Documenta Math. 331 The Projected Single-Particle Dirac Operator for Coulombic Potentials

A sequence of unitary transformations is applied to the one-electron Dirac operator in an external Coulomb potential such that the resulting operator is of the form ¤+A ¤+ + ¤−A ¤− to any given order in the potential strength, where ¤+ and ¤− project onto the positive and negative spectral subspaces of the free Dirac operator. To first order, ¤+A ¤+ coincides with the Brown-Ravenhall operator. Moreover, there exists a simple relation to the Dirac operator trans- formed with the help of the Foldy-Wouthuysen technique. By defining the transformation operators as integral operators in Fourier space it is shown that they are well-defined and that the resulting transformed operator is p-form bounded. In the case of a modified Coulomb po- tential, V = −γx −1+ǫ , ǫ > 0, one can even prove subordinacy of the n-th order term in γ with respect to the n − 1st order term for all n > 1, as well as their p-form boundedness with form bound less than one.

It is well-known that H is not bounded from below.As long as pair creation is neglected, the conventional way to circumvent this deficiency is the introduction of the semibounded operator P + HP + where P + projects onto the positive spectral subspace of H.As a first approximation, Brown and Ravenhall [1] introduced the operator with Λ + projecting onto the positive spectral subspace of the free Dirac operator D 0 , and |D 0 | = D 2 0 is the free energy.In momentum space one has with the electron mass m.By construction, the Brown-Ravenhall operator B is of first order in the potential V and has been shown to be bounded from below for subcritical potential strength γ [5].
An alternative way to derive a semibounded operator from H has been suggested by Douglas and Kroll [4], using the Foldy-Wouthuysen transformation technique [6].The decoupling of the positive and negative spectral subspaces of H to order n in V is achieved by means of n + 1 successive unitary transformations U ′ j , j = 0, 1, ..., n which cast the tranformed operator into a block-diagonal contribution H ′ n plus an error term R n+1 with potential strength given by the n + 1st power of γ.U ′ 0 is the free Foldy-Wouthuysen transformation which block-diagonalises D 0 exactly [14], and for U ′ j , Douglas and Kroll [4] use with antisymmetric operators W j .It should be noted that the choice of U ′ j is not unique, and neither is the resulting operator H ′ n from (1.4) for n > 4 as has been shown by Wolf, Reiher, and Hess [15].Applying the free Foldy-Wouthuysen transformation (1.5) one obtains [4,9] U ′ 0 HU Documenta Mathematica 10 (2005) 331-356 (1.7) where the transformed potential has been split into an even term E 1 (commuting with β) and an odd term O 1 (anticommuting with β, since α k β = −βα k , k = 1, 2, 3).For an exponential unitary transformation, with a symmetric operator S j , the next transformation gives in agreement with [9] e −iS1 U ′ 0 HU ′ −1 0 (1.9) S 1 is defined from the requirement that O 1 is eliminated, hence S 1 is odd and of first order in the potential like O 1 .After each transformation U ′ j , the j + 1st order term in γ of R j+1 is decomposed into even (E j+1 ) and odd (O j+1 ) contributions, and the successive transformation U ′ j+1 = e −iSj+1 is chosen to eliminate O j+1 , which is achieved by the condition (1.10) for the j > 1 under consideration.With this procedure one arrives at the even (and hence block-diagonal) operator The physical quantity of interest is the expectation value of the transformed Dirac operator.For the Brown-Ravenhall operator, consider the expectation value formed with 4-spinors ϕ in the positive spectral subspace of D 0 which in momentum space can be expressed in terms of Pauli spinors u where σ is the vector of the three Pauli matrices.Then, an operator b m acting on H 1/2 (R 3 ) ⊗ C 2 may be defined by [5] (ϕ, Bϕ) =: (u, b m u). (1.13) On the other hand, in case of the Douglas-Kroll transformed operator H ′ n , its upper block corresponds to the particle states (having positive energy) and therefore the expectation value has to be formed with the four-spinor ψ := u 0 with u ∈ H 1/2 (R 3 ) ⊗ C 2 as above.

