Heat kernel estimates and spectral properties of a pseudorelativistic operator with magnetic field
aa r X i v : . [ m a t h - ph ] A p r Heat kernel estimates and spectral properties of apseudorelativistic operator with magnetic field
D. H. Jakubassa-Amundsen
Mathematics Institute, University of MunichTheresienstr. 39, 80333 Munich, Germany
Based on the Mehler heat kernel of the Schr¨odinger operator for a free electron in aconstant magnetic field an estimate for the kernel of E A = | α ( p − e A ) + βm | is derived,where E A represents the kinetic energy of a Dirac electron within the pseudorelativisticno-pair Brown-Ravenhall model. This estimate is used to provide the bottom of theessential spectrum for the two-particle Brown-Ravenhall operator, describing the motionof the electrons in a central Coulomb field and a constant magnetic field, if the centralcharge is restricted to Z ≤ . [J. Math. Phys. 49 (2008) 032305, 1-22]AMS: 81Q10 1. Introduction
Consider two relativistic electrons of mass m in an electromagnetic field whichis generated by a point nucleus of charge Z fixed at the origin, and by a vectorpotential A . The two-particle Coulomb-Dirac operator, introduced by Sucher [26] andaccounting for a magnetic field is defined by H = X k =1 (cid:16) D ( k ) A + V ( k ) (cid:17) + P + , V (12) P + , (1.1) D ( k ) A := α ( k ) ( p k − e A ( x k )) + β ( k ) m, k = 1 , , where P + , projects onto the positive spectral subspace of P k =1 ( D ( k ) A + V ( k ) ). Theunderlying Hilbert space is A ( L ( R ) ⊗ C ) where A denotes antisymmetrizationwith respect to electron exchange. The single-particle and two-particle potentialsare, respectively, V ( k ) = − γx k , V (12) = e | x − x | (1.2)where the field strength γ = Ze and e ≈ / .
04 the fine structure constant. x k = | x k | is the modulus of the spatial coordinate of electron k, k = 1 , . The momentum of electron k is denoted by p k , and β, α = ( α , α , α ) and σ = ( σ , σ , σ ) are the Dirac and Pauli matrices, respectively.Due to the positron degrees of freedom the spectrum of the Coulomb-Diracoperator is unbounded from below. In spectroscopic studies of static ions where pair creation plays no role one can instead work with a semibounded operator,derived from H , which solely describes the electronic states. One of the currenttechniques to construct such an operator is by means of a (unitary) Morse-Feshbachtransformation scheme (see e.g. [16, 6, 7, 13]) which aims at decoupling the positiveand negative spectral subspaces of the electron. For A = it is thereby crucial [18]to include the vector potential in the definition of these subspaces. A decouplingof the spectral subspaces up to second order in e is provided in [14].The first-order transformation leads to the Brown-Ravenhall operator [18] H BR = X k =1 Λ A + , ( D ( k ) A + V ( k ) ) Λ A + , + Λ A + , V (12) Λ A + , (1.3)if the domain is restricted to the positive magnetic spectral subspace H A + , :=Λ A + , ( A ( H ( R ) ⊗ C ) ) of the two electrons where H ( R ) ⊗ C is the domainof D ( k ) A . The projectors are defined byΛ A + , = Λ (1) A, + ⊗ Λ (2) A, + , Λ ( k ) A, + := 12 D ( k ) A E ( k ) A ! ,E ( k ) A := | D ( k ) A | = q ( p k − e A ( x k )) − e σ ( k ) B ( x k ) + m ≥ m (1.4)and B = ∇ × A is the magnetic field. We note that the gauge invariance of thetransformed operator is preserved [19].If the field energy E f = π R R B ( x ) d x is added to H , positivity of theBrown-Ravenhall operator can be established. This relies on the condition k B k < ∞ to render E f finite, which excludes constant magnetic fields. Lieb, Siedentopand Solovej [18] have proven positivity (i.e. stability) for the K -nuclei N -electronBrown-Ravenhall operator, in case of Z ≤ . In the present context E f can bedisregarded, keeping in mind that it just leads to a global shift of the spectrum.The aim of the present work is to provide the HVZ theorem for H BR whichlocalizes the bottom of the essential spectrum [11, 28, 29]. This theorem wasproven for the multiparticle Brown-Ravenhall operator describing electrons in theCoulomb potential of subcritical charge ( Z ≤ A = [15, 22]. Themethod of proof, which we also will adopt here, is based on the work of Lewis,Siedentop and Vugalter [17] for the HVZ theorem concerning the scalar pseudo-relativistic multiparticle (Herbst-type) Hamiltonian.The main difference to the field-free case results from the kinetic energybeing described by an integral operator (instead of a multiplication operator inmomentum space). In order to carry out the necessary computations an estimateof its kernel is needed. If we restrict ourselves to constant magnetic fields we canprofit from the relation to the heat kernel of the Schr¨odinger operator which hasbeen studied extensively ([2, 24, 21, 8] and references therein).Before we prove the HVZ theorem (in sections 5 and 6), Theorem 1 (HVZ theorem) . Let H BR be the two-electron Brown-Ravenhall op-erator from (1.3) where B is a constant magnetic field and where the Coulomb potential strength is restricted to γ < γ c = 0 .
629 ( Z ≤ . Then its essentialspectrum is given by σ ess ( H BR ) = [Σ , ∞ ) (1.5) where Σ is the ground state energy of the one-electron ion, increased by the restmass of the second electron, we provide the relative form boundedness of the potential of H BR with respectto the kinetic energy operator (which requires the restriction γ < γ c ; section 2).For handling the IMS-type localization formula [5, p.28], entering into the proofof the HVZ theorem, we found it convenient to work in a representation of theBrown-Ravenhall operator which invokes the Foldy-Wouthuysen transformation U instead of the projectors Λ A, + (section 3). In section 4 we establish the necessaryestimates for the kernels of E A and U as well as for their commutators with somesimple scaling functions.We call an operator O R -bounded if O is bounded by c/R with some constant c and R ≥ . Relative form boundedness of the electric potential
For ψ + ∈ H A + , we have D ( k ) A ψ + = E ( k ) A ψ + . Therefore, E A,tot := E (1) A + E (2) A can be identified with the kinetic energy operator. We require A ∈ L ,loc ( R ) [2]which guarantees the essential self-adjointness of E ( k ) A , k = 1 , ∇ · A = 0. Lemma 1.
The Brown-Ravenhall operator H BR for A ∈ L ,loc ( R ) is well-definedin the form sense on Λ A + , ( A ( H / ( R ) ⊗ C ) ) and is bounded from below for γ < γ c = 0 . .H BR thus extends to a self-adjoint operator for γ < γ c by means of theFriedrichs extension. Proof.
