)[e^{ikx} ⊗ a_{λ}(k) − e^{−ikx} ⊗ a_{λ}^{∗}(k)]dk .

In Equation (1), σ = (σ_{1}, σ_{2}, σ_{3}) is the 3-component vector of Pauli matrices. The photon field energy operator H_{f} is given by

H_{f} = ∑_{λ=1,2} ∫_{ℝ^{3}} |k|a_{λ}^{∗}(k)a_{λ}(k)dk.

For convenience, in the following, we shall denote

A(x) = A^{−}(x) + A^{+}(x) and B(x) = B^{−}(x) + B^{+}(x)

where

A^{−}(x) := ∑_{λ=1,2} ∫_{ℝ^{3}} \frac{ζ(|k|)}{2π|k|^{1/2}} ε_{λ}(k)e^{ikx} ⊗ a_{λ}(k)dk ,

A^{+}(x) := (A^{−}(x))^{∗},

B^{−}(x) := ∑_{λ=1,2} ∫_{ℝ^{3}} \frac{ζ(|k|)}{2π|k|^{1/2}} k × iε_{λ}(k)e^{ikx} ⊗ a_{λ}(k)dk ,

and B^{+}(x) := (B^{−}(x))^{∗}. The system is translationally invariant, and H commutes with the operator of total momentum

P_{tot} = i∇_{x} ⊗ I_{f} + I_{el} ⊗ P_{f},

where i∇_{x} and P_{f} = ∑_{λ=1,2} ∫ ka_{λ}^{∗}(k)a_{λ}(k)dk denote respectively the electron and the photon momentum operators.

Therefore, if H_{p} ≅ ℂ^{2} ⊗ F denotes the fibre Hilbert space corresponding to conserved total momentum p, for any fixed value p of the total momentum, the restriction of H to the fibre space H_{p} is given by (see e.g. [7])

(2)  H(p) =: (p − P_{f} − \sqrt{α}A(0))^{2} : +\sqrt{α}σ · B(0) + H_{f}.

Henceforth, we will write A^{±} := A^{±}(0) and B^{±} = B^{±}(0). It is known that inf spec(H) = inf spec(H(0)) for small α [9]. Moreover, the ground state energy of the one electron self-energy operator with total momentum p = 0 is an eigenvalue of multiplicity two [7]. The case of spinless electron, without restriction on α, was investigated earlier in [10]. In the sequel, we shall study the operator H(p = 0).

2. Statements of the main results

Consider

(3)  Ω_{0} := λΩ_{f} , with λ ∈ ℂ^{2} such that |λ| = 1 ,

and define

(4)  Γ_{1} := −(H_{f} + P_{f}^{2})^{−1}σ.B^{+}Ω_{0}
(5)  Γ_{2} = −(H_{f} + P_{f}^{2})^{−1} [σ · B^{+}Γ_{1} + 2A^{+} · P_{f}Γ_{1} + A^{+} · A^{+}Ω_{0}] .

On ℂ^{2} ⊗ F, we define the positive bilinear form

(6)  ⟨v, w⟩_{∗} := ⟨v, (H_{f} + P_{f}^{2})w⟩ ,

and its associated semi-norm ∥v∥_{∗} = ⟨v, v⟩_{∗}.

Theorem 2.1 (Ground state energy of H(0)). We have

inf spec(H(0)) = −α∥Γ_{1}∥_{∗}^{2}

+ α^{2} (2∥A^{−}Γ_{1}∥^{2} − ∥Γ_{2}∥_{∗}^{2} + ∥Γ_{1}∥_{∗}^{2} ∥Γ_{1}∥^{2}) + O(α^{3}) . (7)

The proof of the Theorem consists in proving an upper bound obtained with a trial state (see inequality (12) in Section 3), and a lower bound obtained by variational estimates (see (59) in Section 6).

Remark 2.1. Recall that the ground state of H(0) is twice degenerate [7]. The result in Theorem 2.1 does not depend on the choice of Ω_{0}. According to Lemma 7.2, the term of order α is nonzero.

In the remainder, we will need the following notations. For n ∈ ℕ, let P_{n} be the orthogonal projection onto the subspace ℂ^{2} ⊗ F_{n} of the space ℂ^{2} ⊗ F, and P_{≥n} be the orthogonal projection onto the space ℂ^{2} ⊗ (⨁_{k≥n} F_{k}). Let Φ_{0} be a ground state of H(0) with the condition P_{0}Φ_{0} = Ω_{0}. Taking the ⟨ . , . ⟩_{∗}-orthonormal projections of Φ_{0} along the vectors Γ_{1} and Γ_{2}, and denoting by R the component in the ⟨ . , . ⟩_{∗}-orthogonal complement of their span, we get

(8)  Φ_{0} = Ω_{0} + α^{\frac{1}{2}} γ_{1}Γ_{1} + αγ_{2}Γ_{2} + R

Self-energy of an Electron in NRQED 405

where for i = 1, 2 we assume

(9)  ⟨Γ_{i}, R⟩_{∗} = 0 and P_{0}R = 0 .

Theorem 2.2. For Φ_{0} defined by (8) and (9), we have |γ_{1} − 1| = O(α) |γ_{2} − 1| = O(α^{\frac{1}{2}} ) (10) ∥R∥_{∗} = O(α^{\frac{3}{2}} ) ∥R∥ = O(α) . (11) The statement of this theorem follows immediately from the proof of Theorem 2.1.

3. Upper bound to the ground state energy

In this section, we prove the upper bound for the ground state energy inf spec(H(0))

≤ −α∥Γ_{1}∥_{∗}^{2} + α^{2} (2∥A^{−}Γ_{1}∥^{2} − ∥Γ_{2}∥_{∗}^{2} + ∥Γ_{1}∥_{∗}^{2} ∥Γ_{1}∥^{2}) + O(α^{3}) . (12)

Let us define the following trial state

Θ = Ω_{f} + \sqrt{α} Γ_{1} + α Γ_{2} .

