Spectral Analysis of Relativistic Atoms -- Dirac Operators with Singular Potentials

This is the first part of a series of two papers, which investigate spectral properties of Dirac operators with singular poten- tials. We examine various properties of complex dilated Dirac oper- ators. These operators arise in the investigation of resonances using the method of complex dilations. We generalize the spectral analysis of Weder (50) and y (46) to operators with Coulomb type poten- tials, which are not relatively compact perturbations. Moreover, we define positive and negative spectral projections as well as transforma- tion functions between different spectral subspaces and investigate the non-relativistic limit of these operators. We will apply these results in (30) in the investigation of resonances in a relativistic Pauli-Fierz model, but they might also be of independent interest.


Introduction and Definitions
A fascinating question in the mathematical analysis of operators describing atomic systems is the fate of eigenvalues embedded in the continuous spectrum if a perturbation is "turned on".Typically, these eigenvalues "vanish" and one has absolutely continuous spectrum.But the eigenvalues leave a trace: For example, the scattering cross section shows bumps near the eigenvalues, or certain states with energies close to the eigenvalues have an extended lifetime (described by the famous "Fermi Golden Rule" [13, Equation (VIII.2),p. 142] on a certain time scale).These energies are called resonances or resonance energies.Mathematically, resonances are described by poles of a holomorphic continuation of the resolvent (or matrix elements of it) or the scattering amplitude to a second sheet.The generic systems in which resonances occur are many-particle systems.This can be many-electron systems, in which the electron-electron interaction is the perturbation.The corresponding physical effect is called "Auger effect": Excited states ("autoionizing states") relax by emission of electrons.Another typical system in a one-or many-electron atom interacting with the quantized electromagnetic field, in which case excited states can relax by emitting photons.Resonances can also occur in one-particle systems, although this is not typically the case.It is well known (see [8] for example) that for a Schrödinger operator with Coulomb potential the set of resonances is empty.During the last decades numerous results were obtained in the mathematical investigation of resonances so that it seems hopeless to give a complete account of the available literature.Nevertheless we would like to give an overview and mention at least some of the relevant works.
The investigation of resonances as poles of holomorphic continuations of scattering amplitude and resolvent goes back to Weisskopf and Wigner [53] and Schwinger [45].The mathematical theory of resonances was pushed further by Friedrichs [14], Livsic [36], and Howland [27,28].One of the mathematical methods in the spectral analysis is the method of complex dilation, which associates the "vanished" embedded eigenvalue with a non-real eigenvalue of a certain non-selfadjoint operator and was investigated by Aguilar and Combes [2] and Balslev and Combes [6] (see [43] for an overview).Resonances in the case of the Stark effect were investigated by Herbst [24] and by Herbst and Simon [25].Simon [48] initiated the mathematical investigation of the timedependent perturbation theory.This was carried on by Hunziker [32].Herbst [23] proved exponential temporal decay for the Stark effect.The spectral analysis of non-relativistic atoms in interaction with the radiation field was initiated by Bach, Fröhlich, and Sigal [4,5].It was carried on by Griesemer, Lieb und Loss [18], by Fröhlich, Griesemer und Schlein (see for example [15]) and many others (see for example Hiroshima [26], Arai and Hirokawa [3], Dereziński and Gérard [9], Hiroshima and Spohn [12]), Loss, Miyao and Spohn [37] or Hasler and Herbst [21,20]).In particular, Bach, Fröhlich, and Sigal [5] proved a lower bound on the lifetime of excited states in nonrelativistic QED.Later, an upper bound was proven by Hasler, Herbst, and Huber [22] (see also [29]) and by Abou Salem et al. [1].Recently, Miyao and Spohn [38] showed the existence of a groundstate for a semi-relativistic electron coupled to the quantized radiation field.Our overall aim is to show that the lifetime of excited states of a relativistic one-electron atom obeys Fermi's Golden Rule [30] and coincides with the non-relativistic result in leading order in the fine structure constant.We will investigate the necessary spectral properties of a Dirac operator with potential, projected to its positive spectral subspace, coupled to the quantized radiation field.Following Bach et al. [5] and Hasler et al. [22], our main technical tool is complex dilation in connection with the Feshbach projection method.
