Dirac-Coulomb operators with general charge distribution. I. Distinguished extension and min-max formulas
aa r X i v : . [ m a t h . SP ] M a r DIRAC-COULOMB OPERATORS WITH GENERALCHARGE DISTRIBUTIONI. DISTINGUISHED EXTENSION AND MIN-MAXFORMULAS
MARIA J. ESTEBAN, MATHIEU LEWIN, AND ´ERIC S´ER´E
Abstract.
This paper is the first of a series where we study the spectralproperties of Dirac operators with the Coulomb potential generated byany finite signed charge distribution µ . We show here that the operatorhas a unique distinguished self-adjoint extension under the sole condi-tion that µ has no atom of weight larger than or equal to one. Then wediscuss the case of a positive measure and characterize the domain us-ing a quadratic form associated with the upper spinor, following earlierworks [15, 16] by Esteban and Loss. This allows us to provide min-maxformulas for the eigenvalues in the gap. In the event that some eigen-values have dived into the negative continuum, the min-max formulasremain valid for the remaining ones. At the end of the paper we alsodiscuss the case of multi-center Dirac-Coulomb operators correspondingto µ being a finite sum of deltas. Contents
1. Introduction 12. Distinguished self-adjoint extension for a general charge 42.1. Notation 42.2. Distinguished self-adjoint extension 43. Domain and min-max formulas for positive measures 73.1. Description of the domain 73.2. Min-max formulas for the eigenvalues 93.3. Application to (critical and sub-critical) multi-center potentials 104. Proof of Theorem 1 125. Proof of Theorem 2 196. Proof of Theorems 3 and 4 227. Proof of Theorem 6 29References 321.
Introduction
Relativistic effects play an important role in the description of quantumelectrons in molecules containing heavy nuclei, even for not so large valuesof the nuclear charge. Without relativity, gold would have the same coloras silver [18], mercury would be solid at room temperature [5] and batteries
Date : March 10, 2020. would not work [40]. This is due to the very strong Coulomb forces expe-rienced by the core electrons, which can then attain large velocities of theorder of the speed of light.A proper description of such atoms and molecules is based on the Diracoperator [35, 12]. This is an order-one differential operator which has sev-eral famous mathematical difficulties, all associated with important physicalfeatures. For instance the spectrum of the free Dirac operator is not semi-bounded which prevents from giving an unambiguous definition of a “groundstate” and turns out to be related to the existence of the positron [12]. Inaddition, the Dirac operator has a critical behavior with respect to theCoulomb potential 1 / | x | which gives a bound Z
137 on the highest possi-ble charge of atoms in the periodic table, for pointwise nuclei.This paper is the first in a series where we study the spectral properties ofDirac operators with the Coulomb potential generated by any finite (signed)measure µ representing an external charge: D − µ ∗ | x | = − i X j =1 α j ∂ x j + β − µ ∗ | x | . (1)One typical example is when the measure µ describes the M nuclei in amolecule and this corresponds to µ = M X m =1 αZ m δ R m where R m ∈ R and Z m ∈ (0 , ∞ ) are, respectively, the positions and chargesof the M nuclei, and where α ≃ /
