Degenerate perturbation theory for models of quantum field theory with symmetries

We consider Hamiltonians of models describing non-relativistic quantum mechanical matter coupled to a relativistic field of bosons. If the free Hamiltonian has an eigenvalue, we show that this eigenvalue persists also for nonzero coupling. The eigenvalue of the free Hamiltonian may be degenerate provided there exists a symmetry group acting irreducibly on the eigenspace. Furthermore, if the Hamiltonian depends analytically on external parameters then so does the eigenvalue and eigenvector. Our result applies to the ground state as well as resonance states. For our results we assume a mild infrared condition. The proof is based on operator theoretic renormalization. It generalizes the method used in [15] to non-degenerate situations, where the degeneracy is protected by a symmetry group, and utilizes Schur's lemma from representation theory.


Introduction
We consider mathematical models describing non-relativistic quantum mechanical matter interacting with a quantized field consisting of infinitely many bosons. Such models are used to describe atoms or molecules interacting with the surrounding electromagnetic field or particles in solids interacting with lattice excitation, so called phonons.
In this paper we will focus on models describing interaction with the electromagnetic field. In that case the bosons are photons and have a massless relativistic dispersion relation but the electrons and nuclei are treated as non-relativistic quantum mechanical particles. Such type of models are often referred to as non-relativistic qed.
The dynamics as well as the energy of these models is determined by a self-adjoint operator called the Hamiltonian. For these models the Hamiltonian is typically bounded from below and the infimum of its spectrum is called ground state energy. If the ground state energy is an eigenvalue the corresponding eigenvector is called ground state. As a consequence of the massless nature of photons the ground state energy is not isolated from the rest of the spectrum of the Hamiltonian. The question of existence of a ground state is nontrivial. It has been shown that for models of non-relativistic qed a ground state exists [7,13,16,26,30] under natural assumptions.
In this paper we consider models for which the existence of a ground state has been established. We address the question, how the ground state as well as the ground state energy, E, depend on parameters of the system. For example one is interested on its dependence on the coupling constant, on the positions of static nuclei for molecules, or on analytic extensions of dilations and translations. The regularity of E as a function of such parameters is of fundamental importance for Born-Oppenheimer approximation, scattering theory, adiabatic theory, cf. [15].
If E were an isolated eigenvalue, like it is in quantum mechanical description of molecules without radiation, then analyticity of E with respect to any of the aforementioned parameters would follow from regular perturbation theory. But in models of qed describing photons the energy E is not isolated and the analysis of its regularity is a difficult mathematical problem.
The aforementioned question has been adressed in [15]. In that paper, it was shown that if the Hamiltonian of the model depends analytically on some parameter, s, then also the ground state as well as E depend analytically on s. For the proof of the result in [15] a mild infrared regularization was needed. In the special case of the classical spin-boson model analyticity of the ground state and the ground state energy as a function of the coupling constant could be established without the necessity of an infrared regularization [19]. Analyticity of ground states and ground state energies as a function of the coupling parameter has been shown in [18] for atoms in the framework of non-relativistic qed. For models of non-relativistic qed and the spin boson model analytic extensions of dilations have been studied in connection with resonances [5,6,8].
Furthermore, we want to mention related results about translation invariant models of quantum field theory, where the Hamiltonian commutes with the generators of translations. In such a situation one can restrict the Hamiltonian to the generalized eigenspaces corresponding to the eigenvalues p ∈ R 3 of the generators of translations. This restriction, H(p), is called fiber Hamiltonian. Motivated by the construction of scattering states, regularity of the infimum of the spectrum for these fiber Hamiltonians H(p) as a function of p has been intensively investigated for various models [1,4,9,10,11] with results ranging from Hölder continuity up to real analyticity.
A common assumption of the aforementioned analyticity results in [5,6,8,15,18,19] is that the ground state energy of the Hamiltonian describing the massive non-relativistic matter is non-degenerate. However, in many situations this assumption is not met. For example for almost all atoms, except the noble atoms, the valence shell is not fully occupied and therefore by common physical folklore the ground state energy is degenerate by rotation symmetry (we have not found a rigorous proof of this fact but there is almost certain physical evidence corroborating it). Even for molecules, where rotation invariance is broken, degeneracy may occur by the spinorial degrees of freedom.
If an eigenvalue of the Hamiltonian describing the non-relativistic quantum mechanical matter is degenerate, the coupling to the quantized field can lift the degeneracy. It may be lifted completely or there might remain some degeneracy of possibly smaller multiplicity.
The lifting of the degeneracy of an eigenvalue of an atomic Hamiltonian due to the coupling of the electromagnetic field is usually referred to as the Lamb shift. The most prominent example is the spliting of the first excited energy level in the hydrogen atom [24]. For a mathematical discussion of such a phenomenon in the framework of non-relativistic qed, see for example [2] and references therein. The Lamb shift was studied in [20] in a situation where the degeneracy of the ground state energy is lifted at second order formal perturbation theory. It was shown under a mild infrared condition that the ground state as well as the ground state energy are analytic functions of the coupling constant in a sectorial region around the origin. This is in contrast to perturbation theory of isolated eigenvalues, where by general principles analyticity holds on a whole ball around the origin, cf. [27] and references therein.
In [9] the ground state energy of the fiber Hamiltonian H(p) for an electron with spin interacting with the quantized electromagnetic field was studied and its regularity properties as a function of p in a neighborhood of zero were investigated. In this case, the coupling to the quantized electromagnetic field does not lift the spin degeneracy, which can be seen using time reversal symmetry and Kramer's degeneracy theorem [33].
In this paper we consider the situation where the so called atomic Hamiltonian, describing the non-relativistic matter, has a discrete eigenvalue. This eigenvalue may be degenerate, but we assume that there exists an underlying symmetry of the full Hamiltonian, which acts irreducibly on the corresponding eigenspace. In that case the interaction does not lift nor decrease the degeneracy, which turns out to be protected by the symmetry. In particular, we show the existence of an eigenvalue for small but nonzero coupling. Moreover, the main result states that if the Hamiltonian depends analytically on a parameter s, then also the eigenvalue as well as the eigenstate depend analytically on s.
The result is formulated analogously to the main result in [15]. We generalize the main result in that paper to degenerate situations, i.e., we relax the non-degeneracy condition to an irreducibility condition with respect to a symmetry group. Furthermore, we generalize the result in [15] to include general eigenvalues, which may be different from the ground state energy. This allows the treatment of resonance states, by which we understand eigenvectors of an analytically dilated Hamiltonian.
As in [15] we assume that the interaction is linear in the field operator of the quantized field and that there is a mild infrared regularization. In fact, the main part of the proof also applies to situations arising for the standard model of non-relativstic qed, which is quadratic in the field operators. We isolate the part of the proof which applies to general situations as a corollary of the proof in separate theorem within the last section.
The proof of the main result is based on operator theoretic renormalization [6]. This method is based on an iterated application of the Schur complement also called Feshbach map. One can show that this procedure leads to a fix point, provided infrared behaviour of the original operator is not to singular. Using this fixed point one can construct the ground as the limit of a convergent sequence. If the original Hamiltonian is analytic one can show, as in [15], that this approximating sequence is analytic. Analyticity of the eigenvalue as well as the eigenvector will then follow from uniform convergence.
The main difficulty posed by the degeneracy is the iteration procedure of the renormalization analysis. To prove that an iteration step is contracting, one has to control the relevant direction. For this one adjusts the spectral parameter to make vacuum expectations of the n-th renormalized Hamilton operator small. However, in a degenerate situation the vacuum expectation is a matrix. The key idea is to use the symmetry to conclude that this matrix is in fact a multiple of the identity, using irreducibility and Schur's Lemma. This will then turn the analysis of the relevant direction essentially into a one dimensional problem, which can then be handeled with the methods in [15]. Thus our result is based on results from [15] as well as from [3]. To this end we need to show that the symmetry property as well as the irreducibility property are preserved at each iteration step.
Let us give an outline of the paper. In Section 2 we introduce the model and state the main result. In Section 3 we discuss the analysis related to the symmetry which we will need in the proof of the main theorem. In Section 4 we perform a first Feshbach map. Note that details about the Feshbach map can be found in Appendix D. We show that the assumptions needed for the Feshbach map to be applicable are satisfied. In Section 5 we introduce Banach spaces of matrix valued integral kernels, which describe operators on Fock space. Polydiscs in these spaces will later be needed to show that the iteration procedure of the renormalization analysis converges to a fixed point. In Section 6 we show that the first Feshbach map maps the original Hamiltonian into initial polydisc. In Section 7 we give an explicit definition of the renormalization transformation, as a composition of the Feshbach map and a rescaling of the energy. In Section 8 we show that the renormalization transformation preserves analyticity and symmetry. In Section 9 we derive conditions under which an iterated application of the renormalization transformation is possible and converges to a fixed point. Moreover, we show how one can construct the eigenvector, provided the renormalization analysis converges. In Section 10 we provide the proof of the main theorem by combining the results which are discussed in previous sections. In this section we isolate in Theorem 10.1 the part of the renormalization analysis which is not model dependent and can be applied to larger class of Hamiltonians including for example the standard model of non-relativistic qed.
In Appendix A we review basic properties of antilinear maps. In Appendix B we collect properties of eigenprojections of isolated eigenvalues. In Section C we review formal definitions of creation and annihilation operators, and collect identities and estimates of these operators. We plan do consider applications of the main result in a forthcoming paper elaborating on examples discussed in [25].

