The Projected Single-Particle Dirac Operator

for Coulombic Potentials

D. H. Jakubassa-Amundsen

Received: January 22, 2003 Revised: March 24, 2004

Communicated by Ulf Rehmann

Abstract. 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 Λ_{+}A Λ_{+} + Λ_{−}A Λ_{−} to any given order in the potential strength, where Λ_{+} and Λ_{−} project onto the positive and negative spectral subspaces of the free Dirac operator. To first order, Λ_{+}A Λ_{+} coincides with the Brown-Ravenhall operator. Moreover, there exists a simple relation to the Dirac operator transformed 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 potential, V = −γx^{−1+ϵ}, ϵ > 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.

2000 Mathematics Subject Classification: 81Q10, 81Q15, 81V45

1 Introduction

Consider a relativistic electron in the Coulomb field V, described by the Dirac operator (in relativistic units, ℏ = c = 1)

H = D_{0} + V,  D_{0} := −iα ∂/∂x + βm, V (x) := −\frac{γ}{x}  (1.1)

where D_{0} is the free Dirac operator defined in the Hilbert space L_{2}(ℝ^{3}) ⊗ ℂ^{4}. D_{0} is self-adjoint on the Sobolev space H_{1}(ℝ^{3})⊗ℂ^{4} (with H_{σ}(ℝ^{3}) := {φ ∈ L_{2}(ℝ^{3}) : ∫ dp |φ̂(p)|^{2} (1 + p^{2})^{σ} < ∞}), and its form domain is H_{1/2}(ℝ^{3}) ⊗ ℂ^{4}. The potential strength of V is γ := Ze^{2}, Z is the nuclear charge number, e^{2} = (137.04)^{−1} the fine structure constant, α and β the Dirac matrices and x := |x| [14]. A hat on a function denotes its Fourier transform. 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

B := Λ_{+}HΛ_{+},  Λ_{±} := \frac{1}{2} (1 ± \frac{D_{0}}{|D_{0}|})  (1.2)

with Λ_{+} projecting onto the positive spectral subspace of the free Dirac operator D_{0}, and |D_{0}| = \sqrt{D_{0}^{2}} is the free energy. In momentum space one has

D̃_{0}(p) := (\frac{D_{0}}{|D_{0}|}) (p) = \frac{αp + βm}{E_{p}},  E_{p} := \sqrt{p^{2} + m^{2}}  (1.3)

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

(U′_{n} ⋯ U′_{1} · U′_{0}) H (U′_{n} ⋯ U′_{1} · U′_{0})^{−1} =: H′_{n} + R_{n+1}  (1.4)

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],

U′_{0} := A (1 + β \frac{αp}{E_{p} + m}) ,  A := (\frac{E_{p} + m}{2E_{p}})^{\frac{1}{2}} ,  (1.5)

and for U′_{j}, Douglas and Kroll [4] use

U′_{j} = (1 + W_{j}^{2})^{\frac{1}{2}} + W_{j},  j = 1, ..., n  (1.6)

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_{0}^{′−1} = βE_{p} + E_{1} + O_{1}

E_{1} := A (V + \frac{αp}{E_{p}+ m}V \frac{αp}{E_{p}+ m}) A, O_{1} := βA (\frac{αp}{E_{p}+ m}V − V \frac{αp}{E_{p}+ m}) A

(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,

U′_{j} := e^{−iS_{j}},  j = 1, ..., n  (1.8)

with a symmetric operator S_{j}, the next transformation gives in agreement with [9]

e^{−iS_{1}} U′_{0}HU_{0}^{′−1} e^{iS_{1}} = βE_{p} + E_{1} + O_{1} + i[βE_{p}, S_{1}] + i[E_{1} + O_{1}, S_{1}]

−\frac{1}{2} [ [βE_{p}, S_{1}], S_{1}] + R_{3}.  (1.9)

S_{1} is defined from the requirement that O_{1} is eliminated,

i [βE_{p}, S_{j}] = −O_{j},  j = 1  (1.10)

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^{−iS_{j+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

H′_{n} = βE_{p} + E_{1} + ... + E_{n}.  (1.11)

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 ∈ H_{1/2}(ℝ^{3})⊗ℂ^{2},

φ̂(p) = \frac{1}{\sqrt{2E_{p} (E_{p} + m)}} \binom{(E_{p} + m) û(p)}{pσ û(p)}  (1.12)

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 := \binom{u}{0} with u ∈ H_{1/2}(ℝ^{3}) ⊗ ℂ^{2} as above.

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.

(u, b_{m}u) = (\binom{u}{0}, H′_{1}\binom{u}{0}) ,  u ∈ H_{1/2}(ℝ^{3}) ⊗ ℂ^{2},  (1.14)

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

(U_{1} ⋯ U_{n})^{−1}H U_{1} ⋯ U_{n} =: H^{(n)} + R^{(n+1)}

H^{(n)} := Λ_{+} (∑_{k=0}^{n} H_{k}) Λ_{+} + Λ_{−} (∑_{k=0}^{n} H_{k}) Λ_{−}.  (1.15)

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).

Theorem 1.2. Let φ ∈ Λ_{+}(S(ℝ^{3}) ⊗ ℂ^{4}) 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),

(φ, H^{(n)} φ) = (φ, ∑_{k=0}^{n} H_{k} φ) = (\binom{u}{0}, H′_{n}\binom{u}{0}) , n = 1, 2, ... .

(1.16)

This extends the first-order equation (1.14) to arbitrary order n. Actually (1.16) holds also for other types of unitary operators, provided the same type is used in all transformations U_{k} and U′_{k}, k ≥ 1.

2 Proof of Theorem 1.1

2.1 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

\frac{d}{dt}A(t) := \frac{d}{dt} (e^{−iB_{k}t} A e^{iB_{k}t}) = i U_{k}(−t) [A, B_{k}] U_{k}(t)  (2.1)

where the commutator [A, B_{k}] := AB_{k} − B_{k}A. This equation is easily integrated, noting that A(0) = A,

A(t) = U_{k}(−t) A U_{k}(t) = A + i ∫_{0}^{t} dτ U_{k}(−τ) [A, B_{k}] U_{k}(τ).  (2.2)

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

A(1) = A + i[A, B_{k}] + i^{2} ∫_{0}^{1} dτ ∫_{0}^{τ} dt′ U_{k}(−t′) [[A, B_{k}], B_{k}] U_{k}(t′). (2.3)

After n iterations the following representation of A_{1} is obtained,

A(1) = A + i[A, B_{k}] + \frac{1}{2!}i^{2} [[A, B_{k}], B_{k}] + ... + \frac{1}{n!}i^{n} [[...[A, B_{k}], ..., B_{k}] + R

