Table Of Content

Asymptotic completeness for the massless spin-boson model W. DeRoeck1 Institut fu¨r TheoretischePhysik Universita¨tHeidelberg Philosophenweg16, D69120Heidelberg,Germany 3 1 M.Griesemer2 0 2 Departmentof Mathematics n Universita¨tStuttgart a J Pfaffenwaldring57 0 D-70569Stuttgart, Germany 1 ] A. Kupiainen3 h p - Departmentof Mathematics h t Universityof Helsinki a m P.O.Box 68,FIN-00014,Finland [ 1 Abstract v 7 5 We consider generalized versions of the massless spin-boson model. 3 Buildingontherecentworkin[DRK12a]and[DRK12b],weproveasymp- 2 toticcompleteness. . 1 0 3 1 Introduction 1 : v Xi This paper is concerned with the scattering theory for generalized spin-boson r models with massless bosons. That is, we consider a spin system (an “atom”) a coupled to a scalar field of quantized massless bosons. With the help of previously established properties, such as relaxation to the ground state and a uni- form bound on the number of soft bosons, we now show that asymptotic com- pletenessholdsprovidedtheexcitedstatesoftheuncoupledsystemofspinand 1email: [email protected] 2email: [email protected] 3email: [email protected] 1 bosons have finite life times once the interaction is turned on. (Fermi-Golden Rulecondition.) To describe our main result and its proof we now introduce the system in some detail. We confine ourselves to a concrete system satisfying all our assumptions. More general hypotheses are described in the next section. Our model consists of a small system (atom, spin) coupled to a free bosonic field. The Hilbertspace ofthe total system is H = H H S F ⊗ where H = Cn for some n < (S for “small system”) and the field space H S F ∞ isthe symmetric Fock space overL2(R3). The total Hamiltonian isof the form H = H + H +H S F I ⊗1 1⊗ where H is a hermitian matrix with simple eigenvalues only, and H denotes S F the Hamiltonian of the free massless field. The coupling operator H is of the I form H = λD φˆ(k)a∗ +φˆ(k)a dk. I ⊗ZR3 (cid:0) k k(cid:1) Here D = D∗ is a matrix acting on H , λ R is a sufficiently small coup- S ∈ ling constant, and a∗,a are the usual creation and annihilation operators of a k k mode k R3 satisfying the “Canonical Commutation Relations”. The func- ∈ tion φˆ L2(R3) is a “form factor” that imposes some infrared regularity and ∈ ˆ an ultraviolet cutoff: simple and sufficient assumptions are that φ has compact support and that φˆ C3(R3 0 )with ∈ \{ } φˆ(k) = k (α−1)/2, k 1, | | | | ≤ forsomeα > 0. Thisassumptionensures,e.g.,thatH hasauniquegroundstate Ψ . To rule out the existence of excited bound states we assume the Fermi- gs Golden Rule condition stated in the next section. Further spectral input is not needed but our results do certainly have non-trivial consequences for the spectrum of H. Under the time-evolution generated by H it is expected that every excited state relaxes to the ground state by emission of photons whose dynamics is asymptotically free. The existence of excited states with this property is well known [FGS01, GZ09]; they are spanned by products of asymptotic creation operators appliedtothe ground state Ψ , that is, byvectors ofthe form: gs a∗(f )...a∗(f )Ψ = lim ei(H−E)ta∗(f )...a∗(f )Ψ , + 1 + m gs 1,t m,t gs t→∞ 2 where f = e−iωtf and ω(k) = k . Asymptotic completeness of Rayleigh scat- t | | H tering means that the span of these vectors is dense in . In particular, the representation of the CCR given by the asymptotic creation and annihilation is equivalent to the Fock representation; non-Fock representation such as those discussed in [DG04]do notoccur. Asymptotic