The No-Binding Regime of the Pauli-Fierz Model
aa r X i v : . [ m a t h - ph ] A p r The No-Binding Regime of the Pauli-FierzModel
Fumio Hiroshima ,1 , Herbert Spohn ,2 , and Akito Suzuki ,31 Faculty of Mathematics, Kyushu University, Fukuoka,819-0395, Japan Zentrum Mathematik and Physik Department, TU M¨unchen,D-80290, M¨unchen, Germany Department of Mathematics, Faculty of Engineering, ,Shinshu University, Nagano, 380-8553, JapanDecember 17, 2018
Key words : Enhanced binding, ground state, Birman-Schwinger principle,Pauli-Fierz model
Abstract
The Pauli-Fierz model H ( α ) in nonrelativistic quantum electrody-namics is considered. The external potential V is sufficiently shallowand the dipole approximation is assumed. It is proven that there ex-ist constants 0 < α − < α + such that H ( α ) has no ground state for | α | < α − , which complements an earlier result stating that there isa ground state for | α | > α + . We develop a suitable extension of theBirman-Schwinger argument. Moreover for any given δ > V are provided such that α + − α − < δ . Introduction
Let us consider a quantum particle in an external potential described by theSchr¨odinger operator(1.1) H p ( m ) = − m ∆ + V ( x )acting on L ( R d ). If the potential V is short ranged and attractive and if thedimension d ≥
3, then there is a transition from unbinding to binding as themass m is increased. More precisely, there is some critical mass, m c , suchthat H p ( m ) has no ground state for 0 < m < m c and a unique ground statefor m c < m . In fact, the critical mass is given by12 m c = (cid:13)(cid:13) | V | / ( − ∆) − | V | / (cid:13)(cid:13) , see Lemma 3.3. We now couple H p ( m ) to the quantized electromagneticfield with coupling strength α ≥
0. The corresponding Hamiltonian is de-noted by H ( α ). On a heuristic level, through the dressing by photons theparticle becomes effectively more heavy, which means that the critical mass c α ( α ) should be decreasing as a function of α with m c (0) = m c . In par-ticular, if m < m c , then there should be an unbinding-binding transition asthe coupling α is increased. This phenomenon has been baptized enhancedbinding and has been studied for a variety of models by several authors[AK03, BV04, HVV03, HHS05, HS01, HS08]. In case m > m c more generaltechniques are available and the existence of a unique ground state for thefull Hamiltonian is proven in [AH97, BFS99, GLL01, LL03, Ger00, Spo98].The heuristic picture also asserts that the full hamiltonian has a regime ofcouplings with no ground state. This property is more difficult to establishand the only result we are aware of is proved by Benguria and Vougalter[BV04]. In essence they establish that the line m c ( α ) is continuous as α → m eff ( α ) = m + c α with some explicitly computable coefficient c , see Eq. (2.10) below. Thus the most basic guess for m c ( α ) would be m c ( α ) + c α = m c . The corresponding curve is displayed in Fig. 1. Infact the guess turns out to be a lower bound on the true m c ( α ). We will2igure 1: Upper and lower bounds on the critical mass m c ( α ). The dashedline indicates m c ( α )complement our lower bound with an upper bound of the same qualitativeform.The unbinding for the Schr¨odinger operator H p ( m ) is proven by theBirman-Schwinger principle. Formally one has H p ( m ) = 12 m ( − ∆) / (cid:0)
