Spectrum of the semi-relativistic Pauli-Fierz model II

The existence of the ground state of the so-called semi-relativistic Pauli-Fierz model is proven. Let $A$ be a quantized radiation field and $H_{fm}$ the free field Hamiltonians which is the second quantization of $\sqrt{|k|^2+m^2}$. It has been established so far that the semi-relativistic Pauli-Fierz model $$H_{SRPF}=\sqrt{(-i\nabla\otimes 1-A)^2+M^2} +V\otimes1+1\otimes H_{fm}$$ has the unique ground state for $(m,M)\in((0,\infty )\times[0,\infty]) \cup ( [0,\infty ]\times(0,\infty) )$. In this paper the existence of the ground state of $H_{SRPF}$ with $(m,M)\in[0,\infty )\times[0,\infty)$ is shown. We emphasize that our results include a singular case $(m,M)=(0,0)$, i.e., the existence of the ground state of the Hamiltonian of the form: $$|\!-i\nabla\otimes1-A |+V\otimes1+1\otimes H_{f0}$$ is established.


Introduction
In this paper we are concerned with the existence of the ground state of the so-called semirelativistic Pauli-Fierz (it is shorthand as "SRPF" in this paper) model in quantum electrodynamics. This model describes a minimal interaction between a semi-relativistic quantum matter and a quantized radiation field A = (A 1 , A 2 , A 3 ). The matter is governed by the semirelativistic Schrödinger operator defined by √ −∆ + M 2 + V , where M denotes the mass of the matter and V an external potential. On the other hand the free field Hamiltonian is given by H f,m where H f,m is the second quantization of ω(k) = |k| 2 + m 2 and m describes the mass of a boson. Then the decoupled Hamiltonian is defined by √ −∆ + M 2 + V ⊗ 1l + 1l ⊗ H f,m in a product Hilbert space H = L 2 (R 3 ) ⊗ F , where F denotes a boson Fock space. The minimal coupling implies to replace −∆ ⊗ 1l with (p ⊗ 1l − A) 2 , where p µ = −i∇ µ with the generalized differential operator ∇ µ , and A µ (x)dx, µ = 1, 2, 3.
Thus the Hamiltonian of SRPF is given by Note that the boson-mass m should be however zero since the boson physically describes a photon. We show that H m is self-adjoint in [HH13b] and the spectrum of H m is where E m = inf σ(H m ) is the bottom of the spectrum of H m . Eigenvector associated with E m is called a ground state of H m . It is suggested to study the ground state of SRPF Hamiltonian in [GLL01] where the existence of the ground state of the Pauli-Fierz Hamiltonian is proven under the binding condition. Then SRPF Hamiltonian has been studied so far in e.g.[GS12, HH13b, Hir14, HS10, KM13a, KM13b, KMS11a, KMS11b, MS10, MS09, O16, S13a, S13b]. In particular when m = 0 but M > 0, one can also show the existence of ground state. In this case the bottom of the spectrum is the edge of the continuous spectrum and it requires a non-perturbative analysis. This is actually done in [KMS11a,KMS11b]. Results related to biding condition and enhanced binding are also given in [GS12,KM13b,S13a,S13b]. It is also shown that H m has a ground state for m > 0 but M = 0 in [HH13b], where the ground state energy is discrete but the assumption M = 0 produces a singularity. It is emphasized again that E m for m = 0 is the edge of the continuous spectrum and there is no positive gap between E m and inf σ(H m ) \ {E m }. Furthermore E m is simple which is shown in [Hir14, Corollary 6.2]. Then a remaining problem is to study the case of (m, M) = (0, 0), i.e., and we solve this in this paper, i.e., the existence of the ground state of (1.2) is shown.