In this first part of the work, we investigate the necessary properties of oneparticle Dirac operators with singular potentials.In particular, we will derive the necessary properties of complex dilated spectral projections and discuss the non-relativistic limit of complex dilated Dirac operators.This serves mainly as a technical input for the second part of our work [30].However, we believe that some of the results presented in the first part are also of independent interest.Note that the method of complex dilation has already successfully been applied to Dirac operators (see Weder [50] and Šeba [46]).However, these authors assume the relative compactness of the electric potential so that their method does not apply to Coulomb type potentials.Note moreover that Weder [51] considers very general operators including relativistic spin-0-Hamiltonians with potentials with Coulomb singularity.The basic assumption of this work is, however, that the unperturbed operator is sectorial, which is not fulfilled for the Dirac operator.Our results cover a class of Dirac operators which includes Coulomb and Yukawa potentials (with exception of Lemma 11 and Lemma 12 which we prove for the Coulomb case only).Our results about the spectral projections of the dilated Dirac operator can be used to generalize the Douglas-Kroll transformation (see Siedentop and Stockmeyer [47] and Huber and Stockmeyer [31]) to dilated operators.

Definitions and Overview
The free Dirac operator (with velocity of light c > 0) is an operator on the Hilbert space H := L 2 (R 3 ; C 4 ).It is self-adjoint on the domain Dom(D c,0 ) := H 1 (R 3 ; C 4 ) [49, Chapter 1.4].Here α is the vector of the usual Dirac α-matrices, and β is the Dirac β-matrix.
We define for ǫ > 0 the strip S ǫ := {z ∈ C||Im z| < ǫ}.Let χ : R 3 → R a bounded, measurable function.We will suppose that there is a Θ > 0 such that θ → χ(e θ x) admits a holomorphic continuation to θ ∈ S Θ for all x ∈ R 3 .We abbreviate χ θ := χ(e θ •).We will need the following two properties at different places: sup type (A) in the sense of Kato (see Theorem 2).Moreover, we provide a spectral analysis of such operators in Theorem 3. Just as in the case of Schrödinger operator, the real eigenvalues remain fixed under the complex dilation, whereas the essential spectrum swings into the complex plane and thus reveals possible non-real eigenvalues, which correspond to resonances of the original self-adjoint operator (see Figure ??).Note that there are no resonances for the Coulomb potential (see Remark 3).
In Section 5 we extend the notion of positive and negative spectral projections to the complex dilated Dirac operators.The definition of the spectral projections in Formula ( 32) is a straightforward extension of a well known formula from Kato's book (see [33,Lemma VI.5.6]).The rest of this section is devoted to the proof that the operators defined in (32) are actually well defined projections (see Theorem 4), that they commute with the dilated Dirac operator (see Theorem 5), and that their range is what one expects it to be (see Theorem 5 as well), which is not completely obvious in the non-self-adjoint case.Note that the projections themselves are not orthogonal projections.These results enable us to define transformation functions between the positive spectral projections of the dilated and not dilated Dirac operators in Section 6, which is essential in order to show that also the projected Dirac operators are holomorphic families -even if they are coupled to the quantized radiation field.This will be accomplished in [30].Moreover, these results can be used to generalize [47] to complex dilated operators.Transformation functions as defined in Formula (60) are similarity transformations between two (not necessarily orthogonal) projections (see Formula (57) in Theorem 6).Note that our definition requires that the norm difference between the projections be smaller than one, but there are more general approaches.For details on transformation functions we refer the reader to [33,Chapter II.4].In Theorem 7 in Section 7 we prove a resolvent estimate for the dilated Dirac operator projected and restricted onto its positive spectral subspace.In particular, we prove that the norm of the resolvent converges (essentially) to zero as the inverse distance to the right complex half plane.Note that this really requires the restriction of the operator to its positive spectral subspace and that the norm of the resolvent of a non-self-adjoint operator is not bounded from above by the inverse distance of the spectral parameter to the spectrum.
