HYPERFINE SPLITTING IN NON-RELATIVISTIC QED: UNIQUENESS OF THE DRESSED HYDROGEN ATOM GROUND STATE

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1 HYPERFINE SPLITTING IN NON-RELATIVISTIC QED: UNIQUENESS OF THE DRESSED HYDROGEN ATOM GROUND STATE LAURENT AMOUR AND JÉRÉMY FAUPIN Abstract. We consider a free hydrogen atom composed of a spin- 2 nucleus and a spin- 2 electron in the standard model of non-relativistic QED. We study the Pauli-Fierz Hamiltonian associated with this system at a fixed total momentum. For small enough values of the fine-structure constant, we prove that the ground state is unique. This result reflects the hyperfine structure of the hydrogen atom ground state.. Introduction The structure of the spectrum of the Pauli Hamiltonian describing a non-relativistic Hydrogen atom in Quantum Mechanics is well-known see e.g. [0]. Among many properties, a remarkable one is that the interaction between the spins of the nucleus and the electron causes the so-called hyperfine structure of Hydrogen. In particular, in spite of the spins degrees of freedom, the ground state of the Pauli Hamiltonian is unique; A three-fold degenerate eigenvalue appears besides, close to the ground state energy. This phenomenon justifies the famous observed 2-cm Hydrogen line. Mathematically, using standard perturbation theory of isolated eigenvalues, this statement is not difficult to establish. In the framework of non-relativistic QED, due to the absence of mass of photons, the bottom of the spectrum of the Hamiltonian coincides with the bottom of its essential spectrum. Thus the question of the existence and multiplicity of the ground state is much more subtle. In this paper we consider a moving Hydrogen atom in non-relativistic QED. The total system electron, nucleus and photons is translation invariant, hence one can fix the total momentum and study the corresponding fiber Hamiltonian. For sufficiently small values of the total momentum, the bottom of the spectrum is known to be an eigenvalue [3, 20]. Moreover, under simplifying assumptions, the multiplicity of the ground state eigenvalue is also known: If both the electron and nucleus spins are neglected, the ground state is unique [3]. If the electron spin is taken into account and the nucleus spin is neglected, then the ground state is twice degenerate [2]. Now, following the physical prescription, we assume that both the electron and the nucleus have a spin equal to 2. In [], using a contradiction argument, we proved under this assumption that the multiplicity of the ground state is strictly less than 4. This shows that a hyperfine splitting does occur in non-relativistic QED we refer the reader to the introduction of [] for a more detailed discussion on the hyperfine structure of Hydrogen. Refining our previous analysis, we shall prove in the present paper that the ground state of the dressed Hydrogen atom is unique... Definition of the model and main results. We briefly recall the definition of the Hamiltonian associated with a freely moving hydrogen atom at a fixed total momentum P in non-relativistic QED. For more details, we refer to [, Section 2].

2 2 L. AMOUR AND J. FAUPIN The Hamiltonian associated with a hydrogen atom in the standard model of non-relativistic QED acts on the Hilbert space L 2 R 3 ; C 2 L 2 R 3 ; C 2 H ph C 4 L 2 R 6 H ph, where the photon space, [ H ph := C S n L 2 R 3 ] {, 2} n, n= is the symmetric Fock space over L 2 R 3 {, 2} S n denotes the symmetrization operator. In units such that the Planck constant divided by 2π and the velocity of light are equal to, the Hamiltonian of the system, H, is formally given by the expression H := p el α 2 2 AΛα 2x el + p n + α 2 2 AΛα 2x n + V xel, x n + H ph 2m el 2m n α 2 σ el B 2m Λα 2x el + α 2 σ n B el 2m Λα 2x n, n where α = e 2 is the fine-structure constant with e the charge of the electron, x #, p #, m # denote the positions, momenta and masses of the electron and of the nucleus, and V x el, x n = α x el x n is the Coulomb potential. The 3-uples σ el = σ el, σel 2, σel 3 and σn = σ n, σn 2, σn 3 are the Pauli matrices associated with the spins of the electron and the nucleus, respectively. They can be written as σ el = σ n = C A, σel 2 = C A, σn 2 = i i i i i 0 0 i i 0 0 i 0 C A, σel 3 = C A, σn 3 = The symbol H ph stands for the energy of the free photon field see.6 below, and A Λα 2x, B Λα 2x are the vectors of the quantized electromagnetic field in the Coulomb gauge defined in.2.3 below. An ultraviolet cutoff is imposed, which turns off interactions between the charged particles and photons with energies larger than Λα 2, where Λ is an arbitrary large positive parameter. Choosing the energy unit to be 4 Rydberg, with 4Ry = m el α 2, we are led, after the change of scaling x el, x n, k x el /α, x n /α, α 2 k, where k is the photon wave vector, to study the Hamiltonian H g := 2m el p el ga Λ g 2 3 xel 2 + 2m n x el x n + H ph g σ el B Λ g 2 3 xel + 2m el p n + ga Λ g xn C A, C A. g 2m n σ n B Λ g 2 3 xn. Now the coupling parameter g is given by g := α 3/2. Since H g is translation invariant, it can be decomposed into a direct integral, H g R 3 H g P dp, with respect to the total momentum of the system see e.g. [, 2, 20]. The Hilbert space at a fixed total momentum is H fib := C 4 L 2 R 3 H ph,

3 UNIQUENESS OF THE DRESSED HYDROGEN ATOM GROUND STATE 3 and the Hamiltonian we shall study in this paper, acting on H fib, is given by the expression H g P = 2m el mel M P P ph + p r ga m el M g 2 3 r 2 + 2m n mn M P P ph p r + ga m n M g 2 3 r 2 r + H ph g 2m el σ el B m el M g 2 3 r + g 2m n σ n B m n M g 2 3 r.. Here r is the internal position variable of the hydrogen atom, p r := i r is the associated momentum operator, and M := m el + m n is the total mass of the atom. Now we recall the definitions of the operators on Fock space that we consider. As usual, for any h L 2 R 3 {, 2}, we set a h := hk, λa λ kdk, ah := hk, λaλ kdk, R 3 R 3 λ=,2 λ=,2 and Φh := a h + ah, where the creation and annihilation operators, a λ k and a λk, are operator-valued distributions obeying the canonical commutation relations [a λ k, a λ k ] = [a λ k, a λ k ] = 0, [a λ k, a λ k ] = δ λλ δk k. For x R 3, Ax A Λ x and Bx B Λ x are defined by Ax := χ Λ k [ ] ε λ k e ik x a 2π λ=,2 R 3 k λ k + eik x a λ k dk,.2 2 Bx := i k 2 χλ k k 2π R 3 k ελ k [ ] e ik x a λ k eik x a λ k dk,.3 λ=,2 where the polarization vectors are chosen in the following way: ε k := k 2, k, 0 k 2 + k 2 2, ε 2 k := k k ε k = k k 3, k 2 k 3, k 2 + k2 2 k 2 + k2 2 k 2 + k k2 3 In particular, for j {, 2, 3}, we have A j x = Φh A j x and B jx = Φh B j x, with h A j x, k, λ := 2π h B j x, k, λ := i 2π k 2 χλ k χ Λ k ε λ k j ke ik x,.4 2 k k ελ k e ik x..5 j In.2 and.3, χ Λ k denotes an ultraviolet cutoff function which is chosen, for simplicity, as χ Λ k := k Λ k, with Λ > 0. The Hamiltonian and total momentum of the free photon field, H ph and P ph, are defined by H ph := k a λ ka λkdk, P ph := ka λ ka λkdk..6 R 3 R 3 λ=,2 The Fock vacuum is denoted by Ω. Our main result is the following. λ=,2