Applications to asymptotic field and outline of proofs
Let M = 0. In order to avoid the infrared divergence we unitarily transform H m to a regularized Hamiltonian H R m , which is of the form Here A R is given (2.9) and h in (2.8) below. Note that H m is unitarily equivalent to H R m : This transformation is initially used in [BFS99]. In this paper the asymptotic annihilation operator defined by a ±∞ (f, j) = s− lim t→±∞ e −itH R m e itH f,m a(f, j)e −itH f,m e itH R m is applied to prove the existence of ground state. This is established in [AHH99,Hir05] and reviewed in Appendix A. Let x 2 = |x| 2 + 1 as usual. In order to show the existence of the ground state it is enough to check three uniform bounds concerning Φ m for m > 0: (A) spatial decay: sup x∈R 3 x 2 Φ m (x) F < C, (B) the number of bosons: N 1 2 Φ m H < C, where N denotes the number operator, (C) Sobolev norm of n-particle sector: sup 0<m≤m 0 Φ (n) m W 1,p (Ω) for 1 ≤ p < 2 and any finite domain Ω ⊂ R 3 x × R 3n k . We review (A),(B) and (C) below: (A) The spatial exponential decay with some C independent of m is fortunately established in [Hir14,Theorem 5.12]. See also [GLL01,BFS99]. This implies sup x∈R 3 x 2 Φ m (x) F < C ′ . (B) The number of bosons in Φ m can be uniformly estimated in m, i.e., where C is independent of m. To derive this inequality we use the identity: where C j (k) denotes a bounded operator for each k ∈ R 3 . See Lemma 3.13 for the explicit statement. (1.5) can be derived by the Cook method in scattering theory and the fact a ±∞ (f, j)Φ m = 0.
(1.6) It can be also established that the map (1.7) See Proposition 3.8. It is proven in several literatures that this type of argument is very useful to show the existence of the ground state. In order to derive (1.5) we have to compute the commutator: for m > 0. Since |p ⊗ 1l − A| is a non-local operator and not smooth by missing positive mass term M, it is crucial to see (1) and (2) below: (1) to find a dense domain D such that D ⊂ D(|p ⊗ 1l − A R |a(f, j)) ∩ D(a(f, j)|p ⊗ 1l − A R |), (2) to show the boundedness of C j (k).
(1) is needed to guarantee that [|p ⊗ 1l − A|, a(f, j)]Φ m is well-defined, and (2) is used to see (1.7). In this paper we show (1) and (2) in Lemma 3.6 and Lemma 3.9, respectively. We emphasize that it is not trivial to show both of them. Consequently by virtue of (1) and (2) it can be derived that N 1 2 Φ m ≤ C x 2 Φ m and the spatial exponential decay (1.3) yields (1.4).
(C) Let H be decomposed into n-particle sectors: . It is shown that n-particle sector of Φ m satisfies that Φ (n) m ∈ W 1,p (Ω) for any bounded Ω ⊂ R 3 x × R 3n k and We derive this in a different way from [GLL01], where this method is initiated. (1.8) can be also shown by using (1.5) as follows: Let Ψ = (0, · · · , 0, we take the inner product of both sides of (1.5) Thus we have the identity by the integral by parts formula. Hence the right-hand side can be estimated and conclude (1.8) in Lemmas 3.30 and 3.31. Finally combining (A),(B) and (C) we can show that the normalized ground state Φ m strongly converges to a nonzero vector as m → 0, which is nothing but the ground state of H R m , i.e., H m , for m = 0. This paper organized as follows. In Section 2 we give the definition of SRPF Hamiltonian H m as a self-adjoint operator, and introduce a regularized SRPF Hamiltonian H R m . Section 3 is devoted to proving N 1 2 Φ m < C and sup 0<m<m 0 Φ (n) m W 1,p (Ω) < ∞, and then show the main theorem in Theorem 3.33. In Appendix we review asymptotic field used in this paper.