Since

\begin{matrix}H(0) = & H_{f} + P_{f}^{2} + 4\sqrt{α}ReP_{f} · A^{−} + 2αReA^{+} · A^{+} \\ + 2αA^{+} · A^{−} + 2\sqrt{α}Re σ·B^{−} \end{matrix} ; (13)

we obtain

⟨H(0)Θ, Θ⟩ = α∥Γ_{1}∥_{∗}^{2} + α^{2}∥Γ_{2}∥_{∗}^{2} + 2α^{2}Re⟨A^{+} · A^{+}Ω_{f}, Γ_{2}⟩
+ 2αRe⟨σ · B^{−}Γ_{1}, Ω_{f}⟩ + 2α^{2}∥A^{−}Γ_{1}∥^{2} + 4α^{2}Re⟨P_{f} · A^{−}Γ_{2}, Γ_{1}⟩
+ 2α^{2}Re⟨σ · B^{−}Γ_{2}, Γ_{1}⟩ + 2α^{3}∥A^{−}Γ_{2}∥^{2}
= −α^{2}∥Γ_{1}∥_{∗}^{2} − α^{2}∥Γ_{2}∥_{∗}^{2} + 2α^{2}∥A^{−}Γ_{1}∥^{2} + 2α^{3}∥A^{−}Γ_{2}∥^{2}

(14)

where in the last equality, we used

\begin{matrix}α^{2}Re⟨A^{+}·A^{+}Ω_{f}, Γ_{2}⟩+2α^{2}Re⟨P_{f} · A^{−}Γ_{2}, Γ_{1}⟩+α^{2}Re⟨σ · B^{−}Γ_{2}, Γ_{1}⟩ \\ = −α^{2}∥Γ_{2}∥_{∗}^{2} , \end{matrix}

and

2αRe⟨σ · B^{−}Γ_{1}, Ω_{f}⟩ = −2α∥Γ_{1}∥_{∗}^{2} .

The identity ∥Θ∥^{2} = 1 + α∥Γ_{1}∥^{2} + α^{2}∥Γ_{2}∥^{2} together with (14) yields inf spec(H(0)) ≤ \frac{⟨ H(0)Θ, Θ ⟩}{∥Θ∥^{2}} = −α^{2}∥Γ_{1}∥_{∗}^{2} − α^{2}∥Γ_{2}∥_{∗}^{2} + 2α^{2}∥A^{−}Γ_{1}∥^{2} + α^{2}∥Γ_{1}∥_{∗}^{2}∥Γ_{1}∥^{2} + O(α^{3}) .

which concludes the proof of the upper bound (12).

Documenta Mathematica 17 (2012) 401–415

4. A priori estimates

Let Φ_{0} denote the ground state of H(0) with the normalization condition P_{0}Φ_{0} = Ω_{0}, where Ω_{0} is defined by (3). For Γ_{1} defined by (4), we decompose Φ_{0} as

(15)  Φ_{0} = Ω_{0} + (γ_{1}Γ_{1} + R_{1}) + P_{≥2}Φ_{0} with ⟨Γ_{1}, R_{1}⟩_{∗} = 0 , γ_{1} ∈ ℂ .

Proposition 4.1. The following estimate holds

(16)  ∥Φ_{0}∥_{∗} = O(α) .

Proof.

⟨H(0)Φ_{0}, Φ_{0}⟩ = ⟨(H_{f} + P_{f}^{2})Φ_{0}, Φ_{0}⟩ + 4\sqrt{α}Re⟨P_{f} · A^{−}Φ_{0} Φ_{0}⟩

+ 2\sqrt{α}Re⟨σ · B^{−}Φ_{0}, Φ_{0}⟩ + 2αRe⟨A^{−} · A^{−}Φ_{0} Φ_{0}⟩ + 2α∥A^{−}Φ_{0}∥^{2}

≥ ⟨(H_{f} + P_{f}^{2})Φ_{0}, Φ_{0}⟩ − c\sqrt{α}∥Hf^{\frac{1}{2}} Φ_{0}∥ ∥P_{f}Φ_{0}∥

− c\sqrt{α}∥Hf^{\frac{1}{2}} Φ_{0}∥ ∥Φ_{0}∥ − cα∥Hf^{\frac{1}{2}} Φ_{0}∥ (∥Hf^{\frac{1}{2}} ψ∥ + ∥ψ∥)

≥ ⟨(H_{f} + P_{f}^{2})Φ_{0}, Φ_{0}⟩ − (c\sqrt{α} + \frac{1}{4})∥Hf^{\frac{1}{2}} Φ_{0}∥^{2}

− c\sqrt{α}∥P_{f}Φ_{0}∥^{2} − cα∥Φ_{0}∥^{2}

≥ \frac{1}{2}⟨(H_{f} + P_{f}^{2})Φ_{0}, Φ_{0}⟩ − cα = \frac{1}{2}∥Φ_{0}∥_{∗}^{2} − cα

(17)

using in the second inequality of (17) that for all ψ ∈ ℂ^{2} ⊗ F we have

∥A^{−}ψ∥ ≤ c∥Hf^{\frac{1}{2}} ψ∥, ∥B^{−}ψ∥ ≤ c∥Hf^{\frac{1}{2}} ψ∥ and ∥A^{+}ψ∥ ≤ c(∥Hf^{\frac{1}{2}} ψ∥ + ∥ψ∥) (see

e.g. [11, Lemma A4]). The proof of (16) follows from (17) and the fact that ⟨H(0)Φ_{0}, Φ_{0}⟩ ≤ ⟨H(0)Ω_{0}, Ω_{0}⟩ = 0. □

Proposition 4.2. There exists c > 0 such that for all ϕ ∈ ℂ^{2} ⊗ F,

\begin{aligned}& ⟨H(0)ϕ, ϕ⟩ − \frac{1}{2}∥ϕ∥_{∗}^{2} & ≥ −cα∥ϕ∥^{2} . \\ (18) \end{aligned}

Proof. The proof is done by repeating all steps in (17), replacing Φ_{0} by ϕ. □

The next result is a consequence of an a priori photon number bound for the ground state obtained in [8, Proposition 5.1], whose statement is given in Lemma 7.3 for total momentum p = 0.

Proposition 4.3. The following holds

(19)  ∥P_{≥1}Φ_{0}∥^{2} = O(α) .