4 4 L. AMOUR AND J. FAUPIN Theorem.. There exist g c > 0 and p c > 0 such that, for all 0 < g g c and 0 P p c, H g P has a unique ground state, that is Remarks.2. E g P := inf spech g P is a simple eigenvalue of H g P. For convenience, we shall work in the sequel with the Hamiltonian :H g P : obtained from the expression. by Wick ordering. Since :H g P : and H g P only differ by a constant, the statement of Theorem. is equivalent if we replace :H g P : by H g P. From now on, to keep notations simple, we use H g P to designate the Wick-ordered Hamiltonian. 2 With some more work, Theorem. may be proven with the critical value p c = M ε, ε > 0, and g c depending on ε. However, for large values of the total momentum, P > M, due to Cerenkov radiation, one expects that E g P is not an eigenvalue..2. Notations and strategy of the proof. Now we describe the strategy of our proof and introduce corresponding notations. Our main tools will be a suitable infrared decomposition of Fock space combined with iterative perturbation theory. The introduction of an infrared cutoff into the interaction Hamiltonian is a standard step in the analysis of models of nonrelativistic QED []. The idea of considering a sequence of Hamiltonians with decreasing infrared cutoffs and comparing them iteratively by perturbation theory can be traced back to [22]. It was later used successfully in different contexts [5, 9, 2, 8]. Roughly speaking, the method employed in these papers to prove the existence of a unique ground state is as follows: Let H denote the Hamiltonian of the model, and H σ denote the Hamiltonian with an infrared cutoff of parameter σ, acting on the Fock space of particles of energies σ. For large σ s, there is no interaction in H σ and it is easy to verify that H σ has a unique ground state separated by a gap of order Oσ from the rest of the spectrum. Next, using perturbation theory, one shows that, if for some given σ > 0, H σ fulfills this gap property uniqueness of the ground state and gap of order Oσ above it, then the same holds for H σ/2. Proceeding iteratively, one thus obtains the existence of a unique ground state, Φ σ, for any σ > 0. To prove that a ground state persists as the infrared cutoff is removed, one considers the weak limit, Φ := w- lim Φ σj, along some subsequence σ j 0. It is then easy to see that HΦ = EΦ, with E = inf spech, and hence it remains to verify that Φ 0. To this end, using of a pull-through argument, one shows that the expectation in the number of photons, Φ, N Φ, is small assuming that the coupling constant is small. This implies that Φ overlaps with the unperturbed ground state and hence, in particular, that Φ 0. In all the previously cited papers, the ground state of the non-interacting Hamiltonian is unique. In our context, however, it is 4-fold degenerate, so that the method does not directly apply. Let us be more precise. For σ > 0, the infrared fibered Hamiltonian acts on H fib and is defined as H g, P = 2m el : mel + 2m n : M P P ph + p r ga m el M g r : mn M P P ph p r + ga m n r + H ph g 2m el σ el B m el M g 2 3 r + M g r : g σ n B m n 2m n M g 2 3 r,.7

5 UNIQUENESS OF THE DRESSED HYDROGEN ATOM GROUND STATE 5 where, for any j {, 2, 3}, x R 3, k R 3, λ {, 2} and σ 0, A j, x := Φh A j,x with h A j,x, k, λ := k kh A j x, k, λ, B j, x := Φh B j,x with h B j,x, k, λ := k kh B j x, k, λ. The expression.7 is Wick ordered in accordance with Remark.2. For σ > 0, let [ H ph, := C S n L 2 {k, λ R 3 ] {, 2}, k σ} n, H ph, σ := C n= [ S n L 2 {k, λ R 3 ] {, 2}, k σ} n, n= denote the Fock spaces for photons of energies σ, respectively of energies σ. It is wellknown that there exists a unitary transformation mapping H ph to H ph, H ph, σ. H fib, := C 4 L 2 R 3, dr H ph,. Clearly, H fib, identifies with a subset of H fib, and H g, P : H fib, DH g, P H fib,. The restriction of H g, P to H fib, DH g, P is then denoted by In order to avoid any confusion, we also set K g, P := H g, P Hfib, DH g, P. H ph, := H ph Hph, DH ph, P ph, := P ph Hph, DP ph, and the vacuum in H ph, is denoted by Ω. We shall use the decomposition where and K g, P = K 0, P + W g, P, K 0, P :=H 0 P DH0 P H fib, = H r + 2M P P ph, 2 + H ph,,.8 W g, P = g m el m el M P P ph, + p r A m el M g 2 3 r + g m n m n M P P ph, p r A m n M g 2 3 r + g2 In.8, H r denotes the Schrödinger Hamiltonian g2 :A m el 2m el M g 2 3 r 2 : + :A m n 2m n M g 2 3 r 2 : g σ el B m el 2m el M g 2 g 3 r + σ n B m n 2m n M g 2 3 r..9 H r := p2 r 2µ r, where µ is the reduced mass, µ := m m 2 /m + m 2. For any self-adjoint and semi-bounded operator H, we set EH := inf spech and GapH := infspech \ {EH} EH.

6 6 L. AMOUR AND J. FAUPIN We then observe that, for all σ 0, EK 0, P = EH r + P 2 2M =: e 0 + P 2 2M =: E 0P, and that E 0 P is 4-fold degenerate see Lemma A. in the appendix. The lowest eigenvalue of H r is given by e 0 = µ/2. The projection onto the vector space associated with E 0 P is denoted by Π 0, P := {E0 P } K0, P, and Π0, P := Hfib, Π 0,P. Note that Π 0, P is independent of P. Setting Π 0, := Π 0, P, Π 0, := Π 0, P, we have Π 0, = C 4 π 0 Π Ω, where π 0 is the projection onto the vector space associated with the ground state φ 0 of H r, and Π Ω is the projection onto the vacuum sector in H ph,. Moreover, we set E g, P := EK g, P. As mention above, we will analyze the bottom of the spectrum of K g, P iteratively, by letting the infrared cutoff parameter σ 0. Of course, for σ Λ, we have that K g, P = K 0, P and hence the spectrum of K g, P is explicit: It is composed of the 4-fold degenerate eigenvalue E g, P = E 0 P and a semi-axis of absolutely continuous spectrum [E g, P + Cσ, for some positive constant C see Figure. E g, P.. Cσ Figure. Spectrum of K g, P for σ Λ Compared to previous works, the main substantial difficulty we encounter comes from the fact that, as σ becomes strictly less that Λ, E g, P splits into 4 generally distinct eigenvalues. Therefore, in particular, for σ such that g 2 σ < Λ, the gap above E g, P in the spectrum of K g, P, GapK g, P, is negligible compared to σ. To overcome this difficulty, we have to start with analyzing the Hamiltonian K g, P for σ = Cg 2 with C a suitably chosen positive constant. In other words we choose an initial infrared cutoff of order α 3 Ry. Since for these values of the parameters, the perturbation is of the same order as the distance between the ground state and the essential spectrum, one cannot straightforwardly apply usual perturbation theory. We then do second order perturbation theory with the help of the Feshbach-Schur map. The Feshbach-Schur map is a natural tool to study second order perturbation of possibly embedded eigenvalues of self-adjoint operators. In the context of non-relativistic QED, it was introduced in [6] and further developed in [4, 3]. The Feshbach-Schur operator we consider

