On Higher Order Estimates

in Quantum Electrodynamics

Oliver Matte

Received: January 1, 2010

Communicated by Heinz Siedentop

Abstract. We propose a new method to derive certain higher order estimates in quantum electrodynamics. Our method is particularly convenient in the application to the non-local semi-relativistic models of quantum electrodynamics as it avoids the use of iterated commutator expansions. We re-derive higher order estimates obtained earlier by Fröhlich, Griesemer, and Schlein and prove new estimates for a non-local molecular no-pair operator.

2010 Mathematics Subject Classification: 81Q10, 81V10.

Keywords and Phrases: (Semi-relativistic) Pauli-Fierz operator, nopair Hamiltonian, higher order estimates, quantum electrodynamics.

1. Introduction

The main objective of this paper is to present a new method to derive higher order estimates in quantum electrodynamics (QED) of the form

(1.1)  ∥ H_{f}^{n/2} (H + C)^{−n/2} ∥ ⩽ const < ∞ ,
(1.2)  ∥ [H_{f}^{n/2} , H] (H + C)^{−n/2} ∥ ⩽ const < ∞ ,

for all n ∈ ℕ, where C > 0 is sufficiently large. In these bounds H_{f} denotes the radiation field energy of the quantized photon field and H is the full Hamiltonian generating the time evolution of an interacting electron-photon system. For instance, estimates of this type serve as one of the main technical ingredients in the mathematical analysis of Rayleigh scattering. In this context, (1.1) has been proven by Fröhlich et al. in the case where H is the non- or semirelativistic Pauli-Fierz Hamiltonian [4]; a slightly weaker version of (1.2) has been obtained in [4] for all even values of n. Higher order estimates of the form (1.1) also turn out to be useful in the study of the existence of ground states in a no-pair model of QED [8]. In fact, they imply that every eigenvector of the Hamiltonian H or spectral subspaces of H corresponding to some bounded interval are contained in the domains of higher powers of H_{f}. This information is very helpful in order to overcome numerous technical difficulties which are caused by the non-locality of the no-pair operator. In these applications it is actually necessary to have some control on the norms in (1.1) and (1.2) when the operator H gets modified. To this end we shall give rough bounds on the right hand sides of (1.1) and (1.2) in terms of the ground state energy and integrals involving the form factor and the dispersion relation. Various types of higher order estimates have actually been employed in the mathematical analysis of quantum field theories since a very long time. Here we only mention the classical works [5, 11] on P (ϕ)_{2} models and the more recent articles [2] again on a P (ϕ)_{2} model and [1] on the Nelson model. In what follows we briefly describe the organization and the content of the present article. In Section 2 we develop the main idea behind our techniques in a general setting. By the criterion established there the proof of the higher order estimates is essentially boiled down to the verification of certain form bounds on the commutator between H and a regularized version of H_{f}^{n/2}. After that, in Section 3, we introduce some of the most important operators appearing in QED and establish some useful norm bounds on certain commutators involving them. These commutator estimates provide the main ingredients necessary to apply the general criterion of Section 2 to the QED models treated in this article. Their derivation is essentially based on the pull-through formula which is always employed either way to derive higher order estimates in quantum field theories [1, 2, 4, 5, 11]; compare Lemma 3.2 below. In Sections 4, 5, and 6 the general strategy from Section 2 is applied to the non- and semi-relativistic Pauli-Fierz operators and to the no-pair operator, respectively. The latter operators are introduced in detail in these sections. Apart from the fact that our estimate (1.2) is slightly stronger than the corresponding one of [4] the results of Sections 4 and 5 are not new and have been obtained earlier in [4]. However, in order to prove the higher order estimate (1.1) for the no-pair operator we virtually have to re-derive it for the semi-relativistic Pauli-Fierz operator by our own method anyway. Moreover, we think that the arguments employed in Sections 4 and 5 are more convenient and less involved than the procedure carried through in [4]. The main text is followed by an appendix where we show that the semi-relativistic Pauli-Fierz operator for a molecular system with static nuclei is semi-bounded below, provided that all Coulomb coupling constants are less than or equal to 2/π. Moreover, we prove the same result for a molecular no-pair operator assuming that all Coulomb coupling constants are strictly less than the critical coupling constant of the Brown-Ravenhall model [3]. The results of the appendix are based on corresponding estimates for hydrogen-like atoms obtained in [10]. (We remark that the considerably stronger stability of matter of the second kind has been proven for a molecular no-pair operator in [9] under more restrictive assumptions on the involved physical parameters.) No restrictions on the values of the fine-structure constant or on the ultra-violet cut-off are imposed in the present article.

The main new results of this paper are Theorem 2.1 and its corollaries which provide general criteria for the validity of higher order estimates and Theorem 6.1 where higher order estimates for the no-pair operator are established.

Some frequently used notation. For a, b ∈ ℝ, we write a ∧ b := min{a, b} and a ∨ b := max{a, b}. D(T) denotes the domain of some operator T acting in some Hilbert space and Q(T) its form domain, when T is semi-bounded below. C(a, b, …), C′(a, b, …), etc. denote constants that depend only on the quantities a, b, … and whose value might change from one estimate to another.

2. Higher order estimates: a general criterion

The following theorem and its succeeding corollaries present the key idea behind of our method. They essentially reduce the derivation of the higher order estimates to the verification of a certain sequence of form bounds. These form bounds can be verified easily without any further induction argument in the QED models treated in this paper.

Theorem 2.1. Let H and F_{ε}, ε > 0, be self-adjoint operators in some Hilbert space 𝒦 such that H ⩾ 1, F_{ε} ⩾ 0, and each F_{ε} is bounded. Let m ∈ ℕ∪{∞}, let 𝒟 be a form core for H, and assume that the following conditions are fulfilled:

(a) For every ε > 0, F_{ε} maps 𝒟 into Q(H) and there is some c_{ε} ∈ (0, ∞) such that

⟨ F_{ε} ψ | H F_{ε} ψ ⟩ ⩽ c_{ε} ⟨ ψ | H ψ ⟩ , ψ ∈ 𝒟 .

(b) There is some c ∈ [1, ∞) such that, for all ε > 0,

⟨ ψ | F_{ε}^{2} ψ ⟩ ⩽ c^{2} ⟨ ψ | H ψ ⟩ , ψ ∈ 𝒟 .

(c) For every n ∈ ℕ, n < m, there is some c_{n} ∈ [1, ∞) such that, for all ε > 0,

|⟨ H φ_{1} | F_{ε}^{n} φ_{2} ⟩ − ⟨ F_{ε}^{n} φ_{1} | H φ_{2} ⟩|

⩽ c_{n} {⟨ φ_{1} | H φ_{1} ⟩ + ⟨ F_{ε}^{n−1} φ_{2} | H F_{ε}^{n−1} φ_{2} ⟩} , φ_{1}, φ_{2} ∈ 𝒟.

Then it follows that, for every n ∈ ℕ, n < m + 1,

(2.1)  ∥ F_{ε}^{n} H^{−n/2} ∥ ⩽ C_{n} := 4^{n−1} c^{n} ∏_{ℓ=1}^{n−1} c_{ℓ} .

(An empty product equals 1 by definition.)

Proof. We define

T_{ε}(n) := H^{1/2} [F_{ε}^{n−1} , H^{−1}] H^{−(n−2)/2}, n ∈ {2, 3, 4, …}.

T_{ε}(n) is well-defined and bounded because of the closed graph theorem and Condition (a), which implies that F_{ε} ∈ ℒ (Q(H)), where Q(H) = D(H^{1/2}) is equipped with the form norm. We shall prove the following sequence of assertions by induction on n ∈ ℕ, n < m + 1.

A(n) :⇔  The bound (2.1) holds true and, if n > 3, we have

∀ ε > 0 : ∥T_{ε}(n)∥ ⩽ C_{n}/4c^{2} . (2.2)

For n = 1, the bound (2.1) is fulfilled with C_{1} = c on account of Condition (b). Next, assume that n ∈ ℕ, n < m, and that A(1), …, A(n) hold true. To find a bound on ∥F_{ε}^{n+1} H^{−(n+1)/2}∥ we write

F_{ε}^{n+1} H^{−(n+1)/2} = Q_{1} + Q_{2} (2.3)

with

Q_{1} := F_{ε} H^{−1} F_{ε}^{n} H^{−(n−1)/2} ,  Q_{2} := F_{ε} [ F_{ε}^{n} , H^{−1} ] H^{−(n−1)/2} .

By the induction hypothesis we have

(2.4)  ∥Q_{1}∥ ⩽ ∥F_{ε} H^{−1/2}∥ ∥H^{−1/2} F_{ε} ∥ ∥F_{ε}^{n−1} H^{−(n−1)/2}∥ ⩽ c^{2} C_{n−1} ,

where C_{0} := 1. Moreover, we observe that

(2.5)  ∥Q_{2}∥ = ∥F_{ε} H^{−1/2} T_{ε}(n + 1)∥ ⩽ c ∥T_{ε}(n + 1)∥ .

To find a bound on ∥T_{ε}(n + 1)∥ we recall that F_{ε} maps the form domain of H continuously into itself. In particular, since 𝒟 is a form core for H the form bound appearing in Condition (c) is available, for all φ_{1}, φ_{2} ∈ Q(H). Let ϕ, ψ ∈ 𝒟. Applying Condition (c), extended in this way, with

φ_{1} = δ^{1/2} H^{−1/2} ϕ ∈ Q(H) ,  φ_{2} = δ^{−1/2} H^{−(n+1)/2} ψ ∈ Q(H) ,

for some δ > 0, we obtain

|⟨ ϕ | T_{ε}(n + 1) ψ ⟩|

= |⟨ H H^{−1/2} ϕ | F_{ε}^{n} H^{−(n+1)/2} ψ ⟩ − ⟨ F_{ε}^{n} H^{−1/2} ϕ | H H^{−(n+1)/2} ψ ⟩|

⩽ c_{n} inf_{δ>0} { δ ∥ϕ∥^{2} + δ^{−1} ∥{H^{1/2} F_{ε}^{n−1} H^{−n/2}} H^{−1/2} ψ∥^{2} }

⩽ 2 c_{n} ∥{H^{1/2} F_{ε}^{n−1} H^{−n/2}}∥ ∥ϕ∥ ∥ψ∥ .

The operator {⋯ } is just the identity when n = 1. For n > 1, it can be written as

(2.6)  H^{1/2} F_{ε}^{n−1} H^{−n/2} = {H^{−1/2} F_{ε}} F_{ε}^{n−2} H^{−(n−2)/2} + T_{ε}(n) .

Applying the induction hypothesis and c, c_{ℓ} ⩾ 1, we thus get ∥T_{ε}(2)∥ ⩽ 2 c_{1}, ∥T_{ε}(3)∥ ⩽ 6 c c_{1}c_{2}, ∥T_{ε}(4)∥ ⩽ 14 c^{2}c_{1}c_{2}c_{3} < C_{4}/4c^{2}, and

c ∥T_{ε}(n + 1)∥ = c sup { |⟨ ϕ | T_{ε}(n + 1) ψ ⟩| : ϕ, ψ ∈ 𝒟 , ∥ϕ∥ = ∥ψ∥ = 1 }

⩽ 2 c_{n} (c^{2} C_{n−2} + C_{n}/4c) < c_{n} C_{n} = C_{n+1}/4c , n > 3 ,

since c^{2} C_{n−2} ⩽ C_{n}/16, for n > 3. Taking (2.3)–(2.5) into account we arrive at ∥F_{ε}^{2} H^{−1}∥ ⩽ c^{2} + 2c c_{1} < C_{2}, ∥F_{ε}^{3} H^{−3/2}∥ ⩽ c^{3} + 6c^{2}c_{1}c_{2} < C_{3}, and

∥F_{ε}^{n+1} H^{−(n+1)/2}∥ < c^{2} C_{n−2} + C_{n+1}/4c < C_{n+1} , n > 3 ,

which concludes the induction step. □

Corollary 2.2. Assume that H and F_{ε}, ε > 0, are self-adjoint operators in some Hilbert space 𝒦 that fulfill the assumptions of Theorem 2.1 with (c) replaced by the stronger condition

(c’) For every n ∈ ℕ, n < m, there is some c_{n} ∈ [1, ∞) such that, for all ε > 0,

|⟨ H φ_{1} | F_{ε}^{n} φ_{2} ⟩ − ⟨ F_{ε}^{n} φ_{1} | H φ_{2} ⟩|
⩽ c_{n} {∥φ_{1}∥^{2} + ⟨ F_{ε}^{n−1} φ_{2} | H F_{ε}^{n−1} φ_{2} ⟩} , φ_{1}, φ_{2} ∈ 𝒟.

Then, in addition to (2.1), it follows that, for n ∈ ℕ, n < m, [F_{ε}^{n} , H] H^{−n/2} defines a bounded sesquilinear form with domain Q(H) × Q(H) and

(2.7)  ∥ [F_{ε}^{n} , H] H^{−n/2} ∥ ⩽ C′_{n} := 4^{n} c^{n−1} ∏_{ℓ=1}^{n} c_{ℓ} .

Proof. Again, the form bound in (c’) is available, for all φ_{1}, φ_{2} ∈ Q(H), whence

|⟨ H ϕ | F_{ε}^{n} H^{−n/2} ψ ⟩ − ⟨ F_{ε}^{n} ϕ | H H^{−n/2} ψ ⟩|

⩽ c_{n} inf_{δ>0} {δ ∥ϕ∥^{2} + δ^{−1} ∥H^{1/2} F_{ε}^{n−1} H^{−n/2} ψ∥^{2}} ⩽ 2 c_{n} ∥H^{1/2} F_{ε}^{n−1} H^{−n/2}∥ ,

for all normalized ϕ, ψ ∈ Q(H). The assertion now follows from (2.1), (2.6), and the bounds on ∥T_{ε}(n)∥ given in the proof of Theorem 2.1. □

Corollary 2.3. Let H ⩾ 1 and A ⩾ 0 be two self-adjoint operators in some Hilbert space 𝒦 . Let κ > 0, define

f_{ε}(t) := t/(1 + ε t) , t ⩾ 0 , F_{ε} := f_{ε}^{κ}(A) ,

for all ε > 0, and assume that H and F_{ε}, ε > 0, fulfill the hypotheses of Theorem 2.1, for some m ∈ ℕ ∪ {∞}. Then Ran(H^{−n/2}) ⊂ D(A^{κ n}), for every n ∈ ℕ, n < m + 1, and

∥ A^{κ n} H^{−n/2} ∥ ⩽ 4^{n−1} c^{n} ∏_{ℓ=1}^{n−1} c_{ℓ} .