In Section 8 we will investigate the non-relativistic limit of dilated Dirac operators and thereby generalize and extend the results in Thaller's book [49] in various directions.We prove in Theorem 8 and Corollary 2 that complex dilated Dirac operators converge to the corresponding (complex dilated) Schrödinger operators in the sense of norm resolvent convergence as the velocity of light goes to infinity.As in the undilated case, this convergence is needed to gain information about the spectral projections onto the eigenspaces belonging to the real eigenvalues and their behaviour in the nonrelativistic limit (see for example Lemma 7 or Lemma 8).In particular, the complex dilation, restricted to an eigenspace is a bounded operator (uniformly in the dilation parameter and the velocity of light -see Lemma 9) and the projections onto the fine structure components are uniformly bounded as well (see Corollary 5).These statements will be needed in [30].Note that for Schrödinger operators and non-relativistic QED the above mentioned problems are absent, since there is neither a fine structure splitting nor the additional parameter of the velocity of light which has to be controlled.Moreover, we show in Theorem 9 and Theorem 10 that the lower Pauli spinor of a normed eigenfunction of the Dirac operator converges to zero in the sense of the Sobolev space H 1 (R 3 ; C 2 ) and that the upper Pauli spinor is bounded in the sense of H 1 (R 3 ; C 2 ) as the velocity of light tends to infinity.This shows that the notion of "large" and "small" components of a Dirac spinor, which is frequently used by physicists, is also justified for dilated operators.Moreover, it follows that certain expectation values of the Dirac α-matrix vanish as the velocity of light tends to infinity.We will apply this fact in [30].
Note that in the discussion of the non-relativistic limit in Section 8 we need some estimates from Bach, Fröhlich, and Sigal [5] which we cite in Appendix A for the convenience of the reader.

Foldy-Wouthuysen-Transformation
In this section we investigate the complex continuation of the Foldy-Wouthuysen transformation and show some important properties in Theorem 1.We need this as a technical input for the spectral analysis in the following sections.Let us mention that a complex continuation of the Foldy-Wouthuysen transformation was implicitly used by Evans, Perry, and Siedentop [11] for the investigation of the spectrum of the Brown-Ravenhall operator.Also Balslev and Helffer [7] use holomorphic continuations of the Foldy-Wouthuysen transformation.
Theorem 1.Let θ ∈ S π/4 .Then the following statements hold: a) The operator U FW (c; θ) is a bounded operator on L 2 (R 3 ; C 4 ) with bounded inverse V FW (c; θ).There is a constant C FW (independent of c and θ) such that and b) The Foldy-Wouthuysen transformation diagonalizes the Dirac operator: Proof.a) A simple calculation shows We have U FW (c; θ) ≤ sup p∈R 3 ÛF W,c (p; θ) .Thus, it suffices to consider the case c = 1 and Re θ = 0.In view of the identity ÛFW,c (p; θ) Next, observe that where we used the estimate |w| ≥ |Re w| and (5).From (14) it follows that Documenta Mathematica 14 (2009) 297-338

Matthias Huber
Moreover, we have Using ((1 (13) we estimate the denominator by In order to estimate the enumerator we find after some calculations We combine suitable terms in (18): We have Summarizing the estimates ( 13) and ( 15) through (21), we finally obtain where we used that | sin(2ϑ < ∞, equation (22) shows the claim on U FW (c; θ).
The claim on the inverse operator V FW (c; θ) can be proven analogously.

Dilation Analyticity and Spectrum
We show that the operators in equations ( 3) and ( 4) define holomorphic families of closed operators.Since we will be interested in the non-relativistic limit later on, we consider only such values of c and γ which can be dealt with using Hardy's inequality.For θ ∈ S π/2 we define the set Moreover, obviously D c,γ ( θ) * ⊃ D c,γ (θ) holds.Thus, it suffices to prove the inclusion Dom(D c,γ ( θ) * ) ⊂ Dom(D c,γ (θ)) = Ran(D c,γ (θ) −1 ).We adapt a well known strategy from the case of self-adjoint operators (cf.[52,Satz 5.14]).We have Dom(D c,γ (θ Remark 1.Note that if V is the Coulomb potential or the Yukawa potential, then D c,γ (θ) is equal to a multiple of the self-adjoint operator −icα • ∇ + V C up to a bounded operator so that the proof of the above theorem is trivial.Note moreover, that for V = V C , the operator D c,γ (θ) is entire.