Definition of semi-relativistic Pauli-Fierz model
We define the Hamiltonian of SRPF model as a self-adjoint operator on a Hilbert space. As is mentioned in the previous section the Hamiltonian of SRPF model includes nonlocal operator, hence the definition of the self-adjoint operator is not straightforward. The operator consists of a matter part and quantum field part. We firstly introduce the quantum field part.
Let us introduce the boson Fock space. The boson Fock space, F , over Hilbert space where ⊗ n s W denotes the symmetric tensor product of W and ⊗ 0 In particular the Fock vacuum is given by Ω = (1, 0, 0, · · · ) ∈ F .
Let T be a densely defined closable T in W . The second quantization of T is the closed operator in F , which is defined by If T is a non-negative self-adjoint operator in W , then dΓ(T ) turns to be also a non-negative self-adjoint operator. We denote the spectrum (resp. point spectrum) of T by σ(T ) (resp. σ P (T )). The Fock vacuum Ω the eigenvector of dΓ(T ) associated with eigenvalue 0, i.e., dΓ(T )Ω = 0. The number operator, N, is defined by the second quantization of the identity 1l on W : and σ(N) = N ∪ {0}. Let ω(k) = |k| 2 + m 2 , k ∈ R 3 , be a dispersion relation and it can be regarded as the multiplication operator in W . The free field Hamiltonian H f,m is given by the second quanmtization of ω: Then H f,m is a non-negative self-adjoint operator in F , and we see that (2.1) Moreover H f,m Ω = 0. The creation operator a † (f ) smeared by f ∈ W is given by and (a † (f )Ψ) (0) = 0 with the domain: Here S n is the symmetrization operator on ⊗ n W . The annihilation operator smeared by f ∈ W is given by the adjoint of a † (f ): a(f ) = (a † (f )) * . Both a(f ) and a † (f ) are linear in f , and satisfy canonical commutation relations: We formally write a ♯ (f ) = j=1,2 a ♯ (k, j)f (k, j)dk for a ♯ (f ). Let us introduce the finite particle subspace F fin by which is a dense subspace of F . We shall define a quantized radiation field A(x). Let e(·, 1) and e(·, 2) be polarization vectors i.e., e(k, j)·e(k, j ′ ) = δ jj ′ and k ·e(k, j) = 0 for k ∈ R 3 \{0} and j, j ′ = 1, 2, and we choose Note that e(·, j) ∈ C ∞ (R 3 \ {0}) for j = 1, 2. For each x ∈ R 3 the quantized radiation field,

is given by
andφ is a cutoff function. In addition the conjugate momentum is as usual defined by , by Nelson's analytic vector theorem, for each x ∈ R 3 A µ (x) and Π µ (x) are essentially self-adjoint. Then let us introduce assumptions on ultraviolet cutoff functionφ.

Statement
(2) of Assumption 2.1 implies that ω n φ ω ∈ L 2 (R 3 ) for any n ∈ N, which yields together with (1) that SRPF Hamiltonian is self-adjoint and (2) is also used to establish a derivative bound of the massive ground state, which is studied in Section 3.3. Let A µ (x) be the closure of A µ (x), and then it is self-adjoint. We define the self-adjoint operator A µ by ⊕ R 3 A µ (x)dx and we set A = (A 1 , A 2 , A 3 ). We shall explain the particle part. Let p = (p 1 , p 2 , p 3 ) = (−i∇ 1 , −i∇ 2 , −i∇ 3 ) be the momentum operator of particle. Then the particle Hamiltonian under consideration is a relativistic Schrödinger operator given by Here V : R 3 → R denotes an external potential. Finally we define the total Hamiltonian of SRPF model, which is an operator in the Hilbert space and is given by the minimal coupling of the decoupled Hamiltonian We do not write tensor notation ⊗ for notational convenience in what follows. Thus H m can be simply written as Let H fin = C ∞ 0 (R 3 ) ⊗F fin , where ⊗ denotes the algebraic tensor product. Let us introduce classes of external potentials studied in this paper. (2) There exists m 0 > 0 such that

Definition 2.4 (External potentials)
Let us denote the essential spectrum of H m by σ ess (H m ).