Proof. Applying Lemma 7.3, we obtain

\begin{aligned}∥P_{≥1}Φ_{0}∥^{2} & ≤ ⟨P_{≥1}Φ_{0}, N_{f} P_{≥1}Φ_{0}⟩ = ∑_{λ=1,2} ∫ ∥a_{λ}(k)Φ_{0}∥^{2}dk \\ & ≤ c ∫ \frac{α}{|k|^{2}} ζ(|k|)^{2}dk ≤ cα , \end{aligned}

where N_{f} is the photon number operator. □

Corollary 4.1. There exists α_{0} > 0 such that γ_{1} is uniformly bounded in α ∈ [0, α_{0}].

Proof. Since ⟨R_{1}, Γ_{1}⟩_{∗} = 0, then from Proposition 4.1, we have that there exists α_{0} and c such that for all α smaller than α_{0},

(20)  α⟨(H_{f} + P_{f}^{2})γ_{1}Γ_{1}, Γ_{1}⟩ ≤ cα

which implies |γ_{1}|^{2} ≤ c (⟨(H_{f} + P_{f}^{2})Γ_{1}, Γ_{1}⟩)^{−1}. We conclude the proof by applying Lemma 7.2. □

Corollary 4.2. We have

(21)  ∥R_{1}∥_{∗}^{2} = O(α) and ∥R_{1}∥^{2} = O(α) .

Proof. Applying Proposition 4.1 and Corollary 4.1 gives

∥R_{1}∥_{∗} ≤ ∥Π_{1}Φ_{0}∥_{∗} + \sqrt{α} |γ_{1}| ∥Γ_{1}∥_{∗} ≤ c\sqrt{α} .

Similarly, applying Proposition 4.3 and Corollary 4.1 we get

(22)  ∥R_{1}∥ ≤ ∥Π_{1}Φ_{0}∥ + \sqrt{α} |γ_{1}| ∥Γ_{1}∥ ≤ c\sqrt{α} .
□

5. Lower bound up to the order α

In the present section, we derive a sharp lower bound for the ground state energy ⟨H(0)Φ_{0}, Φ_{0}⟩/∥Φ_{0}∥^{2}, up to the order α, with rest of order α^{2}. The proof also implies improved estimates on γ_{1}, ∥R_{1}∥_{∗} and ∥P_{≥2}Φ_{0}∥_{∗}. These results are stated as follows

Proposition 5.1. The following holds

(23) inf spec(H(0)) = −α∥Γ_{1}∥_{∗}^{2} + O(α^{2}) ∥R_{1}∥_{∗} = O(α) (24) ∥P_{≥2}Φ_{0}∥_{∗} = O(α) (25) γ_{1} = 1 + O(\sqrt{α}), Imγ_{1} = O(\sqrt{α}) (26)

Proof. Using the decomposition (15) for Φ_{0}, and the identity (13) for H(0), we get

⟨H(0)Φ_{0}, Φ_{0}⟩
=α|γ_{1}|^{2}∥Γ_{1}∥_{∗}^{2}+∥R_{1}∥_{∗}^{2}+ ∥P_{≥2}Φ_{0}∥_{∗}^{2}+ 4\sqrt{α}Re⟨P_{f} ·A^{−}P_{≥2}Φ_{0}, P_{≥1}Φ_{0}⟩
+ 2\sqrt{α}Re⟨σ · B^{−}(\sqrt{α}γ_{1}Γ_{1} + R_{1} + P_{≥2}Φ_{0}), Φ_{0}⟩ + 2α∥A^{−}Φ_{0}∥^{2}
+ 2αRe⟨A^{−} · A^{−}P_{n≥2}Φ_{0}, Φ_{0}⟩ .

(27)

Applying [11, Lemma A4], Corollary 4.1 and Corollary 4.2, the fourth term in the right hand side of (27) is estimated as

\begin{matrix}4\sqrt{α}Re⟨P_{f} · A^{−}P_{≥2}Φ_{0}, P_{≥1}Φ_{0}⟩ \\ ≥ −ϵ∥Hf^{\frac{1}{2}} P_{≥2}Φ_{0}∥^{2} − c(ϵ)α∥P_{f}P_{1}Φ_{0}∥^{2} − c(ϵ)α∥P_{f}P_{≥2}Φ_{0}∥^{2} \\ ≥ −ϵ∥P_{≥2}Φ_{0}∥_{∗}^{2} − cα^{2} . \end{matrix} ; (28)

In (28), as well as in the sequel, we shall omit the ϵ-dependence of the constants c since ϵ will eventually be given a fixed value independent of α. Similarly, using in addition ⟨σ · B^{−}Γ_{1}, Ω_{0}⟩ = −∥Γ_{1}∥_{∗}^{2} and ⟨σ · B^{−}R_{1}, Ω_{0}⟩ = ⟨R_{1}, Γ_{1}⟩_{∗} = 0, we estimate the fifth term in the right hand side of (27) as

2\sqrt{α}Re⟨σ · B^{−}(\sqrt{α}γ_{1}Γ_{1} + R_{1} + P_{≥2}Φ_{0}), Φ_{0}⟩

= 2αRe⟨σ · B^{−}γ_{1}Γ_{1}, Ω_{0}⟩ + 2\sqrt{α}Re⟨σ · B^{−}R_{1}, Ω_{0}⟩

+ 2\sqrt{α}Re⟨σ · B^{−}P_{≥2}Φ_{0}, P_{≥1}Φ_{0}⟩

≥ −2αReγ_{1}∥Γ_{1}∥_{∗}^{2} − ϵ∥Hf^{\frac{1}{2}} P_{n≥2}Φ_{0}∥^{2} − cα∥P_{≥1}Φ_{0}∥^{2}

≥ −2αReγ_{1}∥Γ_{1}∥_{∗}^{2} − ϵ∥P_{≥2}Φ_{0}∥_{∗}^{2} − cα^{2} (29)

The sixth term in the right hand side of (27) is nonnegative, and with similar arguments as above, the seventh is bounded by

\begin{matrix}2αRe⟨A^{−} · A^{−}P_{≥2}Φ_{0}, Φ_{0}⟩ ≥ −cα∥A^{−}P_{≥2}Φ_{0}∥ ∥A^{+}Φ_{0}∥ \\ ≥ −ϵ∥P_{≥2}Φ_{0}∥_{∗}^{2} − cα^{2} . \end{matrix} ; (30)