completeness of Rayleigh scattering is expected to hold for a large class of models of atoms interacting with quantized radiation. Yet, des- pite considerable efforts [HS95, DG99, FGS02a, Ge´r02], it has so far resisted a rigorous proof. The main stumbling blocks were the lack of time-independent photon bounds and a the lack of a quantitative understanding of a property called relaxation to the ground state (the zero temperature analog of return to equlibrium), which isknown to follow from asymptotic completeness [FGS01]. For the spin-boson model, these problems were solved in the previous papers [DRK12a, DRK12b]. We now use these results and expand on them to give a complete proof of asymptotic completeness for a fairly large class of massless spin-bosons system. Independently from us Faupin and Sigal had embarked on a similar project using results from [DRK12a]. They obtain AC for a class of spin-boson modelsvery similarto ours. Ourproofofasymptoticcompletenessreliesonmethodsandtoolsdeveloped forthepurposeofestablishingtherelaxationtoequilibrium(andnon-equilibrium steady states) of systems at positive temperature. These methods are based on analogues with one-dimensional statistical mechanics. In fact, by a cluster expansion we have previously shown (in [DRK12a]) that a weak form of relaxation to the ground state holds, bywhich we mean that lim Ψ ,OΨ = Ψ ,OΨ (1.1) t t gs gs t→∞h i h i for a suitable C∗ algebra of observables O. Our second key tool, established by the same cluster expansion, states that the number of soft photons emitted in the processofrelaxation isuniformlybound intime. More precisely, there exits a constant κ > 0such that sup Ψ ,eκNΨ < (1.2) t t h i ∞ t where N denotesthe numberoperator on Fock space. Ourthird keyingredient concernsthenumberofbosons inaball x < vtwherev < 1,thespeedoflight. | | Weshow that Ψ ,dΓ(θ )Ψ = Ψ ,dΓ(θ )Ψ + ( t −α) (1.3) t t t gs t gs h i h i O h i whereθ denotesasmoothedcharacteristicfunctionoftheset x R3 x vt t { ∈ | | | ≤ } with some v < 1. Moreover, ifθ isreplaced byθ then (1.3) holds uniformly in t tc 3 t with λ −2 t t. This propagation bound and soft-photon bound were es- c c | | ≤ ≤ tablished in [DRK12b] by a slight variation of the cluster expansion mentioned above(inthecaseofthesoft-photon bound,thisadjustmentwasevenunneces- sary and one could have simply copied the treatment of [DRK12a], as was also remarked in [FS12a]) Our basic strategy for proving asymptotic completeness is the usual one from time-dependent scattering theory [SS87, Gra90]. In the present context this means that we construct a suitable (right-)inverse Z of the wave operator W define by + W a∗(f )...a∗(f )Ω = a∗(f )...a∗(f )Ψ . + 1 m + 1 + m gs The identity W Z = in H shows that W has full range, which is equival- + + 1 ent to asymptotic completeness as explained above. Our tools, however, are certainly not the usual ones: the properties (1.1)-(1.3) augmented by a strong form of the local relaxation (1.1), see Proposition 4.1, are enough for establish- ingboththeexistenceofZ andtheidentityW Z = . Importantstandardtools + 1 fromscatteringtheory,suchaspropagationestimatesandMourreestimates,are not neededin the presentwork. WeconcludethisintroductionwithadiscussionofpreviousworkonRayleigh scattering and asymptotic completeness (AC) in models similar to ours. In ’97 Spohn considered an electron that is bound by a perturbed harmonic po- tential and coupled to the quantized radiation field in dipole approximation [Spo97]. Knowing ACfor the harmonically bound particle, a result due to Arai [Ara83], he