7 UNIQUENESS OF THE DRESSED HYDROGEN ATOM GROUND STATE 7 here is associated with K g, P E g, P and Π 0, and is defined by F g, P :=E 0 P E g, P Π 0, Π 0, W g, P [ K0, P E g, P + Π 0, W g, P Π 0, ] Π0, W g, P Π 0,..0 We shall see in Section 2 that this operator is well-defined for suitable values of the parameters. The main property of F g, P that we shall use is that K g, P E g, P F g, P, see Lemma 2.2. Combined with the min-max principle, this operator inequality appears to be very useful in the context of the present paper. In particular, it will allow us to prove that the spectrum of K g, P has the form pictured in Figure 2 see Section 2: The bottom of the spectrum, E g, P, is an isolated eigenvalue separated by a gap of size δg 2 from three other eigenvalues and a semi-axis of essential spectrum. E g, P δg 2 Figure 2. Spectrum of K g, P for σ = Cg 2, C The key argument that allows the method to be applied is the decomposition of the Feshbach-Schur operator F g, P, viewed as a 4 4 matrix, into three parts, F g, P = dg + g 2 Γ + Remg, where dg is a scalar matrix and Γ is a matrix independent of g whose spectrum exhibits a gap between its first and second eigenvalues. The remainder term, Remg, must be negligible compared to g 2. The rest of the proof borrows ideas from [22, 5, 2]. Namely, we shall prove in Section 3 that if the ground state of K g, P is unique and if GapK g, P ησ, then the same holds for K g, τ P with τ = κσ, 0 < κ <. This will show that for small σ s, the spectrum of K g, P has the form pictured in Figure 3 a non-degenerate eigenvalue separated from a gap of size ησ from the rest of the spectrum. At the end, the parameters η and κ will have to be carefully chosen, in relation with the initial analysis of Section 2. E g, P.. ησ Figure 3. Spectrum of K g, P for σ C g 2, C We conclude this section with introducing a few more notations related to the infrared decomposition, which will be useful in Section 3. For 0 τ σ, let H σ ph, τ := C [ S n L 2 {k, λ R 3 ] {, 2}, τ k σ} n. n=

8 8 L. AMOUR AND J. FAUPIN The vacuum in H σ ph, τ is denoted by Ω σ τ and the projection onto the vacuum sector is denoted by Π Ω σ. The Hilbert spaces H fib, τ and H fib, H σ ph, τ τ sometimes not distinguish between the two of them. As an operator on H fib, τ, we set are isomorphic. We shall W σ g, τ P = g m el : m el M P P ph, τ + p r ga m el M g 2 3 r A σ τ m el M g 2 3 r : + g m n : m n M P P ph, τ p r + ga m n M g 2 3 r A σ τ m n M g 2 3 r : + g2 τ m el :A σ 2m el M g 2 3 r 2 : + g σ el B σ τ 2m m el el M g 2 3 r + g2 2m n :A σ τ m n M g 2 3 r 2 : g σ n B σ τ 2m m n n M g 2 3 r,. where A σ τ and B σ τ are given by the same expressions as A and B respectively, except that the integrals are taken over {k R 3, τ k σ}. Note that H g, τ P = H g, P + W σ g, τ P. Finally, in the case where τ = 0, the subindex 0 is removed from the notations above, that is, for instance, H σ ph := H σ ph, 0, W g σ P := W σ g, 0 P and so on. Throughout the paper, the notation will stand for C where C is a positive constant independent of the parameters. For any vector v, [v] and [v] will denote respectively the subspace spanned by v and its orthogonal complement. 2. Existence of a gap for large enough infrared cutoffs In this section, we investigate the spectrum of the infrared cutoff Hamiltonian K g, P for values of the coupling constant g and of the infrared cutoff parameter σ fixed such that βc2 g2 σ βc g2 for some 0 < β c < β c2 to be determined later. For any small enough g and P, and for any σ, η > 0, let Gapg, P, σ, η denote the following assertion: { i Eg, P is a simple eigenvalue of K g, P, Gapg, P, σ, η ii GapK g, P ησ. The main result of this section is the following. Theorem 2.. There exist p c > 0, β c2 > 0 and δ > 0 such that, for all 0 < β c < β c2, there exists g c > 0 such that, for all 0 P p c, 0 < g g c and β c β β c2, where σ = g 2 β. Gapg, P, σ, δβ holds, The statement of Theorem 2. expresses the fact that, for small enough values of the coupling constant g and total momentum P, there is a gap at least of size δg 2 in the spectrum of K g, P above the non-degenerate ground state eigenvalue E g, P, provided that the infrared cutoff parameter σ obeys βc2 g2 σ βc g2.

9 UNIQUENESS OF THE DRESSED HYDROGEN ATOM GROUND STATE Preliminary lemmas. The proof of Theorem 2. relies on a few lemmas that we shall establish in this preliminary subsection. For the convenience of the reader, some standard estimates used several times in the proofs below are recalled in Appendix A. We begin with verifying that the Feshbach-Schur operator F g, P introduced in.0 is well-defined for suitable values of the parameters. Lemma 2.2. There exist p c > 0 and β c2 > 0 such that, for all 0 < β c < β c2, there exists g c > 0 such that, for all 0 < g g c, 0 P p c and β c β β c2, the Feshbach- Schur operator defined in.0, F g, P where σ = g 2 β, is a bounded operator on RanΠ 0, H fib given by F g, P = E 0 P E g, P Π 0, Π 0, W g, P [ K 0, P E g, P ] Π0, W g, P Π 0, + Rem g, P, β, 2. where Rem g, P, β is a bounded operator on RanΠ 0, satisfying Rem g, P, β g 2 β Moreover the following inequality holds in the sense of quadratic forms on DK g, P : K g, P E g, P F g, P. 2.3 Proof. Let g c, p c, σ c and C W be given by Lemma A.7. Let β c2 be such that β /2 c2 6C W, and let 0 < β c < β c2. Possibly by considering a smaller g c, we can assume that g 2 c β c σ c and hence, for all 0 < g g c and β β c, we have that 0 < σ = g 2 β σ c. In addition we impose that p c M/2. Fix g, P and β as in the statement of the lemma. By Lemmas A. and A.5, we have that K 0, P E g, P is bounded invertible on Ran Π 0, and satisfies K0, P E g, P Π0, E 0 P E g, P + P M σ Π0, σ 2 Π 0,. Using again that E g, P E 0 P by Lemma A.5, it then follows from a straightforward application of the Spectral Theorem that [ K 0, P E g, P ] Π0, K0, P E 0 P + σ [ K 0, P E g, P ] Π0, K0, P E g, P + σ Moreover, by Lemma A.7, we have that [ K 0, P E 0 P + σ ] 2 W g, P [ K 0, P E 0 P + σ ] 2 C W gσ 2 = CW β 2, 2.5 and hence [ K 0, P E g, P ] 2 Π0, W g, P [ K 0, P E g, P ] 2 Π0, 3CW β 2.