In order to show the relative form boundedness of the potential with respectto E A,tot let us assume that B is bounded, k B k ∞ := B < ∞ . For the two-particle potential we have the estimate [3] for ψ ∈ A ( H / ( R ) ⊗ C ) , using Kato’s inequality,( ψ, V (12) ψ ) ≤ e π ψ, p ψ ) ≤ e π ψ, q p + p + 2 m ψ ) . (2.1)Furthermore we employ the diamagnetic inequality ([12], see also [2] and ref-erences therein) which holds in arbitrary dimension. Defining S ( k ) A := [( p k − e A ( x k )) + m ] , it can for N particles be written in the following way [9, 2], | ( P Nk =1 S ( k )2 A ) − /n ψ | ≤ ( P Nk =1 p k + N m ) − /n | ψ | , n = 1 , , , ... . Upon multipli-cation with some function f > k f ( x , ..., x N )( P Nk =1 p k + N m ) − /n k ≤ c n we obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f P Nk =1 S ( k )2 A ) /n ψ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f P Nk =1 p k + N m ) /n | ψ | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ c n k ψ k . (2.2)Let us choose N = 2 , n = 4 , f ( x , x ) = e | x − x | − / and ψ = ( N P k =1 S ( k )2 A ) / ψ + . Then (2.2) turns into( ψ + , V (12) ψ + ) ≤ e π ψ + , ( X k =1 S ( k )2 A ) / ψ + ) ≤ e π ψ + , X k =1 S ( k ) A ψ + ) (2.3)with the constant from (2.1). Using S ( k )2 A = E ( k )2 A + e σ ( k ) B ( x k ) and | σ ( k ) B ( x k ) | = B ( x k ) ≤ B we estimate the r.h.s. of (2.3) further such that( ψ + , V (12) ψ + ) ≤ e π ψ + , E A,tot ψ + ) + e π ( eB ) k ψ + k . (2.4)The relative E ( k ) A -form boundedness of the single-particle potential was shown in[14], | ( ψ + , V ( k ) ψ + ) | ≤ γ π ψ + , E ( k ) A ψ + ) + γ π eB ) k ψ + k (2.5)for k = 1 , . Therefore, we have | ( ψ + , ( V (1) + V (2) + V (12) ) ψ + ) | ≤ (cid:18) γπ e π (cid:19) ( ψ + , E A,tot ψ + ) + C ( B ) k ψ + k , (2.6)with C ( B ) = π ( γ + e )( eB ) / and form bound smaller than one for γ < γ c := π − e ≈ . . The lower bound − C ( B ) of H BR , derived from (2.6), decreaseswith increasing magnetic field strength. (cid:3) Alternative representation of H BR and the heat kernel of E A Following our strategy in the field-free case [15] we use a representation of theBrown-Ravenhall operator where its single-particle contributions do not dependon the coordinate of the second particle. We note that any ψ + ∈ H A + , can berepresented in terms of a single-particle Foldy-Wouthuysen transformation U [6](we will drop the superscript ( k ) throughout when referring to the single-particlecase), U = A E (cid:18) β α ( p − e A ) E A + m (cid:19) ,A E = (cid:18) E A + m E A (cid:19) . (3.1)This operator has the advantage of being unitary and hence norm preserving (incontrast to the projector Λ A, + ). We have [14] ψ + = ( U (1)0 U (2)0 ) −
12 (1 + β (1) ) 12 (1 + β (2) ) u (3.2) where the inverse ( U ) − follows from (3.1) if β is replaced by − β and where u ∈ A ( H ( R ) ⊗ C ) . Since (1 + β ( k ) ) projects onto the upper components ofthe spinor associated with particle k one may without restriction assume that thelower components of u are zero (and omit (1 + β ( k ) ) , k = 1 , . Thus, from (1.3),( ψ + , H BR ψ + ) = ( u, U (1)0 U (2)0 ( X k =1 ( D ( k ) A + V ( k ) ) + V (12) ) ( U (1)0 U (2)0 ) − u )=: ( u, h BR u ) . (3.3)Using U ( k )0 D ( k ) A ( U ( k )0 ) − = β ( k ) E ( k ) A [6] the kinetic energy contribution to (3.3)can be simplified to( u, U (1)0 U (2)0 D ( k ) A ( U (1)0 U (2)0 ) − u ) = ( u, E ( k ) A u ) . (3.4)Therefore we can identify h BR = X k =1 (cid:16) E ( k ) A + U ( k )0 V ( k ) ( U ( k )0 ) − (cid:17) + U (1)0 U (2)0 V (12) ( U (1)0 U (2)0 ) − (3.5)keeping in mind that its quadratic form has to be taken with spinors of vanishinglower components.The defining equation (1.4) for E A reveals the close connection between E A and the Schr¨odinger operator H s := ( p − e A ) for a free electron in a magneticfield. Let us now restrict ourselves to a constant magnetic field, B = B , andchoose the e -direction along B . Using the gauge ∇ · A = 0 we take [2] A ( x ) = 12 ( B × x ) = 12 B ( − x , x , . (3.6)Then the difference between the two operators E A and H s is a constant in space.This allows us to adopt the properties of H s . For a B -field with constant directionthe Schr¨odinger operator H s = p + ( p − eA ) + ( p − eA ) separates, and themagnetic field problem reduces to two dimensions.Let us denote by O ( x , x ′ ) for x , x ′ ∈ R the kernel of an integral operator O .For constant B = B the (Mehler) heat kernel of H s , i.e. the kernel of e − tH s , isknown explicitly (see e.g. [2],[24, p.168],[21]), e − tH s ( x , x ′ ) = 1(4 πt ) eB π sinh( eB t ) e − i eB ( x x ′ − x x ′ ) · e − ( x − x ′ ) / (4 t ) e − eB coth( eB t ) [( x − x ′ ) +( x − x ′ ) ] (3.7)where t > . Since H s is essentially self-adjoint on C ∞ ( R ), its heat kernel satisfies the symmetry property e − tH s ( x , x ′ ) = e − tH s ( x ′ , x ) ∗ . Note that the phase in (3.7) differs in sign from the one given by Loss and Thaller [21]because they consider ˜ H s = ( p + e A ) in place of H s . The heat kernel of E A follows from e − tE A ( x , x ′ ) = e − tm e te σ B e − tH s ( x , x ′ ) . (3.8)For the subsequent estimates we need a series of inequalities for the hyperbolicfunctions, z coth z ≤ z and sinh z ≥ z for z ≥ , (3.9) z coth z ≥ , z e z sinh z ≤ z. It is well known (see e.g. [24, p.35]) that, using (3.9), e − tH s ( x , x ′ ) can be estimatedfrom above by the heat kernel of the free Schr¨odinger operator, (cid:12)(cid:12) e − tH s ( x , x ′ ) (cid:12)(cid:12) ≤ πt ) e − ( x − x ′ ) / (4 t ) = e − tp ( x , x ′ ) . (3.10)For (3.8), a less restrictive estimate of the prefactor is required, since | e te σ B | ≤ e | te σ B | = e teB . Using the last inequality of (3.9) we have (cid:12)(cid:12)(cid:12) e − tE A ( x , x ′ ) (cid:12)(cid:12)(cid:12) ≤ e − tm πt ) (1 + 2 eB t ) e − ( x − x ′ ) / t (3.11)and note in passing that for m = 0 the r.h.s. of (3.11) can be further estimated by c ( B ) e − tp ( x , x ′ ) . Estimate of the kernels of E A and U and their commutators Again we consider only the single-particle case and suppress the superscript( k ) . We have
Lemma 2.
For a magnetic field generated by the vector potential (3.6) the kernelof the kinetic energy E A can be estimated by | E A ( x , x ′ ) | ≤ C ( B ) | x − x ′ | (4.1) where the constant C ( B ) increases quadratically with the field strength B . Weeven have for some < ǫ < m , | E A ( x , x ′ ) | ≤ C ( B ) | x − x ′ | e − ( m − ǫ ) | x − x ′ | . (4.2)We remark that Frank, Lieb and Seiringer [9] have derived a (4.1)-type es-timate for the scalar pseudorelativistic Hamiltonian (where E A is replaced by | p − e A | with A ∈ L ,loc ( R )) in the context of the localization formula, with aconstant C independent of the magnetic field. Proof.