If H and F_{ε}, ε > 0, fulfill the hypotheses of Corollary 2.2, then, for every n ∈ ℕ, n < m, it additionally follows that A^{κ n} H^{−n/2} maps D(H) into itself so that [A^{κ n} , H] H^{−n/2} is well-defined on D(H), and

∥ [A^{κ n} , H] H^{−n/2} ∥ ⩽ 4^{n} c^{n−1} ∏_{ℓ=1}^{n} c_{ℓ} .

Proof. Let U : 𝒦 → L^{2}(Ω, μ) be a unitary transformation such that a = U A U^{∗} is a maximal operator of multiplication with some non-negative measurable function – again called a – on some measure space (Ω, 𝔄, μ). We pick some ψ ∈ 𝒦 , set ϕ_{n} := U H^{−n/2} ψ, and apply the monotone convergence theorem to conclude that

∫_{Ω} a(ω)^{2κ n} |ϕ_{n}(ω)|^{2} dμ(ω) = lim_{ε↘0} ∫_{Ω} f_{ε}^{κ}(a(ω))^{2n} |ϕ_{n}(ω)|^{2} dμ(ω)

= lim_{ε↘0} ∥F_{ε}^{n} H^{−n/2} ψ∥^{2} ⩽ C_{n} ∥ψ∥^{2},

for every n ∈ ℕ, n < m+1, which implies the first assertion. Now, assume that H and F_{ε}, ε > 0, fulfill Condition (c’) of Corollary 2.2. Applying the dominated convergence theorem in the spectral representation introduced above we see that F_{ε}^{n} ψ → A^{κ n} ψ, for every ψ ∈ D(A^{κ n}). Hence, (2.7) and Ran(H^{−n/2}) ⊂ D(A^{κ n}) imply, for n < m and ϕ, ψ ∈ D(H),

|⟨ ϕ | A^{κ n}H^{−n/2} H ψ ⟩ − ⟨ H ϕ | A^{κ n} H^{−n/2} ψ ⟩|

= lim_{ε↘0} |⟨ F_{ε}^{n} ϕ | H H^{−n/2} ψ ⟩ − ⟨ H ϕ | F_{ε}^{n} H^{−n/2} ψ ⟩|

⩽ lim sup_{ε↘0} ∥ [F_{ε}^{n} , H] H^{−n/2} ∥ ∥ϕ∥ ∥ψ∥ ⩽ C′_{n} ∥ϕ∥ ∥ψ∥ .

Thus, |⟨ H ϕ | A^{κ n} H^{−n/2} ψ ⟩| ⩽ ∥ϕ∥ ∥A^{κ n} H^{−n/2}∥ ∥H ψ∥ + C′_{n}∥ϕ∥ ∥ψ∥, for all ϕ, ψ ∈ D(H). In particular, A^{κ n} H^{−n/2} ψ ∈ D(H^{∗}) = D(H), for all ψ ∈ D(H), and the second asserted bound holds true. □

3. Commutator estimates

In this section we derive operator norm bounds on commutators involving the quantized vector potential, A, the radiation field energy, H_{f}, and the Dirac operator, D_{A}. The underlying Hilbert space is

ℋ := L^{2}(ℝ_{x}^{3} × ℤ_{4}) ⊗ ℱ_{b} = ∫_{ℝ^{3}}^{⊕} ℂ^{4} ⊗ ℱ_{b} d^{3}x ,

where the bosonic Fock space, ℱ_{b}, is modeled over the one-photon Hilbert space

ℱ_{b}^{(1)} := L^{2}(A × ℤ_{2}, dk) , ∫ dk := ∑_{λ∈ℤ_{2}} ∫_{A} d^{3}k .

With regards to the applications in [8] we define A := {k ∈ ℝ^{3} : |k| ⩾ m}, for some m ⩾ 0. We thus have

ℱ_{b} = ⨁_{n=0}^{∞} ℱ_{b}^{(n)}, ℱ_{b}^{(0)} := ℂ , ℱ_{b}^{(n)} := S_{n} L^{2}((A × ℤ_{2})^{n}), n ∈ ℕ,

where S_{n} = S_{n}^{2} = S_{n}^{∗} is given by

(S_{n} ψ^{(n)})(k_{1}, …, k_{n}) := \frac{1}{n!} ∑_{π∈𝔖_{n}} ψ^{(n)}(k_{π(1)}, …, k_{π(n)}),

for every ψ^{(n)} ∈ L^{2}((A × ℤ_{2})^{n}), 𝔖_{n} denoting the group of permutations of {1, …, n}. The vector potential is determined by a certain vector-valued function, G, called the form factor.

Hypothesis 3.1. The dispersion relation, ω : A → [0, ∞), is a measurable function such that 0 < ω(k) := ω(k) ⩽ |k|, for k = (k, λ) ∈ A × ℤ_{2} with k ≠ 0. For every k ∈ (A\{0})×ℤ_{2} and j ∈ {1, 2, 3}, G^{(j)}(k) is a bounded continuously differentiable function, ℝ_{x}^{3} ∋ x ↦ G_{x}^{(j)}(k), such that the map (x, k) ↦ G_{x}^{(j)}(k)

is measurable and G_{x}^{(j)}(−k, λ) = \overline{G_{x}^{(j)}(k, λ)}, for almost every k and all x ∈ ℝ^{3} and λ ∈ ℤ_{2}. Finally, there exist d_{−1}, d_{0}, d_{1}, … ∈ (0, ∞) such that

(3.1)  2 ∫ ω(k)^{ℓ} ∥G(k)∥_{∞}^{2} dk ⩽ d_{ℓ}^{2} ,  ℓ ∈ {−1, 0, 1, 2, …} ,
(3.2)  2 ∫ ω(k)^{−1} ∥∇_{x} ∧ G(k)∥_{∞}^{2} dk ⩽ d_{1}^{2} ,

where G = (G^{(1)}, G^{(2)}, G^{(3)}) and ∥G(k)∥_{∞} := sup_{x} |G_{x}(k)|, etc.

Example. In the physical applications the form factor is often given as

(3.3)  G_{x}^{e,Λ}(k) := −e \frac{𝟙_{\{|k|⩽Λ\}}}{2π\sqrt{|k|}} e^{−ik·x} ε(k),

for (x, k) ∈ ℝ^{3} × (ℝ^{3} × ℤ_{2}) with k ≠ 0. Here the physical units are chosen such that energies are measured in units of the rest energy of the electron. Length are measured in units of one Compton wave length divided by 2π. The parameter Λ > 0 is an ultraviolet cut-off and the square of the elementary charge, e > 0, equals Sommerfeld’s fine-structure constant in these units; we have e^{2} ≈ 1/137 in nature. The polarization vectors, ε(k, λ), λ ∈ ℤ_{2}, are homogeneous of degree zero in k such that {k̊, ε(k̊, 0), ε(k̊, 1)} is an orthonormal basis of ℝ^{3}, for every k̊ ∈ S^{2}. This corresponds to the Coulomb gauge for ∇_{x} · G^{e,Λ} = 0. We remark that the vector fields S^{2} ∋ k̊ ↦ ε(k̊, λ) are necessarily discontinuous. ⋄

It is useful to work with more general form factors fulfilling Hypothesis 3.1 since in the study of the existence of ground states in QED one usually encounters truncated and discretized versions of the physical choice G^{e,Λ}. For the applications in [8] it is necessary to know that the higher order estimates established here hold true uniformly in the involved parameters and Hypothesis 3.1 is convenient way to handle this. We recall the definition of the creation and the annihilation operators of a photon state f ∈ ℱ_{b}^{(1)},

(a^{†}(f) ψ)^{(n)}(k_{1}, …, k_{n}) = n^{−1/2} ∑_{j=1}^{n} f(k_{j}) ψ^{(n−1)}(…, k_{j−1}, k_{j+1}, …), n ∈ ℕ,

(a(f) ψ)^{(n)}(k_{1}, …, k_{n}) = (n + 1)^{1/2} ∫ \overline{f}(k) ψ^{(n+1)}(k, k_{1}, …, k_{n}) dk, n ∈ ℕ_{0},

and (a^{†}(f) ψ)^{(0)} = 0, a(f) (ψ^{(0)}, 0, 0, …) = 0, for all ψ = (ψ^{(n)})_{n=0}^{∞} ∈ ℱ_{b} such that the right hand sides again define elements of ℱ_{b}. a^{†}(f) and a(f) are formal adjoints of each other on the dense domain

𝒞_{0} := ℂ ⊕ ⨁_{n=1}^{∞} S_{n} L_{comp}^{∞}((A × ℤ_{2})^{n}) . (Algebraic direct sum.)

For a three-vector of functions f = (f^{(1)}, f^{(2)}, f^{(3)}) ∈ (ℱ_{b}^{(1)})^{3}, we write a^{♯}(f) := (a^{♯}(f^{(1)}), a^{♯}(f^{(2)}), a^{♯}(f^{(3)})), where a^{♯} is a^{†} or a. Then the quantized vector potential is the triplet of operators given by

A ≡ A(G) := a^{†}(G) + a(G) , a^{♯}(G) := ∫_{ℝ^{3}}^{⊕} 𝟙_{ℂ^{4}} ⊗ a^{♯}(G_{x}) d^{3}x .

The radiation field energy is the direct sum H_{f} = ⨁_{n=0}^{∞} dΓ^{(n)}(ω) : D(H_{f}) ⊂ ℱ_{b} → ℱ_{b}, where dΓ^{(0)}(ω) := 0, and dΓ^{(n)}(ω) denotes the maximal multiplication operator in ℱ_{b}^{(n)} associated with the symmetric function (k_{1}, …, k_{n}) ↦ ω(k_{1}) + ⋯ + ω(k_{n}). By the permutation symmetry and Fubini’s theorem we thus have

(3.4) ⟨ H_{f}^{1/2} ϕ | H_{f}^{1/2} ψ ⟩ = ∫ ω(k) ⟨ a(k) ϕ | a(k) ψ ⟩ dk ,  ϕ, ψ ∈ D(H_{f}^{1/2}) ,

where we use the notation

(a(k) ψ)^{(n)}(k_{1}, …, k_{n}) = (n + 1)^{1/2} ψ^{(n+1)}(k, k_{1}, …, k_{n}) , n ∈ ℕ_{0} ,

almost everywhere, and a(k) (ψ^{(0)}, 0, 0, …) = 0. For a measurable function f : ℝ → ℝ and ψ ∈ D(f(H_{f})), the following identity in ℱ_{b}^{(n)},

(a(k) f(H_{f}) ψ)^{(n)} = f(ω(k) + dΓ^{(n)}(ω)) (a(k) ψ)^{(n)} , n ∈ ℕ_{0} ,

valid for almost every k, is called the pull-through formula. Finally, we let α_{1}, α_{2}, α_{3}, and β := α_{0} denote hermitian four times four matrices that fulfill the Clifford algebra relations

(3.5)  α_{i} α_{j} + α_{j} α_{i} = 2 δ_{ij} 𝟙 ,  i, j ∈ {0, 1, 2, 3} .

They act on the second tensor factor in L^{2}(ℝ_{x}^{3} × ℤ_{4}) = L^{2}(ℝ_{x}^{3}) ⊗ ℂ^{4}. As a consequence of (3.5) and the C^{∗}-equality we have

(3.6) ∥α · v∥_{ℒ (ℂ^{4})} = |v| , v ∈ ℝ^{3} ,  ∥α · z∥_{ℒ (ℂ^{4})} ⩽ \sqrt{2} |z| , z ∈ ℂ^{3} ,

where α · z := α_{1} z^{(1)} + α_{2} z^{(2)} + α_{3} z^{(3)}, for z = (z^{(1)}, z^{(2)}, z^{(3)}) ∈ ℂ^{3}. A standard exercise using the inequality in (3.6), the Cauchy-Schwarz inequality, and the canonical commutation relations,

[a^{♯}(f) , a^{♯}(g)] = 0 ,  [a(f) , a^{†}(g)] = ⟨ f | g ⟩ 𝟙 ,  f, g ∈ ℱ_{b}^{(1)} ,

reveals that every ψ ∈ D(H_{f}^{1/2}) belongs to the domain of α · a^{♯}(G) and (3.7) ∥α · a(G) ψ∥ ⩽ d_{−1} ∥H_{f}^{1/2} ψ∥ , ∥α · a^{†}(G) ψ∥^{2} ⩽ d_{−1}^{2} ∥H_{f}^{1/2} ψ∥^{2} + d_{0}^{2} ∥ψ∥^{2}.

(Here and in the following we identify H_{f} ≡ 𝟙⊗H_{f}, etc.) These relative bounds imply that α · A is symmetric on the domain D(H_{f}^{1/2}).

The operators whose norms are estimated in (3.9) and the following lemmata are always well-defined a priori on the following dense subspace of ℋ ,

𝒟 := C_{0}^{∞}(ℝ^{3} × ℤ_{4}) ⊗ 𝒞_{0} . (Algebraic tensor product.)

Given some E ⩾ 1 we set

(3.8)  Ȟ_{f} := H_{f} + E

in the sequel. We already know from [10] that, for every ν ⩾ 0, there is some constant, C_{ν} ∈ (0, ∞), such that

(3.9)  ∥ [α · A , Ȟ_{f}^{−ν}] Ȟ_{f}^{ν} ∥ ⩽ C_{ν}/E^{1/2} ,  E ⩾ 1 .