137 is the Sommerfeld fine structureconstant. For instance for water (H O) we have M = 3, Z = Z = 1and Z = 8. In practice, one should also take into account the Coulombrepulsion between the electrons. In mean-field type models such as Dirac-Fock or Kohn-Sham [17], this is described by a nonlinear potential which isoften more regular than the nuclear attraction. This leads us to consideringthe following class of (signed) measures µ = M X m =1 αZ m δ R m + e µ, where the measure e µ is more regular (for instance absolutely continuouswith respect to the Lebesgue measure).In this paper, we first quickly recall existing results and then prove the ex-istence of a distinguished self-adjoint extension for operators of the form (1),under the sole assumptions that | µ | ( R ) < ∞ and | µ ( { R } ) | < R ∈ R . (2)In particular we allow an infinite number M = + ∞ of atoms but assumethat the total nuclear charge is bounded. We follow well established methodsfor singular Dirac operators [32, 37, 38, 39, 29, 30, 24, 23, 22] but faceseveral difficulties due to the generality of our measure µ . In a second stepwe consider the particular case of a positive measure (or more generally ameasure so that the Coulomb potential µ ∗ | x | − is bounded from below) IRAC-COULOMB OPERATORS WITH GENERAL CHARGE DISTRIBUTION I 3 and we characterize the domain using a method introduced in [15, 16] andrecently generalized in [31]. This method allows us to provide min-maxformulas for the eigenvalues in the gap [ − , µ atfixed maximal charge ν : λ ( ν ) := inf µ > µ ( R ) ν λ (cid:18) D − µ ∗ | x | (cid:19) , that is, we ask what is the lowest possible eigenvalue of all possible chargedistributions with µ ( R ) ν . This problem is indeed our main motivationfor studying Dirac operators of the type (1) with general measures µ . Wewill prove in [14] that there exists a critical coupling constant2 π + π < ν λ ( ν ) > − ν < ν , that is, the first eigenvaluecan never attain the bottom of the spectral gap for any measure µ with µ ( R ) < ν . In addition, for 0 ν < ν we prove the existence of anoptimal minimizing measure for λ ( ν ) and show that it concentrates on a setof Lebesgue measure zero. That the optimal measure is necessarily singularis the main justification for considering general charge distributions.It is well known that the first eigenvalue of the non-relativistic Schr¨odingeroperator is concave in µ , which implies thatinf µ > µ ( R ) ν λ (cid:18) − ∆2 − µ ∗ | x | (cid:19) = λ (cid:18) − ∆2 − ν | x | (cid:19) = − ν . We conjecture that the same holds in the Dirac case, which would imply ν = 1 and λ ( ν ) = √ − ν . We will mention in this paper some physicalimplications that the validity of this conjecture would have for the electroniccontribution to the potential energy surface of diatomic systems and othermolecules.The paper is organized as follows. In the next section we show thatthe operator (1) has a unique distinguished self-adjoint extension under theassumption (2), whereas in Section 3 we discuss the domain and min-maxformulas for the eigenvalues under the additional condition that µ >
0. Therest of the paper is then devoted to the proofs of our main results.
Acknowledgement.
This project has received funding from the EuropeanResearch Council (ERC) under the European Union’s Horizon 2020 researchand innovation programme (grant agreement MDFT No 725528 of M.L.),and from the Agence Nationale de la Recherche (grant agreement molQED).
M.J. ESTEBAN, M. LEWIN, AND ´E. S´ER´E Distinguished self-adjoint extension for a general charge
In this section we give a meaning to the operator D − µ ∗ | x | − for thelargest possible class of bounded measures µ . But first we need to clarifysome notation.2.1. Notation.
We work in a system of units for which m = c = ~ = 1.The free Dirac operator D is given by D = − i α · ∇ + β = − i X k =1 α k ∂ x k + β, (3)where α , α , α and β are Hermitian matrices which satisfy the followinganticommutation relations: α k α ℓ + α ℓ α k = 2 δ kℓ ,α k β + βα k = 0 ,β = 11 . The usual representation in 2 × β = (cid:18) I − I (cid:19) , α k = (cid:18) σ k σ k (cid:19) ( k = 1 , , , where the Pauli matrices are defined as σ = (cid:18) (cid:19) , σ = (cid:18) − ii (cid:19) , σ = (cid:18) − (cid:19) . The operator D is self-adjoint on H ( R , C ) and its spectrum is Sp( D ) =( −∞ , − ∪ [1 , ∞ ) [35].2.2. Distinguished self-adjoint extension.
