Continuous Renormalization Group Analysis of Spectral Problems in Quantum Field Theory

Journal of Functional Analysis 268 (2015) 749–823

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Journal of Functional Analysis www.elsevier.com/locate/jfa

Continuous Renormalization Group Analysis of Spectral Problems in Quantum Field Theory Volker Bach a , Miguel Ballesteros b,∗ , Jürg Fröhlich c a

Institut für Analysis und Algebra, Technische Universität Braunschweig, 38092 Braunschweig, Germany b Department of Mathematical Physics, Applied Mathematics and Systems Research Institute (IIMAS), National Autonomous University of Mexico (UNAM), Campus CU (University City), 01000 Mexico City, Mexico c Institute for Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland

a r t i c l e

i n f o

Article history: Received 31 January 2014 Accepted 21 October 2014 Available online 10 December 2014 Communicated by B. Schlein Keywords: Non-relativistic quantum electrodynamics Spectral problems Renormalization Pauli–Fierz

a b s t r a c t The isospectral renormalization group is a powerful method to analyze the spectrum of operators in quantum field theory. It was introduced in 1995 (see [2,4]) and since then it has been used to prove several results for non-relativistic quantum electrodynamics. After the introduction of the method there have been many works in which extensions, simplifications or clarifications are presented (see [7,11,13]). In this paper we present a new approach in which we construct a flow of operators parametrized by a continuous variable in the positive real axis. While this is in contrast to the discrete iteration used before, this is more in spirit of the original formulation of the renormalization group introduced in theoretical physics in 1974 [22]. The renormalization flow that we construct can be expressed in a simple way: it can be viewed as a single application of the Feshbach–Schur map with a clever selection of the spectral parameter. Another advantage of the method is that there exists a flow function for which the renormalization group that we present is the orbit under this flow of an initial Hamiltonian. This opens

* Corresponding author. E-mail addresses: [email protected] (V. Bach), [email protected] (M. Ballesteros), [email protected] (J. Fröhlich). http://dx.doi.org/10.1016/j.jfa.2014.10.022 0022-1236/© 2014 Elsevier Inc. All rights reserved.

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the possibility to study the problem using different techniques coming from the theory of evolution equations. © 2014 Elsevier Inc. All rights reserved.

1. Introduction 1.1. Historical Context and Description of the Problem The processes of emission and absorption of photons by atoms can be rigorously understood in the low-energy limit, if we neglect the creation and annihilation of electrons. The corresponding theory is frequently referred to as nonrelativistic quantum electrodynamics (NR QED). The description of matter and light from the mathematical point of view relies on the study of eigenvalues of operators, which are immersed in the continuum. The study of eigenvalues immersed in the continuum requires sophisticated constructions that do not fall into the realm of regular perturbation theory used to analyze isolated eigenvalues (see [19]). There are two methods that have been applied to investigate these questions. The first one, introduced in [2–4], is the spectral renormalization group. Inspired by a construction that Feshbach used in [9], the Feshbach–Schur projection map is defined and developed [2–4]. The method is based on a transformation that allows for a localization of the regions of the spectrum that we are interested in. The second method, introduced in [17], produces a sequence of isolated eigenvalues that converges to the desired eigenvalue, which is immersed in the continuum, by including ever more momentum shells into the dynamics. The spectral renormalization group has been extensively used to analyze spectral problems in nonrelativistic quantum electrodynamics. In numerous works this method has been used to prove basic properties of the spectrum of Hamiltonian operators for different models (see for example [1–8] and [10–16]). Although it is a powerful tool to analyze the spectrum of operators, it yet has the disadvantage of being conceptually and technically complicated. Further developments of the original method have been presented in [7,11,13]. In these works, new techniques and methods are presented that simplify the computations and clarify the concepts of the original procedures introduced in [2] and [4]. In this paper we present a new approach to the renormalization group described in the following section. Interestingly, this new approach uses the spatial length scale as a flow parameter and is thus closer to the original renormalization group introduced by Kogut and Wilson in [22] and subsequently improved by Polchinski [18] and later Wieczerkowski [21] and Salmhofer [20]. 1.2. Short Description of the Main Results In this section we describe our main results. We present a short description of the method and the most important theorems without giving precise definitions of the operators and spaces that we use. The definitions are deferred to later sections.

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Our intention is to present the method, rather than the specific complications of each model of nonrelativistic quantum field theory. For this reason, and for the sake of simplicity, we restrict ourselves to a class of operators that is as simple as possible but includes all important ingredients. Our method can easily be applied to a bigger class of operators that includes the Pauli–Fierz and the Spin–Boson models. 1.2.1. The Operators The Hilbert space on which the operators are defined is the symmetric Fock space (see Section 3.1.1) H=

∞  

 ⊗N s L2 R3 .

(1.1)

N =0

˜ the creation and annihilation operators For every k, k˜ ∈ R3 , we denote by a∗ (k) and a(k) (see Section 3.1.2) acting in the Fock space H. The free Hamiltonian is [see (3.13)]  Hf := |k|a∗ (k)a(k). (1.2) R3

We fix a positive number ρ ∈ (0, 1) and use the symbol Dρ/2 := {ζ ∈ C : |ζ| < ρ/2} for the open disc in the complex plane of radius ρ/2 centered at the origin. We denote by w = (wm,n )∞ m,n=0

(1.3)

 3 m  3 n ∀m, n ≥ 0: wm,n : Dρ/2 × R+ × R → C, 0 × R

(1.4)

a family of measurable functions

where R+ := (0, ∞), R+ 0 := [0, ∞), and, for every α ≥ 0, by χα the characteristic function in R of the set [0, e−α ρ]. We identify χα ≡ χα (Hf ),

χα ≡ 1 − χα (Hf ).

We study a class operators of the form H(z) = T (z) + W (z),

(1.5)

T (z) = w0,0 (z, Hf )χ0

(1.6)

where

is a function of Hf defined by functional calculus and  W (z) = χ0 Wm,n (w)χ0 , m+n≥1

(1.7)

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with

 Wm,n (w) =

    dk 1 · · · dk m dk˜1 · · · dk˜n a∗ k1 · · · a∗ km

      × wm,n z; Hf ; k1 , · · · , km ; k˜1 , · · · , k˜n a k˜1 · · · a k˜n .

(1.8)

The specific properties that sequences of functions w satisfy are defined in Section 2. 1.2.2. The Feshbach–Schur Map Our goal is to analyze the spectrum (and eigenvalues) of operators of the form H = T + W, where T and W do not depend of z and are given by (1.6)–(1.8). In particular, we study the spectral points in Dρ/2 of the operator H. For this purpose we define a family of Hamiltonians parametrized by a complex number z ∈ Dρ/2 as follows, Dρ/2  z → H(z) := H + z.

(1.9)

The spectral points of H in Dρ/2 are, thus, the complex numbers −z ∈ Dρ/2 such that H(z) is not bounded invertible, and the eigenvalues are the complex numbers −z ∈ Dρ/2 such that H(z) is not injective (see Remark 1.5). The key mathematical tool that we use is the Feshbach–Schur map    −1 Fα H(z) = χα H(z)χα − χα H(z)χα χα H(z)χα χα H(z)χα .

(1.10)

Eq. (1.10) is well-defined (i.e., exists) if χα H(z)χα is (bounded) invertible, which we rephrase by saying that H(z) belongs to the domain of Fα . The Feshbach–Schur map has the important property of being isospectral in the sense that (a) H(z) is (bounded) invertible if, and only if, Fα (H(z)) is (bounded) invertible. (b) H(z) is not injective if, and only if, Fα (H(z)) is not injective. 1.2.3. The Rescaled Feshbach–Schur Map For every α ∈ R, we denote by u(α) : h → h the unitary (dilation) operators on h defined by       u(α)φ (k) := e−3α/2 φ e−α k , ∀φ ∈ h = L2 R3 . (1.11) We denote by Γα its second quantization, i.e., the operator that results from lifting u(α) to the Fock space. u(α) represents a re-scaling of the photon momenta. We rescale the image under the Feshbach–Schur map which yields another isospectral map by

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    α H(z) := eα Γα Fα H(z) Γα∗ . R

753

(1.12)

Rescaling the Feshbach–Schur map is not necessary, but convenient. It amounts to dilate the photon momentum such that the free photon Hamiltonian χ0 Hf is left invariant, α . i.e., χ0 Hf is a fixed point for R The rescaled Feshbach–Schur has the important property of isospectrality, which means that  α (H(z)) is (bounded) invertible. (a) H(z) is (bounded) invertible if, and only if, R  (b) H(z) is not injective if, and only if, Rα (H(z)) is not injective. Note that the rescaled Feshbach–Schur map is not defined for all operators H and all z,  α ), i.e., for H and and (a) and (b) are valid only for H and z such that H(z) ∈ dom(R z such that χα H(z)χα is (bounded) invertible. Another important property of the rescaled Feshbach–Schur map is that, for H(z) belonging to its domain, there is a unique choice of operators Tα(z) and Wα (z) such that    α H(z) = Tα (z) + Wα (z), R

(1.13)

where Tα (z) ≡ Tα (z, Hf )χ0 is of the form (1.6), and its spectral properties are explicit, while Wα (z) decays exponentially in α (in operator norm), as α tends to infinity. From the latter decay property we obtain increasingly accurate information on the invertibility  α (H(z)) and hence on the invertibility of H(z) = H + z, thanks to the isospectrality of R (a), (b), as α gets large. 1.2.4. Main Results One of the main results that we prove in this paper is the following (see Theorem 8.8; Fig. 1): Theorem 1.1. There exists a family {Es }s≥0 of biholomorphic functions Es : Dρ/2 → Es (Dρ/2 ) ⊂ Dρ/2

(1.14)

such that ∀ζ ∈ Es (Dρ/2 ):

 s ). H(ζ) = H + ζ ∈ dom(R

(1.15)

Using Theorem 1.1 we can define a family {Hs (z)}s≥0 of operators by ∀z ∈ Dρ/2 :



  s H Es (z) , Hs (z) = R

H0 (z) = H + z = T + z + W.

Another result that we prove is the next theorem (see Theorem 8.8)

(1.16)

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 s ), for every ζ ∈ Es (Dρ/2 ). These Fig. 1. This figure shows the sets Es (Dρ/2 ) for which H + ζ ∈ dom(R sets shrink as s grows. Here we draw D(ρ/2) and Esj (Dρ/2 ) for j ∈ {1, · · · , 6}, where sj+1 > sj for every j ∈ {1, · · · , 5}. With increasing j, the sets are depicted increasingly dark.

Theorem 1.2. For every s ≥ 0, there exist operators Ts (z) and Ws (z) such that Hs (z) = Ts (z) + Ws (z),

(1.17)

where the spectrum of Ts (z) can be computed explicitly and there are constants μ > 0 and C > 0 such that   Ws (z) ≤ Ce−μs/4 .

(1.18)

In the case that s = 0 we take T0 (z) = T + z and W0 (z) = W . The operator Ts (z) is actually a function of Hf : Ts (z) = τs (z, Hf ),

(1.19)

for a function τ : Dρ/2 × R+ 0 → C. This implies that Ts (z) has a simple and well-known spectral decomposition. We call the family of functions {Es }s≥0 the renormalization flow (or renormalization group) of the spectral parameter z and the one-parameter family {Hs (z)}s≥0 of operators the renormalization flow (or renormalization group) of operators with initial condition H0 (z) = H(z). The assignment of the name flow (or group) has actually a mathematical meaning in the sense that {Hs (z)}s≥0 is the orbit of H(z) under a flow Φ(·, s). This is the content of the next theorem (see Theorem 8.18).

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Theorem 1.3. For a suitable space of operators H[Wξ ](0) there is a flow map (0) Φ : H[Wξ ](0) × R+ , 0 → H[Wξ ]

(1.20)

such that, for every H(z) ∈ H[Wξ ](0) and every s ≥ 0,   Φ H(z), s = Hs (z).

(1.21)

The function Φ satisfies the flow (or semigroup) property     Φ Hs (z), t = Φ H(z), s + t ,

∀s, t ≥ 0, ∀H(z) ∈ H[Wξ ](0) .

(1.22)

1.2.5. The Differential Equation A natural question to ask is whether the flow {Hs (z)}s≥0 constructed in Theorem 1.3 is associated to a differential equation. The answer is affirmative, at least on a formal level. Namely, it is possible to (formally) compute the derivative

Wt δ(Hf − 1)Wt Ω ∂ ∂ Ht = Ht + [A, Ht ] + −z + Ht ∂t τt (z, 1) ∂z −

Wt δ(Hf − 1)Wt , τt (z, 1)

(1.23)

where δ refers to the Dirac delta, A is the (skew-adjoint) generator of the dilation group (Γα )α∈R : Γα = eαA , and · Ω is the vacuum expectation value. Eq. (1.23) is understood in the sense of quadratic forms on χ0 (H). The quadratic form on the right side of (1.23) is not well-defined for the class of operators that we use. To make sense of it, it is necessary to assume regularity conditions for the initial Hamiltonian H and to prove that this regularity is preserved by the renormalization flow. The rigorous justification of (1.23) is deferred to a future work, since the inclusion of it here would considerably increase the length of this paper. It is important to stress that the operator in (1.23) contains only quadratic terms in Wt . This considerably simplifies the discrete RG flow equations whose right side includes all powers of Wt . In physics terminology, this amounts to having only 1-loop corrections contributing to the β-function. One of the main advantages of the continuous renormalization group over the discrete one is that only 1-loop contributions need to be computed. In Theorem 8.20 we prove the existence of one-parameter families of functions {ws }s≥0 such that

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  ∀s ≥ 0: Hs (z) = H ws (z) , where H(ws (z)) is defined as in (1.5)–(1.8) (here we explicitly exhibit the dependence on ws (z)). Eq. (1.23) additionally yields a differential equation on the sequences of functions {ws }s≥0 , which might actually be more useful that the quadratic-form version of the equation (for the estimates). To make use of the differential equation one should start with an initial condition given by a sequence of functions w for which all elements have a finite number of surfaces of discontinuity, outside of them the functions should be C 1 in the photon variables. We expect that the surfaces of discontinuity are preserved by the renormalization flow and that there are no new discontinuities arising. The discontinuitysurfaces naturally arise from the sharp cutoff function χα in (1.10), their co-dimension is 1. α (H(z)) = eα Γα Fα (H(z))Γ ∗ . Recall from (1.12) that Kα (z) := Tα (z) + Wα (z) = R α The operator Tα (z) actually depends on Hf : Tα (z) ≡ Tα (z, Hf ). Taking the derivative with respect to α, we (formally) obtain the differential equation

Wα (z)δ(Hf − 1)Wα (z) . ∂α Kα (z) = Kα (z) + A, Kα (z) − Tα (z, 1)

(1.24)

Here, Kα (z) + [A, Kα (z)] is the infinitesimal form of the scaling map, and Wα (z)δ(Hf − 1)Tα (z, 1)−1 Wα (z) accounts for integrating over the energy shell Hf = 1, which suppresses the corresponding photon-energy degrees of freedom. The scaling map is chosen as to leave Hf invariant, i.e., Hf + [A, Hf ] = 0. Regarding (1.5)–(1.8), • the relevant term of the operator flow Kα (z) solving (1.24) is of the form Tα (z) Ω , • the marginal term Tα (z) − Tα (z) Ω is a function of Hf , and • all other terms, i.e., Wα (z) are irrelevant, as ∂α Wα (z) = Wα (z) + [A, Wα (z)] decays exponentially. To be able to control the relevant term, it is necessary to renormalize the values of the spectral parameter z in such a way that Tα (z) Ω remains bounded. This is achieved by −1 composing Tα (z) Ω with a natural rescaling map Q−1 α such that Tα (Qα (z)) Ω = z (see also (1.27) and (1.37)–(1.38)), which leads to the additional term −z +

Wt δ(Hf − 1)Wt Ω τt (z, 1)

∂ Ht ∂z

on the right side of (1.23) and can be viewed as a 1-loop correction that renormalizes the spectral variable z. It is remarkable that only the inclusion of this renormalization of the spectral parameter ensures the semi-group (flow) property (1.22). To precisely define the differential equation (1.23) on the sequences of functions {ws }s≥0 , it is crucial to introduce norms with respect to which the derivative is taken.

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Suitable norms are not expected to include a derivative with respect to the variable + r ∈ R+ 0 (here the variable r ∈ R0 represents the second argument of wm,n [see (1.3)–(1.4)]; it actually corresponds to the spectral variable of Hf [see (1.8)]). An intuitive (imprecise) reason is the following: The sequences of functions {ws }s≥0 have at most one derivative with respect to r ∈ [0, ∞). If we could compute the derivative in (1.23) using a norm that already includes one derivative with respect to the parameter + r ∈ R+ 0 , then, as a result, we would have taken two derivatives with respect to r ∈ R0 : One derivative would result from the commutator [A, Ht ] in (1.23) and the other from the referred norm. Remark 1.4. Although the renormalization semi-group of operators {Hs (z)}s≥0 is an explicit solution to the differential equation (1.23), it is interesting in its own right to construct solutions of (1.23) using standard methods of differential equations, as this would generate new alternative techniques to study spectral properties of operators in nonrelativistic quantum electrodynamics. Remark 1.5. We could choose H − z instead of H + z to define H(z) in (1.9). This selection would be more natural from the point of view of spectral theory. Nevertheless, the selection H(z) = H + z is convenient because it simplifies the notation in many places of the paper. As the complex number z is directly related to the spectrum of H, we still call it the spectral parameter. 1.3. Comparison to Previous Result The main interest of the previous works (e.g., [4,7]) lies in the analysis of the spectrum of operators. Starting with an operator H(z) as in Section 1.2, one usually constructs a sequence of operators {Hnα0 (z)}n∈N labeled by a discrete set of numbers {nα0 }n∈N , for which it is necessary to take a sufficiently large, fixed, α0 . This sequence of operators {Hnα0 (z)}n∈N also satisfies (1.17) and (1.18), if we choose s = nα0 . The central fact proven in previous works is that, for every n ∈ N there is a biholomorphic function  n : Dρ/2 → Q  n (Dρ/2 ) ⊂ Dρ/2 Q

(1.25)

that possesses the following properties, referred to as isospectrality:  n (z)) is invertible if, and only if, H(n+1)α (z) is invertible. (a ) Hnα0 (Q 0   n (z)) is not injective if, and only if, H(n+1)α (z) is not injective. (b ) Hnα0 (Q 0 Items (a ) and (b ) together with (1.18) facilitate the analysis of the spectrum of the  n }n∈N . original operator by an iterated composition of the functions {Q