Regularized SRPF Hamiltonians
We transform H m to a certain regular Hamiltonian to avoid the infrared divergence. Let us define the unitary operator U = ⊕ R 3 U(x)dx on H by U(x) = exp (ix · A(0)) , and we set In a similar manner to Proposition 2.2, we can also see that (p − A R ) 2 is essentially self-adjoint on D(−∆) ∩ C ∞ (N), and the closure of (p holds as self-adjoint operators, and Then (2.10) holds on H fin . Since H fin is a core of H m , (2.10) follows from a limiting argument. Furthermore H R m is essentially self-adjoint on U −1 H fin and self-adjoint on U −1 D(H m ).

Infrared singularity
In what follows we study H R m instead of H m . An advantage of studying H R m is that we do not need the infrared regular condition: to show the existence of the ground state. Physically reasonable choice ofφ(0) is nonzero, sinceφ(0) amounts to the charge. Then by the singularity at the origin, R 3 |φ(k)| 2 ω(k) 3 dk = ∞ when m = 0. Actually instead of (2.11) we need the condition: to show the existence of ground state. Thus in the case of m = 0 andφ(0) = 0 we can also show the existence of the ground state. See Theorem 3.33 and Corollary 3.14.

Infrared bounds
Throughout we assume Assumption 2.1. Let Φ m be a normalized ground state of H R m . Note that sup 0<m<m 0 x 2 Φ m < ∞ since UΦ m is a ground state of H R m , and [ x 2 , U] = 0. In this section we shall prove two bounds concerning Φ m by using the so-called pull-through formula. In this section we set M = 0. Then

Stability of a domain
For notational simplicity we set Then The pull-through formula we see later is a useful tool to study the ground state associated with embedded eigenvalues. In order to establish the pull-through formula we begin with establishing that [ T p , a(f, j)] is well defined on some dense domain In order to find D we apply a stochastic method. Let (B t ) t≥0 be the three dimensional Brownian motion on a probability space (W, B(W), P x ). Here P x is the Wiener measure starting from x. The expectation with respect to P x is simply denoted by E x [· · · ]. Let A (F ) be the Gaussian random process indexed by F ∈ ⊕ 3 L 2 (R 3 ) on a probability space The unitary equivalence between L 2 (Q) and F is established and under this equivalence it follows that for We set the right-hand side above as A(F ).
Proposition 3.1 The Feynman-Kac type formula of e −tTp is given by The range of T p restricted on D ∞ is denoted by Proof: Since a ♯ (F ) leavers C ∞ (N) invariant, and A µ (x)Φ for Φ ∈ D is infinitely differentiable with respect to x by virtue of the fact thatφ has a compact support, it follows that D ⊂ D ∞ . Furthermore we can check that By Lemma 3.2, on D operator T p a(F ) is well defined, but it is not clear whether a(F ) T p can be defined on D or not. Hence we shall prove that (1) D is dense, . Hence together with Lemma 3.2 we can conclude that commutator [a(F ), T p ] is well defined on D. In order to prove (1) and (2), we prepare several lemmas. We have , and is a stochastic process. Note that under the identification L 2 (Q) ∼ = F , We set the righthand side above as π(K). Let P µ = p µ ⊗ 1l + 1l ⊗ P f µ , µ = 1, 2, 3, be the total momentum. We can also see the commutation relation between P f ν and e −iA (K) , which is given by is also a stochastic process. Note that We set the right-hand side above as π ν (K).
Lemma 3.3 Let K be ⊕ 3 L 2 (R 3 )-valued stochastic integral given by (3.3). Then (1) and (2) below follow: (1) Let k ∈ N. Then there exists a polynomial P k = P k (x) of degree k such that (2) Let 1 ≤ µ 1 , · · · , µ n ≤ 3. Then there exists a polynomial Q n = Q n (x 1 , · · · , x n ) of degree n such that Proof: Let us show (1) by an induction. For k = 1 it can be seen that By the assumption of the induction it is trivial to see that We can also see that Then statement (1) follows. Statement (2) can be similarly proven.
Lemma 3.4 Suppose Assumption 2.1. Then e −tTp D ∞ ⊂ D ∞ . In particular T p ⌈ D∞ is essentially self-adjoint and D is dense.
Proof: Let Ψ ∈ D ∞ be arbitrary. It is enough to show that e −tTp Ψ ∈ D(N n ) ∩ D(p µ 1 · · · p µm ) for arbitrary n and 1 ≤ µ 1 , · · · , µ m ≤ 3. In order to do that we show bounds below for arbitrary Φ 1 ∈ D(N n ) and Φ 2 ∈ D(p µ 1 . . . p µm ): By the Feynman-Kac formula and the equivalence Π(K) ∼ = π(K), we have Using bounds shown in Lemma 3.3 we have with some constant c proved in [Hir00, Theorem 4.5], we can then get Next we estimate (3.8). Note that [P µ , e −iA (K) ] = 0 for µ = 1, 2, 3. We have Using again bounds shown in Lemma 3.3 we have Thus in a similar manner to (3.7), the BDG-inequality (3.9) yields that and we can show (3.8). Then D ∞ is an invariant domain of e −tTp , which implies that T p ⌈ D∞ is essentially self-adjoint and thus D = T p D ∞ is dense. Let us set for any Φ ∈ D(N) with some constant C independent of Φ. In order to show (3.10) we again apply Feynman-Kac formula for e −tTp . By the definition of T p we have where we set F = T p Ψ, and use the identity We estimate integrands as By BDG inequality (3.9) we can derive that Together with them we have with some constant C. Next we estimate (3.14) It is trivial to see that In a similar manner to (3.13) we can see that We note that with some constant C. Then from (3.13), (3.15) and (3.16), (3.10) follows.
Proof: D ⊂ D( T p a(F )) follows from Lemma 3.2 and D ⊂ D(a(F ) T p ) from Lemma 3.5.