(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,

(U_{1} ⋯ U_{n−1})^{−1} H U_{1} ⋯ U_{n−1} = H^{(n−1)} + H_{n} + R̃^{(n+1)}  (2.5)

where H_{n} and R̃^{(n+1)} are respectively of order γ^{n} and γ^{n+1}, and still couple the spectral subspaces. Decompose H_{n} into

H_{n} = V_{n} + W_{n},  V_{n} := Λ_{+}H_{n}Λ_{+} + Λ_{−}H_{n}Λ_{−}
W_{n} := Λ_{+}H_{n}Λ_{−} + Λ_{−}H_{n}Λ_{+} . (2.6)

The next transformation, U_{n} = e^{iB_{n}}, 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

U_{n}^{−1} (H^{(n−1)} + H_{n}) U_{n} = H^{(n−1)} + V_{n} + W_{n} + i[D_{0}, B_{n}]  (2.7)

+ i [(Λ_{+} ∑_{k=1}^{n−1} H_{k}Λ_{+} + Λ_{−} ∑_{k=1}^{n−1} H_{k}Λ_{−}), B_{n}] + R̃,

where R̃ collects the terms containing at least two commutators with B_{n}. B_{n} is determined from the requirement

W_{n} + i [D_{0}, B_{n}] = 0.  (2.8)

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_{+}H_{k}Λ_{+} + Λ_{−}H_{k}Λ_{−}), B_{n}] are of order γ^{n+k} with k ≥ 1, and R̃ is of order γ^{2n}. Hence, these terms are disregarded (together with the remainder R̃^{(n+1)} from (2.5)) in constructing the transformed operator to order n,

H^{(n)} = H^{(n−1)} + V_{n} = D_{0} + V_{1} + V_{2} + ... + V_{n}.  (2.9)

Particularly interesting are the cases n = 1 and n = 2. For n = 1, we have

H^{(1)} = D_{0} + V_{1} = Λ_{+} (D_{0} + V) Λ_{+} + Λ_{−} (D_{0} + V) Λ_{−}.  (2.10)

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^{iB_{1}}, only. However, the existence of the second transformation, U_{2} = e^{iB_{2}}, has to be established to show that H^{(2)} is indeed the transformed operator, with a remainder of order γ^{3}. We have

U_{1}^{−1} H U_{1} = D_{0} + V_{1} + W_{1} + i [D_{0}, B_{1}] + i [V, B_{1}] − \frac{1}{2}[[D_{0}, B_{1}], B_{1}] + R,

R = − ∫_{0}^{1} dτ ∫_{0}^{τ} dt′ U_{1}(−t′) [[V, B_{1}], B_{1}] U_{1}(t′)  (2.11)

− i ∫_{0}^{1} dτ ∫_{0}^{τ} dt′ ∫_{0}^{t′} dτ′ U_{1}(−τ′) [[[D_{0}, B_{1}], B_{1}], B_{1}] U_{1}(τ′).

Making use of the defining relation for B_{1}, W_{1} + i[D_{0}, B_{1}] = 0, the operator H^{(2)} takes the form

H^{(2)} = D_{0} + V_{1} + Λ_{+}H_{2}Λ_{+} + Λ_{−}H_{2}Λ_{−}  (2.12)

H_{2} := i [V_{1}, B_{1}] + \frac{i}{2}[W_{1}, B_{1}].

2.2 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

(W φ)(x) := \frac{1}{(2π)^{3/2}} ∫ dp e^{ipx} w(x, p) φ̂(p)  (2.13)

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

(W φ)(x) = \frac{1}{(2π)^{3}} ∫ dp e^{ipx} ∫ dq e^{iqx} ŵ(q, p) φ̂(p).  (2.14)

From this, the Fourier transform of W φ is found

(Ŵ φ)(p′) = \frac{1}{(2π)^{3/2}} ∫ dp ŵ(p′ − p, p) φ̂(p).  (2.15)

With φ in (2.14) replaced by Gφ, the symbol of a product W G of two integral operators is derived,

(W G φ)(x) = \frac{1}{(2π)^{3}} ∫ dp′ e^{ip′x} ∫ dp ŵ(p′ − p, p) Ĝφ(p).  (2.16)

Using (2.15) for Ĝφ(p), as well as the definition (2.14) of the Fourier transformed symbol ŵg of W G, one gets

ŵg(q, p) = \frac{1}{(2π)^{3/2}} ∫ dp′ ŵ(q − p′, p + p′) ĝ(p′, p).  (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 W_{1} = Λ_{+}V Λ_{−}+ Λ_{−}V Λ_{+} and (1.2),

−i [D_{0}, B_{1}] = W_{1} = \frac{1}{2} (V − \frac{D_{0}}{|D_{0}|} V \frac{D_{0}}{|D_{0}|}) .  (2.18)

Let ϕ_{1} be the symbol of B_{1}. From (2.14) and with D_{0} from (1.1) one has

(D_{0} B_{1} φ)(x) = \frac{1}{(2π)^{3}} ∫ dp dq D_{0} e^{i(p+q)x} ϕ̂_{1}(q, p) φ̂(p)

= \frac{1}{(2π)^{3}} ∫ dp dq [α(p + q) + βm] e^{i(p+q)x} ϕ̂_{1}(q, p) φ̂(p).  (2.19)

The Fourier transforms of D_{0}φ and of V φ are, respectively, obtained from

(D̂_{0}φ)(p) = (αp + βm) φ̂(p)
(V φ)(x) = \frac{1}{(2π)^{3/2}} ∫ dq e^{iqx} (−\frac{γ}{2π^{2}q^{2}} ) ∫ dp e^{ipx} φ̂(p)  (2.20)

such that the symbol v of V is defined by v̂(q, p) = −\sqrt{2/π} γ/q^{2}. Acting (2.18) on φ and equating the respective symbols leads to the following algebraic equation for ϕ̂_{1} :

[α(p + q) + βm] ϕ̂_{1}(q, p) − ϕ̂_{1}(q, p) [αp + βm] = i ŵ_{1}(q, p) (2.21)

= −\frac{iγ_{0}}{q^{2}} [1 − D̃_{0}(q + p) · D̃_{0}(p)]

with γ_{0} := γ/\sqrt{2π} and D̃_{0}(p) the operator from (1.3) with norm unity. ŵ_{1}(q, p), behaving like q^{−1} for q → 0, is less singular than v̂(q, p), such that the prescription (2.6) for W_{1} implies a regularisation of the potential V.

Lemma 2.1. A solution ϕ̂_{1}(q, p) for the symbol of B_{1} is given by

ϕ̂_{1}(q, p) = −\frac{iγ_{0}}{q^{2}} \frac{1}{E_{p} + E_{|q+p|}} (D̃_{0}(q + p) − D̃_{0}(p))  (2.22)

which satisfies the condition for symmetry of B_{1} [13],

ϕ̂_{1}(−q, p + q)^{∗} = ϕ̂_{1}(q, p).  (2.23)

It is estimated by

|ϕ̂_{1}(q, p)| ≤ \frac{c}{q} \frac{1}{(q + p + 1)^{2}}  (2.24)

with some constant c ∈ ℝ_{+}. B_{1} is a bounded operator on L_{2}(ℝ^{3}) ⊗ ℂ^{4}.