Collecting (27)-(30) gives

\begin{matrix}⟨H(0)Φ_{0}, Φ_{0}⟩ \\ ≥ \frac{1}{2}∥P_{≥2}Φ_{0}∥_{∗}^{2} + α|1 − γ_{1}|^{2}∥Γ_{1}∥_{∗}^{2} − α∥Γ_{1}∥_{∗}^{2} + ∥R_{1}∥_{∗}^{2} − cα^{2} \end{matrix} ; (31)

To prove (23) we first note that from the decomposition (15) of Φ_{0} and Proposition 4.3, we have ∥Φ_{0}∥^{2} = 1 + O(α). Therefore, (23) is a consequence of this equality, the upper bound (12), and the lower bound (31). The estimates (24)-(26) are direct consequences of (31) and the fact that ⟨H(0)Φ_{0}, Φ_{0}⟩ ≤ 0. □

6. Lower bound up to the order α^{2}

Equipped with the estimates of Sections 4 and 5, we are now ready to establish a lower bound up to the order α^{2}, with error term of the order α^{3} for the ground state energy. For Γ_{1} defined by (4) and for γ_{1} given by the decomposition (15) of Φ_{0}, we define

\begin{aligned}& Γ_{2}^{(γ_{1})} = −(H_{f} + P_{f}^{2})^{−1} (γ_{1}σ · B^{+}Γ_{1} + 2γ_{1}A^{+} · P_{f}Γ_{1} + A^{+} · A^{+}Ω_{0}) , \\ (32) \end{aligned}

and we define γ_{2} and R_{2} (depending on γ_{1}) by

(33)  P_{2}Φ_{0} = αγ_{2}Γ_{2}^{(γ_{1})} + R_{2} and ⟨R_{2}, Γ_{2}^{(γ_{1})}⟩_{∗} = 0 .

Thus, we have

(34)  Φ_{0} = Ω_{0} + \sqrt{α}γ_{1}Γ_{1} + R_{1} + αγ_{2}Γ_{2}^{(γ_{1})} + R_{2} + P_{≥3}Φ_{0} .

6.1. Preliminary estimates. Before estimating the ground state energy, we need to prove some estimates for the vectors occurring in the decomposition (34) for Φ_{0}.

Proposition 6.1. There exits α_{1} > 0 and c > 0 such that for all α ∈ (0, α_{1}) and all γ_{1} ∈ (\frac{1}{2}, \frac{3}{2}), we have

|γ_{2}| < c .

Proof. From (25) of Proposition 5.1 we have

(35)  ∥P_{2}Φ_{0}∥_{∗} = α^{2}|γ_{2}|^{2}∥Γ_{2}^{(γ_{1})}∥_{∗}^{2} + ∥R_{2}∥_{∗}^{2} < cα^{2} .

Together with Lemma 7.1, this yields the result. □

Proposition 6.2. There exists α_{2} > 0 and c > 0 such that for all α ∈ (0, α_{2}), for all ϵ > 0, and for all γ_{1} ∈ (\frac{1}{2}, \frac{3}{2})

\begin{matrix}∥R_{1}∥^{2} ≤ & c ϵ^{−1}α^{2} + ϵα^{−1}∥Hf^{\frac{1}{2}} R_{1}∥^{2} , \\ ∥R_{2}∥^{2} ≤ & c ϵ^{−1}α^{2} + ϵα^{−1}∥Hf^{\frac{1}{2}} R_{2}∥^{2} , \\ ∥P_{≥3}Φ_{0}∥^{2} ≤ & c ϵ^{−1}α^{2} + ϵα^{−1}∥Hf^{\frac{1}{2}} P_{≥3}Φ_{0}∥^{2} . \end{matrix} ; (36)

Proof. We have, applying lemma 7.3

(37)  ∥a_{λ}(k)R_{2}∥ ≤ ∥a_{λ}(k)P_{2}Φ_{0}∥ + α |γ_{2}| ∥a_{λ}(k)Γ_{2}^{(γ_{1})}∥ ≤ c\sqrt{α}/|k| .

Therefore, given ϵ > 0, we have

∥R_{2}∥^{2} ≤ ∑_{λ=1,2} ∫ ∥a_{λ}(k)R_{2}∥^{2}dk

= ∑_{λ=1,2} ∫_{ϵ|k|<α} ∥a_{λ}(k)R_{2}∥^{2}dk + ∑_{λ=1,2} ∫_{ϵ|k|≥α} ∥a_{λ}(k)R_{2}∥^{2}dk

≤ ∫_{ϵ|k|<α} |\frac{c\sqrt{α}}{|k|}|^{2} dk + ∫_{ϵ|k|≥α} \frac{ϵ|k|}{α}∥a_{λ}(k)R_{2}∥^{2}dk

≤ c^{2}ϵ^{−1}α^{2} + ϵα^{−1}∥Hf^{\frac{1}{2}} R_{2}∥^{2} .

The bounds for ∥R_{1}∥ and ∥P_{≥3}Φ_{0}∥ can be derived similarly. □

To estimate ⟨H(0)Φ_{0}, Φ_{0}⟩ we use the above decomposition (34) of Φ_{0} and the identity (13).

6.2. Terms involving P_{1}Φ_{0} but not P_{≥2}Φ_{0}. Denoting by (I) the terms in ⟨H(0)Φ_{0}, Φ_{0}⟩ involving P_{1}Φ_{0} but not P_{≥2}Φ_{0} we have

\begin{matrix}(I) = & α|γ_{1}|^{2} ∥Γ_{1}∥_{∗}^{2} + ∥R_{1}∥_{∗}^{2} + 2\sqrt{α}Re⟨σ · B^{−}(\sqrt{α}γ_{1}Γ_{1} + R_{1}), Ω_{0}⟩ \\ + 2α∥A^{−}(\sqrt{α}γ_{1}Γ_{1} + R_{1})∥^{2} . \end{matrix} ; (38)