Commutator estimates and number operator bounds
Let m > 0 throughout this section. In this section we estimate N 1 2 Φ m uniformly in m > 0. In order to do this we apply or suitably modify the method developed in [Hir05].  (1) There exists an operator B j (k) : F → F for each k ∈ R 3 , j = 1, 2, such that D(B j (k)) ⊃ D(H R m ) for almost everywhere k, and [a(f, j), T p ] Proposition 3.8 Suppose (1),(2) and (3) in Proposition 3.7. Then (1) 2 ) if and only if T gj is a Hilbert-Schmidt operator. (4) If T gj is a Hilbert-Schmidt operator. Then the Hilbert-Schmidt norm of T gj is given by Proof: See [Hir05, Lemmas 2.7 and 2.8].
We note that ω ∈ C ∞ (R 3 \ {K}). We set For each k ∈ R 3 let us define the operator I j (k) by I j (k) = and On D, we also have The commutator [a(f, j), T p ] is computed on R t 2 D as Since the Coulomb gauge condition k · e(k, j) = 0, we have T j (k)e −ikx = e −ikx T j (k). Then It is also shown in Lemma 3.10 that with a constant C independent of m and k, and f (k)(|k| + |k| 2 )φ ω (k) is integrable by the fact that φ ω has a compact support. By Fubini's lemma, we can see that Hence (3.20) follows. We can see in Lemma 3.10 that I j (k) is bounded with I j (k) ≤ C(|k| + |k| 2 ). On the other hand, ρ j (k) 1 x 2 ≤ ω(k) |φ ω (k)|. Then for almost every k ∈ R 3 , C j (k) is bounded and (3.19) follows. In particular C j (·)Ψ ∈ L 2 (R 3 ). Then the proof is complete.
Lemma 3.10 For each k ∈ R 3 , I j (k) is a bounded operator such that for Ψ, Φ ∈ D. By Schwarz's inequality, (3.22) Here we used the estimate: where dE λ denotes the spectral measure of T p with respect to Ψ. The diamagnetic inequality yields that (3.24) Then we have by (3.22)