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The novelties of our present approach are the following: • We define a continuous family of operators {Hs (z)}s≥0 parametrized by a real number s ∈ R+ 0. • The renormalization group of operators that we construct has a simple and conceptually clear interpretation: It is only one single application of the (rescaled) Feshbach–Schur map with a clever selection of the spectral parameter given by the continuous flow of functions {Es }s≥0 [see Theorem 1.1]. • In spite of its name, the set of operators {Hnα0 (z)}n∈N is not a (discrete) semigroup (on α0 N) of operators [see Remark 1.6]. It does not satisfy (1.22) for s, t ∈ α0 N, because the renormalization of the spectral parameter (1.25) is not properly chosen [see Remark 1.6]. Here, another function to renormalize the spectral parameter is chosen instead. It is denoted by Q−1 α , where Qα is explicitly given in (1.37). It is possible, nevertheless, that the set of operators {Hnα0 (z)}n∈N satisfies a groupoid property, instead a group semi-property [see Remark 1.7]. • We can define a flow for which the renormalization group is the orbit of the operator H(z) under this flow [see Theorem 1.3]. In our approach we do not use the smooth Feshbach–Schur map introduced in [7] because we don’t know how to make it compatible with the semigroup property. Instead, we use the original Feshbach–Schur map projection method introduced in [2] and [4], inspired by the procedures of [9]. We modify the construction of [4] in order to fulfill (1.22). This is an important input of our method. We also use different seminorms to make the computations mathematically precise. Our results provide a new mathematical structure for the renormalization group. As a semigroup, {Hs (z)}s≥0 can be viewed to be the orbit of a flow with initial condition H(z). This opens a new perspective, in which the renormalization group is regarded as the solution to an (autonomous) evolution equation. This was not possible before because any solution to an evolution equation has to be a flow (although it might be possible that the previous scheme would give rise to a non-autonomous differential equation, see Remark 1.7). The study of the renormalization flow, from the perspective of differential equations, is an interesting and challenging mathematical problem. An important area in evolution equations is the study of stable and unstable manifolds. Although these concepts are already used for the renormalization group (see [4], for example), this analysis is done on a sequence of operators which is not proved to have the structure of a (discrete) flow and is not related, therefore, to an (autonomous) evolution equation. Our results give new examples of evolution equations in which the concepts of fixed point, stable, and unstable manifolds can be applied to the spectral theory and have a clear interpretation. Our results provide a nontrivial example of an explicitly solvable problem which has important applications to mathematical physics. Remark 1.6. To construct a renormalization group of operators the selection (or renormalization) of the spectral parameter is fundamental. The renormalization of the spectral

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parameter localizes the regions in the complex plane in which the Feshbach–Schur map can be applied and it is, therefore, the ingredient that allows to apply the Feshbach–Schur map iteratively, which in turn define the sequence of operators {Hnα0 (z)}n∈N . Without the renormalization of the spectral parameter it is possible to apply the Feshbach–Schur map only once and the sequence of Hamiltonians cannot be constructed, the reason being that multiples of the identity define a relevant direction in the space of operators. In [4], the selection of the spectral parameter is represented by the inverse of the function    −1 = eα0 Hnα (z) , Q n 0 Ω

(1.26)

where (·) Ω denotes vacuum expectation value (see Section 3). It is this choice which ruins the semigroup property, even though the Feshbach–Schur map has this property. In contrast to previous work, we not only define a family of isospectral operators {Hs (z)}s≥0 parametrized by a continuous variable s that satisfies (1.16), but we prove that the family satisfies a semigroup property (see also Remark 1.7). This is a consequence of a new definition of the renormalization of the spectral parameter, which produces a new renormalization semigroup of operators. The selection of the spectral parameter is done in such a way that 



Hs (z)

Ω

= z,

∀s ≥ 0.

(1.27)

Eq. (1.27) is the key input for the proof of the semi-group property. Remark 1.7. Although the set of operators {Hnα0 (z)}n∈N does not satisfy a group property, it is possible that it satisfies a similar property (a groupoid property). This means that if we could construct a continuous set of operators {Hs (z)}s≥0 following the construction of the discrete set {Hnα0 (z)}n∈N , then the continuous set of operators might satisfy a non-autonomous differential equation (with initial condition at 0). 1.4. A Guideline of the Paper In this section we give an outline of our paper. First, we briefly review the operators that we use (a more detailed description is done in Sections 2 and 3). We denote by N0 = N ∪ {0} and Bρ the ball of radius ρ and center 0 in R3 . We use the symbol w to represent a sequence of functions w = (wm,n )m,n∈N0 , such that, for every m, n ∈ N0 , m n wm,n : Dρ/2 × R+ 0 × Bρ × Bρ → C

(1.28)

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is a measurable function, that is analytic with respect to the first variable belonging to Dρ/2 and symmetric with respect to the variables belonging to (R3 )m and (R3 )n (see Section 2, the analyticity is understood with respect to the seminorm (2.13) in the sense of Remark 2.2). The functions wm,n can be regarded as functions defined on 3m Dρ/2 × R+ × R3n , if we extend them by zero. We do this identification without 0 ×R mentioning it later on. If this clarifies the matter, we explicitly display the dependence on the spectral parameter z ∈ Dρ/2 as follows   w = (wm,n )m,n∈N0 ≡ wm,n (z) m,n∈N = w(z). 0

(1.29)

In Section 2.2 we associate to every sequence of functions w different seminorms. The seminorms associated to wm,n , wm,n (∞) ,

wm,n (0) ,

(1.30)

are defined in (2.13)–(2.16). The most relevant semi-norms are introduced in (2.19)–(2.21), they are denoted by (I)

wξ ,

w(Z) ,

w − r(F ) ,

(1.31)

where r := (rm,n )m,n≥0 , with r0,0 (z, r) = r, and rm,n = 0, for all m + n ≥ 1. In (1.31), I stands for “interacting”, Z refers to the derivative with respect to the spectral parameter z that is considered in the definition of the semi-norm, and F stands for “free”. The different semi-norms in (1.31) are important to bound the coefficients in w in the renormalization flow. Recalling (1.5)–(1.8) and (1.17)–(1.18), we point out that the (I) interacting part W (z) is controlled by the norm wξ (this is the irrelevant term that decreases exponentially under the renormalization flow [see explanation below (1.24)]). The marginal term T (z) − T (z) Ω [see explanation below (1.24)] is regarded as a free Hamiltonian (it neither contains creation nor annihilation operators). The semi-norm (F ) associated to it is w − rξ , which measures the distance of T (z) − T (z) Ω from Hf . The semi-norm w(Z) is used to prove that the renormalization flow is differentiable with respect to the spectral parameter z; this is necessary for the definition of the renormalization of z which is crucial for the renormalization flow. We denote by Wξ the space of sequences of functions for which the seminorms (1.31) are finite [see (2.24)]. Here ξ ∈ (0, 1) is a small parameter. In Section 2.3 we additional define restrictions for the sequences of functions w ∈ Wξ . To this end, we define a set E α consisting of triples of positive numbers

= ( I , Z , F ) that satisfy certain properties [see Definition 2.6]. ξ of sequences of functions w ∈ Wξ such that We finally define the set W w(I) ≤ I ,

w(Z) ≤ Z ,

w − r(F ) ≤ F ,

(1.32)

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for some ∈ E α . These definitions completely describe the properties that are important concerning the functions spaces. The operators that we study are constructed from the sequences of functions as follows (see Section 3): For every w ∈ Wξ we define [see (3.18)]  Wm,n (w) =

    dk 1 · · · dk m dk˜1 · · · dk˜n a∗ k1 · · · a∗ km

      × wm,n z; Hf ; k1 , · · · , km ; k˜1 , · · · , k˜n a k˜1 · · · a k˜n

(1.33)

and by [see (3.19)] H(w) =



χ0 Wm,n (w)χ0 ,

(1.34)

m,n

where χ0 is the projection introduced in Definition 3.3. The image under H of Wξ is denoted by HWξ . This is the space of operators that we study. In Section 3.3 we derive some norm estimates on the operators Wm,n (w) in terms of the seminorms of w ∈ Wξ . In Section 3.4 we address the question of whether a composition of two operators in HWξ can be written in the form (1.34). We state the results that we need and refer to [4] for the proofs. Much of the notation that we use throughout the paper is introduced in Section 3.4. ξ In Section 4 we introduce the Feshbach–Schur map [see (4.1)]. We prove that if w ∈ W and z ≤ (ρ/2)e−ια [see (2.9) and (2.25)–(2.27)], then the Feshbach–Schur map    Fα H w(z)

(1.35)

is well-defined (see Lemma 4.2). In Section 5 we define the renormalization map. It consists of a slight modification of the Feshbach–Schur map, which we call rescaled Feshbach–Schur map [see (5.3)], and a renormalization of the spectral parameter (see Section 5.2.2). We denote by     α H w(z) R

(1.36)

the rescaled Feshbach–Schur map applied to H(w(z)), and by        α H w(z) Ω , Qα (z) = Ω  R

(1.37)

where Ω is the vacuum. In Section 5.2.2 we prove that Qα is invertible. Then we define the renormalization operator by        α H w(ζ) , Rα H(w) (z) := R where ζ = Q−1 α (z) is the renormalized spectral parameter.

(1.38)

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Section 6 is the biggest part in this paper where most of the laborious computations ξ there exists a sequence and estimates are done. There we prove that for every w ∈ W of functions w  = (w m,n )m,n∈N0 such that     Rα H(w) (z) = H w  (z) .

(1.39)

We study the seminorms (1.30) for w  . To this purpose we first construct a sequence of (sym) (sym) functions w  = (w m,n )m,n∈N0 such that     (sym)  Fα H w(z) = χα H(w)χα + χα H w  (z) χα .

(1.40)

Notice that w  (sym) contains only the terms in the interaction of quadratic and higher order in w. The analysis of the seminorms (1.30) for w  (sym) is done in Section 6.1. This section requires long computations and is divided into many subsections. The title of each subsection indicates which term or which norm is being estimated. The difficult part of Section 6 is the estimation of the seminorms for w  (sym) . Once this is achieved, the analysis of the seminorms (1.30) for w  is straightforward and is done in Section 6.2. This concludes Section 6. In Section 7 we construct a series of iterations of the renormalization map as follows: Given a sequence of positive real numbers α := {αj }j∈N ξ , we give conditions on I , Z and F [see and an initial sequence of functions wα ∈ W (1.32)] and on the sequence α, in order to assure that the iterated renormalization map (0)

     H w() := (Rα ◦ · · · ◦ Rα1 ) H w(0) α α

(1.41)

is well-defined. () To achieve our purpose, we define a sequence of triples { α }∈N0  () () ()  ()

α := I,α , Z,α , F,α

(1.42)

[see (7.6)–(7.11)] and prove inductively [see Section 7.2] that for every  ∈ N0 a sequence () of functions wα can be constructed, that satisfies (1.41) and  () (Z) () w  ≤ Z,α , α

 ()  w − r(F ) ≤ () , α F,α

 () (I) w  ≤ () . α ξ I,α ()

(1.43)

In particular we obtain that the interacting part, that is controlled by I,α, decreases exponentially to zero as  goes to ∞ [see (7.11)]. In Section 8 we prove our main results. We prove Theorems 1.1, 1.2 and 1.3. We fix (0) an initial sequence of functions w satisfying (1.43) for  = 0 (with w instead of wα ). For every s ∈ R+ 0 we select a sequence α and a number β ∈ [0, α+ ], see (2.25), such that

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αj + β = s,

j=1

for some  ∈ N0 (if  = 0 we omit the sum). We define the operator    Hs = Rβ H w() . α (,β)

We prove that, for  ≥ 1, there exists wα

ξ such that ∈W

 (,β)  Hs = H wα (,β)

(for  = 0 and ξ < 1/4 we can find wα We define

4ξ satisfying the same equality). ∈W

  Ts = wα(,β) 0,0 (z, Hf ),

Ws = Hs − Ts .

Then we have Hs = Ts + Ws . The operators Hs define a family of isospectral operators for which the interacting part Ws satisfies (1.18) [see (8.30)]. In Section 8 we prove that the family of operators is well-defined, in the sense that it neither depends on α nor on β. The key ingredient is the construction of the (continuous) renormalization of the spectral parameter Es : Dρ/2 → Dρ2 , which is a biholomorphic function. It has the following properties:  s (H(w(Es (z)))) is well• For every z ∈ Dρ/2 , the rescaled Feshbach–Schur map R defined. • The following equation holds true      s H w Es (z) . Hs (z) = R

(1.44)

In Section 8.1 we construct the function Es . We prove furthermore (1.44) and the exponential decay of the interacting term. In Section 8.2 we define a set of sequences functions {ws }s≥0 such that Hs = H(ws ). In Section 8.3 we define a space of operators H[Wξ ](0) and a flow

(1.45)

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(0) Φ : H[Wξ ](0) × R+ 0 → H[Wξ ]

whose orbits are the sets {Hs }s≥0 . In particular this proves that the operators Hs satisfy a group property. We furthermore define the corresponding flow in the function spaces. 1.4.1. A Remark for Keeping Track of Constants In this paper several consecutive calculations are done. To keep our estimations short and as simple as possible, we make convenient assumptions on the parameters (see Section 2.3). This, however, might obscure some computations by not explicitly writing the exact dependence on the parameters in our formulas. The key constants we obtain to control the renormalization scheme are A(∞) ( , α) and A(0) ( , α), which are defined in Eq. (6.13) of Theorem 6.7 and Eq. (6.61) of Theorem 6.14, respectively. Once having A(∞) ( , α) and A(0) ( , α), the rest of the estimations are relatively short. To help the readers who wish keeping track of the constants, we include several remarks (Remarks 2.7, 6.5, 6.8, 6.12 and 6.15), where we comment on the dependence of our estimations on the parameters, regarding the derivation of Eqs. (6.13) and (6.61). 2. Function Spaces 2.1. Notation We use the symbol N0 for the set N0 := N ∪ {0}.

(2.1)

The ball in R3 with radius r and center 0 is denoted by   Br := x = (x1 , x2 , x3 ) ∈ R3 : |x| ≤ r ,

(2.2)

 1/2 |x| = x21 + x22 + x23 .

(2.3)

where

(m,n)

For every m ∈ N, Brm is the Cartesian product of m copies of Br and Br Brm × Brn . The disc in the complex plane with radius r and center 0 is represented by   Dr := z ∈ C : |z| ≤ r .

=

(2.4)

We remark that (2.1)–(2.4) were already partially presented in the introduction. Nevertheless we repeat their definition to ease the reading.

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We use the symbol   Δρ := (r, s) ∈ R+ × R+ 0 :r+s≤ρ .

(2.5)

For any set of vectors k1 , k2 , · · · , km , k˜1 , · · · , k˜n ∈ R3 we define    m k(m) := k1 , k2 , · · · , km ∈ R3 ,    n k˜(n) := k˜1 , k˜2 , · · · , k˜n ∈ R3 ,    m  n k(m,n) := k(m) ; k˜(n) ∈ R3 × R3 ,

(2.6)

and        (m)  k  := k1  · k2  · · · k m , m   (m)   i k  := k , 1

 (m,n)     (n)  k  := k(m)  · ˜ k ,

 (m,n)    (n)   k  := k(m)  + ˜ k 1 , 1 1

i=1

dk

(m)

dk (m,n) := dk (m) · dk˜(n) ,

:= dk 1 · dk2 · · · dk m ,

(2.7)

where dk i is the Lebesgue measure in R3 . We furthermore identify k(m,0) := k(m) ,

k(0,n) := k˜(n) ,

(2.8)

for n = 0 or m = 0, respectively, and we omit k(0,0) altogether in case that m = n = 0. 2.1.1. Fixed Parameters We select two positive real numbers μ > 0 and ρ > 0, that are fixed throughout the paper, 0 < μ < 1,

ρ=

1 1 . = 122 144

(2.9)

Eq. (2.9) is used several times in our estimations, without mentioning it always. The value or ρ is used to estimate 4πρ < 1,

6ρ1/2 ≤

1 2

in many proofs throughout the paper. 2.2. The Function Spaces W(m, n) 2.2.1. Definition of the Spaces For any pair (m, n) ∈ N20 we denote by W(m, n) the space of measurable functions (m,n) wm,n : Dρ/2 × R+ →C 0 × Bρ

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that are analytic with respect to the variable z ∈ Dρ/2 (see Remark 2.2), symmetric with respect to the variables belonging to (R3 )m and (R3 )n (separately), and that satisfy the following properties: (i) For any (m, n) ∈ N0 × N0 [recall (2.1)–(2.8)],    1/2−μ/2  ∂ 1/2−μ/2        sup wm,n z; r; k(m,n)  · k(m,n)  +  wm,n z; r; k(m,n)  · k(m,n)  ∂z    (2.10)  z ∈ Dρ/2 , r ≥ 0, k (m,n) ∈ Bρ(m,n) < ∞. (ii) For any (m, n) ∈ N × N0 ∪ N0 × N, 



1 sup r

˜(n) 1 ≤ρ k(m) 1 ,k

     dk (m,n) wm,n z; s + r; k(m,n) − wm,n z; s; k(m,n)  (m,n) 3/2+μ/2 |k |

    z ∈ Dρ/2 , (r, s) ∈ Δρ < ∞.

(2.11)

(iii) For m = n = 0,    1   w0,0 (z; s + r) − w0,0 (z; s)  z ∈ Dρ/2 , (r, s) ∈ Δρ < ∞. sup r 

(2.12)

m Remark 2.1. The elements wm,n ∈ W(m, n) are functions wm,n : Dρ/2 × R+ 0 × Bρ × n Bρ → C. If required, we make the dependence on the variable z ∈ Dρ/2 explicit,

wm,n ≡ wm,n (z). Remark 2.2. The analyticity is understood with respect to the seminorm (2.13) below, ∂ in the sense that there exists a function ∂z wm,n such that (∞)    wm,n (z + h) − wm,n (z) ∂  − w lim  (z) = 0. m,n   h→0 h ∂z 0 Note that this implies pointwise analyticity of |k(m,n) |1/2−μ/2 w(·; r; k(m,n) ), as we are (∞) using the supremum in the definition (2.13) of  · 0 —not the essential supremum. 2.2.2. Semi-Norms In this section we define semi-norms attributed to the space W(m, n) which we list below. Let wm,n ∈ W(m, n) and recall (2.1)–(2.8). We define:

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(i) For any (m, n) ∈ N0 × N0 (a) (∞)

wm,n 0

 1/2−μ/2    := sup wm,n z; r; k(m,n)  · k(m,n)      z ∈ Dρ/2 , r ≥ 0, k(m,n) ∈ Bρ(m,n) ,

(2.13)

(b) (∞)

wm,n (∞) := wm,n 0

 (∞) ∂   + w  ∂z m,n  . 0

(2.14)

(ii) For any (m, n) ∈ N × N0 ∪ N0 × N  wm,n (0) := sup



1 r

˜ (n) 1 ≤ρ k(m) 1 ,k

dk (m,n) |k(m,n) |3/2+μ/2

     · wm,n z; s + r; k(m,n) − wm,n z; s; k(m,n)      z ∈ Dρ/2 , (r, s) ∈ Δρ .