Documenta Mathematica 10 (2005) 331-356
For the first-order term, H ′ 1 , it is easy to show [2] that its expectation value agrees with the expectation value of the Brown-Ravenhall operator, i.e.
such that H ′ n may be considered as the natural continuation of B to higher order in V .While H ′ n is known explicitly up to n = 5 [15], the spectral properties, in particular the boundedness from below, have only been investigated for the Jansen-Hess operator (i.e.n = 2) [2,8].The aim of this work is to prove two theorems.
Theorem 1.1.Let H = D 0 + V be the one-particle Dirac operator acting on S(R 3 ) ⊗ C 4 with S the Schwartz space of smooth strongly localised functions.Let γ be the strength of the Coulomb potential V and p := |p|.Then there exists a sequence of unitary transformations U k = e iB k , k = 1, ..., n, such that the transformed Dirac operator can be written in the following way Here, Λ + projects onto the positive spectral subspace of D 0 , Λ − = 1 − Λ + , and H k is a p-form bounded operator, its form bound being proportional to γ k , k = 1, ..., n.The remainder R (n+1) which still couples the spectral subspaces of D 0 is p-form bounded with form bound O(γ n+1 ) when γ tends to zero.The operators B k are symmetric and bounded, extending to self-adjoint operators on L 2 (R 3 ) ⊗ C 4 .
An operator H k with the properties stated in the theorem is said to be of order γ k .It is shown below that H 0 := D 0 and H 1 := V such that to first order, the Brown-Ravenhall operator B is recovered.(1.15) implies that the transformed Dirac operator can be expressed in terms of projectors to arbitrary order in the potential strength.Similar transformation schemes are known for bounded operators on lattices, see e.g.[3] and [13].
The next theorem states the unitary equivalence of the transformed Dirac operators obtained with either the transformation scheme from Theorem 1.1 or the Douglas-Kroll transformation scheme (1.8) - (1.11).
) be a 4-spinor in the positive spectral subspace of D 0 , which defines a Pauli spinor u by means of (1.12).Let H ′ n be the Douglas-Kroll transformed Dirac operator to n-th order in the potential strength, obtained with exponential unitary operators U ′ j .Then to any order n, its expectation value agrees with the one of the transformed Dirac operator from (1.15), 2 Proof of Theorem 1.1

Derivation of unitary transformations
The sequence of unitary operators U k = e iB k is constructed with the help of an iteration scheme.Following Sobolev [13] we consider U k as an element of the group U k (t) = e iB k t , t ∈ R.
Let A be an arbitrary t-independent operator.The derivative of the transformed operator is given by where the commutator [A, This equation is easily integrated, noting that A(0) = A, Iterating once, i.e. replacing A by the operator [A, B k ] in (2.2) and inserting the resulting equation into the r.h.s. of (2.2), one obtains for t = 1 After n iterations the following representation of A 1 is obtained, (2.4) where the n-th term contains n commutators with B k , and the remainder R is an (n + 1)-fold integral.Let us apply this scheme inductively to the Dirac operator H = D 0 +V.Assume that to order n − 1 the transformation has been achieved with a resulting operator of the form given in Theorem 1.1, where H n and R(n+1) are respectively of order γ n and γ n+1 , and still couple the spectral subspaces.Decompose H n into The next transformation, U n = e iBn , aims at eliminating the term W n which, in contrast to V n , couples the spectral subspaces.This condition will fix B n .We note that from (2.4), the transformation reproduces the operator itself, such that the term H (n−1) , already in the desired form, is preserved.From this it follows that H (n−1) contains the zero-order term Λ + D 0 Λ + + Λ − D 0 Λ − = D 0 (note that Λ ± commutes with D 0 and Λ 2 + + Λ 2 − = 1).We obtain where R collects the terms containing at least two commutators with B n .B n is determined from the requirement Since W n is of order γ n , B n is proportional to γ n (the boundedness of B n is shown later).Moreover, the commutators of the type [(Λ B n ] are of order γ n+k with k ≥ 1, and R is of order γ 2n .Hence, these terms are disregarded (together with the remainder R(n+1) from (2.5)) in constructing the transformed operator to order n, Particularly interesting are the cases n = 1 and n = 2.For n = 1, we have Restricting H (1) to the positive spectral subspace Λ + (S(R 3 ) ⊗ C 4 ), the second term on the r.h.s. of (2.10) vanishes and the remaining term agrees with the Brown-Ravenhall operator.