The study of self-adjoint ex-tensions is a classical subject for Dirac-Coulomb operators. For instance,the one-center Coulomb operator D − ν | x | is known to be essentially self-adjoint on C ∞ c ( R \{ } , C ) for 0 ν √ / H ( R , C ) when 0 ν < √ /
2, whereas it has several possibleself-adjoint extensions for ν > √ /
2. When √ / < ν <
1, there is a uniqueextension which is distinguished by the property that its domain is includedin the ‘energy space’ H / ( R , C ). The domain of the extension is alwayslarger than H ( R , C ) when ν ∈ [ √ / , ν increases. When ν = 1 one can also define adistinguished self-adjoint extension (obtained for instance by taking the limit ν → − ) but its domain is no longer included in H / ( R , C ). There is nophysically-relevant extension for ν >
1. This corresponds to the previouslymentioned property that we should work under the constraint ν = αZ ≤ Z
137 on the maximal possible (integer) pointcharge in the periodic table, within Dirac theory.These relatively simple ODE-type results for the one-center Dirac oper-ator have been generalized in many directions. Investigating how robust isthe distinguished extension with regard to perturbations has indeed been
IRAC-COULOMB OPERATORS WITH GENERAL CHARGE DISTRIBUTION I 5 the object of many works. A survey of known results, mainly in the one-center case, may be found for instance in [13, Sec. 1.3]. A typical result isthat Dirac operators in the form D V = D + V ( x ) with | V ( x ) | ν | x | , ν < H / ( R , C ). Hence a pointwisebound on V is sufficient, which is rather remarkable for operators which arenot semi-bounded. This extension can also be obtained as a norm-resolventlimit by truncating the singularity of the potential V . Note that the criticalcase ν = 1 was handled in [15] for V > H / ( R , C ).There are fewer results about the multi-center case, which is howeveras physically important as the atomic case. The intuitive picture is thatself-adjointness is essentially a local problem at the singularities of the po-tentials, hence the results should be similar for multi-center Coulomb po-tentials. Indeed, Nenciu [30] and Klaus [23] have proved that there is aunique distinguished self-adjoint extension for D + V under the pointwiseassumption that | V ( x ) | M X j =1 ν m | x − R m | (5)with R m = R ℓ for m = ℓ and 0 ν m < m = 1 , ..., M . Note thatwe can add to V any bounded potential (or even a regular potential in thesense of [29, 30]), without changing the domain of self-adjointness. For otherresults on multi-center Dirac operators, see [21, 4].One important tool in these works is the Birman-Schwinger-type formulaof the resolvent [29, 25, 23]( D + V − z ) − = ( D − z ) − − ( D − z ) − p | V | (1+ SK z ) − S p | V | ( D − z ) − (6)where K z = p | V | ( D − z ) − p | V | , (7)and S = sgn( V ). This formula is valid as long as 1 − SK z is invertible withbounded inverse, and it can serve to define D V via its resolvent. In theone-center case | V ( x ) | ν | x | − we have for z = 0 k K k ν (this was conjectured in [29] and then, proved in [39, 22, 1]). This gives thedistinguished self-adjoint extension for 0 ν <
1. In the multi-center caseone cannot always use z = 0 since it can be an eigenvalue, when P Mj =1 ν j is large and the nuclei are close to each other. But the set of problematic z ’s is at most countable, hence the formula also allows one to define thedistinguished self-adjoint extension [30, 23].The above results do not cover the case where V ( x ) = − µ ∗ | x | − forgeneral measures µ . Such potentials indeed diverge like µ ( { R } ) | x − R | − atpoints R ∈ R where µ ( { R } ) >
