Asymptotic Completeness for Rayleigh Scattering arXiv:math-ph/0103048v1 30 Mar 2001 J. Fröhlich1∗, M. Griesemer2†and B. Schlein1‡ 1. Theoretical Physics, ETH–Hönggerberg, CH–8093 Zürich, Switzerland 2. Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294 30 March, 2001 Abstract It is expected that the state of an atom or molecule, initially put into an excited state with an energy below the ionization threshold, relaxes to a groundstate by spontaneous emission of photons which propagate to spatial infinity. In this paper, this picture is established for a large class of models of non-relativistic atoms and molecules coupled to the quantized radiation field, but with the simplifying feature that an (arbitrarily tiny, but positive) infrared cutoff is imposed on the interaction Hamiltonian. This result relies on a proof of asymptotic completeness for Rayleigh scattering of light on an atom. We establish asymptotic completeness of Rayleigh scattering for a class of model Hamiltonians with the features that the atomic Hamiltonian has point spectrum coexisting with absolutely continuous spectrum, and that either an infrared cutoff is imposed on the interaction Hamiltonian or photons are treated as massive particles. We show that, for models of massless photons, the spectrum of the Hamiltonian strictly below the ionization threshold is purely continuous, except for the groundstate energy. 1 Introduction Ever since the inception of the quantum theory of atoms interacting with the quantized radiation field, theoreticians have expected that when an atom (with an infinitely heavy nucleus) in a state where all electrons are bound to the nucleus is targeted by a finite number of photons in such a way that the total energy of the composed system remains ∗ juerg@itp.phys.ethz.ch marcel@math.uab.edu ‡ schlein@itp.phys.ethz.ch † 1 2 below the ionization threshold of the atom the following physical processes unfold: First, some of the electrons in the shells of the atom are lifted into an excited state by absorbing incoming photons; but, since the total energy is below the ionization threshold, they remain bound to the nucleus. As time goes on, the excited state relaxes to a groundstate of the atom by spontaneous emission of photons, which propagate essentially freely to spatial infinity. Thus, asymptotically, the state of the total system, atom plus quantized radiation field, describes an atom in its ground state and a cloud of photons escaping to infinity with the velocity of light. Relaxation of an excited initial state to a groundstate by emission of outgoing radiation is the simplest example of an ”irreversible process”, accompanied by information loss at infinity, occuring in an open quantum system with infinitely many degrees of freedom. It would seem worthwhile to attempt to understand this process mathematically precisely; (see Sect. 10). The picture described above suggests that the scattering operator describing Rayleigh scattering of light off an atom with a static nucleus, i.e., the scattering operator restricted to the subspace of states with energies below the ionization threshold of the atom, is unitary; (see Sect. 9 and 10). If true - one says that ”asymptotic completeness” (AC) is valid for Rayleigh scattering. In attempting to establish this picture mathematically, one faces the problem that, in the scattering of light at an atom, an arbitrarily large number of soft photons of arbitrarily small total energy can, in priciple, be produced (in processes of high order in the feinstructure constant). Perturbative calculations of scattering amplitudes suggest, however, that in the analysis of Rayleigh scattering of light at an atom with a static nucleus, one does not encouter a genuine infrared catastrophe of the kind first described by Bloch and Nordsieck. Yet, the mathematical problems connected with controlling very large numbers of very soft photons in a mathematically rigorous, non-perturbative way are quite substantial and have not been fully mastered, yet. In order to simplify matters to a manageable size, we propose to study Rayleigh scattering and the phenomenon of relaxation to a ground state for models of massive photons and for models of massless photons with an infrared cutoff. In this paper, results in this direction are proven. In order to avoid inessential technical complications, we consider models of ”scalar photons”, bosons. But our analysis can be extended to the quantum electrodynamics of non-relativistic electrons (bound to a static nucleus) interacting with the quantized electromagnetic field, provided we work within the dipole approximation and impose an (arbitrarily tiny) infrared cutoff on the electron-field interaction. We plan to study more difficult scattering problems in similar models and their consequences for ”irreversible phenomena” in future papers. Next, we describe our main results in more detail. To avoid starting with a list of assumptions, we formulate our results for a concrete, simple model, which is physically relevant and captures the main features of the problems we propose to solve. For precise assumptions and other models see Section 3. Consider N non-relativistic electrons subject to a potential V , which may also include two–body interactions. The electrons are linearly coupled to a quantized field of 3 FGSch, 30/Mar/01—Asymptotic Completeness relativistic bosons. The Hamilton operator of this system is H = K ⊗ 1 + 1 ⊗ dΓ(ω) + φ(G) (1) and acts on the Hilbert space H = Hel ⊗ F , where Hel = ∧N L2 (R3 ; C2 ) is the antisymmetric tensor product of N copies of L2 (R3 ; C2 ), and F is the bosonic Fock space over L2 (R3 ). The operator K describes the time evolution of the electrons without radiation and is given by K = −∆ + V , where ∆ denotes the Laplacian on R3N . (In our units, ~ and the electron mass are equal to one.) Electron spin will be neglected henceforth. We assume that V− , the negative part of V , is infinitesimally form–bounded with respect to −∆, that K is essentially self-adjoint on C0∞ (R3N ), and that inf σess (K) > inf σ(K), i.e., that K has bound states. In particular, inf σ(K) is an isolated eigenvalue of K. The operator dΓ(ω) describes the energy of free bosons. Formally Z dΓ(ω) = dk ω(k)a∗ (k)a(k) R3 where a(k) and a∗ (k) are the usual annihilation- and creation√operators (operator–valued distributions) depending on the wave vector k, and ω(k) = k 2 + m2 is the energy of a relativistic particle with momentum k and mass m ≥ 0. The operator Z   ∗ φ(G) = dk Gx (k)a(k) + Gx (k)a (k) describes P the interaction between electrons and bosons. In this introduction, we choose −ik·xi κ(k), and κ(k) ∈ C0∞ (R3 ). If m = 0 we cut off the infrared modes Gx (k) = N i=1 e from the interaction by assuming that κ(k) = 0, for |k| small. Thus, in any case inf k∈supp κ ω(k) > 0. (2) Without this assumption, it is presently not known how to control the number of soft bosons produced in the course of the time evolution. Let Σ denote the ionization threshold. This is the smallest energy the system can reach when one or several electrons have been moved to infinity. States with energy below Σ are exponentially localized w.r. to the electron coordinates. More precisely keα|x| E∆ (H)k < ∞ if ∆ ⊂ (−∞, Σ) and sup ∆ + α2 < Σ. Clearly Σ ≥ inf σ(H), and, for one-electron atoms and if the coupling is weak enough, it is known that Σ > inf σ(H) [GLL00, BFS98]. Of course, one expects Σ > inf σ(H) for all neutral atoms and molecules. Note that Σ = ∞ if V (x) → ∞, for |x| → ∞, i.e., when σ(K) is discrete. This situation is included in our analysis, but one of our main points is to prove results on Rayleigh scattering when Σ < ∞. 4 For states ϕ ∈ H with energy below Σ, that is ϕ = E(−∞,Σ) (H)ϕ, one expects that ϕt = e−iHt ϕ is well approximated, in the distant future, by a linear combination of states of the form a∗ (h1,t ) . . . a∗ (hn,t )e−iEt ϕb (3) where ϕb is a bound state of H, Hϕb = Eϕb , hj,t = e−iωt hj , j = 1, . . . , n, are R one–particle wave functions of freely propagating bosons, and a∗ (h) = h(k)a(k) dk, a(h) := (a∗ (h))∗ . This property is called asymptotic completeness (AC) for Rayleigh scattering. It asserts, in particular, that the asymptotic dynamics of escaping photons is well approximated by their free dynamics. This requires that the strength of the interaction between ballistically moving bosons and electrons decays at an integrable rate. In our model of non-relativistic electrons bound to static nuclei this is true, thanks to the spatial localization of the electrons. For massless bosons, it follows more generally from the fact that the propagation velocity of electrons is strictly smaller than the velocity of the bosons, i.e., the velocity of light, [FGS00]. To give a mathematically more precise formulation of AC, let us introduce asymptotic creation operators a∗+ (h). Let ϕ = Eλ (H)ϕ, hj ∈ L2 (R3 ), j = 1, . . . , n and Mj = sup{ω(k)|hj (k) 6= 0}. Then a∗+ (h1 ) . . . a∗+ (hn )ϕ = lim eiHt a∗ (h1,t ) . . . a∗ (hn,t )e−iHt ϕ t→∞ (4) exists if λ+ n X M(hi ) < Σ. (5) i=1 Asymptotic completeness of Rayleigh scattering is the statement that linear combinations of states of the form (4), (5), with ϕ ∈ Hpp (H), are dense in E(−∞,Σ) (H)H. Assuming (2), we prove AC for all m ≥ 0, with an infrared cutoff imposed when m = 0. If m = 0 one can show more: If the Pauli principle for the electrons is neglected then H has a unique ground state ϕ0 [BFS98, GLL00] but no other stationary states with energy below inf σess (K) − ε, ε > 0, provided g is sufficiently small and the life times of all excited states of Hg=0 , as computed by Fermi’s Golden Rule, are finite. This follows from results of Bach et al. [BFSS99] together with an argument given in the present paper, which excludes eigenvalues close to inf σ(H). As a consequence, states of the form X a∗+ (h1 ) . . . a∗+ (hn )ϕ0 , inf σ(H) + M(hi ) ≤ inf σess (K) − ε (6) are dense in Einf σess (K)−ε (H)H, for some ε > 0 depending on the coupling constant. Moreover, we show that every state ψt ∈ Einf σess (K)−ε (H)H eventually relaxes to the ground state ϕ0 , in the following sense: Let A denote the C ∗ algebra generated by the Weyl operators eiφ(h) where φ(h) = a(h) + a∗ (h), and h ∈ S(R3 ), the Schwartz space of test functions. By taking sums of tensor products of operators in A with arbitrary bounded operators acting on the N–electron Hilbert space one obtains a C ∗ algebra Ã. 5 FGSch, 30/Mar/01—Asymptotic Completeness ”Relaxation of ψt to the ground state ϕ0 ” is the statement that lim hψt , Aψt i = hϕ0 , Aϕ0 i hψ, ψi, t→∞ (7) for all operator A ∈ Ã, and for all ψ ∈ Einf σess (K)−ε (H)H. This is our second main result. It essentially follows from asymptotic completeness and, of course, from the absence of eigenvalues in (inf σ(H), inf σess (K) − ε]. Asymptotic Completeness for massive bosons was previously established by Derezinski and Gérard, for confined electrons (i.e., Σ = ∞) and under a somewhat unphysical short-range assumption [DG99]. From this important paper, and from [DG00], we have learned how to translate techniques from N-body quantum theory to quantum field theory. Before [DG99], Arai had established AC in the standard model of non-relativistic QED in the dipole–approximation and with V (x) = x2 , a model which is explicitly soluble [Ara83]. Later, Spohn extended this result, using the Dyson series, to include potentials which are small perturbations of x2 [Spo97]. Our proof of asymptotic completeness adapts methods and techniques from the scattering theory of N-particle Schrödinger operators to the present situation. In particular, we use a Mourre estimate and propagation estimates, and we rely on localization techniques in bosonic configuration space. As in the more recent papers on N-body quantum scattering, we derive AC from the fact that the mean square diameter of hdΓ(y 2 )iψt of a given state ψt , with y the position operator in bosonic configuration space, diverges like t2 if ψ is away, in energy, from thresholds and eigenvalues. Correspondingly, a central object in our proof is an asymptotic observable W that measures the square of the asymptotic velocities of the escaping photons. That is, hW i = lim hdΓ(y 2 /2t2 )it t→∞ = lim t→∞ d hdΓ(y 2/2t)it dt (8) Thanks to the ballistic escape property mentioned above, W is positive and thus invertible, on suitable spectral subspaces. We construct a Deift-Simon wave operator W+ with the property that W+ W −1 is a right-inverse of an extended wave operator on a dense subspace of Hcont (H) ∩ E(−∞,Σ) (H)H, the orthogonal complement of all eigenvectors. The proof is completed with an inductive argument explained further below. Let us temporarily assume that the interaction φ(G) vanishes, in order to explain the ideas underlying the construction of W and W+ in their purest form. The main observation is that D2 1 y2 = (∇ω − y)2 ≥ 0, 2t t (9) where D denotes the Heisenberg derivative [iω, .] + ∂/∂t. As a consequence the time derivative of the expression (8), whose limit is hW i, is non-negative. Since d/dthdΓ(y 2/2t)it is bounded uniformly in time, it follows that d2 /dt2 hdΓ(y 2/2t)it = hdΓ(D 2 [y 2 /2t])it is integrable. This propagation estimate, with small modifications to accommodate the interaction, proves existence of hW i and suffices to establish existence of W as a strong 6 limit. Existence of W+ requires, in addition, some geometry in bosonic configuration space. Once these asymptotic operators are constructed, AC follows by induction in the number of energy intervals of length m, the smallest energy of a boson. Assuming that AC holds on the spectral subspace of H corresponding to energies below min(Σ, m(n − 1)), we prove AC for energies below min(Σ, mn). Roughly speaking, the positivity of W on suitable spectral subspaces E∆ (H)H, where ∆ ⊂ min(Σ, mn), allows us to show that at least one boson of a given state ϕ ∈ E∆ (H)H escapes to infinity. It thus carries away an energy of at least m. The energy distribution of the remaining system is contained in (−∞, m(n − 1)), where asymptotic completeness holds by assumption, and hence ϕt , for large t, is of the form (3). Obviously the positivity of the boson mass, or condition (2), in the case of more general dispersion relations, is absolutely crucial in this argument. Our strategy and the constructions of W and W+ are strongly inspired by ideas and constructions developed by Graf and Schenker for N-body quantum scattering theory [GS97]. The Mourre estimate we use is essentially the one of Derezinsky and Gérard [DG99]. We follow closely the notation of [DG99]; but, otherwise, there are only few similarities between our approach to AC and the one in [DG99]. Our paper is organized as follows. In Sect. 2, we consider the quantum theory of the bosons. We briefly review the standard formalism of second quantization and introduce some basic notions that are useful in scattering theory. In Sect. 3, we describe the physical systems and define the models studied in this paper. We formulate some basic assumptions on the Hamiltonians generating the dynamics in these models which will be important to gain mathematical control over Rayleigh scattering. We describe some concrete examples of models. In Sect. 4, we review and prove results on spectral properties of the Hamiltonians of our models. In particular, we recapitulate a theorem on the existence of a ground state and on the location of the essential spectrum. We state a Mourre estimate and use it to establish properties of the continuous and point spectrum and to prove a virial theorem. In Sect. 5, we construct the Møller wave operators of our models on spectral subspaces corresponding to bound electrons, using a variant of Cook’s argument; see also [FGS00] for more detailed results. Sect. 6 contains our basic propagation estimates needed for the construction of an asymptotic observable and of a Deift-Simon wave operator. An asymptotic observable, W , is constructed in Sect. 7 and shown to be selfadjoint, positive on appropriate spectral subspaces, and to commute with the Hamiltonian H. In Sect. 8, a Deift-Simon wave operator is constructed and shown to invert, with respect to W , an extended variant of the Møller wave operator on spectral subspaces where W is positive. By combining the results of previous sections, asymptotic completeness for Rayleigh scattering is established in Sect. 9 with the help of an inductive argument in the number of asymptotic bosons. In Sect. 10, models of massless bosons with an infrared cutoff are analyzed, and the phenomenon of relaxation to a groundstate is exhibited. A novel positive-commutator estimate is proven which, together with results in [BFSS99], excludes the existence of 7 FGSch, 30/Mar/01—Asymptotic Completeness point spectrum above the groundstate energy (and below the ionization threshold). Our most difficult and innovative results appear in Sects. 