Remark 2. Theorem 2 and its proof imply that H 1 (R 3 ; C 4 ) is the maximal domain of the operator on L 2 (R 3 ; C 4 ) generated by the differential expression Dc,γ (θ) := −e −θ icα • ∇ + c 2 β − γV (θ).To see this set where the gradient is to be understood in distributional sense.Note that f ∈ As in the proof of Theorem 2 it would follow that there was a 0 The following lemma, whose simple proof we omit, contains a useful fact: Now we need the spectrum of the operator D c,γ (θ).Theorem 1 shows (see Figure 1) In the case of self-adjoint operators the compactness of the difference of free and interacting resolvent would imply that D c,0 (θ) and D c,γ (θ) with γ = 0 have the same essential spectrum.This is however not true for non-self-adjoint operators in general.In particular there exist several different definitions of the essential spectrum, which do not coincide in general and have different invariance properties.In the case of relatively compact perturbations this difficulty can be mastered using the analytic Fredholm theorem [50].Since Coulomb type potentials are not relatively compact, we adapt a strategy invented by Nenciu [40] for the self-adjoint case.We need the following lemma: . Moreover, let B c;θ;+ and B c;θ;− (see Figure 1) the closed subsets of {z ∈ C|Re z > 0} and {z ∈ C|Re z < 0} respectively, which are enclosed be the Note the inequality 1/ cos(Im θ) ≤ C(Im θ).
The following theorem yields a precise description of the spectrum of the operator D c,γ (θ).In particular, outside the set B c,θ the spectra of D c,γ (θ) and D c,γ (0) coincide so that one particle resonances -if any exist -can be located only within the set B c,θ .Let B(L 2 (R 3 ; C 4 )) be the set of bounded and everywhere defined operators on Proof.We denote the r.h.s. of (25) by R c,γ;θ (z).
Remark 3. Note that for V = V C the set of resonances is empty.This follows similarly as for the Schrödinger case (see [8]): If there was a resonance, then D c,γ (π) would have a non-real eigenvalue.
In this section we extend the notion of positive and negative spectral projections to dilated Dirac operators.We define for p ∈ R 3 the matrices Λ c,0 (p; θ).Moreover, one verifies the identity Λ Dc,0(p;θ)−i η .These observations motivate the following definition for the dilated interacting operators: It is well known [33, Chapter VI-5.2, Lemma 5.6] that Equation (32) yields the positive and negative spectral projections for real θ.Note that similar formulas for not necessarily self-adjoint operators are known (see [16,Chapter VX]).These authors use a different definition for the spectral projections, however.First, we show in Theorem 4 that these operators are well defined and bounded projections even if θ / ∈ R. We need the following technical lemma: where C 1 (Im θ) is defined in (24).
Proof.We prove the estimate We estimate the first summand using inequality (5) and Lemma 1.For the second summand we restrict ourselves to the case Im θ < 0. The proof for Im θ > 0 works analogously, and (33) holds obviously if Im θ = 0.Moreover, it suffices to consider Re θ = 0. We investigate the term (34).The claim follows using Theorem 1.
Step 1: The resolvent equation (25) and the estimate (26) yield the convergence of the series Step 2: We show that the expression defines a bounded operator on L 2 (R 3 ; C 4 ).In order to achieve this, we estimate where we used (26) in the first estimate and Lemma 3 in the second estimate.C(Im θ) and C 1 (Im θ) were defined in (24).As in [47, Proof of Lemma 1] we obtain Step 3: The expressions are holomorphic functions of θ ∈ S min{π/4,Θ} .These estimates show the existence of an integrable and summable majorant, independent of θ for θ ∈ M γ/c .Thus, the operator in Equation ( 36) is a holomorphic function of θ [33, Chapter VII-1.1], and the identity Λ Step 4: We show that the limit exists as a strong limit and estimate for g ∈ H 1/2 (R 3 ; C 4 ) as follows: Here we estimated the expression in the square brackets similarly to (26), but used Hardy's inequality instead of Kato's inequality.Moreover, we used the estimate (33) twice.Since σ(D c,0 This estimate shows that the convergence in formula ( 36) is uniform in f ∈ L 2 (R 3 ; C 4 ), which implies the strong convergence [33, Theorem III.1.32and Lemma III.3.5],since Obviously, the identity Λ c,γ (θ), wehre ∔ denotes the direct sum.We call the Λ (±) c,γ (θ) positive and negative spectral projections and H (±) c,γ (θ) positive and negative spectral subspaces, respectively.This is justified because of Theorem 5.The following corollary generalizes [47, Lemma 1] to dilated spectral projections.