Proof. a) Calculation of ϕ̂_{1}.

In order to solve (2.21) the ansatz is made

ϕ̂_{1}(q, p) = −\frac{iγ_{0}}{q^{2}} (c_{1}αq + c_{2}αp + c_{3}β)  (2.25)

and from the properties of the Dirac matrices β^{2} = 1, α_{i}^{2} = 1, βα_{i} = −α_{i}β, i = 1, 2, 3, α_{i}α_{k} = −α_{k}α_{i} (i ≠ k), the following identities are derived

αp · αp = p^{2},  αq · αp = 2pq − αp · αq.  (2.26)

Insertion of (2.25) into (2.21) then leads to an equation of the type

λ_{1} αp · αq + λ_{2} αq · β + λ_{3} αp · β + λ_{4} = 0  (2.27)

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,

c_{1} (q^{2}+2pq) = 1 − \frac{E_{p}}{E_{|q+p|}} ,  c_{2} = 2c_{1} − \frac{1}{E_{p} E_{|q+p|}} ,  c_{3} = c_{2}m

(2.28)

such that

ϕ̂_{1}(q, p)=−\frac{iγ_{0}}{q^{2}} [[(q+2p)α+2βm]\frac{1}{q^{2}+2pq}(1− \frac{E_{p}}{E_{|q+p|}} )− \frac{pα + βm}{E_{p}E_{|q+p|}}] .

(2.29)

It is readily verified that (2.29) can be cast into the form (2.22), proving that ϕ̂_{1}(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 ϕ̂_{1}(q, p). By inspection of (2.22) one finds that ϕ̂_{1}(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^{iB_{1}} and (2.13)) whence ϕ̂_{1}(q, p) is of dimension (momentum)^{−3}, the estimate |ϕ̂_{1}(q, p)| ≤ c/q · (q + p + 1)^{−2} is obtained. The constant c is determined by the maximum of q · |ϕ̂_{1}(q, p)| in q, p ∈ ℝ_{+} (which exists due to continuity) and by the decay constants of ϕ̂_{1}(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,

|∫ dq dp \overline{φ̂(q)} |K(q, p) | φ̂(p) |  (2.30)

≤(∫ dqdp|φ̂(p)|^{2}|K(q, p)| |\frac{f(p)}{f(q)}|^{2})^{\frac{1}{2}}· (∫ dqdp|φ̂(p)|^{2}|K(p, q)| |\frac{f(p)}{f(q)}|^{2})^{\frac{1}{2}}

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,

|ϕ̂_{1}^{∗}(q − p, p)| = |ϕ̂_{1}(p − q, q)| ↦^{q↔p} |ϕ̂_{1}(q − p, p)|.  (2.31)

One then obtains for φ ∈ L_{2}(ℝ^{3}) ⊗ ℂ^{4} applying (2.15) and subsequently (2.30) and (2.31)

|(φ, B_{1}φ)| = |(φ̂, B̂_{1}φ)| ≤ \frac{1}{(2π)^{\frac{3}{2}}} ∫ dq |\overline{φ̂(q)}| ∫ dp |ϕ̂_{1}(q − p, p)| |φ̂(p)|

(2.32)

≤ \frac{1}{(2π)^{\frac{3}{2}}} (∫ dq dp |φ̂(p)|^{2} |ϕ̂_{1}(q − p, p)| |\frac{f(p)}{f(q)}|^{2})^{\frac{1}{2}}

· (∫ dq dp |φ̂(p)|^{2} |ϕ̂_{1}^{∗}(q − p, p)| |\frac{f(p)}{f(q)}|^{2})^{\frac{1}{2}} .

Both symbol and its adjoint can be estimated by the same expression (2.24) (from (2.22) one even has ϕ̂_{1}^{∗}(q, p) = −ϕ̂_{1}(q, p)), yielding

|(φ, B_{1}φ)| ≤ \frac{c}{(2π)^{\frac{3}{2}}} ∫ dp |φ̂(p)|^{2} · I_{1}(p)

I_{1}(p) := ∫ dq \frac{1}{|q − p|} \frac{1}{(|q − p| + p + 1)^{2}} |\frac{f(p)}{f(q)}|^{2} .  (2.33)

Documenta Mathematica 10 (2005) 331–356

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

∫_{−1}^{1} dx \frac{1}{\sqrt{b + ax}} \frac{1}{(\sqrt{b + ax} + p + 1)^{2}} = \frac{2}{a} ∫\begin{matrix}√ & \overline{b+a} \\ √ & \overline{b−a} \end{matrix} \frac{dz}{(z + p + 1)^{2}}

= \frac{2}{a} (\frac{1}{\sqrt{b − a} + p + 1} − \frac{1}{\sqrt{b + a} + p + 1})  (2.34)

identifying |q − p| = \sqrt{q^{2} + p^{2} − 2qpx} =: \sqrt{b + ax}, x := cos ϑ_{q,p}. Choosing f(p) := p^{\frac{1}{2}} , one obtains

I_{1}(p) = 2π ∫_{0}^{∞}dq (\frac{1}{|q − p|+p+1} − \frac{1}{q+2p+1}) = 4π ln \frac{2p + 1}{p + 1} < ∞.

(2.35)

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^{iB_{1}} is unitary.

2.3 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

−i[D_{0}, B_{2}]=W_{2} =iΛ_{+}[[V_{1}, B_{1}]+ \frac{1}{2}[W_{1}, B_{1}]]Λ_{−} + iΛ_{−}[[V_{1}, B_{1}]+ \frac{1}{2}[W_{1}, B_{1}]]Λ_{+}.

(2.36)

With W_{1} from (2.18) and V_{1} = V − W_{1} one obtains

W_{2} = \frac{i}{8} (3 [V, B_{1}] + [D̃_{0}, V D̃_{0} B_{1}] + [D̃_{0}, B_{1}D̃_{0}V] + 3 D̃_{0} [B_{1}, V] D̃_{0}) .

(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 ŵ_{2} of W_{2} is composed of expressions of the type (cf (2.17))

v̂ϕ_{1}(q, p) = −\frac{γ}{2π^{2}} ∫ dp′ \frac{1}{|q − p′|^{2}} ϕ̂_{1}(p′, p)  (2.38)

where v̂ϕ_{1} is the Fourier transformed symbol of the product V B_{1}. This implies that — in contrast to (2.21) — the equation for ϕ̂_{2}(p, q),

[α (p + q) + βm] ϕ̂_{2}(q, p) − ϕ̂_{2}(q, p) [αp + βm] = i ŵ_{2}(q, p), (2.39)

involves an extra integral on the r.h.s. and is solved with the ansatz (using (2.22) for ϕ̂_{1}(p′, p))

ϕ̂_{2}(q, p) = −\frac{iγγ_{0}}{16π^{2}} ∫ dp′ \frac{1}{p^{′2}} \frac{1}{|q − p′|^{2}} (c_{1} + c_{2} αp · βm

+ c_{3} αq · βm + c_{4} αp′ · βm + c_{5} αp′ · αp + c_{6} αq · αp + c_{7} αq · αp′) .