0, but they can diverge at many other points
M.J. ESTEBAN, M. LEWIN, AND ´E. S´ER´E in space where µ is not necessarily a delta. In this paper, we prove the fol-lowing result, which confirms the intuition that only deltas are problematicwith regard to self-adjointness. Theorem 1 (Distinguished self-adjoint extension) . Let µ be any finite signedBorel measure on R , such that | µ ( { R } ) | < for all R ∈ R .Then the operator D − µ ∗ | x | , defined first on H ( R , C ) or on C ∞ c ( R , C ) , has a unique self-adjointextension whose domain is included in H / ( R , C ) . The domain of theextension satisfies D (cid:18) D − µ ∗ | x | (cid:19) ⊂ H R \ K [ j =1 B r ( R j ) (8) for all r > , where R , ..., R K ∈ R are all the points so that | µ ( { R j } ) | > / . This operator is the norm-resolvent limit of D − µ ∗ | x | ( | µ ∗ | x | | n ) when n → ∞ . Its essential spectrum is Sp ess (cid:0) D − µ ∗ | x | − (cid:1) = ( −∞ , − ∪ [1 , + ∞ ) . The proof of Theorem 1 is provided later in Section 4.The property (8) shows that functions in the domain can only be moresingular than H at the finitely many atoms of µ which have a mass largerthan 1 /
2. Note, however, that the domain can in principle be different fromthat of the finitely many point charges, D (cid:0) D − µ ∗ | x | − (cid:1) = D D − K X j =1 µ ( { R j } ) | x − R j | . Although we think that there should be a similar result with a larger spacethan H / ( R , C ) under the weaker condition that | µ ( { R } ) |
1, we havenot investigated this question in the general setting. More about the criticalcase can be read in Section 3.3 where we investigate the particular case of ameasure µ which is a pure sum of deltas, following [13].One important argument of the proof is to show that the operator B R se µ ∗ | x | | p | is compact, for every positive measure e µ with no atom. Here we have usedthe notation p = − i ∇ and B R for the ball of radius R . Then, after separatingthe region about each nucleus from the rest of space, we show that (cid:13)(cid:13)(cid:13)(cid:13)q | V µ | D + is q | V µ | (cid:13)(cid:13)(cid:13)(cid:13) < s large enough, where V µ = µ ∗ | x | − . This is what is needed to applyNenciu’s method [29, Cor. 2.1]. IRAC-COULOMB OPERATORS WITH GENERAL CHARGE DISTRIBUTION I 7 Domain and min-max formulas for positive measures
By following a method introduced in [15, 16] and further developed in [13,31], one can describe the distinguished self-adjoint extension more preciselyin the case of a positive measure: µ > . What we really need in this section is that V µ is bounded from below, butfor simplicity we require the positivity of µ everywhere. We use the notation V µ = µ ∗ | x | for the Coulomb potential induced by µ .3.1. Description of the domain.
Following [13], we introduce a new spacefor the upper component. We define the following norm k ϕ k V µ := (cid:18) ˆ R | σ · ∇ ϕ ( x ) | V µ ( x ) dx + ˆ R | ϕ ( x ) | dx (cid:19) / (9)which is controlled by the H norm of ϕ since (1 + V µ ) −
1. Hence thenorm is well defined on H ( R , C ). Since V µ >
0, we can replace 1 + V µ byany λ + V µ with λ > V µ of H ( R , C )for the norm in (9) is the same as the largest space given by the conditions ϕ ∈ L ( R , C ) , σ · ∇ ϕ (1 + V µ ) / ∈ L ( R , C ) . The following answers this question affirmatively.