Corollary 1.Let θ ∈ S min{π/4,Θ} and suppose that (H1) holds.Then there exists a constant C NR > 0 such that for 2γ c C(Im θ) < 1 the estimate Proof.This follows directly from Equation (37) in the proof of Theorem 4.
The next theorem shows that the spaces H (±) c,γ (θ) are invariant under D c,γ (θ) and describes the spectrum of the restriction of the operator to these spaces.If a part of the spectrum is contained in a Jordan curve, analogous statements can be found in [33, Theorem III-6.17].The following theorem describes a more general situation, but the essential elements of the proof of [33, Theorem III-6.17] can be adapted.For a closed operator A we denote its resolvent set by ρ(A).
On the other side, we have f ∈ H c,γ (θ)f.Using the first resolvent identity, we find for z ∈ C with Re z < 0 respectively Re z > 0 since for z ∈ C with Re z < 0 respectively Re z > 0 the residue theorem implies c,γ (θ) would imply the contradiction z ∈ ρ(D c,γ (θ)).This shows (39) and (40).
Next, we need spectral projections for the eigenvalues: We define for all n ≥ 1 (and n ≤ N max if there only finitely many eigenvalues) the spectral projections where z runs through Γ n (c, γ) in the positive sense.Γ n (c, γ) is chosen such that for all 1 ≤ l ≤ N n the eigenvalues Ẽn,l (c, γ) are located within the contour, but no other elements of the spectrum D c,γ (θ).
For later, we need spectral projections for the fine structure components.We set for n ≥ 1 and 1 where z runs through Γ n,l (c, γ) in the positive sense, and Γ n,l (c, γ) is chosen such that only the eigenvalue Ẽn,l (c, γ) lies within the contour.We denote the corresponding normed eigenfunctions by φ n,l (c, γ; θ).

Transformation Functions
We need transformation functions between the spectral subspaces of dilated and not dilated operators for the resolvent estimate in Section 7 and in order to establish the dilation analyticity of a relativistic Pauli-Fierz model in [30].
Another example for a transformation function is the Douglas-Kroll transformation, which was investigated by Siedentop and Stockmeyer [47] (see also Huber and Stockmeyer [31]).Contrary to the situation there, our spectral projections are not self-adjoint and thus the transformation function is a nonunitary similarity transformation.The estimates in this section can be used to generalize the Douglas-Kroll transformation to complex dilated operators.
In order to prove the existence of the transformation function, we need norm estimates on the difference between the spectral projections.
Documenta Mathematica 14 (2009) 297-338 holds.The operator |D c,0 (0 Then there is a constant C DLS > 0 (independent of c, γ and θ) such that for 2γ c C(Im θ) < q the estimate Proof.We adapt method which was used by Siedentop and Stockmeyer [47] and by Griesemer, Lewis and Siedentop [19] for other choices of projections.We start with the difference of resolvents and note that |e −θ − 1| ≤ B|θ| holds with B = e π/4 for all |θ| ≤ π/4.
Step 3: Proof of (45).We use the expansion Documenta Mathematica 14 (2009) 297-338 and start with the necessary estimates on the differences of the resolvents: Using Hardy's inequality, we obtain as in ( 26) and we find analogously as well as For the terms with the resolvents we use Lemma 3 and Lemma 1 to estimate and (cf.Formula (26)) Formulas ( 52) through (56) show which in turn proves (45).
Step 4: Holomorphicity.This follows as in the proof of Theorem 4, since are holomorphic functions of θ and the above estimates imply the existence of summable and integrable majorant which does not depend on θ.