(2.15)

(iii) For m = n = 0  w0,0 (0) := sup

 

  1  w0,0 (z; s + r) − w0,0 (z; s)   z ∈ Dρ/2 , (r, s) ∈ Δρ . r (2.16)

2.2.3. The Spaces of Sequences of Functions Wξ We fix a parameter ξ ∈ (0, 1).

(2.17)

We use the symbol w to denote a general sequence of functions of the form w := (wm,n )m+n≥0 ,

(2.18)

where wm,n ∈ W(m, n), ∀m, n ∈ N0 , and m + n ≥ 0, denotes (m, n) ∈ N20 . To every such sequence w we associate the quantities (I)

wξ :=

2 (1 − ξ)2

  sup ξ −(m+n) wm,n (∞) + wm,n (0) ,

(2.19)

m+n≥1

   ∂  w(Z) := sup  w0,0 (z, r) : z ∈ Dρ/2 , r ∈ [0, ρ] , ∂z

(2.20)

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with m + n ≥ 1 denoting (m, n) ∈ (N0 × N) ∪ (N × N0 ), w − r(F ) := w0,0 − r(0) ,

(2.21)

r := (rm,n )m,n≥0

(2.22)

where

with r0,0 (z, r) = r,

rm,n = 0,

m + n ≥ 1,

(2.23)

and r is identified with the identity map r → r on [0, ρ]. In our notation “I” stands for interaction and “F ” stands for free. The space Wξ of sequences of functions is defined by     (I) Wξ := w = (wm,n )m+n≥0  wξ + w(Z) + w − r(F ) < ∞ . (2.24) Remark 2.3. The elements w = (wm,n )m+n≥0 ∈ Wξ are sequences of functions wm,n : m n Dρ/2 × R+ 0 × Bρ × Bρ → C that are analytic with respect to z ∈ Dρ/2 . For clarity, we occasionally make the dependence on the variable z explicit: w ≡ w(z),

wm,n ≡ wm,n (z).

We furthermore use the notation ∂ w(z) = ∂z



∂ . wm,n (z) ∂z m+n≥0

2.3. Further Definitions and Parameters 2.3.1. Definitions and Parameters Definition 2.4. We introduce three new nonnegative parameters α+ , α and ι ≡ ι(α+ ), satisfying the following conditions: 6 > 6, μ

(2.25)

0 ≤ α ≤ α+

(2.26)

1 59 < ι := 1 − < 1. 60 10α+

(2.27)

α+ >

and

The constants α+ and, hence, ι are fixed.

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Definition 2.5. For all I , Z , F > 0 such that  −1 1 1 (1 − F ) − e1/10 − I ρe−α+ > , 2 3

(2.28)

and every α ∈ [0, α+ ], we define Gα ( I , Z , F ) := 3

min(α, 1) 2I e−α ρ



 3 2+ ( +

) Z I · e−α ρ

(2.29)

Note that Gα is monotonically increasing in α, I , Z and F , i.e., Gα ( I , Z , F ) ≤ Gα˜ (˜

I , ˜Z , ˜F ), whenever α ≤ α ˜ , I ≤ ˜I , Z ≤ ˜Z , and F ≤ ˜F . It actually depends trivially on F provided that (2.28) is satisfied. Inequality (2.28) is also monotone in F and I in the sense that once it is satisfied for ˜F and ˜I , it is also satisfied for all F and

I obeying F ≤ ˜F and I ≤ ˜I . Definition 2.6. For α ∈ [0, α+ ], denote by E α the set of triples of positive numbers  3

:= ( I , Z , F ) ∈ R+ satisfying (2.28) and the following properties: (i)  −1

I ρe−α <

1 1 , = 4 3·2 48

(2.30)

(ii) Gα ( ) < 1,

(2.31)

1−ι 3eα 2I < ρ. (1 − Gα ( ))e−α ρ 2

(2.32)

(iii)

Remark 2.7. As 1 − 12 e1/10 > 13 , taking F and I small enough ensures that (2.28) is satisfied. Eqs. (2.30) and (2.31) are fulfilled for small I . By the fact that Gα ( ) decreases as I decreases, selecting small I implies (2.32). In Definitions 2.5 and 2.6 we include some simplifications to make the notation shorter. Nevertheless, the functions and conditions stated there do not always appear in that form in the proofs. The next inequalities, that are simple consequences of Definitions 2.5 and 2.6, are useful to follow the arguments in some proofs.

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(i)

min(α, 1)( I ξ)2 (2.33) e−α ρ(1 − F ) − 12 e−ια ρ − I ξ 

 1 · 2+ ( Z + I ξ) ≤ Gα ( I , Z , F ), e−α ρ(1 − F ) − 12 e−ια ρ − I ξ

(ii) 1 (1 − F ) − e(1−ι)α > 0, 2

(2.34)

1

I (ρe−α )−1 1 (1−ι)α < 24 , (1 − F ) − 2 e

(2.35)

1−ι ( I ξ)2 eα < ρ. 1 − Gα ( ) e−α ρ(1 − F ) − 12 e−ια ρ − I ξ 2

(2.36)

(iii)

(iv)

Remark 2.8. The hypotheses stated in Definitions 2.5 and 2.6 are necessary to define the Feshbach–Schur map (see Definition 4.1 and Lemma 4.2) for α ∈ [0, α+ ]. These properties are not used in previous works (see [7] and [4], for example) because there, it is required that α is sufficiently big. We need to define the Feshbach–Schur map for α tending to 0 to be able to construct a continuous renormalization group of operators, in contrast to the discrete one that is used in earlier papers. ξ 2.3.2. The Polydisc W ξ if it satisfies the following properties: Definition 2.9. We say that w ∈ Wξ belongs to W (a) w0,0 (z, 0) = z,

(2.37)

(b) (I)

wξ ≤ I , for some = ( I , Z , F ) ∈ E α .

w(Z) ≤ Z ,

w − r(F ) ≤ F ,

(2.38)

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3. The Space of Operators 3.1. Basic Notions 3.1.1. Fock Space We denote by   h := L2 R3

(3.1)

F (n) = h⊗s n

(3.2)

and by

the space of symmetric functions in L2 (R3n ), for n ∈ N. We furthermore denote by (n)

FS

  := F (n) ∩ S R3n ,

(3.3)

where S(R3n ) is the Schwartz space. The Bosonic Fock space is given by F :=

∞ 

F (n) ,

(3.4)

n=0

where (0)

F (0) ≡ FS := CΩ and Ω is the normalized vacuum vector. We additionally define FS :=

∞ 

(n)

FS

(3.5)

n=0

and Ffin :=

∞  n 

F (j)

(3.6)

n=0 j=0

the subspace of finite vectors, i.e., vectors with finitely many nonvanishing components.

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3.1.2. Creation and Annihilation Operators (n) For every k ∈ R3 , the annihilation operator a(k) takes a function ψ ∈ FS to the (n−1) function a(k)ψ ∈ FS given by a(k)ψ(k1 , · · · , kn−1 ) :=

√

nψ(k, k1 , · · · , kn−1 ).

(3.7)

Using (3.7) we define a(k) as an operator from FS to FS , additionally setting a(k)Ω := 0.

(3.8)

Note that a(k) is densely defined on F, but not closable and hence has no adjoint (as an operator). The creation operator a∗ (k) is the (formal) adjoint of a(k), it takes a function (n) (n+1)  ψ ∈ FS to the tempered distribution a∗ (k)ψ ∈ (FS ) given by a∗ (k)ψ(k1 , · · · , kn+1 ) := √

n+1  1 δ(k − kj )ψ(k1 , · · · , kj−1 , kj+1 , · · · , kn+1 ). (3.9) n + 1 j=1

Let k1 , · · · , kp ∈ R3 and k˜1 , · · · , k˜q ∈ R3 . The product a(k˜1 ) · · · a(k˜q ) is a well-defined operator on FS , but the product of creation operators a∗ (k1 ) · · · a∗ (kp ) is not. We can, however, define the product a∗ (k1 ) · · · a∗ (kp )a(k˜1 ) · · · a(k˜q )

(3.10)

as a quadratic form, namely,    ∗    ψ  a (k1 ) · · · a∗ (kp )a(k˜1 ) · · · a(k˜q )φ := a(k1 ) · · · a(kp )ψ  a(k˜1 ) · · · a(k˜q )φ (3.11) for any ψ, φ ∈ FS . But this is only possible if the creation operators are to the left of the annihilation operators, i.e., for normal-ordered products. The creation and annihilation operators (formally) satisfy the canonical commutation relations

 

a# (k), a# k = 0,

 

  a(k), a∗ k  = δ k − k ,

(3.12)

where a# = a or a∗ . We suggest the reader who is not familiar with these operators to review Section X.7 in [19].

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3.2. The Space of Operators HWξ In this section we define the vector space of operators that we study (see Definiξ of functions tion 3.8). We define a linear map H that associates to every sequence w ∈ W an operator on F (see Definition 3.5). The space of operators we are interested in is the image of Wξ under this linear function. Definition 3.1 (Free Hamiltonian). The free boson Hamiltonian is the self-adjoint operator  Hf := d3 k a∗ (k)|k|a(k). (3.13) It is the operator in F that represents the (positive) quadratic form in FS derived from (3.11). The operator Hf leaves F (n) invariant and maps ψ ∈ F (n) to the function φ = Hf ψ given by   φ(k1 , · · · , kn ) = |k1 | + · · · + |kn | ψ(k1 , · · · , kn ),

(3.14)

provided that φ ∈ F (n) . Definition 3.2. For every k(m) ∈ R3m and k˜(n) ∈ R3n (see Section 2.1), we set       a∗ k(m) = a∗ k1 · · · a∗ km ,

      a k˜(n) = a k˜1 · · · a k˜n .

Definition 3.3. We denote by χ0 : R → R the characteristic function  1, if r ∈ [0, ρ] χ0 (r) := 0, otherwise,

(3.15)

and by χα , χα the functions defined by   χα (r) = χ0 eα r ,

χα (r) := 1 − χα (r),

(3.16)

for every α ∈ R+ 0 . For abbreviation, we use the following identifications χα ≡ χα (Hf ),

χα ≡ χα (Hf ).

(3.17)

Definition 3.4. For every w = (wm,n )m+n≥0 ∈ Wξ we define  Wm,n (w) :=

      dk (m,n) a∗ k(m) wm,n z; Hf ; k(m,n) a k˜(n) .

(3.18)

Bρm+n

Wm,n (w) is the operator in F representing the quadratic form on FS obtained from (3.11). The existence of this operator is a consequence of Theorem X.44 of [19] (see also the proof of Lemma 3.9 below).

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Definition 3.5 (Hamiltonian operators). For any w = (wm,n )m+n≥0 ∈ Wξ , we define H(w) :=



χ0 Wm,n (w)χ0 .

(3.19)

m+n≥0

The series in (3.19) converges in operator norm and, thus, defines a bounded operator ξ → B[F]. This is a consequence of Lemma 3.9 below. on F. In other words, H : W Remark 3.6. If w = (wm,n )m+n≥0 is a series of functions such that ∀m, n ∈ N0 :

(∞)

wm,n 0

< ∞,

(3.20)

then Wm,n (w) defined by (3.18) determines an operator in F, and Lemma 3.9 implies that χ0 Wm,n (w)χ0 is bounded. In this case, we may also define H(w) as a quadratic form on Ffin . Definition 3.7 (Interaction operators). For any w = (wm,n )m+n≥0 ∈ Wξ we define W (w) := H(w) − χ0 W0,0 (w) =



χ0 Wm,n (w)χ0 .

(3.21)

m+n≥1

Definition 3.8. We denote by HWξ the vector space of operators of the form    ξ ⊆ B[F]. HWξ := H(w)  w ∈ W

(3.22)

3.3. Basic Estimates In this section we study the operators Wm,n (w) and give some norm estimates. We prove furthermore that the operators H(w) and W (w) are bounded and provide some bounds for their norms. (Here and henceforth we use  ·  :=  · B[F] to denote the operator norm on B[F].) Lemma 3.9. Suppose that w ∈ Wξ , then χ0 Wm,n (w)χ0 is a bounded operator and the following estimates are satisfied • For m, n ≥ 1,     χα χ0 Wm,n (w)χα χ0  ≤ wm,n (∞) 4πρ2+μ (m+n)/2 α. 0

(3.23)

• For m ≥ 0, n ≥ 1,     χ0 Wm,n (w)χα χ0  ≤ wm,n (∞) 4πρ2+μ (m+n)/2 α1/2 . 0

(3.24)

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775

• For m ≥ 1, n ≥ 0,     χα χ0 Wm,n (w)χ0  ≤ wm,n (∞) 4πρ2+μ (m+n)/2 α1/2 .

(3.25)

    χ0 Wm,n (w)χ0  ≤ wm,n (∞) 4πρ2+μ (m+n)/2 . 0

(3.26)

0

• For m, n ≥ 0

Proof. We suppose that φ = (φj )j∈N0 , ψ = (ψj )j∈N0 ∈ FS . Suppose furthermore that there exists l ∈ N0 such that φj = 0 for all j = n + l and ψj = 0 for all j = m + l. We estimate the left hand side of Eq. (3.23). We take the operators χα and χ0 to the other side of the inner product and compute it by integrating with respect to the variable x() ∈ (R3 ) . We get (remember the definition of Hf in (3.13)):     ψ  χα χ0 Wm,n (w)χα χ0 φ        ≤ dx() dk (m,n)  a k(m) χα χ0 ψ l x()           ·  wm,n z; x() 1 , k(m,n) a k˜(n) χα χ0 φ l x()    −1/2+μ/2   (m)    (∞) · a k χα χ0 ψ l x()  dx() dk (m,n) k(m)  ≤ wm,n 0  (n) −1/2+μ/2   (n)     a k˜ · ˜ k  χα χ0 φ l x() .

(3.27)

The term χα χ0 permits us to restrict the domain of integration and write the right hand side of Eq. (3.27) as follows    X :=  ψ  χα χ0 Wm,n (w)χα χ0 φ     (∞) () ≤ wm,n 0 dx

 −1/2+μ/2 dk (m) k(m) 

e−α ρ≤k(m) 1 +x() 1 ≤ρ

   (m)    ()    · a k χα χ0 ψ l x 

  (n) −1/2+μ/2   (n)    dk˜(n) ˜ ·  a k˜ χα χ0 φ l x()  . k 



· ˜ (n) 1 +x() 1 ≤ρ e−α ρ≤k

(3.28) Using the Cauchy–Schwarz inequality we estimate Eq. (3.28) by (∞)

X ≤ wm,n 0







dx() e−α ρ≤k(m) 1 +x() 1 ≤ρ

 −1/2+μ/2 dk (m) k(m) 

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2 1/2

     ·  a k (m) χα χ0 ψ l x()  

 ·

dx



 (n) −1/2+μ/2 dk˜(n) ˜ k 

() ˜ (n) +x() 1 ≤ρ e−α ρ≤k

    (n)    ()  2 1/2 ˜   · a k χα χ0 φ l x ,

(3.29)

which is estimated again using the Cauchy–Schwarz inequality by (∞) wm,n 0



 ·

dx



()

dk e−α ρ≤k(m) 





1

+x() 

   (m)   2 · a k χα χ0 ψ l x() 

k

1/2

e−α ρ≤k(m) 1 +x() 1 ≤ρ



 ·



 (n)    (n)   2 dk˜(n) ˜ χα χ0 φ l x()  k  ·  a k˜

dx()



1 ≤ρ

 −2+μ dk (m) k(m) 

·



(m)  (m) 



˜ (n) 1 +x() 1 ≤ρ e−α ρ≤k





 (n) −2+μ k  dk˜(n) ˜

·

1/2

˜ (n) 1 +x() 1 ≤ρ e−α ρ≤k

 (∞)

≤ wm,n 0

·

x() 1 ≤ρ

 ·

·

dk 

 −2+μ dk (m) k(m) 

1/2

e−α ρ≤k(m) 1 +x() 1 ≤ρ



 (n) −2+μ k  dk˜(n) ˜

sup x() 1 ≤ρ



·



sup

1/2

˜ (n) 1 +x() 1 ≤ρ e−α ρ≤k



   (m)  2 · a k χα χ0 ψ 

(m)  (m) 

k

 (n)    (n)  2 dk˜(n) ˜ χα χ0 φ  k  ·  a k˜

1/2

1/2 .

(3.30)

From (3.27)–(3.30) we obtain     ψ  χα χ0 Wm,n (w)χα χ0 φ  (∞)

≤ wm,n 0 ·

· φ · ψ · ρ(n+m)/2  

sup x() 1 ≤ρ

e−α ρ≤k(m) 1 +x() 1 ≤ρ

 −2+μ dk (m) k(m) 

1/2

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 ·



 (n) −2+μ dk˜(n) ˜ k 

sup x() 1 ≤ρ

777

1/2 .

(3.31)

˜ (n) 1 +x() 1 ≤ρ e−α ρ≤k

Note that, to obtain (3.31), we estimate the last two lines in (3.30) by     (Hf )n/2 χ0 φ · (Hf )m/2 χ0 ψ  ≤ φ · ψρ(n+m)/2

(3.32)

using (III.17)–(III.20) in [7]. Next, we estimate the integrals in (3.31): 



sup

dk

x() 1 ≤ρ

 

e−α ρ≤k(m) 

≤

1

+x() 

d3 y|y|−2+μ

m−1

1 ≤ρ



k







· sup s∈[0,ρ]

|y|≤ρ



(m)  (m) −2+μ

dk1 e−α ρ≤|k1 |+s≤ρ

|k1 |μ |k1 |2



m−1    m  4π 1 − e−α ρ1+μ ≤ 4πρ1+μ α. ≤ 4πρ1+μ

(3.33)

As ψm+l ∈ FSm+l is arbitrary, it follows from Eqs. (3.31) and (3.33) that     χα χ0 Wm,n (w)χα χ0 φ ≤ wm,n (∞) 4πρ1+μ (m+n)/2 αρ(m+n)/2 φ. 0

(3.34)

For a general η = (ηj )j∈N ∈ F, we define η (j) ∈ F such that its j-th component equals ηj and the others are zero. Then (3.34) is valid for η (j) instead of φ. Using the fact that for j = l, χα χ0 Wm,n (w)χα χ0 η (j) and χα χ0 Wm,n (w)χα χ0 η (l) are orthogonal, we conclude that (3.34) is valid for η instead of φ. This implies (3.23). Eqs. (3.24)–(3.26) are proved in the same way. 2 Theorem 3.10. For every w ∈ Wξ , W (w) and H(w) are bounded operators and   W (w) ≤ ξw(I) ,

  H(w) ≤ w0,0 (∞) + ξw(I) .