Model and Statement of Results
We consider the following model. Let the atomic Hilbert space, H at , be a separable complex Hilbert space. Let h = L 2 (R 3 × Z 2 ) and let denote the Fock space, which is used to describe quantum states of the field. Here S 0 (⊗ 0 h) := C and for n ≥ 1, S n ∈ L(⊗ n h) denotes the orthogonal projection onto the subspace left invariant by all permutation of the n factors of h. We call F n the space of n-particle subspace. A vector ψ ∈ F can be identified with a sequences (ψ n ) n∈N 0 such that ψ n ∈ F n . The vector Ω := (1, 0, 0, ...) ∈ F is called the Fock vacuum. Furthermore, we shall use the following identification where the subsript s indicates that the elements are symmetric with respect to interchange of coordinates. For details we refer the reader to [28] or Appendix C. A unitary operator U ∈ L(h) can be naturally extended to the linear operator Γ(U) in F by An easy calculation shows that Γ(U) is unitary again. For ρ > 0 and f ∈ h define It is straight forward to see that U ρ is a unitary operator on h. The so called dilation operator on F is then given by For a vector z ∈ C N we write |z| = N j=1 |z j | 2 1/2 . To simplify our notation we define for We will identify the tensor product of the Fock space F with a separable Hilbert space H ′ using the canonical identification cf. [28]. For G ∈ L 2 (R 3 × Z 2 ; L(H ′ )) one associates an annihilation operator a(G) as follows. For ψ = (ψ n ) ∞ n=0 ∈ H ′ ⊗ F with the property that ψ n = 0 for all but finitely many n, we define a(G)ψ as a sequence of H ′ -valued measurable functions such that the n-th term satisfies a.e.
[a(G)ψ] n (k 1 , ...., k n ) = (n + 1) 1/2ˆG (k) * ψ n+1 (k, k 1 , ...., k n )dk, (2.2) where the integral on the right hand side is defined as a Bochner integral. Eq. (2.2) defines a closable operator a(G) whose closure is also denoted by a(G). The creation operator a * (G) is defined to be the adjoint of a(G) with respect to the natural scalar product in F . In Appendix C further properties about creation and annihilation operators can be found.
In this paper, we are interested in the dynamics of bosonic particles of mass zero. The energy, ω(k), of such a particle with wave vector k is ω(k) := |k| := |k|.
We define the free-field Hamiltonian, H f , on a vector ψ ∈ H ′ ⊗ F as the sequence of H ′ -valued functions whose n-th term is defined by One verifies that H f with this domain defines a positive, selfadjoint linear operator on H ′ ⊗ F with purely absolutely continuous spectrum, except for an eigenvalue at 0, with eigenspace consisting of all vectors of the form (v, 0, 0, ...) with v ∈ H ′ . Let us now fix an atomic Hilbert space H at . The Hilbert space, describing the atomic degrees of freedom and the quantized field, is given by the tensor product Let X be an open subset of C ν , where ν ∈ N. For each s ∈ X let H at (s) be a densely defined closed operator in H at . For g ≥ 0 and s ∈ X we study the operator where the interaction operator is given by which possibly may be infinite. In the following we formulate Hypotheses, which will be used in the statements of the main results Theorem 2.10.
Hypothesis I. For s ∈ X and j = 1, 2 the mapping s → G j,s is a bounded analytic function that has values in L 2 (R 3 × Z 2 ; L(H at )). Moreover there exists a µ > 0 such that A consequence of this Hypothesis is that the interaction operators W (s) and its adjoint W (s) * are well-defined operators on H at ⊗ D(H f ) which are infinitesimally bounded with respect to H f for all s ∈ X, cf. Lemma C.1. Hence the operator H g (s) is defined on D H at (s) ⊗ D(H f ). Since H at (s) is closed, this space is dense in H and H g (s) is densely defined. Thus the adjoint H g (s) * exists and is closed.
has a densely defined adjoint and is therefore closable [22,Theorem 5.28]. Let us now introduce the notation of a symmetry of an operator. Details can be found in Appendix A. In that case we say that T is symmetric or invariant with respect to S. If T is symmetric with respect to all elements of a set S of symmetries, we say T is symmetric or invariant with respect to S. Remark 2.2. We note that the set of symmetries of an operator form a group. More precisely, if S 1 and S 2 are symmetries, then so are S 1 S 2 and S −1 1 . Thus without loss of generality we can assume that we are given a group of symmetries.
To formulate the second Hypothesis we need the notion of a discrete point in the spectrum of a closed operator. We use the definition as given in [27]. To state it let us first recall the following theorem. We shall make use of the following notation for open balls in the complex plane B r (a) = {z ∈ C : |z − a| < r} , where a ∈ C and r > 0.
exists and is independent of r. Moreover, P λ is a projection, i.e., P 2 λ = P λ . Definition 2.4. Let A be a closed operator. A point λ ∈ σ(A) is called discrete if λ is isolated and P λ , given by Theorem 2.3, is finite dimensional. If P λ is one dimensional we say λ is a nondegenerate eigenvalue. The dimension of P λ is called the algebraic multiplicity. The dimension of Ker(A − λ) is called the geometric multiplicity. If algebraic and geometric multiplicity agree and are finite, we say λ is non-defective.
We can now state the second Hypothesis.