Documenta Mathematica 14 (2009) 297-338
Before we turn to the existence of a transformation function in Theorem 6, we need two operator inequalities, one of which was proven in [19].Since the other inequality can be proven completely analogously, we omit the proof.Let us mention that there exits an improved version of one of these inequalities (see [39]).But since we will be interested in the non-relativistic limit only, it is sufficient to use the original version.
Lemma 5 ([19], Lemma 2).Suppose that ϑ ∈ R and γ c < 1 2 .Then the operator inequalities c,γ (θ).This is necessary for technical reasons, since the latter operates on a fixed space (i.e.Ran Λ (+) c,γ (0)).We will prove in [30] that this operator defines a holomorphic family of operators.Moreover, we will need the transformation function in the proof of the resolvent estimate in Theorem 7.
A first application of the transformation function U DL (c, γ; θ) is the following lemma, which estimates the difference between the dilated Dirac operator and its original version.Lemma 6.Under the assumptions of Theorem 6 b) there is a constant C UD > 0, independent of γ, c and θ, such that Proof.We have which implies the claim, if we use additionally and Theorem 6.Moreover, we used the inequality |e −θ − 1| ≤ B|θ| with B = e π/4 and Kato's inequality in the proof of (62).

A resolvent estimate for the Dirac operator
In the following, we choose an η > 0 such that for some ñ > The following theorem partly generalizes [5,Lemma 3.8] for Dirac operators (see also Theorem A.1).We will slightly extend this theorem in the nonrelativistic limit (see Lemma 7 and Corollary 4).This theorem and Corollary 4 enable us to control the norm of the resolvent of the non-self-adjoint operator D c,γ (θ)| Ran Pdisc,ñ (c,γ;θ) .Note that the usual theorems about the norm of the resolvent of a self-adjoint operator fail in general, and that for the following to hold it is essential that to restrict the operator to (a subspace of) the positive spectral subspace.
Theorem 7. Suppose that the assumptions of Theorem 6 b) hold.Assume additionally that the inequalities C UD |θ|(1+2γ/c) < q and 2γ(1+C FW |Im θ|) < q are fulfilled for some 0 < q < 1.Then the following statements are true: The operator D c,γ (θ)| Ran Pdisc,ñ (c,γ;θ) − z has a bounded inverse for all z ∈ C with Re z ≤ c 2 − 1 .There is a constant C R > 0, independent of c, γ and θ, such that for all z ∈ C with Re z ≤ c 2 − 1 the estimate Proof.We make a case distinction: Case 1: Re z ≤ 0. Theorem 6 implies the inclusion Ran(U DL (c, γ; θ) ).Thus, using Theorem 6 again, it suffices to show As in [5, Proof of Lemma 3.8], we use a resolvent expansion: In order to prove the convergence of the series, we have to estimate the terms in (63).First, we note that holds, since Re z ≤ 0.Moreover, the spectral theorem implies Lemma 6, Lemma 5 and (64) prove the convergence of the series in (63).Using Formula (65), the claim follows for Re z ≤ 0 from (63).
Case 2: 0 ≤ Re z ≤ c 2 − 1.We use the resolvent expansion Hardy's inequality and Theorem 1 yield . In order to control this norm, we estimate as follows: sup , it suffices to consider the case with the minus sign.
We need to find the supremum of the function f c,l : [0, ∞) → R, f c,l (r) := r √ c 2 r 2 +c 4 −l for 0 ≤ l ≤ (c 2 − 1).If we differentiate this function, we find that it attains its maximum at the point r 0 := . Now, we define the function . This function is obviously monotonously increasing in l and therefore attains its maximum at the point Thus, Equation (66) and Theorem 1 yield the estimate , which remains true, if we restrict the resolvent to Ran Pdisc,ñ (c, γ; θ).

Non-relativistic limit
In this section we investigate the non-relativistic limit of complex dilated Dirac operators.We will use these results in [30], where we will discuss the interaction with the second quantized radiation field.Moreover, we can extend the resolvent estimate of Theorem 7 to the region close to the spectrum of the operator and control the norm of the projection occurring there.