10 0 L. AMOUR AND J. FAUPIN Taking into account the choice of β c2, we have 3C W β 2 /2 so that in particular K 0, P E g, P Π 0, + Π 0, W g, P Π 0, 2 K 0,P E g, P Π 0, σ 4 Π 0,. 2.6 Therefore the operator K 0, P E g, P Π 0, + Π 0, W g, P Π 0, is bounded invertible on Ran Π 0,. Using in addition that W g, P is relatively bounded with respect to K 0, P, we obtain that F g, P is indeed a well-defined bounded operator on RanΠ 0,. Next, using again 2.4 and 2.5, we obtain Π 0, W g, P [ K 0, P E g, P ] 2 Π0, 2CW β 2 σ 2 = 2CW g. 2.7 A standard Neumann series decomposition together with the previous estimates then lead to Π 0, W g, P [ K 0, P E g, P + Π 0, W g, P Π 0, ] Π0, W g, P Π 0, = Π 0, W g, P [ K 0, P E g, P ] Π0, W g, P Π 0, + Rem g, P, β, where Rem g, P, β is a bounded operator on RanΠ 0, satisfying 2.2. Finally, to prove 2.3, it suffices to use the following identity where K g, P E g, P = F g, P + R R, 2.8 R := [ K 0, P E g, P + Π 0, W g, P Π 0, ] 2 Π0, K g, P E g, P. We observe that the operator square root appearing in the expression of R is well-defined by 2.6. Equation 2.8 follows from straightforward algebraic computations see e.g. [4, 3]. This concludes the proof of the lemma. Our next task is to extract the second order term from 2.. following three lemmas. It is the purpose of the Lemma 2.3. There exist g c > 0, p c > 0 and σ c > 0 such that, for all 0 g g c, 0 P p c and 0 < σ σ c, Π 0, W g, P [ K 0, P E g, P ] Π0, W g, P Π 0, = Π 0, w r, k, λ [ H r + R 3 2M P k2 + k E g, P ] w r, k, λπ 0, dk where λ=,2 + Rem 2 g, P, σ, w r, k, λ := g m el m el M P P ph, + p r h A m el M g 2 3 r, k, λ g m n m n M P P ph, p r h A m n M g 2 3 r, k, λ g σ el h B 2m m el el M g 2 g 3 r, k, λ + σ n h B 2m m n n M g 2 3 r, k, λ, 2.0 w r, k, λ is given by the same expression except that h A, hb are replaced by their conjugate h A, h B, and Rem 2g, P, σ is a bounded operator on RanΠ 0, satisfying Rem 2 g, P, σ g 3.

11 UNIQUENESS OF THE DRESSED HYDROGEN ATOM GROUND STATE Proof. It suffices to introduce the expression.9 of W g, P into the operator Π 0, W g, P [ K 0, P E g, P ] Π0, W g, P Π 0,, and next to estimate each term separately. An explicit computation then leads directly to the statement of the lemma see the proof of Lemma A.9 in [] for more details. Lemma 2.4. There exist g c > 0, p c > 0 and σ c > 0 such that, for all 0 g g c, 0 P p c, 0 < σ σ c and λ {, 2}, Π 0, w r, k, λ [ H r + R 3 2M P k2 + k E g, P ] w r, k, λπ 0, dk = R 3 Π 0, w 0, k, λ [ H r + 2M P k2 + k E g, P ] w 0, k, λπ 0, dk + Rem 3 g, P, σ, λ, where w 0, k, λ and w 0, k, λ are defined by 2.0, and Rem 3 g, P, σ, λ is a bounded operator on RanΠ 0, satisfying Rem 3 g, P, σ, λ g 8 3. Proof. It follows from the definitions.4.5 of h A j and h B j that h A j, r, k, λ h A j,0, k, λ k k k 2 χλ k r, h B j,r, k, λ h B j,0, k, λ k k k 3 2 χλ k r, for any j {, 2, 3}, λ {, 2}, r R 3 and k R 3. This implies that m el Π 0, M P P ph, + p r h A m el M g 2 3 r, k, λ h A 0, k, λ g 2 3 k k k m el 2 χλ k Π 0, M P P ph, + p r r g 2 3 k k k 2 χλ k. In the last inequality, we used in particular that r p r π 0 <, where, recall, π 0 is the projection onto the ground state of the Schrödinger operator H r. Similarly, Π 0, σ el h B m el M g 2 3 r, k, λ h B 0, k, λ g 2 3 k k k 3 2 χλ k Π 0, σ el r g 2 3 k k k 3 2 χλ k. The same holds if m el is replaced by m n and σ el is replaced by σ n. Besides, using that H r + P 2 /2M E 0 P E g, P by Lemma A.5, we obtain that [ H r + 2M P k2 + k E g, P ] k k k P /M + k 2 /2M + k k k 2 k k k, for P M/2, and hence the statement of the lemma easily follows.

12 2 L. AMOUR AND J. FAUPIN Lemma 2.5. There exist g c > 0, p c > 0 and σ c > 0 such that, for all 0 g g c, 0 P p c, 0 < σ σ c, and λ {, 2}, Π 0, w 0, k, λ [ H r + R 3 2M P k2 + k E g, P ] w 0, k, λπ 0, dk where and = g 2 Γ A,diag P, λ + Γ B P, λ + Rem 4 g, P, σ, λ, Γ A,diag P, λ := µ 2 Π 0, p r h A 0, k, λ [ H r + R 3 2M P k2 + k E 0 P ] Γ B P, λ := Π 0, R 3 C 4 π 0 Hph p r h A 0, k, λπ 0, dk, σ n 2m h B 0, k, λ n σ el 2m h B 0, k, λ + el [ e0 + 2M P k2 + k E 0 P ] σ el h B 2m 0, k, λ + σ n h B el 2m 0, k, λ n Moreover Rem 4 g, P, σ, λ is a bounded operator on RanΠ 0, satisfying Proof. It follows from 2.0 that w 0, k, λ = g µ p r h A 0, k, λ Rem 4 g, P, σ, λ g 4. Π 0, dk. g σ el h B 2m 0, k, λ + g σ n h B el 2m 0, k, λ, n and likewise for w 0, k, λ except that h A, hb are replaced by h A, h B. Observe in particular that the terms proportional to P P ph vanish. This is due to the fact that the charge of the total system we consider vanishes. Moreover, we have that h A 0, k, λ = ha 0, k, λ and This yields where Π 0, p r h A 0, k, λ C 4 π 0 Hph = 0. R 3 Π 0, w 0, k, λ [ H r + 2M P k2 + k E g, P ] w 0, k, λπ 0, dk = g 2 ΓA,diag g, P, λ + Γ B g, P, λ, Γ A,diag g, P, λ := µ 2 Π 0, p r h A 0, k, λ [ H r + R 3 2M P k2 + k E g, P ] C 4 π 0 Hph p r h A 0, k, λπ 0, dk,