6 through 10. Various technical arguments are deferred to appendices, the most important ones being Appendices E through G. Acknowledgements. We thank V. Bach, Chr. Gérard, G.-M. Graf and I.M. Sigal for many useful discussions of problems related to those studied in this paper. 2 Fock Space and Second Quantization The natural Hilbert space of states of the radiation field is the Fock space. Let h be a complex Hilbert space and let ⊗ns h denote the n-fold symmetric tensor product of h. The bosonic Fock space over h F = F (h) = ⊕n≥0 ⊗ns h is the space of sequences ϕ = (ϕn )n≥0 , with ϕ0 ∈ C, ϕn ∈ ⊗ns h, and with the scalar product given by hϕ, ψi := X (ϕn , ψn ), n≥0 where (ϕn , ψn ) denotes the inner product in ⊗n h. The vector Ω = (1, 0, . . . ) ∈ F is called the vacuum. By F0 ⊂ F we denote the dense subspace of vectors ϕ for which ϕn = 0, for all but finitely many n. The number operator N on F is defined by (Nϕ)n = nϕn . 2.1 Creation- and Annihilation Operators The creation operator a∗ (h), h ∈ h, on F is defined by a∗ (h)ϕ = √ n S(h ⊗ ϕ), for ϕ ∈ ⊗sn−1 h, and extended by linearity to F0 . Here S ∈ B(⊗n h) denotes the orthogonal projection onto the subspace ⊗ns h ⊂ ⊗n h. The annihilation operator a(h) is the adjoint of a∗ (h) restricted to F0 . Creation- and annihilation operators satisfy the canonical commutation relations (CCR) [a(g), a∗ (h)] = (g, h), [a# (g), a# (h)] = 0. In particular [a(h), a∗ (h)] = khk2 , which implies that the graph norms associated with the closable operators a(h) and a∗ (h) are equivalent. It follows that the closures of a(h) and a∗ (h) have the same domain. On this common domain we define 1 φ(h) = √ (a(h) + a∗ (h)). 2 (10) 8 The creation- and annihilation operators, and thus φ(h), are bounded relative to the square root of the number operator: ka# (h)(N + 1)−1/2 k ≤ khk k(N + 1)−1/2 a# (h)k ≤ khk. (11) More generally, for any p ∈ R and any integer n k(N + 1)p a# (h1 ) . . . a# (hn )(N + 1)−p−n/2k ≤ Cn,p kh1 k · . . . · khn k. (12) This follows from a∗ (h)N = (N − 1)a∗ (h), a(h)N = (N + 1)a(h), and from (11). 2.2 The Functor Γ Let h1 and h2 be two Hilbert spaces and let b ∈ B(h1 , h2 ). We define Γ(b) : F (h1) → F (h2 ) Γ(b)|` ⊗ns h1 = b ⊗ . . . ⊗ b. In general Γ(b) is unbounded but if kbk ≤ 1 then kΓ(b)k ≤ 1. From the definition of a∗ (h) it easily follows that Γ(b)a∗ (h) = a∗ (bh)Γ(b), Γ(b)a(b∗ h) = a(h)Γ(b), h ∈ h1 . h ∈ h2 . (13) (14) where (14) is derived by taking the adjoint of (13). If b∗ b = 1 on h1 then these equations imply that Γ(b)a(h) = a(bh)Γ(b) Γ(b)φ(h) = φ(bh)Γ(b) 2.3 h ∈ h1 h ∈ h1 . (15) (16) The Operator dΓ(b) Let b be an operator on h. Then dΓ(b) : F (h) → F (h) n X n (1 ⊗ . . . b ⊗ . . . 1). dΓ(b)|` ⊗s h = i=1 For example N = dΓ(1). From the definition of a∗ (h) we get [dΓ(b), a∗ (h)] = a∗ (bh) [dΓ(b), a(h)] =−a(b∗ h), where the second equation follows from the first one by taking the adjoint. If b = b∗ then i[dΓ(b), φ(h)] = φ(ibh). Note that kdΓ(b)(N + 1)−1 k ≤ kbk. (17) 9 FGSch, 30/Mar/01—Asymptotic Completeness 2.4 The Operator dΓ(a, b) Suppose a, b ∈ B(h1 , h2 ). Then dΓ(a, b) : F (h1 ) → F (h2 ) n X n . . . a}). dΓ(a, b)|` ⊗s h = (a . . . a} ⊗b ⊗ |a ⊗{z | ⊗{z j=1 j−1 n−j−1 For a, b ∈ B(h) this definition is motivated by Γ(a)dΓ(b) = dΓ(a, ab), and [Γ(a), dΓ(b)] = dΓ(a, [a, b]). (18) If kak ≤ 1 then kdΓ(a, b)(N + 1)−1 k ≤ kbk. 2.5 The Tensor Product of two Fock Spaces Let h1 and h2 be two Hilbert spaces. We define a linear operator U : F (h1 ⊕ h2 ) → F (h1 ) ⊗ F (h2) by UΩ = Ω ⊗ Ω Ua (h) = [a∗ (h(0) ) ⊗ 1 + 1 ⊗ a∗ (h(∞) )]U ∗ for h = (h(0) , h(∞) ) ∈ h1 ⊕ h2 . (19) This defines U on finite linear combinations of vectors of the form a∗ (h1 ) . . . a∗ (hn )Ω. From the CCRs it follows that U is isometric. Its closure is isometric and onto, hence unitary. It follows that Ua(h) = [a(h(0) ) ⊗ 1 + 1 ⊗ a(h(∞) )]U. (20) Furthermore we note that UdΓ(b) = [dΓ(b0 ) ⊗ 1 + 1 ⊗ dΓ(b∞ )]U if b=  b0 0 0 b∞  (21) For example UN = (N0 + N∞ )U where N0 = N ⊗ 1 and N∞ = 1 ⊗ N. Let Fn = ⊗ns h and let Pn be the projection from F = ⊕n≥0 Fn onto Fn . Then the tensor product F ⊗ F is norm-isomorphic to ⊕n≥0 ⊕nk=0 Fn−k ⊗ Fk , the corresponding isomorphism being given by ϕ 7→ (ϕn,k )n≥0, k=0..n where ϕn,k = (Pn−k ⊗ Pk )ϕ. In this representation of F ⊗ F and with pi (h(0) , h(∞) ) = h(i) , U becomes n  1/2 X n n p ⊗ . . . ⊗ p0 ⊗ p∞ ⊗ . . . ⊗ p∞ . U|` ⊗s (h ⊕ h) = (22) | 0 {z } | {z } k k=0 n−k factors 2.6 k factors Factorizing Fock Space in a Tensor Product Suppose j0 and j∞ are linear operators on h and j : h → h ⊕ h is defined by jh = ∗ ∗ (j0 h, j∞ h), h ∈ h. Then j ∗ (h1 , h2 ) = j0∗ h1 + j∞ h2 and consequently j ∗ j = j0∗ j0 + j∞ j∞ . On the level of Fock spaces, Γ(j) : F (h) → F (h ⊕ h), and we define Γ̆(j) = UΓ(j) : F → F ⊗ F . 10 It follows that Γ̆(j)∗ Γ̆(j) = Γ(j ∗ j) which is the identity if j ∗ j = 1. Henceforth j ∗ j = 1 is tacitly assumed in this subsection. From (13) through (16), (19) and (20) it follows that Γ̆(j)a# (h) = [a# (j0 h) ⊗ 1 + 1 ⊗ a# (j∞ h)]Γ̆(j) Γ̆(j)φ(h) = [φ(j0 h) ⊗ 1 + 1 ⊗ φ(j∞ h)]Γ̆(j). (23) (24) Furthermore, if ω = ω ⊕ ω on h ⊕ h, then by (21) Γ̆(j)dΓ(ω) = UΓ(j)dΓ(ω) = UdΓ(ω)Γ(j) − UdΓ(j, ω j − jω) = [dΓ(ω) ⊗ 1 + 1 ⊗ dΓ(ω)]Γ̆(j) − dΓ̆(j, ω j − jω) (25) where the notation dΓ̆(a, b) = UdΓ(a, b) was introduced. In particular Γ̆(j)N = (N0 + N∞ )Γ̆(j). Finally we remark that, by (22), n  1/2 X n n (26) j ⊗ . . . ⊗ j0 ⊗ j∞ ⊗ . . . ⊗ j∞ . Γ̆(j)|` ⊗s h = |0 {z } | {z } k k=0 2.7 n−k factors k factors The ”Scattering Identification” An important role will be played by the scattering identification I : F ⊗ F → F defined by I(Ω ⊗ Ω) = Ω Iϕ ⊗ a (h1 ) · · · a∗ (hn )Ω = a∗ (h1 ) · · · a∗ (hn )ϕ, ∗ ϕ ∈ F0 , and extended by linearity to F0 ⊗ F0 . (Note that this definition is symmetric with respect to the two factors in the tensor product.) There is a second characterization of I which will often be used. Let ι : h ⊗ h → h be defined by ι(h(0) , h(∞) ) = h(0) + h(∞) . Then I = Γ(ι)U ∗ (27) with U as above. To see this consider states of the form ψ = a∗ (h1 ) · · · a∗ (hm )Ω ⊗ a∗ (g1 ) · · · a∗ (gn )Ω m n Y Y ∗ = [a (hi ) ⊗ 1] [1 ⊗ a∗ (gj )] Ω ⊗ Ω i=1 (28) j=1 in F0 ⊗ F0 . By equations (19) and (13) Γ(ι)U ∗ [a∗ (hi ) ⊗ 1] = Γ(ι)a∗ (hi , 0)U ∗ = a∗ (hi )Γ(ι)U ∗ (29) and similarly Γ(ι)U ∗ [1 ⊗ a∗ (gj )] = a∗ (gj )Γ(ι)U ∗ . Furthermore Γ(ι)U ∗ Ω ⊗ Ω = Ω = I(Ω ⊗ Ω). This shows that Γ(ι)U ∗ ψ = Iψ for ψ given by (28) which proves equation (27). √ Since kιk = 2, the operator I is unbounded. 11 FGSch, 30/Mar/01—Asymptotic Completeness Lemma 1. For each positive integer k, the operator I(N + 1)−k ⊗ χ(N ≤ k) is bounded. Let j : h → h ⊕ h be defined by jh = (j0 h, j∞ h) where j0 , j∞ ∈ B(h). If j0 + j∞ = 1, then Γ̆(j) is a right inverse of I, that is, I Γ̆(j) = 1. (30) Indeed I Γ̆(j) = Γ(ι)U ∗ UΓ(j) = Γ(ιj) = Γ(1) = 1. 3 3.1 Definition of the System and Basic Assumptions The Electron System The dynamics of the electron system (atom) is given by a self-adjoint operator K on L2 (X), where X is a measure space. Typically X = (Rn , dn x) or X = {1, . . . , n} equipped with the counting measure, in which case L2 (X) = Cn . We assume that K is bounded from below and that (H0) inf σess (K) > inf σ(K). In other words, inf σ(K) is an isolated eigenvalue of K. We use |x| to denote the norm of x ∈ X if X is a euclidean space. Otherwise |x| := 0. 3.2 The Radiation Field Pure states of the radiation field are described by vectors in the bosonic Fock space F (h) over h = L2 (Rd , dk). Their time evolution is generated by the Hamiltonian dΓ(ω), where ω denotes multiplication with a real-valued function ω(k) on Rd . For easy reference, we summarize all further properties of ω in the following assumption.   m := inf ω(k) > 0, ω ∈ C ∞ (Rd ), and ∂ α ω is bounded for all α 6= 0, (H1)  ∇ω(k) 6= 0 if k 6= 0. Remarks. (i) The last condition ensures positivity of the Mourre constant away from √ 2 thresholds. (ii) Typical examples we have in mind are ω(k) = k + m2 and smooth dispersions ω that differ from |k| only for small |k|, where ω is chosen such that inf k ω(k) > 0 (see Section 10). (iii) Bosons with non-zero spin or helicity, with h = L2 (Rd , dk) ⊗ Cs , d can also be handled. P We then R dinterpret k as a pair (k, λ) ∈ R × {1, . . . , s} and define s integration over k as λ=1 d k. Throughout this paper, y denotes the position operator in h, i.e., y = i∇k . 3.3 The Composed System The dynamics of the composed system of matter (electron system) interacting with the radiation field is given by the Hamilton operator H = K ⊗ 1 + 1 ⊗ dΓ(ω) + φ(G) = H0 + φ(G), 12 where φ(G) = Z ⊕ φ(Gx ) dx X and φ has been defined in (10). It acts on the Hilbert space H = L2 (X) ⊗ F ∼ = For each x ∈ X, Gx ∈ h, and we assume that R⊕ X F dx. sup kGx k < ∞. (31) x It follows that φ(G)(N + 1)−1/2 and hence that φ(G)(H0 + i)−1/2 are bounded operators. This implies that φ(G) is infinitesimal w.r.to H0 , which shows that H is self-adjoint on D(H0 ) and bounded from below. All further assumptions are listed below and will be cited upon use. (H2) Exponential decay: There exists an ionization threshold Σ > inf σ(H) with the property that keα|x| E∆ (H)k < ∞, for some α > 0, and for any given closed interval ∆ contained in (−∞, Σ). (H3) Fall-off of the form factor: For arbitrary α > 0, sup e−α|x| k(1 + y 2)Gx k < ∞. x (H4) Dispersion of the form-factor: If h ∈ C0∞ (Rd \{0}) then, for arbitrary α > 0,  Z  −α|x| sup e |(Gx , ht )| dt < ∞. x Here ht = e−iωt h. (H5) Short range condition: There exists a µ > 1 such that sup e−α|x| kχ(|y| ≥ R)y n Gx k ≤ Cα R−µ+n x for all R≥0 for all α > 0 and n ∈ {0, 1, 2}. Further hypotheses (IR), (H6), and (H7) are introduced in Section 10.2. Remarks: (i) These hypotheses are satisfied and, with the exception of (H2), easily verified for many concrete systems (see the next section and Sect. 10.3). (ii) Hypothesis (H2) says that the particle system is localized near the origin for small energies. This is the central physical assumption in this paper. Since (H2) can often be derived from (H0) (see [BFS98, GLL00]) it is legitimate to impose (H2). Assumption (H2) makes (H0) obsolete in all of this paper with the exception of Section 10. (iii) In (H5), χ(|y| ≥ R) stands for multiplication with the characteristic function of the set {y : |y| ≥ R}. More 13 FGSch, 30/Mar/01—Asymptotic Completeness generally we use χ(A < c) and χ(A ≤ c) to denote the spectral projections E(−∞,c) (A) and E(−∞,c] (A) of a given self-adjoint operator A. (iv) Hypothesis (H3) follows from (H5). We conclude this section by defining an extended Hilbert space H̃ = H ⊗ F and an extended Hamilton operator H̃ = H ⊗ 1 + 1 ⊗ dΓ(ω), (32) which describes a time evolution where the photons in the auxiliary Fock space do not interact with the particles. With the help of (32) and the scattering identification I the time evolution (3) will be described as a unitary time evolution on the extended Hilbert space in Section 5. Moreover the extended Hamilton operator is invaluable in the spectral analysis of H (see Section 4). In analogy to H̃ we define H̃0 = H0 ⊗ 1 + 1 ⊗ dΓ(ω). 3.4 Examples of Concrete Physical Systems In this section some concrete systems are discussed for which the hypotheses (H0) through (H5) are all satisfied. See also Section 10.3 where we explain how the standard model of non-relativistic QED fits into a slightly expanded version of the general framework introduced above. The example discussed in the introduction, where K is an arbitrary atomic Schrödinger operator, that is, K = −∆ + V on L2 (R3N ), √ satisfying hypothesis (H0), ω(k) = k 2 + m2 , and Gx (k) = g N X e−ik·xj κ(k), j=1 κ ∈ C0∞ (R3 ), satisfies hypotheses (H1) through (H5),Rfor g sufficiently small. In fact, (H2) is proven in [BFS98], with Σ = inf σess (H) − g supx dk |Gx (k)|2 /ω(k), (H3) follows from y 2 = −∆k , (H4) from |(Gx , ht )| ≤ Ct−3/2 , see e.g. [RS79], and (H5) even holds in the strong form sup e−α|x| kχ(|y| ≥ R)y n Gx k ≤ Cn,µR−µ (33) x for arbitrary n, µ ∈ N. To see this, let f (y) = (1 + |y|)µκ̂(y) and put N = 1 for simplicity (this means that we consider the case with only one electron). Then Z −2α|x| n 2 −2α|x| |y n κ̂(x − y)|2 dy e kχ(|y| ≥ R)y Gx k = e |y|≥R Z (1 + |x − y|)−2µ |y nf (x − y)|2 dy. = e−2α|x| |y|≥R This decays exponentially as R → ∞ for |x| ≥ R/2, and if |x| < R/2 it is bounded by (1 + R/2)−2n supx e−2α|x| kχ(|y| ≥ R)y n f (x − ·)k2 which is of order O(R−2n ), because κ̂ ∈ C0∞ (R3 ) by assumption and thus f is rapidly decreasing. 14 The spin-boson models, where K is a hermitian n × n matrix on Cn = L2 (X), with X = {1, . . . , n}, also fits into the our general framework. Suppose ω is as above and Gx ∈ S(R3 ) for all x ∈ X. Then hypotheses (H0) through (H5) are satisfied with the convention that |x| ≡ 0. Hypotheses (H0) and (H2) are trivial, (H4) is seen as in the first example above, and (H3) and (H5) follow from the fact that Gx ∈ S(R3 ), for x = 1, . . . , n. 15 FGSch, 30/Mar/01—Asymptotic Completeness 4 4.1 The Spectrum of Pauli-Fierz Hamiltonians Essential Spectrum and Existence of a Ground State Theorem 2. Assume (H1) and (H2) (Sect. 3), and let E = inf(H). Then inf σess (H) ≥ min{Σ, E + m}. In particular, inf σ(H) is an isolated eigenvalue of H. Proof. Given λ ∈ σess (H) with λ < Σ we need to show that λ ≥ E + m. Let ∆ be an open interval in R containing λ with supp ∆ < Σ and let (ϕn )n≥0 ⊂ E∆ (H)H with k(H − λ)ϕn k → 0, kϕn k = 1 and ϕn ⇀ 0. Then λ = lim hϕn , Hϕn i n→∞ and we estimate the r.h.s from below. Let j0,R , j∞,R ∈ C ∞ (R3 ) be a partition of unity 2 2 defined as in Lemma 32 with j0,R + j∞,R = 1. Pick α > 0 according to (H2) such that α|x| α|x| e E∆ (H) is bounded. Then supn ke ϕn k < ∞ by assumption on ϕn and hence by Lemma 32 and Lemma 31 hϕn , Hϕn i = hϕn , Γ̆(jR )∗ Γ̆(jR )Hϕn i = hϕn , Γ̆(jR )∗ H̃ Γ̆(jR )ϕn i + o(R0 ). (34) uniformly in n. From H ≥ E and dΓ(ω) ≥ m − mχ(N = 0), it is clear that H̃ ≥ (E + m) − mχ(N∞ = 0). (35) 2 2 Since Γ̆(jR )∗ χ(N∞ = 0)Γ̆(jR ) = Γ(j0,R ) and since E∆ (H)Γ(j0,R )E∆ (H) = (E∆ (H)eα|x| )× 2 (e−α|x| Γ(j0,R )E∆ (H)) is compact by Lemma 34 in Appendix E, the equation (34) combined with (35) implies that λ = lim hϕn , Hϕn i ≥ E + m + o(R0 ). n→∞ Letting R → ∞ this proves the theorem. 