(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^{iB_{n}}, 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 ϕ_{n}. Then W_{n} is p-form bounded on H_{1/2}(ℝ^{3}) ⊗ ℂ^{4} by means of

|(φ, W_{n}φ)| ≤ c (φ, p φ)  (2.41)

with c ∈ ℝ_{+}, and B_{n} extends to a bounded operator on L_{2}(ℝ^{3}) ⊗ ℂ^{4}.

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 D̃_{0},

|(φ, W_{1}φ)| ≤ \frac{1}{2} |(φ, V φ)| + \frac{1}{2} |(D̃_{0}φ, V D̃_{0}φ)| ≤ \frac{γπ}{4} (φ, pφ) + \frac{γπ}{4} (φ, pφ)

(2.42)

where in the second term, D̃_{0}pD̃_{0} = 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 D̃_{0} can be disregarded in the context of p-form boundedness since D̃_{0} 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. Without loss of generality one may assume that [·] is symmetric. This guarantees the symmetry property (2.31) for the Fourier transformed symbol ˆ[·](p, q). We estimate | ˆ[·]| ≤ | ˆ[·]| + | ˆ[·]^{∗}| 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

|(φ, [·]φ)|≤ \frac{1}{(2π)^{\frac{3}{2}}} ∫ dp |φ̂(p)|^{2}∫ dq( | ˆ[·](q−p, p)| +| ˆ[·]^{∗}(q−p, p)|)|\frac{f(p)}{f(q)}|^{2}

≤ \frac{1}{(2π)^{3/2}} ∫ dp |φ̂(p)|^{2} c p = c′ (φ, p φ)  (2.43)

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 \frac{1}{2} < λ < \frac{3}{2}. This is true because both ˆ[·](q − p, p) and ˆ[·](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 < \frac{3}{2}, and because ˆ[·] is of dimension (momentum)^{−2}, decreasing like q^{−2} for q → ∞, such that λ > \frac{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 ŵ_{k} in place of ˆ[·]. 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 ϕ̂_{n} is related to the estimate of ŵ_{n} by an equation of the type (2.39), derived from the defining equation (2.8). This equation implies that the behaviour of ϕ̂_{n} for p → 0 and q → 0 is that of ŵ_{n}, while there occurs an extra power of q^{−1} and p^{−1} for q → ∞ and p → ∞, respectively. Therefore

|ϕ̂_{n}(q, p)| ≤ \frac{c}{q + p + 1} |ŵ_{n}(q, p)|.  (2.44)

i) p-form boundedness of [V, B_{n−1}]

From the symmetry of V, B_{n−1} and (2.15) we get

|(φ, [V, B_{n−1}]φ)| ≤ |(V̂ φ, { \atop B_{n−1}φ})| + |({ \atop B_{n−1}φ}, V̂ φ)|

≤ \frac{1}{(2π)^{3}} ∫ dp′dp dq |φ̂(p)| \{|v̂^{∗}(p′ − p, p)| |ϕ̂_{n−1}(p′ − q, q)|

+ |ϕ̂_{n−1}^{∗}(p′ − p, p)| |v̂(p′ − q, q)|} |φ̂(q)|.  (2.45)

Due to the symmetry in p and q, the Lieb and Yau formula can be applied in the same way as in (2.32) and in (2.43). Using f(p) := p for the convergence generating function and estimating v̂ one gets

|(φ, [V, B_{n−1}] φ)| ≤ c ∫ dp |φ̂(p)|^{2} \{∫ dp′ \frac{1}{|p′ − p|^{2}} \frac{p^{2}}{p^{′2}} · ∫ dq

|ϕ̂_{n−1}(p′−q, q)| \frac{p^{′2}}{q^{2}} +∫ dp′ |ϕ̂_{n−1}(p − p′, p′)| \frac{p^{2}}{p^{′2}} ·∫ dq\frac{1}{|p′ − q|^{2}} \frac{p^{′2}}{q^{2}} \} .

(2.46)

The last q-integral is evaluated with the substitution q′ := q/p′ and with e_{p′} := p′/p′,

∫ dq\frac{1}{|q − p′|^{2}} \frac{p^{′2}}{q^{2}} = p′∫_{0}^{∞}dq′∫_{S^{2}}dΩ_{q′} \frac{1}{|q′−e_{p′}|^{2}} = 2πp′∫_{0}^{∞} \frac{dq′}{q′} ln |\frac{1 + q′}{1 − q′} |

= π^{3}p′.  (2.47)

Also ϕ̂_{n−1} is estimated by ŵ_{n−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)

∫dp′|ϕ̂_{n−1}(p−p′, p′)|\frac{p^{2}}{p^{′2}} ·π^{3}p′ ≤ c̃∫ dp′ \frac{1}{|p−p′|+p′+1} |ŵ_{n−1}(p−p′, p′)|\frac{p^{2}}{p′}

≤ c′′p  (2.48)

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

∫ dq |ϕ̂_{n−1}(p′ − q, q)| \frac{p^{′2}}{q^{2}} ≤ \frac{c̃}{p′ + 1} ∫ dq |ŵ_{n−1}(p′ − q, q)| \frac{p^{′2}}{q^{2}}

≤ \frac{c̃}{p′ + 1} · cp′ ≤ c̃c  (2.49)

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, B_{n−1}].

ii) p-form boundedness of [[·], B_{k}]

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

|(φ, [ [·], B_{k}] φ)| ≤ \frac{c̃}{(2π)^{3}} ∫ dp |φ̂(p)|^{2} · \{∫ dp′ | ˆ[·](p − p′, p′)| \frac{p^{2}}{p^{′2}}

·∫ dq |ϕ̂_{k}(p′−q, q)| \frac{p^{′2}}{q^{2}} +∫ dp′|ϕ̂_{k}(p−p′, p′)| \frac{p^{2}}{p^{′2}} ∫ dq | ˆ[·](p′−q, q)| \frac{p^{′2}}{q^{2}}\}.