(3.35)

Proof. The result follows from (2.9) (which implies that 4πρ2+μ < 1), (2.19) and Lemma 3.9. 2 Remark 3.11. Following the arguments used in Lemma 3.9, we can prove that the operator-valued function   z → H w(z) is analytic; actually the derivative is given by

 ∂  ∂ H w(z) = H w(z) . ∂z ∂z

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3.4. Composition Formulae (Wick’s Theorem) Suppose that w ∈ Wξ . A natural question is whether there is a sequence of functions  w = (wm,n )m+n≥0 such that 

  H(w)H(w) = H w . The answer to this and similar questions is well-known and actually results from an application of Wick’s Theorem. In this section we state the results that we need concerning the question above and refer to [7] and [4] for the proofs. A large part of the notation that we use in the paper is introduced here. 3.4.1. Definitions and Notation Definition 3.12. Let w ∈ Wξ . We define the operators m,n Wp,q

  z; r; k(m,n) :=



  dx(p) d˜ x(q) a∗ x(p)

      (q)  ˜(q) a x ˜ . · wm+p,n+q z; Hf + r; k(m) , x(p) ; k˜(n) , x (3.36) Definition 3.13. For every L ∈ N and m, n ∈ N0 we denote by BL (m, n) the set of arrays 4 υ = (m, ¯ n ¯ , p¯, q¯) ∈ (NL 0 ) that satisfy the following properties: • The elements m, ¯ n ¯ , p¯, q¯ belong to NL 0 and their components are denoted by a sub¯ = (m )∈{1,···,L} = (m1 , · · · , mL ). script, for example m • m1 + · · · + mL = m and n1 + · · · + nL = n. • For any  ∈ {1, · · · , L}, m + n + p + q ≥ 1. For every υ = (m, ¯ n ¯ , p¯, q¯) ∈ BL (m, n) we use the following notation:  (m ,n )  (m ) (n )  (m ,n )  ¯ n) k(m,¯ := k1 1 1 , · · · , kL L L , k(m ,n ) = k  ; k˜  ,  (n1 )   (n−1 )   (m )   (m )   ¯ n)  r = r k(m,¯ := ˜ k1 1 + · · · + ˜ k−1 1 + k+1+1 1 + · · · + kL L 1 ,  (n )   ¯ n)  r˜ = r˜ k(m,¯ := r + ˜ k 1 . If m = 0 or n = 0 we omit the corresponding terms. If m + n = 0 then r := r˜ := 0. ¯ = (uJ , · · · , uL ) ∈ NL−J+1 for some J ≤ L, we define Let v¯ = (v1 , · · · , vJ ) ∈ NJ0 , u 0 ¯ n) ˜ (¯ k(m,¯ × y˜ v)  (m ) (n ) (v )   (m ) (n ) (v )  (m ,n ) (m ,n )  := k1 1 ; k˜1 1 , y˜1 1 , · · · , kJ J ; k˜J J , y˜J J , kJ+1J+1 J+1 , · · · , kL L L ,

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¯ n) k(m,¯ × y (¯u)  (m ,n )  (m ) (u ) (n )  (m ,n )  (m ) (u ) (n )  := k1 1 1 , · · · , kJ−1J−1 J−1 , kJ J , yJ J ; k˜J J , · · · , kL L , yL L ; k˜L L . (3.37)

Definition 3.14. Let w ∈ Wξ . Suppose that F : Dρ/2 × R → C

(3.38)

is a bounded (measurable) function. For  ∈ {0, 1, . . . , L}, we define  F :=

F, 1,

if  ∈ {1, · · · , L − 1}, if  ∈ {0, L}.

(3.39)

For every υ = (m, ¯ n ¯ , p¯, q¯) ∈ BL (m, n) we use the following notation      z; r; k(m ,n ) := W m ,n z; r; k(m ,n ) , W p ,q and   VυF z; r; k(m,n)   (m,n) ≡ V(Fm,¯ ¯ n,p,¯ ¯ q ) z; r; k   L    (m,¯  (m ,n )    (m,¯ 

 ¯ n ) ¯ n )  z; r + r k := Ω  W ; k   · F Hf + r + r˜ k Ω . =1

(3.40) ¯ n) We frequently omit to display the dependence of r and r˜ on (k(m,¯ ).

Definition 3.15. For every array M = (M1 , · · · , MN ) ∈ NN 0 and any J ∈ {1, · · · , N } we denote by M ≤J := (M1 , · · · , MJ ),

M ≥J := (MJ , · · · , MN ).

For every υ = (m, ¯ n ¯ , p¯, q¯) ∈ BL (m, n) and any J ∈ {1, · · · , L} we use the symbols υ≤J := (m ¯ ≤J , n ¯ ≤J , p¯≤J , q¯≤J ),

υ≥J := (m ¯ ≥J , n ¯ ≥J , p¯≥J , q¯≥J ).

For any two arrays s¯, t¯ ∈ NN 0 we say that s¯ ≤ t¯ if the inequality holds componentwise. Furthermore, we use the symbol |t¯|1 := t1 + · · · + tN .

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Definition 3.16. Let υ = (m, ¯ n ¯ , p¯, q¯) ∈ BL (m, n) and w ∈ Wξ . Suppose that F satisfies (3.38) and F satisfies (3.39). Suppose furthermore that v¯ = (v1 , · · · , vJ ) ≤ q¯≤J and that u ¯ = (uJ , · · · , uL ) ≤ p¯≥J . We define     ,¯v z; r; k(m ) ; k˜(n ) , y˜(v ) := W m ,n +v z; r; k(m ) ; k˜(n ) , y˜(v ) , W p ,q −v j for  ≤ J, and     ,¯u z; r; k(m ) , y (u ) ; k˜(n ) := W m +u ,n z; r; k(m ) , y (u ) ; k˜(n ) , W p −u ,q j for  ≥ J. We furthermore define   VυF≤J ,¯v z; r; k(m) ; k˜(n) , y˜(|¯v|1 )    J−1  ¯ n)      ˜ y˜(¯v) ; k(m ) ; k˜(n ) , y˜(v ) × W,¯v z; r + r k (m,¯ := Ω  =1

  ¯ n)  ˜ y˜(¯v) · F Hf + r + r˜ k(m,¯ ×    (m,¯  (mJ ) (nJ ) (vJ )  ¯ n ) (¯ v ) J,¯v z; r + rJ k ˜ y˜ ·W × ;k ; k˜ , y˜ Ω ,

(3.41)

  VυF≥J ,¯v z; r; k(m) , y (|¯u|1 ) ; k˜(n)  L    ¯ n)    ,¯u z; r + r k(m,¯ × y (¯u) ; k(m ) , y (u ) ; k˜(n ) W := Ω  =J

   (m,¯ 

¯ n) (¯ u) · F Hf + r + r˜ k ×y Ω .

(3.42)

3.4.2. Composition Formulae Lemma 3.17. Let w ∈ Wξ . Suppose that F satisfies (3.38) and F satisfies (3.39). Then F,sym there exists a sequence of functions (wm,n )m+n≥0 satisfying (3.20) such that    L−1  F,sym   W w(z) F (z; Hf )W w(z) = H wm,n m+n≥0 in the sense of quadratic forms in Ffin . F,sym wm,n is the symmetrization with respect to k(m) and k(n) of the function F w m,n



z; r; k

(m,n)



:=





υ∈BL (m,n)



L  m + p  n + q =1

p

q

  VυF z; r; k(m,n) .

Proof. The result follows from Wick’s theorem (see Theorem 3.6 in [7]). 2

(3.43)

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3.4.3. Recursive Relation Lemma 3.18 (Wick-ordering recursive relation). Let υ = (m, ¯ n ¯ , p¯, q¯) ∈ BL (m, n) and w ∈ Wξ . Suppose that F satisfies (3.38) and F satisfies (3.39). For every J ∈ {1, · · · , L}, the following equality holds true   VυF z; r; k(m,n) 

=

 ! J−1 



v ¯≤¯ q≤(J−1) u ¯≤p¯≥(J+1)

(vj )

d˜ y

j=1

"!  " L qj (uj ) pj dy vj uj j=J+1



 · VυF≤(J−1) ,¯v z; r; k(m) ; k˜(n) , y˜(|¯v|1 )       ˜ y˜(¯v) FJ r + r˜J k(m,n) × y (¯u) · FJ−1 r + r˜J−1 k (m,n) ×  J−1  L   (v )    (m,n)  (mJ ,nJ )    (uj )  j ∗ J z, r + rJ k · a y˜ a y W ;k j

j=1

·

VυF≥(J+1) ,¯u



j

z; r; k

(m)

,y

(|¯ u| 1 )

j=J+1

 ; k˜(n) .

Ω

(3.44)

If, furthermore, 1 ≤ J ≤ L − 1, then   VυF z; r; k(m,n) 



=

 ! J

v ¯≤¯ q≤J u ¯≤p¯≥(J+1)

(vj )

d˜ y

j=1

"!  " L qj (uj ) pj dy vj uj j=J+1



 · VυF≤J ,¯v z; r; k(m) ; k˜(n) , y˜(|¯v|1 )  J  L  (vj )     (m,n)     (¯ u) ∗ (uj ) · FJ r + r˜J k ×y a y˜j a yj ·

VυF≥(J+1) ,¯u



j=1

z; r; k

(m)

,y

(|¯ u| 1 )

˜(n)

;k

 ,

j=J+1

Ω

(3.45)

where 



(·)

Ω

   := Ω  (·)Ω .

(3.46)

Proof. See the proof of Theorem A.5 in [3]. 2 Remark 3.19. The vacuum expectation value appearing in (3.44) and (3.45) contains annihilation operators to the left of creation operators (it is not in normal order). The product of creation and annihilation operators that are not normal-ordered does not make sense as an operator nor as a quadratic form. In this remark we explain what is the meaning of the vacuum expectation value in (3.45). The explanation for (3.44) is similar and is detailed in (6.41)–(6.42).

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We notice that the expectation value in Eq. (3.44) is different from zero only when |¯ u|1 = |¯ v |1 . To analyze (formally) the product J L   (u )   (v )   a y˜j j a∗ yj j , j=1

j=J+1

we use the canonical commutation relations (3.12) to take the creation operators to the left. Then we take the vacuum expectation value, which sets to zero all terms containing creation or annihilation operators. We identify     y (|¯u|1 ) ≡ y 1 , · · · , y |¯u|1 ≡ y (uJ+1 ) , · · · , y (uL ) ,     y˜(|¯v|1 ) ≡ y˜1 , · · · , y˜|¯v|1 ≡ y˜(v1 ) , · · · , y˜(vJ ) .

(3.47)

As a result of the formal computation we obtain 

 J L     (vj )       ∗ (uj ) a y˜j a yj := δ y (|¯u|1 ) − p y˜(|¯v|1 ) ,

j=1

j=J+1

Ω

(3.48)

p∈S|¯ v |1

where     p y˜(|¯v|1 ) := y˜p(1) , · · · , y˜p(|¯v|1 )

(3.49)

and S|¯v|1 is the set of permutations of the first |¯ v|1 natural number. We take Eq. (3.48) as a definition. Eq. (3.48) can be taken as a distribution in the variables of y˜(|¯v|1 ) if we let y (|¯u|1 ) fixed. We understand the integral with respect to y˜(|¯v|1 ) in Eq. (3.45) as an application of the distribution (3.48) (which makes sense even if the integrand is not a test function). 4. The Feshbach–Schur Map In this section we introduce the Feshbach–Schur map. It is a map that takes operators into operators and has the important property of being isospectral in the sense that an operator is invertible if, and only if, its Feshbach–Schur map is invertible, and a similarly one can establish a one-to-one correspondence between eigenvalues and eigenvectors. The advantage of studying the Feshbach–Schur map applied to an operator rather that the operator itself is that the Feshbach–Schur map restricts the domain of the operator to a subspace of energies that are close to the spectral region of interest. We can therefore study the spectral properties of the operators locally and neglect the in uence of the spectral points far away from the region under consideration.

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ξ , the Feshbach–Schur map is Definition 4.1 (Feshbach–Schur map). For every w ∈ W defined by    −1 Fα H(w) := χα H(w)χα − χα H(w)χα χα H(w)χα χα H(w)χα ,

(4.1)

provided that χα H(w)χα is bounded invertible. ξ and z ≤ In the next lemma we establish the invertibility of χα H(w)χα for w ∈ W e ρ/2, which implies that, in this case, the Feshbach–Schur map is well-defined. −ια

ξ – see (3.15)–(3.16), and |z| ≤ e−ια ρ/2. Then, for Lemma 4.2. Suppose that w ∈ W 1 −1 (1−ι)α every κ ∈ [0, 1 − 2 (1 − F ) e ),    χ0 χα (Hf + κe−α ρ)  [(1 − κ)ρe−α ]−1 ≤   (1 − ) − 1 e(1−ι)α .  w0,0 (z; Hf ) F 2(1−κ)

(4.2)

In particular, for k = 0,    χ0 χα (Hf )  −α −1   .  w0,0 (z; Hf )  ≤ 3 ρe

(4.3)

Moreover, the operator χα H(w)χα is invertible and     χα H(w)χα −1  ≤

1 . e−α ρ(1 − F ) − 12 e−ια ρ − I ξ

(4.4)

Proof. First we notice that [see (2.16), (2.34), (2.37) and (2.38)] for ρ ≥ r ≥ e−α ρ − κe−α ρ = (1 − κ)e−α ρ         w0,0 (z, r) ≥ r − r w0,0 (z, r) − w0,0 (z, 0) − 1 − w0,0 (z, 0)   r ≥ r(1 − F ) −

(4.5)

1 reα e−ια . 2(1 − κ)

Eq. (4.5) and the functional calculus imply (4.2); we recall that W0,0 (w) commutes with Hf , it depends only on Hf and z. We use a Neumann series to calculate  −1 χα H(w)χα =



L ∞  χα χ0 χα χ0 · (−1)L W (w) , W0,0 (w) W0,0 (w)

(4.6)

    W (w) χα χ0  < 1.  W0,0 (w) 

(4.7)

L=0

which is well-defined whenever

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Eq. (2.35), Eq. (2.38), Theorem 3.10 and Eq. (4.2) ensure that Inequality (4.7) is accomplished. Eq. (4.4) follows from (2.38), Theorem 3.10, (4.2) and (4.6). Eq. (4.3) follows from (2.25)–(2.28) and (4.2). 2 ξ . Then H(w(z)) is invertible for every real number Corollary 4.3. Suppose that w ∈ W z in the interval

1 −α 1 ρe , ρ . z∈ 4 2 Proof. We notice that (2.37)–(2.38) implies that [see also (2.28)] w0,0 (z, r) ≥ z + r − F r ≥ z and therefore w0,0 (z, Hf ) is invertible. The assertion follows from a norm-convergent Neumann series expansion as in (4.6)–(4.7) and from (2.35). 2 Remark 4.4. Remark 3.11, Lemma 4.2 and Formula (4.1) imply that the operator-valued function 

 z ∈ De−ια ρ/2 → Fα H w(z) ξ . is analytic, provided that w ∈ W 5. The Renormalization Map Rα 5.1. The Dilation Operator For every α ∈ R, we denote by u(α) : h → h the group of dilation operators on h. The unitary operator u(α) is defined by the formula     u(α)φ (k) := e−3α/2 φ e−α k ,

  ∀φ ∈ h = L2 R3 .

(5.1)

We denote by Γα := Γ (u(α)) the operator that results after lifting u(α) to Fock space. We have that   Γα a∗ (k)Γα∗ = e−3α/2 a∗ e−α k ,   Γα a(k)Γα∗ = e−3α/2 a e−α k .

Γα Ω = Ω,

(5.2)

5.2. Definition of the Renormalization Map The renormalization map consists on two parts: a rescaled Feshbach–Schur map (see Section 5.2.1) and a renormalization of the spectral parameter (z) (see Section 5.2.2).

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Rescaling the Feshbach–Schur map is convenient, but not necessary. The renormalization of the spectral parameter is fundamental for the renormalization map. Renormalizing the spectral parameter amounts to finding the values of ζ for which the Feshbach–Schur map can be applied not only once but actually twice. Repeating this procedure we find values of ζ for which one can apply the Feshbach–Schur map N times for any (fixed) natural number N . Each application of the Feshbach–Schur map yields a smaller interacting term of the resulting operator, and we can, therefore, estimate the spectrum more precisely. As the Feshbach–Schur map is isospectral, the more often it can be applied, the more accurate estimates we get on the spectrum. The values ζ for which the Feshbach–Schur map can be applied two times are parametrized by an injective function [see (5.7)] (Qα )−1 : Dρ/2 → e−ια Dρ/2 ,

ζ = (Qα )−1 (z),

which is what we refer as the renormalization of the spectral parameter z. This function is biholomorphic on its range. 5.2.1. The Rescaled Feshbach–Schur Map The next definition is a slight modification of the Feshbach–Schur map. It introduces a rescaling that is convenient (though not necessary) for our analysis. ξ , we define the following Definition 5.1 (Rescaled Feshbach–Schur map). For every w ∈ W operator,     α H(w) := eα Γα Fα H(w) Γ ∗ . R α

(5.3)

α (H(w)) satisfies a semigroup property. In the next lemma we prove that R Lemma 5.2 (Semigroup property). Let α, β ≥ 0 and w ∈ Wξ . Let A be a simply connected open subset of C and f : A → Dρ/2 be an analytic function. Suppose that there is an infinite compact subset C ⊂ A such that H(w(f (z))) is invertible for all z ∈ C. Suppose furthermore that χα H(w(f (z)))χα , χα+β H(w(f (z)))χα+β and α (H(w(f (z))))χβ are invertible for every z ∈ C, then χβ R          β R α H w f (z)  R =R , α+β H w f (z)

(5.4)

for all z ∈ A. Proof. First suppose that H(w(f (z))) is invertible. The isospectrality of the Feshbach–   Schut map, Theorem II.1 in [4], implies that R α+β (H(w(f (z)))), Rα (H(w(f (z)))) and   Rα (Rα (H(w(f (z))))) are invertible and that 

   −1   −1 α H w f (z) R = e−α Γα χα H w f (z) χα Γα∗ ,

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   −1   −1 ∗  R = e−(α+β) Γα+β χα+β H w f (z) χα+β Γα+β , α+β H w f (z)  −1          −1 α H w f (z) α H w f (z) β R = e−β Γβ χβ R χβ Γ ∗ , R β

(5.5)

which implies that 

   −1      −1 α H w f (z) β R  R = R α+β H w f (z)

(5.6)

and thus (5.4) holds true. As Equality (5.6) is valid for z belonging to the infinite compact set C contained in A (in which H(w(f (z))) is invertible) and both sides of the equality are analytic in A [see Remark 3.11, Definition 4.1 and Remark 4.4], it follows that (5.6) holds true for z ∈ A. 2 5.2.2. The Renormalization of the Spectral Parameter ξ and z ∈ De−ια ρ/2 . We define Definition 5.3. Let w ∈ W          α H w(z) Qα (z) := eα Fα H w(z) Ω = R , Ω

(5.7)

where we recall that 

(·)

 Ω

   = Ω  (·) Ω .