4.2 The Mourre Estimate Next we establish a type of Mourre theorem with conjugate operator A = dΓ(a) and 1 a = [iω, y 2/2] = (∇ω · y + y · ∇ω). 2 That is we prove positivity of i[H, A] on spectral subspaces of H away from thresholds and eigenvalues, and, as in N-body quantum theory, we obtain important spectral properties of H as a byproduct. Here the thresholds are the elements of τ := σpp (H) + Nm, (0 6∈ N). The Mourre inequality will allow us to show that hdΓ(y 2)iψt ≥ ct2 , as |t| → ∞, (36) 16 where c > 0 for states separated in energy from thresholds and eigenvalues. This together with the above mentioned spectral properties suffices to derive AC. As (36) can only be true if the particles are spatially confined, our Mourre estimate only holds for energies below Σ. On a suitable dense subspace, i[H, A] = dΓ(|∇ω|2) − φ(iaG). We use this equation to define the operator i[H, A] on ∪µ<Σ Eµ (H)H. Note that φ(−iaG)ϕ makes sense for ϕ ∈ Ran Eµ (H), µ < Σ, thanks to the exponential decay and the boundedness of e−α|x| φ(−iaG)(N + 1)−1/2 . The following virial theorem is an important ingredient in the proof of Theorem 4. Furthermore, in the case of massless bosons and IR-cutoff interaction (see Section 10) it will allow us to prove absence of eigenvalues close to, but different from the ground state energy. Lemma 3 (Virial Theorem). Assume hypotheses (H1), (H2), and (H3). If Hϕ = Eϕ and E < Σ then hϕ, i[H, A]ϕi = 0. The proof of this lemma is deferred to Appendix E. The following theorem is the main result of this section. Theorem 4. Assume (H1), (H2) and (H5). Then (i) For each λ ∈ (−∞, Σ)\τ there exists an open interval ∆ ∋ λ, a positive constant Cλ , and a compact operator E such that E∆ (H)[iH, A]E∆ (H) ≥ Cλ E∆ (H) + E. (ii) Non-threshold eigenvalues in (−∞, Σ) have finite multiplicity and can accumulate only at threshold. Furthermore τ ∩ (−∞, Σ] is closed and countable. (iii) If λ ∈ (−∞, Σ)\τ is not an eigenvalue, then there exists an open interval ∆ ∋ λ and a positive constant Cλ such that E∆ (H)[iH, A]E∆ (H) ≥ Cλ E∆ (H). The proof of this theorem follows the lines of the proof in [DG99] with only minor modifications due to the presence of continuous spectrum in the particle Hamiltonian K. For the sake of completeness we have included a proof of Theorem (4) in this paper, but it is deferred to Appendix E. 17 FGSch, 30/Mar/01—Asymptotic Completeness 5 The Wave Operator Recall from Section 3 that H̃ = H ⊗ 1 + 1 ⊗ dΓ(ω) on H̃ = H ⊗ F , and let PB (H) in (37) denote the orthogonal projector onto Hpp (H), the closure of the space spanned by all eigenvectors of H. The purpose of this section is to establish existence of the wave operator Ω+ = s − lim eiHt Ie−iH̃t PB (H) ⊗ 1 t→∞ (37) on spectral subspaces of H̃ corresponding to compact intervals ∆ ⊂ (−∞, Σ). Furthermore we will see that Ω+ is isometric if restricted to vectors in Hpp (H) ⊗ F . The existence of (37) will essentially follow from the existence of asymptotic field operators a♯+ (h)ϕ = lim eiHt a♯ (ht )e−iHt ϕ t→∞ (38) and the existence of products of such operators, which is established in the next theorem. Theorem 5. Assume hypotheses (H1), (H2) and (H4) are satisfied, and let f, h ∈ L2 (Rd ). i) If ϕ = Eη (H)ϕ for some η < Σ then the limit a♯+ (h)ϕ = lim eiHt a♯ (ht )e−iHt ϕ t→∞ exists. Here ht = e−iωt h. ii) The canonical commutation relations [a+ (g), a∗+ (h)] = (g, h) and [a♯+ (h), a♯+ (g)] = 0, hold true, in form–sense, on χ(H < η)H for all η < Σ. iii) Let m = inf{ω(k) : h(k) 6= 0} and M = sup{ω(k) : h(k) 6= 0}. Then a∗+ (h) Ran χ(H ≤ E) ⊂ Ran χ(H ≤ E + M) a+ (h) Ran χ(H ≤ E) ⊂ Ran χ(H ≤ E − m). iv) Suppose ϕ = Eη (H)ϕ, hi ∈ L2 (Rd ; C) for i = 1, . . . , n and let Mi = sup{ω(k) : hi (k) 6= 0}. Then a∗+ (h1 ) . . . a∗+ (hn )ϕ = lim eiHt a∗ (h1,t ) . . . a∗ (hn,t )e−iHt ϕ t→∞ provided that η + Pn i=1 Mi < Σ. v) Suppose η < Σ and ϕ ∈ Eη (H)Hpp (H). Then a+ (h)ϕ = 0 for all h ∈ L2 (Rd ). 18 √ Remark. For relativistic massive bosons, that is for ω(k) = k 2 + m2 with m > 0, as well as in the case of relativistic electrons and massless bosons, the asymptotic field operators actually exist on a dense subspace of H irrespective of Σ (see [FGS00]). Proof. i) Assume first that h ∈ C0∞ (Rd \{0}). By Cook’s argument it suffices to show that Z ∞ k(Gx , ht )ϕt kdt < ∞. (39) 1 This follows from the assumptions (H2) and (H4). For the proofs of ii), iii) and iv) we refer to [FGS00]. v) It suffices to show that a+ (h)ϕ = 0 if Hϕ = Eϕ. Statement v) then follows from the boundedness of a+ (h)Eη (H). Since ht ⇀ 0 weakly as t → ∞ we have s − limt→∞ a(ht )(H + i)−1/2 = 0. Hence a+ (h)ϕ = lim eiHt e−iEt (E + i)1/2 a(ht )(H + i)−1/2 ϕ = 0. t→∞ (40) Next we prove existence of the extended wave operator Ω̃+ := s − limt→∞ eiHt Ie−iH̃t on a suitable spectral subspace of H̃. Since Ω̃+ agrees with Ω+ for vectors in Hpp (H)⊗F , this will immediately imply existence of Ω+ . Lemma 6. Assume the hypotheses of the theorem above are satisfied. P a) Suppose ψ = ϕ ⊗ a∗ (h1 ) . . . a∗ (hn )Ω where ϕ = Eλ (H)ϕ and λ + ni=1 Mi < Σ, where Mi is defined as in Theorem 5, iv). Then Ω̃+ ψ := limt→∞ eiHt Ie−iH̃t ψ exists and Ω̃+ ψ = a∗+ (h1 ) . . . a∗+ (hn )ϕ. (41) b) Ω̃+ exists on Eµ (H̃)H̃ for all µ < Σ. Proof. Statement a) follows from e−iH̃t ϕ ⊗ a∗ (h1 ) . . . a∗ (hn )Ω = e−iHt ϕ ⊗ a∗ (h1,t ) . . . a∗ (hn,t )Ω (42) the definition of I and Theorem 5, iv). (b) Since eiHt Ie−iH̃t Eµ (H̃) is bounded uniformly in t, it suffices to prove existence of Ω̃+ on a dense subspace of Eµ (H̃)H̃. By Lemma 30 finite linear combinations of vectors of the form described in part a) span such a subspace and hence b) follows from a). The following theorem is the main result of this section. 19 FGSch, 30/Mar/01—Asymptotic Completeness Theorem 7. Assume (H1), (H2) and (H4). Then Ω+ ϕ = s − lim eiHt Ie−iH̃t PB (H) ⊗ 1 t→∞ (43) exists on ∪µ<Σ Eµ (H̃)H̃, kΩ+ k = 1 and hence Ω+ has a unique extension, also denoted by Ω+ , to EΣ (H̃)H̃. Ω+ is isometric on (PB (H) ⊗ 1)EΣ (H̃)H̃ and hence Ran Ω+ is closed. Furthermore eiHt Ω+ = Ω+ eiH̃t . (44) Proof. The existence of Ω+ on ∪µ<Σ Eµ (H̃)H̃ follows from Lemma 6 b). By Lemma 6 a), the CCR in Theorem 5 ii), and by part v) of that theorem, Ω+ is isometric on (PB (H) ⊗ 1) ∪µ<Σ Eµ (H̃)H̃. So Ω+ is a partial isometry and hence kΩ+ k = 1. All these properties carry over to the closure of Ω+ . 20 6 Propagation Estimates This section establishes the propagation estimates needed later on to construct the asymptotic observable W (see Eq. (8)) and the Deift Simon wave operator W+ . To begin with, we define a smooth convex function S(y, t) which modifies y 2 /2t for y in a neighborhood of size tδ , δ ∈ (0, 1), of the origin. The operator W will be constructed using S(y, t) in place of y 2/2t. This will not affect W but allows us to prove its existence. Here and henceforth we use the following notation for the various Heisenberg derivatives. Suppose A is an operator in H. Then we define ∂A ∂t ∂A D0 A := i[H0 , A] + ∂t DA := i[H, A] + (45) (46) Similarly, the Heisenberg derivative d of an operator a on the one boson sector h is defined by da = i[ω, a] + ∂a . ∂t (47) Furthermore, if A(R) is a family of operators on H, and R ∈ R+ , we write A(R) = O(Rm)N p 6.1 if kA(R)N −p k = O(Rm ). Construction of S(y, t) R Pick m ∈ C0∞ (R+ ) with m ≥ 0, supp(m) ⊂ [1, 2] and dσ m(σ) = 1. Set Z S0 (y) = dσ m(σ)χ(y 2 /2 > σ) (y 2/2 − σ). (48) Then S0 is smooth and ( 0 S0 (y) = y 2 /2 + b y 2 /2 ≤ 1 y 2 /2 ≥ 2 R where b = − dσ m(σ)σ. It follows that S0 (y) = y 2/2 + a(y) + b (49) (50) where a ∈ C0∞ (R3 ). This formula will allow us to apply Lemma 27 in Appendix A to [iω, S0 ]. It is important that S0 is convex, which is easy to see from the definition. Next we define a scaled version of S0 by S(y, t) = t−1+2δ S0 (y/tδ ) where 0 < δ < 1. (51) 21 FGSch, 30/Mar/01—Asymptotic Completeness Lemma 8. For all integers n ≥ 0 and all α  
y2 n α = O t−(n+1)+δ(2−|α|) . D ∂ S− 2t Here D = ∂/∂t or D = ∂/∂t + (y/t) · ∇ and ∂ α is any spacial derivative of order |α|. In particular ∇S = y + O(t−1+δ ), t ∂S y2 = − 2 + O(t−2+2δ ). ∂t 2t (52) These estimates hold uniformly in y ∈ Rd . Proof. By definition of S(y, t) and (50), S(y, t) − y 2/2t = t−1+2δ a(yt−δ ) + t−1+2δ b. The second term clearly enjoys the desired property. For the first term we have ∂ α [t−1+2δ a(yt−δ )] = t−1+δ(2−|α|) (∂ α a)(yt−δ ) (53) where the right hand side is of the form tc h(yt−δ ) with h ∈ C0∞ (R3 ). Both ∂/∂t(tc h) and D(tc h) are again of this form with c replaced by c − 1. This proves the lemma. Lemma 9. Suppose ω ∈ C ∞ (Rd ) and ∂ α ω is bounded for all α 6= 0. Then 1 ∂S dS = (∇ω · ∇S + ∇S · ∇ω) + + O(t−1 ), 2 ∂t and for any smooth vector field v(y, t) and Dv = ∂/∂t + v · ∇, d(dS) =(∇ω − v) · S ′′ (∇ω − v) + (Dv ∇S) · (∇ω − v) + (∇ω − v) · (Dv ∇S) + Dv2 S − (Dv v) · ∇S + O(t−1−δ ). (54) (55) (56) Proof. The first part follows from (50) and Lemma 27. By definition of the Heisenberg derivative, d(dS) is given by d(dS) = [iω, [iω, S]] + 2[iω, ∂S/∂t] + ∂2S . ∂t2 By Lemma 8, (50), and Lemma 27 ∂S ∂S +∇ · ∇ω + O(t−2) ∂t ∂t [iω, [iω, S]] = ∇ωS ′′ ∇ω + O(t−1−δ ). 2[iω, ∂S/∂t] = ∇ω · ∇ To prove the second equation an explicit formula for [iω, S] − ∇ω · ∇S is also needed (see the proof of Lemma 27). Since for every smooth vector field v(y, t) ∂S ∂2S ∂S +∇ · ∇ω + 2 ∂t ∂t ∂t ′′ =(∇ω − v) · S (∇ω − v) + (Dv ∇S) · (∇ω − v) + (∇ω − v) · (Dv ∇S) (57) + Dv2 S − (Dv v) · ∇S ∇ω · S ′′ ∇ω + ∇ω · ∇ the lemma follows. 22 6.2 Propagation Estimates We are now ready to state and prove our propagation estimates. Note that these are basic propagation estimates which are well known in other contexts (see [GS97]). Proposition 10. Assume (H1) and (H3), let χ = χ̄ ∈ C0∞ (R), and suppose eα|x| χ(H) is a bounded operator for some α > 0. Let χ = χ(H) and v = y/t. Then, if λ > 0 is large enough, there exists a constant C such that Z ∞ 1 dt hχψt , χ(λ2 ≤ dΓ(v 2 ) ≤ 2λ2 )χψt i ≤ Ckψk2 t 1 for all ψ ∈ H. Here ψt = e−iHt ψ. Remark: This propagation estimate equally holds on H̃ and with H and dΓ(v 2 ) replaced by H̃ and dΓ(v 2 ) ⊗ 1 + 1 ⊗ dΓ(v 2 ). This is needed for the proof of the remark to Proposition 11. Rs Proof. Pick h ∈ C0∞ (1/2, 3) with h(r) = 1 on [1, 2], 0 ≤ h ≤ 1 and set h̃(s) = 0 ds′ h2 (s′ ). Note that h̃(s) = h̃(3) for s ≥ 3. Hence g(s) = h̃(s) − h̃(3) for s ≥ 0 and g(−s) = g(s) define a C0∞ -function g on R. The operator B = −χh̃(dΓ(v 2 /λ2 ))χ is our propagation observable. Since B is bounded the theorem will follow if we show that ∂B C DB := [iH, B] + ≥ χ(H) χ(1 ≤ dΓ(v 2 /λ2 ) ≤ 2) χ(H) + integrable terms ∂t t for some C > 0 if λ is large enough. Henceforth we use the abbreviations h, h̃ and g to denote the operators h(dΓ(v 2 /λ2 )), h̃(dΓ(v 2 /λ2 )) and g(dΓ(v 2/λ2 )), respectively. Clearly 1 2 ∂B = χh2 dΓ(v 2 /λ2 ) χ ≥ χh2 χ. ∂t t t (58) Next −[iH, B] = χ[iH, h̃]χ = χ[idΓ(ω), h̃]χ + χ[iφ(G), h̃]χ. Consider first the second term on the right hand side. By [iφ(G), h̃] = [iφ(G), g], the Helffer–Sjöstrand functional calculus (see Appendix A.2), and by (H3) ke−α|x| [iφ(G), h̃](N + 1)−1/2 k ≤ Cke−α|x| φ(−iv 2 G)(N + 1)−1/2 k (59) C ≤ 2 2 sup e−α|x| kφ(iy 2Gx )(N + 1)−1/2 k = O(t−2 ). λ t x Hence χ[iφ(G), h̃]χ is integrable. By (28) [idΓ(ω), h̃] = [idΓ(ω), g] 1 = g ′ dΓ(∇ω · v/λ + v/λ · ∇ω) + O(t−2 )N λt 1 = hdΓ(∇ω · v/λ + v/λ · ∇ω)h + O(t−2 )N λt C ≤ h(N + 1)h + O(t−2 )N λt 23 FGSch, 30/Mar/01—Asymptotic Completeness Since [χ, h] = O(t−1 ) it follows that χ[idΓ(ω), h̃]χ ≤ C 2 h + O(t−2 )N. λt For C/λ < 1 this in conjunction with (58) proves the proposition. Proposition 11. Assume (H1) and (H5), let χ ∈ C0∞ (R), and suppose eα|x| χ(H) is a bounded operator for some α > 0. Let f ∈ C0∞ (R), with 0 ≤ f ≤ 1, f (x) = 0 for x ≥ 2, and f (x) = 1 for 0 ≤ x ≤ 1. Denote χ = χ(H) and f = f (dΓ(v 2 /λ2 )), where v = y/t and λ ∈ R. Then, for λ large enough Z ∞ dt hψt , χf dΓ ((∇ω − v) · S ′′ (∇ω − v)) f χψt i ≤ Ckψk2 . 1 Remark: This proposition equally holds on H̃ and with H, dΓ(v 2 ) and dΓ(P ) (here P = (∇ω − v) · S ′′ (∇ω − v)), replaced by H̃, dΓ(v 2 ) ⊗ 1 + 1 ⊗ dΓ(v 2 ) and dΓ(P ) ⊗ 1 + 1 ⊗ dΓ(P ). This is needed for the proof of Theorem 15. Proof. Let γ(t) = hψt , χf dΓ(dS)f χψti. From ±dS ≤ const(v 2 + 1) it follows that dΓ(dS)f (N + 1)−1 is a bounded operator and thus that sup|t|≥1 |γ(t)| < ∞ because of the cut-off χ. Next we show that γ ′ (t) ≥hψt , χf dΓ((∇ω − v) · S ′′ (∇ω − v))f χψt i + (integrable w.r. to t) × kψk2 . (60) By the Leibnitz rule γ ′ (t) = hψt , χ(Df )dΓ(dS)f χψt i + h.c. + hψt , χf (DdΓ(dS))f χψti (61) Only the last term will contribute to (60). DdΓ(dS) = D0 dΓ(dS) + [iφ(G), dΓ(dS)] = dΓ(d(dS)) + φ(−idS G) where d(dS) is given by Lemma 9. Since v = y/t the terms in (56) are of order O(t−1−ε ) where ε = min(δ, 2 − 2δ). For the terms of (55) we have ± [(Dv ∇S) · (∇ω − v) + (∇ω − v) · (Dv ∇S)] ≤ t2−δ (Dv ∇S)2 + t−2+δ (∇ω − v)2 = O(t−2+δ )(1 + v 2 ) which, thanks to the cutoffs f and χ, gives an integrable contribution to γ ′ (t). This shows that hψt , χf dΓ(d2 S)f χψt i ≥hψt , χf dΓ((∇ω − v) · S ′′ (∇ω − v))f χψt i + O(t−1−ε )kψk2 24 To estimate the contribution due to φ(−idS G) use kχf φ(−idSG)f χk ≤ constke−α|x| φ(−idSG)(N + 1)−1/2 k (62) and dS = ∇S · ∇ω − i X 2 2 (∂rs S)(∂rs ω) + r,s ∂S + O(t−1−δ ). ∂t This shows that dS = χ(|y| ≥ tδ )dS + O(t−δ−1 ) and in conjunction with Lemma 8 and (H5) it follows that (62) is integrable. To estimate the contribution in (61) due to Df note that Df = D0 f + [iφ(G), f ]. The second term gives a contribution of order O(t−2 ). This is seen in the same way as (59). Next choose g ∈ C0∞ (R) with supp(g) ⊂ (1, 2) and gf ′ = f ′ and denote g = g(dΓ(v 2/λ2 )). Then χ(D0 f )dΓ(dS)f χ = χg(D0 f )dΓ(dS)gf χ + O(t−2 ) (63) c |hψt , χ(D0 f )dΓ(dS)f χψt i| ≤ kg(N + 1)χψt k2 + O(t−2) t (64) and hence which is integrable by Proposition 10. 7 The Asymptotic Observable In this section existence of the asymptotic observable W is proved. An auxiliary version Wλ of W will involve a space cutoff at |y| = λ in the bosonic configuration space. W is then obtained in the limit λ → ∞. To define the space cutoff we pick, once and for all, a function f ∈ C0∞ (R) with 0 ≤ f ≤ 1, f (x) = 0 for x ≥ 2 and f (x) = 1 for 0 ≤ x ≤ 1. The space cutoff is the operator f [dΓ(v 2 /λ2 )] or 1 ⊗ f [dΓ(v 2 /λ2 )] on F or H respectively. Here v = y/t and λ ∈ R. For brevity these operators will also be denoted by f if there is no danger of confusion. Since (dS)2 ≤ const(v 4 + 1), which follows from Lemma 9 and Lemma 27, and since f ∈ C0∞ (R), the operator dΓ(dS)f (N + 1)−1 is bounded. Theorem 12. Assume Hypotheses (H1), (H2), and (H5). If χ ∈ C0∞ (R) with supp χ ⊂ (−∞, Σ), then Wλ = s − lim eiHt χf dΓ(dS)f χe−iHt t→∞ exists, is self-adjoint, and commutes with H. 25 FGSch, 30/Mar/01—Asymptotic Completeness Proof. To prove existence of Wλ ϕ we use Cook’s argument, i.e. we show that Z ∞ d dt hψt , χf dΓ(dS)f χϕt i ≤ Ckψk. dt 1 As in the proof of Proposition 11 one shows that d hψt , χf dΓ(dS)f χϕt i = hψt , χf dΓ[(∇ω − v) · S ′′ (∇ω − v)]f χϕt i dt + O(t−1−ε )kψkkϕk. (65) Since S ′′ ≥ 0 the first term on the right side defines a non-negative sesquilinear form in ψ and ϕ and hence we can apply Schwarz |hψt , χf dΓ[(∇ω − v) · S ′′ (∇ω − v)]f χϕt i| ≤ |hψt , χf dΓ[(∇ω − v) · S ′′ (∇ω − v)]f χψt i|1/2 × |hϕt , χf dΓ[(∇ω − v) · S ′′ (∇ω − v)]f χϕt i|1/2 . This, together with Proposition 11 after an application of Hölder’s inequality, shows that also the first term in (65) is integrable. This proves existence of Wλ ϕ. Clearly Wλ is bounded and symmetric. To prove that Wλ commutes with H it suffices to show that e−iHt Wλ eiHt = Wλ for all t ∈ R. This follows from −iHt e−iHs Wλ eiHs ϕ − Wλ ϕ = lim eiHt χ[f dΓ(dS)f ]t+s ϕ t χe t→∞ −1 because χ[f dΓ(dS)f ]t+s t χ = O(t ). Indeed ∂ ∂f ∂f ∂ (f dΓ(dS)f ) = f dΓ(dS)f + dΓ(dS)f + f dΓ(dS) ∂t ∂t ∂t ∂t where ∂f /∂t = O(t−1 ) and χf dΓ(d(∂S/∂t))f χ = O(t−1). The latter follows from (50) and Lemma 27. In the next step we remove the space cutoff f . This will allow us to prove positivity of W = limλ→∞ Wλ away from thresholds and eigenvalues. Proposition 13. Under the assumptions of Theorem 12, the limit W = limλ→∞ Wλ exists in operator norm sense, and W is given by hϕ, W ψi = lim hϕ, eiHt χdΓ(dS)χe−iHt ψi t→∞ for all ϕ, ψ ∈ D(dΓ(y 2)) ∩ D(N). W commutes with H. Proof. We pick ϕ, ψ ∈ D(dΓ(y 2)) ∩ D(N) and we consider the difference |hϕt , χdΓ(dS)χψt i−hϕt , χf dΓ(dS)f χψt i| ≤ |hϕt , χ(1 − f )dΓ(dS)f χψt i| + |hϕt , χdΓ(dS)(1 − f )χψt i| ≤ k(1 − f )χϕt kkdΓ(dS)f χψt k + k(1 − f )χψt kkdΓ(dS)χϕtk. (66) 26 Since (1 − f (s))2 ≤ s2 , we have k(1 − f )χϕt k ≤ 1 C kdΓ(y 2 /t2 )χϕt k ≤ 2 (1/t2 kdΓ(y 2 + 1)ϕk + kϕk), 2 λ λ (67) and analogously for ϕ replaced by ψ. The second inequality in (67) follows by Lemma 36 in Appendix F. To handle the factor kdΓ(dS)χϕt k on the r.h.s. of (66) use that, by Lemma 8 and Lemma 9, dS = y2 1 (∇ω · y + y · ∇ω) − 2 + O(t−1+δ ). 2t 2t (68) Part (iv) of Lemma 37 (with y replaced by y/t) and Lemma 36 in Appendix F lead then to kdΓ(dS)χϕt k ≤ C (1/t2 kdΓ(y 2 + 1)ϕk + kϕk). (69) Insering (67) and (69) into (66) and using kdΓ(dS)f χψt k ≤ Ckψk we find C (1/t2 kdΓ(y 2 + 1)ϕk + kϕk) λ2 × (1/t2 kdΓ(y 2 + 1)ψk + kψk), |hϕt , χdΓ(dS)χψt i − hϕt , χf dΓ(dS)f χψt i| ≤ for arbitrary ϕ, ψ ∈ D(dΓ(y 2)) ∩ D(N), and for all t ≥ 1. This shows that W exists as a weak limit and that |hϕ, (W − Wλ )ψi| ≤ C kϕk kψk, λ which proves that W also exists as a norm limit. Finally, that W commutes with H follows from the fact that Wλ commutes with H for each λ. Using the Mourre inequality from Theorem 4, we next prove positivity of W away from thresholds and eigenvalues. Note that this is the only place where the Mourre inequality is used. Proposition 14. Assume hypotheses (H1), (H2) and (H5) are satisfied. Assume, moreover, that the energy cutoff χ in the definition of W satisfies supp χ ⊂ (−∞, Σ)\S, where S = σpp (H) + m · (N ∪ {0}) is the set of all eigenvalues and thresholds of H. Then W ≥ d χ2 , for some d > 0. In particular, if ∆ ⊂ (−∞, Σ)\S and χ|`∆ = 1, then W ≥ d on RanE∆ (H). Proof. By the compactness of σ(H) ∩ (−∞, Σ)\S, and since W commutes with H, it is enough if we prove that, for each x ∈ (−∞, Σ)\S, W |`RanEUx (H) = EUx (H)W EUx (H) ≥ dx EUx (H)χ2 , 27 FGSch, 30/Mar/01—Asymptotic Completeness where Ux is an arbitrarily small neighborhood of x, and dx > 0. Let χ̃ = EUx (H) be the spectral projection of H on Ux . If we choose Ux to be sufficiently small, then, by the Mourre estimate (Theorem 4), we have χ̃[iH, A]χ̃ ≥ 2dx χ̃(H), (70) for some dx > 0. Here A = dΓ(a) = dΓ([iω, y 2/2]). Choose now ψ ∈ D(dΓ(y 2 )) ∩ D(N) and let ψt = e−iHt ψ. Then, by Proposition 13, we have hψ, χ̃W χ̃ψi = lim hψt , χ χ̃dΓ(dS)χ̃ χψt i t→∞ d hψt , χ χ̃dΓ(S)χ̃ χψt i, t→∞ dt (71) = lim where the second equality holds because, by Hypothesis (H5), χ̃[iφ(G), dΓ(S)]χ̃ = −χ̃φ(iSG)χ̃ = o(t−1+δ ). From (71) it now follows that 1 hψt , χ χ̃dΓ(S)χ̃ χψt i t→∞ t 1 = lim 2 hψt , χ χ̃dΓ(y 2/2)χ̃ χψt i, t→∞ t hψ, χ̃W χ̃ψi = lim (72) where, in the second equality, we used the definition of the function S(y). Now we have Z t iHt 2 −iHt 2 e χ̃dΓ(y /2)χ̃e = χ̃dΓ(y /2)χ̃ + ds eiHs χ̃[iH, dΓ(y 2 /2)]χ̃e−iHs Z t Z0 t y2 −iHs 2 iHs + ds eiHs χ̃Aχ̃e−iHs = χ̃dΓ(y /2)χ̃ − ds e χ̃φ(i G)χ̃e 2 0 Z0 t 2 y = χ̃dΓ(y 2/2)χ̃ − ds eiHs χ̃φ(i G)χ̃e−iHs 2 Z t 0Z s + tχ̃Aχ̃ + ds dr eiHr χ̃[iH, A]χ̃e−iHr . 0 0 (73) Note that the operator φ(iy 2G)χ̃ is bounded. Moreover the expectation values of χ̃dΓ(y 2 )χ̃ and of χ̃Aχ̃ in the state χψ are finite, because ψ ∈ D(dΓ(y 2)) ∩ D(N), ±A ≤ C dΓ(y 2 + 1) and because of Lemma 35 (see Appendix F). Thus, after division by t2 , only the last term in (73) gives a non-vanishing contribution to (72) in the limit t → ∞. By (70) Z Z s 1 t ds dr hψr , χ χ̃[iH, A]χ̃ χψr i ≥ dx hψ, χ2 χ̃ψi, hψ, χ̃W χ̃ψi = lim 2 t→∞ t 0 0 which proves the proposition because D(dΓ(y 2)) ∩ D(N) is dense in H. 28 8 Inverting the Wave Operator The Deift-Simon wave operator W+ , to be constructed in this section, inverts the extended wave operator Ω̃+ with respect to Wλ in the sense that Wλ = Ω̃+ W+ = lim eiHt Ie−iH̃t W+ . t→∞ (74) On spectral subspaces where Wλ is positive and thus invertible, W+ Wλ−1 is then in fact a right inverse of Ω̃+ . Formally, and when space and energy cutoffs are ignored, then W+ is given by   W+ = s − lim eiH̃t dΓ̆(jt , djt )dΓ(S) + Γ̆(jt )dΓ(dS) e−iHt (75) t→∞ where jt = (j0,t , j∞,t ) and j0,t + j∞,t = 1. By the last identity, I Γ̆(jt ) = 1 and IdΓ̆(jt , djt ) = D0 [I Γ̆(jt )] = 0. Hence (74) is obvious at least on this formal level. The functions j0,t and j∞,t are constructed as follows. Let j0 , j∞ ∈ C ∞ (Rd ) where j0 (y) = 1 for |y| < 1, j0 (y) = 0 for |y| > 2, and let j∞ = 1 − j0 . Next set j♯,t (y) = j♯ (y/ut) where u > 0 is a fixed parameter. By construction of jt and W+ , Eq. (75), photons with velocity u or larger are mapped to the second Fock space where their interaction with the electrons is turned off. First we prove existence of W+ in Theorem 15 and then we prove (74). Theorem 15 together with the Mourre estimate, Theorem 4, is the heart of our proof of AC. Recall from Section 7 that DA, D0 A and da denote Heisenberg derivatives of operators A and a on H and h respectively. If B is an operator on the extended Hilbert space H̃ and if C maps H to H̃, then we set DB := i[H̃, B] + ∂B ∂t D̃C := iH̃C − CiH + ∂C . ∂t The derivatives D0 , and D̃0 are defined in a similar way using H0 and H̃0 rather than H and H̃. Finally the Heisenberg derivative db of an operator b mapping the one-boson sector h to h ⊕ h (that is, bh = (b0 h, b∞ h) with b0,∞ being operators on h) is defined by     ∂b ω 0 db0 b − b iω + db = i . = 0 ω db∞ ∂t The next theorem is the main result of this section. Theorem 15. Assume Hypotheses (H1), (H2), and (H5). If χ ∈ C0∞ (R) with supp χ ⊂ (−∞, Σ), then h i W+ = s − lim eiH̃t χ̃f˜D̃0 Γ̆(jt )dΓ(S) f χe−iHt t→∞ exists. Here χ = χ(H) and χ̃ = χ(H̃). Furthermore e−iH̃t W+ = W+ e−iHt . 29 FGSch, 30/Mar/01—Asymptotic Completeness Proof. By Cook’s argument we need to show that there exists a constant C such that Z ∞ d dt hψ, W+ (t)ϕi ≤ Ckψk (76) dt 1 for all ψ ∈ H, where In form sense W+ (t) = eiH̃t χ̃f˜Qf χe−iHt h i Q = D̃0 Γ̆(jt )dΓ(S) = dΓ̆(jt , djt )dΓ(S) + Γ̆(jt )dΓ(dS). d W+ (t) = eiH̃t χ̃D̃[f˜Qf ]χe−iHt dt D̃[f˜Qf ] = (D f˜)Qf + f˜(D̃Q)f + f˜Q(Df ). The contributions due to D f˜ and Df are dealt with as in Proposition 11 and are integrable due to Proposition 10. The operator D̃Q is the sum D̃Q = D̃0 Q + i(φ(G) ⊗ 1)Q − Qiφ(G) (77) where the last two terms give a contribution of order t−µ because of Hypothesis (H5). To show this write Q =i [dΓ(ω) ⊗ 1 + 1 ⊗ dΓ(ω)] Γ̆(jt )dΓ(S) − Γ̆(jt )dΓ(S)idΓ(ω) ∂ − Γ̆(jt )dΓ(S) ∂t (78) and commute iφ(G) from the right through the terms in (78) using (17) and (24) (before differentiating with respect to t). This is a lengthy computation which leads to   i(φ(G) ⊗ 1)Q − Qiφ(G) =i [φ(j∞ G) ⊗ 1 − 1 ⊗ φ(j∞ G)] D̃0 Γ̆(jt )dΓ(S) − i [φ(d(j0 S)iG) ⊗ 1 + 1 ⊗ φ(d(j∞ S)iG)] Γ̆(jt )   − i [φ(j0 SiG) ⊗ 1 + 1 ⊗ φ(j∞ SiG)] D̃0 Γ̆(jt ) − [φ(d(j0 )G) ⊗ 1 + 1 ⊗ φ(d(j∞ )G)] Γ̆(jt )dΓ(S). Using Hypothesis (H5), Eq. (50) and Lemma 27 one shows that each of these terms is of order t−µ . We demonstrate this for the last term. By Lemma 27 " # X (∂rs ω)(∂rs j∞ ) + ∂t j∞ G + O(t−2 )G. t(dj∞ )G = t ∇ω · ∇j∞ − i (79) r,s Since derivatives of ω are bounded, and since derivatives of j∞ are of order t−1 and live on {|y| ≥ ut}, Equation (79) in conjunction with Hypothesis (H5) implies e−α|x| φ(t(dj∞ )G)(N + 1)−1 = O(t−µ ). (80) 30 On the other hand dΓ(S/t)f (N + 1)−1 is bounded. Hence χ̃f˜ 1 ⊗ φ((dj∞ )G)Γ̆(jt )dΓ(S)f χ   =χ̃eα|x| f˜ 1 ⊗ e−α|x| φ(t(dj∞ )G)(N∞ + 1)−1 (N∞ + 1)(N0 + N∞ + 1)−1 × Γ̆(jt )dΓ(S/t)f (N + 1)−1 (N + 1)2 χ is of order t−µ . To handle the contribution due to D̃0 Q we work in the representation F ⊗ F = ⊕n≥0 ⊕nk=0 Fn−k ⊗ Fk where Fk is the k-boson subspace of F . Since D̃0 Q maps Fn to ⊕nk=0 Fn−k ⊗ Fk one has hψt , χ̃f˜D̃0 Qf χϕt i = n ∞ X X n=0 k=0 hαt,nk , Pn−k ⊗ Pk D̃0 Qβt,n i (81) where αt,nk = Pn−k ⊗ Pk f˜χ̃ψt , βt,n = Pn f χϕt and Pn is the orthogonal projection F → Fn . Note that χ̃ = χ̃ χ(N∞ ≤ n∞ ) for some n∞ large enough, and hence αt,nk = 0 for k > n∞ . Next we estimate |hαt,nk , Pn−k ⊗ Pk D̃0 Qβt,n i| for each given n and k separately. To this end we identify Fn−k ⊗ Fk with a subspace of L(Rdn ). Then, on Fn , the operator Pn−k ⊗ Pk Γ̆(jt )dΓ(S) = Jnk Sn =: Snk acts by multiplication with a function Jnk Sn where Sn (y, t) = Jnk Pn i=1 S(yi , t) and by (26)  1/2 n = j ⊗ . . . ⊗ j0 ⊗ j∞ ⊗ . . . ⊗ j∞ . } | {z } |0 {z k n−k factors (82) k factors In terms of Snk the operator Pn−k ⊗ Pk D̃0 Q is given by   Pn−k ⊗ Pk D̃0 Q = D02 Pn−k ⊗ Pk Γ̆(jt )dΓ(S) = D02 Snk (83) P where D0 Snk = [iΩ, Snk ]+∂Snk /∂t and Ω(k1 , . . . , kn ) = ni=1 ω(ki ). Let V = (y1 , . . . , yn )/t ∈ Rdn and let DV = ∂/∂t + V · ∇ denote the material derivative w.r. to V . In the appendix we show that ′′ D02 Snk =(∇Ω − V ) · Snk (∇Ω − V ) + (DV ∇Snk ) · (∇Ω − V ) + (∇Ω − V ) · (DV ∇Snk )  1/2 2 2 n + DV Snk + n O(t−1−δ ) k (84) for |y|/t ≤ 2λ (Lemma 40) and that DVm ∂ α Snk − Jnk n X i=1 ! yi2/2t m+1 =n  1/2 n O(t−(1+m)+2(δ−|α|) ) k (85) 31 FGSch, 30/Mar/01—Asymptotic Completeness (Lemma 39). These are analogs of Lemma 9 and Lemma 8. The binomial factor in (84) and (85) stems from (82) and will be estimated by nk . From Eq. (85) we get for the last term of (84) DV2 (Jnk Sn ) = Jnk DV2 Sn − n X y2 i i=1 2t ! = nk/2+3 O(t−3+2δ ). (86) where we used that DV f = 0 for any function f (y, t) which only depends on V and DV2 (y 2/2t) = 0. For the second and third term in Eq. (84), Eq. (85) shows that   ± (DV ∇Snk ) · (∇Ω − V ) + (∇Ω − V ) · (DV ∇Snk ) ≤ t2−δ (DV ∇Snk )2 + t−2+δ (∇Ω − V )2 = O(t−2+δ )(n + V 2 )nk+4 (87) Combining (81) and (83) with (84), (86) and (87) we get hψt , χ̃f˜D̃0 Qf ϕt i ≤ n ∞ X X ′′ |hαt,nk , (∇Ω − V ) · Snk (∇Ω − V )βt,n i| n=0 k=0 n ∞ X X + n=0 k=0 k+4 kαt,nk k kβt,n kn −1−ε O(t (88) ) where ε = min(1 − δ, δ) > 0. Here nk+1 ≤ nn∞ +1 because αt,nk = 0 for k > n∞ . Hölder’s inequality in the form ∞ X n X n=0 k=0 Ank Bn ≤ ∞ X n X n=0 k=0 A2nk !1/2 ∞ X (n + 1)Bn2 n=0 !1/2 (89) where Ank , Bn ≥ 0, will be used frequently in the following. It shows that the second term in (88) is bounded from above by kf˜χ̃ψt k k(N + 1)n∞ +5/2 f χϕt k O(t−1−ε ) which is integrable. To deal with the first term in (88) we use that ′′ ±Snk (y, t)  1/2 n Sn′′ (y, t), ≤ const × n k 2 for |y| ≤ 2λt and utδ−1 ≥ 2 by Lemma 38. This allows us to estimate the contribution due ′′ ′′ to Snk χ(|V | ≤ 2λ). The contribution due to Snk χ(|V | > 2λ) is bounded by O(t−2 )(n + 4 1) kαt,nk k kβt,n k thanks to the space cutoff f . Together with (88), the Schwarz inequality, and kαt,nk k = 0 for k > n∞ , this implies that ′′ |hαt,nk , (∇Ω − V ) · Snk (∇Ω − V )βt,n i| ≤ |hαt,nk , (∇Ω − V ) · Sn′′ (∇Ω − V )αt,nk i|1/2 × |hβt,n , (∇Ω − V ) · Sn′′ (∇Ω − V )βt,n i|1/2 nn∞ /2+2 + O(t−2 )(n + 1)4 kαt,nk k kβt,n k 32 δ−1 ′′ for ≥ 2. Now insert this bound into (88), use that (∇Ω − V ) · Snk (∇Ω − V ) = Pn ut ′′ (∇ω(k ) − y /t) · S (∇ω(k ) − y /t), sum over k and n, and apply (89) to see that i i i i i=1 ∞ X n X n=0 k=0 ′′ |hαt,nk , (∇Ω − V ) · Snk (∇Ω − V )βt,n i| 1/2     ≤ hf˜χ̃ψt , dΓ (∇ω − v) · S ′′ (∇ω − v) ⊗ 1 + 1 ⊗ dΓ (∇ω − v) · S ′′ (∇ω − v) f˜χ̃ψt i   1/2 × hf χϕt , N (n∞ +5)/2 dΓ (∇ω − v) · S ′′ (∇ω − v) N (n∞ +5)/2 f χϕt i + O(t−2) kf˜χ̃ψt k k(N + 1)9/2 f χϕt k. This is integrable w. r. to t by Proposition 11. To prove the last statement it suffices to show that e−iH̃s W+ eiHs ϕ = W+ ϕ for all s ∈ R. This follows from −iHt e−iH̃s W+ eiHs ϕ − W+ ϕ = lim eiH̃t χ̃[f˜Qf ]t+s ϕ t χe t→∞ if we prove that χ̃∂/∂t[f˜Qf ]χ = O(t−1 ). The contributions due to ∂t f˜ and ∂t f are easily seen to be of order t−1 . As for ∂t Q note that, by Lemma 39,  1/2 ∂ 2 Snk 2 n Pn−k ⊗ Pk ∂t Q = D0 ∂t Snk = ∇Ω · ∇(∂t Snk ) + +n O(t−2) 2 ∂t k  1/2 n = n2 O(t−1) k for |y/t| ≤ 2λ. Use this, (89), and that k ≤ n∞ thanks to the energy cutoff χ̃. By construction of W+ , Wλ = Ω̃+ W+ as we show in the next lemma. Some minor technical difficulties in its proof are due to the presence of the cutoffs and due to the unboundedness of I. Lemma 16. Suppose Wλ and W+ are defined as in Theorem 12 and Theorem 15. Then, under the assumptions of these theorems, Wλ = s − lim eiHt Ie−iH̃t W+ = Ω̃+ W+ . t→∞ Proof. By definition of W+ we have, for all ϕ ∈ H, Ie−iH̃t W+ ϕ = I χ̃f˜D̃0 [Γ̆(jt )dΓ(S)]f χe−iHt ϕ + o(1), for t → ∞. (90) Note here that e−iH̃t W+ ϕ is in the domain of I, for all t ∈ R, because W+ ϕ = χ′ (H̃)W+ ϕ, for any χ′ ∈ C0∞ (R), with χ′ = 1 on supp χ. From (90) it follows now, if we expand the free Heisenberg derivative, that Ie−iH̃t W+ ϕ = I χ̃f˜dΓ̆(jt , djt )dΓ(S)f χe−iHt ϕ + I χ̃f˜Γ̆(jt )dΓ(dS)f χe−iHt ϕ + o(1) = f I χ̃dΓ̆(jt , djt )dΓ(S)f χe−iHt ϕ + I χ̃Γ̆(jt )f dΓ(dS)f χe−iHtϕ + o(1). (91) FGSch, 30/Mar/01—Asymptotic Completeness 33 To prove the second equality commute f˜ to the left in the first term, using that, by Lemma 28, [χ̃, f˜] = O(t−1), and to the right in the second term, using that f˜Γ̆(jt ) = Γ̆(jt )f . Choose now χ1 ∈ C0∞ (R), with χ1 = 1 on supp χ, and supp χ1 ⊂ (−∞, Σ), and set χ1 = χ1 (H). Then χ1 χ = χ, and thus, by (91), Ie−iH̃t W+ ϕ = f I χ̃ dΓ̆(jt , djt )dΓ(S)f χ1 χe−iHt ϕ + I χ̃ Γ̆(jt )f dΓ(dS)f χ1 χe−iHt ϕ + o(1) = f I χ̃ dΓ̆(jt , djt )χ1 dΓ(S)f χe−iHt ϕ + I χ̃ Γ̆(jt )χ1 f dΓ(dS)f χe−iHt ϕ + o(1), (92) where, in the second equality, we used that [f dΓ(S)f, χ1 ]χ = O(t−1 ), for t → ∞, which easily follows, expanding χ1 in a Hellfer-Sjöstrand integral, by Hypothesis (H3), by Lemma 28 and because, by assumption, eα|x| χ is a bounded operator, for some α > 0. Below we will show that I χ̃ Γ̆(jt ) χ1 = χ + o (1) and I χ̃ dΓ̆(jt , djt ) χ1 = o (t−1 ). (93) (94) Inserting these two equations in (92) it follows, since dΓ(S/t)f χ is uniformly bounded in t, Ie−iH̃t W+ ϕ = χf dΓ(dS)f χe−iHt ϕ + o(1) = e−iHt Wλ ϕ + o(1), which proves the lemma. It only remains to prove (93) and (94). We begin proving (93). To this end we note that I Γ̆(jt ) = 1H (because, by construction of jt , j0,t + j∞,t = 1) and thus, for any n ∈ N, χ = I Γ̆(jt )χ = IE[0,n] (N∞ )Γ̆(jt )χ + IE(n,∞) (N∞ )Γ̆(jt )χ, (95) where E∆ denotes the characteristic function of the set ∆. Now we claim that the norm of the second term on the r.h.s. of the last equation can be made arbitrarily small by choosing n sufficiently large. This follows because IE(n,∞) (N∞ )Γ̆(jt )χ = IE(n,∞) (N∞ )Γ̆(jt )E(n,∞) (N)χ, which implies, since kIE(n,∞) (N∞ )Γ̆(jt )k ≤ 1 for all n ∈ N, that kIE(n,∞) (N∞ )Γ̆(jt )χk ≤ kIE(n,∞) (N∞ )Γ̆(jt )k kE(n,∞) (N)(N + 1)−1 k k(N + 1)χk (96) C . ≤ n+2 On the other hand the first term on the r.h.s. of (95) can be written as IE[0,n] (N∞ )Γ̆(jt )χ = I(N0 + 1)−n E[0,n] (N∞ )(N0 + 1)n Γ̆(jt )χχ1 = I(N0 + 1)−n E[0,n] (N∞ )(N0 + 1)n χ̃Γ̆(jt )χ1 + o(1), (97) where the second equality follows because of Lemma 32, and because I(N0 +1)−n E[0,n] (N∞ ) is a bounded operator (see Lemma 1). Now, for n ∈ N sufficiently large, E[0,n] (N∞ )χ̃ = χ̃. This remark, together with (95), (96) and (97) shows that, for any ε > 0, kχ − I χ̃Γ̆(jt )χ1 k < ε, for t sufficiently large. This proves (93). Eq.