We use the integral representation [2]1 E A = 1 √ π Z ∞ dt √ t e − tE A (4.3) to write E A = lim t → − e − tE A tE A = lim t → √ π t Z ∞ dτ √ τ (cid:16) e − τE A − e − ( t + τ ) E A (cid:17) . (4.4)Since for t > , e − tE A ( x , x ′ ) is analytic in t (see (3.8) with (3.7)) we can expand e − ( t + τ ) E A ( x , x ′ ) in powers of t for small t . Using the Taylor formula, f ( t + τ ) = f ( τ ) + tf ′ ( τ ) + O ( t ) we need the derivative ddτ e − τE A ( x , x ′ ) = e − τE A ( x , x ′ ) (cid:26) − eB coth( eB τ ) − τ − m + e σ B + 14 τ ( x − x ′ ) + ( eB ) ( eB τ ) (cid:2) ( x − x ′ ) + ( x − x ′ ) (cid:3)(cid:27) . (4.5)An estimate of the derivative of the heat kernel of E A for a wide class of smoothvector potentials is provided by Ueki [27]. However, due to the presence of the e σ B -term in E A the given estimate increases exponentially for t → ∞ and cannotbe used in the present context.In order to get an estimate for the kernel of E A we employ further the esti-mates (3.9) and (3.11) (as well as the triangle inequality). This leads to | E A ( x , x ′ ) | = (cid:12)(cid:12)(cid:12)(cid:12) − √ π Z ∞ dτ √ τ (cid:18) ddτ e − τE A ( x , x ′ ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π Z ∞ dττ e − τm e − ( x − x ′ ) / (4 τ ) (cid:26) τ + 5 eB + m + eB τ ( x − x ′ ) + 14 τ ( x − x ′ ) + ( m + 2 eB ) 2 eB τ (cid:27) . (4.6)Performing the integrals with the help of Appendix A, we get, setting z ′ := x ′ − x , | E A ( x , x ′ ) | ≤ π (cid:26) K ( mz ′ ) (cid:20) mz ′ + 2 m + 7 meB z ′ (cid:21) + K ( mz ′ ) (cid:20) m z ′ + 2 eB ( m + eB ) (cid:21)(cid:27) . (4.7)According to the behaviour of the modified Bessel functions K ν (see Appendix A),the function in curly brackets diverges like 1 /z ′ for z ′ → m =0) like e − mz ′ /z ′ / for z ′ → ∞ . Based on the continuity of this function for z ′ ∈ R + one gets the estimate | E A ( x , x ′ ) | ≤ c | x − x ′ | where c increases quadratically with B . From the less restrictive estimate | E A ( x , x ′ ) | ≤ c ( B ) z ′ (1 + z ′ / ) e − mz ′ oneobtains an exponential decay for 0 < ǫ < m (since z ′ ν e − ǫz ′ is bounded for any ν > (cid:3) For the bounded operator U the singularity of the kernel is weaker. Lemma 3.
For a constant magnetic field and < ǫ < m the kernel of the Foldy-Wouthuysen transformation U can be estimated by | U ( x , x ′ ) | ≤ c ( B ) | x − x ′ | e − ( m − ǫ ) | x − x ′ | , (4.8) where c ( B ) increases quadratically with the strength B of the magnetic field.Proof. From the definition (3.1) we have U = A E + β √ α ( p − e A ) √ E A ( E A + m ) and weestimate the kernels of the two summands separately.Concerning the second summand, (4.3) leads to1 √ E A √ E A + m = 1 π Z ∞ dt ′ √ t ′ Z ∞ dτ √ τ e − τm e − ( τ + t ′ ) E A . (4.9)Abbreviating t := τ + t ′ we make use of the integral representation [9] e − tE A = 1 √ π Z ∞ dτ ′ √ τ ′ e − τ ′ e − ( t / τ ′ ) E A . (4.10)Thus α ( p − e A ) p E A ( E A + m ) ( x , x ′ ) = 1 π Z ∞ dt ′ √ t ′ Z ∞ dτ √ τ e − τm (4.11) · Z ∞ dτ ′ √ τ ′ e − τ ′ ( − i α ∇ x − α e A ) e − ( t / τ ′ ) E A ( x , x ′ ) . For the derivative we have from (3.8) with (3.7) − i α ∇ x e − tH s ( x , x ′ ) = (cid:26) i t α ( x − x ′ ) + i eB eB t ) · [ α ( x − x ′ ) + α ( x − x ′ )] − eB α x ′ − α x ′ ] (cid:27) e − tH s ( x , x ′ ) (4.12)and further − α e A e − tH s ( x , x ′ ) = eB α x − α x ) e − tH s ( x , x ′ ) (4.13)which have to be inserted into (4.11).In the following we make use of some relations. From α i = 1 , σ i = 1 , i =1 , , , and α i α k = − α k α i for i = k we have | α ( x − x ′ ) + α ( x − x ′ ) | = p ( x − x ′ ) + ( x − x ′ ) . Moreover, employing | β | = 1, (3.9) and (3.11) we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ( p − e A ) p E A ( E A + m ) ( x , x ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π | x − x ′ | Z ∞ dt ′ √ t ′ Z ∞ dτ √ τ e − τm t · Z ∞ dτ ′ e − τ ′ e − ( x − x ′ ) τ ′ /t e − t m / τ ′ ( τ ′ t + 3 eB τ ′ + ( eB ) t ) . (4.14) The dτ ′ -integral can be evaluated with the help of (A.1), resulting in Z ∞ dτ ′ · · · = t (cid:26) m K ( mξ ) ξ + 32 eB m K ( mξ ) ξ + ( eB ) m K ( mξ ) ξ (cid:27) (4.15)where ξ := p t + ( x − x ′ ) . For the further estimate we note that the modifiedBessel functions K ν are monotonously decreasing in (0 , ∞ ), as are the inversepowers of ξ . Therefore, (4.15) is estimated from above if ξ is replaced by y := p τ + t ′ + ( x − x ′ ) ≤ ξ. With the additional estimate e − τm ≤ τ and t ′ such that t /t = τ + t ′ can be replaced by 2 t ′ . With thesemanipulations the dτ -integral can be evaluated analytically by (A.2). The resultis, defining a := t ′ + ( x − x ′ ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ( p − e A ) p E A ( E A + m ) ( x , x ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π | x − x ′ | Z ∞ dt ′ √ t ′ − Γ( ) m · ( m K ( ma ) a + 32 eB m K ( ma ) a + ( eB ) m K ( ma ) a ) . (4.16)Applying again (A.2) for the remaining integral turns the r.h.s. of (4.16) into √ π (cid:26) m K ( m | x − x ′ | ) | x − x ′ | + 32 eB m K ( m | x − x ′ | )+ ( eB ) | x − x ′ | K ( m | x − x ′ | ) (cid:27) . (4.17)Following the argumentation below (4.7) one obtains the result | α ( p − e A ) √ E A ( E A + m ) ( x , x ′ ) | ≤ c ( B ) | x − x ′ | e − ( m − ǫ ) | x − x ′ | . It is shown in Appendix C that thekernel of A E obeys the same estimate. Thus | U ( x , x ′ ) | ≤ | x − x ′ | e − ( m − ǫ ) | x − x ′ | (cid:20) ˜ c ( B ) + k β √ k c ( B ) (cid:21) (4.18)which completes the proof. (cid:3) We remark that with the tools from the proof of Lemma 3 it is straightfor-ward to show that a (4.8)-type estimate holds also for the kernel of the projector, | Λ A, + ( x , x ′ ) | ≤ δ ( x − x ′ ) + C ( B ) | x − x ′ | e − ( m − ǫ ) | x − x ′ | .Let us now turn to the estimates of the commutators which are needed inthe context of the proof of the HVZ theorem. In part they are based on a lemma(with its corollary), proven in [15, section 8]. Lemma 4.
Let O be a single-particle operator the kernel of which satisfies |O ( x , x ′ ) | ≤ c | x − x ′ | with some constant c . Let g ∈ C ∞ ( R ) (or g ∈ C ( R ) ∩ C ( R \{ } ) and g (0) = 0) be a real function of x and let its derivative be bounded.Then for ϕ ∈ C ∞ ( R ) ⊗ C and ψ ∈ L ( R ) ⊗ C one has | ( ψ, [ O , g ] 1 x ϕ ) | ≤ c k ψ k k ϕ k (4.19) with a constant c (depending on c ) where [ O , g ] := O g − g O . If g is a function of x /R , where R > is a scaling parameter, but otherwisewith the same properties as g then | ( ψ, [ O , g ] 1 x ϕ ) | ≤ cR k ψ k k ϕ k . (4.20) Corollary 1.