13 and UNIQUENESS OF THE DRESSED HYDROGEN ATOM GROUND STATE 3 Γ B g, P, λ := Π 0, R 3 σ el 2m h B 0, k, λ + el [ e0 + 2M P k2 + k E g, P ] σ n 2m h B 0, k, λ n 2m el σ el h B 0, k, λ + 2m n σ n h B 0, k, λ For any P M/2 and k σ, we have that [ H r + 2M P k2 + k E 0 P ] C 4 π 0 Hph and hence also that [ H r + 2M P k2 + k E g, P ] C 4 π 0 Hph Π 0, dk. e e 0, e e 0, by Lemma A.5. Therefore, using the first resolvent equation together with the facts that E 0 P E g, P g 2 see Lemma A.5 and h A 0, k, λ k k k /2 χ Λ k, we get where Γ A,diag g, P, λ = Γ A,diag P, λ + Rem A g, P, σ, Rem A g, P, σ g 2 Likewise, for any P M/2 and k σ, we have that R 3 σ k Λ dk k g2. [ e 0 + 2M P k2 + k E g, P ] 2 k, and since h B 0, k, λ k k k /2 χ Λ k, we thus obtain that Γ B g, P, λ = Γ B P, λ + Rem B g, P, σ, where Hence the lemma is proven. Rem B g, P, σ g 2 R 3 σ k Λ dk k g2. To conclude this subsection, we estimate the size of the splitting induced by the second order term in 2.2. Since the matrix Γ A,diag P, λ of Lemma 2.5 is scalar, only Γ B P, λ is responsible for this splitting. From now on, to simplify a few computations and since the system is rotation invariant, we choose the total momentum P to be directed along ɛ 3. Lemma 2.6. There exist p c > 0 and σ c > 0 such that, for all P = P 3 ɛ3 satisfying 0 P p c, for all 0 < σ σ c, and λ {, 2}, Γ B P, λ = Γ B,diag P, λ + Γ B,# P, λ,

14 4 L. AMOUR AND J. FAUPIN where Γ B,diag and P, λ is the scalar operator on Ran Π 0, given by Γ B,diag P, λ := 4m 2 + el 4m 2 n R 3 Γ B,# P, λ := 2m el m n j=,2,3 R 3 h B 0, k, λ 2 dk k k P /M + k 2 /2M Π 0,, h B j, 0, k, λ 2 dk k k P /M + k 2 /2M Π 0,σ el j σ n j Π 0,. Proof. Using standard properties of the Pauli matrices see Lemma A.0, since for any k R 3 and λ {, 2}, h B 0, k, λ = hb 0, k, λ, we have that σ el h B 0, k, λ σ el h B 0, k, λ = h B 0, k, λ 2 = h B 0, k, λ 2, and likewise with σ n replacing σ el. Next, we observe that for P = P 3 ɛ3, λ {, 2} and j, j {, 2, 3}, j j, h B j, 0, k, λhb j,0, k, λ k k P /M + k 2 dk = 0. /2M R 3 The lemma then follows straightforwardly from the expression of Γ B P, λ given in the statement of Lemma 2.5. Lemma 2.7. There exist p c > 0 and σ c > 0 such that, for all P = P 3 ɛ3 satisfying 0 P p c and for all 0 < σ σ c, the eigenvalues of the operator are given by γ 0 P := 8m el m n π 2 γ j P := 8m el m n π 2 Γ B,# P := λ=,2 R 3 R 3 Γ B,# P, λ k σ k Λk k k P /M + k 2 /2M dk, k σ k Λk k k P /M + k 2 /2M kj 2 dk, j =, 2, 3. k 2 Proof. It directly follows from the properties of the Pauli matrices see Lemma A.. Remarks 2.8. For P = 0, we observe that γ 0 = γ2 0 = γ3 0. It may however not be the case for P 0. 2 The gap above the lowest eigenvalue γ 0 P is non-vanishing. More precisely, letting δ P := min γ P, γ2 P, γ3 P γ 0 P, and we have δ δ := 8m el m n π 2 R 3 inf δ P, 0 P p c,0 σ Λ/2 k Λ/2 k Λk + p c /M k + k 2 dk > 0. /2M

15 UNIQUENESS OF THE DRESSED HYDROGEN ATOM GROUND STATE Proof of Theorem 2.. Proof of Theorem 2.. Let p c be fixed as the minimum of the p c s given by Lemmas and A.5, and let C W be given by Lemma A.7. Let σ c = β c2 = ε where ε > 0 is a small, fixed parameter smaller, in particular, than the minimum of the σ c s given by Lemmas and than the β c2 s given by Lemma 2.2. Let now 0 < β c < β c2 and let g c be fixed smaller than the minimum of the g c s given by Lemmas We recall in particular from the proofs of Lemma 2.2 that gc 2 β c σ c, which implies that for all 0 < g g c and β β c, we have that 0 < σ g,β σ c, with σ g,β = g 2 β. Let 0 P p c, 0 < g g c, β c β β c2. By rotation invariance, we can assume that P = P 3 ɛ3. Before starting the proof we introduce a few more notations to simplify expressions. Combining Lemmas , we can write F g, P =E 0 P E g, P Π 0, + g 2 d P Π 0, + g 2 Γ B,# P + Remg, P, β, where d P is the scalar bounded operator on RanΠ 0, defined by d P := Γ A,diag P, λ + Γ B,diag P, λ, λ=,2 and Remg, P, β is a bounded operator on RanΠ 0, satisfying Remg, P, β g 2 β 2 + g + 2g g 2 g 2 ε Here we used that, by assumption, β and g are smaller than ε. We also introduce the operator F 2 :=F g, P + E g, P Π 0, Remg, P, β = E 0 P + g 2 d P Π 0, + g 2 Γ B,# P. Identifying d P with a scalar quantity, the lowest eigenvalue of F 2 is, according to Lemma 2.7, given by e 0 P := E 0P + g 2 d P + γ 0 P. Moreover, it follows again from Lemma 2.7 that e 0 P is simple and that GapF 2 g 2 δ P g 2 δ, 2.2 on RanΠ 0,, where δ P and δ > 0 are given by Remark Let φ 0 P RanΠ 0, denote a normalized eigenstate associated with the eigenvalue e 0 P of F 2. Step Let us prove that Let ψ 0 P be the following trial state: e 0 P E g,p g2 δ ψ 0 P := φ0 P [ K 0, P E g, P + Π ] 0, W g, P Π 0, Π 0, W g, P φ 0 P.