(94) follows in a very similar way by IdΓ̆(jt , djt ) = D0 (I Γ̆(jt )) = D0 (1H ) = 0, using that kIE(n,∞) (N∞ )dΓ̆(jt , djt )(N + 1)−1 k ≤ const and applying Lemma 33 (see Appendix D). 34 So far the positive parameter u in the construction of W+ was arbitrary. The next lemma now shows that, by choosing u small, we can neglect the possibility to find zero ”escaping photons” in states in the range of W+ . This will be important in the proof of asymptotic completeness, Theorem 19. Lemma 17. Assume Hypotheses (H1), (H2) and (H5) hold, and let the Deift-Simon wave operator W+ be defined as in Theorem 15. Then k(1 ⊗ E{0} (N))W+ k ≤ 2 u2k(N + 1)1/2 χ(H)k2 . Proof. For ϕ ∈ H and ψ ∈ H ⊗ F we define ϕt = e−iHt ϕ respectively ψt = e−iH̃t ψ. Then by definition of W+ , we have n o hψ, W+ ϕi = lim hψ, eiH̃t χ̃f˜D̃0 Γ̆(jt )dΓ(S) f e−iHt χϕi t→∞ n o (98) = lim hψt , χ̃f˜D̃ Γ̆(jt )dΓ(S) f χϕt i. t→∞ The last equality follows because, by assumption, the operator eα|x| χ is bounded for some α > 0 and because the norm of the operator n o (99) e−α|x| (iφ(G) ⊗ 1)Γ̆(jt )dΓ(S) − Γ̆(jt )dΓ(S)iφ(G) f (N + 1)−2 tends to 0, as t → ∞. To see this write the operator in (99) as n o e−α|x| (iφ(G) ⊗ 1)Γ̆(jt ) − Γ̆(jt )iφ(G) dΓ(S)f (N + 1)−2 −α|x| + Γ̆(jt )e −2 (100) [iφ(G), dΓ(S)]f (N + 1) . Now the operator in the first line equals, by (24), −ie−α|x| {φ((j0,t − 1)G) ⊗ 1 + 1 ⊗ φ(j∞,t G)} (N0 + N∞ + 1)−1 Γ̆(jt ) dΓ(S)f (N + 1)−1 , and tends to 0 as t → ∞. This follows because dΓ(S)f (N + 1)−1 = O(t), while, by Hypothesis (H5), the factor on the left of Γ̆(jt ) is of order O(t−µ ), for some µ > 1. To handle the operator in the second line of (100) use that [iφ(G), dΓ(S)] = φ(−iSG), and that, by (H5), sup e−α|x| kφ(−iSGx )(N + 1)−1 k = sup e−α|x| kφ(χ(|y| ≥ tδ )iSGx )(N + 1)−1 k = O(t−1+δ ). x x This implies that also the term in the second line of (100) tends to zero as t → ∞. From (98) it follows now, because terms involving the Heisenberg derivative of f , or f˜ give, by Lemma 28, a vanishing contribution in the limit t → ∞, that d 1 hψt , χ̃f˜Γ̆(jt )dΓ(S)f χϕt i = lim hψt , χ̃f˜Γ̆(jt )dΓ(S)f χϕt i t→∞ dt t 1 = lim hΓ̆(jt )∗ χ̃ψt , f dΓ(y 2/2t)f χϕt i, t→∞ t hψ, W+ ϕi = lim t→∞ (101) 35 FGSch, 30/Mar/01—Asymptotic Completeness where, in the last equality, we used Lemma 8 to replace S by y 2/2t. Consider in particular the case ψ = (1 ⊗ E{0} (N))ψ; then there exists some α ∈ H with ψ = α ⊗ Ω, and Γ̆(jt )∗ χ̃ψt = I (Γ(j0,t ) ⊗ Γ(j∞,t )) ((χαt ) ⊗ Ω) = Γ(j0,t )χαt . For such ψ = α ⊗ Ω, (101) reads 1 hαt , χf Γ(j0,t )dΓ(y 2 )f χϕt i. t→∞ 2t2 hψ, W+ ϕi = lim (102) Now we note that |hαt , χf Γ(j0,t )dΓ(y 2)f χϕt i| = |hαt , χf dΓ(j0,t , j0,t y 2 )f χϕt i| ≤ k(N + 1)1/2 χαt k k(N + 1)1/2 χϕt k kj0,t y 2 k 2 2 ≤ 4u t kαk kϕk k(N + 1) 1/2 (103) 2 χk , where, in the last step, we used that kj0,t y 2 k ≤ sup|y|≤2u t y 2 ≤ 4t2 u2 . Since kαk = kψk, it follows from (103) and (102) that |hψ, (1 ⊗ E{0} (N))W+ ϕi| ≤ 2u2 k(N + 1)1/2 χ(H)k2 kψk kϕk. 36 9 Asymptotic Completeness As explained in the introduction we prove asymptotic completeness by induction in the energy measured in units of m. The first step is the following, essentially trivial lemma. The idea is that AC on Ran Eη (H) as explained in the introduction in Eq. 3, implies the same property for Ie−iH̃t on Ran Eη (H) ⊗ F , the photons from F merely contributing to the asymptotically free radiation. Lemma 18. Assume that hypotheses (H1), (H2) and (H4) are satisfied, and let the wave operators Ω̃+ and Ω+ be defined as in Lemma 6 and in Theorem 7, respectively. Suppose Ran Ω+ ⊃ Eη (H)H and µ < Σ. Then for every ϕ ∈ Ran Eµ (H̃) there exists a vector ψ ∈ Ran Eµ (H̃) such that Ω̃+ (Eη (H) ⊗ 1)ϕ = Ω+ ψ. If ϕ ∈ E∆ (H̃)H̃ then ψ ∈ E∆ (H̃)H̃. Proof. By Lemma 30 every given ϕ ∈ Eµ (H̃)H̃ can be approximated by a sequence of vectors ϕn ∈ Eµ (H̃)H̃ which are finite linear combinations of vectors of the from ∗ ∗ γ = α ⊗ a (h1 ) . . . a (hn )Ω, λ+ n X Mi < µ (104) i=1 where α = Eλ (H)α and Mi = {|k| : hi (k) 6= 0}. Let γ ∈ H̃ be of the form (104). Then Ie−iH̃t (Eη (H) ⊗ 1) γ = Ie−iH̃t Eη (H)α ⊗ a∗ (h1 ) . . . a∗ (hn )Ω = a∗ (h1,t ) . . . a∗ (hn,t ) e−iHt Eη (H)α. (105) By assumption Eη (H)α = Ω+ β for some β ∈ H̃ and we may assume β = Eη (H̃)β by the intertwining relation for Ω+ . From (105) it follows that Ie−iH̃t (Eη (H) ⊗ 1) γ = a∗ (h1,t ) . . . a∗ (hn,t )e−iHt Ω+ β = a∗ (h1,t ) . . . a∗ (hn,t )Ie−iH̃t (PB ⊗ 1) β (106) n o ∗ ∗ −iHt −iH̃t + a (h1,t ) . . . a (hn,t ) e Ω+ β − Ie (PB ⊗ 1) β , where PB denotes the orthogonal projector onto Hpp (H). After inserting a factor (N + 1)−n/2 (N + 1)n/2 in the second factor on the r.h.s. of last equation, just in front of the braces, we get kIe−iH̃t (Eη (H) ⊗ 1) γ − Ie−iH̃t (PB ⊗ 1) (1 ⊗ a∗ (h1 ) . . . a∗ (hn ))βk n o ≤ ka∗ (h1,t ) . . . a∗ (hn,t )(N + 1)−n/2 k k(N + 1)n/2 e−iHt Ω+ β − Ie−iH̃t (PB ⊗ 1)β k. (107) The first factor on the r.h.s. is bounded by a finite constant, uniformly in t. The second factor converges to zero as t → ∞. To see this use that it stays bounded for 37 FGSch, 30/Mar/01—Asymptotic Completeness all integers n, a fact which follows from the boundedness of (H + i)n/2 Ω+ Eµ (H̃) and of (N + 1)n/2 I Eµ (H̃). Since (1 ⊗ a∗ (h1 ) . . . a∗ (hn ))β ∈ Eµ (H̃)H̃ it follows that Ω̃+ (Eη (H) ⊗ 1)γ = Ω+ (1 ⊗ a∗ (h1 ) . . . a∗ (hn ))β. Hence to each ϕn , as defined at the very beginning, there exists a vector ψn ∈ Eµ (H)H̃ such that Ω̃+ (Eη (H) ⊗ 1)ϕn = Ω+ ψn . The left side converges to Ω̃+ (Eη (H) ⊗ 1)ϕ as n → ∞, and hence the right side converges as well. Since Ω+ is isometric on Hpp (H) ⊗ F it follows that (PB ⊗ 1)ψn is Cauchy and hence has a limit ψ ∈ Eµ (H)H̃. Thus Ω̃+ (Eη (H) ⊗ 1)ϕ = Ω+ ψ which proves the first part of the lemma. The second part follows from the intertwining relations for Ω̃+ and Ω+ . Theorem 19 (Asymptotic Completeness). Assume hypotheses (H1) through (H5) are satisfied. Then Ran Ω+ ⊃ E(−∞,Σ) (H)H. Proof. The proof is by induction. We show that Ran(Ω+ ) ⊃ E(−∞,km) (H)E(−∞,Σ)(H)H (108) holds for all integers k with (k − 1)m ≤ Σ. For k < 0 and |k| large enough (108) is trivially correct because H is bounded from below. Hence we may assume (108) holds for k = n − 1. To prove it for k = n it suffices to show that Ran Ω+ ⊃ E∆ (H)H (109) for compact intervals ∆ ⊂ (−∞, nm) ∩ (−∞, Σ)\S because, by Theorem 4, the union of such subspaces is dense in E(−∞,nm) (H)E(−∞,Σ) (H)H and because Ran Ω+ is closed. Now choose χ ∈ C0∞ (R) such that χ = 1 on ∆ and supp χ ⊂ (−∞, Σ)\S. Let λ, u > 0 and define Wλ and W+ in terms of χ, λ and u as in Theorem 12 and 15. Define moreover the asymptotic observable W = limλ→∞ Wλ as in Proposition 13. By Proposition 14, the operator W : E∆ (H)H → E∆ (H)H is onto and hence for every given ϕ = E∆ (H)ϕ there exists a ψ ∈ Ran E∆ (H) such that ϕ = W ψ. Given ε > 0 we pick u, λ small, respectively large enough so that kΩ̃+ χ(N∞ = 0)W+ ψk < ε, k(W − Wλ )ψk < ε, (110) by using Lemma 17, Proposition 13, and the boundedness of Ω̃+ E∆ (H̃). Then by Lemma 16 Wλ ψ = Ω̃+ W+ ψ = Ω̃+ χ(N∞ > 0)W+ ψ + Ω̃+ χ(N∞ = 0)W+ ψ. (111) The vector χ(N∞ > 0)W+ ψ has at least one boson in the outer Fock space and thus an energy of at most (n − 1)m in the inner one. More precisely χ(N∞ > 0)W+ ψ = χ(H < (n − 1)m) ⊗ χ(N > 0)W+ ψ. 38 Hence by induction hypothesis and Lemma 18 there exist γ ∈ E∆ (H)H̃ such that Ω̃+ χ(N∞ > 0)W+ ψ = Ω+ γ. This equality together with (110) and (111) shows that kϕ − Ω+ γk = kW ψ − Ω̃+ χ(N∞ > 0)W+ ϕk < 2ε which proves the theorem. 39 FGSch, 30/Mar/01—Asymptotic Completeness 10 Massless Photons This section is devoted to the case where the bosons are massless photons, but the soft modes do not interact with the particles (electrons). That is H = K ⊗ 1 + 1 ⊗ dΓ(|k|) + φ(G) = H0 + φ(G) (112) and (IR) Gx (k) = 0 if |k| < m for some m > 0. As before, we assume that Gx ∈ h for each x ∈ X and that supx kGx k < ∞. Then φ(G)(H0 + 1)−1/2 is again bounded and hence H is self-adjoint on D(H0 ). The key idea is to compare H with the modified Hamiltonian Hmod = K ⊗ 1 + 1 ⊗ dΓ(ω) + φ(G) (113) where K and G are as above but the dispersion ω(k) = |k| is modified for |k| < m. We choose ω in such a way that (H1) is satisfied (with m/2 instead of m) and ω(k) = |k| for |k| ≥ m. 10.1 Asymptotic Completeness Asymptotic completeness for H is essentially a corollary of Theorem 19. Let H̃ = H ⊗ 1 + 1 ⊗ dΓ(|k|) and let Ω+ and Ω+,mod be defined in terms of H, Hmod , and H̃mod = Hmod ⊗ 1 + 1 ⊗ dΓ(ω). Theorem 20 (Asymptotic Completeness). Assume (IR), (H2), and (H5) for the system defined by (112). Then the wave operator Ω+ exists on E(−∞,Σ) (H̃)H̃ and Ran(Ω+ ) ⊃ E(−∞,Σ) (H)H. Proof. We split the Fock space into two Fock spaces, one with interacting photons the other one with non-interacting photons. Henceforth the subindices i and s refer to interacting and soft respectively. Let Ki = {k ∈ Rd : |k| ≥ m} Ks = {k ∈ Rd : |k| < m} hi = L2 (Ki ) (114) hs = L2 (Ks ). (115) Then h = hi ⊕ hs and correspondingly F (h) is isomorphic to F (hi ) ⊗ F (hs ) with an isomorphism U as given in Section 2.5. By assumption on G and ω and by (21) UHmod U ∗ = Hi ⊗ 1 + 1 ⊗ dΓ(ωs ) UHU ∗ = Hi ⊗ 1 + 1 ⊗ dΓ(|k|s ) with respect to the factorization H = (L2 (X) ⊗ F (hi )) ⊗ F (hs ). Here Hi = K ⊗ 1 + 1 ⊗ dΓ(|k|) + φ(G) on Hi := L2 (X) ⊗ F (hi ). It follows that Hmod and H have the same 40 eigenvectors, they are of the form U ∗ (ϕi ⊗ Ωs ) where ϕi is an eigenvector of Hi . Hence PB (H) = PB (Hmod ). Furthermore eiHt Ie−iH̃t = eiHmod t e−idΓ(ω−|k|)t Iei[dΓ(ω−|k|)⊗1+1⊗dΓ(ω−|k|)]t e−iH̃mod t = eiHmod t Ie−iH̃mod t because Hmod and dΓ(ω − |k|) commute. This shows that Ω+ = Ω+,mod and hence that Ω+ exists on χ(H̃mod ≤ µ)H̃ and that Ran Ω+ ⊃ χ(Hmod ≤ µ)H for all µ < Σ, by Theorem 19. To reformulate these results in terms of Hi we consider U ⊗ U as a map from F ⊗ F to Fi ⊗ Fi ⊗ Fs ⊗ Fs . Then UΩ+ (U ∗ ⊗ U ∗ ) exists on (U ⊗ U)χ(H̃mod ≤ µ)H̃ ⊃ χ(H̃i ≤ µ)H̃i ⊗Ωs ⊗Ωs , where H̃i = Hi ⊗1+1⊗dΓ(|k|) on H̃i = Hi ⊗Fi , and Ran UΩ+ (U ∗ ⊗U ∗ ) ⊃ Uχ(Hmod ≤ µ)H ⊃ χ(Hi ≤ µ)Hi ⊗ Ωs . Furthermore UΩ+ (U ∗ ⊗ U ∗ ) = Ωi ⊗ (Is (χ(Ns = 0) ⊗ 1s )) where Ωi = s − limt→∞ eiHi t Ii e−iH̃i t (PB (Hi ) ⊗ 1). In fact UI(U ∗ ⊗ U ∗ ) = Ii ⊗ Is and UPB (H)U ∗ = PB (Hi ) ⊗ χ(Ns = 0). It follows that UΩ+ (U ∗ ⊗ U ∗ ) exists on χ(H̃i ≤ µ)H̃i ⊗ Fs ⊗ Fs ⊃ (U ⊗ U)χ(H̃ ≤ µ)H̃ and that its range contains χ(Hi ≤ µ)Hi ⊗ Fs ⊃ Uχ(H ≤ µ)H. Hence Ω+ exists on χ(H̃ ≤ µ)H̃ and by the intertwining relation for Ω+ , the range of Ω+ |`χ(H̃ ≤ µ)H̃ contains χ(H ≤ µ)H. The theorem now follows because µ < Σ was arbitrary and kΩ+ k = 1. 10.2 Relaxation to the Ground State With the help of AC established in the last section we next show that states below the ionization threshold relax to the ground state in the sense (7) under the dynamics generated by the Hamilton operator (112). We begin by summarizing results due to Bach et al. [BFSS99], [BFS98] on the point spectrum of H that are needed in this section. No infrared cutoff is assumed in the following discussion. Consider the Hamilton operator Hg = K ⊗ 1 + 1 ⊗ dΓ(|k|) + gφ(G) (116) where K = −∆ + V on L2 (Rn ) and V is operator-bounded w.r.t. −∆ with relative bound zero. This assumption allows for typical N-body Schrödinger operators [HS00]. We assume that   Z 1 2 + 1 dk < ∞ (117) sup |Gx (k)| |k| x to ensure self adjointness of Hg on D(Hg=0). Following [BFSS99] we furthermore assume that Z |(k · ∇k )2 Gx (k)|2 2 −M/2 dk <∞ (H6) sup(1 + |x| ) |k| x 41 FGSch, 30/Mar/01—Asymptotic Completeness for some M ≥ 0. All exited bound states of Hg will be unstable if their life time as given by Fermi’s Golden Rule is finite. To state this condition we have to introduce some notation. Suppose E0 < E1 < · · · < inf σess (K) are the isolated eigenvalues of K with finite multiplicity. Let mj be the multiplicity of the eigenvalue Ej , and let ϕj,l ∈ L2 (Rn ), for l = 1, . . . mj be an orthonormal base of the eigenspace of K to the eigenvalue Ej . Then for each 0 ≤ i < j and for each k ∈ Rd we define the mi × mj matrix Aij (k) := hϕi,r , Gx (k)ϕj,si. Now, for each j ≥ 0 we define the mj × mj matrix XZ Γj = A∗ij (k)Aij (k)δ(ω(k) − Ej + Ei )dk (118) i:i<j The eigenvalues of this matrix are then the resonance widths in second order perturbation theory corresponding to the eigenvalues Ej . To show that Hg has no eigenvalues in neighborhoods of the eigenvalues of the unperturbed Hamiltonian H0 , we therefore need the following assumption: (H7) Fermi Golden Rule. For each j ≥ 1 we have Γj > 0. The following theorem summarizes results from [BFSS99] and [BFS98]. Theorem 21. i) (Exponential decay, [BFS98]) Suppose µ < inf σess (K) and ε > 0. Then there exists a constant M = M(ε) such that keα|x| χ(Hg < µ)k < M R for all α, g with inf σess (K) − µ − α2 − g 2 supx dk |Gx (k)|2 /|k| > ε. (119) kPψg − Pϕ0 ⊗Ω k → 0, (120) ii) (Existence and uniqueness of the ground state, [BFS98, GLL00]) If inf σ(K) < R inf σess (K) − g 2 supx dk |Gx (k)|2 /|k| and Gx (−k) = Gx (k) then Eg := inf σ(Hg ) is a non-degenerate eigenvalue of Hg . Moreover if ψg is an eigenvector of Hg corresponding to the eigenvalue Eg , then as g → 0, where Pψg and Pϕ0 ⊗Ω denote the orthogonal projections onto the spaces spanned by the ground states ψg and ϕ0 ⊗ Ω, respectively. iii) (Absence of exited eigenstates, [BFSS99]) Assume (H6) and (H7). Set ∆ = [E0 + ε, µ], for fixed ε > 0 and µ < inf σess (K). Then σpp (Hg ) ∩ ∆ = ∅ (121) for g > 0 sufficiently small. If the infrared cutoff (IR) is imposed, then assumption (117) simplifies to supx kGx k < ∞ and all results of the above theorem then hold for g sufficiently small. Note that m must be small in order for (H7) to hold, because transitions between energy levels Ej with separation less than m are suppressed by the infrared cutoff. Next we prove absence of eigenvalues above and close to Eg for g small, and assuming (IR). As in [BFSS99] we argue by contradiction and prove a virial theorem as well as the positivity of [iH, A] on a spectral interval (Eg , E1 − ε] and for a suitable conjugate operator A. 42 Lemma 22 (Virial RTheorem). Assume (IR) and (H3). inf σess (K) − g 2 supx dk |Gx (k)|2 /|k| then ϕ ∈ D(N) and If Hϕ = Eϕ and E < hϕ, (N − gφ(iaG)) ϕi = 0. Proof. With the notation of the proof of Theorem 20 the eigenvector ϕ is of the form U ∗ (ϕi ⊗ Ωs ). It follows that ϕ is an eigenvector of Hmod and thus in D(Hmod ) ⊂ D(N). By Theorem 21, part i), eα|x| ϕ ∈ H for some α > 0 and hence Lemma 3 applies to ϕ and Hmod . This shows that hϕ, [dΓ(|∇ω|2) − gφ(iaG)]ϕi = 0 which proves the theorem because, by the form of ϕ, dΓ(|∇ω|2)ϕ = Nϕ. Theorem 23 (Positive commutator). Assume (IR), (H0), and (H3). Set ∆ = (Eg , E1 − ε], for some fixed ε > 0 (here E1 is the first point in the spectrum of K above inf σ(K)). Then there is a constant C > 0 such that E∆ (Hg ) (N − g φ(iaG)) E∆ (Hg ) ≥ CE∆ (Hg ), for all g > 0 sufficiently small. In particular, by Lemma 22, it follows that Hg has no eigenvalue in ∆, if g > 0 is small enough. Proof. Using N ≥ 1 − 1 ⊗ PΩ we get E∆ (Hg ) (N − g φ(iaG)) E∆ (Hg ) ≥ E∆ (Hg ) (1 − 1 ⊗ PΩ − g φ(iaG)) E∆ (Hg ) = E∆ (Hg ) − E∆ (Hg ) (1 ⊗ PΩ ) E∆ (Hg ) − gE∆ (Hg )φ(iaG)E∆ (Hg )     −α|x| α|x| 1/2 ≥ E∆ (Hg ) 1 − g sup e kiaGx k kE∆ (Hg )e k k(Ni + 1) E∆ (Hg )k (122) x − E∆ (Hg ) (1 ⊗ PΩ ) E∆ (Hg ) R where Ni = |k|>m dk a∗ (k)a(k) is the operator counting the number of interacting bosons (which is bounded w.r.t. Hg ), and where PΩ is the orthogonal projector onto Ω. By Hypothesis (H3), and because kaGx k ≤ const (kyGx k + kGx k), the number in the parenthesis in the first term on the r.h.s. of the last equation is larger than C, for any C < 1, if g > 0 is small enough. It remains to show that the last term in (122) converges to 0 as g → 0. To do this we split it into two parts, according to