Let g and g be as in Lemma 4 with x := x − x or x = ( x , x ) ∈ R (i.e. g, g ∈ C ∞ ( R )) . Then one has ( ψ, | x − x | [ O , g ] ϕ ) | ≤ c k ψ k k ϕ k (4.21) and | ( ψ, | x − x | [ O , g ] ϕ ) | ≤ cR k ψ k k ϕ k . (4.22)Note that an operator and its adjoint have the same bound.Since according to Lemma 3 the operator U satisfies | U ( x , x ′ ) | ≤ c ( B ) | x − x ′ | , we can apply Lemma 4 with a suitable function g to obtain the commutatorestimate (with a generic constant c ), | ( ψ, [ U , g ] 1 x ϕ ) | ≤ c ( B ) R k ψ k k ϕ k . (4.23)We note that the same estimate is valid if U is replaced by its inverse ( U ) − . Also the other three estimates, (4.19), (4.21) and (4.22) hold for the operators U and ( U ) − .It is more involved to obtain a commutator estimate for the kinetic energyoperator, because the singularity of its kernel is of the fourth power of | x − x ′ | − . Lemma 5.
Let g ∈ C ∞ ( R ) be a real function of x /R with bounded first andsecond derivative and a scaling parameter R > . Then its commutator with thekinetic energy operator E A (for a constant magnetic field) satisfies k [ E A , g ] ϕ k ≤ c ( B ) R (4.24) for ϕ ∈ C ∞ ( R ) ⊗ C , k ϕ k = 1 , where the constant c depends on the field strength B .Proof. We have( [ E A , g ] ϕ )( x ) = Z R d x ′ E A ( x , x ′ ) (cid:26) g (cid:18) x ′ R (cid:19) − g (cid:16) x R (cid:17)(cid:27) ϕ ( x ′ ) . (4.25) Let us define ω := eB x , ω := − eB x , ω := 0 and E ω =0 A ( x ′ − x ) by meansof E A ( x , x ′ ) = E ω =0 A ( x ′ − x ) e i ω ( x ′ − x ) where E ω =0 A is an even function and wherewe have used that ω x ′ = ω ( x ′ − x ) . We aim at isolating the leading singularity of E ω =0 A at x ′ = x . With E ω =0 A ( x ′ − x ) = − √ π Z ∞ dτ √ τ (cid:18) ddτ e − τE ω =0) A ( x ′ − x ) (cid:19) (4.26)and (4.5) the integrand is analytic in x ′ − x and in τ for τ > . Therefore,the singular behaviour of E ω =0 A in x ′ − x = (which results from the factor e − ( x ′ − x ) / τ /τ n , n >
1) is found from an expansion of the integrand near τ = 0.Setting z ′ := x ′ − x one obtains for the leading term − π τ e − z ′ / τ (cid:16) z ′ τ − τ (cid:17) . Performing the τ -integral over a small interval (0 , ǫ ) one thus gets E ω =0 A ( z ′ ) = − π z ′ + O ( 1 z ′ ) for z ′ → . (4.27)Therefore we decompose E A ( z ′ ) = E ω =0 A ( z ′ ) + e − mz ′ π z ′ ! e i ω z ′ − e − mz ′ π z ′ e i ω z ′ . (4.28)In order to check the boundedness of an operator O the kernel of which canbe estimated by a positive function k , viz. |O ( x , x ′ ) | ≤ k ( x , x ′ ) , the Lieb andYau formula [20] can be applied. This formula is related to the Schur test for theboundedness of integral operators and tells us that O is bounded if the followingestimates hold for k , I ( x ) := R R d x ′ k ( x , x ′ ) f ( x ) f ( x ′ ) ≤ CJ ( x ′ ) := R R d x k ( x , x ′ ) f ( x ′ ) f ( x ) ≤ C (4.29)for all x ∈ R and x ′ ∈ R , respectively, where f > x > C some constant.We will apply this formula to the contribution to (4.25) which arises frominsertion of the first term of (4.28). For the function g ∈ C ∞ ( R ) we can applythe mean value theorem to get (cid:12)(cid:12)(cid:12)(cid:12) g ( x ′ R ) − g ( x R ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) x − x ′ R (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ( ∇ x R g )( ξ R ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c R | x − x ′ | (4.30)with some constant c for ξ on the line between x and x ′ . Then we obtain for theintegral I (which is identical to J for the choice f = 1), I ( x ) = Z R d x ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ω =0 A ( z ′ ) + e − mz ′ π z ′ ! e i ω z ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) g (cid:18) x ′ R (cid:19) − g (cid:16) x R (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R d z ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ω =0 A ( z ′ ) + e − mz ′ π z ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c R z ′ ≤ c ( B ) R . (4.31)Since | E ω =0 A ( z ′ ) + e − mz ′ π z ′ | behaves like z ′ near z ′ = 0, is analytic for z ′ > I ( x ) is finite and its boundis independent of x .In order to treat the second contribution to E A ( z ′ ) we need the Taylor ex-pansion of g up to second order. We use the notation g ( x /R ) =: g R ( x ) . Then g R ( x ′ ) − g R ( x ) = ( x ′ − x ) ∇ g R ( x ) + 12 X k,l =1 ( x ′ k − x k )( x ′ l − x l ) ∇ k ∇ l g R ( ξ ) (4.32)with ξ on the line between x and x ′ .The insertion of the second term of (4.28) together with the second-orderterm of (4.32) into (4.25) provides the kernel of a bounded operator (with bound cR ) according to the Schur test (again with f = 1). The argumentation is thesame as given in the context of (4.31) supplemented by the 1 /R -boundedness ofthe second derivatives of g . Therefore it is sufficient to consider only the first termof (4.32) in the remaining proof of the 1 /R -boundedness of [ E A , g ] ϕ .For ω = 0 denote the corresponding kernel by k ( z ′ ) := c R e − mz ′ z ′ z ′ (4.33)where c /R := ∇ g R ( x ) /π is bounded and independent of z ′ . The kernel k definesa bounded operator according to Stein’s theorem [25, § i ) | k ( z ′ ) | ≤ cz ′ ( ii ) | ∇ k ( z ′ ) | ≤ cz ′ (4.34)( iii ) Z R 52 ; ω m + ω ) (4.43)which is finite for all ω = 0 . We note that for m = 0 (or equivalenly, ω → ∞ ) onegets directly [10, p.684] Z ∞ dξξ / J / ( ξ ) = √ π √ S k ( m = 0) = π i ω k ω (which agrees with (4.43) at m = 0 [10, p.1042]).Thus (4.39) can be estimated by cR k ψ k k ϕ k which completes the proof. (cid:3) Proof of Theorem 1 (easy part) Let us denote with j = 1 , j = 0 the nucleuswhich generates the electric field. We define the two-cluster decompositions of h BR , h BR = E A,tot + a j + r j , j = 0 , , , (5.1)where E A,tot = E (1) A + E (2) A is the kinetic energy operator and a j collects thepotential terms not involving particle j . These three decompositions representthe cases where either