General Theory
We extend some statements from [49] to the non-self-adjoint case.We define β ± := 1 2 (1 ± β) as well as M := {z ∈ C| − 1 ≤ Re z < 0, |Im z| ≤ 1} and fix a γ > 0 such that D c,γ (θ) − c 2 has no eigenvalues E with Re E ≤ −1.This is at least true for 0 ≤ γ < 1 in the case of V = V C , which can be seen, for example, using the explicit formula for the eigenvalues, see [35].We define as operators on L 2 (R 3 ; C 4 ): as well as First, we generalize [49, Theorem 6.1 and Theorem 6.4] to dilated operators.
As in [49], Theorem 8 is the starting point for the investigation of the nonrelativistic limit.
Corollary 2. Suppose that |θ| < θ 0 , where θ 0 is sufficiently small (see Appendix A), and θ ∈ S min{π/4,Θ} as well as 2γ c C(Im θ) < 1. Suppose moreover that (H1) holds.Then the resolvent expansion holds for all z ∈ M η,ǫ and all sufficiently large c.The series converges in norm, uniformly in θ and z.In particular, uniformly in θ and z.
Proof.First, we need an estimate on the resolvent of D ∞,γ (θ).We split the resolvent according to Corollary A.1.Thus, we have for sufficiently small 1/c (dependent on ǫ) and all z ∈ M η,ǫ the expansion Hardy's inequality implies for ] f with a sufficiently small a > 0. It follows that holds with C 1 , C 2 > 0 (independent of γ, c and θ), which implies that the last factor in (68) has a norm convergent series expansion in 1/c for 1/c small enough.
Lemma 7. Suppose that the assumptions of Corollary 2 hold.Then there is a constant C P,n > 0 (independent of c and θ) such that for sufficiently large c the estimate Proof.This follows immediately from Corollary 2.
The following two corollaries extend Theorem 7.
Lemma 8. Suppose that the assumptions of Corollary 2 and the inequality C P,n /c < q < 1 hold for some 0 < q < 1.Then the mapping U NR (c, γ; θ) is bounded with bounded inverse V NR (c, γ; θ).The relations and hold with a constant C NRP > 0 independent of c and θ.U NR (c, γ; θ) is a holomorphic function of θ.
Proof.Using Lemma 7 this can be proven in the same way as Theorem 6.
Remark 5.As in [49] we obtain by Remark 4 that in the series expansion of U NR (c, γ; θ) the operators occurring with even powers of 1/c are even and the operators occurring with odd powers of 1/c are odd.In particular, where U N R,g (c, γ; θ) and U N R,ug (c, γ; θ) are even and odd operators holomorphic in 1/c.
The following theorem generalizes [49,Theorem 6.7] and shows that the lower component of an eigen-spinor of the Dirac operator converges to zero as c → ∞.
We use these statements to prove that eigenfunctions are bounded in the norm of H 1 (R 3 ; C 4 ).Proof.We follow Esteban and Séré [10, Proof of Lemma 7 and Theorem 3], who considered the non-relativistic limit of self-adjoint Dirac-Fock operators.Since D c,γ (θ) is not self-adjoint, there are some additional difficulties.To simplify the notation, we suppress the dependence of φ n,l (c, γ; θ) on c, γ and θ.We have where we used Hardy's inequality.Since E n,l (c, γ) 2 − c 4 ≤ 0, it follows that for sufficiently large c, where C > 0 does not depend on c.Note that the term proportional to c 2 in (82) does not occur for Im θ = 0, which implies immediately the boundedness of ∇φ n,l in this case.To circumvent this difficulty, we write the Dirac equation in its components, where (in abuse of notation) φ n,l,± denotes the upper and, respectively, lower components of φ n,l : Dividing (83) by c, using Hardy's inequality and the boundedness of E n,l (c, γ)− c 2 , Formula (82) implies for some C > 0 independent of c, i.e. ∇φ n,l,− is bounded in c.Dividing (84) by c, we obtain for some C > 0 independent of c, where we used Theorem 9 and Equation (85).This shows (80).Inserting (86) in (85), Equation (81) follows.
Remark 6.Their validity of Theorem 9 and Theorem 10 in the Coulomb case could be derived from the explicit form of the eigenfunctions (see the proof of Lemma 11).
Moreover, we need a bound on the norm of the dilation operator U(θ), restricted to the spaces Ran P n (c, γ; θ).