Let us now consider n = 2. From (2.9) it follows that the transformed Dirac operator in second order is determined by the first transformation, U 1 = e iB1 , only.However, the existence of the second transformation, U 2 = e iB2 , has to be established to show that H (2) is indeed the transformed operator, with a remainder of order γ 3 .We have Making use of the defining relation for 2) takes the form 12)

Integral operators in Fourier space and the determination of B 1
Since D 0 is a multiplication operator in momentum space, it is convenient to set up the calculus in Fourier space.Let ϕ ∈ S(R 3 )⊗C 4 .We define an integral operator W acting on ϕ by means of where here and in the following the (three-dimensional) momentum integrals extend over the whole R 3 .This agrees with the formal definition of a pseudodifferential operator [13] and we will call w(x, p) the symbol of W . Introducing the Fourier transform ŵ(q, p), W ϕ takes the form 3  dp e ipx dq e iqx ŵ(q, p) φ(p). (2.14) From this, the Fourier transform of W ϕ is found With ϕ in (2.14) replaced by Gϕ, the symbol of a product W G of two integral operators is derived, (2.17) For the goal of determining the transformation U 1 , its exponent B 1 is considered as an integral operator.One has to solve (2.8) for n = 1, using

.18)
Let φ 1 be the symbol of B 1 .From (2.14) and with D 0 from (1.1) one has The Fourier transforms of D 0 ϕ and of V ϕ are, respectively, obtained from such that the symbol v of V is defined by v(q, p) = − 2/π γ/q 2 .Acting (2.18) on ϕ and equating the respective symbols leads to the following algebraic equation for φ1 : with γ 0 := γ/ √ 2π and D0 (p) the operator from (1.3) with norm unity.ŵ1 (q, p), behaving like q −1 for q → 0, is less singular than v(q, p), such that the prescription (2.6) for W 1 implies a regularisation of the potential V .Lemma 2.1.A solution φ1 (q, p) for the symbol of B 1 is given by which satisfies the condition for symmetry of B 1 [13], φ1 (−q, p + q) * = φ1 (q, p). (2.23) It is estimated by Proof.a) Calculation of φ1 .
In order to solve (2.21) the ansatz is made and from the properties of the Dirac matrices Insertion of (2.25) into (2.21) then leads to an equation of the type where the λ k , k = 1, ..., 4, are scalars depending on p and q. (2.27) must hold for p, q ∈ R 3 whence λ k = 0, k = 1, ..., 4. The resulting system of 4 equations for the c i , i = 1, 2, 3 has a unique solution, (2.29) It is readily verified that (2.29) can be cast into the form (2.22), proving that φ1 (q, p) is continuous in both variables except for q = 0.
b) The symmetry condition (2.23) follows immediately from (2.22) using the self-adjointness of α and β.
c) We define the class of our integral operators (2.14) by means of the estimate of their symbols in the six-dimensional space (q, p) ∈ R 6 .This estimate determines the convergence properties of integrals without the precise knowledge of the symbols themselves, and it is an easy way to deal with products of integral operators in proofs of boundedness or p-form boundedness.
In order to estimate a symbol by its asymptotic behaviour for q, p → 0 and q, p → ∞, it must be a continuous function of the two variables in R + × R + .This condition is fulfilled for φ1 (q, p).By inspection of (2.22) one finds that φ1 (q, p) is finite = 0 for p = 0 while it behaves ∼ 1/q, q → 0, ∼ 1/q 3 , q → ∞ and ∼ 1/p 2 , p → ∞.Taken into consideration that φ 1 (x, p) is dimensionless as is B 1 (cf.U 1 = e iB1 and (2.13)) whence φ1 (q, p) is of dimension (momentum) −3 , the estimate | φ1 (q, p)| ≤ c/q • (q + p + 1) −2 is obtained.The constant c is determined by the maximum of q • | φ1 (q, p)| in q, p ∈ R + (which exists due to continuity) and by the decay constants of φ1 (q, p) for q → ∞ and p → ∞, respectively (maximised in the second variable).
d) We present here only the proof of the form boundedness of B 1 ; the operator boundedness can be shown along the same lines.
The basic ingredient is the Lieb and Yau formula which is a consequence of the Schur test [12] and which also can be derived from Schwarz's inequality [11].We give it in a slightly generalised form, with f (p) > 0 for p > 0 a smooth convergence generating function.For a symmetric kernel, K(p, q) = K(q, p), (2.30) simplifies to the conventional form [11,5].