Theorem 2 (The upper-spinor space V µ ) . Let µ > be any finite Borelmeasure on R so that µ ( { R } ) < for all R ∈ R .We have k ϕ k H / ( R , C ) max (cid:0) , µ ( R ) (cid:1) k ϕ k V µ k ϕ k H ( R , C ) (10) for all ϕ ∈ H ( R , C ) . We call V µ the completion of H ( R , C ) for thenorm k · k V µ , a space which is continuously embedded in H / ( R , C ) . Itcoincides with the completion of C ∞ c ( R , C ) for the same norm and is givenby V µ = n ϕ ∈ L ( R , C ) : ∃ g ∈ L ( R , C ) , σ · ∇ ϕ = (1 + V µ ) / g o (11) where σ · ∇ ϕ is understood in the sense of distributions. The proof of Theorem 2 is provided later in Section 5. The first part ofthe theorem says that there is a Hardy-type inequality ˆ R | σ · ∇ ϕ ( x ) | V µ ( x ) dx + ˆ R | ϕ ( x ) | dx > (cid:13)(cid:13)(cid:13) ( − ∆) ϕ (cid:13)(cid:13)(cid:13) max (cid:0) , µ ( R ) (cid:1) , (12) M.J. ESTEBAN, M. LEWIN, AND ´E. S´ER´E for ϕ ∈ H ( R , C ) and all positive measures µ . The second part saysthat the space of functions ϕ ∈ L ( R , C ) such that (1 + V µ ) − / σ · ∇ ϕ ∈ L ( R , C ) (this being interpreted as in (11)), which could a priori be largerthan V µ , is indeed equal to V µ . This is an important property for whatfollows and it allows one to extend the inequality (12) to all such functions ϕ .With regard to (11), we remark that V µ ∈ L ( R ) for every Radon measure µ , so that (1 + V µ ) / g ∈ L ( R , C ) is a distribution when g ∈ L ( R , C ).Moreover, we prove later in Lemma 9 that ∇ (1 + V µ ) − / ∈ L ( R ) for every µ . This implies that (1 + V ) − / σ · ∇ ϕ makes sense as a distribution and,in (11), it is then equivalent to requiring that this distribution belongs to L ( R , C ).In the special case of µ = M X m =1 ν m δ R m with 0 ν m <
1, we have by [13] V µ = ( ϕ ∈ L ( R , C ) : M X m =1 ˆ R | x − R m | | x − R m | | σ · ∇ ϕ ( x ) | dx < ∞ ) . In other words, the functions in V µ must be in H ( R \ { R , ..., R M } , C )and behave as stated close to the singularities. In general the space V µ depends on the size and location of the singularities of the potential V µ ,which are not necessarily produced by the atomic part of µ . Recall that thepotential V e µ of the non-atomic part e µ of µ can still diverge in some places,namely at all the points R ∈ R so that ˆ R d e µ ( x ) | x − R | = + ∞ . At each of these points, the norm is affected because 1 / (1 + V µ ) tends tozero, allowing thereby | σ · ∇ ϕ | to diverge a bit.We can now describe the domain of the distinguished self-adjoint exten-sion using the space V µ . Theorem 3 (Domain of the distinguished self-adjoint extension for µ > . Let µ > be any finite Radon measure on R so that µ ( { R } ) < for all R ∈ R .Then the domain of the self-adjoint extension from Theorem 3 is explicitlygiven by D ( D − V µ ) = (cid:26) Ψ = (cid:18) ϕχ (cid:19) ∈ L ( R , C ) : ϕ ∈ V µ , D Ψ − V µ Ψ ∈ L ( R , C ) (cid:27) (13) where in the last condition D Ψ and V µ Ψ are understood in the sense ofdistributions. Moreover, this extension is the unique one included in V µ × L ( R , C ) . We have D ( D − V µ ) ⊂ V µ × V µ ⊂ H / ( R , C ) . IRAC-COULOMB OPERATORS WITH GENERAL CHARGE DISTRIBUTION I 9
In addition, the Birman-Schwinger principle holds: λ ∈ ( − , is aneigenvalue of D − V µ if and only if is an eigenvalue of the bounded self-adjoint operator K λ = p V µ D − λ p V µ . The condition µ > V µ > V µ . If V µ is bounded-below, then the exact same result is validwith 1 + V µ replaced by C + V µ with a large enough constant C everywhere.The proof of Theorem 3 is provided below in Section 6.3.2. Min-max formulas for the eigenvalues.