16 6 L. AMOUR AND J. FAUPIN We observe that ψ 0 P, φ0 P = φ0 P, φ0 P =. Proceeding as in the proof of Lemma 2.2, using 2.6 and 2.7, we find that ψ 0 P φ0 P 4 2C W β 2 ε 2, 2.4 where we used that β /2 ε /2. Moreover, using the properties of the Feshbach operator, we obtain that and Therefore we deduce that Π 0, Kg, P E g, P ψ 0 P = F g, P φ 0 P = e 0 P E g,p φ 0 P + Remg, P, βφ0 P, Π 0, Kg, P E g, P ψ 0 P = Π 0, W g, P φ 0 P Π 0, W g, P φ 0 P = 0. 0 ψ 0 P, K g, P E g, P ψ 0 P = e 0 P E g,p + ψ 0 P, Remg, P, βφ0 P e 0 P E g,p + ψ 0 P Remg, P, β e 0 P E g,p + Cg 2 ε 2, where in the last inequality we used 2.4 and 2.. Hence, choosing ε /2 δ, 2.3 is proven. Step 2 Let µ 2 denote the second point above E g, P in the spectrum of K g, P. The min-max principle implies that µ 2 inf ψ DK g, P, ψ =, ψ [φ 0 P ] ψ, K g, P ψ. Now, for ψ as above, using inequality 2.3 of Lemma 2.2, we can write ψ, K g, P ψ ψ, F g, P ψ + E g, P ψ, F 2 ψ + ψ, Remg, P, βψ. Since F 2 Π0, = 0 and since ψ [φ 0 P ], it follows from 2.2 that ψ, F 2 ψ e 0 P + g2 δ Π 0, ψ 2 e 0 P + g2 δ. In the last inequality, we used that E 0 P = P 2 /2M + e 0 = P 2 /2M µ/2 is negative for P p c small enough, and hence that e 0 P + g2 δ is also negative for g and P small enough. Moreover, 2. yields ψ, Remg, P, βψ Cg 2 ε 2.

17 UNIQUENESS OF THE DRESSED HYDROGEN ATOM GROUND STATE 7 Thus, applying Step, we obtain that µ 2 E g, P g2 δ Cg 2 ε 2 Eg, P + 2 g2 δ, provided ε is small enough. Changing notations δ/2 δ, this implies the statement of the theorem. 3. Proof of Theorem. The main theorem of this section, which will allow us to prove Theorem., is the following. Theorem 3.. There exist g c > 0, p c > 0 and 0 < η /4 such that, for all 0 < g g c and 0 P p c, there exist σ c > 0 such that for all 0 < σ σ c, Gapg, P, σ, η holds. We need the following two lemmas before starting the proof of Theorem 3.. Lemma 3.2. Let g c > 0, σ c > 0 and p c > 0 be fixed small enough and let 0 g g c, 0 σ σ c and 0 P p c be such that Gapg, P, σ, η holds. Let Φ g, P be a normalized ground state of K g, P. For all 0 τ σ, E g, P is a simple eigenvalue of H g, P Hfib, τ associated with the normalized ground state Φ g, P Ω σ τ, and GapH g, P Hfib, τ minησ, τ 4. Proof. First, one can readily verifies that Φ g, P Ω σ τ is an eigenstate of H g,p Hfib, τ associated with the eigenvalue E g, P. Since and since H σ Π ph, τ Ω σ τ Hfib, τ Π g, P Π Ω σ τ = Hfib, Π g,p Π Ω σ τ + Hfib, H σ Π ph, τ Ω σ, 3. τ commutes with H g, P Hfib, τ, we have that inf ψ DH g, P Hfib, τ, ψ =, ψ [Φ g, P Ω σ min τ ] inf ψ DH g, P Hfib, τ, ψ =, ψ [Φ g, P ] [Ω σ τ ] inf ψ DH g, P Hfib, τ, ψ =, ψ H fib, [Ω σ τ ] The assumption that GapK g, P ησ yields that inf ψ DH g, P Hfib, τ, ψ =, ψ [Φ g, P ] [Ω σ τ ] ψ, H g, P Hfib, τ ψ ψ, H g, P Hfib, τ ψ, ψ, H g, P Hfib, τ ψ. 3.2 ψ, H g, P Hfib, τ ψ E g, P + ησ. To estimate from below the second term in the right-hand-side of 3.2, we use the fact that H g, P Hfib, τ commutes with the number operator Hfib, N σ ph, τ defined in A.4.

18 8 L. AMOUR AND J. FAUPIN We then consider ψ DH g, P Hfib, τ such that ψ = and Hfib, N σ ph, τ ψ = nψ with n. This state ψ may be seen as an element of L 2 {k, λ, τ k σ}; H fib, L 2 {k, λ, τ k σ} n, and we have that ψ, H g, P Hfib, τ ψ = ψk, λ, Hg, P k Hfib, L 2 {k,λ,τ k σ} n + k ψk, λ dk λ=,2 inf τ k σ τ k σ Eg, P k + k E g, P + τ 4, where the last inequality is a consequence of Lemma A.2. Therefore we obtain that inf ψ DH g, P Hfib, τ, ψ =, ψ H fib, [Ω σ τ ] which concludes the proof of the lemma. ψ, H g, P Hfib, τ ψ E g, P + τ 4, Lemma 3.3. There exists p c > 0 such that, for all η and κ such that 0 < 4η κ <, there exists g c > 0 such that, for all 0 g g c, 0 P p c and σ > 0 Gapg, P, σ, η Gapg, P, κσ, η. Proof. Assume that Gapg, P, σ, η is satisfied for some σ > 0. Let Φ g, P be a ground state of K g, P. Let τ = κσ and let µ 2 denote the first point above E g, τ P in the spectrum of K g, τ P. By the min-max principle, µ 2 inf ψ DK g, τ P, ψ = ψ [Φ g, P Ω σ τ ] ψ, K g, τ P ψ, where [Φ g, P Ω σ τ ] denotes the orthogonal complement of the vector space spanned by Φ g, P Ω σ τ in H fib, H σ ph, τ H fib, τ. It follows from Lemma A.8 that for any ρ > 0, ψ, K g, τ P ψ = ψ, H g, P Hfib, τ ψ + ψ, W σ g, τ P ψ [ C W gσ /2 ρ /2] ψ, H g, P Hfib, τ ψ + C W gσ /2 ρ /2 E g, P C W gσ /2 ρ /2. Next, by Gapg, P, σ, η, Lemma 3.2 and the fact that τ = κσ 4ησ, we obtain that for any ψ in [Φ g, P Ω σ τ ], ψ =, ψ, H g, P Hfib, τ ψ E g, P + ησ, provided that g is sufficiently small. Hence for any ρ > 0 such that ρ /2 > C W gσ /2, [ ψ, K g, τ P ψ E g, P + C W gσ /2 ρ /2] ησ C W gσ /2 ρ /2. Choosing ρ /2 = 2 κ C W gσ /2 and using Lemma A.6, we get ψ, K g, τ P ψ E g, P + + κ ησ 2 κ C 2 W g 2 σ 2 E g, τ P + ητ + κ ησ 2 κ C 2 W g 2 σ. 2