Hypothesis II.
(i) The mapping s → H at (s) is an analytic family in the sense of Kato.
(ii) There exists s 0 ∈ X such that E at (s 0 ) is a non-defective, discrete element of the spectrum of H at (s 0 ).
(iii) If E at (s 0 ) is degenerate, there exists a group of symmetries, S, such that H at (s) ⊗ 1 F , H f , and W (s) are symmetric with respect to S for all s ∈ X. Each element of S can be written in the form S 1 ⊗ S 2 , where S 1 is a symmetry in H at and S 2 is a symmetry in F . Furthermore, the set of symmetries in H at acts irreducibly on the eigenspace of H at (s 0 ) with eigenvalue E at (s 0 ). Each element of S 2 := {S 2 : S 1 ⊗ S 2 ∈ S} leaves the Fock vacuum as well as the one particle subspace invariant and commutes with the operator of dilations, cf. (2.1).
By Hypothesis II and the Kato-Rellich theorem of analytic perturbation theory, [27], together with a symmetry argument one can show the following lemma, which will be needed to formulate the third hypothesis. We note that parts (a) and (b) are well known results and can be found in [27]. The proof of (c) will require a symmetry argument. We will provide a proof in Section 3.
Lemma 2.5. Suppose the situation is as in Hypothesis II. Then there exists an ǫ > 0 sufficiently small and a neighborhood N ⊂ X of s 0 , such that the following holds.
defines a projection valued analytic function and the dimension of the range is finite and constant. In particular, p at (s 0 ) projects onto the eigenspace of E at (s 0 ). For s ∈ N the point e at (s) ∈ C is the only point in the spectrum of H at (s) in a neighborhood of E at (s 0 ). The number e at (s) is a non-defective, discrete element of the spectrum of H at (s). Furthermore, e at (s 0 ) = E at (s 0 ).
If Hypothesis II holds, it follows from a repeated application of Lemma 2.5, that there exists a connected open neighborhood X 1 ⊂ X of s 0 , an analytic projection valued function P at on X 1 , and an analytic function E at on X 1 extending E at (s 0 ) such that the following holds. For all s ∈ X 1 the number E at (s) is in the discrete spectrum of H at (s) and it is non-defective, moreover For any s 1 ∈ X 1 the there exists an ǫ 1 > 0 and a neighborhood N 1 ⊂ X 1 of s 1 such that for all s ∈ N 1 {z ∈ C : |z − E at (s 1 )| = ε 1 } ⊂ ρ(H at (s 1 )) and P at (s) = − 1 2πi Henceforth, we denote by P at and E at any mappings having the properties stated above on an open connected neighborhood X 1 ⊂ X of s 0 .
Remark 2.6. In principle one could use Lemma 2.5 to obtain a maximal analytic extension of P at and E at . This will not be needed as it does not necessarily improve the main result.
To formulate the third Hypothesis, we use the notion of a reduced resolvent, which is introduced in Remark 2.7, below.
Remark 2.7. Let A : D(A) ⊂ X → X be a densely defined closed linear operator and let P be a bounded projection in X such for P = 1 − P Then it is reasonable to study the densely defined operator A| RanP ∩D(A) in RanP . If z ∈ ρ(A| RanP ∩D(A) ) we shall use the notation (A − z) −1 P := ((A − z)| RanP ∩D(A) ) −1 P , and refer to this expression as the reduced resolvent.
The third Hypothesis will be used to invert for z close to E at (s 0 ) the operator H at (s) −z when restricted to the range of P at (s) := 1 Hat − P at (s).
Aforementioned we formulate this in terms of the reduced resolvent. For this, we note that it follows from well known properties about projections (2.6), c.f. [27] or part (a) of Lemma B.1 in the appendix, that the assumptions (2.9), i.e., RanP at (s) is closed , RanP at (s) ∩ D(H at (s)) is dense in RanP at (s) H at (s) RanP at (s) ∩ D(H at (s)) ⊂ RanP at (s).
are satisfied for s ∈ X 1 . Thus the reduced operator H at (s)| P at (s)∩D(Hat(s))) is a densely defined operator in RanP at (s).
Hypothesis III. Hypothesis II holds and there exists a neighborhood U ⊂ X 1 × C of (s 0 , E at (s 0 )) such that for all (s, z) ∈ U we have |E at (s)−z| < 1/2, sup (s,z)∈U P at (s) < ∞, and Remark 2.8. We note that one can show that Hypothesis III follows from Hypothesis I and II and the additional assumption that H g (s) is an analytic family of type (A) and that a semiboundedness condition holds, see [15].
When dealing with the ground state, we can assume the following additional Hypothesis. It will ensure that in the limit, as the interaction strength tends to zero, the ground state of the interacting system converges to the ground state of the non-interacting system. For a subset Ω ⊂ C n we write Ω * := {z : z ∈ Ω}.
Hypothesis IV. The following holds.
(i) We have X = X * and for all s ∈ X the identities G 1,s = G 2,s and H at (s) * = H at (s) hold.
Definition 2.9. Let H 0 be a Hilbert space and let X ⊂ C d with X * = X. For each x ∈ X let a densely defined operator T (x) in the Hilbert space H 0 be given. We say that T is With theses Hypotheses at hand we can now state the main result. Then there exists a neighborhood X b ⊂ X of s 0 and a positive constant g b such that for all s ∈ X b and all g ∈ [0, g b ] the operator H g (s) has an eigenvalue E g (s) with d linearly independent eigenvectors ψ g,j (s), j = 1, ..., d, with the following properties.
(i) The functions s → E g (s) and s → ψ g,j (s) for j = 1, ..., d are analytic functions on X b .
If in addition Hypothesis IV holds, then X b = X * b and (iii) for all s ∈ X b ∩ R ν it holds that E g (s) = inf σ(H g (s)) , (iv) for all s ∈ X b it holds that E g (s) = E g (s).
Remark 2.11. In case that the irreducibility assumptions of Hypothesis II (iii) is not met the eigenspace of the ground-state eigenvalue is expected to split at higher order in perturbation theory. This phenomenon is known as the Lamb shift and has been considered in the literature [17,23]. It is natural to assume that degeneracies of eigenvalues are lifted at some order in perturbation theory until they are protected by a set of symmetries. Analyticity questions for degenerate ground-state eigenvalues which are lifted in second order perturbation theory where investigated in [20] in the framework of generalized Spin-Boson models.
We note that the above result can be used to obtain analyticity in the coupling constant. We note that this will immediately improve the continuity statement, Part (ii), in Theorem 2.10. This will be the content of the following corollary. To state the result first recall that W (s) is infinitesimally H f bounded, cf. Lemma C.1. Thus for each s ∈ X the map on C is an analytic family of type (A). It follows that (g, s) → H g (s) is an analytic family, since the weak analyticity of the resolvent implies strong analyticity of the resolvent and to show jointly weak analyticity we can use Hartog's theorem, cf. [21].