(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

∫ dp′|ϕ̂_{k}(p − p′, p′)|\frac{p^{2}}{p′} ≤ c ∫ dp′ \frac{1}{|p −p′| +p′ +1}|ŵ_{k}(p − p′, p′)|\frac{p^{2}}{p′} ≤ c′p

(2.51)

according to (2.48) because ŵ_{k} is p-form bounded. By (2.49), the other q-integral is estimated by a constant c̃, 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 D̃_{0}. We have therefore proven that to arbitrary order n,

|(φ, (V_{1} + ... + V_{n}) φ)| ≤ c (φ, p φ)  (2.52)

for φ ∈ H_{1/2}(ℝ^{3}) ⊗ ℂ^{4}.

b) Boundedness of B_{n}

This is a consequence of the p-form boundedness of W_{n}. From (2.43) with [·] replaced by B_{n} one gets

|(φ, B_{n}φ)| ≤ \frac{c}{(2π)^{3/2}} ∫ dp |φ̂(p)|^{2} ∫ dq |ϕ̂_{n}^{∗}(q − p, p)| \frac{p^{2}}{q^{2}}  (2.53)

where the convergence generating function is again chosen as f(p) := p. From (2.49) the q-integral is estimated by a constant. Hence,

|(φ, B_{n}φ)| ≤ const (φ, φ).  (2.54)

Remark. Due to logarithmic divergencies occurring in the estimates of ŵ_{n}(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

|(φ, U_{k}(−τ) [·] U_{k}(τ) φ)| = |(U_{k}(τ) φ, [·] U_{k}(τ) φ)|
≤ c (U_{k}(τ) φ, p U_{k}(τ) φ) = c (φ, U_{k}(−τ) p U_{k}(τ) φ).  (2.55)

Since U_{k}(τ) = e^{iB_{k}τ} with B_{k} a bounded operator, we can Taylor expand

(φ, e^{−iB_{k}τ} p e^{iB_{k}τ} φ) ≤ ∑_{n,m=0}^{∞} \frac{τ^{n}}{n!} \frac{τ^{m}}{m!} |(φ, B_{k}^{n} p B_{k}^{m} φ)|.  (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) pB_{k}, (ii) pB_{k}^{m} (by induction), (iii) B_{k}pB_{k}^{m}, (iv) B_{k}^{n}pB_{k}^{m}. 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 ŝ_{A}. For B_{k}, we have boundedness from (2.49),

I_{B_{k}} := \frac{1}{(2π)^{3/2}} ∫ dq |ϕ̂_{k}(p − q, q)| \frac{p^{2}}{q^{2}} ≤ c_{k}.  (2.57)

p-form boundedness is proven by showing that the integrals I_{A} (with A := B_{k}^{n}pB_{k}^{m}) are proportional to p. (i)

I_{pB_{k}} = \frac{1}{(2π)^{3/2}} ∫ dq p |ϕ̂_{k}(p − q, q)| \frac{p^{2}}{q^{2}} ≤ p c_{k}.  (2.58)

(ii) Our induction hypothesis is I_{pB_{k}^{m}} ≤ pc_{k}^{m}. We decompose pB_{k}^{m+1} = pB_{k}^{m} · B_{k} and use (2.17) for the symbol of a product of operators. Then with (2.57),

I_{pB_{k}^{m+1}} = \frac{1}{(2π)^{3/2}} ∫ dq′ |ŝ_{pB_{k}^{m+1}}(p − q′, q′)| \frac{p^{2}}{q^{′2}}  (2.59)

≤ \frac{1}{(2π)^{3}} ∫ dq |ŝ_{pB_{k}^{m}}(p−q, q)| \frac{p^{2}}{q^{2}} ·∫ dq′ |ϕ̂_{k}(q−q′, q′)| \frac{q^{2}}{q^{′2}} ≤ p c_{k}^{m} ·c_{k} = p c_{k}^{m+1}.

(iii) Decomposing B_{k}pB_{k}^{m} = B_{k} · pB_{k}^{m}, one has from (2.51)

I_{B_{k}pB_{k}^{m}} ≤ \frac{1}{(2π)^{3}} ∫ dq |ϕ̂_{k}(p − q, q)| \frac{p^{2}}{q^{2}} · ∫ dq′ |ŝ_{pB_{k}^{m}}(q − q′, q′)| \frac{q^{2}}{q^{′2}}

≤ c_{k}^{m} \frac{1}{(2π)^{3/2}} ∫ dq |ϕ̂_{k}(p − q, q)| \frac{p^{2}}{q} ≤ c_{k}^{m} c′_{k} p.  (2.60)

(iv) We claim I_{B_{k}^{n}pB_{k}^{m}} ≤ p c_{k}^{′n} c_{k}^{m}. Then, using (2.60)

I_{B_{k}^{n+1}pB_{k}^{m}} ≤ \frac{1}{(2π)^{3}} ∫ dq |ϕ̂_{k}(p − q, q)| \frac{p^{2}}{q^{2}} · ∫ dq′ |ŝ_{B_{k}^{n}pB_{k}^{m}}(q − q′, q′)| \frac{q^{2}}{q^{′2}}

≤ c_{k}^{′n} c_{k}^{m} · c′_{k} p = p c_{k}^{′n+1} c_{k}^{m}.  (2.61)

Thus we obtain from the Lieb and Yau formula applied to (2.56)

∑_{n,m=0}^{∞} \frac{τ^{n}}{n!} \frac{τ^{m}}{m!} |(φ, B_{k}^{n} p B_{k}^{m} φ)| ≤ c ∑_{n,m=0}^{∞} \frac{τ^{n}}{n!} \frac{τ^{m}}{m!} c_{k}^{′n} c_{k}^{m} (φ, p φ)

= c e^{c′_{k}τ+c_{k}τ} (φ, p φ)  (2.62)

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,

V (x) := −\frac{γ}{x^{1−ϵ}} ,  v̂(q) = −γ\sqrt{\frac{2}{π}} \frac{f_{ϵ}}{q^{2+ϵ}} , ϵ > 0  (3.1)

where v̂(q) is the Fourier transform and f_{ϵ} := cos \frac{πϵ}{2} · Γ(1 + ϵ) → 1 for ϵ → 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, ϵ < \frac{1}{k+1}, the k-th order potential term V_{k} is p-form bounded with form bound less than 1.

(ii) Let φ ∈ S(ℝ^{3})⊗ℂ^{4}. Let μ_{k} > 0 for k ∈ ℕ be the infimum of the constant c in the estimate |(φ, V_{k}φ)| ≤ c (φ, p^{1−kϵ}φ). Then V_{k+1} is subordinate to V_{k} in the sense

|(φ, V_{k+1} φ)| ≤ δ |(φ, V_{k}φ)| + C (φ, φ)  (3.2)

with 0 < δ < 1 and C ∈ ℝ_{+} a constant depending on δ.

(iii) Let R^{(n+1)} := U_{n}^{∗} ⋯ U_{1}^{∗}HU_{1} ⋯ U_{n} − H^{(n)} 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.