From the condition (2.23) we have the following symmetry with respect to interchange of q and p, (2.31) One then obtains for ϕ ∈ L 2 (R 3 ) ⊗ C 4 applying (2.15) and subsequently (2.30) and (2.31) .
Both symbol and its adjoint can be estimated by the same expression (2.24) (from (2.22) one even has φ * 1 (q, p) = − φ1 (q, p)), yielding (2.33) For the form boundedness of B 1 it remains to prove that I 1 (p) is bounded for p ∈ R + .The angular integration is performed with the help of the formula As a consequence of the boundedness on L 2 (R 3 ) ⊗ C 4 and the symmetry, B 1 is self-adjoint and U 1 = e iB1 is unitary.

Existence of B 2 and the transformations of higher order
Let φ n denote the symbol of B n .It is briefly indicated how φ 2 can be obtained explicitly, but for B n , n ≥ 2, the calculus with operator classes is applied instead.B 2 is defined by (2.37) (2.36) is, like the corresponding equation for B 1 , solved in momentum space by introducing the respective symbols of the operators.The Fourier transformed symbol ŵ2 of W 2 is composed of expressions of the type (cf (2.17)) where vφ 1 is the Fourier transformed symbol of the product V B 1 .This implies that -in contrast to (2.21) -the equation for φ2 (p, q), involves an extra integral on the r.h.s. and is solved with the ansatz (using (2.22) for φ1 (p ′ , p)) (2.40) The scalar coefficients c j , j = 1, ..., 7 (depending on q, p and p ′ ) are uniquely determined.The matter of interest is, however, not the explicit form of B 2 or generally, of B n , n ≥ 2, but the existence of the potentials W n and V n on the form domain H 1/2 (R 3 ) ⊗ C 4 of the free Dirac operator, such that the expectation value of H (n) from (2.9) is well defined.Lemma 2.2.Let U n = e iBn , n ≥ 1, be the transformations from Theorem 1.1.Let φ n be the symbol of B n and W n the potential in the defining equation for with c ∈ R + , and B n extends to a bounded operator on From the inductive definition of the transformation scheme it is easily seen that W n is symmetric.As a consequence of (2.8), B n is symmetric and hence self-adjoint, such that U n is unitary for all n ≥ 1. Proof.

a) p-form boundedness of W n
The proof is by induction.The p-form boundedness of W 1 , eq. (2.18), follows from Kato's inequality [10] and from the self-adjointness and boundedness of D0 , Without loss of generality one may assume that [•] is symmetric.This guarantees the symmetry property (2.31) for the Fourier transformed symbol [•](p, q).We estimate | and apply the Lieb and Yau formula for this symmetrised kernel.Then, using that a symbol can be estimated by its adjoint and vice versa, p-form boundedness can be expressed in the following way with some constant c > 0. The inequality in the second line of (2.43) restricts the convergence generating function to f (p) := p λ with 1 2 < λ < 3 2 .This is true because both [•](q − p, p) and [•](p − q, q) are regular for q → 0 (since all operators of which [•] is composed have symbols which are regular when the second variable tends to zero), restricting λ < 3  2 , and because [•] is of dimension (momentum) −2 , decreasing like q −2 for q → ∞, such that λ > 1 2 is required.These properties hold also for the symmetric operator W k .Thus for k < n when W k is p-form bounded, (2.43) therefore is also valid for its Fourier transformed symbol ŵk in place of [•].Another ingredient in the proof of p-form boundedness of W n is the fact that the symbol φ n can be estimated by w n .First note that the estimate of φn is related to the estimate of ŵn by an equation of the type (2.39), derived from the defining equation (2.8).This equation implies that the behaviour of φn for p → 0 and q → 0 is that of ŵn , while there occurs an extra power of q −1 and p −1 for q → ∞ and p → ∞, respectively.Therefore From the symmetry of V, B n−1 and (2.15) we get the convergence generating function and estimating v one gets (2.46) The last q-integral is evaluated with the substitution q ′ := q/p ′ and with e p ′ := p ′ /p ′ , dq (2.47) Also φn−1 is estimated by ŵn−1 via (2.44).One obtains for the second term in curly brackets of (2.46) with the help of the p-form boundedness of W n−1 by means of (2.43) where |p − p ′ | + p ′ + 1 ≥ p ′ has been used.Further we note that the factor (|p ′ − q| + q + 1) −1 is bounded for all q ≥ 0 and hence can be estimated by its value at q = 0. We thus get for the other q-integral such that by means of (2.47), the first term in the curly brackets of (2.46) is estimated by c ′ p with some constant c ′ ∈ R + .This proves the p-form boundedness of [V, In (2.45), V and B n−1 are replaced with [•] and B k , respectively, and the expression in curly brackets (integrated over p ′ ) is taken as the kernel K(q, p) in the Lieb and Yau formula (2.30).Then, estimating the symbol by its adjoint and vice versa, one arrives at Documenta Mathematica 10 (2005) 331-356 (2.50) By (2.43), the last q-integral is bounded by cp ′ such that the second term in the curly brackets can be estimated by 51) according to (2.48) because ŵk is p-form bounded.By (2.49), the other q-integral is estimated by a constant c, such that the first term of (2.50) in curly brackets is with (2.43) estimated by p • c ′′ with some constant c ′′ .This proves the p-form boundedness of [[•], B k ], k < n − 1, and hence together with (i) the p-form boundedness of W n .