concludes AC in his perturbed model by summing a Dyson series for the inverse wave operator. This is the first proof of AC in a non-solvable model with massless bosons. AC in more general models of a bound particle coupled to massive bosons was established by [DG99] and by [FGS02a]. These works adaptthe methods from many-body quantum scattering, i.e. Mourre estimates and propagation estimates, to non-relativistic QFT. Consequently the Mourre estimate for massless bosons was a main concern of subsequent work [Ski98, GGM04, FGS08]. For interesting partial results on massless boson scattering assuming aMourre estimate we referto [Ge´r02]. Most recently, in [FS12b, FS12a], Faupin and Sigal succeeded in establish- ing AC for a large class of models of atoms interacting with quantized radiation. They assume a time-independent bound on Ψ ,dΓ(1/ k )2Ψ . Based on t t h | | i this assumption they establish AC for a large variety of models including non- relativistic QED and spin-boson models. Following the strategy in [DRK12a], a bound on Ψ ,dΓ(1/ k )2Ψ can be obtained in the same way as a bound on t t h | | i Ψ ,N2Ψ uponslightlystrenghtening theinfrared assumption,asisremarked t t h i in[FS12a]. Foracertainclassofspin-bosonmodels(similartoours),Faupinand 4 SigalhavethusprovenACpriortous. However,giventhemethodsandresults from [DRK12a], which are neededby them aswell, we believe our approach to ACfor spin-boson modelsissimpler. Acknowledgements M.G. thanks I.M. Sigal for explanations on his work with Faupin. W.D.R. ac- knowledges the support of the DFG and A.K. is supported by the ERC and the AcademyofFinland. 2 Assumptions and Results Wenowdescribe the classofspin-boson modelsconsidered inthis paperalong with all assumptions. It is instructive to do this in d rather than 3 space dimen- sion, although we shall later confine ourselves to 3 dimensions for simplicity. Thissection also servesusfor collecting frequently used notations. 2.1 Notations ThesymmetricFock-space overaone-particlespacehisdenotedbyΓ(h)which isdefined by ∞ Γ(h) = P h⊗n (2.1) S n⊕=0 with P is the projection to symmetric tensors and h⊗0 C. Hence H = Γ(h) S F ≡ with h = L2(Rd). Wedefinethe finite-photon space fin := n∈N N≤nHF, (2.2) D ∪ 1 H and wewrite also for . fin S fin D ⊗D Bya∗(f)anda(f)wedenotethesmearedcreationandannihilationoperators in H thatare related to a∗ anda by F k k a∗(f) = f(k)a∗dk, a(f) = f(k)a dk. k k Z Z Theirsumistheself-adjointfieldoperatorΦ(f) = a∗(f)+a(f),whichgenerates the Weyl-operator (f) = eiΦ(f). (2.3) W Ourassumptionsonformfactorandinitialstatesareconvenientlyexpressedin terms ofthe following spaces: 5 Definition 2.1. For 0 < α < 1 the subspace h h consists of all ψ h such α ⊂ ∈ that ψˆ C3(Rd 0 ) has compact support, and, for all multi-indices m with ∈ \ { } m 3, | | ≤ ∂mψˆ(k) C k (β−d+2)/2−|m| (2.4) | k | ≤ | | forsomeβ > α andC < . Byaneasyapplication ofLemmaA.1of[DRK12b], ∞ anyψ h satisfies α ∈ ψ(x) C(1+ x )−min{d+β2+2,3} (2.5) | | ≤ | | for some β > α. H The subspace isdefinedby α D ⊂ := Span ψ (f)Ω ψ H , f h . (2.6) α S s S α D { ⊗W | ∈ ∈ } Itisdense in H becauseh isdensein h. α The following notation is very convenient and often used in this paper: If Ψ H = H H andγ H , then Ψ = γ Ψ H isdefinedby a b a γ b ∈ ⊗ ∈ h | ∈ η γ Ψ = η γ Ψ, for η H . a ⊗h | | ih |⊗1 ∈ (cid:0) (cid:1) Itfollows that η Ψ ,Φ = Ψ ,Φ and γ Ψ 2 = Ψ,( γ γ )Ψ . γ γ η h ⊗ i h i kh | k h | ih |⊗1 i 2.2 Model and Assumptions Recall from the introduction that the system we consider is described by a Hamiltonian H on H = H H thatisof the form S F ⊗ H = H + H +H . (2.7) S