1l + 2 m ( − ∆) − / V ( − ∆) − / (cid:1) ( − ∆) / . If m is sufficiently small, then 2 m ( − ∆) − / V ( − ∆) − / is a strict contraction.Hence the operator 1l + 2 m ( − ∆) − / V ( − ∆) − / has a bounded inverse and H p ( m ) has no eigenvalue in ( −∞ , [ m , ∞ ) ( V / ( − ∆) − V / ) ≥ dim1l ( −∞ , ( H p ( m )) . For small m the left hand side equals 0 and thus H p ( m ) has no eigenvaluesin ( −∞ , H ( α ) isdefined on the Hilbert space H = L ( R d ) ⊗ F , where F denotes the bosonFock space. Transforming H ( α ) unitarily by U one arrives at U − H ( α ) U = H ( α ) + W + g (1.3) 3s the sum of the free Hamiltonian(1.4) H ( α ) = − m eff ( α ) ∆ ⊗
1l + 1l ⊗ H f , involving the effective mass of the dressed particle and the Hamiltonian H f of the free boson field, the transformed interaction(1.5) W = T − ( V ⊗ T, and the global energy shift g . m eff ( α ) is an increasing function of α . We willshow that (1.3) has no ground state for sufficiently small | α | by means ofa Birman-Schwinger type argument such as (1.2). In combination with theresults obtained in [HS01] we provide examples of external potentials V suchthat for some given δ > < α − < α + satisfying(1.6) δ > α + − α − > H ( α ) has no ground state for | α | < α − but has a ground state for | α | > α + .Our paper is organized as follows. In Section 2 we define the Pauli-Fierzmodel and in Section 3 we prove the absence of ground states. Section 4 listsexamples of external potentials exhibiting the unbinding-binding transition. We assume a space dimension d ≥ c = 1 and the Planck constant divided 2 π , ~ = 1. TheHilbert space H for the Pauli-Fierz Hamiltonian is given by H = L ( R d ) ⊗ F , where F = ∞ M n =0 (cid:2) ⊗ ns ( ⊕ d − L ( R d )) (cid:3) denotes the boson Fock space over the ( d − ⊕ d − L ( R d ). LetΩ = { , , , ... } ∈ F denote the Fock vacuum. The creation operator and theannihilation operator are denoted by a ∗ ( f, j ) and a ( f, j ), j = 1 , . . . , d − f ∈ L ( R d ), respectively, and they satisfy the canonical commutation relations[ a ( f, j ) , a ∗ ( g, j ′ )] = δ jj ′ ( f, g )1l , [ a ( f, j ) , a ( g, j ′ )] = 0 = [ a ∗ ( f, j ) , a ∗ ( g, j ′ )]4ith ( f, g ) the scalar product on L ( R d ). We write(2.1) a ♯ ( f, j ) = Z a ♯ ( k, j ) f ( k ) dk, a ♯ = a, a ∗ , The energy of a single photon with momentum k ∈ R d is(2.2) ω ( k ) = | k | . The free Hamiltonian on F is then given by(2.3) H f = d − X j =1 Z ω ( k ) a ∗ ( k, j ) a ( k, j ) dk. Note that σ ( H f ) = [0 , ∞ ), and σ p ( H f ) = { } . { } is a simple eigenvalue of H f and H f Ω = 0.Next we introduce the quantized radiation field. The d -dimensional po-larization vectors are denoted by e j ( k ) ∈ R d , j = 1 , . . . , d −
1, which satisfy e i ( k ) · e j ( k ) = δ ij and e j ( k ) · k = 0 almost everywhere on R d . The quantizedvector potential then reads(2.4) A ( x ) = d − X j =1 Z p ω ( k ) e j ( k ) (cid:0) ˆ ϕ ( k ) a ∗ ( k, j )e − ikx + ˆ ϕ ( − k ) a ( k, j )e ikx (cid:1) dk for x ∈ R d with ultraviolet cutoff ˆ ϕ . Conditions imposed on ˆ ϕ will be suppliedlater. Assuming that V is centered, in the dipole approximation A ( x ) isreplaced by A (0). We set A = A (0). The Pauli-Fierz Hamiltonian H ( α ) inthe dipole approximation is then given by(2.5) H ( α ) = 12 m ( p ⊗ − α ⊗ A ) + V ⊗
1l + 1l ⊗ H f , where α ∈ R is the coupling constant, V the external potential, and p =( − i∂ , ..., − i∂ d ) the momentum operator. For notational convenience we omitthe tensor notation ⊗ in what follows. Assumption 2.1
Suppose that V is relatively bounded with respect to − m ∆ with a relative bound strictly smaller than one, and (2.6) ˆ ϕ/ω ∈ L ( R d ) , √ ω ˆ ϕ ∈ L ( R d ) .