one electron has moved to infinity ( j = 1 , 2) or whereboth electrons have moved far away from the nucleus ( j = 0). Then the residualinteraction ( r j ) is expected to tend to zero. For example, for j = 1, a = U (2)0 V (2) ( U (2)0 ) − , r = U (1)0 V (1) ( U (1)0 ) − + U (1)0 U (2)0 V (12) ( U (1)0 U (2)0 ) − . (5.2)This allows us to define Σ by means ofΣ := min j inf σ ( E A,tot + a j ) . (5.3)The proof of the HVZ theorem consists of two parts, conventionally called the’easy part’, [Σ , ∞ ) ⊂ σ ess ( H BR ) = σ ess ( h BR ) , and the ’hard part’, σ ess ( h BR ) ⊂ [Σ , ∞ ). Following Morozov and Vugalter [22] we shall work in coordinate spaceonly. For the easy part we use the strategy of Weyl sequences [5, 15]. Let λ ∈ [Σ , ∞ ) . Without restriction we can assume that Σ = inf σ ( E A,tot + a j ) for j = 1 . This relies on the symmetry of the operator under electron exchange, while for j = 0 we have a = U (1)0 U (2)0 V (12) ( U (1)0 U (2)0 ) − ≥ a ≤ that inf σ ( E A,tot + a ) ≥ inf σ ( E A,tot + a ) . Since E A,tot + a does not containany electron-electron interaction it can be written as a sum of operators acting ondifferent particles. This leads to a decomposition of the spectrum, σ ( E A,tot + a ) = σ ( E (1) A ) + σ ( E (2) A + U (2)0 V (2) ( U (2)0 ) − ) . (5.4)The spectrum of E (1) A is continuous and extends to infinity since the magneticfield does not affect the electronic motion along B (which is the e -direction).Therefore, σ ( E A,tot + a ) is also continuous such that λ ∈ σ ( E A,tot + a ) with thedecomposition λ = λ + λ according to (5.4).Let ( ϕ (1) n ) n ∈ N be a Weyl sequence for λ consisting of normalized functionswith ϕ (1) n w ⇀ k ( E (1) A − λ ) ϕ (1) n k → n → ∞ . (5.5)According to Appendix B (Lemma 6, which also holds for single-particle operatorsof the form E ( k ) A + w ( k ) ) we can in addition assume that ϕ (1) n ∈ C ∞ ( R \ B n (0)) ⊗ C .Let ( φ (2) n ) n ∈ N be a defining sequence for λ with φ (2) n ∈ C ∞ ( R ) ⊗ C satisfying k φ (2) n k = 1 and k ( E (2) A + U (2)0 V (2) ( U (2)0 ) − − λ ) φ (2) n k → n → ∞ . This implies,for a given ǫ > , the existence of N ∈ N such that k ( E (2) A + U (2)0 V (2) ( U (2)0 ) − − λ ) φ (2) N k < ǫ. (5.6)We claim that a subsequence of the sequence ( A ψ n ) n ∈ N with ψ n := ϕ (1) n φ (2) N isa Weyl sequence for λ obeying k ( h BR − λ ) A ψ n k → n → ∞ such that λ ∈ σ ess ( h BR ) . Disregarding for the moment the antisymmetrization, we estimate k ( h BR − λ ) ϕ (1) n φ (2) N k ≤ k ( E A,tot + a − λ ) ϕ (1) n φ (2) N k + k r ϕ (1) n φ (2) N k . (5.7)For the first term we have k ( E (1) A − λ + E (2) A + U (2)0 V (2) ( U (2)0 ) − − λ ) ϕ (1) n φ (2) N k≤ k φ (2) N ( E (1) A − λ ) ϕ (1) n k + k ϕ (1) n ( E (2) A + U (2)0 V (2) ( U (2)0 ) − − λ ) φ (2) N k (5.8) ≤ k φ (2) N k ǫ + k ϕ (1) n k ǫ by assumption for n sufficiently large.We will now show that the remainder r in (5.7) is n -bounded. This is equiv-alent to proving the following estimates, | ( ψ, U (1)0 V (1) ( U (1)0 ) − ψ n ) | ≤ c ( B ) n k ψ k k ψ n k| ( ψ, U (1)0 U (2)0 V (12) ( U (1)0 U (2)0 ) − ψ n ) | ≤ c ( B ) n k ψ k k ψ n k (5.9)for all ψ ∈ ( C ∞ ( R ) ⊗ C ) , with c ( B ) some constant. Using that ϕ (1) n is local-ized outside the ball B n (0) we define a smooth auxiliary function χ ∈ C ∞ ( R )mapping to [0 , χ (cid:16) x n (cid:17) = (cid:26) , x < Cn/ , x ≥ Cn, (5.10) and set C = 1 . Then ϕ (1) n = χ ϕ (1) n and we have | ( ψ, U (1)0 V (1) ( U (1)0 ) − χ ϕ (1) n φ (2) N ) | ≤ | (( U (1)0 ) − ψ, ( γx χ ) ( U (1)0 ) − ϕ (1) n φ (2) N ) | + | (( U (1)0 ) − ψ, γx [( U (1)0 ) − , χ ] ϕ (1) n φ (2) N ) | . (5.11)The first term is bounded by 2 c/n since supp χ requires x ≥ n/ . According tothe note below (4.23) the second term can be estimated by cn k ψ k k ϕ (1) n k k φ (2) N k . For the two-particle potential we define χ ( x − x n ) as in (5.10) with C = . Since φ (2) N ∈ C ∞ ( R ) ⊗ C there exists an R > x < R on supp φ (2) N . If one chooses n such that n > R , then | x − x | ≥ x − x > n on supp ψ n . Thus ϕ (1) n φ (2) N = χ ϕ (1) n φ (2) N . In the decomposition | ( ψ, U (1)0 U (2)0 V (12) ( U (1)0 U (2)0 ) − χ ϕ (1) n φ (2) N ) | ≤ | (( U (1)0 U (2)0 ) − ψ, e | x − x | χ · ( U (1)0 U (2)0 ) − ϕ (1) n φ (2) N ) | + | (( U (1)0 U (2)0 ) − ψ, e | x − x | n [( U (1)0 ) − , χ ] ( U (2)0 ) − + ( U (1)0 ) − [( U (2)0 ) − , χ ] o ϕ (1) n φ (2) N ) | (5.12)the operator in the first term is 1 /n -bounded since χ = 0 only if | x − x | ≤ /n. The operator containing | x − x | [( U (1)0 ) − , χ ] is 1 /n -bounded according to thenote below (4.23). For the last term we use a decomposition and subsequent esti-mate as indicated in (B.4) and (B.5). This proves the assertion (5.9) and therefore, k r ψ n k ≤ c ( B ) n k ψ n k < ǫ (5.13)for n sufficiently large. Hence the sequence ( ψ n ) n ∈ N obeys k ( h BR − λ ) ψ n k < ǫ. The consideration of the antisymmetry of the sequence as well as its normal-izability for sufficiently large n can be done in the same way as in the absence of amagnetic field [15]. Collecting results, this shows that a subsequence of ( A ψ n ) n ∈ N is a Weyl sequence for λ and verifies that λ ∈ σ ess ( h BR ) . Proof of Theorem 1 (hard part) Let us introduce the Ruelle-Simon partition of unity ( φ j ) j =0 , , ∈ C ∞ ( R )subordinate to the two-cluster decompositions (5.1) [5, 17]. It is defined on theunit sphere and has the following properties, X j =0 φ j = 1 , φ j ( λ x ) = φ j ( x ) for x = 1 and λ ≥ , supp φ j ∩ R \ B (0) ⊂ { x ∈ R \ B (0) : | x − x | ≥ Cx and x j ≥ Cx } , j = 1 , , supp φ ∩ R \ B (0) ⊂ { x ∈ R \ B (0) : x k ≥ Cx ∀ k ∈ { , } } , (6.1) where x = ( x , x ) , x = | x | and C is a positive constant. The (IMS-type)localization formula for h BR is written in the following way( u, h BR u ) = X j =0 ( φ j u, h BR φ j u ) − X j =0 ( φ j u, [ h BR , φ j ] u ) . (6.2)For reasons which will become clear shortly we can assume that the support