Corollary 2.12. Suppose Hypotheses I, II, III hold and let
Then there exists a neighborhood X b ⊂ X of s 0 and a positive constant g b such that for all s ∈ X b and all g ∈ B g b (0) the operator H g (s) has an eigenvalue E g (s) with d linearly independent eigenvectors ψ g,j (s), j = 1, ..., d, with the following property.
The functions (s, g) → E g (s) and (s, g) → ψ g,j (s) for j = 1, ..., d are analytic func- Proof. First we extend the parameter spaceX = X × B 1 (0) and define for (s, s ′ ) ∈X and g ≥ 0Ĥ (2.10) Now one easily verfies that (s, s ′ ) →Ĥ g (s, s ′ ) satisfies the assumptions I, II, III. Thus it follows from Theorem 2.10 that there exists a g b > 0 such thatĤ g b (s, s ′ ) has an eigenvalue E g b (s, s ′ ) and an eigenvector ψ g b (s, s ′ ) both depending analytically on (s, s ′ ). Now in view of (2.10) we see that they are also eigenvalue and eigenvector of H (s ′ g b ) (s). This shows the corollary.
We note that one can formulate the result in Theorem 2.10 in terms of so called eigenprojections.
A densely defined operator H in a Hilbert space with the property that for some antiunitary operator J is called complex-selfadjoint with respect to J . To formulate the next corollary we make another hypothesis.
Hypothesis V. Hypothesis II holds. For all g ≥ 0 and s ∈ X the operator H g (s) is complex-selfadjoint with respect to a antiunitary operator J . The bilinear form J : Corollary 2.13. Suppose Hypotheses I, II, III hold and let d = dim ker(H at (s 0 ) − E at (s 0 )). Assume that Hypothesis IV or Hypothesis V holds. Then there exists a neighborhood X b ⊂ X of s 0 and a positive constant g b such that for all s ∈ X b and all g ∈ [0, g b ] there exists a complex number E g (s) and a projection P g (s) with rank d such that with the following properties.
Proof. Let the situation be as in Theorem 2.10. First we assume that Hypothesis IV holds. By possibly restricting to the intersection of X b and X * b we can assume without loss that these sets are equal and nonzero, since both contain By linear independence of the ψ g,j (s) and continuity we can assume without loss that M is invertible for all s ∈ X b (by possible making X b smaller, by intersecting it with a neighborhood of the real line). We define It is straightforward to verify that this is a projection Furthermore, since ψ g,a are eigenvectors we find H g (s)P g (s) = E g (s)P g (s) and with Theo-rem 2.10 (iv) It is now straight forward using Parts (i) and (ii) of Theorem 2.10 that Parts (i) and (ii) of Corollary 2.13 hold. Now assume that Hypothesis V holds. In that case we argue analogously. Define the matrix N a,b (s) = J ψ a (s), ψ b (s) , a, b = 1, ..., d, for s ∈ X b . Again by linear independence of the ψ g,j (s) and Hypothesis V we find that N a,b (s) is invertible for s = s 0 and g = 0. Now by continuity in s and (ii) of Theorem 2.10 we can assume without loss that N is invertible for all s ∈ X b (by possible making X b as well as g b > 0 smaller). It is now again straightforward to verify using (i) and (ii) of Theorem 2.10 that has the claimed properties. To show the first relation in (2.12) we observe that using (2.11) we find