5y this assumption H ( α ) is self-adjoint on D ( − ∆) ∩ D ( H f ) and boundedbelow for arbitrary α ∈ R [Ara81, Ara83]. We need in addition some technicalassumptions on ˆ ϕ which are introduced in [HS01, Definition 2.2]. We listthem as Assumption 2.2
The ultraviolet cutoff ˆ ϕ satisfies (1)-(4) below.(1) ˆ ϕ/ω / ∈ L ( R d ) ;(2) ˆ ϕ is rotation invariant, i.e. ˆ ϕ ( k ) = χ ( | k | ) with some real-valued func-tion χ on [0 , ∞ ) ; and ρ ( s ) = | χ ( √ s ) | s ( d − / ∈ L ǫ ([0 , ∞ ) , ds ) for some < ǫ ,and there exists < β < such that | ρ ( s + h ) − ρ ( s ) | ≤ K | h | β for all s and < h ≤ with some constant K ;(3) k ˆ ϕω ( d − / k ∞ < ∞ ;(4) ˆ ϕ ( k ) = 0 for k = 0 . The Hamiltonian H ( α ) with V = 0 is quadratic and can therefore be diag-onalized explicitly, which is carried out in [Ara83, HS01]. Assumption 2.2ensures the existence of a unitary operator diagonalizing H ( α ).Let D + ( s ) = m − α d − d Z | ˆ ϕ ( k ) | s − ω ( k ) + i dk, s ≥ . We see that D + (0) = m + α d − d k ˆ ϕ/ω k > D + ( s )is α d − d πS d − ρ ( s ) = 0 for s = 0, where ρ is defined in (2) of Assumption 2.2and S d − is the volume of the d − D + ( s ) satisfies that lim s →∞ ℜ D + ( s ) = m >
0. These properties followsfrom Assumption 2.2. In particular(2.7) inf s ≥ | D + ( s ) | > . Define(2.8) Λ µj ( k ) = e µj ( k ) ˆ ϕ ( k ) ω / ( k ) D + ( ω ( k )) . Then k Λ µj k ≤ C k ˆ ϕ/ω / k for some constant C .6 roposition 2.3 Under the assumptions 2.1 and 2.2, for each α ∈ R , thereexist unitary operators U and T on H such that both map D ( − ∆) ∩ D ( H f ) onto itself and (2.9) U − H ( α ) U = − m eff ( α ) ∆ + H f + T − V T + g, where m eff ( α ) and g are constants given by m eff ( α ) = m + α (cid:18) d − d (cid:19) k ˆ ϕ/ω k , (2.10) g = d π Z ∞−∞ t α (cid:0) d − d (cid:1) k ˆ ϕ/ ( t + ω ) k m + α (cid:0) d − d (cid:1) k ˆ ϕ/ √ t + ω k dt. (2.11) Here U is defined in (4.29) of [HS01] and T by (2.12) T = exp (cid:18) − i αm eff ( α ) p · φ (cid:19) , where φ = ( φ , ..., φ d ) is the vector field φ µ = 1 √ d − X j =1 Z (cid:16) Λ µj ( k ) a ∗ ( k, j ) + Λ µj ( k ) a ( k, j ) (cid:17) dk. Proof:
See [HS01, Appendix]. ✷ Let h = − ∆. We assume that V ∈ L ( R d ) and V is relatively form-bounded with respect to h with relative bound a <
1, i.e., D ( | V | / ) ⊃ D ( h / ) and(3.1) || V | / ϕ k ≤ a k h / ϕ k + b k ϕ k , ϕ ∈ D ( h / ) , with some b >
0. Then the operators(3.2) R E = ( h − E ) − / | V | / , E < , R ∗ E = | V | / ( h − E ) − / isbounded and thus R E is closable. We denote its closure by the same symbol.Let(3.3) K E = R ∗ E R E . Then K E ( E <
0) is a bounded, positive self-adjoint operator and it holds K E f = | V | / ( h − E ) − | V | / f, f ∈ C ∞ ( R d ) . Now let us consider the case E = 0. Let(3.4) R = h − / | V | / . The self-adjoint operator h − / has the integral kernel h − / ( x, y ) = a d | x − y | d − , d ≥ , where a d = √ π ( d − / / Γ(( d − /