F I ⊗1 1⊗ The free field operator H andthe interaction operator H are given by F I H = dΓ(ω) = ω(k)a∗a dk, ω(k) = k , F Z k k | | Rd H = λD Φ(φ). I ⊗ InthefollowingwedescribeourassumptionsonthematricesH ,D andthe S form factor φ. These choices and assumptions are assumed to hold throughout the article withthe exception ofthe presentSection 2,where wewillstate some earlierresultsthatdonotrequirethesestrongassumptions. Theinfrared(small Fourier mode k) behavior of the form factor determines temporal correlations in the model andsome regularity neark = 0 isneeded: Assumption 2.1 (α-Infrared regularity). The form factor φ is in h and the di- α mension d = 3. 6 Ofcourse,therestriction tod = 3isnotreallynecessaryprovidedonemodi- fiesslightlythe infrared assumption butweprefertokeepitforsimplicity. Our second assumption ensures that the couplingis effective: Assumption 2.2 (Fermi Golden Rule). We assume that the spectrum of H is S non-degenerate (all eigenvaluesare simple)and we let e := minσ(H ) (atomic 0 S ground state energy). Most importantly, we assume that for any eigenvalue e σ(H ),e = e , there isa set e n ofeigenvaluessuch that ∈ S 6 0 { i}i=1 e = e > e > ... > e > e , (2.8) n n−1 1 0 and for alli = 0,...,n 1, − Tr[P DP DP ] dkδ( k e e ) φˆ(k) 2 > 0. (2.9) i i+1 i i i+1 Z | |− − | | Rd HereP denotestheprojection ontotheeigenspaceassociatedwithe . Theright i i ˆ hand sideis well-definedsince φ iscontinuous. 2.3 Results Thefollowingresultsonthegroundstateanditspropertiesareprerequisitesfor our definition of the wave operator. Theorem 2.3 ([BFS98, Ge´r00]). Let the form factor φ satisfy φ/ω h. Then the ∈ ground-state energy E = inf Ψ,HΨ (2.10) gs kΨk=1h i isfinite; E infH , and there isavectorΨ H such that gs S gs ≤ ∈ HΨ = E Ψ (2.11) gs gs gs Ifthecouplingconstant λ issufficientlysmall,thensuchaΨ isunique (uptoscalar gs | | multiplication). The uniqueness of the ground state for small λ follows from the simplicity | | of the eigenvalue e := minσ(H ) by an overlap argument that is probably due 0 S to [BFS98]. Additionally,weneedanexponentiallocalizationpropertyofphotons,namely: Lemma 2.4. Assume Assumptions 2.1 and 2.2 and let λ be sufficiently small. Then, | | for someκ > 0, Ψ ,eκNΨ < . (2.12) gs gs h i ∞ 7 This property is proven in this paper, at the end of Section 4. Earlierresults, exhibiting somewhat weakerlocalization, can be found in [Hir03]. To describe scattering, let us introduce the standard identification operator I : H H F → Ia∗(f )a∗(f )...a∗(f )Ω := a∗(f )a∗(f )...a∗(f )Ψ (2.13) 1 2 m 1 2 m gs for f ,...,f such that (1 + k −1)f h, and then extended to a closed oper- 1 m j | | ∈ ator, see e.g. [HS95]. Note that the adjoint of I is densely defined and hence I is closable. The well-known Cook’s argument yields the following standard result: Theorem 2.5. Thewave operators W Ψ := lim W Ψ, W Ψ = e∓itHIe±it(Egs+HF)Ψ (2.14) ± t t t→∞ exist for Ψ in a dense subset of . They satisfy W Ψ = Ψ and therefore W fin ± ± D k k k k H H extendsto anisometry . F → Existence of wave operators for a large class of models with massive and masslessphotons isestablished in [HK69, DG99, FGS01,GZ09]. Our main result complementsthe former statement byestablishing Theorem 2.6 (Asymptotic completeness). Assume Assumptions 2.1 and 2.2. Then thereis aλ > 0such thatfor any couplingstrength λ satisfying0 < λ λ ; 0 0 | | ≤ RanW = H . (2.15) ± Remark2.7. WewillonlydiscussW . TotreatW ,itsufficestodefineatime-reversal + − operator (antilinear involution) Θ = Θ Θ (2.16) S F ⊗ where Θ is complex conjugation in a basis diagonalizing H , and Θ = Γ(θ ) with S S F F θ ψ(q) = ψ( q). Then we check that, if H satisfies our assumptions, then ΘHΘ F − satisfiesthemas well. 3 Technical tools In this section, we list all the tools that we use which are not proven in the presentpaper,oronlyintheappendix. Bysmooth indicatorsθ,wemeanspher- ically symmetric functions in C∞(Rd), i.e. θ(x) = θ( x ), with 0 θ 1, and 0 | | ≤ ≤ wewriteθ (x) = θ(x/t). Allthefollowinglemmataholdforsufficientlysmallλ, t 8 uniformly in λ. From now on, we always assume that Assumptions 2.1 and 2.2 hold. Moreover, we write throughout this section Ψ = e−itHΨ with Ψ . t 0 0 α ∈ D The first tool isthe weakrelaxation to the ground state. Lemma 3.1 (Weak relaxation [DRK12a]). Let the C∗-algebra be generated by α W S (H ) and (ψ),ψ h . For any O , wehave S α α ∈ B W ∈ ∈ W lim Ψ ,OΨ = Ψ ,OΨ (3.1) t t gs gs t→∞h i h i Wewillalsoneedthetime-independentboundonthetotalnumberofphotons. Lemma3.2(Photon bound [DRK12a]). Thereis aκ > 0 suchthat sup Ψ ,eκNΨ C < (3.2) t t h i ≤ ∞ t with C depending onΨ and κ. 0 α ∈ D The following result describes the localization of photons in the ground state. Lemma 3.3 (Localization of ground state photons). Let θ be a smooth indicator such thatSuppθ 0 < x . Then ⊂ { | |} Ψ ,dΓ(θ )Ψ = ( t −α) (3.3) gs t gs h i O h i Thenextlemmaboundsthenumberofphotonsaftertimetinspatialregions around the atom. Lemma 3.4 (Propagation bound [DRK12b]). Let θ be a smooth indicator such that Suppθ x < 1 . Thenforanytimet λ −2thelimita(θ,t ) := lim Ψ ,dΓ(θ )Ψ ⊂ {| | } c ≥ | | c t→∞h t tc ti existsand moreover Ψ ,dΓ(θ )Ψ = a(θ,t )+ ( t −α) (3.4) h t tc ti c O h i uniformlyfor t [ λ −2,t]. c ∈ | | In Section 4, wewill identify a(θ,t) = Ψ ,dΓ(θ )Ψ (3.5) gs t gs h i Note thatinthe casewhere0 Suppθ,wecancombine thispropagation bound 6∈ and (3.5)with Lemma3.3 toconclude that Ψ ,dΓ(θ )Ψ = ( t −α) (3.6) t t t h i O h i Finally, weneed amore detailedphoton bound Lemma3.5(Soft photon bound [DRK12b]). For any ǫ > 0,we have sup Ψ ,dΓ( )Ψ Cǫα/2 (3.7) t |k|≤ǫ t h 1 i ≤ t with C depending onΨ and α,but not onǫ. 0 α ∈ D 9 4 Strong local relaxation In Lemma 3.1, we saw that the system relaxes to the ground state in a weak sense. The following result expresses that, locally in space, it also relaxes in a strong sense. This will also allow us to identify the numbers a(θ,t) in Lemma 3.4. Let B(r) R3 be the ball with radius r in position space and Bc(r) = R3 ⊂ \ B(r). Wewill often split H = H H (4.1) B(r) Bc(r) ⊗ whereitisunderstoodthatthesmallsystemisincorporatedintheinnersphere, i.e. H := H Γ(L2(B(r))), H := Γ(L2(Bc(r))) (4.2) B(r) S Bc(r) ⊗ Let us then write in general Tr and Tr for the partial traces over H B(r) Bc(r) B(r) H and ,respectively. Bc(r) Proposition 4.1 (Strong local relaxation). Fix r > 0. Then lim Tr Ψ Ψ = Tr Ψ Ψ (4.3) Bc(r) t t Bc(r) gs gs t→∞ | ih | | ih | Proof. Let∆ betheLaplacianontheballB(r)withNeumannboundarycon- B(r) ditions. Then,for anyψ in its form domain, ψ,∆ ψ = dx ψ(x) 2 (4.4) −h B(r) iL2(B(r)) Z |∇ | B(r) and this form domain is in fact the Sobolev space W2,1(B(r)) (completion of C∞(B(r)) w.r.t. the norm ψ + ψ ). Hence, in the sense of quadratic forms k k k∇ k on the form domain of ∆ , Rd ∆ ∆ . (4.5) B(r) B(r) B(r) Rd −1 1 ≤ − For Ψ Dom(H ), we then get F ∈ Ψ,H2Ψ Ψ,dΓ( ∆ )Ψ Ψ,dΓ( ∆ )Ψ (4.6) h F i ≥ h − Rd i ≥ h −1B(r) B(r)1B(r) i Using the relative boundedness of H w.r.t to H , the boundedness of H and I F S the fact thatH2 isconserved, we obtain sup Ψ ,H2Ψ C Ψ 2 + HΨ 2. (4.7) h t F ti ≤ k 0k k 0k t LetN = dΓ(1 ), then bythe photon numberbound wehave B(r) B(r) sup Ψ ,N2 Ψ C. (4.8) h t B(r) ti ≤ t 10