In our first lemma we derive a generalization of (3.9). Its proof resembles the one of (3.9) given in [10]. Since we shall encounter many similar but slightly different commutators in the applications it makes sense to introduce the numerous parameters that obscure its statement (but simplify its proof).

Lemma 3.2. Assume that ω and G fulfill Hypothesis 3.1. Let ε ⩾ 0, E ⩾ 1, κ, ν ∈ ℝ, γ, δ, σ, τ ⩾ 0, such that γ + δ + σ + τ ⩽ 1/2, and define

(3.10)  f_{ε}(t) := \frac{t + E}{1 + εt + εE} ,  t ∈ [0, ∞) .

Then the operator Ȟ_{f}^{ν+γ} f_{ε}^{σ}(H_{f}) [α · A , f_{ε}^{κ}(H_{f})] Ȟ_{f}^{−ν+δ}f_{ε}^{−κ+τ} (H_{f}), defined a priori on 𝒟, extends to a bounded operator on ℋ and

∥ Ȟ_{f}^{ν+γ} f_{ε}^{σ}(H_{f}) [α · A , f_{ε}^{κ}(H_{f})] Ȟ_{f}^{−ν+δ} f_{ε}^{−κ+τ} (H_{f}) ∥

⩽ |κ| 2^{(ρ+1)/2} (d_{1} + d_{ρ}) E^{γ+δ+σ+τ−1/2} , (3.11)

where ρ is the smallest integer greater or equal to 3 + 2|κ| + 2|ν|.

Proof. We notice that all operators Ȟ_{f}^{s} and f_{ε}^{s}(H_{f}) leave the dense subspace 𝒟 invariant and that α · a^{♯}(G) maps 𝒟 into D(Ȟ_{f}^{s}), for every s ∈ ℝ. Now, let φ, ψ ∈ 𝒟. Then

⟨ φ | Ȟ_{f}^{ν+γ} f_{ε}^{σ}(H_{f}) [α · A , f_{ε}^{κ}(H_{f})] Ȟ_{f}^{−ν+δ} f_{ε}^{−κ+τ} (H_{f}) ψ ⟩

= ⟨ φ | Ȟ_{f}^{ν+γ} f_{ε}^{σ}(H_{f}) [α · a(G) , f_{ε}^{κ}(H_{f})] Ȟ_{f}^{−ν+δ} f_{ε}^{−κ+τ} (H_{f}) ψ ⟩ (3.12)

− ⟨ f_{ε}^{−κ+τ} (H_{f}) Ȟ_{f}^{−ν+δ} [α · a(G) , f_{ε}^{κ}(H_{f})] f_{ε}^{σ}(H_{f}) Ȟ_{f}^{ν+γ} φ | ψ ⟩ . (3.13)

For almost every k, the pull-through formula yields the following representation,

Ȟ_{f}^{ν+γ} f_{ε}^{σ}(H_{f}) [a(k) , f_{ε}^{κ}(H_{f})] Ȟ_{f}^{−ν+δ} f_{ε}^{−κ+τ} (H_{f}) ψ = F (k; H_{f}) a(k) Ȟ_{f}^{−1/2} ψ , 216 Oliver Matte where

F (k; t) := (t + E)^{ν+γ} f_{ε}^{σ}(t) (f_{ε}^{κ}(t + ω(k)) − f_{ε}^{κ}(t))
· (t + E + ω(k))^{−ν+δ+1/2} f_{ε}^{−κ+τ} (t + ω(k))

= (\frac{t + E}{t + E + ω(k)})^{ν}(t + E)^{γ} (t + E + ω(k))^{δ+1/2}

· ∫_{0}^{1} \frac{d}{ds}f_{ε}^{κ}(t + s ω(k)) ds \frac{f_{ε}^{σ}(t) f_{ε}^{τ} (t + ω(k))}{f_{ε}^{κ}(t + ω(k))} ,

for t ⩾ 0. We compute

(3.14)  \frac{d}{ds}f_{ε}^{κ}(t + s ω(k)) = \frac{κ ω(k) f_{ε}^{κ}(t + s ω(k))}{(t + s ω(k) + E)(1 + ε t + ε s ω(k) + ε E)} .

Using that f_{ε} is increasing in t ⩾ 0 and that

(t + ω(k) + E)/(t + s ω(k) + E) ⩽ 1 + ω(k) , s ∈ [0, 1] ,

thus

f_{ε}^{κ}(t + s ω(k))/f_{ε}^{κ}(t + ω(k)) ⩽ (1 + ω(k))^{−(0∧κ)} , s ∈ [0, 1] ,

it is elementary to verify that

|F_{ε}(k; t)| ⩽ |κ| ω(k) (1 + ω(k))^{δ+τ−(0∧κ)−(0∧ν)+1/2} E^{γ+δ+σ+τ−1/2} ,

for all t ⩾ 0 and k. We deduce that the term in (3.12) can be estimated as |⟨ φ | Ȟ_{f}^{ν+γ} f_{ε}^{σ}(H_{f}) [α · a(G) , f_{ε}^{κ}(H_{f})] Ȟ_{f}^{−ν+δ} f_{ε}^{−κ+τ} (H_{f}) ψ ⟩| ⩽ ∫ ∥φ∥ ∥α · G(k) Ȟ_{f}^{ν+γ} f_{ε}^{σ}(H_{f}) [a(k) , f_{ε}^{κ}(H_{f})] Ȟ_{f}^{−ν+δ} f_{ε}^{−κ+τ} (H_{f}) ψ∥ dk ⩽ \sqrt{2} ∫ ∥φ∥ ∥G(k)∥_{∞} ∥F_{ε}(k; H_{f})∥ ∥a(k) Ȟ_{f}^{−1/2} ψ∥ dk ⩽ |κ| \sqrt{2}( ∫ ω(k) (1 + ω(k))^{2(δ+τ)−(0∧2κ)−(0∧2ν)+1} ∥G(k)∥_{∞}^{2} dk)^{1/2}

· ( ∫ ω(k) ∥ a(k) Ȟ_{f}^{−1/2} ψ ∥^{2} dk)^{1/2} ∥φ∥ E^{γ+δ+σ+τ−1/2}

⩽ |κ| 2^{(ρ−1)/2} (d_{1} + d_{ρ}) ∥φ∥ ∥ H_{f}^{1/2} Ȟ_{f}^{−1/2} ψ ∥ E^{γ+δ+σ+τ−1/2} . (3.15)

In the last step we used δ + τ ⩽ 1/2 and applied (3.4). (3.15) immediately gives a bound on the term in (3.13), too. For we have

f_{ε}^{−κ+τ} (H_{f}) Ȟ_{f}^{−ν+δ} [α · a(G) , f_{ε}^{κ}(H_{f})] f_{ε}^{σ}(H_{f}) Ȟ_{f}^{ν+γ} φ
= Ȟ_{f}^{−ν+δ} f_{ε}^{τ} (H_{f}) [f_{ε}^{−κ}(H_{f}) , α · a(G)] Ȟ_{f}^{ν+γ} f_{ε}^{κ+σ}(H_{f}) φ ,

which after the replacements (ν, κ, γ, δ, σ, τ) ↦ (−ν, −κ, δ, γ, τ, σ) and φ ↦ −ψ is precisely the term we just have treated. □

Documenta Mathematica 15 (2010) 207–234

Lemma 3.2 provides all the information needed to apply Corollary 2.3 to nonrelativistic QED. For the application of Corollary 2.3 to the non-local semirelativistic models of QED it is necessary to study commutators that involve resolvents and sign functions of the Dirac operator,

D_{A} := α · (−i∇_{x} + A) + β .

An application of Nelson’s commutator theorem with test operator −Δ+H_{f} +1 shows that D_{A} is essentially self-adjoint on 𝒟. The spectrum of its unique closed extension, again denoted by the same symbol, is contained in the union of two half-lines, σ[D_{A}] ⊂ (−∞, −1] ∪ [1, ∞). In particular, it makes sense to define

R_{A}(iy) := (D_{A} − iy)^{−1} , y ∈ ℝ ,

and the spectral calculus yields

∥R_{A}(iy)∥ ⩽ (1 + y^{2})^{−1/2}, ∫_{ℝ} ∥ |D_{A}|^{1/2} R_{A}(iy) ψ ∥^{2} \frac{dy}{π} = ∥ψ∥^{2}, ψ ∈ ℋ .

The next lemma is a straightforward extension of [10, Corollary 3.1] where it is also shown that R_{A}(iy) maps D(H_{f}^{ν}) into itself, for every ν > 0.

Lemma 3.3. Assume that ω and G fulfill Hypothesis 3.1. Then, for all κ, ν ∈ ℝ, we find k_{i} ≡ k_{i}(κ, ν, d_{1}, d_{ρ}) ∈ [1, ∞), i = 1, 2, such that, for all y ∈ ℝ, ε ⩾ 0, and E ⩾ k_{1}, there exist Υ_{κ,ν}(iy), Υ̃_{κ,ν}(iy) ∈ ℒ (ℋ ) satisfying

(3.16)  R_{A}(iy) Ȟ_{f}^{−ν} f_{ε}^{−κ}(H_{f}) = Ȟ_{f}^{−ν} f_{ε}^{−κ}(H_{f}) R_{A}(iy) Υ_{κ,ν}(iy)

= Ȟ_{f}^{−ν} f_{ε}^{−κ}(H_{f}) Υ̃_{κ,ν}(iy) R_{A}(iy) , (3.17)

on D(Ȟ_{f}^{−ν}), and ∥Υ_{κ,ν}(iy)∥, ∥Υ̃_{κ,ν}(iy)∥ ⩽ k_{2}, where ρ is defined in Lemma 3.2.

Proof. Without loss of generality we may assume that ε > 0 for otherwise we could simply replace ν by ν +κ and f_{0}^{κ} by f_{0}^{0} = 1. First, we assume in addition that ν ⩾ 0. We observe that

T_{0} := [ Ȟ_{f}^{−ν} f_{ε}^{−κ}(H_{f}) , α · A ] Ȟ_{f}^{ν} f_{ε}^{κ}(H_{f}) = T_{1} + T_{2}

on 𝒟, where

T_{1} := [Ȟ_{f}^{−ν} , α · A] Ȟ_{f}^{ν} ,  T_{2} := Ȟ_{f}^{−ν} [f_{ε}^{−κ}(H_{f}) , α · A] f_{ε}^{κ}(H_{f}) Ȟ_{f}^{ν}.

Due to (3.9) (or (3.11) with ε = 0) the operator T_{1} extends to a bounded operator on ℋ and ∥T_{1}∥ ⩽ C_{ν}/E^{1/2}. According to (3.11) we further have ∥T_{2}∥ ⩽ C_{κ,ν}(d_{1} + d_{ρ})/E^{1/2}. We pick some ϕ ∈ 𝒟 and compute

[ R_{A}(iy) , Ȟ_{f}^{−ν} f_{ε}^{−κ}(H_{f}) ] (D_{A} − iy) ϕ = R_{A}(iy) [ Ȟ_{f}^{−ν} f_{ε}^{−κ}(H_{f}) , D_{A} ] ϕ
= R_{A}(iy) T_{0} Ȟ_{f}^{−ν} f_{ε}^{−κ}(H_{f}) ϕ

= R_{A}(iy) \overline{T}_{0} Ȟ_{f}^{−ν} f_{ε}^{−κ}(H_{f}) R_{A}(iy) (D_{A} − iy) ϕ . (3.18)

Since (D_{A} −iy) 𝒟 is dense in ℋ and since Ȟ_{f}^{−ν} and f_{ε}^{κ}(H_{f}) are bounded (here we use that ν ⩾ 0 and ε > 0), this identity implies

R_{A}(iy) Ȟ_{f}^{−ν} f_{ε}^{−κ}(H_{f}) = (𝟙 + R_{A}(iy) \overline{T}_{0}) Ȟ_{f}^{−ν} f_{ε}^{−κ}(H_{f}) R_{A}(iy) .

Taking the adjoint of the previous identity and replacing y by −y we obtain

Ȟ_{f}^{−ν} f_{ε}^{−κ}(H_{f}) R_{A}(iy) = R_{A}(iy) Ȟ_{f}^{−ν} f_{ε}^{−κ}(H_{f}) (𝟙 + T_{0}^{∗} R_{A}(iy)) .

In view of the norm bounds on T_{1} and T_{2} we see that (3.16) and (3.17) are valid with Υ_{κ,ν}(iy) := ∑_{ℓ=0}^{∞}{−T_{0}^{∗} R_{A}(iy)}^{ℓ} and Υ̃_{κ,ν}(iy) := ∑_{ℓ=0}^{∞}{−R_{A}(iy) T_{0}^{∗}}^{ℓ}, provided that E is sufficiently large, depending only on κ, ν, d_{1}, and d_{ρ}, such that the Neumann series converge. Now, let ν < 0. Then we write T_{0} on the domain 𝒟 as

T_{0} = Ȟ_{f}^{−ν} f_{ε}^{−κ}(H_{f}) [α · A , Ȟ_{f}^{ν} f_{ε}^{κ}(H_{f})] ,

and deduce that

R_{A}(iy) Ȟ_{f}^{ν} f_{ε}^{κ}(H_{f}) (𝟙 + \overline{T}_{0} R_{A}(iy)) = Ȟ_{f}^{ν} f_{ε}^{κ}(H_{f}) R_{A}(iy)

by a computation analogous to (3.18). Taking the adjoint of this identity with y replaced by −y we get

(𝟙 + R_{A}(iy) T_{0}^{∗}) Ȟ_{f}^{ν} f_{ε}^{κ} R_{A}(iy) = R_{A}(iy) Ȟ_{f}^{ν} f_{ε}^{κ}(H_{f}) .