(5.8)

ξ and z ∈ De−ια ρ/2 . The following inequality holds true Lemma 5.4. Let w ∈ W   ∂       −1     χα H w(z) Ω  ≤ Gα ( ),  ∂z H w(z) χα χα H w(z) χα where we recall that Gα ( ) is defined in (2.29). Proof. We compute     −1   ∂   H w(z) χα χα H w(z) χα χα H w(z) Ω ∂z # $   ∂      −1      W w(z) χα χα W w(z) χα ≤ χα W w(z)  ∂z Ω # $         −1   ∂  +  W w(z) χα χα H w(z) χα χα W w(z)  ∂z Ω #

      −1 ∂    +  W w(z) χα χα H w(z) χα χα H w(z) χα ∂z $     −1     · χα H w(z) χα χα W w(z)  Ω

1 2 ≤ 2 min(α, 1)( I ξ) e−α ρ(1 − F ) − 12 e−ια ρ − I ξ

(5.9)

V. Bach et al. / Journal of Functional Analysis 268 (2015) 749–823

2 1 + min(α, 1)( I ξ) e−α ρ(1 − F ) − 12 e−ια ρ − I ξ 

   ∂    · χ H w(z) χ α α   ∂z

1 2 ≤ 2 min(α, 1)( I ξ) e−α ρ(1 − F ) − 12 e−ια ρ − I ξ

2 1 + min(α, 1)( I ξ)2 −α ( Z + I ξ), e ρ(1 − F ) − 12 e−ια ρ − I ξ

787



2

(5.10)

where we used Lemma 3.9 and (4.4). Notice that we use the proof of (3.24) to estimate the operator W (w) to the left in the inner product because the terms containing Wm,n (w) with n = 0 [see (3.19)–(3.21)] vanish after taking the inner product. Similarly, we use (3.25) to estimate the operator W (w) to the right in the inner product. We conclude using (2.33) and (5.10). 2 ξ and z ∈ De−ια ρ/2 . Then Qα : Q−1 (Dρ/2 ) ∩ (e−ια Dρ/2 ) → Dρ/2 Lemma 5.5. Let w ∈ W α is a bijection. It follows furthermore that    d 1  ≤ e−α . sup  Q−1 (z) α  dz 1 − G α ( ) z∈D

(5.11)

ρ/2

−ια The set Q−1 Dρ/2 ) can, for every ρ˜ ≤ ρ, be estimated in the following way: ˜ ) ∩ (e α (Dρ/2

(I) Every z ∈ e−ια Dρ/2 such that eα |z| + eα Gα ( ) <

ρ˜ , 2

(5.12)

belongs to Q−1 ˜ ). α (Dρ/2 (II)  −1  Qα (ζ) − e−α ζ  ≤

1 ( I ξ)2 min(α, 1) 1 − Gα ( )

1 , · e−α ρ(1 − F ) − 12 e−ια ρ − I ξ

(5.13)

−ια where Q−1 Dρ/2 whose image under Qα is ζ. α (ζ) denotes the unique point in e

Proof. We consider the function, h(z) = z + e−α ζ − e−α Qα (z),

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for an arbitrary ζ ∈ Dρ/2 . We search for a fixed point of h. First we notice that, as a consequence of (2.31), (2.37) and Lemma 5.4, h is a contraction [see (2.38)]:    ∂  h(z) ≤ Gα ( ) < 1.   ∂z We define a sequence {zn }n∈N0 recursively in the following way. We take z0 := e−α ζ and define zn+1 := h(zn ),

∀zn ∈ e−ια Dρ/2 .

We notice that [see (2.37) and (3.21)]    |z1 − z0 | = e−α ζ − e−α Qα e−α ζ    −1  =  W (w)χα χα H(w)χα χα W (w) 

1 2 , ≤ ( I ξ) min(α, 1) −α e ρ(1 − F ) − 12 e−ια ρ − I ξ

(5.14)

where we used Lemmata 3.9 and 4.2. Notice that we use (3.24) to estimate the operator W (w) to the left in the inner product because the terms containing Wm,n (w) with n = 0 [see (3.19)–(3.21)] vanish after taking the inner product. Similarly we use (3.25) to estimate the operator W (w) to the right in the inner product. By the contraction property of h we have that |zn − z0 | ≤

∞ 

Gα ( )j |z1 − z0 |

j=0

≤

1 1 . ( I ξ)2 min(α, 1) −α 1 − Gα ( ) e ρ(1 − F ) − 12 e−ια ρ − I ξ

(5.15)

The desired property |zn | < 12 e−ια ρ is satisfied if

 ρ  −ια 1 1 −α < ( I ξ)2 α −α e − e , 1 1 − Gα ( ) 2 e ρ(1 − F ) − 2 e−ια ρ − I ξ which is accomplished because (2.36) holds true. We conclude that lim zn = z ∈ e−ια Dρ/2

n→∞

exists and that Qα (z) = ζ. The injectivity of Qα restricted to e−ια Dρ/2 follows from the unicity of the fixed point for the contraction h. Eq. (5.11) follows from (2.37), (5.7) and (5.9). Item (I) follows from (5.7) and (5.9), since (5.12) implies that |Qα (z)| < ρ2˜ . Item (II) is a consequence of (5.15). 2

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5.2.3. The Renormalization Map ξ we define the renormalization group by Definition 5.6. For every w ∈ W       α H w(ζ) , Rα H(w) (z) := R where ζ = Q−1 α (z), for all z ∈ Dρ/2 . ξ the following holds true Lemma 5.7. For every w ∈ W   Rα H(w) Ω = z.



Proof. The desired result follows from the following computation: 

    

  α H w(ζ) = Qα (ζ) = Qα (Qα )−1 (z) = z. Rα H(w) Ω = R Ω

2 (5.16)

6. Analysis of the Renormalization Map Rα ξ there is a w In this section we prove that for every w ∈ W  = (w m,n )m+n≥0 satisfying (3.20) such that Rα (w) = H( w)

(6.1)

in the sense of quadratic forms in Ffin . We furthermore do some estimations for the norms w m,n (∞) , w m,n (0) , and w 0,0 − r(0) . In the next section (Section 7) we give (I) ξ and we prove furthermore that w conditions to have w  ∈W  ξ decreases exponentially in α. In this case clearly (6.1) holds as an operator equality. 6.1. Analysis of the Feshbach–Schur Map Fα ξ and every z ∈ e−ια Dρ/2 there exists In this section we prove that for every w ∈ W (sym) w  (sym) = (w m,n )m+n≥0 satisfying (3.20) such that  (sym)  Fα (w) = χα H(w)χα + χα H w  χα

(6.2) (sym)

in the sense of quadratic forms in Ffin . We furthermore estimate the norms w m,n (∞) , (sym) (α) (sym) (∞) w m,n  , and w 0,0 − r . 6.1.1. Definition of w  (sym) 6.1.1.1. Notation Definition 6.1. We fix the function F appearing in (3.38) to be F (z; r) := χα (r)χ0 (r)

1 . w0,0 (z, r)

(6.3)

V. Bach et al. / Journal of Functional Analysis 268 (2015) 749–823

790

Notice that term in the denominator is well-defined due to (4.5). The superscripts (·)F appearing in several objects in Section 3.4 are omitted for the rest of the paper, in the understanding that the corresponding function F used is given by (6.3). In some proofs we omit writing the dependence on z of w0,0 and take w0,0 (z, r) ≡ w0,0 (r), if keeping track of the variable z is not necessary for the argument. ξ and z ∈ e−ια Dρ/2 there exists Lemma 6.2. For every w ∈ W  (sym)  w  (sym) = w m,n m+n≥0 satisfying (3.20) such that   (sym)   Fα H(w) (z) = χα H(w)χα + χα H w  (z) χα . (sym)

w m,n

(6.4)

is the symmetrization with respect to k(m) and k(n) of the functions   w m,n z; r; k(m,n) :=

∞ 

L−1

(−1)

L=2





υ∈BL (m,n)



L  m + p  n + q =1

p

q

  Vυ z; r; k(m,n) .

(6.5)

Proof. The result follows from the Neumann expansion (4.6), Definitions 3.12–3.14 and Lemma 3.17. The only missing part is to prove that the functions (6.5) satisfy (3.20). We defer the proof of this to further sections. It is proved in Theorem 6.7. 2 Definition 6.3. For any (m, n) ∈ N × N0 ∪ N0 × N and α ≥ 0 we define wm,n 

(α)

 1 := sup r

 ˜(n) 1 ≤e−α ρ k(m) 1 ,k

dk (m,n) |k (m,n) |3/2+μ/2

     · wm,n z; s + r; k(m,n) − wm,n z; s; k(m,n)     −α

−α

  z ∈ Dρ/2 , r ∈ 0, e ρ − s , s ∈ 0, e ρ .

(6.6)

In the case that m = 0 or n = 0 we omit the corresponding variable. For m = n = 0 and α ≥ 0 we denote by w0,0 

(α)

  1  := sup  w0,0 (z; s + r) − w0,0 (z; s)  r    −α

−α

  z ∈ Dρ/2 , r ∈ 0, e ρ − s , s ∈ 0, e ρ .

(6.7)

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6.1.2. Estimates for w m,n (∞) Lemma 6.4. Suppose that L ≥ 2, m + n ≥ 0 and that υ ∈ BL (m, n). Suppose furthermore ξ and that z ∈ e−ια Dρ/2 . Then the following estimate holds true that w ∈ W  (n) 1/2−μ/2      Vυ z; r; k(m,n)  · k(m) 1/2−μ/2 · ˜ k   L    −α −1 L−1 (|p| ¯ 1 +|¯ q |1 )/2 ≤ 3 ρe ·ρ wm +p ,n +q (∞) .

(6.8)

=1

Proof. Eqs. (3.39), (4.3) and (6.3) imply that     F (Hf + r + r˜ ) ≤ 3 e−α ρ −1 ,

 ∈ {1, · · · , L − 1}.

(6.9)

The result follows from (2.9) (we estimate 4πρ1+μ < 1), similar arguments as in Lemma 3.9 and        1 z; r + r1 ; k(m1 ,n1 ) · χα χ0 (Hf + r + r˜1 ) Vυ z; r; k(m,n)  ≤ χ0 W     L−1 χα χ0 (Hf + r + r˜−1 ) · F1 (Hf + r + r˜1 ) · =2

      z; r + r ; k(m ,n ) χα χ0 (Hf + r + r˜ ) · F (Hf + r + r˜ ) ·W     L z; r + rL ; k(mL ,nL ) χ0 . · χα χ0 (Hf + r + r˜L−1 ) · W

2 (6.10)

Remark 6.5. In the proof of Lemma 6.4 we use that 4πρ1+μ < 1 to simplify the formulas. For the readers who wish to keep track of the constants it might be useful to ¯ 1 +|¯ q |1 )/2 stress that without using the bound above the term ρ(|p| would be substituted 2+μ (|p| ¯ 1 +|¯ q |1 )/2 by (4πρ ) . In (6.9) we also use specific bounds for the parameters. This, however, is a direct consequence of (2.25)–(2.28), see Lemma 4.2. The same applies to Lemma 6.6 below. Lemma 6.6. Suppose that L ≥ 2, m + n ≥ 0 and that υ ∈ BL (m, n). Suppose furthermore ξ and that z ∈ e−ια Dρ/2 . Then the following estimate holds true [see (2.38)] that w ∈ W   ∂  (n) 1/2−μ/2      Vυ z; r; k(m,n)  · k (m) 1/2−μ/2 · ˜ k   ∂z   L    −α −1 L (|p| ¯ 1 +|¯ q |1 )/2 ≤ L( Z + 1) 3 e ρ ρ wm +p ,n +q (∞) . =1

(6.11)

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Proof. The proof is similar to the one of Lemma 6.4. Here we use the Leibniz rule to compute the derivative. The new ingredient is the term d 1 = dz w0,0



1 w0,0

2

d w0,0 , dz

which produces the extra  −1 3 e−α ρ

Z factor [relative to (6.8)]. 2 Theorem 6.7. Suppose that L ≥ 2, m +n ≥ 0 and that υ ∈ BL (m, n). Suppose furthermore ξ and that z ∈ e−ια Dρ/2 . Then the following estimate holds true that w ∈ W w m,n (∞) ≤ 2I ξ m+n 4m+n

(1 − ξ)2 (∞) A ( , α), 2

(6.12)

where A(∞) ( , α) := (2 + Z )

3[(3(e−α ρ)−1 )]2 (24 )2 . (1 − [ I (3(e−α ρ)−1 )]24 )2

(6.13)

Proof. We use (2.19), (2.38) and Lemmata 6.2, 6.4 and 6.6 to obtain that w m,n (∞) ≤

∞   −1 L (1 − ξ)2 m+n ξ (2 + Z ) L 3 I e−α ρ 2 L=2  L  m + p n + q   p +q ξρ1/2   . · p q

Now we use that

M  N

(6.14)

=1

υ∈B(m,n)

≤ 2M and (2.9) to compute





υ∈B(m,n)

≤4



L  m + p  n + q p

=1



m+n

q



≤4

m+n

∞  

 1/2 j

2ξρ

j=0

 L ≤ 24 · 4m+n ,

L  

2ξρ1/2

p +q

2−(m +n )

=1

υ∈B(m,n)



 (p +q ) ξ p +q ρ1/2  

2L  ∞ 

2L

2

−j

j=0

(6.15)

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793

where we used (2.9) to estimate 2ξρ1/2 < 12 . Using (6.14) and (6.15) we obtain that [see also (2.30)]   −1  4 (1 − ξ)2 m+n ξ (2 + Z ) I 3 e−α ρ 2 2 ∞    −1  L−1  4 L−1 2 · L I 3 e−α ρ

w m,n (∞) ≤ 4m+n

L=2

= 2I 4m+n

3[(3(e−α ρ)−1 )]2 (24 )2 (1 − ξ)2 m+n ξ (2 + Z ) · , 2 (1 − [ I (3(e−α ρ)−1 )]24 )2

(6.16)

where we used that −x2 + 2x = (1 − x)2



d 1 dx 1 − x

%

−1=

∞ d  L x dx

& −1=

L=0

∞ 

LxL−1 ,

L=2

and that for x ∈ (0, 1) 1 | − x2 + 2x| ≤ 3|x| . 2 (1 − x) (1 − x)2

2

Remark 6.8. In the proof of Theorem 6.7 we use that 4πρ1+μ < 1 and 2ξρ1/2 < 1/2 to simplify the formulas. For the readers who wish to keep track of the constants it might be useful to stress that without using the bounds above the term A(∞) ( , α) would be substituted by

(2 + Z )

1 4 3[(3(e−α ρ)−1 )]2 (2 1−2ξ(4πρ 2+μ )1/2 ) 1 2 2 (1 − [ I (3(e−α ρ)−1 )](2 1−2ξ(4πρ 2+μ )1/2 ) )

.

We additionally use (6.9), for which some specific bounds for the parameters are necessary. This, however, is a direct consequence of (2.25)–(2.28), see Lemma 4.2. 6.1.3. Estimates for w m,n (α) In this section we analyze the norm w m,n (α) and give some estimates for this norm that are similar to (6.12). The analogue of Theorem 6.7 is the goal of this section. The proofs are considerably more involved. As in the proof of Theorem 6.7, the analysis of Vυ (z; r; k(m,n) ) is essential, but in this case, due to (6.6) and (6.7), we have to estimate the difference    1   Vυ z; r + s; k(m,n) − Vυ z; s; k(m,n) , r

(6.17)

which is more difficult to handle because of the singularity of the factor 1r , as r tends to 0. To overcome this problem we make use of the recursive relation (Lemma 3.18).

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In the first part of this section we study the term (6.17). This part is technical. In the second part of this section we conclude using (6.5) as in the proof of Theorem 6.7. 6.1.3.1. Estimates for 1r |Vυ (z; r + s; k(m,n) ) − Vυ (z; s; k(m,n) )| For every s˜ ∈ [0, e−α ρ], Vυ (z; s˜; k(m,n) ) is the expectation value of a product consisting of L factors of the form  and L − 1 factors of the form F . We use Leibniz’ rule for finite differences to rewrite W  (6.17) as a sum of 2L − 1 terms: L terms related to the differences of the operators W and L − 1 terms containing differences of operators F . These two types of terms are studied differently. Both types of terms are analyzed using the recursive relation (see Lemma 3.18). The analysis of the terms containing differences of operators F is done in Lemmata 6.9 and 6.10. In this case, the step from s to r + s [see (6.17)] is divided in smaller steps [see (6.18)–(6.19) and (6.34)]. The size of the steps is taken to zero as in a Riemann integral (see the proof of Lemma 6.10).  is done in The analysis of the terms containing differences of the operators W Lemma 6.11. The main result of this section is Lemma 6.13, in which we add all estimates from Lemmata 6.9–6.11. Lemma 6.9. Suppose that L ≥ 2, m + n ≥ 0 and that υ ∈ BL (m, n). Suppose furthermore ξ and that z ∈ e−ια Dρ/2 . Let s ∈ [0, e−α ρ), r ∈ (0, e−α ρ − s], and t, t˜ ≥ 0. that w ∈ W Define κς :=

r e−α ρς

,

(6.18)

and r rς := e−α ρκς = , ς for sufficiently big ς ∈ N, such that κς ∈ [0, 1 − 3(1 − F ), see (2.25)–(2.28)]. The following estimate holds true 

y (|¯v|1 ) dy (|¯u|1 ) d˜



˜(n) 1 ≤e−α ρ k(m) 1 ,k

(6.19)

e(1−ι)α 2(1−F ) )

dk (m) |k(m) |3/2+μ/2

[this is accomplished if ς >

dk˜(n) |k˜(n) |3/2+μ/2

    1  · Vυ≤J ,¯v z; t; k(m) ; k˜(n) , y˜(|¯v|1 )  w0,0 (brς + s + r˜J (k(m,n) × y (¯u) ))    1 χα χ0 (b + 1)rς + s + r˜J k(m,n) × y (¯u) rς   

− χα χ0 brς + s + r˜J k(m,n) × y (¯u)

·

V. Bach et al. / Journal of Functional Analysis 268 (2015) 749–823

 ·

J L   (u )   (v )   a y˜j j a∗ yj j j=1



j=J+1



Vυ≥(J+1) ,¯u z; t˜; k

(m)

,y

(|¯ u|1 )

˜(n)

;k

Ω

795

   

 −1 16π ≤ (4πρ)n+m+|¯u|1 −1 e−(m+n)α ρ|¯u|1 e−α ρ u!   −1 L−2 ((1 − κς )e−α ρ)−1 · 3 e−α ρ 1 (1 − F ) − 2(1−κ e(1−ι)α ς) ·

J 

(ρ)(p +q −v )/2 wm +p ,n +q (∞)

=1 L 

·

(ρ)(p +q −u )/2 wm +p ,n +q (∞) ,

(6.20)

=J+1

for every b ∈ {0, · · · , ς − 1}. u|1 ≥ 1, because if it is zero then Proof. First we notice that we may assume m + n + |¯    χα χ0 (b + 1)rς + s + r˜J k (m,n) × y (¯u)    = 0 = χα χ0 brς + s + r˜J k (m,n) × y (¯u)

(6.21)

(notice that r + s ≤ e−α ρ and (b + 1)rξ ≤ r). We use Remark 3.19 to define the vacuum expectation value of J L   (v )    (u )  a y˜j j a∗ yj j , j=1

j=J+1

which is different from zero only when |¯ u|1 = |¯ v |1 . We identify as in Remark 3.19     y (|¯u|1 ) ≡ y 1 , · · · , y |¯u|1 ≡ y (uJ+1 ) , · · · , y (uL ) ,     y˜(|¯v|1 ) ≡ y˜1 , · · · , y˜|¯v|1 ≡ y˜(v1 ) , · · · , y˜(uJ ) . We have that 

J L   (vj )    (u )  a y˜j a∗ yj j j=1

j=J+1

 :=



   δ y (|¯u|1 ) − p y˜(|¯v|1 )

(6.22)

(6.23)

p∈S|¯ v |1

where     p y˜(|¯v|1 ) := y˜p(1) , · · · , y˜p(|¯v|1 ) and S|¯v|1 is the set of permutations of the first |¯ v |1 integers.