E∆ (Hg ) (1 ⊗ PΩ ) E∆ (Hg ) = E∆ (Hg ) E{E0 } (K) ⊗ PΩ E∆ (Hg )
(123) + E∆ (Hg ) E[E1 ,∞) (K) ⊗ PΩ E∆ (Hg ), where K is the particle Hamiltonian, and E0 and E1 are its ground state energy and its first exited eigenvalue. Since Pψg E∆ (Hg ) = 0 the first term in Eq. (123) can be written as

(124) E∆ (Hg ) E{E0 } (K) ⊗ PΩ E∆ (Hg ) = E∆ (Hg ) Pϕ0 ⊗Ω − Pψg E∆ (Hg ), FGSch, 30/Mar/01—Asymptotic Completeness 43 which converges to zero by Theorem 21, part ii). Consider now the second term on the r.h.s. of (123). Choose χ ∈ C∞ 0 (R), such that 0 ≤ χ ≤ 1, χ(s) = 0 if s > E1 − ε/2, and χ(s) = 1 if s ∈ ∆ (this is a smooth version of the characteristic function E∆ ). Then we have
on the one hand χ(Hg )E∆ (Hg ) = E∆ (Hg ) and on the other hand E[E1 ,∞) (K) ⊗ PΩ χ(H0 ) = 0. Thus we get

(125) E[E1 ,∞) (K) ⊗ PΩ E∆ (Hg ) = E[E1 ,∞) (K) ⊗ PΩ (χ(Hg ) − χ(H0 )) E∆ (Hg ). Now if χ̃ is an almost analytic extension of χ, in the sense of the Helffer–Sjöstrand functional calculus (see Appendix A.2), then we have Z g χ(Hg ) − χ(H0 ) = − dxdy ∂z̄ χ̃(z − Hg )−1 φ(G)(z − H0 )−1 . (126) π This implies, since χ has a compact support, that kχ(Hg ) −
χ(H0 )k ≤ Cg, for some constant C > 0, and thus, by (125), that k E[E1 ,∞)(K) ⊗ PΩ E∆ (Hg )k → 0 as g → 0. This completes the proof of the theorem. The last Theorem, together with Theorem 21, proves the following corollary. Corollary 24. Assume Hypotheses (IR), (H3), (H6), and (H7). If µ < inf σess (K), then σpp (Hg ) ∩ (Eg , µ) = ∅, for all g > 0 sufficiently small. With the help of this corollary and Theorem 20 we next prove relaxation to the groundstate in the sense of the following theorem. To define the algebra of observables let A denote the C ∗ algebra generated by all Weyl operators W (h) = exp(iφ(h)), with h ∈ L2 (Rd , dk). By taking tensor products of operators in A with bounded operators acting on the Hilbert space Hel = L2 (Rn , dx) of the electrons one obtains a C ∗ algebra, which we denote by Ã. Theorem 25 (Relaxation to the ground state). Assume Hypotheses (IR), (H0), and (H3) through (H7). Choose µ < inf σess (K). Then, for sufficiently small values of the coupling constant g > 0, the Hamiltonian Hg exhibits the property of relaxation to the ground state for states with energy less than µ. This means that, if g > 0 is sufficiently small, then, for all A ∈ Ã and for all ψ ∈ Ran(H ≤ µ), we have lim hψt , Aψt i = hψg , Aψg i hψ, ψi, t→∞ (127) where ψt = e−iHt ψ and ψg denotes the groundstate of Hg . Proof. Since the C ∗ algebra A is generated by the Weyl-operators W (h) = eiφ(h) , and because products of Weyl-operators are again Weyl-operator (up to some unimportant phase) it is enough if we prove (127) for A = B ⊗ W (h), where B is a bounded operator on Hel and h ∈ S(Rd ). 44 By Theorem 20 we know that the system we are considering is asymptotically complete. On the other hand we know, from Corollary 24, that the ground state ψg is the only eigenstate of Hg , which lies in the range of the spectral projection χ(Hg ≤ µ). These two results imply that each ψ ∈ Ranχ(Hg ≤ µ) can be written as limit of a sequence of finite linear combinations of states like a∗+ (f1 ) . . . a∗+ (fm )ψg , with fi ∈ S(Rd ). Since we are dealing only with bounded operators, it follows that it is enough to prove (127) for A = B ⊗ W (h) and for ψ= N X ci a∗+ (f1i )a∗+ (f2i ) . . . a∗+ (fni i )ψg . (128) i=1 In this case we have −iHt lim he t→∞ −iHt ψ, Ae ψi = N X i,j=1 = N X i,j=1 c̄i cj lim he−iHt t→∞ c̄i cj lim h t→∞ Qni l=1 Q ni ∗ i −iHt l=1 a+ (fl )ψg , Ae i a∗ (fl,t )ψg , A Qnj m=1 Qnj m=1 j a∗+ (fm )ψg i j a∗ (fm,t )ψg i, (129) where we used the definition of the asymptotic fields a∗+ (h). Notice now that the ground state ψg of Hg is in the domain of arbitrary powers of the field-Hamiltonian dΓ(|k|). Moreover we know that the Weyl operators leave D(dΓ(|k|)n ) invariant. This follows by the commutation relations [W (h), dΓ(|k|)] = −φ(i|k|h)W (h) + 1/2 Re (|k|h , h) W (h). These remarks imply that we can rewrite the limit in the r.h.s. of the last equation as Qnj Q i ∗ i j a∗ (fm,t lim h nl=1 a (fl,t )ψg , A m=1 )ψg i t→∞ Qni Qnj j i = lim hψg , l=1 a(fl,t ) (B ⊗ W (h)) m=1 a∗ (fm,t )ψg i t→∞ Q i Qnj j i = lim hA∗ ψg , nl=1 a(fl,t ) m=1 a∗ (fm,t )ψg i (130) t→∞  Qni Q n j j i ∗ + lim hψg , B ⊗ m=1 a (fm,t )ψg i. l=1 a(fl,t ), W (h) t→∞ If we expand the commutator in the last equation, we get a sum of ni terms. Each i of these terms contains a contraction (fl,t , h)L2 . Now, since we have assumed that i d fl , hr ∈ S(R ), we have Z i (fl,t , hr ) = dkfli (k)hr (k)ei|k|t → 0 as t → ∞, and thus the second term on the r.h.s. of (130) vanishes. To handle the first term on the Qn r.h.s. of (130) we use that, by Lemma 26, limt→∞ l=1 a(fl,t )ψg = 0. Assuming ni ≥ nj , this implies that the first term on the r.h.s. of (130) vanishes if ni > nj , and that lim t→∞ ni Y l=1 i a(fl,t ) nj Y m=1 j a∗ (fm,t )ψg = lim ψg hψg , t→∞ Qni l=1 i a(fl,t ) Qnj m=1 j a∗ (fm,t )ψg i Q i Qnj j = ψg hψg , nl=1 a+ (fli ) m=1 a∗+ (fm )ψg i Q nj Qni ∗ i j )ψg i, = ψg h l=1 a+ (fl )ψg , m=1 a∗+ (fm 45 FGSch, 30/Mar/01—Asymptotic Completeness for all ni ≥ nj . Using the antisymmetry of the inner product it is analogously proven that the r.h.s. of (130) vanishes if ni < nj . Thus, for arbitrary ni , nj , lim h t→∞ Qni l=1 i a∗ (fl,t )ψg , A Qnj m=1 j a∗ (fm,t )ψg i = hA∗ ψg , ψg i h Qni ∗ i l=1 a+ (fl )ψg , Qnj m=1 j a∗+ (fm )ψg i. The theorem now follows if we insert this result into (129) and compute the sum over i, j. Lemma 26. Suppose N R∋ n > 0 and ϕ ∈ D((Hg + i)n/2 ). Then, if h1 , . . . , hn ∈ L2 (Rd , dk) with khi k2ω = dk |hi (k)|2 (1 + 1/|k|) < ∞, we have lim t→∞ Proof. Notice that k k n Y j=1 a(hj,t )ϕk ≤ k( ≤ Qn j=1 n X j=1 +k ≤C a(hj,t )ϕ = 0. (131) j=1 a∗ (hi )(dΓ(|k|) + 1)−n/2 k ≤ C i=1 n Y n Y a(hj,t ) − n Y a(h̃j,t ))(H + i) j=1 −n/2 Qn i=1 khi kω . This implies that kk(H + i) n/2 ϕk + k n Y a(h̃j,t )ϕk j=1   k a(h1,t ) . . . a(hj,t − h̃j,t) . . . a(h̃n,t ) (H + i)−n/2 kk(H + i)n/2 ϕk n Y a(h̃j,t )ϕk j=1 n X j=1 kh1 kω . . . khj − h̃j kω . . . khn kω + k n Y a(h̃j,t )ϕk, j=1 (132) where we used that k(dΓ(|k|) + 1)n/2 (Hg + i)−n/2 k < ∞ (see [FGS00]). Because of the last equation it is enough to prove (131) when hj ∈ C0∞ (Rd \{0}). In this case we have M = minj inf{|k| : hj (k) 6= 0} > 0, and there exists f ∈ C ∞ (Rd ) with f (k) = 0 if |k| < M/2, and f (k) = 1 if |k| ≥ M. Then, on the one hand, dΓ(f ) is bounded w.r.t. Hg (and higher powers of dΓ(f ) are bounded w.r.t. corresponding powers of Hg ). This Q implies that ϕ ∈ D((dΓ(f ) + 1)n/2 ). On the other hand nj=1 a(hj,t ) is bounded w.r.t. (dΓ(f ) + 1)−n/2 . Thus, with ψ = (dΓ(f ) + 1)+n/2 ϕ, we have n Y j=1 a(hj,t )ϕ = n Y a(hj,t )(dΓ(f ) + 1)−n/2 ψ. (133) j=1 Q Since nj=1 a(hj,t )(dΓ(f ) + 1)−n/2 is uniformly bounded in t, it is enough if we show that the r.h.s. of the last equation converges to 0, as t → ∞ for ψ = α ⊗ a∗ (f1 ) . . . a∗ (fm )Ω. 46 To this end we write n Y a(hj,t )(dΓ(f ) + 1) j=1 −n/2 m Y ∗ a (fi )Ω = (dΓ(f ) + 1 + n) −n/2 i=1 n Y a(hj,t ) j=1 = (dΓ(f ) + 1 + n)−n/2 " n Y j=1 m Y a∗ (fi )Ω i=1 a(hj,t ) , m Y i=1 # a∗ (fi ) Ω. (134) Expanding the commutator, we find a sum of terms containing n contractions (hj,t , fi )L2 . The lemma now follows, because all these contractions converge to 0 as t → ∞. 10.3 QED in Dipole Approximation As mentioned in the introduction, our methods can be extended to prove AC, as well as our further main results, for atoms described by ”non-relativistic QED” in the dipole approximation. In this section we briefly explain how this is accomplished. For an introduction to the standard model of non-relativistic QED and for the justification of the dipole approximation we refer to [BFS98]. Here we merely show how this model fits into our general framework. We consider a non-relativistic electron interacting with the quantized radiation field. (The generalization to N electrons is straightforward.) States of this system are described by vectors in the Hilbert space H = Hat ⊗ F , where Hat = L2 (R3 , dx), and F is the bosonic Fock space over h = L2 (R3 , C2 ). The Hamilton operator is H = K + dΓ(|k|) + φ(G), (135) where K = −∆ + V is assumed to satisfy Hypothesis (H0). To describe QED in the dipole approximation, we set Gx (λ, k) = κ(k)g(x)x · ελ (k) where x ∈ R3 is the position of the electron, and ελ (k), λ = 1, 2, are the polarization vectors orthogonal to k. As above κ ∈ C0∞ (R3 ), and κ(k) = 0 if |k| < m, for some m > 0. The factor g ∈ C0∞ (R3 ) is a space cutoff necessary to make Gx bounded as a function of x. From a physical point of view this simplification is legitimate, because the electrons are exponentially localized near the origin (see also [BFS98]). The Hamilton operator (135) clearly satisfies assumptions (H1), (H2), and (H4). The problem is that the polarization vectors ελ (k), as functions of k, cannot be chosen in such a way that they are twice differentiable on the unit sphere (an easy application of a famous theorem due to H. Hopf), and hence hypotheses (H3) and (H5) cannot be satisfied. In order to circumvent this problem, we introduce two systems of polarization S vectors, the ”north”-system εN λ (k) and the ”south”-system ελ (k). The north-system N 3 ελ (k) depends smoothly on k, for k ∈ R \ZS , where ZS is an open neighborhood of the negative z-axis {k ∈ R3 : k1 = k2 = 0, and k3 ≤ 0}, whereas the south-system εSλ (k) depends smoothly on k, for k ∈ R3 \ZN , where ZN is an open neighborhood of the 47 FGSch, 30/Mar/01—Asymptotic Completeness positive z-axis {k ∈ R3 : k1 = k2 = 0, and k3 ≥ 0}. The sets ZN and ZS are chosen such that ZN ∩ ZS ⊂ Bm/2 (0) = {k ∈ R3 : |k| < m/2}. For each k 6= 0, {εN λ (k)}λ=1,2 and S {ελ (k)}λ=1,2 are orthogonal bases in the plane perpendicular to k. Hence there exists a matrix R(k) ∈ O(2) such that X εSµ (k)Rµλ (k). εN λ (k) = µ=1,2 Correspondingly, if f ∈ L2 (R3 , C2 ) is a wave function describing a photon with respect to the north base, then the same photon is described by X (Rf )(k, λ) = Rλµ (k)f (k, µ), µ=1,2 with respect to the south base εSλ (k). The Hamilton operator (135) depends on the choice of polarization vectors, but two Hamilton operators corresponding to two different choices are unitarily equivalent, and hence we may make a choice by convenience. We choose the north system and denote the corresponding Hamilton operator by HN . As a reminder of this choice we also attach a subindex N to the Hilbert space HN = Hat ⊗ FN and to its Fock space FN . The idea is now to split each photon into two parts, a part supported in the northern half space and a part supported in the southern half space. The parts in the south will then be mapped into an auxiliary Fock space, where they are described with respect to the south base. The Hamilton operator describing the time evolution in this split representation then turns out to involve only interactions with smooth coupling functions. In order to separate the north-photons from the south-photons, we introduce two 2 functions jN , jS ∈ C ∞ (R3 ), with jN (k) + jS2 (k) = 1, if |k| > m, and with supp jN ⊂ 3 3 R \ZS , and supp jS ⊂ R \ZN (such functions exist because of the assumption ZN ∩ZS ⊂ Bm/2 (0)). Next we define an isometry u : h → h ⊕ h by f 7→ (jN f, RjS f ). This induces another isometry Γ̆(u) : FN → F ⊗ F , which is characterized by Γ̆(u)Ω = Ω ⊗ Ω and Γ̆(u)a♯ (f ) = (a♯ (jN f ) ⊗ 1 + 1 ⊗ a♯ (RjS f ))Γ̆(u). States of the total system are now described by vectors in the new Hilbert space Hnew = Hat ⊗ F ⊗ F ; however, only vectors in the subspace Hu := Hat ⊗ Ran Γ̆(u) correspond to physical states. On Hnew we define the new Hamiltonian Hnew = K ⊗ 1 ⊗ 1 + 1 ⊗ (dΓ(|k|) ⊗ 1 + 1 ⊗ dΓ(|k|)) + φ(jN GN ) ⊗ 1 + 1 ⊗ φ(RjS GN ). (136) The operators φ(jN GN ) ⊗ 1 and 1 ⊗ φ(RjS GN ) couple the north- and the south photons, respectively, to the electron. The new Hamiltonian Hnew leaves the subspace Hu of physical states invariant, and its restriction to Hu , denoted by Hu := Hnew |`Hu , is unitarily equivalent to the Hamiltonian HN , acting on the Hilbert space HN . In fact Hu = Γ̆(u)HN Γ̆(u)∗ . Most importantly, both form factors (jN GN )x (k, λ) = jN (k)κ(k)g(x)x · εN λ (k) and (RjS GN )x (k, λ) = (jS GS )x (k, λ) = jS (k)κ(k)g(x)x · εSλ (k) 48 in (136) are smooth on the entire k-space, and hence they satisfy Hypotheses (H3) and (H5). In order to describe asymptotically free photons, we introduce the new extended Hilbert space H̃new := Hat ⊗ F ⊗ F ⊗ F ⊗ F . Vectors in the first two copies of F describe interacting photons, while photons in the third and fourth copies of F are asymptotically free and live in the north and in the south of k-space, respectively. Physical states with asymptotically free photons are contained in the subspace H̃u = Hat ⊗ Ran Γ̆(u) ⊗ Ran Γ̆(u) of H̃new . The Hamiltonian generating the dynamics on the extended Hilbert space H̃new is given by H̃new = Hnew ⊗ 1F ⊗F + 1Hnew ⊗ (dΓ(|k|) ⊗ 1 + 1 ⊗ dΓ(|k|)). This operator leaves the subspace H̃u invariant, and H̃u := H̃new |`H̃u is unitary equivalent to the extended Hamiltonian H̃N = HN ⊗ 1 + 1 ⊗ dΓ(|k|), acting on the extended Hilbert space H̃N = Hat ⊗ FN ⊗ FN . On the spectral subspace of H̃new where H̃new < Σ we define the new wave operator Ωnew + : χ(H̃new < Σ)H̃new → χ(Hnew < Σ)Hnew iHnew t Ωnew Inew e−iH̃new t (PB (Hnew ) ⊗ 1F ⊗F ), + := s − lim e t→∞ where Inew takes all bosons from the third and the fourth Fock space and puts them into the first and into the second Fock space, respectively. The restriction of Ωnew to χ(H̃u < + u N Σ)H̃u , denoted by Ω+ , maps H̃u to Hu and is unitary equivalent to Ω+ : H̃N → HN , the usual wave operator defined in terms of HN and H̃N . In this new description, asymptotic completeness means that RanΩu+ = Ranχ(Hu < Σ). But this follows if we show that RanΩnew + = Ranχ(Hnew < Σ), (137) because Ωnew maps H̃u into Hu and its orthogonal complement H̃u⊥ ⊂ H̃new into HU⊥ . + Eq. (137) can be proved by extending, in an obvious manner, the methods of Sections 6 to 9 to Hnew on Hnew , to the new extended Hilbert space H̃new , and to the new wave operator Ωnew + . A Pseudo Differential Calculus and Functional Calculus This appendix collects our main tools for computing commutators. A.1 Pseudo Differential Calculus on Fock Space Lemma 27. Suppose f ∈ S(Rd ), g ∈ C n (Rd ) and sup|α|=n k∂ α gk∞ < ∞. Let p = −i∇. Then X (−i)|α| (∂ α f )(x)(∂ α g)(p) + R1,n i[g(p), f (x)] = i α! 1≤|α|≤n−1 = (−i) X 1≤|α|≤n−1 i|α| α (∂ g)(p)(∂ α f )(x) + R2,n α! 49 FGSch, 30/Mar/01—Asymptotic Completeness where α kRj,n k ≤ Cn sup k∂ gk∞ |α|=n Z dk |k|n |fˆ(k)|. In particular, and most importantly, if n = 2 in the limit ε → 0 i[g(p), f (εx)] = ε∇g(p) · ∇f (εx) + O(ε2 ) = ε∇f (εx) · ∇g(p) + O(ε2 ). Proof. Let f (x) = R dk eikx fˆ(k). The first equation follows from   g(p)eikx − eikx g(p) = eikx e−ikx g(p)eikx − g(p) = eikx [g(p + k) − g(p)] (138) and Taylor’s formula g(p + k) − g(p) = X (∂ α g)(p) 1≤|α|≤n−1 +n Z 0 kα α! 1 dt (1 − t)n−1 X (∂ α g)(p + tk)k α /α!. |α|=n To obtain the second equation write g(p)eikx − eikx g(p) = −[g(p − k) − g(p)]eikx instead of (138). Lemma 28. Suppose ω is in ∈ C ∞ (Rd ) and has bounded derivatives. If f ∈ C0∞ (R), then 1 [idΓ(ω), f ] = f ′ dΓ(∇ω · y/t + y/t · ∇ω) + O(t−2 )N t where f = f (dΓ(v 2 )), v = y/t and f ′ = f ′ (dΓ(v 2 )). Proof. The operators dΓ(ω) and f commute with N, hence it suffices to prove the equation on ⊗ns h. On this subspace " # n n X X i[dΓ(ω), f ] = ω(kj ), f ( vi2 ) . j=1 i=1 P P To evaluate [ω(kj ), f ( ni=1 vi2 )] for given fixed j ∈ {1, . . . , n} consider f ( vi2 ) as a function of yj only, and apply Lemma 27. It follows that " # n X 1 vi2 ) = 2 f ′ 2yj · ∇ω(kj ) + Rj,t ω(kj ), f ( t i=1 50 where −2 kRj,t k ≤ t α sup k∂ ωk∞ |α|=2 Z |k|2 |fˆj (k)|dk and fˆj (k) = (2π)−d Z e−ik·yj f yj2 + n X ! yl2 dk. l=1,l6=j It is easy to see that |fˆj (k)| ≤ Cp (1 + |k|)p where Cp only depends on f but not on j or and hence that on ⊗ns h for all Pn l=1,l6=j p∈N yl2. It follows that kRj,t k ≤ Ct−2 n 1 ′ X Cn k[idΓ(ω), f ] − f 2 yj /t · ∇ω(kj )k ≤ 2 . t t j=1 (139) The same equation holds with ∇ω(kj ) · yj /t instead of yj /t · ∇ω(kj ), as follows from [∇ω(kj ), yj /t] = ∆ω(kj )/t = O(t−1). In conjunction with (139) this proves the lemma. A.2 Helffer-Sjöstrand Functional Calculus Suppose f ∈ C0∞ (R; C) and A is a self-adjoint operator. A convenient representation for f (A), which is often used in this paper, is then given by Z ∂ f˜ 1 dxdy (z) (z − A)−1 , z = x + iy, f (A) = − π ∂ z̄ which holds for any extension f˜ ∈ C0∞ (R2 ; C) of f with |∂z̄ f˜| ≤ C|y|,   ˜ ∂ f ∂f 1 ∂f f˜(z) = f (z) and (z) = 0 for all (z) = +i ∂ z̄ 2 ∂x ∂y z ∈ R. (140) Such a function f˜ is called an almost analytic extension of f . A simple example is given by f˜(z) = (f (x) + iyf ′ (x)) χ(z) where χ ∈ C0∞ (R2 ) and χ = 1 on some complex neighborhood of supp f . Sometimes we need faster decay of |∂z̄ f˜| as |y| → 0 in the form |∂z̄ f˜| ≤ C|y|n . In that case we work with the almost analytic extension ! n k X (iy) χ(z) f˜(z) = f (k) (x) k! k=0 where χ is as above. We call this an almost analytic extension of order n. For more details and extensions of this functional calculus the reader is referred to [HS00] or [Dav95]. 51 FGSch, 30/Mar/01—Asymptotic Completeness B Representation of States in χ(H̃ < c)H̃ The representation of states in Ran χ(H̃ < c) proved in this section is used in Lemma 18 and Theorem 20. Lemma 29. Suppose ω(k) = |k| or ω satisfies (H1), and let c > 0. Then the space ∗ ∗ 2 d of Pnlinear combinations of vectors of the form a (h1 ) . . . a (hn )Ω with hi ∈ L (R ) and i=1 sup{ω(k) : k ∈ supp(hi )} < c is dense in χ(dΓ(ω) < c)F . Proof. Let Dc denote the space specified in the lemma. Clearly Dc ⊂ χ(dΓ(ω) < c)F . Since the span of vectors of the form χ(dΓ(ω) < c)a∗ (h1 ) . . . a∗ (hn )Ω with hi and n arbitrary is dense in χ(dΓ(ω) < c)F , it suffices to show that such vectors can be approximated by vectors in Dc . Let n ∈ N and h1 , . . . , hn ∈ L2 (Rd ) be fixed and let ψ = χ(dΓ(ω) < c)a∗ (h1 ) . . . a∗ (hn )Ω. Then X 1 X ψ(k1 , . . . , kn ) = χ( ω(ki ) < c) √ h1 (kσ1 ) · · · hn (kσn ) (141) n! σ where the sum extends over all permutations σ of {1, . . . , n}. The set T = {t ∈ n Pn R+ | i=1 ti < c} can be written as a countable union of cubes Qα with disjoint interiors and edges parallel to the coordinate axis. That is T = ∪α Qα , Q◦α ∩ Q◦β = ∅ and Q Qα is the cartesian product ni=1 Iαi of n intervals Iαi in R+ (α = (α1 , . . . , αn )). It follows that ( ) n X S = (k1 , . . . , kn ) : ω(ki ) < c i=1 = [ α = {(k1, . . . , kn ) : (ω(k1), . . . , ω(kn )) ∈ Qα } n [\ {(k1 , . . . , kn ) : ω(ki ) ∈ Iαi } α i=1 and correspondingly χS (k1 , . . . , kn ) = n XY α i=1 χ(ω(ki ) ∈ Iαi ) a.e. in Rnd . (142) P Q This shows that χS ϕ = α ni=1 χ(ω(ki ) ∈ Iαi )ϕ for any function ϕ ∈ L2 (Rnd ) where the sum converges in L2 -norm by the monotone convergence theorem. Now let Jαi (k) = χ(ω(k) ∈ Iαi ). By (141) and (142) and the symmetry of χS (k1 , . . . , kn ) with respect to permutation of the variables we get 1 X (χS ψ)(k1 , . . . , kn ) = √ χS (kσ1 , . . . , kσn )h1 (kσ1 ) · · · hn (kσn ) n! σ X 1 X √ = (Jα1 h1 )(kσ1 ) · · · (Jαn hn )(kσn ). n! σ α P Hence ψ = α a∗ (Jα1 h1 ) · · · a∗ (Jαn hn )Ω ∈ Dc which proves the lemma. 52 Lemma 30. Suppose ω(k) = |k| or ω satisfies (H1), and let c > 0. Then the set of all linear combinations of vectors of the form ϕ ⊗ a∗ (h1 ) . . . a∗ (hn )Ω, λ+ N X Mi < c (143) i=1 where ϕ = χ(H < λ)ϕ for some λ < c, n ∈ N and Mi = sup{ω(k) : hi (k) 6= 0}, is dense in χ(H̃ < c)H̃. Proof. Obviously the set of vectors χ(H̃ < c)ϕ ⊗ a∗ (h1 ) . . . a∗ (hn )Ω with ϕ ∈ H, hi ∈ h and n arbitrary is dense in χ(H̃ < c)H̃. Thus it suffices to approximate such vectors by vectors of the form (143). In the sense of a strong Stieltjes integral Z χ(H̃ < c) = dEλ (H) ⊗ χ(dΓ(ω) < c − λ). This means that χ(H̃ < c)ϕ ⊗ a∗ (h1 ) . . . a∗ (hn )Ω = lim sup |∆i |→0 X i E∆i (H)ϕ ⊗ χ(dΓ(ω) < c − λi )a∗ (h1 ) . . . a∗ (hn )Ω where ∆i = (λi−1 , λi ]. The lemma now follows from Lemma 29 applied to χ(dΓ(ω) < c − λi )a∗ (h1 ) . . . a∗ (hn )Ω. C Number–Energy Estimates. Thanks to the positivity of the boson mass, assumption (H1), one has the operator inequality N ≤ aH + b, (144) for some constants a and b. The purpose of this Section is to prove that also higher powers of N are bounded with respect to the same powers of H. This easily follows from (144) if the commutator [N, H] is zero, that is, for vanishing interaction. Otherwise it follow, as we will see, from the boundedness of adkN (H)(H + i)−1 for all k. Lemma 31. Assume the hypotheses (H1) and (H2) and suppose m ∈ N ∪ {0}. i) Then uniformly in z, for z in a compact subset of C, k(N + 1)−m (z − H)−1 (N + 1)m+1 k = O(| Im z|−m ). ii) (N + 1)m (H + i)−m is a bounded operator. iii) If χ ∈ C ∞ (R) with supp χ ⊂ (−∞, Σ), then N m eα|x| χ(H) is a bounded operator, provided α > 0 is small enough. 53 FGSch, 30/Mar/01—Asymptotic Completeness Proof. i) This is proved by induction in m. For m = 0 it suffices to show that N(H + i)−1 is bounded, because k(H + i)(H − z)−1 k = O(| Im(z)|−1 ) for z in a compact subset of C. The operator N(H0 +i)−1 is bounded because of the positivity of the boson mass, assumption (H1). Since φ(G) is infinitesimally small with respect to H0 , it follows that H0 (H + i)−1 is bounded and hence so is N(H + i)−1 . Next assume that the assertion in i) holds for any non–negative integer less than a given m ∈ N. To prove it for m we commute (z − H)−1 with (N + 1)m+1 and get (N+1)−m (z − H)−1 (N + 1)m+1 = (N + 1)(z − H)−1 + (N + 1)−m (z − H)−1 [H, (N + 1)m+1 ](z − H)−1 = (N + 1)(z − H)−1 m+1 X  m + 1 −m −1 (−1)l (N + 1)m+1−l adlN (H)(z − H)−1 + (N + 1) (z − H) l l=1 = (N + 1)(z − H)−1 m+1 X m + 1 −1 (N + 1)−(m−1) (z − H)−1(N + 1)m+1−l il φ(il G)(z − H)−1 . + (N + 1) | {z } l l=1 O(| Im z|−m+1 ) (145) By induction hypothesis and because φ(il G)(z − H)−1 is of order O(| Im z|−1 ), for z in a compact set in C, it follows that the r.h.s. of the last equation is of order O(| Im z|−m ). ii) Follows directly from i) if we put z = −i and write (N + 1)m (H + i)−m = (N + 1)m (H + i)−1 (N + 1)−m+1 × (N + 1)m−1 (H + i)−1 (N + 1)−m+2 . . . . . . (N + 1)(H + 1)−1 . iii) We begin to write N m eα|x| χ(H) = N m eα|x| (H + i)−m e−α|x| e+α|x| χ(H)(H + i)m = N m (i + e+α|x| He−α|x| )−m e+α|x| χ(H)(H + i)m (146) = N m (i + H + δHα )−m e+α|x| χ(H)(H + i)m , where δHα = 2iαp·x/|x|−2α/|x|−α2 , and α > 0 is so small that kδHα (H +i)−1 k < 1 (this ensures that (i + H + δHα ) = (1 + δHα (H + i)−1 )(H + i) is invertible, with a bounded inverse given by (H + i)−1 (1 + δHα (H + i)−1 )−1 ). The last equation implies iii), since N m (i + H + δHα )−m is a bounded operator. This can be shown in the same way we showed that N m (H + i)−m is a bounded operator, using that N commutes with δHα . 54 D Commutator Estimates 2 Let j0 , j∞ ∈ C ∞ (Rd ) be real-valued with j02 +j∞ ≤ 1, j0 (y) = 1 for |y| ≤ 1 and j0 (y) = 0 for |y| ≥ 2. Given R > 0 set j#,R (y) = j# (y/R) and let jR : h → h ⊕ h denote the operator defined by jR h = (j0,R h, j∞,R h). Lemma 32. Assume hypotheses (H1), (H2), (H5). Suppose α > 0, m ∈ N ∪ {0}, and jR is as above. Suppose also that χ, χ′ ∈ C0∞ (R), with supp χ′ ⊂ (−∞, Σ). Then, for R → ∞,   i) e−α|x| (N0 + N∞ + 1)m Γ̆(jR )H − H̃ Γ̆(jR ) (N + 1)−m−1 = O(R−1 ),   ii) (N0 + N∞ + 1)m χ(H̃)Γ̆(jR ) − Γ̆(jR )χ(H) χ′ (H) = O(R−1). Proof. i) From the intertwining relations (24), and (25) we have that Γ̆(jR )H − H̃ Γ̆(jR ) = − dΓ̆(jR , adω (jR )) + [φ((j0,R − 1)G) ⊗ 1 + 1 ⊗ φ(j∞,R G)]Γ̆(jR ). By Lemma 27 adω (jR ) = O(R−1). Hence (N0 + N∞ + 1)m dΓ̆(jR , adω (jR ))(N + 1)−m−1 = O(R−1 ). To see that the other two terms lead to contributions of order O(R−1 ) write (N0 +N∞ + 1)m [φ((j0,R − 1)G) ⊗ 1 + 1 ⊗ φ(j∞,R G)] = [φ((j0,R − 1)G) ⊗ 1 + 1 ⊗ φ(j∞,R G)](N0 + N∞ + 1)m m   X m (−i)l [φ(il (j0,R − 1)G) ⊗ 1 + 1 ⊗ φ(il j∞,R G)](N0 + N∞ + 1)m−l , + l l=1 then use (N0 + N∞ + 1)j Γ̆(jR ) = Γ̆(jR )(N + 1)j and use Hypothesis (H5). ii) Let χ̃ be an almost analytic extension of χ of order m, as defined in Sect. A.2. Then we have (N0 + N∞ + 1)m (χ(H̃)Γ̆(jR ) − Γ̆(jR )χ(H))χ′ (H) Z 1 = − dxdy ∂z̄ χ̃(N0 + N∞ + 1)m (z − H̃)−1 e−α|x| (H̃ Γ̆(jR ) − Γ̆(jR )H) π × eα|x| χ′ (H)(z − H)−1 . Then the statement follows by i) because (N0 + N∞ + 1)m (z − H̃)−1 (N0 + N∞ + 1)−m+1 = O(| Im z|−m ), (147) and because (N + 1)m eα|x| χ′ (H) is a bounded operator, provided α > 0 is sufficiently small. 55 FGSch, 30/Mar/01—Asymptotic Completeness Lemma 33. Assume hypotheses (H1), (H2), (H5). Suppose α > 0, m ∈ N ∪ {0}. Let jR be as above and set djR = [iω, jR ] + ∂jR /∂R. Suppose also that χ, χ′ ∈ C0∞ (R), with supp χ′ ⊂ (−∞, Σ). Then   i) e−α|x| (N0 + N∞ + 1)m dΓ̆(jR , djR )H − H̃dΓ̆(jR , djR ) (N + 1)−m−2 = O(R−2),   ii) (N0 + N∞ + 1)m dΓ̆(jR , djR )χ(H) − χ(H̃)dΓ̆(jR , djR ) χ′ (H) = O(R−2 ). Proof. We begin proving part i). To this end note that dΓ̆(jR , djR )H − H̃dΓ̆(jR , djR ) = dΓ̆(jR , djR )dΓ(ω) − (dΓ(ω) ⊗ 1 + 1 ⊗ dΓ(ω))dΓ̆(jR , djR ) + dΓ̆(jR , djR )φ(G) − (φ(G) ⊗ 1)dΓ̆(jR , djR ). (148) Consider the term on the first line on the r.h.s. of the last equation. We have dΓ̆(jR , djR )dΓ(ω)−(dΓ(ω) ⊗ 1 + 1 ⊗ dΓ(ω))dΓ̆(jR , djR )|` ⊗ns h " X =U (jR ⊗ · · · ⊗ djR ⊗ · · · ⊗ [jR , ω] ⊗ · · · ⊗ jR ) i6=j + n X i=1 (149) # (jR ⊗ · · · ⊗ [djR , ω] ⊗ · · · ⊗ jR ) . Each one of the n2 terms on the r.h.s. of the last equation is of order O(R−2 ), by Lemma 27. Hence  m (N0 + N∞ +1) dΓ̆(jR , djR )dΓ(ω)  −(dΓ(ω) ⊗ 1 + 1 ⊗ dΓ(ω))dΓ̆(jR , djR ) (N + 1)−m−2 = O(R−2). Consider now the term on the second line on the r.h.s. of (148). This is given by dΓ̆(jR , djR )φ(G) − (φ(G) ⊗ 1)dΓ̆(jR , djR ) = (φ((j0,R − 1)G) ⊗ 1 + 1 ⊗ φ(j∞,R G)) dΓ̆(jR , djR ) + (φ(dj∞,R G) ⊗ 1 + 1 ⊗ φ(dj∞,R G)) Γ̆(jR ). By Hypothesis (H5) it follows, in the same way as in the proof of Lemma 32, that   e−α|x| (N0 + N∞ + 1)m dΓ̆(jR , djR )φ(G) − (φ(G) ⊗ 1)dΓ̆(jR , djR ) (N + 1)−m−2 = O(R−µ−1 ). This completes the proof of part i). Part ii) of the Lemma follows in the same way as in Lemma 32. 56 E Mourre Estimate This section contains the proofs of Lemma 3 (the Virial Theorem) and of Theorem 4 (the Mourre Estimate). Recall that A stands for dΓ(a) with a = 1/2(∇ω · y + y · ∇ω). Proof of Lemma 3, (Virial Theorem). First we prove the virial theorem for a regularized variant, Aε , of A = Aε=0 defined on D(H), and then we let ε → 0. Let aε = 1/2(∇ω · yε + yε · ∇ω) where yε = y(1 + εy 2)−1 and let Aε = dΓ(aε ). Then   ihHϕ, Aε ϕi − ihAε ϕ, Hϕi = hϕ, dΓ(i[ω, aε ]) − φ(iaε G) ϕi (150) for all ϕ ∈ D(K) ⊗ F0 , which is a core for H. Since all operators in (150) are Hbounded this equation extends to D(H). If ϕ is an eigenvector H then the left side of (150) vanishes because Aε ⊂ A∗ε . Thus it suffices to prove that the right side converges to hϕ, i[H, A]ϕi, as ε → 0, for ϕ ∈ Eµ (H)H, µ < Σ. We show that and φ(iaε G)ϕ → φ(iaG)ϕ, dΓ(i[ω, aε ])ϕ → dΓ(|∇ω|2)ϕ (151) (152) as ε → 0. Eq. (151) follows from kaε G − aGk → 0 and supx e−α|x| kφ((a − aε )G)(N + 1)−1/2 k < ∞ by Lebesgue’s dominated convergence theorem if α > 0 is chosen in such a way that eα|x| (N + 1)1/2 ϕ belongs to H. Here we used (H2) and (H3). To prove (152) note that i[ω, yε ] → ∇ω strongly and hence that i[ω, aε ] → |∇ω|2 strongly. Since moreover supε ki[ω, aε ]k < ∞ this proves (152) for all ϕ ∈ D(N). The property proved in the following lemma has a well known analog, called local compactness, in the theory of Schrödinger operators. We used it in the proof of Theorem 2 and will need it again in the subsequent proof of the Mourre theorem, Thm. 4. Lemma 34. Assume (H1) and suppose f ∈ L∞ (X) and g ∈ L∞ (Rd , dy) with kgk∞ ≤ 1. If f (x) → 0, as |x| → ∞, and g(y) → 0, as |y| → ∞ in Rd , then f (x) ⊗ Γ(g)(H + i)−1/2 is compact. This lemma follows from the fact that p2 and dΓ(ω) are form bounded w.r. to H and from the positivity of the boson mass. Proof. From m > 0 it follows that χ(N > n)Γ(g)(dΓ(ω) + 1)−1/4 → 0 as n → ∞. Furthermore Γ(g)(dΓ(ω) + 1)−1/4 |` ⊗ns h = g(y1) . . . g(yn)[ω(k1 ) + . . . + ω(kn ) + 1]−1/4 (153) which is compact. It follows that Γ(g)(dΓ(ω) + 1)−1/4 is a compact operator on F . Since f (x)(p2 + 1)−1/4 is compact on L2 (X) and (p2 + 1)1/4 ⊗ (dΓ(ω) + 1)1/4 (H + i)−1/2 is a bounded operator by ?, the operator f ⊗ Γ(g)(H + i)−1/2 is compact. 