Related to the above char-acterization of the domain are min-max formulas for eigenvalues [19, 8, 9,10, 11, 27, 28, 13, 31]. Our main result is the following
Theorem 4 (Min-max formulas) . Let µ > be any finite non-trivial Radonmeasure on R so that µ ( { R } ) < for all R ∈ R .Define the min-max values λ ( k ) := inf W subspace of F + dim W = k sup Ψ ∈ ( W ⊕ F − ) \{ } h Ψ , ( D − V µ ) Ψ ik Ψ k , k > , (14) where F is any chosen vector space satisfying C ∞ c ( R , C ) ⊆ F ⊆ H / ( R , C ) , and F + := (cid:26) Ψ = (cid:18) ϕ (cid:19) ∈ F (cid:27) , F − := (cid:26) Ψ = (cid:18) χ (cid:19) ∈ F (cid:27) . Then we have(i) λ ( k ) is independent of the chosen space F ;(ii) λ ( k ) ∈ [ − , for all k ;(iii) it is a non-decreasing sequence converging to 1: lim k →∞ λ ( k ) = 1 . (15) Let k be the first integer so that λ ( k ) > − . Then ( λ ( k ) ) k > k are all the eigenvalues of D − V µ in non-decreasing order,repeated in case of multiplicity, which are larger than − : Sp (cid:18) D − µ ∗ | x | (cid:19) ∩ ( − ,
1) = { λ ( k ) , λ ( k +1) , · · · } . Finally, if µ ( R ) π/ /π ≃ . , (16) then we have λ (1) > and there is no eigenvalue in ( − , . The min-max formula (14) for the eigenvalues of D − V µ is based on adecomposition of the four-component wavefunction into its upper and lowerspinors, Ψ = (cid:18) ϕχ (cid:19) . That one can obtain the eigenvalues by maximizing over χ and minimiz-ing over ϕ was suggested first in the Physics literature by Talman [34] andDatta-Devaiah [6]. The min-max is largely based on the fact that the energy h Ψ , ( D − V µ )Ψ i is concave in χ and more or less convex in ϕ (up to finitelymany directions corresponding to the indices k < k ). This approach is rem-iniscent of the Schur complement formula, which is an important ingredientin the proof. Indeed, writing the eigenvalue equation in terms of ϕ and χ ,solving the one for χ and inserting it in the equation of ϕ , one formally findsthat (cid:18) − σ · ∇
11 + λ + V µ σ · ∇ + 1 − λ − V µ (cid:19) ϕ = 0 . (17)The formal operator on the left is associated with the quadratic form q λ ( ϕ ) := ˆ R | σ · ∇ ϕ | λ + V µ dx + ˆ R (1 − λ − V µ ) | ϕ | (18)and most of the work is to show that it is bounded from below, and equiv-alent to the V µ –norm. This allows one to give a meaning to the operatorin (17) by means of the Riesz-Friedrichs method, hence to transform the(strongly indefinite) Dirac eigenvalue problem into an elliptic eigenvalueproblem, nonlinear in the parameter λ . In the proof of Theorems 3 and 4we explain how to relate an information on the operator K z in (7) to thaton the quadratic form q λ and we then directly apply [15, 16, 13, 31].3.3. Application to (critical and sub-critical) multi-center poten-tials.
Let us now discuss the special case of a positive measure made of afinite sum of deltas, µ = M X m =1 κ m δ R m , ν m , which describes the nuclear density of a molecule. We always assume thatthe R m are all distinct from each other.When ¯ κ := max κ m <
1, Theorem 1 provides the self-adjointness of thecorresponding multi-center Dirac-Coulomb operator. This was proved beforein [30, 23]. Theorem 3 gives the domain in terms of the space V µ which, aswe have already mentioned, is equal to V µ = ( ϕ ∈ L ( R , C ) : M X m =1 ˆ R | x − R m | | x − R m | | σ · ∇ ϕ ( x ) | dx < ∞ ) . This is proved by localizing about each nucleus. One can also give a simpleexplicit lower bound on the quadratic form q λ in terms of the κ m and of the R m , which remains valid in the critical case max κ m = 1. IRAC-COULOMB OPERATORS WITH GENERAL CHARGE DISTRIBUTION I 11