Using the definition of Γ_{1} and ⟨Γ_{1}, R_{1}⟩_{∗} = 0, we get that the third term in the right hand side equals −2αReγ_{1}∥Γ_{1}∥_{∗}^{2}. Using the bound (26) on γ_{1}, we get that the fourth term in the right hand side of (38) is estimated as

\begin{matrix}2α^{2}|γ_{1}|^{2}∥A^{−}Γ_{1}∥^{2} + 4α^{\frac{3}{2}} Re⟨A^{−}γ_{1}Γ_{1}, A^{−}R_{1}⟩ + 2α∥A^{−}R_{1}∥^{2} \\ ≥ 2α^{2}|γ_{1}|^{2}∥A^{−}R_{1}∥^{2} − ϵ∥Hf^{\frac{1}{2}} R_{1}∥^{2} + O(α^{3}) . \end{matrix} ; (39)

The above thus implies

(I) ≥ α∥Γ_{1}∥_{∗}^{2}(|γ_{1}|^{2}−2Reγ_{1}) + (1−ϵ)∥R_{1}∥_{∗}^{2}
+ 2α^{2}|γ_{1}|^{2}∥A^{−}Γ_{1}∥^{2}+ O(α^{3})
=−α^{2}∥Γ_{1}∥_{∗}^{2} + α|1 − γ_{1}|^{2}∥Γ_{1}∥_{∗}^{2} + 2α^{2}(|γ_{1}|^{2} − 1)∥A^{−}Γ_{1}∥^{2}
+ 2α^{2}∥A^{−}Γ_{1}∥^{2} + (1 − ϵ)∥R_{1}∥_{∗}^{2} + O(α^{3})

This yields

(I) ≥ −α^{2}∥Γ_{1}∥_{∗}^{2} + 2α^{2}∥A^{−}Γ_{1}∥^{2} + \frac{1}{2}α|1−γ_{1}|^{2}∥Γ_{1}∥_{∗}^{2}

+ \frac{1}{2}α|1−γ_{1}|^{2}∥Γ_{1}∥_{∗}^{2} − cα^{2} ||γ_{1}|^{2} − 1| ∥Γ_{1}∥_{∗}^{2} + (1 − ϵ)∥R_{1}∥_{∗}^{2} + O(α^{3})

≥ −α^{2}∥Γ_{1}∥_{∗}^{2} + 2α^{2}∥A^{−}Γ_{1}∥^{2} + \frac{1}{2}α|1 − γ_{1}|^{2}∥Γ_{1}∥_{∗}^{2}

+α∥Γ_{1}∥_{∗}^{2}(\frac{1}{2}(1−Reγ_{1})^{2} + \frac{1}{2}(Imγ_{1})^{2} − cα |Reγ_{1}−1|− cα(Imγ_{1})^{2})

+ (1 − ϵ)∥R_{1}∥_{∗}^{2} + O(α^{3})

≥ −α^{2}∥Γ_{1}∥_{∗}^{2} + 2α^{2}∥A^{−}Γ_{1}∥^{2} + \frac{1}{2}α|1 − γ_{1}|^{2}∥Γ_{1}∥_{∗}^{2}

+ (1 − ϵ)∥R_{1}∥_{∗}^{2} + O(α^{3}) , (40)

using \frac{1}{2}(1−Reγ_{1})^{2} −cα |Reγ_{1} − 1| ≥ −c′α^{2} and \frac{1}{2}(Imγ_{1})^{2} −cαImγ_{1} ≥ −c′α^{2}.

6.3. Terms involving P_{2}Φ_{0} but not P_{≥3}Φ_{0}. Denoting by (II) the terms in ⟨H(0)Φ_{0}, Φ_{0}⟩ involving P_{2}Φ_{0} but not P_{≥3}Φ_{0} we have

(II) = α^{2}|γ_{2}|^{2}∥Γ_{2}^{(γ_{1})}∥_{∗}^{2} + ∥R_{2}∥_{∗}^{2}
+ 4α^{2}Reγ_{2}\overline{γ_{1}}⟨P_{f} · A^{−}Γ_{2}^{(γ_{1})}, Γ_{1}⟩ + 4α^{\frac{3}{2}} Reγ_{2}⟨P_{f} · A^{−}Γ_{2}^{(γ_{1})}, R_{1}⟩
+ 4αRe\overline{γ_{1}}⟨P_{f} · A^{−}R_{2}, Γ_{1}⟩ + 4\sqrt{α}Re⟨P_{f} · A^{−}R_{2}, R_{1}⟩
+ 2α^{2}Reγ_{2}\overline{γ_{1}}⟨σ · B^{−}Γ_{2}^{(γ_{1})}, Γ_{1}⟩ + 2α^{\frac{3}{2}} Reγ_{2}⟨σ · B^{−}Γ_{2}^{(γ_{1})}, R_{1}⟩
+ 2αRe\overline{γ_{1}}⟨σ · B^{−}R_{2}, Γ_{1}⟩ + 2\sqrt{α}Re⟨σ · B^{−}R_{2}, R_{1}⟩
+ 2α^{2}Reγ_{2}⟨A^{−} · A^{−}Γ_{2}^{(γ_{1})}, Ω_{0}⟩ + 2αRe⟨A^{−} · A^{−}R_{2}, Ω_{0}⟩
+ 2α^{3}|γ_{2}|^{2}∥A^{−}Γ_{2}^{(γ_{1})}∥^{2} + 2α∥A^{−}R_{2}∥ + 4α^{2}Reγ_{2}⟨A^{+} · A^{−}Γ_{2}^{(γ_{1})}, R_{2}⟩ .