Lemma 3.2. For 0 < (n + 1)ϵ < 1, c_{0} ∈ ℝ_{+}, n ∈ ℕ and every φ ∈ H_{1/2}(ℝ^{3}) ⊗ ℂ^{4} one has

c_{0} (φ, p^{1−(n+1)ϵ}φ) ≤ c (φ, p^{1−nϵ}φ) + C (φ, φ)  (3.3)

with c < 1 and C ∈ ℝ_{+}. For n = 0 this implies p-form boundedness with form bound < 1,

c_{0} (φ, p^{1−ϵ}φ) ≤ c (φ, p φ) + C (φ, φ).  (3.4)

Proof. We use an elementary inequality from analysis,

a · b ≤ \frac{a^{λ}}{λ} + \frac{b^{μ}}{μ} ,  a, b > 0, \frac{1}{λ} + \frac{1}{μ} = 1,  (3.5)

choose λ := \frac{1−nϵ}{1−(n+1)ϵ} > 1, μ = \frac{1−nϵ}{ϵ} and 0 < δ < 1 to be specified later. We decompose

p^{1−(n+1)ϵ} = (δp^{1−(n+1)ϵ}) · \frac{1}{δ} ≤ \frac{1 − (n + 1)ϵ}{1 − nϵ} δ^{\frac{1−nϵ}{1−(n+1)ϵ}} p^{1−nϵ} + \frac{ϵ}{1 − nϵ}(\frac{1}{δ})^{\frac{1−nϵ}{ϵ}}.

(3.6)

Then, estimating further (using δ^{λ} < δ),

c_{0} (φ, p^{1−(n+1)ϵ}φ) ≤ c_{0} δ (φ, p^{1−nϵ}φ) + c_{0} \frac{ϵ}{1 − nϵ} δ^{− \frac{1−nϵ}{ϵ}} (φ, φ). (3.7)

With the choice δ := min\{\frac{1}{2c_{0}} , \frac{1}{2}\}, (3.3) is verified.

Proof of Proposition.

We start by showing that |(φ, V_{k}φ)| ≤ c (φ, p^{1−kϵ}φ) with some constant c > 0, such that the definition of μ_{k} in (ii) makes sense. Since 0 < (φ, p^{1−kϵ}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 v̂_{1}(q, p) = v̂(q) − ŵ_{1}(q, p) = −\frac{γ_{0}}{q^{2+ϵ}} f_{ϵ} (1 + D̃_{0}(q + p) · D̃_{0}(p)) . Since the multiplier of q^{−(2+ϵ)} is a bounded operator which is estimated by a constant, one finds according to (2.32) and (2.33) with f(p) := p

|(φ, V_{1} φ)| ≤ \frac{1}{(2π)^{3/2}} ∫ dp |φ̂(p)|^{2} I_{1}(p),  (3.8)

I_{1}(p) := ∫ dq |v̂_{1}(q − p, p)| |\frac{f(p)}{f(q)}|^{2} ≤ c_{1} ∫ dq \frac{1}{|q − p|^{2+ϵ}} · \frac{p^{2}}{q^{2}} .

With the substitution p q′ := (q − p) one obtains for the latter integral

p^{3} ∫ dq′ \frac{1}{(pq′)^{2+ϵ}} \frac{p^{2}}{|pq′ + p|^{2}} = 2π p^{1−ϵ} ∫_{0}^{∞} \frac{dq′}{q^{′1+ϵ}} ln |\frac{q′ + 1}{q′ − 1}| ≤ c · p^{1−ϵ}

(3.9)

resulting in |(φ, V_{1}φ)| ≤ c_{0} (φ, p^{1−ϵ}φ) where c_{1}, c, c_{0} ∈ ℝ_{+} are constants. Equation (3.4) completes the proof of (i) for k = 1. Note that since μ_{1} = inf c_{0} > 0, we have |(φ, V_{1}φ)| > \frac{μ_{1}}{2} (φ, p^{1−ϵ}φ). The proof of (ii) is by induction. First we show (ii) for k = 1. From (2.12) we have V_{2} = i[V_{1}, B_{1}] + \frac{i}{2} [W_{1}, B_{1}] − W_{2} with W_{2} from (2.37). Following the argumentation given in section 2.3 one can disregard the bounded operators D̃_{0} 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 ŵ_{1}(q, p) according to (2.21), is estimated by |ϕ̂_{1}(q, p)| ≤ \frac{c}{q^{1+ϵ}} \frac{1}{(q+p+1)^{2}} . The estimate of |(φ, [V, B_{1}] φ)| is obtained from (2.46) by means of interchanging ϕ̂_{n+1} with ϕ̂_{1} and |p′−p|^{2}, |p′−q|^{2} with |p′−p|^{2+ϵ}, |p′−q|^{2+ϵ}. Recalling that |ϕ̂_{1}| can be replaced by its adjoint |ϕ̂_{1}^{∗}| and substituting q′ := q/p′ in the first integral one obtains

|(φ, [V, B_{1}]φ)| ≤ c ∫ dp |φ̂(p)|^{2} {I_{11}(p) + I_{12}(p)}  (3.10)

I_{11}(p) = ∫ dp′ \frac{1}{|p′ − p|^{2+ϵ}} \frac{p^{2}}{p^{′2}} · ∫ dq′ \frac{1}{q^{′2}} \frac{1}{p^{′ϵ}|q′−e_{p′}|^{1+ϵ}(|q′−e_{p′}| + 1 + \frac{1}{p′})^{2}}

I_{12}(p) = ∫ dp′ \frac{1}{|p′ − p|^{1+ϵ}(|p′ − p| + p + 1)^{2}} \frac{p^{2}}{p^{′2}} · ∫ dq \frac{1}{|p′ − q|^{2+ϵ}} \frac{p^{′2}}{q^{2}} .

For I_{11}, the q′-integral is estimated by dropping \frac{1}{p′} in the last factor of the denominator. Using (2.47) together with the substitution k := q′ − e_{p′}, one finds

∫ \frac{dq′}{q^{′2}} \frac{1}{|q′ − e_{p′}|^{1+ϵ}} \frac{1}{(|q′ − e_{p′}| + 1)^{2}} = ∫_{0}^{∞} dk \frac{k^{1−ϵ}}{(k + 1)^{2}} · \frac{2π}{k} ln |\frac{k + 1}{k − 1}| < ∞.

(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

p^{2} ∫ dp′ \frac{1}{|p′ − p|^{2+ϵ}} \frac{1}{p^{′2+ϵ}} = p^{1−2ϵ} ∫_{0}^{∞} dp′′ \frac{1}{p^{′′ϵ}} \frac{2π}{ϵp′′} (\frac{1}{|p′′ − 1|^{ϵ}} − \frac{1}{|p′′ + 1|^{ϵ}} )

≤ c · p^{1−2ϵ}  (3.12)

with a constant c.