Lemma 5 (Lower bound on q λ for multi-center) . Let µ = P Mm =1 κ m δ R m with κ m and where the R m are all distinct. Then we have q λ ( ϕ ) > (1 − ¯ ν ) ˆ R | σ · ∇ ϕ | V µ dx − (cid:18) λ + 2( M − νd + Cd (1 + λ ) (cid:19) ˆ R | ϕ | dx. for every ϕ ∈ H ( R , C ) , where ¯ ν = max( κ m ) and d = max k = ℓ | R k − R ℓ | . By arguing as in [15, 16, 13], this allows us to find a self-adjoint extensiondistinguished from the property that its domain satisfies D ( D − V µ ) ⊂ W µ × L ( R , C )where W µ is the space obtained after closing the quadratic form q λ . Thisspace is larger than V µ when max( κ m ) = 1. A simple localization as in theproof of Lemma 5 allows to apply the results of [13] and deduce that W µ = ϕ ∈ L ( R , C ) : M X m =1 ˆ R (cid:12)(cid:12)(cid:12) σ · ∇| x − R m | ϕ ( x ) (cid:12)(cid:12)(cid:12) | x − R m | (cid:0) | x − R m | (cid:1) dx < ∞ . (19)Arguing like in [13], one can prove that the distinguished self-adjoint exten-sion is the norm-resolvent limit of the operators obtained after truncatingthe potential or after decreasing the critical nuclear charges. One can indeedtreat any potential V so that 0 V V µ but then the space W µ has to be modified accordingly. The arguments areexactly the same as in [13].In chemistry one is interested in the potential energy surface which, bydefinition, is the graph of the first eigenvalue of the multi-center Dirac-Coulomb operator, seen as a function of the locations of the nuclei at fixed κ m and including the nuclear repulsion:( R , ..., R M ) λ D − M X m =1 κ m | x − R m | ! + X m<ℓ M κ m κ ℓ | R m − R ℓ | . The following is an extension of similar results proved before for M = 2in [23, 20, 4], which now includes the critical case. Theorem 6 (Molecular case) . Assume that R , . . . , R M are M distinctpoints in R , and that µ = P Mm =1 κ m δ R m with < κ m < . Let λ D − M X m =1 κ m | x − R m | ! be the first min-max level as in (14) . Then, ( i ) the map ( R , ..., R M ) λ (cid:16) D − P Mm =1 κ m | x − R m | (cid:17) is a continuous func-tion on the open set Ω defined as Ω = (cid:26) ( R , ..., R M ) ∈ ( R ) M : λ D − M X m =1 κ m | x − R m | ! > − (cid:27) . ( ii ) Moreover, lim min k = ℓ | R k − R ℓ |→∞ λ D − M X m =1 κ m | x − R m | ! = p − max κ m . (20)( iii ) If in addition P Mm =1 κ m < π/ /π ) − then lim max k = ℓ | R k − R ℓ |→ λ D − M X m =1 κ m | x − R m | ! = vuut − M X m =1 κ m ! . (21)The part ( iii ) of the theorem actually holds for P Mm =1 κ m < ν where thecritical number ν is defined in the second part [14] of this work. If ν = 1 aswe believe, then ( iii ) holds for all P Mm =1 κ m <
1. In [4] the result is claimedfor M = 2 and κ = κ < / M = 2 it is a famous conjecture that the energy of a diatomic moleculeis always greater than the single atom with the two nuclei merged. Thisproperty was conjectured for two-atoms Dirac operators by Klaus in [23,p. 478] and by Briet-Hogreve in [4, Sec. 2.4]. Numerical simulations from [2,26] seem to confirm the conjecture for M = 2, even for large values of thenuclear charges. We make the stronger conjecture that the same holds forany M . Conjecture 1.
We have λ D − M X m =1 κ m | x − R m | ! > λ D − P Mm =1 κ m | x | ! = vuut − M X m =1 κ m ! (22) for all M > , all R , ..., R M ∈ R and all κ m > so that P Mm =1 κ m . In [14] we discuss a stronger conjecture which implies Conjecture 1. Notethat Conjecture 1 has been shown in the non-relativistic case, as recalled inthe introduction. 4.