Symmetry Considerations
In this section we consider consequences of the symmetries which will be used for the renomormalization analysis. Elementary definitions and properties are collected in Ap-pendix A. First we discuss Schur's Lemma for symmetries of an operator. This will be needed to show that certain matrix valued vacuum expectations, occurring in the renormalization analysis, are multiples of the identity. Then we consider general properties of symmetries of analytic family of operators. We will apply these properties to the Hamiltonian defined in Section 2. As a main result, see Lemma 3.6, we will be able to assume without loss of generality that P at (s) is a constant function of s. Moreover, in Lemma 3.8 at the end of this section we prove a crucial property of the Feshbach operator which will be important later during the renormalization procedure.
Definition 3.1. Let V be a subspace of a Hilbert space H and let S be a set whose elements are unitary or antiunitary operators on H. We say that S ∈ S acts irreducibly on V if for The next two lemmas are versions of the well-known Lemma of Schur [29]. The first lemma is for self-adjoint operators. Since analytic continuations of the Hamiltonian are in general non-self-adjoint we need a second lemma for ordinary linear operators, as well. Then there exists a number λ ∈ R such that T = λ 1 V .
Proof. First observe that T has a real eigenvalue, say λ. Thus T − λ has a nonvanishing kernel. Now S leaves the space Ker(T − λ) invariant since λ is real. Thus by irreducibility we see that Ker(T − λ) = V . This yields the claim.
Now we want to extend the above lemma to non-self-adjoint operators. Then there exists a number λ ∈ C such that T = λ 1 V .
Proof. Note that there exits a unique decomposition with Y and Z self-adjoint operators on V. Then it follows from Eq. (3.1) that for S unitary/antiunitary The uniqueness of the decomposition (3.2) and Lemma A.2 (c) implies for all S ∈ S. Thus Z and Y are multiples of the identity by Lemma 3.2.
The next proposition will allow us to work with the constant projection P at (s 0 ) instead of the s dependent projection P at (s), by means of an invertible analytic family. This is a standard method used in analytic perturbation theory. The theorem below is a version of Theorem XII.12 in [27] incorporating in addition a symmetry property.
Theorem 3.4. Let H be a Hilbert space. Let P (s) ∈ L(H) be a projection-valued analytic function on a connected, simple connected region of the complex plane X. For s 0 ∈ X there exists an analytic family U(s) of bounded and invertible operators on X with the following properties: (c) If S is a symmetry of P (s), then one can choose U(s) to satisfy For the proof we use as in [27] the following lemma.
Lemma 3.5. Let R be a connected, simply connected subset of C with β 0 ∈ R and let A(β) be an analytic function on R with values in the bounded operators on some Banach space X . Then for any For a proof of the lemma we refer the reader to [27].
Proof of Theorem 3.4. The detailed proofs of (a) and (b) can be found in Theorem XII.12 of [27]. Here we merely give a sketch. Let Q(s) = P ′ (s)P (s) − P (s)P ′ (s), where P ′ (s) = d ds P (s). Then a calculation shows that We now use Lemma 3.5 with X = L(H). Let U(s) is the unique solution of the initial value problem and let V (s) be the unique solution of the initial value problem On the other hand if F = UV , then F solves the differential equation F ′ = [Q, F ] with initial condition F (s 0 ) = 1. Since F = 1 solves the same initial value problem it follows by uniqueness that It follows that U is invertible.
By uniqueness of the initial value problem, Lemma 3.5, we conclude SU(s)S * = U(s).
Now let us suppose that S is an antiunitary symmetry of P (s). Then we have by assumption SP (s)S * = P (s) * , and hence taking the adjoint we find SP (s) * S * = P (s). Differentiating we find d ds P (s) = S d ds P (s) * S * . A calculation now shows that where we used (3.8) in the last identity. Now from (3.5) we conclude by uniqueness of the initial value problem, Lemma 3.5. Since V (s) = U(s) −1 , by (3.6) and (3.7), the identity in (c) for antiunitary symmetries is now also shown.
Next we shall give a proof of Lemma 2.5 about the eigenprojection of P at stated in the introduction.
Proof of Lemma 2.5. By Hypothesis II(ii) we can pick ǫ > 0 such that the only point of is open (Theorem XII.7 in [27]), we can find a δ > 0 so that z ∈ ρ(H at (s)) if |z −E at (s 0 )| = ǫ and |s − s 0 | ≤ δ. Thus (a) holds for the set In case p at (s 0 ) = 1, we can use that the dimension of the projection is constant, i.e. , dim Ranp at (s) = dim Ranp at (s 0 ) = 1. In that case (3.9) now follows since H at (s) leaves the range of p at (s) invariant. In case p at (s 0 ) > 1 we will use the symmetry property of Hypothesis II (iii). Since S 1 is a symmetry of H at (s) it follows from the integral representation (2.7) that it is also a symmetry of p at (s). By Theorem 3.4 there exists an analytic family U(s) for s ∈ N of bounded invertible operators satisfying the assertions of Theorem 3.4 for the projection p at (s). In particular, and for antiunitary S ∈ S 1 that Thus by the Lemma of Schur and the irreducibility condition of Hypothesis II (iii), there exists a function e at : N → C such that H at (s)p at (s 0 ) = e at (s)p at (s 0 ).
By (3.10) this implies H at (s)p at (s) = e at (s)p at (s), for all s ∈ N, i.e., (3.9). Now the analyticity of e at (s) follows from the analyticity of p at (s) and H at (s) and by calculating an inner product with a nonzero vector in the range of p at (s). Furthermore, it follows from (a) and Theorem B.1 (c) that for all s ∈ N we have This and (3.9) imply that for s ∈ N the point e at (s) ∈ C is the only point in the spectrum of H at (s) in B ǫ (E at (s 0 )). Thus e at (s) is isolated from the rest of the spectrum. Furthermore it follows, by deforming the contour and Cauchy's theorem that for Thus (3.9) implies that the number e at (s) is a non-defective, discrete element of the spectrum of H at (s). Finally, it follows for s = s 0 from the definition of p at (s) and (3.9) that e at (s 0 ) = E at (s 0 ).
In Lemma 3.6, below, we show that in the proof of the main theorem, Theorem 2.10, we can assume without loss of generality that the following Hypothesis holds.
Hypothesis VI. Hypothesis II holds and P at (s) = P at (s 0 ) for all s ∈ X.
Lemma 3.6. Theorem 2.10 holds, if its assertion holds under the additional Assumption of Hypothesis VI.
Proof. Suppose that Hypotheses I, II, and III hold for some s 0 ∈ X and some symmetry group S. By restricting to a smaller neighborhood of s 0 we can assume without loss of generality that X is open, connected, simply connected. Then by Theorem 3.4 there exists an analytic family U(s) of bounded invertible operators on X such that We now defineĤ Thus if G j,s satisfy Hypothesis I, then alsoĜ j,s satisfies Hypothesis I on any subset X 0 ⊂ X on which U(s) and its inverse are uniformly bounded operator valued functions (by continuity any bounded open X 0 with closure contained in X will work). By analyticity of U(s) it follows thatĤ at (s) is an analytic family in the sense of Kato, and hence part (i) of Hypothesis II holds. NowĤ at (s) satisfies part (ii) of Hypothesis II by the invertibility of U(s). Next we consider part (iii) of Hypothesis II. Since by assumption S 1 is a symmetry group for H at (s) it follows from the integral representation of P at (s), cf. (2.8), that it is also a symmetry of the latter. ThusĤ at (s) satisfies also Part (iii) of Hypothesis II. Similarly one shows thatŴ (s) satisfies Part (iii) of Hypothesis II. Finally, if H at (s) satisfies Hypothesis III, then by invertibility of U(s) alsoĤ at satisfies Hypothesis III on any subset X 0 ⊂ X on which U(s) and its inverse are uniformly bounded operator valued functions. Thus we have shown thatĤ g (s) satisfies Hypothesis I, II, and III on an open set X 0 containing s 0 . Furthermore, Hypothesis VI holds forĤ g (s) by construction. Thus by assumption the assertion of the main result, Theorem 2.10, holds for the operatorĤ g (s). We conclude that there exists a neighborhood X b ⊂ X 0 of s 0 and a positive constant g b such that for all g ∈ [0, g b ) and s ∈ X b the operatorĤ g (s) has an eigenvalueÊ g (s) with d := dim ker(Ĥ at (s 0 ) − E at (s 0 )) = dim ker(H at (s 0 ) − E at (s 0 )) linearly independent eigenvectorŝ ψ g,j (s), j = 1, ..., d, all depending analytically on s ∈ X b . By the invertibility of U(s) we see that the operator H g (s) has the eigenvalue E g (s) :=Ê g (s) with d linearly independent eigenvectors ψ g,j (s) := (U(s) ⊗ 1)ψ g,j (s), j = 1, ..., d. They also depend analyticaly on s, since U(s) and its inverse depend by Theorem 3.4 analytically on s. This shows (i) of Theorem 2.10. Similarly one verifies (ii) of Theorem 2.10 by using the uniform boundedness of U(s) and U(s) −1 . Finally, suppose that the operator H g (s) satisfies Hypothesis IV. Then by Theorem 3.4 (b) we can choose the family of invertible operators U(s) to be unitary for real s such that U(s) * = U(s) −1 for all s ∈ X. Thus alsoĤ g (s) satisfies Hypothesis IV and moreover it is isospectral to H g (s) for real s. In that case we have for real s ∈ R ν ∩ X b that E g (s) =Ê g (s) = inf σ(Ĥ g (s)) = inf σ (H g (s)).
This implies (iii) of Theorem 2.10.
Thus we have shown that the assertion of Theorem 2.10 also holds for the original operator H g (s).
The next lemma will be used to show that the so called relevant direction in the renormalization analysis is one dimensional. For this, let us introduce the following definition. For V a finite dimensional complex vector space and a bounded operator T ∈ B(V ⊗ F ) define T Ω as the unique operator on V such that for all v 1 , v 2 ∈ V . Note that it is straight forward to see that Lemma 3.7. Let V be a finite dimensional complex vector space and let T ∈ B(V ⊗ F ). Assume that T is symmetric with respect to a set of symmetries S such that every element can be written in the form S 1 ⊗S 2 , where S 1 is a symmetry in V and S 2 is a symmetry in F leaving the Fock vacuum invariant. Assume that S 1 := {S 1 : S 1 ⊗ S 2 ∈ S} acts irreducibly on V . Then there exists a number c ∈ C such that Proof. For all S 1 ⊗ S 2 ∈ S we have the following symmetry property. For A an operator or a number let A # stand for A or A * whether the symmetry S 1 ⊗ S 2 is unitary or antiunitary, respectively. Moreover, we write where in the last line we used (3.12). Thus The claim now follows from Schur's Lemma 3.3 and the irreduciblity assumption.
To conclude this section we show that the Feshbach transformation preserves symmetry properties. A detailed review of the properties of the Feshbach-Schur map, which was introduced in [3] is given in Appendix D. Proof. This follows from the definition of the Feshbach operator given in Eq. (D.1). Let S ∈ S be a symmetry and let A # stands for A or A * if S is unitary or antiunitary, respectively. Then inserting S * S = 1, we find

The initial Hamiltonian
The first step of the operator-theoretic renormalization analysis is to prove that H g (s) and H 0 (s) are a Feshbach pair for a suitable choice for the projection operator, see (4.3) below. This is the content of Theorem 4.1. For a definition as well as the properties of Feshbach pairs we refer to Appendix D. Moreover we will show in this section, that the associated Feshbach operator, cf. (D.1), is an analytic function of s and the spectral parameter z and that it inherits the symmetry property of the original operator. This will be shown in Theorem 4.7.
We choose smooth functions χ, χ ∈ C ∞ (R; [0, 1]) such that χ 2 + χ 2 = 1 and The following theorem gives us the conditions for which we can define the so called first Feshbach operator.
Proposition 4.1. Suppose Hypothesis I, II, and III hold, and let U ⊂ X 1 × C be given by Hypothesis III. Then there is a g b > 0 such that for all g ∈ [0, g b ) and all (s, z) ∈ U, the pair (H g (s) − z, H 0 (s) − z) is a Feshbach pair for χ(s). Furthermore one has the absolutely convergent expansion on U For the proof of this proposition we make use the following lemma.