2) and Γ( · ) the Gamma function. It holdsthat (cid:12)(cid:12)(cid:12) ( h − / g, | V | / f ) (cid:12)(cid:12)(cid:12) ≤ a d k g k k| V | / f k d/ ( d +2) for f, g ∈ C ∞ ( R ) by the Hardy-Littlewood-Sobolev inequality. Since f ∈ C ∞ ( R ) and V ∈ L ( R ), one concludes k| V | / f k d/ ( d +2) < ∞ . Thus | V | / f ∈ D ( h − / ) and R is densely defined. Since V is relatively form-bounded with respect to h , R ∗ is also densely defined, and R is closable.We denote the closure by the same symbol. We define(3.5) K = R ∗ R . Next let us introduce assumptions on the external potential V . Assumption 3.1 V satisfies that (1) V ≤ and (2) R is compact. Lemma 3.2
Suppose Assumption 3.1. Then(i) R E , R ∗ E and K E ( E ≤ ) are compact.(ii) k K E k is continuous and monotonously increasing in E ≤ and it holdsthat (3.6) lim E →−∞ k K E k = 0 , lim E ↑ k K E k = k K k . roof: Under (2) of Assumption 3.1, R ∗ and K are compact. Since(3.7) ( f, K E f ) ≤ ( f, K f ) , f ∈ C ∞ ( R d ) , extends to f ∈ L ( R ), K E , R E and R ∗ E are also compact. Thus (i) is proven.We will prove (ii). It is clear from (3.7) that K E is monotonously increas-ing in E . Since R is bounded, (3.7) holds on L ( R d ) and K E = R ∗ (cid:0) ( h − E ) − h (cid:1) R , E ≤ . (3.8)From (3.8) one concludes that k K E − K E ′ k ≤ k K k | E − E ′ || E ′ | for E, E ′ <
0. Hence k K E k is continuous in E <
0. We have to provethe left continuity at E = 0. Since k K E k ≤ k K k ( E < E ↑ k K E k ≤ k K k . By (3.8) we see that K = s- lim E ↑ K E and k K f k = lim E ↑ k K E f k ≤ (cid:18) lim inf E ↑ k K E k (cid:19) k f k , f ∈ L ( R d ) . Hence we have k K k ≤ lim inf E ↑ k K E k and lim E ↑ k K E k = k K k . It re-mains to prove that lim E →−∞ k K E k = 0. Since R ∗ is compact, for any ǫ >
0, there exists a finite rank operator T ǫ = P nk =1 ( ϕ k , · ) ψ k such that n = n ( ǫ ) < ∞ , ϕ k , ψ k ∈ L ( R d ) and k R ∗ − T ǫ k < ǫ . Then it holds that k K E k ≤ ( ǫ + k T ǫ h ( h − E ) − k ) k R k . For any f ∈ L ( R d ), we have k T ǫ h ( h − E ) − f k ≤ n X k =1 k h ( h − E ) − ϕ k kk ψ k k ! k f k and lim E →−∞ k T ǫ h ( h − E ) − k = 0, which completes (ii). ✷ Let(3.9) H p ( m ) = − m ∆ + V. By (ii) of Lemma 3.2, we have lim E →−∞ k| V | / ( h − E ) − / k = 0. Therefore V is infinitesimally form bounded with respect to h and H p ( m ) is the self-adjoint operator associated with the quadratic form f, g m ( h / f, h / g ) + ( | V | / f, | V | / g )9or f, g ∈ D ( h / ). Note that the domain D ( H p ( m )) is independent of m .Under (2) of Assumption 3.1, the essential spectrum of H p ( m ) coincideswith that of − m ∆, hence σ ess ( H p ( m )) = [0 , ∞ ). Next we will estimatethe spectrum of H p ( m ) contained in ( −∞ , ( O ) ( T ), O ⊂ R , be thespectral resolution of self-adjoint operator T and set(3.10) N O ( T ) = dim Ran1l O ( T ) . The Birman-Schwinger principle [Sim05] states that(3.11) (