Next, we invert 𝟙 + R_{A}(iy) T_{0}^{∗} by means of the same Neumann series as above. As a result we obtain

Ȟ_{f}^{ν}f_{ε}^{κ}(H_{f}) R_{A}(iy) = R_{A}(iy) Υ_{κ,ν}(iy) Ȟ_{f}^{ν}f_{ε}^{κ}(H_{f}) = Υ̃_{κ,ν}(iy) R_{A}(iy) Ȟ_{f}^{ν}f_{ε}^{κ}(H_{f}),

where the definition of Υ_{κ,ν} and Υ̃_{κ,ν} has been extended to negative ν. It follows that R_{A}(iy) Υ_{κ,ν}(iy) = Υ̃_{κ,ν}(iy) R_{A}(iy) maps D(Ȟ_{f}^{−ν}) = D(Ȟ_{f}^{−ν} f_{ε}^{−κ}(H_{f})) = Ran(Ȟ_{f}^{ν} f_{ε}^{κ}(H_{f})) into itself and that (3.16) and (3.17) still hold true when ν is negative. □

In order to control the Coulomb singularity 1/|x| in terms of |D_{A}| and H_{f} in the proof of the following corollary, we shall employ the bound [10, Theorem 2.3]

(3.19)  \frac{2}{π} \frac{1}{|x|} ⩽ |D_{A}| + H_{f} + k d_{1}^{2} ,

which holds true in sense of quadratic forms on Q(|D_{A}|) ∩ Q(H_{f}). Here k ∈ (0, ∞) is some universal constant. We abbreviate the sign function of the Dirac operator, which can be represented as a strongly convergent principal value [6, Lemma VI.5.6], by

(3.20)  S_{A} ψ := D_{A} |D_{A}|^{−1} ψ = lim_{τ→∞} ∫_{−τ}^{τ} R_{A}(iy) ψ \frac{dy}{π} .

We recall from [10, Lemma 3.3] that S_{A} maps D(H_{f}^{ν}) into itself, for every ν > 0. This can also be read off from the proof of the next corollary.

Corollary 3.4. Assume that ω and G fulfill Hypothesis 3.1. Let κ, ν ∈ ℝ. Then we find some C ≡ C(κ, ν, d_{1}, d_{ρ}) ∈ (0, ∞) such that, for all γ, δ, σ, τ ⩾ 0 with γ + δ + σ + τ ⩽ 1/2 and all ε ⩾ 0, E ⩾ k_{1},

(3.21)  ∥ Ȟ_{f}^{ν} f_{ε}^{κ}(H_{f}) S_{A} Ȟ_{f}^{−ν} f_{ε}^{−κ}(H_{f}) ∥ ⩽ C,
(3.22)  ∥ |D_{A}|^{1/2} Ȟ_{f}^{ν+γ} f_{ε}^{σ}(H_{f}) [S_{A} , f_{ε}^{κ}(H_{f})] Ȟ_{f}^{−ν+δ} f_{ε}^{−κ+τ} (H_{f}) ∥ ⩽ C,
(3.23)  ∥ |x|^{−1/2} Ȟ_{f}^{ν} f_{ε}^{σ}(H_{f}) [S_{A} , f_{ε}^{κ}(H_{f})] Ȟ_{f}^{−ν−σ−τ} f_{ε}^{−κ+τ} (H_{f}) ∥ ⩽ C.

(k_{1} is the constant appearing in Lemma 3.3, Ȟ_{f} is given by (3.8), f_{ε} by (3.10).)

Proof. First, we prove (3.22). Using (3.20), writing

[R_{A}(iy) , f_{ε}^{κ}(H_{f})] = R_{A}(iy) [f_{ε}^{κ}(H_{f}) , α · A] R_{A}(iy)

on 𝒟 and employing (3.16), (3.17), and (3.11) we obtain the following estimate, for all φ, ψ ∈ 𝒟, and E ⩾ k_{1},

|⟨ |D_{A}|^{1/2} φ | Ȟ_{f}^{ν+γ} f_{ε}^{σ}(H_{f}) [S_{A} , f_{ε}^{κ}(H_{f})] Ȟ_{f}^{−ν+δ} f_{ε}(H_{f})^{−κ+τ} ψ ⟩|
⩽ ∫_{ℝ} |⟨ Ȟ_{f}^{ν+γ} |D_{A}|^{1/2} φ | f_{ε}^{σ}(H_{f}) [f_{ε}^{κ}(H_{f}) , R_{A}(iy)] ×

× Ȟ_{f}^{−ν+δ} f_{ε}^{−κ+τ} (H_{f}) ψ ⟩| \frac{dy}{π}

= ∫_{ℝ} |⟨ φ | |D_{A}|^{1/2} R_{A}(iy) Υ_{σ,ν+γ}(iy) Ȟ_{f}^{ν+γ} f_{ε}^{σ}(H_{f}) [f_{ε}^{κ}(H_{f}) , α · A] ×

× Ȟ_{f}^{−ν+δ} f_{ε}^{−κ+τ} (H_{f}) Υ̃_{κ−τ,ν−δ}(iy) R_{A}(iy) ψ ⟩| \frac{dy}{π}

⩽ C_{κ,ν} (d_{1} + d_{ρ}) E^{γ+δ+σ+τ−1/2} sup_{y∈ℝ}{∥Υ_{σ,ν+γ}(iy)∥ ∥Υ̃_{κ−τ,ν−δ}(iy)∥}

· ( ∫_{ℝ} ∥ |D_{A}|^{1/2} R_{A}(iy) φ ∥^{2} \frac{dy}{π} )^{1/2}( ∫_{ℝ} ∥ R_{A}(iy) ψ ∥^{2} \frac{dy}{π} )^{1/2}

⩽ C_{κ,ν,d_{1},d_{ρ}} E^{γ+δ+σ+τ−1/2} ∥φ∥ ∥ψ∥ .

This estimate shows that the vector in the right entry of the scalar product in the first line belongs to D((|D_{A}|^{1/2})^{∗}) = D(|D_{A}|^{1/2}) and that (3.22) holds true. Next, we observe that (3.23) follows from (3.22) and (3.19). Finally, (3.21) follows from ∥X∥ ⩽ const(ν, κ, d_{1}, d_{ρ}), where X := Ȟ_{f}^{ν} f_{ε}^{κ}(H_{f}) [S_{A} , Ȟ_{f}^{−ν} f_{ε}^{−κ}(H_{f})]. Such a bound on ∥X∥ is, however, an immediate consequence of (3.22) (where we can choose ε = 0) because

X = [Ȟ_{f}^{ν} , S_{A}] Ȟ_{f}^{−ν} + Ȟ_{f}^{ν} [f_{ε}^{κ}(H_{f}) , S_{A}] f_{ε}^{−κ}(H_{f}) Ȟ_{f}^{−ν}

on the domain 𝒟. □

4. Non-relativistic QED

The Pauli-Fierz operator for a molecular system with static nuclei and N ∈ ℕ electrons interacting with the quantized radiation field is acting in the Hilbert space

(4.1)  ℋ_{N} := A_{N}L^{2}((ℝ^{3} × ℤ_{4})^{N}) ⊗ ℱ_{b} ,

where A_{N} = A_{N}^{2} = A_{N}^{∗} denotes anti-symmetrization,

(A_{N} Ψ)(X) := \frac{1}{N!} ∑_{π∈𝔖_{N}}(−1)^{π} Ψ(x_{π(1)}, ς_{π(1)}, …, x_{π(N)}, ς_{π(N)}) ,

for Ψ ∈ L^{2}((ℝ^{3} × ℤ_{4})^{N}) and a.e. X = (x_{i}, ς_{i})_{i=1}^{N} ∈ (ℝ^{3} × ℤ_{4})^{N}. a priori it is defined on the dense domain

𝒟_{N} := A_{N}C_{0}^{∞}((ℝ^{3} × ℤ_{4})^{N}) ⊗ 𝒞_{0} ,

the tensor product understood in the algebraic sense, by

(4.2)  H_{nr}^{V} ≡ H_{nr}^{V}(G) := ∑_{i=1}^{N}(D_{A}^{(i)})^{2} + V + H_{f} .

A superscript (i) indicates that the operator below is acting on the pair of variables (x_{i}, ς_{i}). In fact, the operator defined in (4.2) is a two-fold copy of the usual Pauli-Fierz operator which acts on two-spinors and the energy has been shifted by N in (4.2). For (3.5) implies

(4.3)  D_{A}^{2} = T_{A} ⊕ T_{A} , T_{A} := (σ · (−i∇_{x} + A))^{2} + 1 .

Here σ = (σ_{1}, σ_{2}, σ_{3}) is a vector containing the Pauli matrices (when α_{j}, j ∈ {0, 1, 2, 3}, are given in Dirac’s standard representation). We write H_{nr}^{V} in the form (4.2) to maintain a unified notation throughout this paper. We shall only make use of the following properties of the potential V.

Hypothesis 4.1. V can be written as V = V_{+} − V_{−}, where V_{±} ⩾ 0 is a symmetric operator acting in A_{N}L^{2}((ℝ^{3}×ℤ_{4})^{4}) such that 𝒟_{N} ⊂ D(V_{±}). There exist a ∈ (0, 1) and b ∈ (0, ∞) such that V_{−} ⩽ a H_{nr}^{0}+b in the sense of quadratic forms on 𝒟_{N}.

Example. The Coulomb potential generated by K ∈ ℕ fixed nuclei located at the positions {R_{1}, …, R_{K}} ⊂ ℝ^{3} is given as

(4.4)  V_{C}(X) := − ∑_{i=1}^{N} ∑_{k=1}^{K} \frac{e^{2} Z_{k}}{|x_{i} − R_{k}|} + ∑_{{i,j=1 \atop i<j}}^{N} \frac{e^{2}}{|x_{i} − x_{j}|} ,

for some e, Z_{1}, …, Z_{K} > 0 and a.e. X = (x_{i}, ς_{i})_{i=1}^{N} ∈ (ℝ^{3} × ℤ_{4})^{N}. It is well-known that V_{C} is infinitesimally H_{nr}^{0}-bounded and that V_{C} fulfills Hypothesis 4.1. ⋄

It follows immediately from Hypothesis 4.1 that H_{nr}^{V} has a self-adjoint Friedrichs extension – henceforth denoted by the same symbol H_{nr}^{V} – and that 𝒟_{N} is a form core for H_{nr}^{V}. Moreover, we have

(4.5)  (D_{A}^{(1)})^{2}, …, (D_{A}^{(N)})^{2}, V_{+}, H_{f} ⩽ H_{nr}^{V_{+}} ⩽ (1 − a)^{−1} (H_{nr}^{V} + b)

on 𝒟_{N}. In [4] it is shown that D((H_{nr}^{V})^{n/2}) ⊂ D(H_{f}^{n/2}), for every n ∈ ℕ. We re-derive this result by means of Corollary 2.3 in the next theorem where

E_{nr} := inf σ[H_{nr}^{V}] ,  H′_{nr} := H_{nr}^{V} − E_{nr} + 1 .

Theorem 4.2. Assume that ω and G fulfill Hypothesis 3.1 and that V fulfills Hypothesis 4.1. Assume in addition that

(4.6)  2 ∫ ω(k)^{ℓ} ∥∇_{x} ∧ G(k)∥_{∞}^{2} dk ⩽ d_{ℓ+2}^{2} ,
(4.7)  ∫ ω(k)^{ℓ} ∥∇_{x} · G(k)∥_{∞}^{2} dk ⩽ d_{ℓ+2}^{2} ,

for all ℓ ∈ {−1, 0, 1, 2, …}. Then, for every n ∈ ℕ, we have D((H_{nr}^{V})^{n/2}) ⊂ D(H_{f}^{n/2}), H_{f}^{n/2} (H′_{nr})^{−n/2} maps D(H_{nr}^{V}) into itself, and

∥ H_{f}^{n/2} (H′_{nr})^{−n/2} ∥ ⩽ C(N, n, a, b, d_{−1}, d_{1}, d_{5+n}) (|E_{nr}| + 1)^{(3n−2)/2} ,
∥ [H_{f}^{n/2} , H_{nr}^{V}] (H′_{nr})^{−n/2} ∥ ⩽ C′(N, n, a, b, d_{−1}, d_{1}, d_{5+n}) (|E_{nr}| + 1)^{(3n−1)/2} .

Proof. We pick the function f_{ε} defined in (3.10) with E = 1 and verify that the operators F_{ε}^{n} := f_{ε}^{n/2}(H_{f}), ε > 0, n ∈ ℕ, and H′_{nr} fulfill the conditions (a), (b), and (c’) of Theorem 2.1 and Corollary 2.2 with m = ∞. Then the assertion follows from Corollary 2.3. We set Ȟ_{f} := H_{f} + E in what follows. By means of (4.5) we find

(4.8)  ⟨ Ψ | F_{ε}^{2} Ψ ⟩ ⩽ ⟨ Ψ | Ȟ_{f} Ψ ⟩ ⩽ \frac{E_{nr} + b + E}{1 − a} ⟨ Ψ | H′_{nr} Ψ ⟩ ,

for all Ψ ∈ 𝒟_{N}, which is Condition (b). Next, we observe that F_{ε} maps 𝒟_{N} into itself. Employing (4.5) once more and using −V_{−} ⩽ 0 and the fact that V_{+} ⩾ 0 and F_{ε} act on different tensor factors we deduce that

⟨ F_{ε} Ψ | (V + H_{f}) F_{ε} Ψ ⟩ ⩽ ∥f_{ε}∥_{∞} ⟨ Ψ | (V_{+} + H_{f}) Ψ ⟩

⩽ ∥f_{ε}∥_{∞} \frac{E_{nr} + b + E}{1 − a} ⟨ Ψ | H′_{nr} Ψ ⟩ , (4.9)

for every Ψ ∈ 𝒟_{N}. Thanks to (3.11) with κ = 1/2, ν = γ = δ = σ = τ = 0, and (4.5) we further find some C ∈ (0, ∞) such that

∥ D_{A}^{(i)} F_{ε} Ψ ∥^{2} ⩽ 2 ∥f_{ε}∥_{∞} ∥D_{A}^{(i)} Ψ∥^{2} + 2 ∥f_{ε}∥_{∞} ∥ F_{ε}^{−1} [α · A , F_{ε}] ∥^{2} ∥Ψ∥^{2}

⩽ C ∥f_{ε}∥_{∞} ⟨ Ψ | H′_{nr} Ψ ⟩ , (4.10)

for all Ψ ∈ 𝒟_{N}. (4.9) and (4.10) together show that Condition (a) is fulfilled, too. Finally, we verify the bound in (c’). We use

[α · (−i∇_{x}) , α · A] = Σ · B − i (∇_{x} · A) ,

where B := a^{†}(∇_{x} ∧ G) + a(∇_{x} ∧ G) is the magnetic field and the j-th entry of the formal vector Σ is −i ϵ_{jkℓ} α_{k} α_{ℓ}, j, k, ℓ ∈ {1, 2, 3}, to write the square of the Dirac operator on the domain 𝒟 as

D_{A}^{2} = D_{0}^{2} + Σ · B − i (∇_{x} · A) + (α · A)^{2} + 2 α · A α · (−i∇_{x}) .