From the p-form boundedness of W n , n ≥ 1, on H 1/2 (R 3 ) ⊗ C 4 , proven above, follows immediately the p-form boundedness of V n , n ≥ 1 since both operators differ only by factors D0 .We have therefore proven that to arbitrary order n, This is a consequence of the p-form boundedness of W n .From (2.43) with [•] replaced by B n one gets where the convergence generating function is again chosen as f (p) := p. From (2.49) the q-integral is estimated by a constant.Hence, (2.54) Remark.Due to logarithmic divergencies occurring in the estimates of ŵn (q, p), n ≥ 1, the proof of boundedness of B n cannot be based on the algebra of symbol estimates, a powerful method in the case of periodic potentials [13].

2.4
The Remainder R (n+1) and its p-form boundedness From its definition as remainder after multiple iterations of (2.3)-type equations (see e.g.(2.11)), R (n+1) is composed of a finite number of compact integrals over a unitary transform of the same multiple commutators [•] which would contribute to the n + 1st order term V n+1 after one additional transformation (for the commutator involving D 0 , use (2.8)).These commutators are p-form bounded according to the proof of Lemma 2.2, and it remains to show that the unitary transform preserves the p-form boundedness.Consider (2.55) Since U k (τ ) = e iB k τ with B k a bounded operator, we can Taylor expand (2.56) The sum on the r.h.s.represents a symmetric operator such that its kernel has the required symmetry property to apply the Lieb and Yau formula (with convergence generating function f (p) = p).Our proof proceeds in 4 steps: We prove p-form boundedness of (i) According to (2.32) we establish boundedness of an operator A by means of boundedness of the integral I A over its Fourier transformed symbol ŝA .For B k , we have boundedness from (2.49), and use (2.17) for the symbol of a product of operators.Then with (2.57), Then, using (2.60) Thus we obtain from the Lieb and Yau formula applied to (2.56) with c a constant resulting from using the same estimate for symbol and its adjoint.This shows that (ϕ, U k (−τ )pU k (τ )ϕ) is p-form bounded and completes the proof since exp(c ′ k τ + c k τ ) is a continuous function of τ.With the same reasoning, any multiple finite-dimensional compact integral over multiple unitary transforms of p-form bounded commutators is therefore again p-form bounded.
3 Subordinacy of the higher-order contributions Since to any order n the p-form bound of H (n) is proportional to γ n while for the remainder R (n+1) it is proportional to γ n+1 , one gets convergence of the expansion in the strength of the Coulomb field for γ → 0. However, for larger γ < 1, the p-form bounds of V n obtained with the above estimates can in general not be restricted to numbers less than 1.In this section we will consider a slight modification of the Coulomb potential, where v(q) is the Fourier transform and f ǫ := cos πǫ 2 • Γ(1 + ǫ) → 1 for ǫ → 0. All quantities defined previously will now pertain to the modified potential (3.1).Our results are collected in the following proposition.Proposition 3.1.For the modified Coulomb potential (3.1) we have (i) For every k ∈ N, ǫ < 1 k+1 , the k-th order potential term V k is p-form bounded with form bound less than 1.
be the remainder of order n + 1 in the potential strength.Then R (n+1) is subordinate to V n .