E < N ( −∞ , Em ] ( H p ( m )) = N [ m , ∞ ) ( K E ) , ( E = 0) N ( −∞ , ( H p ( m )) ≤ N [ m , ∞ ) ( K ) . Now let us define the constant m c by the inverse of the operator norm of K ,(3.12) m c = k K k − . Lemma 3.3
Suppose Assumption 3.1.(1) If m < m c , then N ( −∞ , ( H p ( m )) = 0 .(2) If m > m c , then N ( −∞ , ( H p ( m )) ≥ .Proof: It is immediate to see (1) by the Birman-Schwinger principle (3.11).Suppose m > m c . Then, using the continuity and monotonicity of E → k K k ,see Lemma 3.2, there exists ǫ > m c < k K − ǫ k − ≤ m . Since K − ǫ ispositive and compact, k K − ǫ k ∈ σ p ( K − ǫ ) follows and hence N [ m , ∞ ) ( K − ǫ ) ≥ ✷ Remark 3.4
By Lemma 3.3, the critical mass at zero coupling m c (0) = m c . In the case m > m c , by the proof of Lemma 3.3 one concludes that thebottom of the spectrum of H p ( m ) is strictly negative. For ǫ > m ǫ = k K − ǫ k − . Corollary 3.5
Suppose Assumption 3.1 and m > m ǫ . Then (3.14) inf σ ( H p ( m )) ≤ − ǫm . Proof:
The Birman-Schwinger principle states that 1 ≤ N ( −∞ , − ǫm ] ( H p ( m )),since 1 /m < k K − ǫ k , which implies the corollary. ✷ .2 The case of the Pauli-Fierz model In this subsection we extend the Birman-Schwinger type estimate to thePauli-Fierz Hamiltonian.
Lemma 3.6
Suppose Assumption 3.1. If m < m c , then the zero couplingHamiltonian H p ( m ) + H f has no ground state.Proof: Since the Fock vacuum Ω is the ground state of H f , H p ( m ) + H f hasa ground state if and only if H p ( m ) has a ground state. But H p ( m ) has noground state by Lemma 3.3. Therefore H p ( m ) + H f has no ground state. ✷ From now on we discuss U − H ( α ) U with α = 0. We set(3.15) U − H ( α ) U = H ( α ) + W + g, where H ( α ) = − m eff ( α ) ∆ + H f ,W = T − V T. (3.16)
Theorem 3.7
Suppose Assumptions 2.1, 2.2 and 3.1. If m eff ( α ) < m c , then H ( α ) + W + g has no ground state.Proof: Since g is a constant, we prove the absence of ground state of H ( α ) + W . Since V is negative, so is W . Hence inf σ ( H ( α )+ W ) ≤ inf σ ( H ( α )) = 0.Then it suffices to show that H ( α ) + W has no eigenvalues in ( −∞ , E ∈ ( −∞ ,