19 UNIQUENESS OF THE DRESSED HYDROGEN ATOM GROUND STATE 9 Hence µ 2 E g, τ P +ητ provided that g 2 4C 2 W κ 2 η, which proves the lemma. Let us now prove Theorem 3.. Proof of Theorem 3.. Let p c be the minimum of the p c s given by Theorem 2. and Lemma 3.3. Let β c2 and δ be given by Theorem 2.. Note that, possibly by considering a smaller β c2, we can assume that 4δ < βc2. Fix β c such that 0 < β c < β c2. Let η := δβ c and κ := β c βc2. In particular we have that 0 < 4η κ <. Let g c be the minimum of the g c s given by Theorem 2. and Lemma 3.3. Let 0 < g g c and 0 P p c. By rotation invariance, we can assume that P = P 3 ɛ3. Let σ c := g 2 βc. By Theorem 2., we know that, for all σ such that g2 βc2 σ g2 βc, Gapg, P, σ, η holds. Using Lemma 3.3, this implies that, for all σ such that g 2 βc2 σ g 2 βc and for all n N 0, Gapg, P, κ n σ, η holds. Since κ = β c βc2, we deduce that Gapg, P, σ, η is satisfied for all 0 < σ g 2 βc, which concludes the proof of the theorem. We are now able to prove the main theorem of this paper. Proof of Theorem.. Let g c > 0, p c > 0 and 0 < η /4 be given by Theorem 3. and let C W be given by Lemma A.8. Let 0 < g g c and 0 P p c. By rotation invariance, we can assume without loss of generality that P = P 3 ɛ3. Let σ c > 0 be given by Theorem 3.. For 0 < σ σ c, let Φ g, P denote a normalized ground state of K g, P. Recall from Lemma 3.2 that Φ g, P Ω σ is then a normalized ground state of H g, P. By the Banach-Alaoglu theorem, there exists a sequence Φ g,n P Ω σn n N which converges weakly as n, with 0 < σ n minσ c, and σ n 0. Let Φ g P denote the corresponding limit. One easily verifies that H g P Φ g P = E g P Φ g P. To prove that Φ g P 0, we decompose = Φ g,n P Ω σn, Φ g,n P Ω σn = Φ g,n P Ω σn, C 4 π 0 Π Ωn Φ g,n P Ω σn + Φ g,n P Ω σn, C 4 π 0 Π Ωn Φ g,n P Ω σn + Φ g,n P Ω σn, C 4 L 2 R 3 Π Ωn Φ g,n P Ω σn. By Lemma A.9, the second term in the r.h.s. is bounded by Og 2. Likewise, since Π Ωn N ph,n ΠΩn, Lemma A.3 implies that the last term in the r.h.s. is also bounded by Og 2. Letting α j, j =,..., 4, denote an orthonormal basis of C 4, and taking the limit n, we deduce that j=,...,4 Φ g P, α j φ 0 Ω 2 Og 2. Hence Φ g P 0, and it follows that Φ g P is a normalized ground state of H g P. In order to prove Theorem., it now suffices to follow [5]. Assume by contradiction that there exists another normalized ground state, say Φ gp, such that Φ g P, Φ gp = 0. We

20 20 L. AMOUR AND J. FAUPIN write Φg P, Φ gp 2 = lim n = lim n = lim n Φg,n P Ω σn, Φ gp 2 Φ g P, {Eg,n P } H g,n P Φ gp Φ g P, Hfib {Eg,n P } H g,n P Φ gp. 3.3 Notice that, in the second equality, we used the fact that, by Theorem 3. and Lemma 3.2, E g,n P is a simple eigenvalue of H g,n P. We decompose Hfib {Eg,n P }H g,n P = Hfib,n {Eg,n P }K g,n P Π Ω σn + Hfib,n H σn ph Π Ω σn, 3.4 and use that, by Lemma A.4, Φ g P, Hfib,n H σn Π Ω σn Φ gp Φ gp, N σn ph Φ gp g 2 σn 2 g ph On the other hand, by Theorem 3., we can write Φ g P, Hfib,n {E g,n P } K g,n P Π Ω σn Φ gp Φ ησ g P, K g,n P E g,n P Π Ω σn Φ gp n = Φ ησ g P, H g,n P E g,n P Π Ω σn Φ gp n Φ ησ g P, H g,n P E g,n P Φ gp n = ησ n Φ g P, E g P E g,n P W σn g P Φ gp, where in the last equality we used that H g P E g P Φ gp = 0 and that H g P = H g,n P + Wg σn P. By Lemmas A.6 and A.8, this implies that Φ g P, Hfib, n {Eg,n P } K g,n P Π Ω σn Φ gp Cg2 σ n η + C W g η g, 3.6 since σ n. Combining 3.4, 3.5 and 3.6, we obtain that, for g small enough, 3.3 > 0, which is a contradiction. Hence the theorem is proven. Appendix A. Known results and standard estimates This appendix contains several already known results and fairly standard estimates which were used above. Lemma A.. For all 0 P < M, and σ 0, we have that K 0, P E 0 P P Hph,. A. M Moreover, the spectrum of K 0, P satisfies speck 0, P { E 0 P } [ E 0 P + P σ, +, A.2 M

21 and Proof. Recall that UNIQUENESS OF THE DRESSED HYDROGEN ATOM GROUND STATE 2 dim Ker K 0, P E 0 P = 4. K 0, P = H r + P 2 2M M P P ph, + 2M P ph, 2 + H ph,, on the Hilbert space H fib,. For Φ DK 0, P, we have that Φ, P P ph, Φ P Φ, H ph, Φ. Therefore, K 0, P e 0 + P 2 2M + P Hph,, M which proves A.. In order to prove A.2 and A.3, it suffices to observe that K 0, P y φ 0 Ω = E 0 P y φ 0 Ω, A.3 for any y C 4, and that, by A., if ψ DK 0, P satisfies ψ = and ψ, y φ 0 Ω = 0 for all y C 4, then ψ, K 0, P E 0 P ψ P σ. M Hence the lemma follows. The next result is proven in [3, Lemma 4.3]. This type of inequality is an important ingredient in the proof of the existence of a ground state for translation invariant models in non-relativistic QED see e.g. [, 22, 3, 20]. Lemma A.2. There exist g c > 0, p c > 0 and σ c > 0 such that, for all 0 g g c, 0 P p c, 0 σ σ c and k R 3, E g, P k E g, P 3 4 k. The number operator of particles of energies σ is defined by N ph, := a λ ka λkdk. λ=,2 k The following two lemmas are easy generalizations of [3, Lemma 4.5] based on the combination of a pull-through formula together with a Pauli-Fierz transformation see also [6, 4]. We do not recall the proofs. Lemma A.3. There exist g c > 0, p c > 0 and σ c > 0 such that, for all 0 g g c, 0 P p c and 0 σ σ c, KerK g, P E g, P DN /2 ph,, and the following holds: For all Φ g, P KerK g, P E g, P, Φ g, P =, we have that In the next lemma, we use the notation N σ ph, τ := Φ g, P, N ph, Φ g, P g 2. λ=,2 τ k σ a λ ka λkdk. A.4