Proof.Surely U(θ)| Ran Pn(∞,γ;0) : Ran P n (∞, γ; 0) → Ran P n (∞, γ; θ) is well defined for all θ ∈ C with |θ| ≤ min{π/4, Θ} (see [2,6]) and (as a mapping between finite-dimensional vector spaces) bounded.Since the operator is a holomorphic function of θ for |θ| ≤ min{π/4, Θ}, there is a bound C ′ > 0 (independent of θ) on its norm.Let f ∈ Ran P n (c, γ; 0).Then there is a f ∈ Ran P n (∞, γ; 0) with f = U NR (c, γ; 0) f , and for real θ f (θ , where f (θ) := U el (θ) f .By holomorphic continuation we obtain for complex θ the equality f The following corollary shows that also the projections on the fine structure components are bounded uniformly in c.This follows from the fact the dilated projections are similar to the corresponding orthogonal projections belonging to the corresponding self-adjoint Dirac operators because of Lemma 9. Note that in general such projections are not uniformly bounded in the perturbation parameter (see [33, Chapter II-1.5]).
Corollary 5. Let 1 ≤ n ≤ ñ and suppose that the assumptions of Lemma 9 hold.Then P n,l (c, γ; θ) ≤ C for some C > 0 independent of n, l, c and θ.

Application to expectation values of Dirac matrices
We are now in the position to investigate expectation values of the matrices α.Since these matrices are odd, such expectation values involve scalar products of the upper component of one spinor with the lower component of the other spinor.Therefore, one expects that such expectation values converge to zero like 1/c as c → ∞ uniformly in a set of suitable spinors.We show in the following that this is true, if one of the spinors is in the set of eigenstates (in the positive part of the gap) and the other state is an arbitrary state from the positive spectral subspace.Note that this is not true, if both states are arbitrary states from the positive spectral subspace.At least for the free spectral subspaces this can be seen from the explicit form of the projections (see Section 5).We will apply this result in [30].
Lemma 10.Suppose that the assumptions of Lemma 9 hold and let ñ as in Section 7. Then there is a constant C > 0, independent of c and θ, such that for all 1 ≤ n, n ′ ≤ ñ, 1 ≤ l ≤ n, 1 ≤ l ′ ≤ n ′ and k 1 , k 2 ∈ R 3 Here the radial quantum number fulfills n r ∈ N 0 if κ < 0 and n r ∈ N if κ > 0, and κ ∈ ±N is the eigenvalue of the spin-orbit operator (see [49,Chapter 4.6]).F denotes the confluent hypergeometric function, which reduces to a polynomial in 2λr here (see [35,Abschnitt 36] and [34,Abschnitt d]).Moreover, γ := κ 2 − γ 2 /c 2 and λ := 1 − E 2 n,l /c 4 .Thus, the radial parts f ± (r) of the upper respectively lower components of f are Using the explicit formula (see [35]) for the eigenvalues, we see that cλ is a function bounded in c with cλ −→ γ/n for c → ∞.Moreover, obviously γ → |κ| holds.This shows the claim.
Remark 7. At this point we make use of the explicit from of the eigenfuntions of the Coulomb Dirac operator.There do not seem any results to be available in the literature about exponential decay of eifenfunctions of the Dirac operator uniformly in the velocity of light.
Lemma 12. Suppose that the assumptions of Lemma 9 are fulfilled and let ñ as in Section 7. Let moreover f : C → C with |f (z)| ≤ |z|.Then there is a constant C > 0, independent of c, such that for all 1 ≤ n ≤ ñ, 1 ≤ l ≤ n and k 1 , k 2 ∈ R 3 Proof.Lemma 11 implies that xP n,l (c, γ; 0) is uniformly bounded in c, in particular (using the notation of Theorem 9) xφ n,l,+ (c, γ; 0).Now the claim follows exactly as in Lemma 10.
The following theorem generalizes Lemma 10.Note that the statement of Lemma 10 is not completely obvious, since not even the lower component of the free positive spectral projection converges to zero in norm as c → ∞.This is, however, compensated by the fact that the H 1 -norm of the upper component of bound states is bounded uniformly in c (Theorem 10).