57 FGSch, 30/Mar/01—Asymptotic Completeness The proof of Theorem 4 is by induction in the number of energy steps of size m, the minimal energy of a free boson. Assuming that the theorem holds for λ < min(Σ, (n − 1)m) we prove it for λ < min(Σ, nm). To this end we need a partition of unity in 2 the bosonic configuration space. Let j0 , j∞ ∈ C ∞ (Rd ), with j02 + j∞ = 1, and with 0 ≤ j0 ≤ 1, j0 (y) = 1 if |y| ≤ 1 and j0 (y) = 0 if |y| ≥ 2. Given R > 0 we set j#,R (y) = j# (y/R) and qR (y) = q(y/R). Each boson h ∈ h will be split into the two parts j0,R h and j∞,R localized near the origin and near infinity respectively. Recall from Section 2.6 that Γ̆(jR ) does the corresponding localization in Fock space. In Appendix D we saw how to move Γ̆(jR ) through H or any bounded function of H (see Lemma 32). Since [iH, A] has a structure similar to H, with ω and G replaced by |∇ω|2 and by iaG, respectively, it is easy to show that an analogue of Lemma 32 also holds for [iH, A]. In particular we use that n o −α|x| 2 e Γ̆(jR )[iH, A] − ([iH, A] ⊗ 1 + 1 ⊗ dΓ(|∇ω| ))Γ̆(jR ) (N + 1)−1 = o(R0 ), (154) as R → ∞. Proof of Theorem 4 (Mourre Estimate). First of all we introduce the Mourre constants d(λ) := ˜ := d(λ) Ω⊥ inf dΓ(|∇ω(k)|2) (155) inf dΓ(|∇ω(k)|2), (156) σpp (H)+dΓ(ω(k))=λ σpp (H)+dΓ(ω(k))=λ where the superscript Ω⊥ in the definition of d means that we exclude the vacuum sector to compute the infimum. Note that d vanishes only on thresholds, while d˜ vanishes on thresholds and on eigenvalues of H. We introduce, moreover, the smeared out versions ˜ For κ > 0, we set ∆κ = [λ − κ, λ + κ], and then we define of the functions d, d. λ ˜ By definition of these functions we have dκ (λ) = inf µ∈∆κλ d(µ) and d˜κ (λ) = inf µ∈∆κλ d(µ). the inequality  Ω⊥  inf d˜κ (λ − dΓ(ω(k))) + dΓ(|∇ω(k)|2) ≥ dκ (λ). (157) For all n ∈ N we will show the following statements. H1 (n): Let ε > 0 and λ ∈ [E0 , E0 + nm) ∩ (−∞, Σ). Then there exists an open interval ∆ ∋ λ and a compact operator E such that E∆ (H)[iH, A]E∆ (H) ≥ (d(λ)−ε)E∆ (H)+E. H2 (n): Let ε > 0 and λ ∈ [E0 , E0 + nm) ∩ (−∞, Σ). Then there exists an open interval ˜ − ε)E∆ (H). ∆ ∋ λ such that E∆ (H)[iH, A]E∆ (H) ≥ (d(λ) H3 (n): Let κ, ε0 , ε > 0. Then there exists δ > 0 such that for all λ ∈ [E0 , E0 + nm − ε0 ] ∩ (−∞, Σ), one has E∆κλ (H)[iH, A]E∆κλ (H) ≥ (d˜κ (λ) − ε)E∆κλ (H). S1 (n): τ is a closed and countable set in [E0 , E0 + nm) ∩ (−∞, Σ). S2 (n): For any closed intervall I ⊂ [E0 , E0 + nm) ∩ (−∞, Σ), with I ∩ τ = ∅, one has dim Ran EI∩σpp (H) < ∞. Note here that all the claims of the theorem follow from these statements, if we prove them for any n ∈ N. Actually, the statements give no new information if n is 58 so large that E0 + nm > Σ. We will prove these statements by induction in n. Since H1 (1) and S1 (1) are obvious, all the statements, for any n ∈ N, follow if we prove the implications: H1 (n) ⇒ H2 (n), H2 (n) ⇒ H3 (n), H1 (n) ⇒ S2 (n), S2 (n − 1) ⇒ S1 (n), and S1 (n) ∧ H3 (n − 1) ⇒ H1 (n). Now the implication S2 (n − 1) ⇒ S1 (n) is trivial. Moreover the implications H1 (n) ⇒ H2 (n), H2 (n) ⇒ H3 (n) and H1 (n) ⇒ S2 (n) are proved by standard abstract arguments, which are typical in the proof of any Mourre inequality. It only remains to prove that S1 (n) together with H3 (n − 1) imply the statement H1 (n). To this end we fix λ ∈ [E0 , E0 + nm) ∩ (−∞, Σ) and ε > 0. Choose now χ ∈ C0∞ (R) with supp χ ⊂ [λ − δ, λ + δ] ∩ (−∞, Σ), where δ is some positive constant which will be fixed later on. Then we have χ(H)[iH, A]χ(H) = Γ̆(jR )∗ E{0} (N∞ )Γ̆(jR )χ(H)[iH, A]χ(H) + Γ̆(jR )∗ E[1;∞)(N∞ )Γ̆(jR )χ(H)[iH, A]χ(H) = Γ(qR )χ(H)[iH, A]χ(H)  + Γ̆(jR )∗ χ(H̃) [iH, A] ⊗ 1 + 1 ⊗ dΓ(|∇ω|2) χ(H̃)E[1;∞) (N∞ )Γ̆(jR ) + o(R0 ), (158) where we used Lemma 32 and Eq. (154) to commute Γ̆(jR ) to the right. The first term on the r.h.s. of (158) is compact, by Lemma 34. To handle the second term we want to diagonalize 1⊗dΓ(ω) and 1⊗dΓ(|∇ω|2) on the range of E[1;∞) (N∞ ). Using the induction hypothesis S1 (n) we find κ > 0 such that dκ (λ) ≥ d(λ) − ε/3. Then, by H3 (n − 1), we know that if δ > 0 is small enough we have  E∆δλ (H + dΓ(ω(k))) [iH, A] ⊗ 1 + 1 ⊗ dΓ(|∇ω(k)|2) E∆δλ (H + dΓ(ω(k)))E[1;∞)(N∞ ) n εo E[1;∞)(N∞ ) ≥ E∆δλ (H + dΓ(ω(k))) d˜κ (λ − dΓ(ω(k))) + dΓ(|∇ω(k)|2) − 3 ε ≥ (dκ (λ) − )E∆δλ (H + dΓ(ω(k)))E[1;∞)(N∞ ) 3 2ǫ ≥ (d(λ) − )E∆δλ (H + dΓ(ω(k)))E[1;∞) (N∞ ), 3 where, in the second inequality we used (157). It follows, since supp χ ⊂ ∆δλ , that  2ǫ χ(H̃) [iH, A] ⊗ 1 + 1 ⊗ dΓ(|∇ω|2 ) χ(H̃)E[1;∞) ≥ (d(λ) − )χ2 (H̃)E[1;∞) (N∞ ). 3 Now we insert this in the second term on the r.h.s. of (158) and then we commute, using again Lemma 32 and Eq. (154), Γ̆(jR ) back to the left. Using that Γ̆(jR )∗ E[1,∞) (N∞ )Γ̆(jR ) = 1−Γ(qR ) and that, by Lemma 34, Γ(qR )χ(H) is a compact operator, we find, from (158), χ(H)[iH, A]χ(H) ≥ (d(λ) − 2ε 2 )χ (H) + E + o(R0 ). 3 δ/2 The statement H1 (n) then follows if we choose χ such that χ = 1 on ∆λ , if we multiply (158) from the right and from the left with E∆δ/2 (H) and if we choose R sufficiently λ large. FGSch, 30/Mar/01—Asymptotic Completeness F 59 Invariance of Domains In this section the invariance of the domain of dΓ(y 2 ) with respect to action of χ(H) for smooth functions χ is proven. Moreover we prove in Lemma 36 that the norm of dΓ(y 2/t2 )χ(H)e−iHt ϕ remains uniformly bounded for all t ≥ 1, if ϕ ∈ D(dΓ(y 2 + 1)). These results are used in Proposition 13 to prove the existence of the asymptotic observable W . Lemma 35. Assume Hypotheses (H1), (H2) and (H3) are satisfied. Suppose moreover that ϕ ∈ D(dΓ(y 2 )) ∩ D(N) and that χ ∈ C ∞ (R) with supp χ ⊂ (−∞, Σ). Then we have kdΓ(y 2 )χ(H)ϕk ≤ C(kdΓ(y 2 + 1)ϕk + kϕk). Proof. Find χ1 ∈ C0∞ (R) with χχ1 = χ, and supp χ1 ⊂ (−∞, Σ). Put χ = χ(H) and χ1 = χ1 (H). Then we have dΓ(y 2)χ = dΓ(y 2 )χχ1 = χdΓ(y 2 )χ1 + [dΓ(y 2 ), χ]χ1 (159) Now expand the χ in the commutator in an integral, according to the Helffer–Sjöstrand functional calculus (see Appendix A.2). We get Z 2i 2 dxdy ∂z̄ χ̃ (z − H)−1 dΓ(a)(z − H)−1 χ1 [dΓ(y ), χ]χ1 = − π Z (160) i −1 2 −1 + dxdy ∂z̄ χ̃ (z − H) φ(iy G)χ1 (z − H) , π where χ̃ is an almost analytic extension of χ, in the sense of the Helffer-Sjöstrand functional calculus, and a = [iω(k), y 2/2] = 1/2(∇ω · y + y · ∇ω). The second term on the r.h.s. of the last equation is bounded, because, by hypotheses (H2) and (H3), kφ(y 2G)χ1 k < ∞. To handle the first term on the r.h.s. of (160) we commute the factor dΓ(a) to the right of the resolvent (z − H)−1 . Using [dΓ(a), H] = idΓ(|∇ω|2) − iφ(iaG), we see that the contributions arising from this commutator are bounded, because |∇ω| < const and because, using again (H2) and (H3), kφ(iaG)χk < ∞ is a bounded operator. Thus we have, from (159) dΓ(y 2)χ = χdΓ(y 2)χ1 + C χ′ (H)dΓ(a)χ1 + bounded, (161) where χ′ is the first derivative of χ. Now we commute the two operators dΓ(y 2 ) and dΓ(a) in the two terms on the r.h.s. of the last equation to the right of χ1 . For example , for the term χdΓ(y 2)χ1 , we find χdΓ(y 2 )χ1 = χχ1 dΓ(y 2 ) + χ[dΓ(y 2 ), χ1 ] Z 1 2 dxdy ∂z̄ χ̃1 χ(z − H)−1 [dΓ(y 2 ), H](z − H)−1 = χdΓ(y ) − π Z 2i 2 = χdΓ(y ) − 2 dxdy ∂z̄ χ̃1 χ(z − H)−1dΓ(a)(z − H)−1 πt Z i + dxdy ∂z̄ χ̃1 (z − H)−1 χφ(iy 2 G)(z − H)−1 . π (162) 60 The third term on the r.h.s. of the last equation is bounded by (H2) and (H3). To handle the second term we commute dΓ(a) to the right. We get Z Z 2i 2i −1 −1 − dxdy ∂z̄ χ̃1 χ(z − H) dΓ(a)(z − H) = − dxdy ∂z̄ χ̃1 χ(z − H)−2 dΓ(a) π π Z 2 + dxdy ∂z̄ χ̃1 (z − H)−2χdΓ(|∇ω|2 )(z − H)−1 π Z 2 dxdy ∂z̄ χ̃1 (z − H)−2χφ(iaG)(z − H)−1 . − π The first term on the r.h.s. of the last equation is proportional to χχ′1 dΓ(a), where χ′1 is the first derivative of χ1 , and thus vanishes, since χ1 is constant on supp χ. The other two terms on the r.h.s. of the last equation are bounded. It follows, by (162), that χdΓ(y 2 )χ1 = χdΓ(y 2 ) + bounded. Similarly we find, that χ′ dΓ(a)χ1 = χ′ dΓ(a) + bounded. These two results imply, by (161), that  kdΓ(y 2 )χϕk ≤ C kdΓ(y 2 )ϕk + kdΓ(a)ϕk + kϕk . (163) The Lemma now follows using Lemma 37, part (iv), to estimate the second term on the r.h.s. of the last equation. Lemma 36. Assume hypotheses (H1), (H2) and (H3) are satisfied. Suppose moreover that ϕ ∈ D(dΓ(y 2 )) ∩ D(N) and that χ ∈ C ∞ (R) with supp χ ⊂ (−∞, Σ). Then we have kdΓ(y 2 )e−iHt χ(H)ϕk ≤ C(kdΓ(y 2 + 1)ϕk + t2 kϕk), for all t ≥ 1. Proof. We begin by noting, that iHt e 2 −iHt dΓ(y )e Z 2 t χ(H) − dΓ(y )χ(H) = ds eiHs [iH, dΓ(y 2/t2 )] χ(H)e−iHs 0 Z t Z t iHs −iHs =2 ds e dΓ(a) χ(H)e − ds eiHs φ(iy 2G) χ(H)e−iHs , 0 0 (164) where a = [iω, y 2/2] = 1/2(∇ω · y + y · ∇ω). The second integral on the r.h.s. of the last equation is bounded, with norm of order t, because, by (H2) and (H3), kφ(iy 2G)χk < ∞. To handle the first integral on the r.h.s. of the last equation use the expansion Z t Z t Z s iHs −iHs 2 ds e dΓ(a)χ(H)e = 2dΓ(a)χ(H) + 2 ds dr eiHr [iH, dΓ(a)] χ(H)e−iHr 0 0 0 Z t Z s = 2dΓ(a)χ(H) + 2 ds dr eiHr dΓ(|∇ω|2) χ(H)e−iHr 0 0 Z t Z s −2 ds dr eiHr φ(iaG) χ(H)e−iHr . 0 0 (165) 61 FGSch, 30/Mar/01—Asymptotic Completeness Since |∇ω| is bounded and kφ(iaG)χ(H)k < ∞, both the integrals on the r.h.s. of the last equation are bounded and of order t2 . This implies, by (164), that kdΓ(y 2)e−iHt χ(H)ϕk ≤ kdΓ(y 2 )χ(H)ϕk + 2t kdΓ(a)χ(H)ϕk + Ct2 kϕk = kdΓ(y 2)χ(H)ϕk + 2t2 kdΓ(a/t)χ(H)ϕk + Ct2 kϕk. (166) By Lemma 37, part (iv), with y replaced by y/t, we have kdΓ(a/t)χϕk ≤ kdΓ(y 2 /t2 + 1)χϕk ≤ kdΓ(y 2/t2 )χϕk + C kϕk. (167) The Lemma now follows inserting the last equation into (166) and applying Lemma 35. Lemma 37. (i) For any operator A A2 + (A∗ )2 ≤ AA∗ + A∗ A. (ii) If a = 1/2(y · ∇ω + ∇ω · y) and ω satisfies (H1) then a2 ≤ const(y 2 + 1). (iii) If a is a symmetric operator in h then dΓ(a)2 ≤ NdΓ(a2 ) (iv) With a as in (ii) we have dΓ(a)2 ≤ const dΓ(y 2 + 1)2 . Proof. (i) This follows from 0 ≤ (iA − iA∗ )2 = −A2 − (A∗ )2 + AA∗ + A∗ A. (ii) By (i), (∇ω · y + y · ∇ω)2 ≤ 2[(∇ω · y)(y · ∇ω) + (y · ∇ω)(∇ω · y)] where (y · ∇ω)(∇ω · y) ≤ k∇ωk2∞ y 2 (∇ω · y)(y · ∇ω) ≤ 2(k∇ωk2∞ + k∆ωk2∞ )(y 2 + 1) (iii) It suffices to prove ϕn P in Ran χ(N = n). For such vectors hϕn , dΓ(a)2 ϕn i = Pnthis for states 2 2 kdΓ(a)ϕn k ≤ ( i=1 kai ϕn k) ≤ n ni=1 kai ϕn k2 = hϕn , NdΓ(a2 )ϕn i, where ai denotes the operator a acting on the i-th boson. (iv) This follows from (ii), (iii), and N ≤ dΓ(y 2 + 1). 62 G Some Technical Parts of Theorem 15 Lemma 38. There exists a constant C = C(λ, u) such that  1/2 ′′ 2 n Sn′′ (y, t) ±(Jnk Sn ) (y, t) ≤ Cn k for |y| ≤ 2λt and ut1−δ ≥ 2. Here y = (y1 , . . . , yn ), yi ∈ Rd . Proof. We Pndrop the combinatorial factor in the definition of Jnk for convenience. Let S0 (y) = i=1 S0 (yi ). From (51) and the scaling factor 1/ut in the arguments of Jnk we get tSn′′ (y, t) = S0′′ (y/tδ ) and ′′ t(Jnk Sn )′′ (y, t) = Jnk (z) · S0′′ (w) + α−2 Jnk (z)S0 (w) −1 + α ∇Jnk (z) ⊗ ∇S0 (w) + α−1 ∇S0 (w) ⊗ ∇Jnk (z) (168) where w = yt−δ , z = y/ut, α = ut1−δ and hence w = αz, α ≥ 2 and |z| ≤ 2λ/u. Proving the lemma thus amounts to showing that the r.h.s. of (168) (and its negative) are bounded by const × n2 S0′′ (w). The first term in (168) enjoys the bound. To estimate the other three terms we need some ”N-body geometry”. For each a ∈ {0, ∞}n , let 1a : Rnd → Rnd denotes the orthogonal projection onto the subspace {w ∈ Rnd |wi = 0 if ai = 0} and let wa = 1a w. Set Y Y χa (σ, w) = χ(wr2 /2 ≤ σ) χ(ws2 /2 > σ). r:ar =0 Then P a s:as =∞ χa ≡ 1. Hence, by (48) and since χa (σ, w) χ(wi2/2 > σ) = χa (σ, w)δai ,∞ , Z S0 (w) = dσ m(σ)S0 (σ, w) (169) S0 (σ, w) = n X i=1 χ(wi2 /2 > σ)(wi2/2 − σ) = X a χa (σ, w)(wa2/2 − σa ) where σa = σ · #{i : ai = ∞}. Furthermore X ∇S0 (σ, w) = χa (σ, w)wa (170) (171) a S0′′ (σ, w) R ≥ X χ(σ, w)1a (172) a R P in the sense that ∇S0 (w) = dσ m(σ) χa (σ, w)wa and S0′′ (w) ≥ dσ m(σ) χa (σ, w)1a . By (169) it suffices to estimate the last three terms of (168) for S0 (w) replaced by S0 (σ, w) uniformly in σ. If wi2 /2 ≤ 1 and α ≥ 2 then |zi | ≤ 1 and hence ∂ β jε (zi ) = 0 for all derivatives of non-zero order β and ε ∈ {0, ∞}. Hence for w such that χa (σ, w) 6= 0 P ′′ ′′ (z)1a (z) = 1a Jnk Jnk ∇Jnk (z) = 1a ∇Jnk (z). (173) (174) 63 FGSch, 30/Mar/01—Asymptotic Completeness From (170), (173) and (172) we get ′′ ′′ (z)S0 (σ, w) ≤ a−2 w 2 /2 kJnk (z)k ±a−2 Jnk ≤ Cn2 S0′′ (σ, w). X χa (w)1a a The last two terms in (168) are estimated similarly using (171), (174) and (172). Lemma 39.  1/2   y2 m+1 n =n O(t−(1+m)+δ(2−|α|) ) D ∂ Snk (y, t) − Jnk (y, t) k t m α Proof. The difference 2 −1+2δ Snk (y, t) − Jnk (y, t)y /2t = Jnk (y, t)t n X [a(yi t−δ ) + b] (175) i=1 is of the form t−c f (y/t)h(yt−δ ) where y = (y1 , . . . , yn ) and f, h ∈ C ∞ ∩ L∞ (Rnd ). Under differentiation (175) becomes a sum of terms of the same form where in each term the exponent of t is decreased by at least δ or 1 for differentiation w.r. to y or t (or D) respectively. This accounts for the power in t. From the explicit form the right hand side of (175) and (82) we see that every derivative w.r. yi (at most) doubles the number of terms, while every derivative w.r. to t multiplies it by n + 1. Lemma 40. For |y|/t ≤ 2λ we have ′′ D02 Snk =(∇Ω − V ) · Snk (∇Ω − V ) + (D∇Snk ) · (∇Ω − V ) + (∇Ω − V ) · (D∇Snk )  1/2 2 2 n + DV Snk + n O(t−1−δ ) k where V = y/t ∈ Rnd . Proof. We drop the factor nk equation. By definition of D0 ,
1/2 since it appears in all terms on both sides of the D02 Snk = [iΩ, [iΩ, Snk ]] + 2[iΩ, ∂Snk /∂t] + ∂ 2 Snk . ∂t2 (176) To evaluate the double commutator we use Snk = Jnk Sn and the Leibnitz rule and get for |y/t| ≤ 2λ [iΩ, [iΩ, Snk ]] =[iΩ, [iΩ, Jnk ]]Sn + 2[iΩ, Jnk ][iΩ, Sn ] + Jnk [iΩ, [iΩ, Sn ]] ′′ =∇Ω · [Jnk Sn + ∇Jnk ⊗ ∇Sn + ∇Sn ⊗ ∇Jnk + Jnk Sn′′ ] ∇Ω + n2 O(t−1−δ ) ′′ =∇Ω · Snk ∇Ω + n2 O(t−1−δ ) (177) 64 where we used [iΩ, Jnk ] = ∇Ω · ∇Jnk + nO(t−2 ), [iΩ, Sn ] = ∇Ω · ∇Sn + nO(t−1 ) (and the same with the order in the dot products reversed) as well as [iΩ, [iΩ, Sn ]] = ∇Ω· Sn′′ ∇Ω+ ′′ nO(t−1−δ ) (see Lemma9), [iΩ, [iΩ, Jnk ]] = ∇Ω·Jnk ∇Ω+n2 O(t−3) and |∂ α Sn | = O(t1−|α| ) for |y/t| ≤ 2λ and |α| ≤ 2. In a similar way one shows that [iΩ, ∂Snk /∂t] = ∇Ω · ∇ ∂Snk + n2 O(t−2 ) ∂t ∂Snk · ∇Ω + n2 O(t−2 ). =∇ ∂t (178) The lemma now follows from (176), (177) and (178) as Lemma 9 did from analogous equations. References [Ara83] Asao Arai. Rigorous theory of spectra and radiation for a model in quantum electrodynamics. J. Math. Phys., 24(7):1896–1910, 1983. [BFS98] V. Bach, J. Fröhlich, and I.M. Sigal. Quantum electrodynamics of confined nonrelativistic particles. Adv. Math., 137(2):299–395, 1998. [BFSS99] V. Bach, J. Fröhlich, I.M. Sigal, and A. Soffer. Positive commutators and spectrum of Pauli–Fierz Hamiltonian of atoms and molecules. Comm. Math. Phys., 207(3):557–587, 1999. [Dav95] E.B. Davies. The functional calculus. J. London Math. Soc. (2), 52(1):166– 176, 1995. [DG99] J. Dereziński and C. Gérard. Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians. Rev. Math. Phys., 11(4):383–450, 1999. [DG00] J. Dereziński and C. Gérard. Spectral scattering theory of spatially cut-off P (φ)2 Hamiltonians. Comm. Math. Phys., 213(1):39–125, 2000. [FGS00] J. Fröhlich, M. Griesemer, and B. Schlein. Asymptotic electromagnetic fields in models of quantum-mechanical matter interacting with the quantized radiation field. x, x(x):x, x 2000. preprint, Los Alamos arXiv math-ph/0009033. [GLL00] M. Griesemer, E.H. Lieb, and M. Loss. Ground states in non–relativistic QED. preprint, Los Alamos Preprint Archive, math-ph/0007014, 2000. [GS97] Gian Michele Graf and Daniel Schenker. Classical action and quantum Nbody asymptotic completeness. In Multiparticle quantum scattering with applications to nuclear, atomic and molecular physics (Minneapolis, MN, 1995), pages 103–119. Springer, New York, 1997. FGSch, 30/Mar/01—Asymptotic Completeness 65 [HS00] W. Hunziker and I.M. Sigal. The quantum N–body problem. J. Math. Phys., 41(6):3448–3510, 2000. [RS79] M. Reed and B. Simon. Methods of modern mathematical physics: Scattering Theory. Volume 3. Academic Press, 1979. [Spo97] H. Spohn. Asymptotic completeness for Rayleigh scattering. J. Math. Phys., 38(5):2281–2296, 1997.