22 22 L. AMOUR AND J. FAUPIN Lemma A.4. There exist g c > 0, p c > 0 and σ c > 0 such that, for all 0 g g c, 0 P p c and 0 τ σ σ c, KerK g, τ P E g, τ P DN σ ph, τ /2, and the following holds: For all Φ g, τ P KerK g, τ P E g, τ P, Φ g, τ P =, we have that Φ g, τ P, N σ ph, τ Φ g, τ P g 2 σ 2. The next lemma is proven in the same way as Lemma A.6 in []. We do not reproduce the proof here. Lemma A.5. There exist g c > 0, p c > 0, σ c > 0 and C 0 > 0 such that, for all 0 g g c, 0 P p c, and 0 σ σ c, Similarly we have the following: E g, P E 0 P E g, P + C 0 g 2. Lemma A.6. There exist g c > 0, p c > 0, σ c > 0 and C 0 > 0 such that, for all 0 g g c, 0 P p c, and 0 τ σ σ c, E g, τ P E g, P E g, τ P + C 0 g 2 σ 2. A.5 Proof. Again, the proof is similar to the one of Lemma A.6 in []. We only sketch the main differences here. Recall that in H fib, τ, where W σ g, τ K g, τ P = H g, P Hfib, τ + W σ g, τ P, A.6 P is given in.. By Lemma 3.2, H g, P Hfib, τ Φg, P Ω σ τ Thus, the first inequality in A.5 follows from = Eg, P Φ g, P Ω σ τ. E g, τ P Φ g, P Ω σ τ, K g, τ P Φ g, P Ω σ τ = Eg, P, since W σ g, τ P is Wick ordered. In order to prove the second inequality in A.5, we use that, by A.6, E g, P Φ g, τ P, H g, P Hfib, τ Φ g, τ P E g, τ P Φ g, τ P, W σ g, τ P Φ g, τ P. Decomposing W σ g, τ P into its expression. and estimating each term separately, we are able to conclude the proof thanks to Lemma A.4. The next two lemmas are proven in the same way as Lemma A.8 in []. Lemma A.7. There exist g c > 0, p c > 0, σ c > 0 and C W > 0 such that, for all 0 g g c, 0 P p c, 0 σ σ c and 0 < ρ <, the following holds: [K 0, P E 0 P + ρ] 2 Wg, P [K 0, P E 0 P + ρ] 2 C W gρ 2. Lemma A.8. There exist g c > 0, p c > 0, σ c > 0 and C W > 0 such that, for all 0 g g c, 0 P p c, 0 τ σ σ c and 0 < ρ <, the following holds: [H g, P Hfib, τ E g, P + ρ] 2 W σ g, τ P [H g,p Hfib, τ E g, P + ρ] 2 C W gσ 2 ρ 2.

23 UNIQUENESS OF THE DRESSED HYDROGEN ATOM GROUND STATE 23 The next lemma is an easy consequence of Lemma A.3. It follows in the same way as [, Lemma A.7]. Lemma A.9. There exist g c > 0, p c > 0 and σ c > 0 such that, for all 0 g g c, 0 P p c and 0 σ σ c, the following holds: for all Φ g, P KerK g, P E g, P, Φ g, P =, we have that Φ g, P, C 4 π 0 Π Ω Φ g, P g 2. A.7 We conclude with two easy lemmas concerning Pauli matrices. Their proofs are left to the reader. Lemma A.0. Let a = a, a 2, a 3 denote a 3-vector of complex numbers. We have that a σ el a σ el = a 2 + a a 2 3 = a σ n a σ n. Lemma A.. Let a = a, a 2, a 3 denote a 3-vector of complex numbers. The eigenvalues of the 4 4 matrix a σ elσn + a 2σ2 elσn 2 + a 3σ3 elσn 3 are the following: a + a 2 a 3, a a 2 + a 3, a + a 2 + a 3, a a 2 a 3. References. L. Amour and J. Faupin, Hyperfine splitting of the dressed hydrogen atom ground state in non-relativistic QED, Rev. Math. Phys., 23, 20, L. Amour, J. Faupin, B. Grébert and J.-C. Guillot, On the infrared problem for the dressed non-relativistic electron in a magnetic field, In Spectral and Scattering Theory for Quantum Magnetic Systems, vol. 500 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2009, L. Amour, B. Grébert and J.-C. Guillot, The dressed mobile atoms and ions, J. Math. Pures Appl., 86, 2006, V. Bach, T. Chen, J. Fröhlich and I.M. Sigal, Smooth Feshbach map and operator-theoretic renormalization group methods, J. Funct. Anal., 203, 2003, V. Bach, J. Fröhlich and A. Pizzo, Infrared-finite algorithms in QED: the groundstate of an atom interacting with the quantized radiation field, Comm. Math. Phys., 264, 2006, V. Bach, J. Fröhlich and I. M. Sigal, Quantum electrodynamics of confined non-relativistic particles, Adv. in Math., 37, 998, V. Bach, J. Fröhlich and I. M. Sigal, Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field, Comm. Math. Phys., 207, 999, J.-M. Barbaroux and J.-C. Guillot, Spectral theory for a mathematical model of the weak interaction: The decay of the intermediate vector bosons W ±. I, Advances in Mathematical Physics, T. Chen, J. Fröhlich and A. Pizzo, Infraparticle scattering states in non-relativistic QED. II. Mass shell properties, J. Math. Phys., 50, 0203, C. Cohen-Tannoudji, B. Diu, F. Laloë, Mécanique quantique II, Hermann, Paris, J. Fröhlich, On the infrared problem in a model of scalar electrons and massless, scalar bosons, Ann. Inst. H. Poincaré Sect. A, 9, 973, J. Fröhlich and A. Pizzo, Renormalized Electron Mass in Nonrelativistic QED, Comm. Math. Phys., 294, 200, M. Griesemer and D. Hasler, On the smooth Feshbach-Schur map, J. Funct. Anal., 254, 2008, M. Griesemer, E.H. Lieb and M. Loss, Ground states in non-relativistic quantum electrodynamics, Invent. Math., 45, 200, D. Hasler and I. Herbst, Uniqueness of the ground state in the Feshbach renormalization analysis, preprint, arxiv: F. Hiroshima. Multiplicity of ground states in quantum field models: application of asymptotic fields, J. Funct. Anal., 224, 2005, F. Hiroshima and J. Lorinczi, Functional integral representations of nonrelativistic quantum electrodynamics with spin /2, J. Funct. Anal., 254, 2008,

24 24 L. AMOUR AND J. FAUPIN 8. F. Hiroshima and H. Spohn, Ground state degeneracy of the Pauli-Fierz Hamiltonian with spin, Adv. Theor. Math. Phys., 5, 200, T. Kato, Perturbation Theory for Linear Operators, second edition Springer-Verlag, Berlin, M. Loss, T. Miyao and H. Spohn, Lowest energy states in nonrelativistic QED: atoms and ions in motion, J. Funct. Anal., 243, 2007, M. Loss, T. Miyao and H. Spohn, Kramers degeneracy theorem in nonrelativistic QED, Lett. Math. Phys., 89, 2009, A. Pizzo, One-particle improper states in Nelson s massless model, Ann. Henri Poincaré, 4, 2003, M. Reed and B. Simon, Methods of modern mathematical physics I-IV, New York, Academic Press H. Spohn, Dynamics of charged particles and their radiation field, Cambridge University Press, Cambridge, L. Amour Laboratoire de Mathématiques, Université de Reims, Moulin de la Housse, BP 039, 5687 REIMS Cedex 2, France, et FR-CNRS address: laurent.amour@univ-reims.fr J. Faupin Institut de Mathématiques de Bordeaux, UMR-CNRS 525, Université de Bordeaux, 35 cours de la libération, Talence Cedex, France address: jeremy.faupin@math.u-bordeaux.fr