(41)

The sum of the fifth, the ninth and the twelfth terms in the right hand side of (41) is equal to −2α⟨R_{2}, Γ_{2}^{(γ_{1})}⟩_{∗} = 0; the sum of the third, seventh and eleventh terms is equal to −2α^{2}γ_{2}∥Γ_{2}^{(γ_{1})}∥_{∗}^{2}; the sum of the fourth and the sixth terms is bounded below by

\begin{matrix}−c\sqrt{α}∥Hf^{\frac{1}{2}} P_{2}Φ_{0}∥ ∥Hf^{\frac{1}{2}} R_{1}∥ ≥ −ϵ∥R_{1}∥_{∗}^{2} − c(ϵ)α∥P_{2}Φ_{0}∥_{∗}^{2} \\ ≥ −ϵ∥R_{1}∥_{∗}^{2} − cα^{3} , \end{matrix}

according to (25) of Proposition 5.1. Applying Proposition 6.2, yields

\begin{matrix}2\sqrt{α}Re⟨σ · B^{−}R_{2}, R_{1}⟩ ≥ −ϵ∥R_{2}∥_{∗}^{2} − cα∥R_{1}∥^{2} \\ ≥ −ϵ∥R_{2}∥_{∗}^{2} − ϵ∥R_{1}∥_{∗}^{2} − cα^{3} . \end{matrix}

The term 2α^{\frac{3}{2}} Reγ_{2}⟨σ · B^{−}Γ_{2}^{(γ_{1})}, R_{1}⟩ in (41) is estimated by

\begin{matrix}2α^{\frac{3}{2}} Reγ_{2}⟨σ · B^{−}Γ_{2}^{(γ_{1})}, R_{1}⟩ = 2α^{\frac{3}{2}} Reγ_{2}⟨H_{f}^{− \frac{1}{2}} σ · B^{−}Γ_{2}^{(γ_{1})}, Hf^{\frac{1}{2}} R_{1}⟩ \\ ≥ −ϵ∥R_{1}∥_{∗}^{2} − cα^{3} , \end{matrix} ; (42)

since the norm of H_{f}^{− \frac{1}{2}} σ ·B^{−}Γ_{2}^{(γ_{1})} is uniformly bounded in γ_{1} ∈ (\frac{1}{2}, \frac{3}{2}). Finally we have

(43)  4α^{2}Reγ_{2}⟨A^{+} · A^{−}Γ_{2}^{(γ_{1})}, R_{2}⟩ ≥ −ϵ∥R_{2}∥_{∗}^{2} − cα^{4} .

Collecting all the above estimates in (41) thus gives

\begin{aligned}(II) & ≥ α^{2}∥Γ_{2}^{(γ_{1})}∥_{∗}^{2}(|γ_{2}|^{2} − 2γ_{2}) + (1 − ϵ)∥R_{2}∥_{∗}^{2} − ϵ∥R_{1}∥_{∗}^{2} − cα^{3} \\ & ≥ −α^{2}∥Γ_{2}^{(γ_{1})}∥_{∗}^{2} + (1 − ϵ)∥R_{2}∥_{∗}^{2} − ϵ∥R_{1}∥_{∗}^{2} − cα^{3} . \end{aligned} ; (44)

6.4. Remaining terms. We collect in (III) all the terms in ⟨H(0)Φ_{0}, Φ_{0}⟩ that have not been treated in subsections 6.2 and 6.3. This yields

\begin{aligned}(III) & = ⟨H(0)P_{≥3}Φ_{0}, Φ_{0}⟩ \\ & = ⟨H(0)P_{≥3}Φ_{0}, P_{≥3}Φ_{0}⟩ + ⟨H(0)P_{≥3}Φ_{0}, (1 − P_{≥3})Φ_{0}⟩ \end{aligned} ; (45)

Applying Proposition 6.2, the first term in the right hand side of (45) is bounded below using the following estimate

\begin{aligned}⟨H(0)P_{≥3}Φ_{0}, P_{≥3}Φ_{0}⟩ − \frac{1}{2}∥P_{≥3}Φ_{0}∥_{∗}^{2} & ≥ −cα∥P_{≥3}Φ_{0}∥^{2} \\ & ≥ −cϵ^{−1}α^{3} − ϵ∥P_{≥3}Φ_{0}∥_{∗}^{2} . \end{aligned} ; (46)

The second term in the right hand side of (45) is

⟨H(0)P_{≥3}Φ_{0}, (1−P_{≥3})Φ_{0}⟩= 2\sqrt{α}Re⟨σ·B^{−}P_{3}Φ_{0}, P_{2}Φ_{0}⟩
+ 4\sqrt{α}Re⟨P_{f} · A^{−}P_{3}Φ_{0}, P_{2}Φ_{0}⟩ + 2αRe⟨A^{−}·A^{−}P_{3}Φ_{0}, P_{1}Φ_{0}⟩
+ 2αRe⟨A^{−}·A^{−}P_{4}Φ_{0}, P_{2}Φ_{0}⟩

(47)

We have

\begin{matrix}2αRe⟨A^{−} · A^{−}P_{3}Φ_{0}, P_{1}Φ_{0}⟩ ≥ −ϵ∥Hf^{\frac{1}{2}} P_{3}Φ_{0}∥_{∗}^{2} − cα^{2}∥A^{+}P_{1}Φ_{0}∥^{2} \\ ≥ −ϵ∥Hf^{\frac{1}{2}} P_{3}Φ_{0}∥_{∗}^{2} − cα^{2}∥P_{1}Φ_{0}∥^{2} ≥ −ϵ∥Hf^{\frac{1}{2}} P_{3}Φ_{0}∥_{∗}^{2} − cα^{3} , \end{matrix} ; (48)

where in the last inequality, we used Proposition 4.3.

Similarly, we get

\begin{aligned}& 2αRe⟨A^{−} · A^{−}P_{4}Φ_{0}, P_{2}Φ_{0}⟩ ≥ −ϵ∥Hf^{\frac{1}{2}} P_{4}Φ_{0}∥_{∗}^{2} − cα^{3} \\ (49) \end{aligned}

and

\begin{matrix}2αRe⟨σ · B^{−}P_{3}Φ_{0}, P_{2}Φ_{0}⟩ ≥ −ϵ∥Hf^{\frac{1}{2}} P_{3}Φ_{0}∥_{∗}^{2} − cα∥P_{2}Φ_{0}∥^{2} \\ ≥ −ϵ∥Hf^{\frac{1}{2}} P_{3}Φ_{0}∥_{∗}^{2} − cα^{3}|γ_{2}|^{2}∥Γ_{2}^{(γ_{1})}∥^{2} − cα∥R_{2}∥^{2} \\ ≥ −ϵ∥Hf^{\frac{1}{2}} P_{3}Φ_{0}∥_{∗}^{2} − cα^{3} − ϵ∥Hf^{\frac{1}{2}} R_{2}∥^{2} , \end{matrix} ; (50)

where in the last inequality we applied Propositions 6.1 and 6.2. The last term we have to estimate in (47) is

\begin{aligned}& 4\sqrt{α}Re⟨P_{f} ·A^{−}P_{3}Φ_{0}, P_{2}Φ_{0}⟩ ≥ −c\sqrt{α} (∥Hf^{\frac{1}{2}} P_{3}Φ_{0}∥^{2}+∥Hf^{\frac{1}{2}} P_{2}Φ_{0}∥^{2}) . \\ (51) \end{aligned}