(6.24)

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Eq. (6.23) can be taken as a distribution in the variables of y˜(|¯v|1 ) if we let y (|¯u|1 ) fixed. We understand the integral with respect to y˜(|¯v|1 ) in Eq. (6.20) as an application of the distribution (6.23). We set   −1 L−2 X = |¯ u|1 ! 3 e−α ρ ·

J 

((1 − κς )e−α ρ)−1 1 (1 − F ) − 2(1−κ e(1−ι)α ς)

(ρ)(p +q −v )/2 wm +p ,n +q (∞)

=1 L 

·

(ρ)(p +q −u )/2 wm +p ,n +q (∞)

(6.25)

=J+1

and  Y = B m+n −α e

ρ

dk (m) dk˜(n) |k(m) |2 |k˜(n) |2



 −1+μ dy (|¯u|1 ) y (|¯u|1 ) 

¯ 1 )  ≤ρ y (|u| 1

   1  χα χ0 (b + 1)rς + s + r˜J k(m,n) × y (¯u) rς    − χα χ0 brς + s + r˜J k (m,n) × y (¯u) . ·

(6.26)

Integrating out the y˜(|¯v|1 ) variable in Eq. (6.20) and applying (4.2)–(4.3) and (6.8), we find that the left side of (6.20) is bounded by X ·Y

(6.27)

[see also (2.9), which implies that 4πρ < 1]. Here we use that we can take y (|¯u|1 ) 1 ≤ ρ, otherwise the factors χ0 in the third and the forth lines of Eq. (6.20) would set everything to zero. To obtain the factor (6.25) we used (4.2) and that we can substitute 1 w0,0 (brς + s + r˜J (k(m,n) × y (¯u) ))

(6.28)

χ0 (brς + s + r˜J (k(m,n) × y (¯u) ))χα ((b + 1)rς + s + r˜J (k(m,n) × y (¯u) )) w0,0 (brς + s + r˜J (k(m,n) × y (¯u) ))

(6.29)

by

in the second line of (6.20). We now estimate the integral in (6.26). Suppose first that u ¯ = ¯0, then m + n ≥ 0 (see Definition 3.13). The integral can be bounded by

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797



   (n)  1     k  χα χ0 (b + 1)rς + s + r˜J k(m,n) dk (m) d˜ rς    (m,n)    ≤ 2(4π)n+m e−α ρ m+n−1 . − χα χ0 brς + s + r˜J k

Y ≤ (4π)n+m

(6.30)

Now, we suppose that u ¯ = ¯0. We obtain 

 n+m Y ≤ 2 4πe−α ρ (4π)|¯u|1

1+μ r11+μ · · · r|¯ u|1 −1 0≤r1 +···+r|u| ¯ 1 −1 ≤ρ

 n+m = 2 4πe−α ρ (4π)|¯u|1 ρ(2+μ)(|¯u|1 −1)

Γ (2 + μ)|¯u|1 −1 Γ ((2 + μ)(|¯ u|1 − 1) + 1)

n+m  (4π)|¯u|1 ρ(2+μ)(|¯u|1 −1) ≤ 2 4πe−α ρ

Γ (2)|¯u|1 −1 Γ (2(|¯ u|1 − 1) + 1)

 n+m ≤ 2 4πe−α ρ (4π)|¯u|1 ρ(2+μ)(|¯u|1 −1)

2 (|¯ u|1 !)2

(6.31)

where we used Lemma C.2 in [4] to compute the integral in terms of the Gamma function. Finally, Eq. (6.20) follows from (6.25), (6.26), and (6.30)–(6.31). 2 Lemma 6.10. Suppose that L ≥ 2, m +n ≥ 0 and that υ ∈ BL (m, n). Suppose furthermore ξ and that z ∈ e−ια Dρ/2 . Let s ∈ [0, e−α ρ), r ∈ (0, e−α ρ − s], and t, t˜ ≥ 0. that w ∈ W The following estimate holds true 

y (|¯v|1 ) dy (|¯u|1 ) d˜



˜(n) 1 ≤e−α ρ k(m) 1 ,k

dk (m)

dk˜(n)

|k(m) |3/2+μ/2

|k˜(n) |3/2+μ/2

     1 χα χ0 (s + r + r˜J (k(m,n) × y (¯u) ))  · Vυ≤J ,¯v z; t; k(m) ; k˜(n) , y˜(|¯v|1 )  r w0,0 (s + r + r˜J (k(m,n) × y (¯u) ))    J L  (vj )     χα χ0 (s + r˜J (k(m,n) × y (¯u) )) ∗ (uj ) − · a y˜j a yj w0,0 (s + r˜J (k(m,n) × y (¯u) )) j=1 j=J+1 Ω     · Vυ≥(J+1) ,¯u z; t˜; k(m) , y (|¯u|1 ) ; k˜(n)   ≤

  −1 L   32π (4πρ)n+m+|¯u|1 −1 e−(m+n)α ρ|¯u|1 3 e−α ρ w − r(F ) + 1 u! ·

J 

(ρ)(p +q −v )/2 wm +p ,n +q (∞)

=1

·

L 

(ρ)(p +q −u )/2 wm +p ,n +q (∞) .

=J+1

(6.32)

798

V. Bach et al. / Journal of Functional Analysis 268 (2015) 749–823

Proof. Define as in Lemma 6.9 κς :=

r , e−α ρς

and r rς = e−α ρκς = , ς (1−ι)α

e for sufficiently big ς ∈ N, such that κς ∈ [0, 1 − 2(1− ), see (2.34). F) We adopt some notation for the proof of this lemma. We denote by F1 (s, r) the left hand side of Eq. (6.32), by F2 (brς + s, rς ) and R1 (κς ) the left and the right side of Eq. (6.20), respectively. We furthermore denote by F3 (s, r) the following

 F3 (s, r) :=

dy

(|¯ u| 1 )



(|¯ v |1 )

d˜ y

˜ (n) 1 ,k(m) ≤e−α ρ k

dk (m) dk˜(n) |k (m) |3/2+μ/2 |k˜(n) |3/2+μ/2

        · Vυ≤J ,¯v z; t; k(m) ; k˜(n) , y˜(|¯v|1 ) χα χ0 r + s + r˜J k(m,n) × y (¯u)   1 1 · r w0,0 (r + s + r˜J (k(m,n) × y (¯u) ))    J L  (vj )     1 ∗ (uj ) · a y˜j a yj − w0,0 (s + r˜J (k(m,n) × y (¯u) )) j=1 j=J+1 Ω     (6.33) · Vυ≥(J+1) ,¯u z; t˜; k(m) , y (|¯u|1 ) ; k˜(n) .  Then we have that F1 (s, r) ≤

ς−1 rς  F1 (brς + s, rς ) r b=0

1 ≤ ς

ς−1  b=0

1 F2 (brς + s, rς ) + F3 (brς + s, rς ). ς ς−1

b=0

Now we do the following estimation using (2.12), (2.21) and (4.2)–(4.3)      χα χ0 (b + 1)rς + s + r˜J k (m,n) × y (¯u)   1 1 · rς w0,0 ((b + 1)rς + s + r˜J (k(m,n) × y (¯u) ))   1  − (m,n) (¯ u ) w (br + s + r˜ (k × y ))  0,0

ς

J

(6.34)

V. Bach et al. / Journal of Functional Analysis 268 (2015) 749–823

   χα χ0 ((b + 1)rς + s + r˜J (k(m,n) × y (¯u) ))    ≤  w0,0 (brς + s + r˜J (k(m,n) × y (¯u) ))    χα χ0 ((b + 1)rς + s + r˜J (k(m,n) × y (¯u) ))    · w0,0 ((b + 1)rς + s + r˜J (k(m,n) × y (¯u) ))     1  ·  w0,0 (b + 1)rς + s + r˜J k(m,n) × y (¯u) rς     − w0,0 brς + s + r˜J k(m,n) × y (¯u) 

 −α −1   ((1 − κς )ρe−α )−1 ≤ 3 ρe w − r(F ) + 1 . 1 (1−ι)α (1 − F ) − 2(1−κς ) e

799

(6.35)

Following the proof of Lemma 6.9 and using (6.35) we obtain that   F3 (brς + s, rς ) ≤ w − r(F ) + 1 3R1 (κς ).

(6.36)

Taking the limit ς → ∞ in (6.34) we obtain, using Lemma 6.9 and (6.36),   F1 (s, r) ≤ 2 w − r(F ) + 1 3R1 (0), 2

which together with (2.25)–(2.28) imply (6.32).

Lemma 6.11. Suppose that L ≥ 2, m +n ≥ 0 and that υ ∈ BL (m, n). Suppose furthermore ξ and that z ∈ e−ια Dρ/2 . Let s ∈ [0, e−α ρ) and r ∈ (0, e−α ρ−s]. The following that w ∈ W estimate holds true  dy

(|¯ u| 1 )

(|¯ v |1 )



d˜ y

˜(n) ≤e−α ρ k(n) ,k

dk (m) dk˜(n) |k(m) |3/2+μ/2 |k˜(n) |3/2+μ/2

     · Vυ≤(J−1) ,¯v z; r + s; k(m) ; k˜(n) , y˜(|¯v|1 )        ˜ y˜(¯v) FJ r + r˜J k(m,n) × y (¯u) · FJ−1 r + s + r˜J−1 k (m,n) ×  J−1   (v )      1 J z; (r + s) + rJ k (m,n) ; k(mJ ,nJ ) a y˜j j W · r j=1

 L   (m,n)  (mJ ,nJ )     ∗ (uj )  a yj − WJ z; s + rJ k ;k j=J+1

   (m) (|¯ u|1 ) ˜ (n)  · Vυ≥(J+1) ,¯u z; s; k , y ;k  

  −1 L−1  m+n−mJ −nJ ≤ 3 e−α ρ 4πe−α ρ

Ω

V. Bach et al. / Journal of Functional Analysis 268 (2015) 749–823

800

|¯ v |1 ! ρ(1+μ)pJ ρ(1+μ)qJ |¯ u|1 ! (|¯ u|1 − qJ )! (|¯ v |1 − pJ )! p(1+μ)pJ q (1+μ)qJ J J    (pl +ql )/2 (∞) · (ρ) · wm +p ,n +q  ·

∈NL \{J}

·

(4π)(|¯v|1 −pJ ) ρ(2+μ)(|¯v|1 −pJ ) · wmJ +pJ ,nJ +pJ (0) . (|¯ v |1 − pJ )!

(6.37)

Proof. First we notice that if m + n = 0, we can take |¯ u|1 , |¯ v |1 ≥ 1, because if one of them is zero then either    ˜ y˜(¯v) = 0 FJ−1 s + r + r˜J−1 k (m,n) ×

(6.38)

   FJ r + r˜J k(m,n) × y (¯u) = 0.

(6.39)

or

For every natural number N , we define NN := {1, · · · , N },

(6.40)

and SM,N the set of injective functions from NM → NN . For any two sets A, B, we denote by S(A, B) the set of injective functions from A to B. We use the notations (3.47) for y (|¯u|1 ) and for y˜(|¯v|1 ) (with J − 1 instead of J). We furthermore write   x(pJ ) = x1 , · · · , xpJ ,

 1  x ˜(qJ ) = x ˜ ,···,x ˜ qJ .

(6.41)

We recall Eq. (3.36) and Definition 3.14 to remark that the vacuum expectation value appearing in (6.37) gives rise to a term of the form % J−1 &   (v )    j a∗ x(pJ ) a y˜j j=1

    · wmJ +pJ ,nJ +qJ z; (r + s) + Hf + rJ k(m,n) ; k(mJ ) , x(pJ ) ; k˜(nJ ) , x ˜(qJ )    

− wmJ +pJ ,nJ +qJ z; s + Hf + rJ k(m,n) ; k(mJ ) , x(pJ ) ; k˜(nJ ) , x ˜(qJ ) % L &   (qJ )    ∗ (uj ) ·a x ˜ a yj . (6.42) j=J+1

Ω

V. Bach et al. / Journal of Functional Analysis 268 (2015) 749–823

801

Eq. (6.42) is not well-defined as a quadratic form, because it contains creation operators to the right of annihilation operators. To define it properly we first notice that it is different from zero only if |¯ v |1 ≥ pJ , |¯ u|1 ≥ qJ and |¯ v |1 − pJ = |¯ u|1 − qJ . A formal computation using the canonical commutation relations (3.12) and the pull-through formulas   Hf a∗ (k) = a∗ (k) Hf + |k| ,



 Hf + |k| a(k) = a(k)Hf

(6.43)

permit us to write (6.42) in the following way (that we take as a definition) 







π1 ∈SpJ ,|¯ v |1 π2 ∈SqJ ,|u| ¯ 1 π3 ∈S(N|¯ v |1 \π1 (NpJ ),N|u| ¯ 1 \π2 (NqJ ))

pJ qJ  



j1 =1 j2 =1 j3 ∈N|¯ v |1 \π1 (NpJ )

    j2    · δ y˜π1 (j1 ) − xj1 δ x ˜ − y π2 (j2 ) δ y˜j3 − y π3 (j3 ) · wmJ +pJ ,nJ +qJ z; (r + s)

+



 j    y˜ 4  + rJ k(m,n) ; k(mJ ) , x(pJ ) ; k˜(nJ ) , x ˜(qJ )

j4 ∈N|¯ v |1 \π1 (NpJ )

× z; s +



− wmJ +pJ ,nJ +qJ

 j    y˜ 4  + rJ k(m,n) ; k(mJ ) , x(pJ ) ; k˜(nJ ) , x ˜(qJ )

 .

(6.44)

j4 ∈N|¯ v |1 \π1 (NpJ )

The product of terms of the form δ(˜ y π1 (j1 ) − xj1 ) together with the operator FJ−1 in the third line of Eq. (6.37) imply that   (m )   k J  + x(pJ )  ≤ ρ 1 1

(6.45)

 (q )   (n )  ˜ ˜ J 1 ≤ ρ, k J 1 + x

(6.46)

    (r + s) + y˜v¯ 1 + rJ k(m,n) ≤ ρ.

(6.47)

and similarly we obtain

and

Eq. (6.44) defines a distribution. The integral with respect to the variables y˜(|¯v|1 ) and y (|¯u|1 ) denotes the application of this distribution to the integrand, which is well-defined even though the integrand is not a test function. Using Lemma 6.4 and (6.40)–(6.46) we bound the l.h.s. of Eq. (6.37) by   −α −1 L−1 |¯ v |1 ! |¯ u|1 ! (|¯ v |1 − pJ )! 3 e ρ (|¯ u|1 − qJ )! (|¯ v |1 − pJ )!    · (ρ)(pl +ql )/2 wm +p ,n +q (∞) ∈NL \{J}

B

m +n e−α ρ

 dk (m ) dk˜(n ) |k (m ) |2 |k˜(n ) |2

V. Bach et al. / Journal of Functional Analysis 268 (2015) 749–823

802

 · v |1 −pJ )  ≤ρ ˜ y (|¯ 1

d˜ y (|¯v|1 −pJ ) |˜ y (|¯v|1 −pJ ) |1−μ 

· ˜ (nJ ) 1 ≤ρ x(pJ ) 1 +k(mJ ) 1 ,˜ x(qJ ) 1 +k

·

dx(pJ )

d˜ x(qJ )

|x(pJ ) |1/2−μ/2

|x(qJ ) |1/2−μ/2

dk (mJ ) dk˜(nJ ) |k(mJ ) |3/2+μ/2 |k˜(nJ ) |3/2+μ/2

   1  ·  wmJ +pJ ,nJ +qJ z; (r + s) + y˜(|¯v|1 −pJ ) 1 r    + rJ k(m,n) ; k(mJ ) , x(pJ ) ; k˜(nJ ) , x ˜(qJ )        ˜(qJ ) . − wmJ +pJ ,nJ +qJ z; s + y˜(|¯v|1 −pJ ) 1 + rJ k (m,n) ; k(mJ ) , x(pJ ) ; k˜(nJ ) , x (6.48) Lemma C.3 in [4] implies that given g : RpJ +qJ → C and t ≥ 0  x(pJ ) 1 ,˜ x(qJ ) 1 ≤ρ−t

≤

dx(pJ )

d˜ x(qJ )

|x(pJ ) |1/2−μ/2

|x(qJ ) |1/2−μ/2



ρ(1+μ)pJ ρ(1+μ)qJ (1+μ)pJ

pJ

(1+μ)qJ

qJ

x(pJ ) 1 ,˜ x(qJ ) 1 ≤ρ−t

  g x(pJ ) , x ˜ qJ

  dx(pJ ) d˜ x(qJ ) g x(pJ ) , x ˜ qJ . |x(pJ ) |3/2+μ/2 |x(qJ ) |3/2+μ/2 (6.49)

Using Eqs. (6.6), (6.47) and (6.49), we bound (6.48) by m+n−mJ −nJ   −α −1 L−1  3 e ρ 4πe−α ρ   ρ(1+μ)pJ ρ(1+μ)qJ |¯ v |1 ! |¯ u|1 !  |¯ v |1 − pJ ! (1+μ)p (1+μ)q J J v |1 − pJ )! |¯ u|1 − qJ ! (|¯ pJ qJ    · (ρ)(pl +ql )/2 · wm +p ,n +q (∞) ·

∈NL \{J}



d˜ y (|¯v|1 −pJ ) wmJ +pJ ,nJ +pJ (0) . |˜ y (|¯v|1 −pJ ) |1−μ

(6.50)

(4π)(|¯v|1 −pJ ) ρ(2+μ)(|¯v|1 −pJ ) d˜ y (|¯v|1 −pJ ) ≤ , (|¯ v |1 − pJ )!2 |˜ y (|¯v|1 −pJ ) |1−μ

(6.51)

· v |1 −pJ )  ≤ρ ˜ y (|¯ 1

We conclude (6.37) using  v |1 −pJ )  ˜ y (|¯

1 ≤ρ

which is proved as in (6.31). 2

V. Bach et al. / Journal of Functional Analysis 268 (2015) 749–823

803

Remark 6.12. In the proof of Lemmas 6.9–6.10 we use that 4πρ1+μ < 1 to simplify the formulas. For the readers who wish to keep track of the constants it might be useful to stress that without using the bound above the terms of the form ρ(p +q −v )/2 would be substituted by (4πρ2+μ )(p +q −v )/2 and the same thing happens to the expressions of the form ρ(p +q −u )/2 . Similarly, in Lemma 6.11 the terms of the form ρ(pl +ql )/2 would be substituted by (4πρ2+μ )(pl +ql )/2 . Additionally the restrictions (2.25)–(2.28) are used to derive the bounds in Lemmas 6.9–6.10. They are only used when applying Lemma 4.2, specifically Formulas (4.2) and (4.3) which are direct consequences of (2.25)–(2.28). In Lemma 6.13 below we use additionally that 4πρ ≤ 1 and w − r(F ) ≤ 1. For this lemma we do not give an expression of the form the estimations have without these assumptions because we need bounds that are independent of the combinatoric sums in Lemma 3.18. Lemma 6.13. Suppose that L ≥ 2, m +n ≥ 0 and that υ ∈ BL (m, n). Suppose furthermore ξ and that z ∈ e−ια Dρ/2 . Let s ∈ [0, e−α ρ) and r ∈ (0, e−α ρ−s]. The following that w ∈ W estimate holds true  ˜(n) 1 ≤e−α ρ k(m) 1 ,k

dk (m) |k(m) |3/2+μ/2

dk˜(n) |k˜(n) |3/2+μ/2

   1  · Vυ z; s + r; k(m) ; k˜(n) − Vυ z; s; k(m) ; k˜(n)  r −1 L   ¯ 1 +|¯ q |1 )/2 |p| 3 ¯ 1 +|¯q|1 3 e−α ρ ≤ 64πLρ(|p|  L  · wm +p ,n +q (∞) + wm +p ,n +q (0) .