I_{12} is treated in a similar way. The q-integral is the one from (3.8) which is estimated by c p^{′1−ϵ}. The remaining integral is estimated by replacing |p′ − p| + p + 1 with |p′ − p| + p. With the substitution p′ =: pk + p and with the techniques from (2.34) one obtains

p^{2} ∫ dp′ \frac{1}{|p′ − p|^{1+ϵ} (|p′ − p| + p)^{2} p^{′1+ϵ}}

= p^{1−2ϵ} ∫_{0}^{∞} \frac{k^{1−ϵ}dk}{(k + 1)^{2}} · \frac{2π}{k(1 − ϵ)} [(k + 1)^{1−ϵ} − |k − 1|^{1−ϵ}] ≤ c p^{1−2ϵ} (3.13)

with some constant c. Using (3.7) and the definition of μ_{k} one thus obtains

|(φ, V_{2}φ)| ≤ c_{0} (φ, p^{1−2ϵ}φ) ≤ c_{0} δ (φ, p^{1−ϵ}φ) + C (φ, φ)  (3.14)

< \frac{2c_{0} δ}{μ_{1}} |(φ, V_{1}φ)| + C(φ, φ)

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))

∫ dq ( | ˆ[·]_{m}(q − p, p)| + | ˆ[·]_{m}^{∗}(q − p, p)| ) (\frac{p}{q})^{2λ} ≤ c p^{1−mϵ} (3.15)

where λ can be chosen in the interval ]\frac{1}{2}, \frac{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)

|ϕ̂_{m}(q, p)| ≤ \frac{c}{q + p + 1} | ˆ[·]_{m}(q, p)| ≤ \frac{c}{q + 1} | ˆ[·]_{m}(q, p)| .  (3.16)

Then from (2.50), one has with some c_{0} ∈ ℝ_{+},

|(φ, [[·]_{m}, B_{k−m+1}] φ)| ≤ \frac{c_{0}}{(2π)^{3}} ∫ dp |φ̂(p)|^{2} (I_{00} + I_{01})  (3.17)

I_{00} :=∫ dp′| ˆ[·]_{m}(p − p′, p′)|\frac{p^{2}}{p^{′2}} ·∫ dq\frac{1}{|q−p′| + p′ + 1}| ˆ[·]_{k−m+1}(q − p′, p′)|\frac{p^{′2}}{q^{2}}

I_{01} :=∫ dp′ \frac{1}{|p − p′| + p′ + 1}| ˆ[·]_{k−m+1}(p − p′, p′)|\frac{p^{2}}{p^{′2}} ·∫ dq | ˆ[·]_{m}(p′ − q, q)|\frac{p^{′2}}{q^{2}}

Documenta Mathematica 10 (2005) 331–356

where in the term denoted by I_{00}, |ϕ̂_{k−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−mϵ}. Further one has with a ≥ 0 and δ > 0

∫ dq \frac{1}{a + q + 1} | ˆ[·]_{n}(p − q, q)| \frac{p^{2}}{q^{2}} · q^{1−δ} ≤ p^{−δ} ∫ dq | ˆ[·]_{n}(p − q, q)| \frac{p^{2+δ}}{q^{2+δ}}

≤ c p^{1−nϵ−δ}  (3.18)

if 1 + \frac{δ}{2} < \frac{3}{2}. Then with δ := mϵ, I_{01} ≤ c̃p^{1−(k−m+1)ϵ} p^{−mϵ} = c̃p^{1−(k+1)ϵ}. 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^{′−(k−m+1)ϵ}. With λ := 1+ \frac{k−m+1}{2} ϵ (for (k−m+1)ϵ < 1) in (3.15) we obtain I_{00} ≤ c̃p^{−(k−m+1)ϵ}·p^{1−mϵ} = c̃p^{1−(k+1)ϵ}. Therefore

|(φ, [[·]_{m}, B_{k−m+1}] φ)| ≤ c_{0} (φ, p^{1−(k+1)ϵ}φ)  (3.19)

which proves (3.15) for k + 1. The same estimate can be shown for |(φ, [V, B_{k}] φ)|. Hence

|(φ, V_{k+1}φ)| ≤ c′ (φ, p^{1−(k+1)ϵ}φ) ≤ c′ δ \frac{2}{μ_{k}} |(φ, V_{k}φ)| + C (φ, φ) (3.20)

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)

|(φ, V_{k+1}φ)| ≤ δ|(φ, V_{k}φ)| + C(φ, φ) ≤ δ (c_{1}(φ, p φ) + C_{1}(φ, φ)) + C(φ, φ).

(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 V_{1} + ... + V_{n}. 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)ϵ}-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, p^{1−kϵ}φ) < \frac{2}{μ_{k}} |(φ, V_{k}φ)| for k = n + 1 and with (3.2),

|(φ, R^{(n+1)} φ)| ≤ const·|(φ, V_{n+1}φ)| ≤ const·δ |(φ, V_{n}φ)| + C′ (φ, φ) (3.22)

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,

I_{B_{1}} ≤ \frac{c}{(2π)^{3/2}} ∫ dq \frac{1}{|p − q|^{1+ϵ}} \frac{1}{(|p − q| + q + 1)^{2}} · \frac{p^{1−ϵ}}{q^{1−ϵ}}

Documenta Mathematica 10 (2005) 331–356

≤ \frac{c}{(2π)^{3/2}} ∫ dq′ \frac{1}{q^{′1+ϵ}} \frac{1}{(q′ + 1)^{2}} · \frac{p^{1−ϵ}}{|p + q′|^{1−ϵ}} ≤ c′  (3.23)

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 f(p) = p^{\frac{1−ϵ}{2}} 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_{B_{1}}, one can also take f(p) = 1). For B_{2}, we use the estimate (2.44) of ϕ̂_{2} by ŵ_{2} and recall that W_{2} is determined from the commutator [V, B_{1}]. Consider the symbol of V B_{1} via (2.17),

|v̂ϕ_{1}(q, p)| ≤ \frac{c}{(2π)^{3/2}} ∫ dp′ \frac{1}{|q − p′|^{2+ϵ}} · \frac{1}{p^{′1+ϵ}(p′ + p + 1)^{2}} .  (3.24)

It is found that |v̂ϕ_{1}(q, p)| = const for p = 0, ∼ 1/p^{2} for p → ∞ and ∼ 1/q^{2+ϵ} 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, ϵ ≠ 0,

|v̂ϕ_{1}(q, p)| ≤ \frac{c̃}{q} ∫_{0}^{∞} \frac{dp′}{p^{′ϵ}} \frac{1}{(p′ + p + 1)^{2}} (\frac{1}{|q − p′|^{ϵ}} − \frac{1}{|q + p′|^{ϵ}} ) . (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^{1−2ϵ} for q → 0. B_{1}V is in the same operator class such that we obtain

|ŵ_{2}(q, p)| ≤ c \frac{1 + q^{ϵ}}{q^{2ϵ}(q + p + 1)^{2}} .  (3.26)

By induction, one can show that for k > 2, one has |ŵ_{k}(q, p)| ≤ \frac{c}{(q+p+1)^{2+ϵ}} . 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 ϵ → 0 cannot be carried out since in (3.7), \frac{ϵ}{1−nϵ} δ^{− \frac{1−nϵ}{ϵ}} → ∞ as ϵ → 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].