Collecting (45)-(51) yields

\begin{aligned}& (III) ≥ \frac{1}{4}∥P_{≥3}Φ_{0}∥_{∗}^{2} − ϵ∥P_{2}Φ_{0}∥_{∗}^{2} − cα^{3} . \\ (52) \end{aligned}

6.5. Proof of the lower bound. The estimates (40), (44) and (52) for (I), (II) and (III) give

\begin{aligned}⟨H(0)Φ_{0}, Φ_{0}⟩ & ≥ −α∥Γ_{1}∥_{∗}^{2} + \frac{1}{2}α|1 − γ_{1}|^{2}∥Γ_{1}∥_{∗}^{2} + 2α^{2}∥A^{−}Γ_{1}∥^{2} \\ & − α^{2}∥Γ_{2}^{(γ_{1})}∥_{∗}^{2}+ \frac{1}{4} (∥R_{1}∥_{∗}^{2}+∥R_{2}∥_{∗}^{2}+∥P_{≥3}Φ_{0}∥_{∗}^{2}) − cα^{3}. \end{aligned} ; (53)

Next, we replace Γ_{2}^{(γ_{1})} by Γ_{2} in the above expression and estimate the difference. For that sake, we estimate

|∥Γ_{2}^{(γ_{1})}∥^{2} − ∥Γ_{2}∥^{2}| ≤ c |∥Γ_{2}^{(γ_{1})}∥ − ∥Γ_{2}∥| ≤ c∥Γ_{2}^{(γ_{1})} − Γ_{1}∥
≤ c|γ − 1| ∥Γ_{2} + (H_{f} + P_{f}^{2})^{−1}A^{+} · A^{+}Ω_{0}∥ ≤ c|γ − 1| .

where in the first inequality we applied Lemma 7.1. Applying this inequality, and Corollary 4.1, we thus can estimate in (53) the following two terms

(54)  \frac{1}{2}|1 − γ_{1}|^{2}α∥Γ_{1}∥_{∗}^{2} − α^{2}∥Γ_{2}^{(γ_{1})}∥_{∗}^{2} ≥ −α^{2}∥Γ_{2}∥_{∗}^{2} − cα^{3} .

Thus, (54) and (53) give

\begin{aligned}⟨H(0)Φ_{0}, Φ_{0}⟩ ≥ & − α∥Γ_{1}∥_{∗}^{2} + 2α^{2}∥A^{−}Γ_{1}∥^{2} − α^{2}∥Γ_{2}∥_{∗}^{2} \\ & + \frac{1}{4} (∥R_{1}∥_{∗}^{2} + ∥R_{2}∥_{∗}^{2} + ∥P_{≥3}Φ_{0}∥_{∗}^{2}) − cα^{3} . \end{aligned} ; (55)

Eventually, we compute

(56) \frac{⟨H(0)Φ_{0}, Φ_{0}⟩}{∥Φ_{0}∥^{2}} = \frac{⟨H(0)Φ_{0}, Φ_{0}⟩}{1 + ∥P_{1}Φ_{0}∥^{2} + ∥P_{≥2}Φ_{0}∥^{2}} ≥ \frac{⟨H(0)Φ_{0}, Φ_{0}⟩}{1 + ∥P_{1}Φ_{0}∥^{2}}

Documenta Mathematica 17 (2012) 401–415

since ⟨H(0)Φ_{0}, Φ_{0}⟩ ≤ 0. Now using from proposition 4.3 that ∥P_{≥1}Φ_{0}∥^{2} = O(α), we obtain, applying (55) and (56)

\begin{aligned}& \frac{⟨H(0)Φ_{0}, Φ_{0}⟩}{∥Φ_{0}∥^{2}} = ⟨H(0)Φ_{0}, Φ_{0}⟩ + ∥P_{1}Φ_{0}∥^{2}α∥Γ_{1}∥_{∗}^{2} + O(α^{3}) . \\ (57) \end{aligned}

Since ∥P_{1}Φ_{0}∥^{2}α∥Γ_{1}∥_{∗}^{2}

= α^{2}∥Γ_{1}∥^{2}∥Γ_{1}∥_{∗}^{2} + α∥R_{1}∥^{2}∥Γ_{1}∥_{∗}^{2} + 2α^{\frac{3}{2}} ∥Γ_{1}∥_{∗}^{2}Re⟨H_{f}^{− \frac{1}{2}} Γ_{1}, Hf^{\frac{1}{2}} R_{1}⟩

≥ α^{2}∥Γ_{1}∥^{2}∥Γ_{1}∥_{∗}^{2} − cα^{\frac{3}{2}} ∥Γ_{1}∥_{∗}^{2}∥H_{f}^{− \frac{1}{2}} Γ_{1}∥ ∥R_{1}∥_{∗}

≥ α^{2}∥Γ_{1}∥^{2}∥Γ_{1}∥_{∗}^{2} − cα^{3} − ϵ∥R_{1}∥_{∗}^{2} , (58)

we obtain, together with (55) and (57)

\frac{⟨H(0)Φ_{0}, Φ_{0}⟩}{∥Φ_{0}∥^{2}}

≥ −α∥Γ_{1}∥_{∗}^{2} + 2α^{2}∥A^{−}Γ_{1}∥^{2} − α^{2}∥Γ_{2}∥_{∗}^{2} + α^{2}∥Γ_{1}∥^{2}∥Γ_{1}∥_{∗}^{2} + O(α^{3}) .

(59)

7. Appendix

Lemma 7.1. There exists δ_{1} > 0, δ_{2} > 0, and α_{0} > 0 such that for all γ_{1} ∈ (\frac{1}{2}, \frac{3}{2}) and all α ∈ (0, α_{0}), ∥Γ_{2}^{(γ_{1})}∥_{∗} ∈ (δ_{1}, δ_{2}).