Thus we have
By the Hardy-Rellich inequality [Yaf99], we have for all u ∈ W |(u, Zu)| ≤ C |x| 2 x 2 u + |x| 1 2 x 2 u |x| 3 2 Thus Z is a bounded operator on L 2 (R 3 ). Then we obtain that which decomposition is often used in what follows. Thus

Let us estimate I
(1) 2 (k). Note that Since for all a > 0, √ a , we see that where dE µ and dẼ µ are spectral measures of T p and T p+k with respect to Ψ and Φ, respectively. Thus we obtain that (3.32) Next let us estimate I Thus we obtain that (3.33) Finally we estimate I (3) 2 (k). By Schwarz's inequality again, it can be seen that Corollary 3.11 There exists a constant C such that for any Φ ∈ H , (3.37) Proof: (3.36) can be derived from (3.24) and (3.26). We show (3.37). Let q = |p| 2 + t 2 . We fix µ and write x and p for x µ and p µ for notational simplicity in this proof. Then [x, q] = 2ip and x 2 q = qx 2 + 2i(px + xp). We extend (3.25). From this we have Directly we can see that We set f = −8 p 2 q 3 + 2 q 2 . Hence we have 1 q x 2 = x 2 1 q + 4ix p q 2 + f and then By a similar argument as the proof of the boundedness of Z mentioned in the proof of Lemma 3.10, we can get the desired results.
Proof: Let 1 ≤ µ ≤ 3 be fixed. We note that Since C j (k) = 4 π I j (k)φ ω (k) + ρ j (k) 1 x 2 , the integral is divided as We estimate (3.39). Integral by parts formula yields that . We now estimate (3.38). Then integral by parts formula also yields that We shall see that is integrable with respect to k. In order to see it we estimate We can estimate I and II in a similar manner to the proof of Lemma 3.10. Hence I and II are bounded with with some constant C independent of t. Let us investigate III. We have III = III 1 + III 2 , where In a similar way to the proof of Lemma 3.10 again, III 1 can be also estimated as We estimate III 2 . We have and we divide the integral as L 1 + L 2 where L 1 = 1 0 ds · · · and L 2 = ∞ 1 ds · · · . We can see that where we used that s R s 2 T j (k) ≤ s T p R s 2 ≤ 1 2 and s R s 2 (p µ −A Rµ ) ≤ s T p R s 2 ≤ 1 2 . We also see that Thus III 2 is also bounded with (3.42) Then we can conclude that By Fubini's lemma we can exchange integrals dk and ds and we see that Since kµ ω(k) f (k)φ ω (k)e ν (k, j) ∈ C ∞ 0 (R 3 k \ {0}) for µ = 1, 2, 3 by the assumption onφ and f . We also note that sup x∈R 3 |ξ We already see that 1 0 dss T p R s 2 e it(H R m −Em) Ψ 2 is finite in (3.23), and moreover by the Hardy-Rellich inequality. Similarly we can see that Next we can estimate ∞ 1 · · · ds. we have In order to estimate −ix µ ξ (2) t (p − A R )R s 2 we compute the commutation relation: Estimates of the remaining terms are straightforward: (1) and the left-hand side above is integrable with respect to t. follows.
Proof: By the general proposition, Proposition 3.7, the proof can be proven under the identifications: C j (k) x 2 and B j (k) in Proposition 3.7, by Lemmas 3.9 and 3.12.
Corollary 3.14 Suppose Assumptions 2.1 and with some constant C independent of m.
Proof: By the assumption we can check (3.44). Then the corollary follows from Lemma 3.13.
Lemma 3.18 Suppose that 2|h µ | ≤ |k µ | for µ = 1, 2, 3, and suppφ ⊂ {k ∈ R 3 ||k| ≤ 2Λ} for some Λ. Then it follows that for each k ∈ R 3 \ K, (3.57) Proof: By the definition of C j (k) we have Let us estimate the first term of the right-hand side of (3.58). Note that for j = 1, 2, |∇ · e(k, j)| ≤ C √ k 2 1 +k 2 2 , and that (3.59) See Appendix B. Furthermore it is straightforward to see that (3.60) Then we obtain that (3.61) Thus follows. Next we shall show that We have (3.64) The second term of the right-hand side of (3.64) can be estimated as We can also show that (3.65) (3.65) is proven by Lemmas 3.19 below. Hence the lemma follows.
Note that Ψ, Then it can be estimated as (3.69) Next we consider the second term of the right-hand side of (3.68). Set For all Ψ, Φ ∈ H , we have Similar to (3.30) and (3.31) we can see that K (1) 2 ≤ C(1 + |h| + |k|). We can also see that Lemma 3.20 Suppose that 2|h µ | ≤ |k µ | for µ = 1, 2, 3, and suppφ ⊂ {k ∈ R 3 ||k| ≤ 2Λ} for some Λ. Then it follows that for each k ∈ R 3 \ K, In particular it is satisfied that for k ∈ R 3 \ K, Here K is defined in (2) of Proposition 3.7.