This yields

[H′_{nr}, F_{ε}^{n}] = ∑_{i=1}^{N} [(D_{A}^{(i)})^{2}, F_{ε}^{n}] = ∑_{i=1}^{N} { [Σ · B^{(i)} , F_{ε}^{n}] − i [(∇_{x} · A^{(i)}) , F_{ε}^{n}]

+ α · A^{(i)} [α · A^{(i)} , F_{ε}^{n}] + [α · A^{(i)} , F_{ε}^{n}] (2D_{A}^{(i)} − α · A^{(i)} − 2β) }

on 𝒟_{N}. For every i ∈ {1, …, N}, we further write

[α · A^{(i)} , F_{ε}^{n}] D_{A}^{(i)} = Q_{ε,n}^{(i)} ( D_{A}^{(i)} F_{ε}^{n−1} − Q_{ε,n−1}^{(i)} F_{ε}^{n−2} )

on 𝒟_{N}, where

(4.11)  Q_{ε,n}^{(i)} := [α · A^{(i)} , F_{ε}^{n}] F_{ε}^{1−n}, n ∈ ℕ ,  Q_{ε,0}^{(i)} := 0 .

According to (3.11) we have ∥Q_{ε,n}^{(i)}∥ ⩽ n 2^{(n+2)/2} (d_{1} + d_{3+n}), ∥Ȟ_{f}^{1/2} Q_{ε,n}^{(i)} Ȟ_{f}^{−1/2}∥ ⩽ n 2^{(n+3)/2}(d_{1} + d_{4+n}). Likewise, we write

[α · A^{(i)} , F_{ε}^{n}] α · A^{(i)} = Q_{ε,n}^{(i)} ( {α · A^{(i)} Ȟ_{f}^{−1/2}} Ȟ_{f}^{1/2} F_{ε}^{n−1} − Q_{ε,n−1}^{(i)} F_{ε}^{n−2} )

on 𝒟_{N}, where ∥α·A Ȟ_{f}^{−1/2}∥^{2} ⩽ 2 d_{0}^{2} +4 d_{−1}^{2} by (3.7). Furthermore, we observe that Lemma 3.2 is applicable to Σ · B as well instead of α · A; we simply have to replace the form factor G by ∇_{x} ∧ G and to notice that ∥Σ · v∥_{ℒ (ℂ^{4})} = |v|, v ∈ ℝ^{3}, in analogy to (3.6). Note that the indices of d_{ℓ} are shifted by 2 because of (4.6). Finally, we observe that Lemma 3.2 is applicable to ∇_{x} · A, too. To this end we have to replace G by (∇_{x} · G, 0, 0) and d_{ℓ} by some universal constant times d_{2+ℓ} because of (4.7). Taking all these remarks into account we arrive at

|⟨ Ψ_{1} | [H′_{nr} , F_{ε}^{n}] Ψ_{2} ⟩| ⩽ ∑_{i=1}^{N} { ∥Ψ_{1}∥ ∥ [Σ · B^{(i)} , F_{ε}^{n}] F_{ε}^{1−n} ∥ ∥F_{ε}^{n−1} Ψ_{2}∥
+ ∥Ψ_{1}∥ ∥ [div A^{(i)} , F_{ε}^{n}] F_{ε}^{1−n} ∥ ∥F_{ε}^{n−1} Ψ_{2}∥
+ ∥Ψ_{1}∥ ∥α · A Ȟ_{f}^{−1/2}∥ ∥ Ȟ_{f}^{1/2} Q_{ε,n}^{(i)} Ȟ_{f}^{−1/2} ∥ ∥Ȟ_{f}^{1/2} F_{ε}^{n−1} Ψ_{2}∥
+ ∥Ψ_{1}∥ ∥Q_{ε,n}^{(i)}∥ ( 2 ∥D_{A}^{(i)} F_{ε}^{n−1} Ψ_{2}∥ + ∥α · A Ȟ_{f}^{−1/2}∥ ∥Ȟ_{f}^{1/2} F_{ε}^{n−1} Ψ_{2}∥ )

+ 3 ∥Ψ_{1}∥ ∥Q_{ε,n}^{(i)}∥ ∥Q_{ε,n−1}^{(i)}∥ ∥F_{ε}^{n−2} Ψ_{2}∥ + 2 ∥Ψ_{1}∥ ∥Q_{ε,n}^{(i)}∥ ∥β∥ ∥F_{ε}^{n−1} Ψ_{2}∥ } ,

for all Ψ_{1}, Ψ_{2} ∈ 𝒟_{N}. From this estimate, Lemma 3.2, and (4.5) we readily infer that Condition (c’) is valid with c_{n} = (|E_{nr}| + 1) C′′(N, n, a, b, d_{−1}, …, d_{5+n}).

□

5. The semi-relativistic Pauli-Fierz operator

The semi-relativistic Pauli-Fierz operator is also acting in the Hilbert space ℋ_{N} introduced in (4.1). It is obtained by substituting the non-local operator |D_{A}| for D_{A}^{2} in H_{nr}^{V}. We thus define, a priori on the dense domain 𝒟_{N},

H_{sr}^{V} ≡ H_{sr}^{V} (G) := ∑_{i=1}^{N} |D_{A}^{(i)}| + V + H_{f} ,

where V is assumed to fulfill Hypothesis 4.1 with H_{nr}^{0} replaced by H_{sr}^{0}. To ensure that in the case of the Coulomb potential V_{C} defined in (4.4) this yields a well-defined self-adjoint operator we have to impose appropriate restrictions on the nuclear charges.

Example. In Proposition A.1 we show that H_{sr}^{V_{C}} is semi-bounded below on 𝒟_{N} provided that Z_{k} ∈ (0, 2/πe^{2}], for all k ∈ {1, …, K}. Its proof is actually a straightforward consequence of (3.19) and a commutator estimate obtained in [10]. If all atomic numbers Z_{k} are strictly less than 2/πe^{2} we thus find a ∈ (0, 1) and b ∈ (0, ∞) such that

(5.1)  ∑_{i=1}^{N} ∑_{k=1}^{K} \frac{e^{2} Z_{k}}{|x_{i} − R_{k}|} ⩽ a H_{sr}^{0} + b

in the sense of quadratic forms on 𝒟_{N}. In particular, V_{C} fulfills Hypothesis 4.1 with H_{nr}^{0} replaced by H_{sr}^{0} as long as Z_{k} ∈ (0, 2/πe^{2}), for k ∈ {1, …, K}. ⋄

For potentials V as above H_{sr}^{V} has a self-adjoint Friedrichs extension which we denote again by the same symbol H_{sr}^{V}. Moreover, 𝒟_{N} is a form core for H_{sr}^{V} and we have the following analogue of (4.5),

(5.2)  |D_{A}^{(1)}|, …, |D_{A}^{(N)}|, V_{+}, H_{f} ⩽ H_{sr}^{V_{+}} ⩽ (1 − a)^{−1} (H_{sr}^{V} + b)

on 𝒟_{N}. In order to apply Corollary 2.3 to the semi-relativistic Pauli-Fierz operator we recall the following special case of [7, Corollary 3.7]:

Lemma 5.1. Assume that ω and G fulfill Hypothesis 3.1. Let τ ∈ (0, 1]. Then there exist δ > 0 and C ≡ C(δ, τ, d_{1}) ∈ (0, ∞) such that

(5.3) C + |D_{A}| + τ H_{f} ⩾ δ (|D_{0}| + H_{f}) ⩾ δ (|D_{0}| + τ H_{f}) ⩾ δ^{2} |D_{A}| − δ C

in the sense of quadratic forms on 𝒟.

In the next theorem we re-derive the higher order estimates obtained in [4] for the semi-relativistic Pauli-Fierz operator by means of Corollary 2.3. (The second estimate of Theorem 5.2 is actually slightly stronger than the corresponding one stated in [4].) The estimates of the following proof are also employed in Section 6 where we treat the no-pair operator. We set

E_{sr} := inf σ[H_{sr}] , H′_{sr} := H_{sr}^{V} − E_{sr} + 1 .

Theorem 5.2. Assume that ω and G fulfill Hypothesis 3.1 and that V fulfills Hypothesis 4.1 with H_{nr}^{0} replaced by H_{sr}^{0}. Then, for every m ∈ ℕ, it follows that D((H_{sr}^{V})^{m/2}) ⊂ D(H_{f}^{m/2}), H_{f}^{m/2} (H′_{sr})^{−m/2} maps D(H_{sr}^{V}) into itself, and

∥ H_{f}^{m/2} (H′_{sr})^{−m/2} ∥ ⩽ C(N, m, a, b, d_{1}, d_{3+m}) (|E_{sr}| + 1)^{(3m−2)/2} ,
∥ [H_{f}^{m/2} , H_{sr}^{V}] (H′_{sr})^{−m/2} ∥ ⩽ C′(N, m, a, b, d_{1}, d_{3+m}) (|E_{sr}| + 1)^{(3m−1)/2} .

Proof. Let m ∈ ℕ. We pick the function f_{ε} defined in (3.10) with E = k_{1} ∨ C. (k_{1} is the constant appearing in Lemma 3.3 with κ = m/2, ν = 0, and depends on m, d_{1}, and d_{3+m}; C is the one in (5.3).) We fix some n ∈ ℕ, n ⩽ m, and verify Conditions (a), (b), and (c’) of Theorem 2.1 and Corollary 2.2 with F_{ε} = f_{ε}^{1/2}(H_{f}), ε > 0. The estimates (4.8) and (4.9) are still valid without any further change when the subscript nr is replaced by sr. Employing (5.3) twice and using (5.2) we obtain the following substitute of (4.10),

⟨ F_{ε} Ψ | |D_{A}| F_{ε} Ψ ⟩ ⩽ δ^{−1}∥ |D_{0}|^{1/2} F_{ε} Ψ∥^{2} + δ^{−1} ∥Ȟ_{f}^{1/2} F_{ε} Ψ∥^{2}

⩽ δ^{−1} ∥f_{ε}∥_{∞} (∥ |D_{0}|^{1/2} Ψ∥^{2} + ∥Ȟ_{f}^{1/2} Ψ∥^{2}) ⩽ C′ ∥f_{ε}∥_{∞} ⟨ Ψ | H′_{sr} Ψ ⟩ ,

for all Ψ ∈ 𝒟_{N}. Altogether we see that Conditions (a) and (b) are satisfied. In order to verify (c’) we set

(5.4) U_{ε,n}^{(i)} := [S_{A}^{(i)} , F_{ε}^{n}] F_{ε}^{1−n} = F_{ε}^{n} [F_{ε}^{−n} , S_{A}^{(i)}] F_{ε} ,  i ∈ {1, …, N} .

By virtue of (3.22) we know that the norms of U_{ε,n}^{(i)} and U_{ε,n}^{(i)} |D_{A}^{(i)}|^{1/2} are bounded uniformly in ε > 0 by some constant, C ∈ (0, ∞), that depends only on n, d_{1}, and d_{3+n}. We employ the notation (4.11) and (5.4) to write

[H′_{sr} , F_{ε}^{n}] = ∑_{i=1}^{N} [ |D_{A}^{(i)}| , F_{ε}^{n} ] = ∑_{i=1}^{N} [ S_{A}^{(i)} D_{A}^{(i)} , F_{ε}^{n} ]

= ∑_{i=1}^{N} {{U_{ε,n}^{(i)} |D_{A}^{(i)}|^{1/2}} S_{A}^{(i)} |D_{A}^{(i)}|^{1/2} F_{ε}^{n−1}

− U_{ε,n}^{(i)} Q_{ε,n−1}^{(i)} F_{ε}^{n−2} + S_{A}^{(i)} Q_{ε,n}^{(i)} F_{ε}^{n−1}}.

The previous identity, (5.2), and |D_{A}| ⩾ 1 permit to get

|⟨ Ψ_{1} | [H′_{sr} , F_{ε}^{n}] Ψ_{2} ⟩| ⩽ ∑_{i=1}^{N} ∥Ψ_{1}∥ {C ∥ |D_{A}^{(i)}|^{1/2} F_{ε}^{n−1} Ψ_{2}∥
+ C ∥Q_{ε,n}^{(i)}∥ ∥F_{ε}^{n−2} Ψ_{2}∥ + ∥Q_{ε,n}^{(i)}∥ ∥F_{ε}^{n−1} Ψ_{2}∥}
⩽ c_{n} { ∥Ψ_{1}∥^{2} + ⟨ F_{ε}^{n−1} Ψ_{2} | H′_{sr} F_{ε}^{n−1} Ψ_{2} ⟩ } ,

for all Ψ_{1}, Ψ_{2} ∈ 𝒟_{N} and some constant c_{n} = C′′(n, a, b, d_{1}, d_{3+n}) (|E_{sr}| + 1). So (c’) is fulfilled also and the assertion follows from Corollary 2.3. □

6. The no-pair operator

We introduce the spectral projections

(6.1)  P_{A}^{+} := E_{[0,∞)}(D_{A}) = \frac{1}{2} 𝟙 + \frac{1}{2} S_{A} ,  P_{A}^{−} := 𝟙 − P_{A}^{+} .