Proof. For the sake of simplicity, we shall fix here Ω_{0} = ( \begin{matrix}1 \\ 0 \end{matrix} ) Ω_{f}. The state-

ment of the Lemma remains true for all Ω_{0} defined as in (3). We have

\begin{matrix}Γ_{2}^{(γ_{1})} = γ_{1}(H_{f} + P_{f}^{2})^{−1}[σ · B^{+}(H_{f} + P_{f}^{2})^{−1}σ · B^{+} \\ + 2A^{+} · P_{f}(H_{f} + P_{f}^{2})^{−1}σ · B^{+}]Ω_{0} − (H_{f} + P_{f}^{2})^{−1}A^{+} · A^{+}Ω_{0} . \end{matrix} ; (60)

In order to prove the bound below, it is sufficient to show that there exists a region J ⊂ ℝ^{3} × ℝ^{3} with strictly positive Lebesgue measure, and δ > 0 such that for all α small enough independent of δ , for all γ ∈ (\frac{1}{2}, \frac{3}{2}) and all (k, k′) ∈ J, and for given λ, μ ∈ {1, 2}, we have |Γ_{2}^{(γ_{1})}(k, λ; k′, μ)| > δ. For that sake, we shall prove that the third vector in the right hand side of (60) has a stronger singularity at the origin than the fist two vectors. For all λ, μ in \{1, 2\}, we have, for all k and k′ in the region S_{1} := \{(k, k′) | \frac{|k′|}{2} ≤ |k| ≤ 2|k′|}

|(σ · B^{+}(H_{f} + P_{f}^{2})^{−1}Ω_{0}) (k, λ; k′, μ)|

= \frac{1}{\sqrt{2}} |σ · \frac{ik′ ∧ ϵ_{μ}(k′) ζ(|k|)}{2π|k′|^{\frac{1}{2}}} \frac{1}{|k| + |k|^{2}} σ · \frac{ik ∧ ϵ_{μ}(k)}{2π|k|^{\frac{1}{2}}} + symmetric |

≤ c (\frac{|k′|^{\frac{1}{2}}}{|k|^{\frac{1}{2}}} + \frac{|k|^{\frac{1}{2}}}{|k′|^{\frac{1}{2}}} ) ≤ c ,

(61) where the symmetric expression is with respect to (k, λ) and (k′, μ), and the constants c are independent of the variables and of the parameters α and γ_{1}. On the other hand, we have

Documenta Mathematica 17 (2012) 401–415

|(A^{+} · P_{f}(H_{f} + P_{f}^{2})^{−1}σ · B^{+}Ω_{0}) (k, λ; k′, μ)|

= \frac{1}{\sqrt{2}} |\frac{ϵ_{μ}(k′) · k′}{2π|k′|^{\frac{1}{2}}} \frac{1}{|k| + |k|^{2}} σ · \frac{ik ∧ ϵ_{λ}(k)}{2π|k|^{\frac{1}{2}}} + symmetric | . (62)

Picking S_{2} = \{(k, k′) | \frac{k′_{2}k_{2}+k′_{1}k_{1}}{\sqrt{k_{1}^{2}+k_{2}^{2}}\sqrt{k′_{1}^{2}+k′_{2}^{2}}} \}, where For k = (k_{1}, k_{2}, k_{3}) and

k′ = (k′_{1}, k′_{2}, k′_{3}), we obtain, for λ = μ = 1 and for k and k′ in S_{2},

\begin{matrix}|(A^{+} · P_{f}(H_{f} + P_{f}^{2})^{−1}σ · B^{+}Ω_{0}) (k, 1; k′, 1)| \\ ≥ c |\frac{1}{|k′|^{\frac{1}{2}} |k|^{\frac{1}{2}}} \frac{k′_{2}k_{2} + k′_{1}k_{1}}{\sqrt{k_{1}^{2} + k_{2}^{2}}\sqrt{k′_{1}^{2} + k′_{2}^{2}}} | ≥ c\frac{1}{|k′|^{\frac{1}{2}} |k|^{\frac{1}{2}}} , \end{matrix} ; (63)

where again the constants are independent of k, k′, γ_{1} and α. Therefore, for any δ > 0 there exists ϵ > 0 such that for S_{3}(ϵ) = {(k, k′) | |k| ≤ ϵ, |k′| ≤ ϵ}, we have, for all α small enough and all γ_{1} ∈ (\frac{1}{2}, \frac{3}{2}), that for all (k, k′) ∈ S_{1} ∩ S_{2} ∩ S_{3}(ϵ), which is of positive Lebesgue measure, |Γ_{2}^{(γ_{1})}(k, 1; k′, 1)| > δ. This concludes the proof of the existence of the uniform lower bound δ_{1} for ∥Γ_{2}^{(γ_{1})}∥_{∗}. The proof of the upper bound for ∥Γ_{2}^{(γ_{1})}∥_{∗} is straightforward. □

Lemma 7.2. We have

∥Hf^{\frac{1}{2}} Γ_{1}∥ < ∞ , 0 < ∥Γ_{1}∥_{∗} and 0 < ∥Γ_{1}∥ .

Proof. Straightforward computations. □

We end this appendix by recalling a useful result due to Chen and Fröhlich [8], which we reproduce below in the case of total momentum p = 0, which is the case of interest for us.

Lemma 7.3. [8, Proposition 5.1] For any normalized state ψ in the eigenspace associated to the ground state energy of H(0), there exists c > 0 such that

∥a_{λ}(k)ψ∥ ≤ \frac{c\sqrt{α}}{|k|}ζ(|k|) .

Acknowledgements. J.-M. B. gratefully acknowledges financial support from Agence Nationale de la Recherche, via the project HAM-MARK ANR-09-BLAN-0098-01. S.A. V. was supported by the DFG Project SFB TR 12-3. The authors thank the referee for valuable remarks.

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Jean-Marie Barbaroux Centre de Physique Théorique UMR7332 Campus de Luminy Case 907 13288 Marseille cedex 9 France and Département de Mathématiques Université du Sud Toulon-Var BP20132 83957 La Garde Cedex France Semjon A. Vugalter Mathematisches Intitüt Universität München Theresienstrasse 39 80333 München Germany 416

Documenta Mathematica 17 (2012)