(6.52)

=1

Proof. We use the Leibniz formula and (3.40) to compute   1      Vυ z; r + s; k(m,n) − Vυ z; s; k(m,n)   r #    (m,¯  (m1 ,n1 )    ¯ n)  (m1 ,n1 ) 

 1   ¯ n) 1 z; s + r1 k (m,¯ ≤  Ω  W ;k −W ;k 1 z; r + s + r1 k r $    (m,¯    (m,¯  (mL ,nL )  ¯ n) ¯ n)  · F1 Hf + r + s + r˜1 k · · · WL z; r + s + rL k ;k Ω  #    (m,¯  (m1 ,n1 )   ¯ n) +  Ω  W ;k 1 z; s + r1 k  ¯ n)    ¯ n) 

1  F1 Hf + r + s + r˜1 k(m,¯ − F1 Hf + s + r˜1 k(m,¯ r $    ¯ n)  (mL ,nL )  L z; r + s + rL k(m,¯ ···W ;k Ω  ·

+ ···

V. Bach et al. / Journal of Functional Analysis 268 (2015) 749–823

804

#  +  Ω

  ¯ n)  (m1 ,n1 )   ;k  W1 z; s + r1 k(m,¯

  ¯ n)  1   ¯ n)  (mL ,nL )  L z; r + s + rL k(m,¯ · W ;k · · · FL−1 Hf + s + r˜L−1 k (m,¯ r $    ¯ n)  (mL ,nL ) 

L z; s + rL k(m,¯ (6.53) −W ;k Ω . Next we apply Lemma 3.18 several times. We apply (3.44) with  ¯ n)  (mJ ,nJ )    ¯ n)  (mJ ,nJ ) 

1   J z; s + rJ k (m,¯ WJ z; r + s + rJ k(m,¯ ;k −W ;k (6.54) r ¯ n) J (z; r + rJ (k(m,¯ instead of W ); k(mJ ,nJ ) ), and we apply (3.45) with

 ¯ n)    ¯ n) 

1  FJ Hf + r + s + r˜J k(m,¯ − FJ Hf + s + r˜J k(m,¯ r

(6.55)

¯ n) instead of FJ (Hf + r + r˜J (k(m,¯ )). We estimate the terms of the form (6.54) using Lemma 6.11 and the terms of the form (6.55) using Lemma 6.10. We estimate the right hand side of Eq. (6.32) in Lemma 6.10 using that w−r(F ) ≤ 1 and (2.9), which implies that 4πρ < 1, by [see (2.34) and (2.38)]

 L      −1 L ¯ 1 +|¯ q |1 )/2 64πρ(|p| wm +p ,n +q (∞) 3 ρe−α

(6.56)

=1

(notice that u ¯ ≤ p¯, v¯ ≤ q¯ and |¯ u|1 = |¯ v |1 ). In Eq. (6.37) in Lemma 6.11 we bound |¯ v |1

|¯ u|1 ! (1+μ)qJ

(|¯ u|1 −qJ )!qJ

by 2|¯u|1 and

|¯ v |1 ! (1+μ)pJ

(|¯ v |1 −pJ )!pJ

by 2 . We use that |¯ v |1 − pJ = |¯ u|1 − qJ (otherwise the left hand side of the equation is zero) and (2.9) to bound the left hand side of (6.37) by   −1 L−1 ¯ 1 +|¯ q |1 )/2 |¯ ρ(|p| 2 v|1 +|¯u|1 3 e−α ρ  L  · wm +p ,n +q (∞) wmJ +pJ ,nJ +qJ (0) .

(6.57)

=1,=J

Next we bound the terms in (6.53) containing (6.55) using Lemmata 3.18 and 6.10, (6.56) 'N   N by and the fact that n=0 N n =2 64π2

|p| ¯ 1 +|¯ q |1 (|p| ¯ 1 +|¯ q |1 )/2

ρ

 L   −α −1 L  3 ρe wm +p ,n +q (∞) .

(6.58)

=1

We bound the terms in (6.53) containing (6.54) using Lemmata 3.18 and 6.11, (6.57) 'N   n N by and the fact that n=0 N n 2 =3

V. Bach et al. / Journal of Functional Analysis 268 (2015) 749–823

  −1 L−1 ¯ 1 +|¯ q |1 )/2 |p| ρ(|p| 3 ¯ 1 +|¯q|1 3 e−α ρ  L  · wm +p ,n +q (∞) wmJ +pJ ,nJ +qJ (0) .

805

(6.59)

=1,=J

We finally obtain (6.52) from (6.53) and (6.58)–(6.59). 2 6.1.3.2. Estimates for w m,n (α) Theorem 6.14. For m + n ≥ 0 the following estimate holds true (1 − ξ)2 (0) A ( , α), 2

(6.60)

3[(3(e−α ρ)−1 )]2 (24 )2 . (1 − [ I (3(e−α ρ)−1 )]24 )2

(6.61)

w m,n (α) ≤ 2I ξ m+n 4m+n where A(0) ( , α) := 128π

Proof. We use Lemmata 6.2 and 6.13 [see also (2.19), (2.38) and (6.5)–(6.7)] to get the following w m,n (α) ≤ 64πξ m+n (1 − ξ)2

∞    −1 L L 3 I e−α ρ L=2



·





L  m + p  n + q

(|¯ q |1 +|¯ q |1 )/2 |p| ¯ 1 +|¯ q |1

ρ

3

=1

υ∈B(m,n)

p

q

· ξ p +q . (6.62)

We compute as in (6.15) using (2.9) (which implies that 6ρ1/2 ≤ 1/2) and (2.30) to obtain 



(|¯ q |1 +|¯ q |1 )/2 |p| ¯ 1 +|¯ q |1

ρ

3



L  m + p  n + q p

=1

υ∈B(m,n)



≤ 4m+n

q

· ξ p +q

¯ 1 +|¯ q |1 |p| ρ(|¯q|1 +|¯q|1 )/2 6|p| ξ ¯ 1 +|¯q|1 2−m−n

υ∈B(m,n)

% ≤4

m+n

∞   1/2 j 6ρ ξ j=0

&2L %

∞ 

&2L 2

−j

j=0

As in (6.16) we conclude using (6.62)–(6.63) that

 L ≤ 4m+n 24 .

(6.63)

806

V. Bach et al. / Journal of Functional Analysis 268 (2015) 749–823

w m,n (α) ≤ 64πξ m+n 4m+n (1 − ξ)2

∞    −1 L  4 L L 3 I e−α ρ 2 L=2

≤ 2I ξ m+n 4m+n (1 − ξ)2 64π

3[(3(e−α ρ)−1 )]2 (24 )2 . (1 − [ I (3(e−α ρ)−1 )]24 )2

2

(6.64)

Remark 6.15. In Theorem 6.14 we utilize the bounds for the parameters necessary for Lemma 6.13 (see Remark 6.12): It is used that 4πρ ≤ 1 and w − r(F ) ≤ 1. The restrictions (2.25)–(2.28) are necessary to derive the bounds in Lemmas 6.9–6.10. They are only used when applying Lemma 4.2, specifically Formulas (4.2) and (4.3) which are direct consequences of (2.25)–(2.28). Additionally, in the proof if Theorem 6.14 we use that 6ρ1/2 ≤ 1/2; without this assumption A(0) ( , α) would be substituted by

128π

3[(3(e−α ρ)−1 )]2 (2 1−6ρ11/2 ξ )4 (1 − [ I (3(e−α ρ)−1 )](2 1−6ρ11/2 ξ )2 )2

.

6.2. Analysis of the Renormalization Map Rα : Definition and Properties of w  ξ there is a w In this section we prove that for every w ∈ W  = (w m,n )m+n≥0 satisfying (3.20) such that Rα (w) = H( w)

(6.65)

in the sense of quadratic forms in Ffin . We furthermore derive some estimations for the norms w m,n (∞) , w m,n (0) , w 0,0 − r(0) , and  w(Z) . (sym) As we already constructed w  in Lemma 6.2 satisfying    (sym)  Fα H(w) (z) = χα H(w)χα + χα H w  (z) χα ,

(6.66)

and established some important properties in Theorems 6.7 and 6.14, the definition and analysis of w  is straightforward and follows from (5.3) and Definition 5.6. ξ . Then there is a sequence of functions w Theorem 6.16. Let w ∈ W  = (w m,n )m+n≥0 satisfying (3.20) such that Rα (w) = H( w).

(6.67)

The functions w m,n are given by     −α w m,n z, r, k (m,n) = e−(3/2)(m+n)α eα wm,n Q−1 r; e−α k(m,n) α (z); e  −1  (sym) + e−(3/2)(m+n)α eα w Qα (z); e−α r; e−α k(m,n) . (6.68) m,n

V. Bach et al. / Journal of Functional Analysis 268 (2015) 749–823

807

Proof. The result follows from (3.19), (5.3), Definition 5.6 and (6.4). 2 ξ . We define Definition 6.17. Let w ∈ W Rα (w) := w .

(6.69)

ξ . For every m + n ≥ 0 the following estimate holds true Theorem 6.18. Let w ∈ W (1 − ξ)2 −(1+μ/2)(m+n)α α e e 2



e−α m+n (∞) . · 1 + I 4 A ( , α) 1 + 1 − Gα ( )

w m,n (∞) ≤ ξ m+n I

(6.70)

Proof. It follows from (2.14), (5.11) and (6.68) that w m,n 

(∞)

 (sym) (∞)

(∞)   m,n e wm,n  + w 1+

−(1+μ/2)(m+n)α α

≤e

e−α 1 − Gα ( )

(1 − ξ)2 −(1+μ/2)(m+n)α α e e 2



e−α m+n (∞) , · 1 + I 4 A ( , α) 1 + 1 − Gα ( )

≤ ξ m+n I

where we used Theorem 6.7 [see also (2.19)].

(6.71)

2

ξ . For every m + n ≥ 1 the following estimate holds true Theorem 6.19. Let w ∈ W w m,n (0) ≤ ξ m+n I

(1 − ξ)2 −μ(m+n)α/2 e 1 + 4m+n I A(0) ( , α) . 2

(6.72)

Proof. It follows from (6.6) and (6.68) that  (sym) (α)

w m,n (0) ≤ e−μ(m+n)α/2 wm,n (α) + w m,n  ≤

(1 − ξ)2 −μ(m+n)α/2 e + 4m+n e−μ(m+n)α/2 I A(0) ( , α) I ξ m+n , 2 (6.73)

where we used Theorem 6.14 [see also (2.19)].

2

ξ . The following estimate holds true Theorem 6.20. Let w ∈ W w 0,0 − r(0) ≤ w0,0 − r(0) + 2I

(1 − ξ)2 (0) A ( , α). 2

(6.74)

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808

Proof. The result follows from Theorem 6.14 and (6.68) as follows       1  −α −α w 0,0 − r(0) ≤ sup eα w0,0 Q−1 (r + s) − w0,0 Q−1 s − r α (z), e α (z), e r      1  −α −α 0,0 Q−1 (r + s) − w 0,0 Q−1 s  + eα w α (z), e α (z), e r    z ∈ D , s ∈ [0, ρ], r ∈ (0, ρ − s]  ρ/2 0,0 (α) ≤ w0,0 − r(0) + w ≤ w0,0 − r(0) + 2I

(1 − ξ)2 (0) A ( , α). 2

2

(6.75)

ξ . We recall that [see (2.20)] Theorem 6.21. Let w ∈ W    ∂  0,0 (z, r) : z ∈ Dρ/2 , r ∈ [0, ρ] . w  (Z) = sup  w ∂z

(6.76)

It follows that w 

(Z)

  2 1 (Z) 2 (1 − ξ) (∞) A ( , α) . ≤ w + I 2 1 − Gα ( )

(6.77)

Proof. The result follows from (5.11), (6.12), and (6.68). 2 7. Iterated Applications of the Renormalization Map Given a sequence of positive real numbers α := {αj }j∈N (0) ξ , we give conditions on I , Z and F [see and an initial sequence of functions wα ∈ W (2.38)] and on the sequence α, in order to assure that the iterated renormalization map

 (0)  w() α := Rα ◦ · · · ◦ Rα1 w

(7.1)

is well-defined. () To achieve our purpose, we define a sequence of triples { α }∈N0 :  () () ()  ()

α := I,α , Z,α , F,α .

(7.2) ()

In Section 7.1 we prove that for every β ∈ [0, α+ ] [see (2.25)], α ∈ E β (see Defini() () () tion 2.6). We additionally analyze the numbers Gβ ( α ), A(∞) ( α , β), and A(0) ( α , β).

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809

Section 7.1 consists on a series of definitions and numerical computations that are used in Section 7.2. () In Section 7.2 we construct the sequence of functions wα satisfying (7.1). We proceed inductively applying the numerical computations obtained in Section 7.1 to Theorems 6.18, 6.19, 6.20 and 6.21. We prove furthermore that  ()  w − r(F ) ≤ () , α F,α

 () (Z) () w  ≤ Z,α , α

 () (I) w  ≤ () . α ξ I,α

(7.3)

()

In particular we obtain that the interacting part, which is controlled by I,α decreases exponentially to zero, as  tends to ∞ [see (7.11)]. 7.1. Assumptions and Analysis of the Parameters Definition 7.1. We fix a real number α− ≥ denote by S(α− , α+ ) the set of sequences

6 μ

and suppose that α+ > α− [see (2.25)]. We

α := {αj }j∈N such that ∀j ∈ N:

α− ≤ αj ≤ α+ .

(7.4)

For every α ∈ S(α− , α+ ) and every j ∈ N we define |α|j := μ(α1 + · · · + αj ).

(7.5)

()

7.1.1. The Sequence { α }∈N0 (0)

(0)

(0)

Definition 7.2. We assume that Z,α , F,α and I,α are positive numbers that satisfy the following properties (i) (0)

Z,α = 1.

(7.6)



1 1 1 1 − e 10 . 10 2

(7.7)

1 1 −2α+ 2 · e ρ . 2 107

(7.8)

(ii) (0)

F,α ≤ (iii) (0)

I,α ≤

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For every  ≥ 1, we define (1) ()

Z,α :=

−1  j=0

+

1 1−

10−12 e−|α|j /2

−1 −1 1  −|α|j /2  1 e . 7 −12 10 j=0 1 − 10 e−|α|l /2

(7.9)

l=j

(2) ()

(0)

F,α := F,α +

−1 1  −|α|j /2 e . 107 j=0

(7.10)

(3)

I,α := I e−|α| /4 .

(7.11)

 () () ()  ()

α := I,α , Z,α , F,α .

(7.12)

()

(0)

We denote

Theorem 7.3. Let α ∈ S(α− , α+ ). For every  ∈ N0 and every β ∈ [0, α+ ], ()

α ∈ Eβ

(7.13)

 ()  (e−α+ ρ)2 Gβ α ≤ e−|α| /2 , 1012    −β −2 4 A(∞) () , α , β ≤ 3 · 10 ρe     −2 () A(0) α , β ≤ 3 · 106 ρe−β .

(7.14)

(see Definition 2.6). It follows furthermore that

Proof. It follows from (7.7), (7.8), (7.10) and (7.11) that

−1 1 1 1 1 9 () ()  1 − e1/10 − > , 1 − F,α − e1/10 − I,α ξ ρe−α+ ≥ 2 10 2 100 3

(7.15)

which implies (2.28). Eq. (2.30) follows from (7.8) and (7.11). Eq. (7.9) implies that

V. Bach et al. / Journal of Functional Analysis 268 (2015) 749–823

Z,α ≤ 2,

811

(7.16)

for all  ∈ N0 , which together with (7.8) and (7.11) imply that [see (2.29)]   (e−α+ ρ)2 Gβ () ≤ e−|α| /2 < 1. 1012

(7.17)

Similarly one verifies that (2.32) is satisfied for all . Using (6.13), (7.8), (7.11), (7.15) and (7.16) we get    −β −2 4 A(∞) () . α , β ≤ 3 · 10 ρe

(7.18)

In the same way we obtain, additionally using (6.61), that  ()   −2 A(0) α , β ≤ 3 · 106 ρe−β .

2

(7.19)

7.2. Inductive Construction 7.2.1. Induction Basis (0) In this section we assume that wα ∈ Wξ satisfies (a0 )  (0) (Z) (0) w  ≤ Z,α . α

(7.20)

 (0)  w − r(F ) ≤ (0) . α F,α

(7.21)

 (0) (I) wα  ≤ (0) . I,α ξ

(7.22)

(b0 )

(c0 )

Remark 7.4. By Theorem 7.3,

(0) α ∈ Eβ

(7.23)

(0) ξ [see (7.20)–(7.22)], which implies that Rβ (w(0) and therefore wα ∈ W α ) is well-defined for every β ∈ [0, α+ ].

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7.2.2. Induction Step (0)

(j)

Theorem 7.5. Let wα ∈ Wξ . Suppose that for every j ∈ {0, · · · , } exists wα ∈ Wξ satisfying (a)  (j) (Z) (j) w  ≤ Z,α , α

(7.24)

 (j)  w − r(F ) ≤ (j), α F,α

(7.25)

 (j) (I) wα  ≤ (j) . I,α ξ

(7.26)

(b)

(c)

Suppose furthermore that for j ∈ {0, · · · ,  − 1}   Rαj+1 w(j) = w(j+1) . α α

(7.27)

  w(+1) := Rα+1 w() α α

(7.28)

If follows that

is well-defined and satisfies (7.24)–(7.26) with  + 1 instead of j. If we additionally suppose that ξ < 14 , β ∈ [0, α+ ] and we define   (,β) wα := Rβ w() α , (,β)

it follows that wα

∈ W4ξ and  (,β) (Z) (+1) w  ≤ Z,α , α  (,β) (F ) (+1) w − r ≤ F,α , α 2  (,β) (I) w  ≤ () (1 − ξ) . α I,α 4ξ (1 − 4ξ)2

(7.29)

(j) α ∈ Eβ

(7.30)

Proof. By Theorem 7.3,

(j) ξ [see (7.20)–(7.22)], which implies that Rβ (w(j) and therefore wα ∈ W α ) is well-defined for every j ∈ {0, · · · , } and every β ∈ [0, α+ ].