4 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^{iB_{j}}, 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^{−iS_{j}}, 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

βE_{p} = U′_{0} D_{0} U_{0}^{′−1},  E_{k} = U′_{0} V_{k} U_{0}^{′−1},  O_{k} = U′_{0} W_{k} U_{0}^{′−1}
S_{k} = U′_{0} B_{k} U_{0}^{′−1},  k = 1, ..., n  (4.1)

with U′_{0} from (1.5).

The key observation is the relation between the spinor ψ = \binom{u}{0} and the spinor φ in the positive spectral subspace of D_{0},

φ = U_{0}^{′−1} ψ.  (4.2)

In momentum space, this equation is easily verified from U_{0}^{′−1} = (\frac{αp}{E_{p}+m} β + 1) A and from the explicit form (1.12) of φ̂(p). Then with the help of (2.6), the assertion (1.16) of Theorem 2.1 reads

(φ, (D_{0} + ∑_{k=1}^{n} V_{k}) φ) = (U_{0}^{′−1} ψ, ∑_{k=0}^{n} H_{k} U_{0}^{′−1} ψ) = (ψ, H′_{n} ψ). (4.3)

Identifying terms of fixed order k ≤ n and using (1.11), the assertion (4.3) is a consequence of

U′_{0} V_{k} U_{0}^{′−1} = E_{k},  k = 1, 2... and U′_{0} D_{0} U_{0}^{′−1} = βE_{p}  (4.4)

and hence of Lemma 4.1.

Proof of Lemma 4.1.

a) Verification of (4.1) up to first order in γ

The equality U′_{0}D_{0}U_{0}^{′−1} = 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)

U′_{0}W_{1}U_{0}^{′−1}= \frac{1}{\sqrt{2}} (1 + β \frac{αp}{p}) · \frac{1}{2} (V − \frac{αp}{p}V \frac{αp}{p}) · \frac{1}{\sqrt{2}} (1 + \frac{αp}{p}β)

= \frac{1}{2} [β \frac{αp}{p} V − βV \frac{αp}{p}] = O_{1}  (4.5)

and similarly, U′_{0}V_{1}U_{0}^{′−1} = E_{1} with V_{1} = \frac{1}{2} (V + \frac{αp}{p} V \frac{αp}{p}) . For the m ≠ 0

case, one needs the relation \frac{m}{E_{p}} + \frac{p^{2}}{E_{p}(E_{p}+m)} = 1.

In order to prove S_{1} = U′_{0}B_{1}U_{0}^{′−1} we first show that S_{1} is uniquely determined by (1.10). Representing S_{1} and O_{1} by their respective symbols s_{1} and o_{1} via (2.14) and noting that βE_{p} is a multiplication operator in Fourier space, one obtains from (1.10)

−ô_{1}(q, p) = iβE_{|p+q|} ŝ_{1}(q, p) − iŝ_{1}(q, p) βE_{p}
= iβ (E_{|p+q|} + E_{p}) ŝ_{1}(q, p)  (4.6)

which can be uniquely solved for ŝ_{1}(q, p). We now transform the defining equation (2.8) for B_{1} with U′_{0}

U′_{0} W_{1} U_{0}^{′−1} = −i (U′_{0}D_{0}U_{0}^{′−1})(U′_{0}B_{1}U_{0}^{′−1}) + i (U′_{0}B_{1}U_{0}^{′−1})(U′_{0}D_{0}U_{0}^{′−1})

⟺  O_{1} = −i [βE_{p}, U′_{0}B_{1} U_{0}^{′−1}] .  (4.7)

From the uniqueness of the solution it follows from (4.7) and (1.10) that U′_{0}B_{1}U_{0}^{′−1} = S_{1} and hence the uniqueness of the operator B_{1}.

b) Proof of (4.1) by induction for arbitrary order n in γ

We assume that to order n − 1 the assertion of Lemma 4.1 holds. Then, the relation between the Dirac operators transformed to n − 1st order, asserted by Theorem 1.2, is also true, i.e. (including the n-th order terms)

βE_{p} + E_{1} + E_{2} + ... + E_{n−1} + E_{n} + O_{n}
= U′_{0} (D_{0} + V_{1} + ... + V_{n−1} + V_{n} + W_{n}) U_{0}^{′−1}  (4.8)

since E_{n} and O_{n} only depend on βE_{p}, E_{j}, O_{j}, S_{j}, j = 1, ..., n − 1, with the identical dependence of V_{n} and W_{n} on D_{0}, V_{j}, W_{j}, B_{j}, j = 1, ..., n − 1. From (4.1) for j = 1, ..., n − 1 it therefore follows that E_{n} = U′_{0}V_{n}U_{0}^{′−1} and O_{n} = U′_{0}W_{n}U_{0}^{′−1}. Carrying out the n-th transformation one gets

U′_{n} ⋯ U′_{0}HU_{0}^{′−1} ⋯ U_{n}^{′−1} =βE_{p} + E_{1} + ... + E_{n−1}+ E_{n} + O_{n} + i[βE_{p}, S_{n}] + R_{n+1}

U_{n}^{∗}⋯ U_{1}^{∗}HU_{1}⋯ U_{n}=D_{0} +V_{1} +...+V_{n−1}+V_{n} +W_{n} +i[D_{0}, B_{n}]+R^{(n+1)}. (4.9)

B_{n} is obtained from W_{n} = −i [D_{0}, B_{n}], or transformed with U′_{0},

U′_{0} W_{n} U_{0}^{′−1} = O_{n} = −iU′_{0} [D_{0}, B_{n}] U_{0}^{′−1} = −i [βE_{p}, U′_{0}B_{n}U_{0}^{′−1}]. (4.10)

Since the solution S_{n} to O_{n} = −i[βE_{p}, S_{n}] is unique, one gets U′_{0}B_{n}U_{0}^{′−1} = S_{n}. □

From the correspondence of the two transformation schemes it follows that for n = 2, H^{(2)} from (2.12) when acting on the positive spectral subspace of D_{0}, reduces to Λ_{+}(D_{0} + V + \frac{i}{2} [W_{1}, B_{1}])Λ_{+} since i[V_{1}, B_{1}] corresponds to an odd operator which vanishes upon projection.

Acknowledgment

I should like to thank H. Kalf and H. Siedentop for enlightening discussions. I am particularly grateful to A. Sobolev for suggesting the unitary transformation sequence applied in this work and for his openness and patience during many discussions. I should also like to thank A. Sobolev and the University of Sussex for their hospitality during my visits in the course of this work. Partial support by the Volkswagen Foundation is gratefully acknowledged.

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D. H. Jakubassa-Amundsen Mathematics Institute University of Munich Theresienstr. 39 80333 Munich Germany doris.jaku@lrz.uni-muenchen.de