For the proof, a lemma is needed.

Proof of Proposition.
We start by showing that |(ϕ, V k ϕ)| ≤ c (ϕ, p 1−kǫ ϕ) with some constant c > 0, such that the definition of µ k in (ii) makes sense.Since 0 < (ϕ, p 1−kǫ ϕ) < ∞ (for ϕ = 0), µ k = 0 implies (ϕ, V k ϕ) = 0 which means that in this case V k does not contribute to the expectation value of the transformed Dirac operator (2.9) and hence can be disregarded.First we estimate the expectation value of V 1 .According to (2.21), the symbol of V 1 is given by v1 (q, p) = v(q) − ŵ1 (q, p) = − γ0 q 2+ǫ f ǫ (1 + D0 (q + p) • D0 (p)) .Since the multiplier of q −(2+ǫ) is a bounded operator which is estimated by a constant, one finds according to (2.32) and (2.33) with f (p) := p Documenta Mathematica 10 (2005) 331-356 With the substitution p q ′ := (q − p) one obtains for the latter integral 2 (ϕ, p 1−ǫ ϕ).The proof of (ii) is by induction.First we show (ii) for k = 1.From (2.12) we have Following the argumentation given in section 2.3 one can disregard the bounded operators D0 in the estimates of p-form boundedness and consider V 2 as being represented by the commutator [V, B 1 ].With the modified Coulomb potential the symbol of B 1 which is proportional to ŵ1 (q, p) according to (2.21), is estimated by For I 11 , the q ′ -integral is estimated by dropping 1 p ′ in the last factor of the denominator.Using (2.47) together with the substitution k := q ′ − e p ′ , one finds (3.11)The p ′ -integral results with the substitution p p ′′ := p ′ and with the same techniques for the angular integration as applied in (2.34), in with some constant c.Using (3.7) and the definition of µ k one thus obtains with 2c 0 δ/µ 1 < 1 for a suitably chosen δ.This proves (ii) of Proposition 3.1 for k = 1.
The proof of the induction step from k to k + 1 proceeds along the same lines as applied in section 2.3 to show the p-form boundedness of V n and W n .By induction hypothesis commutators of order m ≤ k in the potential strength, denoted by [•] m , have the following symbol estimates (compare (2.43)) where λ can be chosen in the interval ] 1 2 , 3 2 [ .We demonstrate the proof for the commutator [[•] m , B k−m+1 ] which contributes to V k+1 .For the commutator [V, B k ] which also contributes to V k+1 the proof is similar.Since the symbol classes of W m and [•] m are equal, it follows from (2.44) Then from (2.50), one has with some .17) where in the term denoted by I 00 , | φk−m+1 (p ′ − q, q)| was estimated by its adjoint before applying (3.16).In I 01 , the q-integral is by (3.15) estimated by c p ′ 1−mǫ .Further one has with a ≥ 0 and δ > 0 Then with δ := mǫ, I 01 ≤ c p 1−(k−m+1)ǫ p −mǫ = c p 1−(k+1)ǫ .In I 00 we estimate in the denominator |q−p ′ |+p ′ +1 by p ′ and subsequently use (3.15) to estimate the q-integral by c p which proves (3.15) for k + 1.The same estimate can be shown for which completes the proof of Proposition 3.1 (ii).The proof of (i) for k > 1 is again by induction.Assume V k is p-form bounded with form bound c 1 < 1.Then we have from (ii) (3.21) Since δ can be chosen arbitrarily small, one has δc 1 < 1.A consequence of (3.21) is the p-form boundedness (with form bound < 1) of every finite sum For the proof of (iii) we have to show that all B k are bounded operators.Then we can proceed as in section 2.4 to show that a unitary transform U k = e iB k τ preserves the p 1−(n+1)ǫ -form boundedness of the commutators of order n + 1 in the potential strength of which R (n+1) is consisting.Consequently, one has with (ϕ, with const • δ < 1 for a suitably chosen δ.This shows the subordinacy with respect to V n .It remains to prove the boundedness of B k .We will show this directly by using the algebra of symbol estimates.For B 1 , from (2.57) with the substitution q ′ := q − p, since the integral is finite for p → 0 and for p → ∞ and the singularity of the last factor at p = −q ′ is integrable.The