5)
and sup (s,z)∈U Proof. We recall that by definition, cf. Eq. (4.2) and (4.3) , χ(s) = P at (s) ⊗ 1 + P at (s) ⊗ χ(H f ). First we estimate (4.6). Applying the triangle inequality we obtain (4.8) We estimate (4.8) by the spectral theorem and find where the right hand side is finite by Hypothesis III. To estimate (4.7) we use again the spectral theorem and find where the last bound follows from Hypothesis III. This shows (4.6). Next we similarly show (4.5). Using the triangle inequality, we find We obtain for the second term in (4.9) by the spectral theorem where the right hand side is again finite by Hypothesis III. To estimate the first term in (4.9) we use again the spectral theorem and find from Hypothesis III This completes the proof.
Now we are ready to prove Proposition 4.1. We will use the following notation. Note that H at (s) leaves the ranges of P at (s) and P at (s) invariant, cf. Theorem B.1. Thus by the spectral theorem H 0 (s) leaves the range of P at (s) ⊗ 1 invariant. Moreover where we used that by Hypothesis III we have |E at (s) − z| < 1/2 and the last inquality of (4.12). On the other hand by the spectral theorem and Hypothesis III we find In particular, for normalized ϕ ∈ D(H at (s))⊗D(H f ) we obtain using the triangle inequality together with (4.12) and (4.13) (4.14) Combining (4.10) and (4.14) we see that, for all φ ∈ D(H at (s)) ⊗ D(H f ) and ǫ > 0 with constants C 0 , C 1 , C 2 . This shows that W (s) is infinitesimally bounded with respect to H 0 (s) and thus we have shown that H g (s) = H 0 (s) + gW (s) is closed on D(H 0 (s)) for all g > 0.
Remark 4.5. We note that if Hypothesis II holds, then it is straight forward to see using (2.8) that χ ρ and χ ρ commute with the group of symmetries S given by Hypothesis II (iii).
Provided the right hand side exists, i.e. the Feshbach pair property holds, cf. Proposition 4.1, we define the so called first Feshbach operator Note that by the choice of the projection χ(s) it follows that (4.15) and (4.16) leave the range of P at (s) ⊗ 1 H f ≤1 invariant. Furthermore, we define the following restrictions, which are for the isospectrality property sufficient to study, cf. Theorem D.2, Note that as operators acting on the range of P at (s) ⊗ 1 H f ≤1 we have We shall refer to (4.19) as the first Feshbach operator as well. Henceforth we shall assume Hypothesis VI and so H Remark 4.6. Note that the notation introduced in (4.15) -(4.18) is similar to the one in [15] but not exactly the same.
In the following theorem we show that the first Feshbach operator H (0) g [s, z] is analytic on a suitable subset of X × C. Moreover we show that this operator is isospectral to H g (s) − z, in the sense of Theorem D.2. Furthermore the first Feshbach operator commutes with the set of symmetries S from Hypothesis II. Note that in the theorem below we make use of the auxiliary operator Q χ defined in Eq. (D.2). Theorem 4.7. Suppose Hypothesis I, II, and III hold, and let U ⊂ X 1 × C be given by Hypothesis III. Then there is a g b > 0 such that for all g ∈ [0, g b ) and all (s, z) ∈ U, the pair (H g (s) − z, H 0 (s) − z) is a Feshbach pair for χ(s) and the following holds on U.    In addition, if Hypothesis IV is valid, we have for (s, z) ∈ U ∩ U * that g (s, z) will follow provided (s, z) → W (0) g (s, z) is analytic. Since that function can be obtained by a restriction to a subspace of the function (s, z) →W (0) g (s, z) the analyticity of the former will follow from the analycity of the latter. To show that the latter is analytic we use the absolutely convergent expansion given in (4.4), which is granted by Proposition 4.1. Since absolutely convergent sequences of analytic functions have an analytic limit, it remains to show that each summand in the following series is analytic in s and z where in the last equality we used associativity of composition and that H f commutes with χ(s) and χ(s). First observe that by Lemma 4.8, W (s)(H f + 1) −1 is analytic. Hence to establish analyticity of (4.20) it remains to prove analyticity of To this end, we observe that from the definition of χ(s) we can write The analyticity of the second term in (4.21) follows by means of the spectral theorem from the fact that for every r ≥ 0 the function (s, z) → (r + 1)(E at (s) + r − z) −1 χ 1 (r) is analytic on U (by Hypothesis III we have on U that |E at (s) − z| < 1/2 and so the denominator does not vanish for r ≥ 0 for which χ 1 (r) = 0) and is uniformly bounded in r ≥ 0. The analyticity of the first term on the r.h.s of (4.21) follows by means of the spectral theorem from the fact that the function (s, z) → (r+1)(H at (s)+r−z) −1 P at (s) is bounded uniformly in r ≥ 0 by the estimate in Hypothesis III and for every r ≥ 0 the function is analytic on U by Proposition B.2. This concludes the proof that H Let us now show Part (f ). First observe that without loss the neighborhood X 1 ⊂ X of s 0 on which P at is defined satisfies X * 1 = X 1 (otherwise take the intersection of the two sets). Now for s ∈ R ∩ X 1 close to s 0 we find from (2.8) with s 1 = s 0 and E at (s 0 ) ∈ R using Hypothesis IV (i), that P at (s) * = P at (s) (4.22) Since both sides of (4.22) are analytic functions of s on X 1 , we conclude that (4.22) holds for all s ∈ X 1 (cf. the unique continuation property of analytic functions, e.g. [21]). for all s ∈ X. Now we recall that for any densely defined, closed operator A in H and z ∈ ρ(A) we find z ∈ ρ(A * ) and This follows directly from [32,Theorem 4.17(b)] as is shown in the proof of Theorem 5.12 in [32]. Using the fact thatH g (s, z) leaves the range of P at (s 0 ) ⊗ 1 H f ≤1 invariant we find for (s, z) ∈ U ∩ U * that where the second to last identity can be seen by taking the adjoint of (4.4) and using (4.22), (4.23), and (4.24).

Banach Space of Hamiltonians
To control the renormalization transformation, in particular proving its convergence, it is convenient to introduce suitable Banach spaces of integral kernels, cf. [3,15]. A generalization to matrix-valued integral kernels is a canonical choice to accommodate degenerate situations. In this section we follow closely the definition and notation given in [15]. The renormalization transformation is defined on a subset of L(H red ) that will be parameterized by vectors of a Banach space W ξ = ⊕ m,n≥0 W m,n . We begin with the definition of this Banach space.
Let L(C d ) denote the space of linear maps A from C d to C d equipped with the operator norm A op := sup{|Ax| : |x| ≤ 1}. The Banach space W 0,0 is the space of continuously differentiable functions where w ′ (r) := ∂ r w(r). For m, n ∈ N with m + n ≥ 1 and µ > 0 we set That is, W m,n is the space of measurable functions w m,n : B m+n → W 0,0 that are symmetric with respect to all permutations of the m arguments from B m and the n arguments from B n , respectively, such that w m,n µ is finite. We note that the notation · µ introduced in (5. 2) also appears in (2.5). Which of the definitions is meant should be clear from the context. For given ξ ∈ (0, 1) and µ > 0 we define a Banach space The formal definition of the operator valued distributions a * (k) and a(k) in (5.3) can be found in Appendix C. By the continuity established in the following proposition, the mapping w → H(w) has a unique extension to a bounded linear transformation on W ξ .
In particular, the mapping w → H(w) is continuous.
(iii) When restricted to Proof. Statement (ii) follows immediately from the triangle inequality and (i) since ξ ≤ 1. For (i) we refer to the proof of [3], Theorem 3.1. which generalizes trivially to C d with d ≥ 1 from d = 1.
(iii) For a proof see the proof of [19,Theorem 5.4], which generalizes straight forward to C d .