The no-pair operator acts in the projected Hilbert space

ℋ_{N}^{+} ≡ ℋ_{N}^{+}(G) := P_{A,N}^{+} ℋ_{N} , P_{A,N}^{+} := ∏_{i=1}^{N} P_{A}^{+,(i)} ,

and is a priori defined on the dense domain P_{A,N}^{+} 𝒟_{N} by

H_{np}^{V} ≡ H_{np}^{V}(G) := P_{A,N}^{+} { ∑_{i=1}^{N} D_{A}^{(i)} + V + H_{f} } P_{A,N}^{+} .

Notice that all operators D_{A}^{(1)}, …, D_{A}^{(N)} and P_{A}^{+,(1)}, …, P_{A}^{+,(N)} commute in pairs owing to the fact that the components of the vector potential A^{(i)}(x), A^{(j)}(y), x, y ∈ ℝ^{3}, i, j ∈ {1, 2, 3}, commute in the sense that all their spectral projections commute; see the appendix to [9] for more details. (Here we use the assumption that G_{x}(−k, λ) = \overline{G_{x}(k, λ)}.) So the order of the application of the projections P_{A}^{+,(i)} is immaterial. In this section we restrict the discussion to the case where V is given by the Coulomb potential V_{C} defined in (4.4). To have a handy notation we set

v_{i} := − ∑_{k=1}^{K} \frac{e^{2} Z_{k}}{|x_{i} − R_{k}|} , w_{ij} := \frac{e^{2}}{|x_{i} − x_{j}|} ,

for all i ∈ {1, …, N} and 1 ⩽ i < j ⩽ N, respectively. Thanks to [10, Proof of Lemma 3.4(ii)], which implies that P_{A}^{+} maps 𝒟 into D(|D_{0}|) ∩ D(H_{f}^{ν}), for every ν > 0, and Hardy’s inequality, we actually know that H_{np}^{V_{C}} is well-defined on 𝒟_{N}. In order to apply Corollary 2.3 to H_{np}^{V_{C}} we extend H_{np}^{V_{C}} to a continuously invertible operator on the whole space ℋ_{N}: We pick the complementary projection,

P_{A,N}^{⊥} := 𝟙 − P_{A,N}^{+} ,

abbreviate

P_{A}^{+,(i,j)} := P_{A}^{+,(i)} P_{A}^{+,(j)} = P_{A}^{+,(j)} P_{A}^{+,(i)} , 1 ⩽ i < j ⩽ N ,

and define the operator H̃_{np} a priori on the domain 𝒟_{N} by

H̃_{np} := ∑_{i=1}^{N} \{ |D_{A}^{(i)}| + P_{A}^{+,(i)} v_{i} P_{A}^{+,(i)} \} + ∑_{{i,j=1 \atop i<j}}^{N} P_{A}^{+,(i,j)} w_{ij} P_{A}^{+,(i,j)}

+ P_{A,N}^{+} H_{f} P_{A,N}^{+} + P_{A,N}^{⊥} H_{f} P_{A,N}^{⊥} . (6.2)

Evidently, we have [H̃_{np} , P_{A,N}^{+}] = 0 and H̃_{np} P_{A,N}^{+} = H_{np}^{V_{C}} P_{A,N}^{+} on 𝒟_{N}. In Proposition A.2 we show that the quadratic forms of the no-pair operator H_{np}^{V_{C}} and of H̃_{np} are semi-bounded below on 𝒟_{N} provided that the atomic numbers Z_{1}, …, Z_{K} ⩾ 0 are less than the critical one of the Brown-Ravenhall model determined in [3],

(6.3)  Z_{np} := (2/e^{2})/(2/π + π/2) .

Therefore, both H_{np}^{V_{C}} and H̃_{np} possess self-adjoint Friedrichs extensions which are again denoted by the same symbols in the sequel. 𝒟_{N} is a form core for H̃_{np} and we have the bound

(6.4) H̃_{np} −∑_{i=1}^{N} P_{A}^{+,(i)} v_{i} P_{A}^{+,(i)} ⩽ \frac{Z_{np} + |𝒵 |}{Z_{np} − |𝒵 |} (H̃_{np} +C(N, 𝒵 , ℛ, d_{−1}, d_{1}, d_{5}))

on 𝒟_{N}, where |𝒵 | := max{Z_{1}, …, Z_{K}} < Z_{np}. Moreover, it makes sense to define

E_{np} := inf σ[H_{np}^{V_{C}}] ,

so that

H′_{np} := H̃_{np} − E_{np} P_{A,N}^{+} + 𝟙 ⩾ 𝟙 .

Theorem 6.1. Assume that ω and G fulfill Hypothesis 3.1 and let N, K ∈ ℕ, e > 0, 𝒵 = (Z_{1}, …, Z_{K}) ∈ [0, Z_{np})^{K}, and ℛ = {R_{1}, …, R_{K}} ⊂ ℝ^{3}, where Z_{np} is defined in (6.3). Then D((H′_{np})^{m/2}) ⊂ D(H_{f}^{m/2}), for every m ∈ ℕ, and

∥ H_{f}^{m/2}↾_{ℋ_{N}^{+}} (H_{np} − (E_{np} − 1) 𝟙_{ℋ_{N}^{+}} )^{−m/2} ∥_{ℒ (ℋ_{N}^{+},ℋ_{N})} ⩽ ∥ H_{f}^{m/2} (H′_{np})^{−m/2} ∥

⩽ C(N, m, 𝒵 , ℛ, e, d_{−1}, d_{1}, d_{5+m}) (1 + |E_{np}|)^{(3m−2)/2} < ∞ .

Proof. Let m ∈ ℕ. Again we pick the function f_{ε} defined in (3.10) and set F_{ε} := f_{ε}^{1/2}(H_{f}), ε > 0. This time we choose E = max{k d_{1}^{2}, k_{1}, C} where k is the constant appearing in (3.19), C ≡ C(d_{1}) is the one in (5.3), and k_{1} the one appearing in Lemma 3.3 with |κ| = (m + 1)/2, |ν| = 1/2. Thus k_{1} depends only on m, d_{1}, and d_{5+m}. On account of Corollary 2.3 it suffices to show that the conditions (a)–(c) of Theorem 2.1 are fulfilled. To this end we observe that on 𝒟_{N} the extended no-pair operator can be written as H′_{np} = H_{sr}^{0} + 𝟙 + W, where

W := ∑_{i=1}^{N} P_{A}^{+,(i)} v_{i} P_{A}^{+,(i)} + ∑_{{i,j=1 \atop i<j}}^{N} P_{A}^{+,(i,j)} w_{ij} P_{A}^{+,(i,j)}

− E_{np} P_{A,N}^{+} − 2Re [P_{A,N}^{+} H_{f} P_{A,N}^{⊥}] .

The semi-relativistic Pauli-Fierz operator H_{sr}^{0} has already been treated in the previous section and the bound

(6.5)  H_{f} ⩽ 2P_{A,N}^{+} H_{f} P_{A,N}^{+} + 2P_{A,N}^{⊥} H_{f} P_{A,N}^{⊥}

together with (6.4) implies

(6.6)  H_{sr}^{0} ⩽ 2 H̃_{np} − 2 ∑_{i=1}^{N} P_{A}^{+,(i)} v_{i} P_{A}^{+,(i)} ⩽ C′ (1 + |E_{np}|) H′_{np}

on 𝒟_{N}, for some C′ ≡ C′(N, 𝒵 , ℛ, d_{−1}, d_{1}, d_{5}) ∈ (0, ∞). Hence, it only remains to consider the operator W.

We fix some n ∈ ℕ, n ⩽ m. When we verify (a) we can ignore the potentials v_{i} since they are negative. Using [F_{ε}^{n}, P_{A,N}^{⊥}] = [P_{A,N}^{+}, F_{ε}^{n}] we obtain

|2Re ⟨ P_{A,N}^{+} F_{ε} Ψ | H_{f} P_{A,N}^{⊥} F_{ε} Ψ ⟩|
⩽ ∥ H_{f}^{1/2} P_{A,N}^{+} F_{ε} Ψ ∥^{2} + ∥ H_{f}^{1/2} P_{A,N}^{⊥} F_{ε} Ψ ∥^{2}
⩽ 2 ∥f_{ε}∥_{∞} ∥H_{f}^{1/2} P_{A,N}^{+} Ψ∥^{2} + 2 ∥f_{ε}∥_{∞} ∥H_{f}^{1/2} P_{A,N}^{⊥} Ψ∥^{2}
+ 4 ∥ H_{f}^{1/2} [P_{A,N}^{+} , F_{ε}] Ȟ_{f}^{−1/2} ∥ ∥Ȟ_{f}^{1/2} Ψ∥^{2},

for every Ψ ∈ 𝒟_{N}, where, for all n ∈ ℕ and ν ∈ ℝ,

Ȟ_{f}^{ν} [P_{A,N}^{+} , F_{ε}^{n}] Ȟ_{f}^{−ν} F_{ε}^{1−n} = ∑_{i=1}^{N} { ∏_{j=1}^{i−1} Ȟ_{f}^{ν} P_{A}^{+,(j)} Ȟ_{f}^{−ν}} ×

× {Ȟ_{f}^{ν} [P_{A}^{+,(i)}, F_{ε}^{n}] Ȟ_{f}^{−ν} F_{ε}^{1−n}}{ ∏_{k=i+1}^{N} Ȟ_{f}^{ν} F_{ε}^{n−1} P_{A}^{+,(k)} Ȟ_{f}^{−ν} F_{ε}^{1−n}}

on 𝒟_{N}. On account of Corollary 3.4 we thus have, for |ν| ⩽ 1/2,

(6.7)  sup_{ε>0} ∥ H_{f}^{ν} [P_{A,N}^{+} , F_{ε}^{n}] Ȟ_{f}^{−ν} F_{ε}^{1−n} ∥ ⩽ C(N, n, d_{1}, d_{4+n}) .

Likewise we have

|⟨ F_{ε} Ψ | P_{A}^{+,(i,j)} w_{ij} P_{A}^{+,(i,j)} F_{ε} Ψ ⟩| ⩽ 2 ∥f_{ε}∥ ∥ w_{ij}^{1/2} P_{A}^{+,(i,j)} Ψ ∥^{2}

+ 4 ∥ w_{ij}^{1/2} [P_{A}^{+,(i,j)} , F_{ε}] Ȟ_{f}^{−1/2} ∥^{2} ∥Ȟ_{f}^{1/2} Ψ∥^{2}, (6.8)

where the first norm in the second line of (6.8) is bounded (uniformly in ε > 0) due to Lemma 6.2. Taking these remarks, v_{i} ⩽ 0, (6.4), and (6.5) into account we infer that

⟨ F_{ε} Ψ | H′_{np} F_{ε} Ψ ⟩ ⩽ c_{ε}⟨ Ψ | H′_{np} Ψ ⟩ , Ψ ∈ 𝒟_{N} ,

showing that (a) is fulfilled. The condition (b) with some constant c^{2} = C(N, 𝒵 , ℛ, d_{−1}, d_{1}, d_{5})(1+|E_{np}|) follows immediately from F_{ε}^{2} ⩽ Ȟ_{f} ⩽ H_{sr}^{0}+E on 𝒟_{N} and (6.6). Finally, we turn to Condition (c). To this end let P_{A,N}^{♯} and P_{A,N}^{♭} be P_{A,N}^{+} or P_{A,N}^{⊥}. On 𝒟_{N} we clearly have

[ P_{A,N}^{♯} H_{f} P_{A,N}^{♭} , F_{ε}^{n} ] = ± [P_{A,N}^{+} , F_{ε}^{n}] H_{f} P_{A,N}^{♭} ± P_{A,N}^{♯} H_{f} [P_{A,N}^{+} , F_{ε}^{n}] . (6.9)

For Ψ_{1}, Ψ_{2} ∈ 𝒟_{N}, we thus obtain

|⟨ Ψ_{1} | [ P_{A,N}^{♯} H_{f} P_{A,N}^{♭} , F_{ε}^{n} ] Ψ_{2} ⟩|

⩽ ∥Ȟ_{f}^{1/2} Ψ_{1}∥ ∥ Ȟ_{f}^{−1/2} [P_{A,N}^{+} , F_{ε}^{n}] H_{f}^{1/2}F^{1−n} ∥ ∥H_{f}^{1/2} F_{ε}^{n−1} P_{A,N}^{♭} Ψ_{2}∥

(6.10) + ∥H_{f}^{1/2} P_{A,N}^{♯} Ψ_{1}∥ ∥H_{f}^{1/2} [P_{A,N}^{+} , F_{ε}^{n}] Ȟ_{f}^{−1/2} F_{ε}^{1−n}∥ ∥Ȟ_{f}^{1/2} F_{ε}^{n−1} Ψ_{2}∥ , where we can further estimate

∥H_{f}^{1/2} F_{ε}^{n−1} P_{A,N}^{♭} Ψ_{2}∥
⩽ {1 + ∥H_{f}^{1/2} F_{ε}^{n−1} P_{A,N}^{+} Ȟ_{f}^{−1/2} F_{ε}^{1−n}∥} ∥Ȟ_{f}^{1/2} F_{ε}^{n−1} Ψ_{2}∥

⩽ {1 + ∥H_{f}^{1/2} F_{ε}^{n−1} P_{A}^{+} Ȟ_{f}^{−1/2} F_{ε}^{1−n}∥^{N}} ∥Ȟ_{f}^{1/2} F_{ε}^{n−1} Ψ_{2}∥ , (6.11)

and, of course,

(6.12) ∥Ȟ_{f}^{1/2} F_{ε}^{n−1} Ψ_{2}∥ ⩽ ∥Ȟ_{f}^{1/2} P_{A,N}^{+} F_{ε}^{n−1} Ψ_{2}∥ + ∥Ȟ_{f}^{1/2} P_{A,N}^{⊥} F_{ε}^{n−1} Ψ_{2}∥ .