convergence generating function was chosen to allow for a (2.60)-type estimate when showing that the presence of U k plays no role (but to prove boundedness of I B1 , one can also take f (p) = 1).For B 2 , we use the estimate (2.44) of φ2 by ŵ2 and recall that W 2 is determined from the commutator [V, B 1 ].Consider the symbol of V B 1 via (2.17), (3.24) It is found that | vφ 1 (q, p)| = const for p = 0, ∼ 1/p 2 for p → ∞ and ∼ 1/q 2+ǫ for q → ∞ while it diverges for q → 0. The behaviour near q = 0 is obtained by performing the angular integration with the help of a (2.34)-type formula such that one gets for q = 0, ǫ = 0, (3.25) Since the divergence at q = 0 results from the behaviour of the integral near p ′ = 0, it is sufficient to reduce the integration region to [0, 1] and estimate (p ′ + p + 1) −2 ≤ 1.The resulting integral can be performed analytically with the help of hypergeometric functions [7, p.284], and it behaves ∼ q 1−2ǫ for q → 0. B 1 V is in the same operator class such that we obtain | ŵ2 (q, p)| ≤ c 1 + q ǫ q 2ǫ (q + p + 1) 2 . (3.26) By induction, one can show that for k > 2, one has | ŵk (q, p)| ≤ c (q+p+1) 2+ǫ .Thus one obtains regularisation upon increasing k, resulting in bounded operators B k , k > 1. Proposition 3.1 provides justification for representing the transformed Dirac operator in terms of a series expansion in the potential strength.Note, however, that the limit ǫ → 0 cannot be carried out since in (3.7), ǫ 1−nǫ δ − 1−nǫ ǫ → ∞ as ǫ → 0, which implies C → ∞ in (3.3).Therefore, this limit cannot be used to prove for the Coulomb potential the p-form boundedness of V k , 1 ≤ k ≤ n with form bound < 1.On the other hand, it has been shown with different tools that this property holds for V 1 and V 1 + V 2 in case of the Coulomb field [2,8].

Proof of Theorem 1.2
The link between the two transformation schemes under consideration is provided by the following lemma.Lemma 4.1.Let H = D 0 + V and U j = e iBj , j = 1, ..., n be the transformation scheme from section 2.1, where the potential term of k-th order in γ is decomposed into V k + W k .Let U ′ 0 , U ′ j = e −iSj , j = 1, ..., n be the Douglas-Kroll transformation scheme with the decomposition E k + O k of the k-th order potential term.Then one has the identification with U ′ 0 from (1.5).
The key observation is the relation between the spinor ψ = u In momentum space, this equation is easily verified from U ′ −1 0 = ( αp Ep+m β + 1) A and from the explicit form (1.12) of φ(p).Then with the help of (2.6), the assertion (1.16) of Theorem 2.1 reads Identifying terms of fixed order k ≤ n and using (1.11), the assertion (4.3) is a consequence of = βE p is a consequence of (1.7) for zero potential.By means of explicit calculation (which for the sake of simplicity is only presented for the massless case m = 0), one gets from (2.18) and similarly, .42) where in the second term, D0 p D0 = p has been used.Let n ≥ 2. According to the transformation scheme outlined in section 2.1,W n is composed of multiple commutators of V with B k , k < n.In particular for n = 2, [V, B 1 ] contributes (see (2.37)) and for n = 3, one needs [[V, B 1 ], B 1 ] and [V, B 2 ].The additionally occurring factors D0 can be disregarded in the context of p-form boundedness since D0 is a bounded multiplication operator in momentum space.In the general case, the product of all factors B k , k ≤ n−1, which enter into a given commutator contributing to W n must be proportional to γ n−1 since W n is of the order γ n .By induction hypothesis, W k is p-form bounded on H 1/2 (R 3 ) ⊗ C 4 for k ≤ n − 1.This means that all commutators of smaller order than n in γ are p-form bounded.In the induction step one has to show that [V, B n−1 ] and [[•], B k ], k < n − 1, are p-form bounded where [•] denotes a p-form bounded multiple commutator.

0
and the spinor ϕ in the positive spectral subspace of D 0 ,