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Theorems 6.18 and 7.3 and Eq. (7.16) imply that for m + n ≥ 1

2 (∞) e−α+1 m+n () (1 − ξ)  1+ ≤ξ

I,α m,n 2 1 − 10−12   −|α| /4 −μα+1 /2 m+n −(m+n)μα+1 /2 3e × e +4 e 1000

 (+1)   wα

≤

1 m+n −μα+1 /4 −|α| /4 (0) (1 − ξ)2 ξ , e e

I 2 2

(7.31)

where we used that 4m+n e−(m+n)μα+1 /4 ≤ 1 and that e−μα+1 /2 ≤ e−3/2 e−μα+1 /4 (here we use that μα− ≥ 6). Theorems 6.19 and 7.3 imply that  (+1)   wα

  2 −|α| /4 (0)  ≤ ξ m+n () (1 − ξ) e−μα+1 /2 + 4m+n e−(m+n)μα+1 /2 3e I,α m,n 2 20 ≤

1 −μα+1 /4 (1 − ξ)2 m+n −|α| /4 (0) e ξ e

I,α , 2 2

(7.32)

where we used that 4m+n e−(m+n)μα+1 /4 ≤ 1 and that e−μα+1 /2 ≤ e−3/2 e−μα+1 /4 (here we use that μα− ≥ 6). Theorems 6.20 and 7.3 imply that  (+1) (F )  () 2 (1 − ξ)2 (0)  ()  () w A − r ≤ F,α + I,α

α , α+1 α 2  2 1 (1 − ξ)2 () . ≤ F,α + 7 e−α+ ρ e−|α| /2 10 10

(7.33)

Theorems 6.21 and 7.3 imply that    (+1) (Z)  () (Z)  () 2 (1 − ξ)2 (∞)  ()  1 w    A ≤ wα + I,α

α , α+1 α () 2 1 − Gα ( α )   2 1 1  (1 − ξ)2 () ≤ Z + 7 e−α+ ρ e−|α| /2 . (7.34) 10 10 1 − e−12 e−|α| /2 Eqs. (7.24)–(7.26) with  + 1 instead of j follows from (2.19)–(2.21), (7.9)–(7.11) and (7.31)–(7.34). Eq. (7.29) is proved similarly. 2 8. The Renormalization Flow We fix an initial sequence of functions w satisfying (7.20)–(7.22) (with w instead of (0) wα ). For every s ∈ R+ 0 we select a sequence α ∈ S(α− , α+ ) and a number β ∈ [0, α+ ] such that

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V. Bach et al. / Journal of Functional Analysis 268 (2015) 749–823  

αj + β = s,

j=1

for some  ∈ N0 (if  = 0 we omit the sum). We define the operator    , Hs = Rβ H w() α (0)

()

where we take wα = w to construct wα (see Theorem 7.5). The operators Hs define a family of isospectral operators for which the interacting part decays exponentially in s, as s tends to ∞ [see (8.30)]. In this section we prove that the family of operators is well-defined, in the sense that it does not depend on α and β. The key ingredient is the construction of the (continuous) renormalization of the spectral parameter Es : Dρ/2 → Dρ/2 , which is an analytic open injective function with analytic inverse. It has the following properties:  s (H(w(Es (z)))) is well• For very z ∈ Dρ/2 , the rescaled Feshbach–Schur map R defined [see (5.3)]. • The following equation holds true      s H w Es (z) . Hs (z) = R

(8.1)

 s imply that the analysis of the Eq. (8.1) and the isospectrality of the operator R spectrum of the original operator H(w(ζ)) is equivalent to the analysis for the operators Hs (z) for ζ = Es (z). As the interacting part of the operator Hs exponentially tends to zero, as s tends to infinity, Hs (z) is easier to analyze than the original Hamiltonian. If we take s → ∞, the spectrum of the resulting Hamiltonian is explicit, in fact. In Section 8.1 we construct the function Es . We prove furthermore (8.1) and the exponential decay of the interacting term. In Section 8.2 we define a set of sequences functions {ws }s≥0 such that Hs = H(ws ).

(8.2)

In Section 8.3 we define a space of operators H[Wξ ](0) and a flow (0) Φ : H[Wξ ](0) × R+ 0 → H[Wξ ]

whose associated orbits are the sets {Hs }s≥0 . In particular, this proves that the operators Hs satisfy a group property. We furthermore define the corresponding flow in the function

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815

spaces. There we have to construct an equivalence class of function spaces, due to fact that the mapping w → H(w) is not injective. The use of equivalence classes of functions is also applied to (8.2). 8.1. The Renormalization Flow of Operators Lemma 8.1. Let O be a bounded operator defined on a Hilbert space H and let P1 and P2 be commuting projections such that P1 P2 = P2 . Suppose furthermore that P 1 OP 1 , is invertible (with bounded inverse), where P i = 1 − Pi ,

i = 1, 2.

The Feshbach–Schur map is defined as in (4.1): FP1 (O) = P1 OP1 − P1 O(P 1 OP 1 )−1 OP1 .

(8.3)

It follows that if P 2 FP1 (O)P 2 is invertible with bounded inverse, then also P 2 OP 2 is invertible with bounded inverse. Proof. As P 1 P 2 = P 1 , we can define the Feshbach–Schur map as in (8.3) with P 2 OP 2 instead of O. FP1 (P 2 OP 2 ) = P1 P 2 OP 2 P1 − P1 P 2 O(P 1 OP 1 )−1 OP 2 P1 = P 2 FP1 (O)P 2 .

(8.4)

As by assumption P 2 FP1 (O)P 2 is invertible with bounded inverse it follows by the basic properties of the Feshbach–Schur map (see [4]) that P 2 OP 2 is invertible with bounded inverse. 2 Theorem 8.2. Assume the hypotheses and notation of Theorem 7.5. We define for every β ∈ [0, α+ ] [see (5.7)]

E (,α,β)

     (,α)  β H w (z) Qβ (z) := R , α Ω  (0,α) −1  (−1,α) −1  (,α) −1 := Qα1 ◦ · · · ◦ Qα ◦ Qβ .

(8.5)

Let z ∈ Dρ/2 and take ζ = E (,α,β) (z). Then   χ|α| /μ+β H w0 (ζ) χ|α| /μ+β

(8.6)

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is invertible (with bounded inverse) and  (,β)      |α| /μ+β H w(0) (ζ) . H wα (z) = R α 

(8.7)

Proof. For simplicity we prove the assertion only for β = 0. If β = 0 the same argument works but is more complicated notationally. We use induction in . For  = 1 the result follows from the definition of Rα1 (see Definition 5.6). We suppose that (8.7) is valid for  − 1, and we prove it for . (−1,α) −1 Let z  = (Qα ) (z) ∈ e−ια Dρ/2 (see Lemma 5.5). By assumption         |α| /μ H w(0) (ζ) . H w(−1) z =R α α −1

(8.8)

Lemma 4.2 and Theorem 7.5 imply that [see (5.3)]       (0)   |α| χα H w(−1) z χα = χα R H wα (ζ) χα α (−1) /μ = e|α|(−1) /μ Γ|α|(−1) /μ χ|α| /μ    ∗ · F|α|(−1)/μ H w(0) α (ζ) χ|α| /μ Γ|α|(−1) /μ

(8.9)

(0)

is invertible. Lemma 8.1 implies that χ|α|/μ H(wα (ζ))χ|α|α/μ is invertible. We apply Lemma 5.2 with f = E (,α,0) , using Corollary 4.3 and Theorem 7.3 to prove the existence of the set C, to conclude that    (0)      |α|  |α| /μ H w(0) (ζ) , α R H wα (ζ) =R R α   (−1)/μ

(8.10)

which together with Definition 5.6 and (8.8) accomplish the induction step. 2 Lemma 8.3. Let α, α ˜ ∈ S(α− , α+ ). Suppose that there exist , ˜ ∈ N0 such that  

αj =

j=1

˜ 

α ˜j

(8.11)

j=1

and ˜

E (,α,0) = E (,˜α,0) .

(8.12)

Suppose furthermore that there exist β, β˜ ∈ [0, α+ ] and natural numbers  >  and  ˜  > ˜ such that 

  j=+1



αj + β =

˜  ˜ j=+1

α ˜ j + β˜ = b0 < 2α+ .

(8.13)

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Then 

˜

˜

E ( ,α,β) = E ( ,˜α,β) .

(8.14)

Proof. First we notice that Theorem 8.2 implies that    ˜  H wα (z) = H wα˜ (z) := Hin (z).

(8.15)

We denote  −1   −1,α) −1  ( ,α) −1 Ein := Q(,α) ◦ · · · ◦ Q( ◦ Qβ α+1 α and  (,α)  (˜ −1,˜α) −1  (˜ ,˜α) −1 ˜ −1 E˜in := Qα˜ +1 ◦ · · · ◦ Qα˜ ˜ ◦ Qβ˜ . Now we iterate (5.12) to obtain that z belongs to Ein (Dρ/2 ) whenever  ()  ρ > eα+1 +···+α +β |z| + eα+1 +···+α +β Gα+1 α 2  (+1)   ) + eα+2 +···+α +β Gα+2 α + · · · + eβ Gβ ( . α

(8.16)

Eqs. (7.14) and (8.13) imply that if |z| is sufficiently small, it belongs to Ein (Dρ/2 ). Using similar arguments we conclude that Ein (Dρ/2 ) ∩ E˜in (Dρ/2 ) contains an open (not empty) ball. Now we define the functions    b Hin (ζ) g(ζ) = R , 0 Ω    b Hin (ζ) , g˜(ζ) = R 0 Ω

ζ ∈ Ein (Dρ/2 ), ζ ∈ E˜in (Dρ/2 ).

(8.17)

That the functions g and g˜ exist is a consequence of the proof of Theorem 8.2. Lemma 5.7 and Theorem 8.2 imply that for every z ∈ Dρ/2 g ◦ Ein (z) = g˜ ◦ E˜in (z) = z.

(8.18)

As Ein and E˜in are bijective over their images, Eq. (8.18) implies that they coincide in g(Ein (Dρ/2 ) ∩ E˜in (Dρ/2 )) (actually g restricted to Ein (Dρ/2 ) ∩ E˜in (Dρ/2 ) is the (analytic) bijective inverse of Ein and E˜in ). As Ein and E˜in are analytic, they coincide in Dρ/2 , which implies (8.14). 2 (0)

(0)

Theorem 8.4. Suppose that α, α ˜ ∈ S(α− , α+ ), with α+ ≥ 2α− , and that wα = wα˜ . Suppose furthermore that there are , ˜ ∈ N0 and β, β˜ ∈ [0, α+ ] such that   j=1

αj + β =

˜  j=1

α ˜ j + β˜ = a,

(8.19)

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818

then ˜

˜

E (,α,β) = E (,˜α,β) .

(8.20)

Proof. In the proof of this lemma we construct four sequences α, α ˜ , α(1) , α ˜ (1) belonging to S(α− , α+ ). To simplify the formulas we define the following notation that is used only for this proof [see (7.5)]: ∀ ∈ N:

a =

1 |α| , μ

a ˜ =

1 |˜ α| , μ

(1)

a

=

1  (1)  α , μ

(1)

a ˜

=

1  (1)  α ˜ . μ

(8.21)

Define 0 := 12 (α+ − α− ). We use induction on a. For a ≤ 0 the result follows from Lemma 8.3. We suppose that the result is valid for a ≤ L 0 and we prove it for a ≤ (L + 1) 0 . Suppose that (8.19) holds for a ≤ (L + 1) 0 . Let J be the minimal j such that a − 2α+ ≤ aj and J˜ the minimal j such that a − 2α+ ≤ a ˜j . Suppose, without loss of generality, that aJ ≤ a ˜J˜. It follows that a − a ˜J˜ > α+ . Let α ˜ (1) ∈ S(α− , α+ ) be a (1) ˜ sequence that coincides with α ˜ in the first J˜ entries and such that α ˜ j = α− for j > J. (1) (1) We denote by ˜(1) the maximal j such that a ˜ ≤ a and by β˜(1) = a − a . It follows ˜(1)

j

that (1)

(1)

a−a ˜J+1 > 0 , ˜

a ˜J+1 − aJ ≥ α− . ˜

(8.22)

The fact that α+ ≥ 2α− implies that there is a sequence α(1) ∈ S(α− , α+ ) that coincides (1) (1) with α in the first J entries and such that aJ (1) = a ˜J+1 for some J (1) ∈ N. We denote ˜ (1)

(1)

by (1) the maximal j such that aj ≤ a and by β (1) = a − a(1) . Lemma 8.3 (and the induction hypothesis) implies that ˜

˜

˜(1) ,˜ α(1) ,β˜(1) )

E (,˜α,β) = E ( E (,α,β) = E ( E (

(1)

,α

(1)

,β

(1)

)

=E

(1)

,α

(1)

,β

(1)

)

, ,

(˜(1) ,˜ α(1) ,β˜(1) )

.

2

(8.23)

Theorems 8.2 and 8.4 imply the following: (0)

(0)

Theorem 8.5. Suppose that α, α ˜ ∈ S(α− , α+ ), with α+ ≥ 2α− , and that wα = wa˜ . Suppose furthermore that there are , ˜ ∈ N0 and β, β˜ ∈ [0, α+ ] such that   j=1

αj + β =

˜ 

α ˜ j + β˜ = s,

(8.24)

j=1

then  (,β)   (, ˜ β) ˜  H wα = H wα˜ .

(8.25)

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ξ satisfy (7.20)–(7.22). Suppose that α+ ≥ 2α− . For every Definition 8.6. Let w(0) ∈ W s ≥ 0 we denote by  (,β)  Hs ≡ Hs (z) := H wα , (,β)

(0)

where wα is the sequence of functions constructed in Theorem 7.5 such that wα = w(0) , α is an element of S(α− , α+ ) and  

αj + β = s.

(8.26)

Es := E (,α,β) ,

(8.27)

j=1

We additionally define

and  (,β)  Ts ≡ Ts (z) := wα (Hf ), 0,0 Ws ≡ Ws (z) := Hs − Ts .

(8.28)

Remark 8.7. The fact that α+ ≥ 2α− implies that if s ≥ α− we can take β = 0 in (8.26). ξ , Theorem 8.8 (Contraction property). For every s ≥ 0 and every w(0) ∈ W     s H0 Es (z) , Hs = R

(8.29)

and for s ≥ α− Ws  ≤

1 −2α+ 2 −μs/4 e ρ e . 107

(8.30)

Proof. The result follows from Theorems 7.5, 8.2, 8.4 and 8.5. 2 8.2. Renormalization Flow on the Function Spaces Definition 8.9. We say that two elements w = (wm,n )m+n≥0 ,

   w = wm,n ∈ Wξ m+n≥0

are equivalent if for every m, n ∈ N0   (m)         χ0 r + k(m) 1 χ0 r + ˜ k 1 wm,n z; r; k(m,n)   (m)          k 1 wm,n z; r; k(m,n) . = χ0 r + k (m) 1 χ0 r + ˜

(8.31)

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We denote by [Wξ ] the quotient space of Wξ induced by the equivalence relation (8.31). The elements of [Wξ ] are denoted by [w] = ([wm,n ])m+n≥0 , where [w] is the equivalent class of w ∈ Wξ . The quantities (2.19)–(2.21) are defined for [w] ∈ [W ξ ] as usual for the quotient spaces, taking the infimum over the elements in the equivalence class. Remark 8.10. Suppose that w and w are equivalent. Then Rα (w) and Rα (w ) are equivalent. Proof. First we notice that [see (3.36) and Definition 3.14]   ¯ n)    ¯ n)  −α (m ,n )   z; r + e−α r k (m,¯ χ0 Hf + e−α r + e−α r˜−1 k (m,¯ W ;e k      ¯ n) · χ0 Hf + e−α r + e−α r˜ k (m,¯      ¯ n)   p   = dx(p ) d˜ x(q ) a∗ x(p ) χ0 Hf + e−α r + e−α r˜−1 k(m,¯ + x  1   ¯ n)   −α (m ) (p )   −α (n ) (q )  ˜  ; e k  , x  ; e k˜  , x · wm +p ,n +q z; Hf + e−α r + e−α r k (m,¯         ¯ n) · χ0 Hf + e−α r + e−α r˜ k (m,¯ + x ˜ q  1 a x ˜(q ) (8.32) does not depend of the values of wwm +p ,n +q (z; t; y (m +p ,n +q ) ) with t+y (m +p ) 1 > ρ or t + ˜ y (n +q ) 1 > ρ and, therefore, if we change w for w it does not change. This (and similar arguments) implies that [see (3.40)]      (m)    −α −α (m,n)   χ0 r + k (m) 1 χ0 r + ˜ k 1 Vυ ζ, e r, e k

(8.33)

does not change if we substitute w for w (notice that we can take p1 = 0 = qL , otherwise everything is zero). The desired result is an easy consequence of (6.4), (6.67), (6.68), (8.33) and similar computations. 2 Remark 8.10 permits us define the following Definition 8.11. For every [w] ∈ [Wξ ] we define the operator   H [w] := H(w) and the renormalization map  

Rα [w] := Rα (w) . Definition 8.12. It follows from [7] that the map   [w] ∈ [Wξ ] → H [w]

(8.34)

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is an injection. We denote by

[ws ] ≡ ws (z) the sequence of functions that satisfy   Hs = H [ws ] . Theorem 8.13. Let ξ < 14 . For every s ≥ α−   [w ](I) ≤ 1 e−2α+ ρ2 e−μs/4 , s ξ 107

 (F ) 2 1 1/10 1 [ws − r] 1− e + 7, ≤ 10 2 10   [w ](Z) ≤ 2.

(8.35)

2   [w ](I) ≤ 1 e−2α+ ρ2 (1 − ξ) , s 4ξ 7 10 (1 − 4ξ)2

  [w − r](F ) ≤ 1 1 − 1 e1/10 + 2 , s 10 2 107  (Z) [ws ] ≤ 2.

(8.36)

s

For every s < α− .

Proof. The result is a direct consequence of Theorem 7.5.

2

8.3. The Flow Operator and the Semigroup Property 8.3.1. Definitions and Notation Definition 8.14. We denote by E (∞) the set [see (7.12)]  ()    α ∈ S(α− , α+ ), α+ > 2α− ,  ≥ 0 , E (∞) := α

(8.37)

and by (∞)

Wξ

(8.38) (j)

the set of sequences w ∈ Wξ satisfying (7.24)–(7.26) (with w instead of wα ), for some

(j) ∈ E (∞) . Definition 8.15. We denote by [Wξ ](0) the following set [Wξ ](0) :=



  (∞) Rα (w)  w ∈ Wξ , α ∈ [α− , α+ ] .

(8.39)

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Definition 8.16. We denote by H[Wξ ](0) the set of operators H of the form   H = H [w] ,

(8.40)

for some [w] ∈ [Wξ ](0) . 8.3.2. The Flow Operator Definition 8.17 (Flow operator). We define the flow function Φ : H[Wξ ](0) × R+ 0 → H[Wξ ](0) through the formula Φ(H, t) := Ht ,

(8.41)

where H = H([w]) for some [w] ∈ [Wξ ](0) and Ht is introduced in Definition 8.6 (with w instead of w(0) ). Theorem 8.18 (Semigroup property). For every t, s ≥ 0 and every H ∈ H[Wξ ](0) Φ(H, s + t) = Φ(Hs , t).

(8.42)

Proof. The result is a consequence of Theorem 8.4 and (8.29). 2 8.3.3. The Flow Operator on the Function Spaces (0) Definition 8.19 (Flow operator). We define the flow function Φ : [Wξ ](0) × R+ 0 → [Wξ ] through the formula

  Φ [w], t := [wt ],

(8.43)

where [wt ] is introduced in Definition 8.12 (we take [w(0) ] = [w]). Theorem 8.20 (Semigroup property). For every t, s ≥ 0, and every [w] ∈ [Wξ ](0) ,     Φ [w], s + t = Φ [ws ], t .

(8.44)

Proof. The result is a consequence of Theorem 8.18. 2 Acknowledgments This project started in collaboration with Israel Michael Sigal. We are grateful to Israel Michael Sigal for helpful discussions at various stages of this project and for his review of the first draft of this manuscript which considerably improved the paper. We additionally thank the referee for helpful remarks that substantially improved the presentation of this paper.

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