0] and set(3.17) K E = | W | / ( H ( α ) − E ) − | W | / , where | W | / is defined by the functional calculus. We shall prove now that if H ( α ) + W has eigenvalue E ∈ ( −∞ , K E has eigenvalue 1. Supposethat ( H ( α ) + W − E ) ϕ = 0 and ϕ = 0, then K E | W | / ϕ = | W | / ϕ. Moreover if | W | / ϕ = 0, then W ϕ = 0 and hence ( H ( α ) − E ) ϕ = 0, but H ( α ) has no eigenvalue by Lemma 3.6. Then | W | / ϕ = 0 is concluded and K E has eigenvalue 1. Then it is sufficient to see kK E k < ( α ) + W has no eigenvalues in ( −∞ , − m eff ( α ) ∆ and T commute, and (cid:13)(cid:13)(cid:13) ( − ∆) / ( H ( α ) − E ) − ( − ∆) / (cid:13)(cid:13)(cid:13) ≤ m eff ( α ) . Then we have kK E k ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | V | / (cid:18) − m eff ( α ) ∆ (cid:19) − / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = m eff ( α ) k K k = m eff ( α ) m c < ✷ In this section we establish the absence, resp. existence, of a ground stateof the Pauli-Fierz Hamiltonian H ( α ) + W . Let κ > V κ ( x ) = V ( x/κ ) /κ by(4.1) H κ = 12 m ( p − αA ) + V κ + H f . We also define K κ by H ( α ) with a ♯ replaced by κa ♯ . Then(4.2) K κ = 12 m ( p − καA ) + V + κ H f .H κ and κ − K κ are unitarily equivalent,(4.3) H κ ∼ = κ − K κ . Let m < m c and ǫ >
0. We define the function α ǫ = ( d − d k ˆ ϕ/ω k ) − / √ m ǫ − m, ǫ > α = ( d − d k ˆ ϕ/ω k ) − / √ m c − m, (4.5)where we recall that m ǫ = k K − ǫ k − for ǫ ≥
0. Note that(1) | α | < α if and only if m eff ( α ) < m c ;122) | α | > α ǫ if and only if m eff ( α ) > m ǫ .Note that α < α ǫ because of m ǫ > m c . Since lim ǫ ↓ m ǫ = m c , it holdsthat lim ǫ ↓ α ǫ = α . We furthermore introduce assumptions on the externalpotential V and ultraviolet cutoff ˆ ϕ . Assumption 4.1
The external potential V and the ultraviolet cutoff ˆ ϕ sat-isfies:(1) V ∈ C ( R d ) and ∇ V ∈ L ∞ ( R d ) ;(2) ˆ ϕ/ω / ∈ L ( R d ) . We briefly comment on (1) of Assumption 4.1. We know that H ( α ) + W = − m eff ( α ) ∆ + V + H f + V ( · − αm eff ( α ) φ ) − V. The term on the right-hand side above, H int = V ( ·− αm eff ( α ) φ ) − V , is regardedas the interaction, and H int ∼ αm eff ( α ) ∇ V ( · ) · φ. By (1) of Assumption 4.1, we have k H int Φ k ≤ C k ( H f + 1) / Φ k with some constant C independent of α . This estimate follows from thefundamental inequality k a ♯ ( f )Φ k ≤ k f / √ ω kk ( H f + 1) / Φ k . Then the inter-action has a uniform bound with respect to the coupling constant α . Sincethe decoupled Hamiltonian − m eff ( α ) ∆ + V + H f has a ground state for suf-ficiently large α , it is expected that H ( α ) + W also has a ground state forsufficiently large α . This is rigorously proven in (1) of Theorem 4.2 below.Now we are in the position to state the main theorem. Theorem 4.2