The operator norms in (6.10) can be estimated by means of (6.7) with ν = ±1/2, the one in the last line of (6.11) is bounded by some C(n, d_{1}, d_{3+n}) ∈ (0, ∞) due to (3.21). In a similar fashion we obtain, for all i, j ∈ {1, …, N}, i < j, and Ψ_{1}, Ψ_{2} ∈ 𝒟_{N},

|⟨ Ψ_{1} | [P_{A}^{+,(i,j)} w_{ij}P_{A}^{+,(i,j)} , F_{ε}^{n}] Ψ_{2} ⟩|

⩽ ∥ F_{ε}^{1−n} w_{ij}^{1/2} [F_{ε}^{n} , P_{A}^{+,(i,j)}] Ȟ_{f}^{−1/2} ∥ ∥Ȟ_{f}^{1/2} Ψ_{1}∥ ∥F_{ε}^{n−1} w_{ij}^{1/2} P_{A}^{+,(i,j)} Ψ_{2}∥

+ ∥ w_{ij}^{1/2} P_{A}^{+,(i,j)} Ψ_{1}∥ ∥ w_{ij}^{1/2} [P_{A}^{+,(i,j)}, F_{ε}^{n}] F_{ε}^{1−n} Ȟ_{f}^{−1/2}∥ ∥Ȟ_{f}^{1/2} F_{ε}^{n−1} Ψ_{2}∥. (6.13)

Here we can further estimate

∥ w_{ij}^{1/2} F_{ε}^{n−1} P_{A}^{+,(i,j)} Ψ_{2}∥ ⩽ ∥ w_{ij}^{1/2} P_{A}^{+,(i,j)} F_{ε}^{n−1} Ψ_{2}∥

+ ∥ w_{ij}^{1/2} [F_{ε}^{n−1} , P_{A}^{+,(i,j)}] Ȟ_{f}^{−1/2}F_{ε}^{1−n} ∥ ∥Ȟ_{f}^{1/2} F_{ε}^{n−1} Ψ_{2}∥ . (6.14)

Lemma 6.2 below ensures that all operator norms in (6.13) and (6.14) that involve w_{ij}^{1/2} are bounded uniformly in ε > 0 by constants depending only on e, n, d_{1}, and d_{5+n}. Furthermore, it is now clear how to treat the terms involving v_{i} or E_{np}. (In order to treat v_{i} just replace P_{A}^{+,(i,j)} by P_{A}^{+,(i)}, w_{ij} by v_{i}, and w_{ij}^{1/2} by |v_{i}|^{1/2} in (6.13) and (6.14).) Combining (6.9)–(6.14) and their analogues for the remaining operators in W we arrive at

|⟨ Ψ_{1} | [W , F_{ε}^{n}] Ψ_{2} ⟩|

⩽ C ∑_{♯∈{+,⊥}}{⟨ Ψ_{1} | P_{A,N}^{♯} H_{f} P_{A,N}^{♯} Ψ_{1} ⟩ + ⟨ F_{ε}^{n−1} Ψ_{2} | P_{A,N}^{♯} H_{f} P_{A,N}^{♯} F_{ε}^{n−1} Ψ_{2} ⟩}

+ C ∑_{{i,j=1 \atop i<j}}^{N} \{⟨ Ψ_{1} | P_{A}^{+,(i,j)}w_{ij}P_{A}^{+,(i,j)}Ψ_{1} ⟩

+ ⟨ F_{ε}^{n−1}Ψ_{2} | P_{A}^{+,(i,j)}w_{ij}P_{A}^{+,(i,j)}F_{ε}^{n−1}Ψ_{2} ⟩}

+ C ∑_{i=1}^{N} {⟨ Ψ_{1} | P_{A}^{+,(i)} |v_{i}| P_{A}^{+,(i)} Ψ_{1} ⟩ + ⟨ F_{ε}^{n−1} Ψ_{2} | P_{A}^{+,(i)} |v_{i}| P_{A}^{+,(i)} F_{ε}^{n−1} Ψ_{2} ⟩}

+ C (1 + |E_{np}|) {∥Ψ_{1}∥^{2} + ∥F_{ε}^{n−1} Ψ_{2}∥^{2}} ,

for all Ψ_{1}, Ψ_{2} ∈ 𝒟_{N} and some ε-independent C ≡ C(N, n, e, d_{1}, d_{5+n}) ∈ (0, ∞). Employing successively (3.19), which implies |v_{i}| ⩽ (πe^{2}|𝒵 |/2)(|D_{A}^{(i)}| + Ȟ_{f}), after that (3.21), which yields ∥Ȟ_{f}^{1/2} P_{A}^{+,(i)} Ψ∥^{2} ⩽ C(d_{1}, d_{4})(∥Ȟ_{f}^{1/2} P_{A,N}^{+} Ψ∥^{2} +

∥Ȟ_{f}^{1/2} P_{A,N}^{⊥} Ψ∥^{2}), and finally (6.4) we conclude that Condition (c) is fulfilled with c_{n} = C(N, n, 𝒵 , ℛ, e, d_{−1}, d_{1}, d_{5+n})(1 + |E_{np}|). □

Lemma 6.2. For all i, j ∈ {1, …, N}, i < j, n ∈ ℤ, and σ, τ ⩾ 0 with σ+τ ⩽ 1,

sup_{ε>0}∥F_{ε}^{σ−n} w_{ij}^{1/2} [F_{ε}^{n} , P_{A}^{+,(i,j)}] Ȟ_{f}^{−1/2} F_{ε}^{τ} ∥

= sup_{ε>0} ∥ w_{ij}^{1/2} F_{ε}^{σ} [P_{A}^{+,(i,j)} , F_{ε}^{−n}] Ȟ_{f}^{−1/2} F_{ε}^{n+τ} ∥ ⩽ e C(n, d_{1}, d_{5+n}) < ∞ .

Proof. We write

w_{ij}^{1/2} F_{ε}^{σ} [P_{A}^{+,(i)} P_{A}^{+,(j)} , F_{ε}^{−n}] Ȟ_{f}^{−1/2}F_{ε}^{n+τ} = Y_{1} + w_{ij}^{1/2} Y_{2} + Y_{3} ,

where

Y_{1} := {w_{ij}^{1/2} F_{ε}^{σ} [P_{A}^{+,(i)} , F_{ε}^{−n}] Ȟ_{f}^{−1/2} F_{ε}^{n+τ} }{Ȟ_{f}^{1/2} F_{ε}^{−n−τ} P_{A}^{+,(j)} Ȟ_{f}^{−1/2} F_{ε}^{n+τ} },
Y_{2} := P_{A}^{+,(i)} F_{ε}^{σ} [P_{A}^{+,(j)} , F_{ε}^{−n}] Ȟ_{f}^{−1/2} F_{ε}^{n+τ} ,
Y_{3} := w_{ij}^{1/2} [F_{ε}^{σ} , P_{A}^{+,(i)}] [P_{A}^{+,(j)} , F_{ε}^{−n}] Ȟ_{f}^{−1/2} F_{ε}^{n+τ} .

Applying Corollary 3.4 we immediately see that ∥Y_{1}∥ ⩽ e C(n, d_{1}, d_{5+n}) and that

∥Y_{3}∥ ⩽ ∥ w_{ij}^{1/2} [F_{ε}^{σ} , P_{A}^{+,(i)}] F_{ε}^{−σ}∥ ∥ F_{ε}^{σ} [P_{A}^{+,(j)} , F_{ε}^{−n}] F_{ε}^{n+τ} ∥ ⩽ e C(n, d_{1}, d_{3+n})

uniformly in ε > 0. Employing (3.19) (with respect to the variable x_{j} for each fixed x_{i}) and using [|D_{A}^{(j)}|^{1/2}, P_{A}^{+,(i)}] = 0, we further get

∥ w_{ij}^{1/2} Y_{2} Ψ∥^{2}

⩽ (πe^{2}/2) ∥P_{A}^{+,(i)}∥^{2} ∥ |D_{A}^{(j)}|^{1/2} F_{ε}^{σ} [P_{A}^{+,(j)} , F_{ε}^{−n}] F_{ε}^{n+τ} ∥^{2} ∥Ȟ_{f}^{−1/2}∥^{2}

+ (πe^{2}/2) ∥Ȟ_{f}^{1/2} P_{A}^{+,(i)} Ȟ_{f}^{−1/2}∥^{2} ∥Ȟ_{f}^{1/2} F_{ε}^{σ} [P_{A}^{+,(i)} , F_{ε}^{−n}] Ȟ_{f}^{−1/2} F_{ε}^{n+τ} ∥^{2}.

By Corollary 3.4 all norms on the right hand side are bounded uniformly in ε > 0 by constants depending only on n, d_{1}, and d_{4+n}. □

Appendix A. Semi-boundedness of H_{sr}^{V_{C}} and H_{np}^{V_{C}}

In this appendix we verify that the semi-relativistic Pauli-Fierz and no-pair operators with Coulomb potential are semi-bounded below for all nuclear charges less than the critical charges without radiation fields. We do not attempt to give good lower bounds on their spectra since this is not the topic addressed in this paper. Our aim here is essentially only to ensure that these operators possess self-adjoint Friedrichs extensions. We recall that the stability of matter of the second kind has been proven for the no-pair operator in [9] under certain restrictions on the fine-structure constant, the ultra-violet cut-off, and the nuclear charges. The stability of matter of the second kind is a much stronger property than mere semi-boundedness. It says that the operator is bounded below by some constant which is proportional to the total number of nuclei and electrons and uniform in the nuclear positions. The restrictions imposed on the physical parameters in [9] do, however, not allow for all atomic numbers less than Z_{np}. First, we consider the semi-relativistic Pauli-Fierz operator. The following proposition is a simple generalization of the bound (3.19) proven in [10] to the case of N ∈ ℕ electrons and K ∈ ℕ nuclei.

Proposition A.1. Assume that ω and G fulfill Hypothesis 3.1 and let N, K ∈ ℕ, e > 0, 𝒵 = (Z_{1}, …, Z_{K}) ∈ (0, 2/πe^{2}]^{K}, and ℛ = {R_{1}, …, R_{K}} ⊂ ℝ^{3}. Then

(A.1)  ∑_{i=1}^{N} |D_{A}^{(i)}| + V_{C} + δ H_{f} ⩾ −C(δ, N, 𝒵 , ℛ, d_{1}) > −∞ ,

for every δ > 0 in the sense of quadratic forms on 𝒟_{N}.

Proof. In view of (3.19) we only have to explain how to localize the non-local kinetic energy terms. To begin with we recall the following bounds proven in [10, Lemmata 3.5 and 3.6]: For every χ ∈ C^{∞}(ℝ_{x}^{3}, [0, 1]),

∥ [χ, S_{A}] ∥ ⩽ ∥∇χ∥_{∞} ,  ∥D_{A} [χ , [χ, S_{A}] ]∥ ⩽ 2 ∥∇χ∥_{∞}^{2} . (A.2)

Now, let B_{r}(z) denote the open ball of radius r > 0 centered at z ∈ ℝ^{3} in ℝ^{3}. We set ϱ := min{|R_{k} − R_{ℓ}| : k ≠ ℓ}/2 and pick a smooth partition of unity on ℝ^{3}, {χ_{k}}_{k=0}^{K}, such that χ_{k} ≡ 1 on B_{ϱ/2}(R_{k}) and supp(χ_{k}) ⊂ B_{ϱ}(R_{k}), for k = 1, …, K, and such that ∑_{k=0}^{K} χ_{k}^{2} = 1. Then we have the following IMS type localization formula,

(A.3)  |D_{A}| = ∑_{k=0}^{K} \{ χ_{k} |D_{A}| χ_{k} + \frac{1}{2} [χ_{k} , [χ_{k}, |D_{A}|] ] \}

on 𝒟, for every i ∈ {1, …, N}. A direct calculation shows that

(A.4)  [χ_{k} , [χ_{k}, |D_{A}|] ] = 2 iα · (∇χ_{k}) [χ_{k}, S_{A}] + D_{A} [χ_{k} , [χ_{k}, S_{A}] ]

on 𝒟. By virtue of (3.6) and (A.2) we thus get

(A.5)  ∥ [χ_{k} , [χ_{k}, |D_{A}|] ] ∥ ⩽ 4 ∥∇χ∥_{∞}^{2} ,

for all k ∈ {0, …, K}. Since we are able to localize the kinetic energy terms and since, by the choice of the partition of unity, the functions ℝ^{3} ∋ x ↦ |x − R_{k}|^{−1} χ_{ℓ}^{2}(x) are bounded, for k ∈ {1, …, K}, ℓ ∈ {0, …, K}, k ≠ ℓ, the bound (A.1) is now an immediate consequence of (3.19) (with δ replaced by δ/N). (Here we also make use of the fact that the hypotheses on G are translation invariant.) □

Next, we turn to the no-pair operator discussed in Section 6. The semiboundedness of the molecular N-electron no-pair operator is essentially a consequence of the following inequality [10, Equation (2.14)], valid for all ω and G fulfilling Hypothesis 3.1, γ ∈ (0, 2/(2/π + π/2)), and δ > 0,

(A.6)  P_{A}^{+} (D_{A}^{(i)} − γ/|x| + δ H_{f}) P_{A}^{+} ⩾ P_{A}^{+} (c(γ) |D_{0}| − C) P_{A}^{+} ,

in the sense of quadratic forms on P_{A}^{+} 𝒟. Here C ≡ C(δ, γ, d_{−1}, d_{0}, d_{1}) ∈ (0, ∞) and c(γ) ∈ (0, ∞) depends only on γ.

Proposition A.2. Assume that ω and G fulfill Hypothesis 3.1 and let N, K ∈ ℕ, e > 0, 𝒵 = (Z_{1}, …, Z_{K}) ∈ (0, Z_{np})^{K}, and ℛ = {R_{1}, …, R_{K}} ⊂ ℝ^{3}, where Z_{np} is defined in (6.3). Then the quadratic form associated with the operator H̃_{np} defined in (6.2) is semi-bounded below,

H̃_{np} ⩾ −C(N, K, 𝒵 , ℛ, d_{−1}, d_{1}, d_{5}) > −∞ ,

in the sense of quadratic forms on 𝒟_{N}.