First Transformation
In the following we denote by d = dim RanP at (s 0 ) (6.1) the dimension of the eigenspace corresponding to the eigenvalue E at (s 0 ) of H at (s 0 ).
Theorem 6.1. Suppose Hypothesis I holds for some µ > 0 , Hypothesis II holds, Hypothesis III holds for some U ⊂ C ν × C, and Hypothesis VI holds. Then, for all ξ ∈ (0, 1) and arbitrarily positive constants α 0 , β 0 and γ 0 , there exits a positive constant g 1 such that for all g ∈ [0, g 1 ) and all (s, z) ∈ U, (H g (s) − z, H 0 (s) − z) is a Feshbach pair for χ(s), and Proof. Using Proposition 4.1 we directly obtain that the Feshbach property is satisfied for sufficiently small g. Hence to prove the theorem it remains to construct a sequence of integral kernels w ∈ W ξ such that H Remark 6.2. We note that a result for matrix-valued integral kernels similar as in Theorem 6.1 can be found with a detailed proof in [20].

RG Transformation
By abuse of notation we shall denote the following operators on H red again by χ ρ and χ ρ , respectively, recalling the notation (6.1). It should be clear from the context which of the expressions is considered.
The proof of the lemma follows from a straight forward generalization of the proof given in Lemma 15 in [15]. Moreover a similar proof can be found in [12].

This implies the bounded invertibility of
The other conditions on a Feshbach pair are now also satisfied, since H(w) − H 0,0 (w) is bounded on H red .
The renormalization transformation we use is a composition of a Feshbach transformation and a unitary scaling that puts the operator back on the original Hilbert space H red . Unlike the renormalization transformation of Bach et al [3], there is no analytic transformation of the spectral parameter.
Given ρ ∈ (0, 1), let H ρ = 1 C d ⊗Ranχ(H f ≤ ρ). Let w ∈ W ξ and suppose (H(w), H 0,0 (w)) is a Feshbach pair for χ ρ . Then F χρ (H(w), H 0,0 (w)) : H ρ → H ρ is iso-spectral with H(w) in the sense of Theorem D.2. In order to get a isospectral operator on H red , rather than H ρ , we use the linear isomorphism introduced in (2.1). Note that Γ ρ H f Γ * ρ = ρH f , and hence Γ ρ χ ρ Γ * ρ = χ 1 . The renormalization transformation R ρ maps bounded operators on H red to bounded linear operators on H red and is defined on those operators H(w) for which (H(w), H 0,0 (w)) is a Feshbach pair with respect to χ ρ . Explicitly, which is a bounded linear operator on H red . The following theorem describes the action of the renormalization transformation on the polydiscs B(α, β, γ). For its statement we recall the notation (3.11).

Renormalization preserves analyticity and symmetry
In this section we show that the renormalization transformation preserves analyticity, symmetry with respect to a group of symmetries S and reflection symmetry. We study these properties on the level of the operators. In principle one could also study the symmetry property on the level of the integral kernels.
In [15,Proposition 17], Griesemer and Hasler proved that analyticity is preserved under renormalization. The following proposition is a straight forward generalization of their result.
Proof. Follows from [15,Proposition 17] and an obvious change of notation to accommodate the matrix valued integral kernels.
The property in Proposition 8.1 together with Proposition 8.2, below, will be one of the main ingredients in the proof of part (i) of Theorem 2.10.
Proposition 8.2. Let X be an open subset of C ν+1 with ν ≥ 0. Assume that for each σ ∈ X we are given an operator H(w σ ) in the polydisc B(α, β, γ).
(a) Let S be a group of symmetries acting on H red leaving the Fock vacuum and the one particle subspace invariant. Assume that it commutes with Γ ρ and H f . Let σ ∈ X.
Suppose that H(w σ ) is symmetric with respect to S.
(i) Then H 0,0 (w σ ) is symmetric with respect to S.
Proof. We first show how one can recover w 0,0 (r) from H(w). We follow the argument in We pick a function f ∈ C ∞ c (B 1 ; [0, ∞) with´|f (x)| 2 dx = 1, and define f ǫ,k := ǫ −3/2 f (ǫ −1 (x − k)). Then we find from (8 This term tends to v 1 , w 0,0 (|k|)v 2 since On the other hand we find from (8 This term tends to 0, because f ǫ,k → 0, weakly in L 2 (B 1 ). Thus from (8.1) -(8.6) we conclude using that w 0,0 is continuous that (a) Since this part does not depend on σ we drop it in the notation. Now since S ∈ S 2 leaves the one photon space invariant, there is a map p 1 (S) such that If S is unitary or antiunitary, it follows that p 1 (S) is unitary or antiunitary, respectively. Now let S = S 1 ⊗ S 2 ∈ S by a symmetry. If S is unitary we write (·) # = (·) and if it is antiunitary we write (·) # = (·) * . Thus we find from (8.7) that where in (8.8) we made use of (8.1), (8.3) and the fact that p 2 (S * )f k,ǫ converges to zero. In (8.9) we used that H f is symmetric with respect to S 2 . In the last line we used (8.6) and (8.5). We conclude that S 1 w 0,0 (r)S * 1 = w 0,0 (r) for all r ∈ [0, 1]. This shows part (i) of (a). This shows (i).
(ii) Then from (i) we know that H 0,0 (w) is symmetric with respect to S. Thus it follows that also W := H(w) − H 0,0 (w) is symmetric. Now the claim for the Feshbach operator follows from Lemma 3.8. Since the symmetry commutes with dilations the claim follows also for the renormalized expression.
(b) Suppose now X = X * and σ → H(w σ ) is reflection symmetric. Then by (8.7) it follows that Thus for r ∈ [0, 1] we find w σ 0,0 (r) = w σ 0,0 (r) * . This shows part (i) of (b). To show (ii) we write T σ = H 0,0 (w σ ) and observe that W σ = H(w σ ) − T (w σ ) is also reflection symmetric as well as χ = χ ρ . We find This shows the claim for the Feshbach operator. Since the symmetry commutes with dilation the claim follows also for the renormalized expression.