Weak derivative with respect to field variables
In what follows we shall see the explicit form of ∇ kµ Φ (n+1) m by using Corollary 3.15. Let Here ∇ µ I j (k) = ∞ 0 dt∇ µ I j (k, t) and (3.73) Then we divide (3.73) as The third term of the right-hand side is again divided as Here ∇ µ φ ω (k) = φ µ ω (k) and ∇ µ ρ j (k) = ρ µ j (k). Furthermore the fourth term on the right-hand side is again divided as where ξ(k) = e −ikx − 1. We conclude that (3.73) can consequently be divided into the eight terms such as In the definitions of T µ j (k) and ρ µ j (k), partial derivative of e(k, j), ∇ µ e(k, j), appears. The lemma below is useful to estimate T µ j (k) and ρ µ j (k).
Proof: The proof is straightforward. Then we show it in Appendix B. We shall estimate G 1 , · · · , G 8 in the following lemmas.
Lemma 3.23 It follows that lim Then we can see that 1l |k|≤Λ √ m x 2 Φ m and the right-hand side is in L 2 (R 3 k ). Then the lemma follows.
Proof: We note that and then there exists 0 ≤ θ = θ(h, k) ≤ 1 such that Then the right-hand side is in L 2 (R 3 k ) and the lemma follows.
In particular it follows that .

Weak derivative with respect to particle variables
We consider the weak derivative of Φ (n) m (x, k 1 , ..., k n ) with respect to x.

Existence of ground states
Now we state the main theorem.
Theorem 3.33 Let m = 0. Suppose that R 3 |k|+|k| 2 ω(k) 2φ (k) 2 ω(k) dk < ∞. Then H R 0 has the ground state and it is unique up to multiple constant. In particular H m for m = 0 has the ground state.

A Asymptotic field and number operator
We quickly reviews a Hilbert space-valued integral operator which is the so-called Carleman operator for the self-consistency of the paper. See [Hir05] for explicit statement and conditions. Let a t (f, j) = e −itH R m e itH f,m a(f, j)e −itH f,m e itH R m . Since we can see that s− lim Here κ j (k) = (H R m − E m + ω(k)) −1 C j x 2 . Let T gj : L 2 (R 3 ) → H be defined by Here T gj is a H -valued integral operator, and a(f, j)Φ m = −T gj f . Adjoint is called a Carleman operator [Wei80, p.141] with D(T * gj ) = {Φ ∈ F |(κ j (·), Φ) H ∈ L 2 (R 3 )}.