Suppose Assumptions 2.1, 2.2, 3.1 and 4.1. Then (1) and(2) below hold.(1) For any ǫ > , there exists κ ǫ such that for all κ > κ ǫ , H κ has a uniqueground state for all α such that | α | > α ǫ ,(2) H κ has no ground state for all κ > and all α such that | α | < α . roof: Let U κ (resp. T κ ) be defined by U (resp. T ) with ω and ˆ ϕ replacedby κ ω and κ ˆ ϕ . Then(4.6) U − κ K κ U κ = H p ( m eff ( α )) + κ H f + δV κ + g, where δV κ = T − κ V T κ − V . Note that g is independent of κ . Since U − κ K κ U κ is unitary equivalent to κ H κ , we prove the existence of a ground state for U − κ K κ U κ . Let N = P d − j =1 R a ∗ ( k, j ) a ( k, j ) dk be the number operator. Since H p ( m eff ( α )) has a ground state by the assumption | α | > α ǫ , i.e., m eff ( α ) >m c , it can be shown that U − κ K κ U κ + νN with ν > U − κ K κ U κ + νN by Ψ ν = Ψ ν ( κ ). Since the unit ball in a Hilbert space isweakly compact, there exists a subsequence of Ψ ν ′ such that the weak limitΨ = lim ν ′ → Ψ ν ′ exists. If Ψ = 0, then Ψ is a ground state [AH97]. Let P = 1l [Σ , ( − m eff ( α ) ∆ + V ) ⊗ { } ( H f ) and Σ = inf σ ( H p ( m eff ( α ))). Adoptingthe arguments in the proof of [HS01, Lemma 3.3], we conclude(4.7) (Ψ , P Ψ) ≥ − | α | ε k ˆ ϕ/ω / k κ m eff ( α ) − Dκ κ ( | Σ | − Dκ ) , where ε > D are constants independent of κ and α . Since m eff ( α ) >m ǫ > m ǫ/ ,(4.8) Σ ≤ inf σ ( H p ( m ǫ )) ≤ − ǫ m ǫ by Corollary 3.5. By (4.8) and (4.7) we have(4.9) (Ψ , P Ψ) ≥ κ − (cid:18) ρ ( κ ) − ε k ˆ ϕ/ω / k | α | m eff ( α ) (cid:19) , where ρ ( κ ) = κ − κξκ − ξ = ǫ m ǫ D . Then there exists κ ǫ > κ > κ ǫ and all α ∈ R . Actuallya sufficient condition for the positivity of the right-hand side of (4.9) is(4.10) ρ ( κ ) > ε k ˆ ϕ/ω / k √ m k ˆ ϕ/ω k , since sup α | α | m eff ( α ) = (2 √ m k ˆ ϕ/ω k ) − . Then Ψ = 0 for all κ > κ ǫ . Thus theground state exists for all | α | > α ǫ and all κ > κ ǫ and (1) is complete.14e next show (2). Notice that U − κ H κ U κ = − m eff ( α ) ∆ + H f + T − V κ T + g. Define the unitary operator u κ by ( u κ f )( x ) = k d/ f ( x/κ ). Then we infer V κ = κ − u κ V u − κ , − ∆ = κ − u κ ( − ∆) u − κ and k| V κ | / ( − ∆) − | V κ | / k = κ − k u κ | V | / u − κ ( − ∆) − u κ | V | / u − κ k = k K k . (2) follows from Theorem 3.7. ✷ Corollary 4.3
Let arbitrary δ > be given. Then there exists an externalpotential ˜ V and constants α + > α − such that(1) < α + − α − < δ ;(2) H ( α ) has a ground state for | α | > α + but no ground state for | α | < α − .Proof: Suppose that V satisfies Assumption 3.1. For δ > ǫ > α ǫ − α < δ . Take a sufficiently large κ such that (4.10) is fulfilled, andset ˜ V ( x ) = V ( x/κ ) /κ . Define H ( α ) by the Pauli-Fierz Hamiltonian withpotential ˜ V . Then H ( α ) satisfies (1) and (2) with α + = α ǫ and α − = α . ✷ Remark 4.4 (Upper and lower bound of m c ( α ) ) Corollary 4.3 impliesthe upper and lower bounds(4.11) m − ( α ) ≤ m c ( α ) ≤ m + ( α ) ,m c (0) = m c , where m − ( α ) = m − α d − d k ˆ ϕ/ω k ,m + ( α ) = m ǫ − α d − d k ˆ ϕ/ω k . Fix the coupling constant α . If m < m − ( α ), then there is no ground state,and if m > m + ( α ), then the ground state exists, compare with Fig. 1. Remark 4.5 ( m c ( α ) for sufficiently large α ) Let ( d − d k ˆ ϕ/ω k ) − m ǫ < α .Then by Remark 4.4, H ( α ) has a ground state for arbitrary m >