Proof. We again employ the parameter ϱ > 0 and the partition of unity introduced in the paragraph succeeding (A.2). Thanks to [10, Proof of Lemma 3.4(ii)] we know that P_{A}^{+} maps D(D_{0} ⊗ H_{f}^{ν}) into D(D_{0} ⊗ H_{f}^{ν−1}), for every ν ⩾ 1. The IMS localization formula thus yields

P_{A}^{+,(i)} v_{i} P_{A}^{+,(i)}

= ∑_{k=0}^{K} \{χ_{k}^{(i)} P_{A}^{+,(i)} v_{i} P_{A}^{+,(i)} χ_{k}^{(i)} + \frac{1}{2} [χ_{k}^{(i)} , [χ_{k}^{(i)} , P_{A}^{+,(i)} v_{i} P_{A}^{+,(i)}] ]\}

on D(D_{0} ⊗ H_{f}), where a superscript (i) indicates that χ_{k} = χ_{k}^{(i)} depends on the variable x_{i}. Using v_{i} ⩽ 0, we observe that

[χ_{k}^{(i)} , [χ_{k}^{(i)} , P_{A}^{+,(i)} v_{i} P_{A}^{+,(i)}] ]
= −2 [χ_{k}^{(i)} , P_{A}^{+,(i)}] v_{i} [P_{A}^{+,(i)} , χ_{k}^{(i)}]
+ 2 Re { P_{A}^{+,(i)} v_{i} [χ_{k}^{(i)} , [χ_{k}^{(i)} , P_{A}^{+,(i)}] ] }

⩾ 2 Re { P_{A}^{+,(i)} v_{i} [χ_{k}^{(i)} , [χ_{k}^{(i)} , P_{A}^{+,(i)}] ] } . (A.7)

We recall the following estimate proven in [10, Lemma 3.6], for every χ ∈ C^{∞}(ℝ_{x}^{3}, [0, 1]),

∥\frac{1}{|x|} [χ , [χ , P_{A}^{+}] ] Ȟ_{f}^{−1/2} ∥ ⩽ 8^{3/2} ∥∇χ∥_{∞}^{2} ,

where Ȟ_{f} = H_{f} + E with E ⩾ 1 ∨ (4d_{1})^{2}. Together with (A.7) it implies

⟨ Ψ | [χ_{k}^{(i)} , [χ_{k}^{(i)} , P_{A}^{+,(i)} v_{i} P_{A}^{+,(i)}] ] Ψ ⟩
⩾ −δ ⟨ Ψ | Ȟ_{f} Ψ ⟩ − (8^{3} ∥∇χ_{k}∥_{∞}^{4}/δ) ∥Ψ∥^{2},

for all k ∈ {0, …, K}, i ∈ {1, …, N}, δ > 0, and Ψ ∈ D(D_{0} ⊗ H_{f}). Next, we pick cut-off functions, ζ_{1}, …, ζ_{K} ∈ C_{0}^{∞}(ℝ_{x}^{3}, [0, 1]), such that ζ_{k} = 1 in a neighborhood of R_{k} and supp(ζ_{k}) ⊂ B_{ϱ/4}(R_{k}), for k ∈ {1, …, K}. By construction, supp(ζ_{k}) ∩ supp(χ_{ℓ}) = ∅, for all k ∈ {1, …, K} and ℓ ∈ {0, …, K} with k ≠ ℓ. Denoting \overline{ζ}_{k} := 1 − ζ_{k} and using the superscript (i) to indicate that ζ_{k} = ζ_{k}^{(i)} is a function of the variable x_{i}, we obtain

⟨ Ψ | χ_{k}^{(i)} P_{A}^{+,(i)} v_{i} P_{A}^{+,(i)} χ_{k}^{(i)} Ψ ⟩

= −⟨ Ψ | χ_{k}^{(i)} P_{A}^{+,(i)} \frac{e^{2} Z_{k}}{|x_{i} − R_{k}|} P_{A}^{+,(i)} χ_{k}^{(i)} Ψ ⟩

− ∑_{{ℓ=1 \atop ℓ≠k}}^{K} ⟨ Ψ | χ_{k}^{(i)} P_{A}^{+,(i)} \frac{e^{2} Z_{ℓ} ζ_{ℓ}^{(i)}}{|x_{i} − R_{ℓ}|} P_{A}^{+,(i)} χ_{k}^{(i)} Ψ ⟩ (A.8)

− ∑_{{ℓ=1 \atop ℓ≠k}}^{K} ⟨ Ψ | χ_{k}^{(i)} P_{A}^{+,(i)} \frac{e^{2} Z_{ℓ} \overline{ζ}_{ℓ}^{(i)}}{|x_{i} − R_{ℓ}|} P_{A}^{+,(i)} χ_{k}^{(i)} Ψ ⟩, (A.9)

for all Ψ ∈ D(D_{0} ⊗ H_{f}). The operators appearing in the scalar products in (A.9) are bounded by definition of \overline{ζ}_{ℓ}. Their norms depend only on ℛ since e^{2} Z_{ℓ} < 1. Furthermore, by virtue of Lemma A.3 below the term in (A.8) is bounded from below by −δ ⟨ Ψ | H_{f} Ψ ⟩ − C_{δ} ∥Ψ∥^{2}, for all δ > 0 and some C_{δ} ≡ C_{δ}(ℛ, d_{1}, d_{4}) ∈ (0, ∞); see (A.11). Taking all the previous remarks into account, using (A.3)–(A.5), w_{ij} ⩾ 0, |D_{A}^{(i)}| ⩾ P_{A}^{+,(i)} D_{A}^{(i)} P_{A}^{+,(i)}, and writing

H_{f} = \frac{1}{N} ∑_{i=1}^{N} ∑_{k=0}^{K} χ_{k}^{(i)} (P_{A}^{+,(i)} + P_{A}^{−,(i)}) H_{f} χ_{k}^{(i)} ,

we deduce that

H̃_{np}

⩾ (1 − 3δ) P_{A,N}^{+} H_{f} P_{A,N}^{+} + (1 − 3δ) P_{A,N}^{⊥} H_{f} P_{A,N}^{⊥}

+ ∑_{♯∈\{+,⊥\}} ∑_{k=0}^{K} P_{A,N}^{♯} \{ ∑_{i=1}^{N} χ_{k}^{(i)} P_{A}^{+,(i)} (D_{A}^{(i)} − \frac{e^{2} Z_{k}}{|x_{i} − R_{k}|} + \frac{δ}{N} H_{f}) P_{A}^{+,(i)} χ_{k}^{(i)}

+ \frac{δ}{N} ∑_{i=1}^{N} ( χ_{k}^{(i)} P_{A}^{−,(i)} H_{f} P_{A}^{−,(i)} χ_{k}^{(i)} + ∑_{♭=±} χ_{k}^{(i)}P_{A}^{♭,(i)} [P_{A}^{♭,(i)}, H_{f}] χ_{k}^{(i)} ) \} P_{A,N}^{♯}

− const(N, ℛ, d_{1}, d_{4})

on 𝒟_{N}, for every δ > 0. Thanks to Corollary 3.4 (with ε = 0) we know that [P_{A}^{♭,(i)}, H_{f}] Ȟ_{f}^{−1/2} extends to an element of ℒ (ℋ_{N}) whose norm is bounded by some constant depending only on d_{1} and d_{5}, whence

\frac{δ}{N} ∑_{i=1}^{N} ∑_{k=0}^{K}⟨ χ_{k}^{(i)} P_{A,N}^{♯} Ψ | P_{A}^{♭,(i)} [P_{A}^{♭,(i)}, H_{f}] χ_{k}^{(i)} P_{A,N}^{♯} Ψ ⟩

⩾ −(δ/2) ∥Ȟ_{f}^{1/2} P_{A,N}^{♯} Ψ∥^{2} − (δ/2) ∥[P_{A}^{♭,(i)}, H_{f}]Ȟ_{f}^{−1/2}∥^{2} ∥Ψ∥^{2},

for every Ψ ∈ 𝒟_{N}, ♯ ∈ {+, ⊥}, and ♭ = ±. For a sufficiently small choice of δ > 0, the assertion of the proposition now follows from the semi-boundedness of P_{A}^{+,(i)} (D_{A}^{(i)} − e^{2} Z_{k}/|x_{i} − R_{k}| + (δ/N) H_{f}) P_{A}^{+,(i)} ensured by (A.6) and the condition Z_{k} < Z_{np}. □

Lemma A.3. Let ζ ∈ C_{0}^{∞}(ℝ^{3}, [0, 1]), χ ∈ C^{∞}(ℝ^{3}, [0, 1]), such that 0 ∈ supp(ζ) and supp(ζ) ∩ supp(χ) = ∅. Set Ȟ_{f} := H_{f} + E, where E ⩾ k_{1} ∨ d_{1}^{2}. Then

(A.10)  ∥ D_{A} H_{f}^{1/2} ζ P_{A}^{+} χ Ȟ_{f}^{−1/2} ∥ ⩽ C(ζ, χ, d_{1}, d_{4}) ,
(A.11)  ∥ \frac{ζ}{|x|} P_{A}^{+} χ Ȟ_{f}^{−1/2} ∥ ⩽ C′(ζ, χ, d_{1}, d_{4}) .

Proof. We pick some χ̃ ∈ C^{∞}(ℝ^{3}, [0, 1]) such that supp(χ̃) ∩ supp(ζ) = ∅ and χ̃ ≡ 1 on supp(∇χ). Using ζ χ = 0 = ζ χ̃ we infer that, for all φ, ψ ∈ 𝒟,

|⟨ D_{A} φ | H_{f}^{1/2} ζ P_{A}^{+} χ Ȟ_{f}^{−1/2} ψ ⟩| = |⟨ D_{A} φ | H_{f}^{1/2} ζ [P_{A}^{+} , χ] Ȟ_{f}^{−1/2} ψ ⟩|

⩽ ∫_{ℝ} |⟨ D_{A} φ | H_{f}^{1/2} ζ [R_{A}(iy) , χ̃] iα · ∇χ R_{A}(iy) Ȟ_{f}^{−1/2} ψ ⟩|\frac{dy}{2π}

= ∫_{ℝ} |⟨ D_{A} φ | H_{f}^{1/2} ζ R_{A}(iy) iα · ∇χ̃ R_{A}(iy) iα · ∇χ R_{A}(iy) Ȟ_{f}^{−1/2} ψ ⟩|\frac{dy}{2π}

= ∫_{ℝ} |⟨ ζ D_{A} φ | R_{A}(iy) Υ_{0,1/2}(iy) iα · ∇χ̃ R_{A}(iy) Υ_{0,1/2}(iy) ×

× iα · ∇χ R_{A}(iy) Υ_{0,1/2}(iy) ψ ⟩|\frac{dy}{2π} .

In the last step we repeatedly applied (3.16). Commuting ζ and D_{A} and using ∥D_{A} R_{A}(iy)∥ ⩽ 1, ∥R_{A}(iy)∥^{2} ⩽ (1 + y^{2})^{−1}, and the fact that ∥Υ_{0,1/2}(iy)∥ is uniformly bounded in y ∈ ℝ, we readily deduce that

|⟨ D_{A} φ | H_{f}^{1/2} ζ P_{A}^{+} χ Ȟ_{f}^{−1/2} ψ ⟩| ⩽ C(ζ, χ, χ̃, d_{1}, d_{4}) ∥φ∥ ∥ψ∥ ,

which implies (A.10). The bound (A.11) follows from (A.10) and the inequality

∥ |x|^{−1} φ ∥^{2} ⩽ 4 ∥ D_{A} φ ∥^{2} + 4 ∥Ȟ_{f}^{1/2} φ∥ , φ ∈ D(D_{0} ⊗ H_{f}^{1/2}) ,

which is a simple consequence of standard arguments (see, e.g., [10, Equation (4.7)]). □

Acknowledgement. It is a pleasure to thank Martin Könenberg and Edgardo Stockmeyer for interesting discussions and helpful remarks.

References

[1] Zied Ammari, Asymptotic completeness for a renormalized nonrelativistic Hamiltonian in quantum field theory: the Nelson model, Math. Phys. Anal. Geom. 3 (2000), no. 3, 217–285.

[2] Jan Dereziński and Christian Gérard, Spectral scattering theory of spatially cut-off P (ϕ)_{2} Hamiltonians, Comm. Math. Phys. 213 (2000), no. 1, 39–125.

[3] William Desmond Evans, Peter Perry, and Heinz Siedentop, The spectrum of relativistic one-electron atoms according to Bethe and Salpeter, Comm. Math. Phys. 178 (1996), no. 3, 733–746.

[4] Jürg Fröhlich, Marcel Griesemer, and Benjamin Schlein, Asymptotic electromagnetic fields in models of quantum-mechanical matter interacting with the quantized radiation field, Adv. Math. 164 (2001), no. 2, 349–398.

[5] James Glimm and Arthur Jaffe, A λ ϕ^{4} quantum field theory without cutoffs I, Phys. Rev. (2) 176 (1968), 1945–1951.

[6] Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, reprint of the 1980 edition.

[7] Martin Könenberg, Oliver Matte, and Edgardo Stockmeyer, Existence of ground states of hydrogen-like atoms in relativistic quantum electrodynamics I: The semi-relativistic Pauli-Fierz operator, preprint, arXiv:0912.4223, 2009.

[8] Martin Könenberg, Oliver Matte, and Edgardo Stockmeyer, Existence of ground states of hydrogen-like atoms in relativistic quantum electrodynamics II: The no-pair operator, in preparation.

[9] Elliott H. Lieb and Michael Loss, Stability of a model of relativistic quantum electrodynamics, Comm. Math. Phys. 228 (2002), no. 3, 561–588.

[10] Oliver Matte and Edgardo Stockmeyer, Exponential localization of hydrogen-like atoms in relativistic quantum electrodynamics, Comm. Math. Phys., Online First, doi:10.1007/s00220-009-0946-6, 2009.

[11] Lon Rosen, The (ϕ^{2n})_{2} quantum field theory: higher order estimates, Comm. Pure Appl. Math. 24 (1971), 417–457.

Oliver Matte Institut für Mathematik TU Clausthal Erzstraße 1 38678 Clausthal-Zellerfeld Germany On leave from: Mathematisches Institut Ludwig-Maximilians-Universität Theresienatraße 39 80333 München Germany matte@math.lmu.de