of u is outside the ball B R (0) . First we show that in this case the commutator in the last term of (6.2) tendsto zero as R → ∞ . Explicitly, we have to prove | X j =0 ( φ j u, [ E ( k ) A , φ j ] u ) | ≤ c ( B ) R k u k | ( φ j u, [ U ( k )0 V ( k ) ( U ( k )0 ) − , φ j ] u ) | ≤ c ( B ) R k u k (6.3) | ( φ j u, [ U (1)0 U (2)0 V (12) ( U (1)0 U (2)0 ) − , φ j ] u ) | ≤ c ( B ) R k u k , where c ( B ) is a generic constant depending on the magnetic field. Let us introducethe smooth function χ ∈ C ∞ ( R ) mapping to [0 , 1] by means of χ (cid:16) x R , x R (cid:17) = (cid:26) , x < R/ , x ≥ R (6.4)where x = ( x , x ) , x = p x + x and R > . Then χ is unity on the supportof u . The resulting property u = χ u allows us to replace in (6.2) the commutatorform ( φ j u, [ O , φ j ] u ) with ( φ j u, [ O , φ j χ ] u ) where the (arbitrary) operator O canbe identified with a constituent of h BR . For the kinetic energy operator of particle1 (or for any single-particle operator O ) one has the identity [9] X j =0 ( φ j u, [ E (1) A , φ j χ ] u )= − X j =0 Z R d x Z R d x ′ u ( x ) E (1) A ( x , x ′ ) [( φ j χ )( x ′ , x ) − ( φ j χ )( x )] u ( x ′ , x ) . (6.5)From the mean value theorem we obtain | ( φ j χ )( x ′ , x ) − ( φ j χ )( x , x ) | ≤ | ( x − x ′ ) ( ∇ x φ j χ )( ξ , x ) | (6.6)with ξ on the line between x and x ′ . Since χ is supported outside B R/ (0)with R/ > φ j obeys the scaling property φ j ( x ) = φ j ( x R/ ) fromthe first line of (6.1) on supp χ . Furthermore, φ j and χ have a bounded deriva-tive since χ ′ ∈ C ∞ ( R ) and since φ j ∈ C ∞ ( R ) is defined on the compact unitsphere, being homogeneous of degree zero outside the unit ball. Therefore we have | ∇ x φ j χ )( ξ , x ) | ≤ c/R. With Lemma 2 for E (1) A ( x , x ′ ) , the 1 /R -boundedness of the kernel of theoperator in (6.5) is established by the Schur test (4.29) (using f = 1), I ( x , x ) = 12 X j =0 Z R d x ′ C ( B ) | x − x ′ | e − ( m − ǫ ) | x − x ′ | | x − x ′ | c R ≤ ˜ C ( B ) R . (6.7)This proves the first inequality of (6.3).For the single-particle potential contribution to [ h BR , φ j χ ] we have[ U ( k )0 V ( k ) ( U ( k )0 ) − , φ j χ ] u = − γ [ U ( k )0 , φ j χ ] 1 x k ( U ( k )0 ) − u − γ U ( k )0 x k [( U ( k )0 ) − , φ j χ ] u. (6.8)The R -boundedness of these two terms is guaranteed by (4.23) and the note below.The two-particle commutator can be treated in the same way, by using (B.4)and (B.5) together with the note following (4.23). This establishes the remaininginequalities of (6.3).In a next step we employ Persson’s theorem (see e.g. [5, Thm 3.12]) statingthat inf σ ess ( h BR ) = lim R →∞ inf k u k =1 ( u, h BR u ) (6.9)if u ∈ A ( C ∞ ( R \ B R (0)) ⊗ ( C ) ) . The assumptions for Persson’s theorem tohold are the relative form boundedness of the potential with respect to the kineticenergy operator (proven in Lemma 1) and the existence of a Weyl sequence ( u n ) n ∈ N to λ ∈ σ ess ( h BR ) where u n is supported outside the ball B n (0). The proof of thelatter item is given in Appendix B.Inserting (6.2) with (6.3) into (6.9) we obtaininf σ ess ( h BR ) = lim R →∞ inf k u k =1 X j =0 ( φ j u, ( E A,tot + a j ) φ j u ) + X j =0 ( φ j u, r j φ j u ) (6.10)where h BR = E A,tot + a j + r j was used.In the final step it remains to show that the second term in the curly bracketsof (6.10) also tends uniformly to zero as R → ∞ . If this is true then, recalling thedefinition (5.3) of Σ , one can estimateinf σ ess ( h BR ) = lim R →∞ inf k u k =1 2 X j =0 ( φ j u, ( E A,tot + a j ) φ j u ) ≥ lim R →∞ inf k u k =1 2 X j =0 Σ ( φ j u, φ j u ) = Σ (6.11)which completes the proof of the hard part. To provide the missing link let us start by taking j = 1 and consider thesingle-particle contribution to r . If in the auxiliary function χ ( x /R ) from (5.10) one takes C equal to theconstant from the partition of unity (6.1) and R > χ = 1 on the supportof φ u (note that supp φ and supp u require x ≥ Cx and x ≥ R , respectively).We decompose | ( φ u, U (1)0 V (1) ( U (1)0 ) − χ φ u ) | ≤ | (( U (1)0 ) − φ u, γx χ ( U (1)0 ) − φ u ) | + | (( U (1)0 ) − φ u, γx [( U (1)0 ) − , χ ] φ u ) | . (6.12)The first contribution can be estimated by γCR k φ u k since supp χ requires x ≥ CR/ . The R -boundedness of the second contribution follows from the notebelow (4.23).For handling the two-particle contribution to r we again introduce the func-tion χ (( x − x ) /R ) from (5.10), its argument being now the difference betweenthe single-particle coordinates. Again, χ = 1 on the support of φ u . With a (6.12)-type decomposition (where U (1)0 is replaced by U (1)0 U (2)0 and γ/x by e / | x − x | )it is easy to see that the first term is bounded by 2 e / ( CR ) k φ u k . The secondcontribution is given by (cid:12)(cid:12)(cid:12)(cid:12) (( U (1)0 U (2)0 ) − φ u, e | x − x | n [( U (1)0 ) − , χ ]( U (2)0 ) − +( U (1)0 ) − [( U (2)0 ) − , χ ] o φ u ) (cid:12)(cid:12)(cid:12) . (6.13)The R -boundedness of this contribution is established by the note following (4.23)(with the help of (B.4)- and (B.5)-type decompositions).The case j = 2 follows from the symmetry of h BR under particle exchange.For j = 0 we have r = P k =1 U ( k )0 V ( k ) ( U ( k )0 ) − and we introduce also herethe function χ ( x k /R ) from (5.10). Since supp φ u ⊂ { x ∈ R : x ≥ R, x ≥ CR, x ≥ CR } , we have φ u = χ φ u for both values of k . The proof of the 1 /R -boundedness of ( φ u, r φ u ) is therefore the same as for the j = 1 single-particlecase. In conclusion, this shows that (6.10) reduces to (6.11) which completes theproof.We remark that the present proof is valid for field strengths 0 ≤ B < ∞ and thus covers the case A = as well. Appendix A (Integral formulae) For convenience we cite some general formulae. We have [10, p.340] Z ∞ dt t ν e − γt e − β/t = 2 (cid:18) βγ (cid:19) ν +12 K ν +1 (2 p βγ ) , β, γ > . (A.1) Moreover [10, p.705], Z ∞ dt t µ +1 ( √ t + a ) ν K ν ( α p t + a ) = 2 µ Γ( µ + 1) α µ +1 a ν − µ − K ν − µ − ( αa ) ,α > , a > , µ > − . (A.2)We also provide the asymptotic formulae for the modified Bessel functions[1, p.374], K ( z ) ∼ − ln z, K ( z ) ∼ z , K ν ( z ) ∼ Γ( ν )2 ν − z ν , ν > , for z → K ν ( z ) ∼ r π z e − z , ν ≥ , for z → ∞ (A.3)and recall that K − ν ( z ) = K ν ( z ), as well as K ν +1 ( z ) = K ν − ( z ) + νz K ν ( z ) . Appendix B (Existence of Weyl sequence outside balls) Lemma 6. Let h BR = E A,tot + w , B a constant magnetic field, and let w berelatively form bounded with respect to E A,tot . If λ ∈ σ ess ( h BR ) there exists aWeyl sequence ( u n ) n ∈ N to λ with the additional property u n ∈ A ( C ∞ ( R \ B n (0)) ⊗ ( C ) ) , where B n (0) is a ball of radius n centered at the origin. The proof follows closely the one given in [5] and in [15]. For λ ∈ σ ess ( h BR )there exists a Weyl sequence ψ n ∈ A ( C ∞ ( R ) ⊗ C ) , k ψ n k = 1 with ψ n w ⇀ k ( h BR − λ ) ψ n k → n → ∞ . Thus for any ǫ := n > N ( n ) > n such that k ( h BR − λ ) ψ N ( n ) k < ǫ. In order to construct the Weyl sequence ( u n ) n ∈ N , which consists of functionslocalized outside B n (0), we define a smooth function χ n ∈ C ∞ ( R ) which issymmetric (with respect to the interchange of x and x ) and maps to [0 , χ n ( x ) := χ (cid:16) x n (cid:17) = (cid:26) , x ≤ n , x > n . (B.1)Then we claim that a subsequence of ( χ n ψ N ( n ) ) n ∈ N with χ n := 1 − χ n is thedesired sequence ( u n ) n ∈ N .In order to show that k ( h BR − λ ) χ n ψ N ( n ) k → n → ∞ , we decompose k ( h BR − λ ) χ n ψ N ( n ) k ≤ k χ n ( h BR − λ ) ψ N ( n ) k + k [ h BR , χ n ] ψ N ( n ) k . (B.2)The first term goes to zero for n → ∞ by assumption since χ n is bounded. Asregards the second term, the kinetic energy contribution k [ E ( k ) A , χ n ] ψ N ( n ) k ≤ c ( B ) n by Lemma 5 since χ n as a C ∞ -function has bounded derivatives. Thecoordinate of the second particle, x ¯ k (with ¯ k ∈ { , }\ k ) , in χ n can be treated asa parameter.For the contribution from the single-particle potential we have[ U ( k )0 V ( k ) ( U ( k )0 ) − , χ n ] = − γ [ U ( k )0 , χ n ] 1 x k ( U ( k )0 ) − − γU ( k )0 x k [( U ( k )0 ) − , χ n ] . (B.3) Each of the two terms is n -bounded according to (4.23) and the note below.The commutator with the two-particle potential is written in the followingway, [ U (1)0 U (2)0 V (12) ( U (1)0 U (2)0 ) − , χ n ]= e [ U (1)0 , χ n ] 1 | x − x | (cid:18) | x − x | U (2)0 | x − x | (cid:19) ( U (1)0 U (2)0 ) − + e U (1)0 · [ U (2)0 , χ n ] 1 | x − x | ( U (1)0 U (2)0 ) − + e U (1)0 U (2)0 | x − x | [( U (1)0 ) − , χ n ] ( U (2)0 ) − (B.4)+ e U (1)0 U (2)0 (cid:18) | x − x | ( U (1)0 ) − | x − x | (cid:19) | x − x | [( U (2)0 ) − , χ n ] . All commutators (including the factor | x − x | ) are n -bounded by Corollary 1 toLemma 4 and by the note following (4.23). The boundedness of the factors inround brackets follows from the decomposition1 | x − x | ( U (1)0 ) − | x − x | = ( U (1)0 ) − + 1 | x − x | [( U (1)0 ) − , | x − x | ] (B.5)where | x − x | satisfies the requirements of Corollary 1 for the function g .It remains to prove that χ n ψ N ( n ) is normalizable for sufficiently large n . Tothis aim we show that k χ n ψ N ( n ) k → n → ∞ . Then k χ n ψ N ( n ) k = k ψ N ( n ) − χ n ψ N ( n ) k → k ψ N ( n ) k = 1 for all n . Due to the semibound-edness of h BR there exists µ > h BR + µ has a bounded inverse. Weestimate k χ n ψ N ( n ) k = k χ n ( h BR + µ ) − [ ( h BR − λ ) + ( µ + λ ) ] ψ N ( n ) k≤ k χ n ( h BR + µ ) − k k ( h BR − λ ) ψ N ( n ) k + | µ + λ | k χ n ( h BR + µ ) − ψ N ( n ) k . (B.6)The first summand is bounded by ǫ times a constant by assumption. The secondterm tends to zero provided we can show that χ n ( h BR + µ ) − is a compactoperator which turns the weakly convergent sequence ( ψ N ( n ) ) n ∈ N into a stronglyconvergent one. Consider the decomposition χ n ( h BR + µ ) − ψ N ( n ) = χ n X k =1 S ( k )2 A ! − · X k =1 S ( k )2 A ! ( E A,tot + µ ) − (cid:16) ( E A,tot + µ ) ( h BR + µ ) − (cid:17) ( h BR + µ ) − ψ N ( n ) . (B.7)From the diamagnetic inequality-based relation (2.2) it immediately follows that(for fixed n ) the operator in curly brackets is compact since χ n ( P k =1 ( p k + m )) − / is compact. For the boundedness of the operator in square brackets we invoke an estimateproven by Balinsky, Evans and Lewis [4] for the Pauli operator (i.e. for the case m = 0 in the lemma below). Lemma 7. For a single particle let S A = (cid:2) ( p − e A ) + m (cid:3) and E A from (1.4)with A ∈ L ,loc ( R ) . Then the following estimate holds, E A ≥ δ m ( B ) S A , (B.8) where δ m ( B ) = inf k f k =1 k (1 − S ∗ m S m ) f k > with S m := ( eB ) ( E A + eB ) − (B.9) and f ∈ L ( R ) ⊗ C . We note that S m S ∗ m = eB ( E A + eB ) − B ≤ eB ( m + eB ) − B < S ∗ m S m < 1) since for m = 0 zero modes are absent irrespectiveof B . Therefore for all f with k f k = 1 one has 1 > ( f, S m S ∗ m f ) = k S ∗ m f k andhence 1 > sup f k S ∗ m f k = k S ∗ m k . Thus the proof of the lemma can be copiedfrom [4]. For a constant magnetic field B , S ∗ m S m ≤ eB m + eB . This leads to δ m ( B ) ≥ m m + eB . The required boundedness of the operator in (B.7) is based on the existenceof a constant c such that for ϕ ∈ A ( L ( R ) ⊗ C ) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k =1 S ( k )2 A ! ( E A,tot + µ ) − ϕ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k =1 S ( k ) A ! ( E A,tot + µ ) − ϕ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ c k ϕ k . (B.11)Defining ˜ ϕ := ( E A,tot + µ ) − ϕ this is equivalent to proving (for µ ≥ ϕ, X k =1 S ( k ) A ˜ ϕ ) ≤ c ( ˜ ϕ, ( E A,tot + µ ) ˜ ϕ ) , (B.12)which is assured by Lemma 7 with c = δ m ( B ) . The boundedness of the next term in (B.7) makes use of the relative formboundedness of the potential (2.6). Defining φ n := ( h BR + µ ) − ˜ ψ n with ˜ ψ n :=( h BR + µ ) − ψ N ( n ) , we have to show k ( E A,tot + µ ) ( h BR + µ ) − ˜ ψ n k = ( φ n , ( E A,tot + µ ) φ n ) ≤ c ( φ n , ( h BR + µ ) φ n ) . (B.13)Using (2.6) the r.h.s. of (B.13) can be estimated,( φ n , ( h BR + µ ) φ n ) ≥ ( φ n , ( E A,tot + µ ) φ n ) − | ( φ n , w φ n ) |≥ (1 − c ) ( φ n , ( E A,tot + µ ) φ n ) − C ( B ) k φ n k + c µ ( φ n , φ n ) (B.14) with c := γπ/ e π/ < . If µ is chosen larger than C ( B ) /c then theterms proportional to k φ n k can be dropped, such that the second line of (B.14)is ≥ c ( φ n , ( E A,tot + µ ) φ n ) with c := − c . This proves (B.13).To complete the proof of the compactness of χ n ( h BR + µ ) − we keep n fixed.From the discussion above there exists to the given ǫ = 1 /n an N ( n ) > n suchthat k χ n ( P k =1 S ( k )2 A ) − / · B ψ N ( n ) k < ǫ. B comprises the bounded operators in(B.7) to the right of the one in curly brackets, and N ( n ) has to be chosen largeenough to satisfy the previous condition k ( h BR − λ ) ψ N ( n ) k < ǫ as well. Thisproves k χ n ψ N ( n ) k ≤ c ǫ + | µ + λ | ǫ and hence the normalizability of the Weylsequence under consideration. Appendix C (Estimate for the kernel of A E ) For √ A E we use the representation r E A + mE A = lim t → − e − t ( E A + m ) t √ E A √ E A + m . (C.1)With the integral formula (4.3) for each of the factors in the denominator weobtain, using the Taylor formula to first order, r E A + mE A = lim t → tπ Z ∞ dt ′ √ t ′ Z ∞ dτ √ τ e − τm n e − ( τ + t ′ ) E A − e − tm e − ( τ + t ′ + t ) E A o = − π Z ∞ dt ′ √ t ′ Z ∞ dτ √ τ e − τm (cid:26) ddτ (cid:16) e − ( τ + t ′ ) E A (cid:17) − m e − ( τ + t ′ ) E A (cid:27) . (C.2)As a next step we estimate the kernel of e − tE A and its derivative. Using the integralrepresentation (4.10) and estimating the kernel of e − tE A with the help of (3.11)leads to | e − tE A ( x , x ′ ) | ≤ π t Z ∞ τ dτ e − τ e − t m / τ e − ( x − x ′ ) τ/t (cid:18) eB t τ (cid:19) . (C.3)From (A.1) one obtains, abbreviating ξ := p t + ( x − x ′ ) , | e − tE A ( x , x ′ ) | ≤ m π (cid:26) mtξ K ( mξ ) + eB tξ K ( mξ ) (cid:27) . (C.4)For the derivative we have with ̺ := t / (4 τ ) ,ddt e − tE A ( x , x ′ ) = 1 √ π Z ∞ dτ √ τ e − τ dd̺ (cid:16) e − ̺E A ( x , x ′ ) (cid:17) · t τ . (C.5)With the help of (4.5) and its estimate (4.6) one obtains (cid:12)(cid:12)(cid:12)(cid:12) ddt e − tE A ( x , x ′ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ π t Z ∞ dτ e − τ e − m t / τ e − ( x − x ′ ) τ/t · (cid:20) τt + 5 eB + m + 2 eB ( x − x ′ ) τt + ( x − x ′ ) τ t + ( eB m + 2 e B ) t τ (cid:21) (C.6)which, using the integral formula (A.1), reduces to (cid:12)(cid:12)(cid:12)(cid:12) ddt e − tE A ( x , x ′ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ π (cid:26) [3 + eB ( x − x ′ ) ] m ξ K ( mξ ) + (5 eB + m ) m ξ K ( mξ ) + m ξ ( x − x ′ ) K ( mξ ) + ( eB m e B ) K ( mξ ) (cid:27) . (C.7)Insertion of (C.4) and (C.7) into (C.2) with t := τ + t ′ results in (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)r E A + mE A ( x , x ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π Z ∞ dτ √ τ e − τm Z ∞ dt ′ √ t ′ n (3 + eB ( x − x ′ ) · m ξ K ( mξ ) + (5 eB + m ) m ξ K ( mξ ) + m ξ ( x − x ′ ) K ( mξ ) (C.8)+ ( eB m e B ) K ( mξ ) + m τ + t ′ ) (cid:20) mξ K ( mξ ) + eB K ( mξ ) ξ (cid:21)(cid:27) . Now we follow the strategy given below (4.15), i.e. we estimate e − τm ≤ ξ − ν K ν ( mξ ) ≤ y − ν K ν ( my ) with y := p τ + t ′ + ( x − x ′ ) in (C.8). Thenthe double integral can be performed analytically by (A.2). The result is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)r E A + mE A ( x , x ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c K / ( m | x − x ′ | ) | x − x ′ | / + c ( B ) | x − x ′ | / K / ( m | x − x ′ | )+ c ( B ) K / ( m | x − x ′ | ) | x − x ′ | / + c K / ( m | x − x ′ | ) | x − x ′ | / + c ( B ) | x − x ′ | / K / ( m | x − x ′ | ) + 2 c K ( m | x − x ′ | ) | x − x ′ | + 2 c ( B ) K ( m | x − x ′ | ) . (C.9)From (C.9) follows the estimate | A E ( x , x ′ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)r E A + m E A ( x , x ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ˜ c ( B ) | x − x ′ | e − ( m − ǫ ) | x − x ′ | (C.10)for ǫ ∈ (0 , m ) . We note that by (C.4) this proof provides an estimate for the heat kernel e − tE A ( x , x ′ ) of the Brown-Ravenhall operator for a free electron in a constantmagnetic field. Appendix D (Erratum to Reference 14) We collect the changes due to the replacement of ( ϕ, ( p − e A ) ϕ ) ≥ ( ϕ, p ϕ )(which is not generally valid) by (2.2). We emphasize that this only affects theproofs, but not the results of [14].Eqs. (3.1), (3.2) should be deleted. Instead, one has k x ϕ k ≤ k p ( p − e A ) + m ϕ k , ( ϕ, x ϕ ) ≤ π ϕ, p ( p − e A ) + m ϕ ) , (3.1)valid for m ≥ . Eq. (3.3) should be replaced by the two separate estimates,tr (cid:20) p + e σ B µ (cid:21) d − ≤ L d, Z R (cid:18) e | B | µ (cid:19) d + d x , tr [ µ ( p − e A ) + e σ B ] d − ≤ µ d L d, Z R (cid:18) e | B | µ (cid:19) d + d x . (3.3)The paragraph on p.7512 starting with ’The strategy ...’ should be replaced by:The strategy to show the compactness of K is to start with the operator K := χ ( p + m ) − which is compact as a product of bounded functions f ( x ) , g ( p ) , each of which tending to zero as x , respectively p , go to infinity (see, e.g., [31,Lemma 7.10]). Then K := χ [( p − e A ) + m ] − is also compact [5, p.117]. Inthe following, bounded operators O , O are constructed such that K ·O ·O = K. Let O := p ( p − e A ) + m /E A . For showing the boundedness of O let ψ := E − A ϕ. Then from (4.6), kO ϕ k = ( ψ, (( p − e A ) + m ) ψ ) ≤ ( ψ, ( E A + e | B | ) ψ ) ≤ − κe k ϕ k + eC κ − κe k E − A k k ϕ k , (5.14)the rhs being obviously bounded. With O := E A Λ A, + ( H + µ ) − ≤ K . Acknowledgment I would like to thank L.Erd¨os for clarifying discussions and S.Morozov forcritical comments. 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