0. It is anopen problem to establish whether this is an artifact of the dipole approxi-mation or in fact holds also for the Pauli-Fierz operator.15
Examples of external potentials
In this section we give examples of potentials V satisfying Assumption 3.1.The self-adjoint operator h − has the integral kernel h − ( x, y ) = b d | x − y | d − , d ≥ , with b d = 2Γ(( d/ − /π ( d/ − . It holds that(5.1) ( f, K f ) = Z dx Z dyf ( x ) K ( x, y ) f ( y ) , where(5.2) K ( x, y ) = b d | V ( x ) | / | V ( y ) | / | x − y | d − , d ≥ , is the integral kernel of operator K . We recall the Rollnik class R of poten-tials is defined by R = (cid:26) V (cid:12)(cid:12)(cid:12) Z R d dx Z R d dy | V ( x ) V ( y ) || x − y | < ∞ (cid:27) . By the Hardy-Littlewood-Sobolev inequality, R ⊃ L p ( R ) ∩ L r ( R ) with1 /p + 1 /r = 4 /
3. In particular, L / ( R ) ⊂ R . Example 5.1 ( d = 3 and Rollnik class) Let d = 3. Suppose that V isnegative and V ∈ R . Then K ∈ L ( R × R ). Hence K is Hilbert-Schmidtand Assumption 3.1 is satisfied.The example can be extended to dimensions d ≥ Example 5.2 ( d ≥ and V ∈ L d/ ( R d ) ) Let L pw ( R d ) be the set of Lebesguemeasurable function u such that sup β> β (cid:12)(cid:12) { x ∈ R d || u ( x ) > β } (cid:12)(cid:12) /pL < ∞ , where | E | L denotes the Lebesgue measure of E ⊂ R d . Let g ∈ L p ( R d ) and u ∈ L pw ( R d ) for 2 < p < ∞ . Define the operator B u,g by B u,g h = (2 π ) − d/ Z e ikx u ( k ) g ( x ) h ( x ) dx. It is shown in [Cwi77, Theorem, p.97] that B u,g is a compact operator on L ( R d ). It is known that u ( k ) = 2 | k | − ∈ L dw ( R d ) for d ≥
3. Let F denoteFourier transform on L ( R d ), and suppose that V ∈ L d/ ( R d ). Then B u, | V | / is compact on L ( R d ) and then R ∗ = F B u,V / F − is compact. Thus R isalso compact. 16ssume that V ∈ L d/ ( R d ). Let us now see the critical mass of zerocoupling m c = m . By the Hardy-Littlewood-Sobolev inequality, we have(5.3) | ( f, K f ) | ≤ D V k f k , where(5.4) D V = √ π Γ(( d/ − d/
2) + 1) (cid:18) Γ( d )Γ( d/ (cid:19) /d k V k d/ , a constant in (5.4) is proved by Lieb [Lie83]. Then(5.5) k K k ≤ D V . By (5.5) we have m c ≥ D − V . In particular in the case of d = 3,(5.6) m c ≥ √ π / / k V k − / . Acknowledgments:
FH acknowledges support of Grant-in-Aid for Science Research (B) 20340032from JSPS and Grant-in-Aid for Challenging Exploratory Research 22654018from JSPS. SA acknowledges support of Grant-in-Aid for Research Activitystart-up 22840022. We are grateful to Max Lein for helpful comments on themanuscript.
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