Iterating the Renormalization Transformation
In this section we follow closely, Section 8 in [15], and generalize the results given there to the non-degenerate situation. In particular the two lemmas stated below are almost identical to the main results stated in Lemma 18, Lemma 19, Corollary 20, and Proposition 21 of [15].
In Part (c) of Theorem 4.7 we have reduced, for small |g|, the problem of finding an eigenvalue of H g (s) in the neighborhood U 0 (s) := {z ∈ C : (s, z) ∈ U} of E at (s) to finding an z ∈ C such that H (0) [s, z] has a non-trivial kernel. We now use the renormalization map to define a sequence of operators on H red , which, by Theorem D.2, are isospectral in the sense that KerH (n+1) [s, z] is isomorphic to KerH (n) [s, z]. The main purpose of the present section is to show that for every n ∈ N the operator H (n) [s, z] is well-defined for all z in a non-empty set U n (s) with the following properties. We have U n+1 (s) ⊂ U n (s) and In Section 9 we will show that H (n) [s, z ∞ (s)] has a non-trivial kernel and hence z ∞ (s) is an eigenvalue of H g (s). The construction of the sets U n (s) is based on Theorem 4.7 and Theorem 6.1, but not on the explicit form of H (0) [s, z] as given by (4.17). Moreover, this construction is pointwise in s and g, all estimates being uniform in s ∈ X and |g| < g b for some g b > 0. We therefore drop these parameters from our notations and we now explain the construction of H (n) [z] making only the following assumption: for n = 0, . . . , N − 1. They are obviously satisfied for all n ∈ N if C γ ρ µ < 1 and if β 0 , γ 0 are sufficiently small. If this is the case we define Since the renormalization transformation R ρ preserves the symmetry by Proposition 8.2, it follows by induction from Assumption (A) that each H (n) [z] is symmetric with respect to the elements of S. Since the symmetries leave the vacuum invariant it follows from Lemma 3.7 that the linear map T (n) 0 (z) is multiple of the identity. That is, there exists a function for n = 0, . . . , N − 1. Since |E (n) (z)| = T (n) 0 (z) op this is achieved by adjusting the admissible values of z step by step. We define recursively, for all n ≥ 1, If z ∈ U N , H (0) (z) ∈ B(∞, β 0 , γ 0 ), and ρ, β 0 , γ 0 are small enough, as explained above, then the operators H (n) (z) for n = 1, . . . , N are well defined by (9.1). In addition we know from Theorem 7.2 that H (n) (z) ∈ B(∞, β n , γ n ), and that This latter information will be used in the proof of Lemma 9.2 to show that the sets U n are not empty. The subsequent lemma is a summary of the above construction.
(a) For n ≥ 0, E (n) : U n → C is analytic in U • n and a conformal map from U n+1 onto D ρ/2 . In particular, E (n) has a unique zero, z n , in U n . Moreover, (b) The limit z ∞ := lim n→∞ z n exists and for ǫ : Then there exists an a < z ∞ such that has a bounded inverse for all x ∈ (a, z ∞ ).
Proof of Lemma 9.2. The Lemma follows as a consequence of Lemma 9.1 and the property of the Feshbach map, cf. Theorem D.2. The details of the proof are the same as the proofs of Lemma 19, Corollary 20, and Proposition 21 in [15].
Let us now discuss the construction of an eigenvector ϕ (0) such that H (0) [z ∞ ]ϕ (0) = 0. The same construction has been used in [3,5,6,15]. The result which we use is from [15]. In order to formulate the result we define the following auxiliary operator for z ∈ U n where W (n) [z] and H

Analyticity of Eigenvalues and Eigenvectors
This section is devoted to the proof of Theorem 2.10. It is essential for this proof, that a neighborhoods V 0 ⊂ V of s 0 and a positive bound, g 1 , on g can be determined in such a way that the renormalization analysis of Sections 9, and in particular the choices of ρ and ξ are independent of s ∈ V 0 and g ≤ g 1 . Once V 0 and g 1 are found, the assertions of Theorem 2.10 are derived from Proposition 8.1 and 8.2 as well as the uniform bounds of Sections 9.
Proof of Theorem 2.10. First let us recall that by Lemma 3.6 we can assume without loss that Hypothesis VI holds and P at (s) = P at (s 0 ) for all s ∈ X. Furthermore by choosing a suitable basis we can assume that RanP at (s 0 ) = C d .
The Theorem now follows for E g (s) = z ∞ (s).
If we neglect the first Feshbach map in the above proof, we obtain the following theorem, which is independent of the explicit structure of the Hamiltonian.

A Symmetries
In this section we introduce anti-linear operators and symmetries in a Hilbert space H. for all x, y ∈ H.
In the following lemma we collect a few properties of anti-linear and antiunitary operators.
Lemma A.2. Let H be a complex Hilbert space. Then the following holds.
In that case, we also say that T is symmetric or invariant with respect to S.
We note that it is elementary to show that the set of symmetries of an operator form a group.
Lemma A.4. Let H be a complex Hilbert space. Then the set of symmetries of an operator in H form a group.
Proof. If S 1 and S 2 are symmetries, then we see from Lemma A.2 (c), (d), and (e) that also S 1 S 2 and S −1 1 are symmetries.

B Eigenprojections and their properties
In this appendix we recall well-known properties about isolated points of the spectrum. For a detailed treatment we refer the reader to the discussion in [27] surrounding Theorems XII.4 and XII.5.
(ii) If |z − λ| > r, then (A − z)| RanP is invertible and (d) If λ is an isolated element of the spectrum σ(A) its algebraic multiplicity is greater or equal to its geometric multiplicity.
For G ∈ L 2 (R 3 × Z 2 ; L(H ′ )) the creation operator a * (G) is by defnition the adjoint of a(G), cf. (2.2). The domain of the creation operator contains the so called finite particle vectors ψ = (ψ n ) ∞ n=0 ∈ H ′ ⊗ F with the property that ψ n = 0 for all but finitely many n, and a * (G)ψ is a sequence of H ′ -valued measurable functions such for n-th term [a * (G)ψ] n (k 1 , ...., k n ) = n −1/2 n j=1ˆG (k j )ψ n−1 (k 1 , ..., k j , .., k n )dk, (C. 1) where means that this variable is to be omitted and the integral on the right hand side is defined as a Bochner integral. A straight forward calculation using (2.2) and (C.1) shows that on finite particle vectors we have the commutation relations [a(F ), a * (G)] =ˆF * (k)G(k)dk, [a(F ), a(G)] = 0, [a * (F ), a * (G)] = 0, which extend to their natural domains. Next we express the creation and annihilation operator in terms of so called operator valued distributions, a * (k) and a(k). For an element ψ ∈ H ′ ⊗ F we define a(k)ψ for a.e. k ∈ R 3 × Z 2 as the sequence of H ′ -valued measurable functions such that the n-th term satisfies a.e.
Using (C.2) we can express the free field energy in terms of the following identity on vectors ϕ, ψ ∈ D(H f ) ϕ, H f ψ =ˆω(k) a(k)ϕ, a(k)ψ dk.
Lemma C.2. Let f : R + → C be a bounded measurable function. Then for all k ∈ R 3 × Z 2 a(k) f (H f ) = f (H f + ω(k)) a(k) .
In order to define field operators that depend on the free field energy we consider measurable functions w m,n on R + × X n+m with values in the bounded linear operators of H ′ . To such a function we associate the sesquilinear form q wm,n (ϕ, ψ) :=X m+n a(k (m) )ϕ, w m,n (H f , K (m,n) ) a(k (n) )ψ dK (m,n) , (C.5) defined for all ϕ and ψ in H ′ ⊗F , for which the integrand on the right hand side is integrable.
Here the r.h.s. of (C.5) is defined by means of an interated application of (C.2). If the integral kernel w m,n has sufficient regularity and decay, one can show that the sesquilinear form (C.5) defines a closed linear operator which we denote bŷ X m+n a * (k (m) ) w m,n (H f , K (m,n) ) a(k (n) )dK (m,n) . (C. 6) In particular, in the case where w m,n ∈ W m,n , cf. Then for all finitely many particle vecotors ϕ, ψ ∈ H ′ ⊗ F | q wm,n (ϕ, ψ) | ≤ w m,n ♯ ϕ ψ .

(C.7)
If w m,n ♯ < ∞, the form q wm,n determines uniquely a bounded linear operator h wm,n such that q wm,n (ϕ, ψ) = ϕ, h wm,n ψ , for all ϕ, ψ in H ′ ⊗ F and h wm,n ≤ w m,n ♯ .