Proof of Theorem 1
The proof relies on two preliminary lemmas which we first state and show,before we turn to the actual proof of the theorem.Loosely speaking, the first lemma asserts that g ( x ) | p | − s f ( x ) is compactwhen f and g have disjoint supports. Recall that everywhere p = − i ∇ . IRAC-COULOMB OPERATORS WITH GENERAL CHARGE DISTRIBUTION I 13
Lemma 7 (Compactness for disjoint supports) . Let f ∈ L ( R d ) with supportin a compact set B ⊂ R d . Let Ω be a (bounded or unbounded) set in R d sothat d( B, Ω) > and let g ∈ L ( R d ) supported on Ω , such that ˆ Ω | g ( x ) | (1 + | x | ) d − s ) dx < ∞ where < s < d . Then the operator K = g ( x ) 1 | p | s f ( x ) is compact and its norm can be estimated by k K k C k f k L ( B ) (cid:18) ˆ Ω | g ( x ) | | x | d − s ) dx (cid:19) . (23) where C only depends on s , on the dimension, on d( B, Ω) and on sup x ∈ B | x | .Proof. The kernel of the operator K is given by K ( x, y ) = κ Ω ( x ) g ( x ) f ( y ) | x − y | d − s for some constant κ . Since the two functions f and g have supports at a finitedistance from each other, we have | x − y | > d( B, Ω) > B is compact, we even have | x − y | > | x | − R where R = max y ∈ B | y | . In particular, we conclude that | x − y | > c ( | x | + 1) for all x ∈ Ω and all y ∈ B . Therefore, the kernel of K is pointwise bounded by | K ( x, y ) | κc d − s | g ( x ) | (1 + | x | ) d − s | f ( y ) | . The right side is a rank-one operator which is bounded under the conditionthat f ∈ L and g (1 + | x | ) s − d ∈ L . This shows that K is bounded asin (23).To prove the compactness of K we can approximate f and g by func-tions in C ∞ c and use classical compactness results. But we can also arguedirectly as follows. Let u n ⇀ k u n k L = 1. We remark that( Ku n )( x ) = κ g ( x ) ˆ B f ( y ) u n ( y ) | x − y | d − s dy converges to 0 almost everywhere, since y f ( y ) | x − y | s − d belongs to L for all x ∈ Ω. In addition, we have the pointwise bound | ( Ku n )( x ) | κc d − s g ( x )(1 + | x | ) d − s k f k L ( B ) which, by the dominated convergence theorem, implies that k Ku n k → (cid:3) Using Lemma 7 we can show the following result.
Lemma 8 (Local compactness in the absence of atoms) . Let e µ > be afinite Radon measure on R , with no atom. Then B R se µ ∗ | x | | p | is a compact operator for every finite R > . Note that the operator p V e µ | p | − / is not compact since at infinity itessentially behaves like pe µ ( R ) | x | − / | p | − / which is not compact. Thecharacteristic function B R is really necessary. In addition, the correspond-ing result cannot hold for a measure which comprises some deltas, due tothe lack of compactness at the corresponding centers. Proof of Lemma 8.
First we write e µ = e µ B N + e µ R \ B N where B N is theball of radius N , and remark that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) se µ ∗ | x | − s ( e µ B N ) ∗ | x | ! | p | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)s ( e µ R \ B N ) ∗ | x | | p | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r π e µ ( R \ B N ) . (24)The first inequality holds because p V µ − p V µ p V µ pointwise, for µ = µ + µ . The second uses Kato’s inequality1 | x | π | p | (25)which implies that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)s µ ∗ | x | | p | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | p | s µ ∗ | x | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r π p µ ( R )for every bounded measure µ . The right side of (24) tends to zero when N → ∞ and this shows that we may assume for the rest of the proof that e µ has compact support.For r >
0, let us consider two tilings of the whole space R with cubes C j = 3 r ( j + [ − / , / ) and C ′ k = r ( k + [ − / , / ) of side length 3 r and r , respectively, where k ∈ Z . For every k , we call j k the index of the largecube C j k of which C ′ k is exactly at the center. Let ε >
0. By compactnessof the support of e µ and the fact that it has no atom, we can find r > e µ ( C j ) ε for every j .We then write1 | p | B R (cid:18)e µ ∗ | x | (cid:19) | p | = X k | p | B R ∩ C ′ k (cid:18)e µ ( C jk + R \ C jk ) ∗ | x | (cid:19) | p | . Recall that the sum is finite. The sets C ′ k and R \ C j k are at a distance atleast equal to r from each other. Then (cid:12)(cid:12)(cid:12)(cid:12) C ′ k (cid:18)e µ R \ C jk ∗ | x | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) cr . In particular this function is in L p ( C ′ k ) for all 1 p ∞ and the operator1 | p | B R ∩ C ′ k (cid:18)e µ R \ C jk ∗ | x | (cid:19) | p | IRAC-COULOMB OPERATORS WITH GENERAL CHARGE DISTRIBUTION I 15 is compact. This is true for instance because | p | − / f ( x ) | p | − / is compactunder the condition that f ∈ L ( R ), by Cwikel’s inequality [33]. Hence, atthis step we have written1 | p | B R (cid:18)e µ ∗ | x | (cid:19) | p | = X k | p | B R ∩ C ′ k (cid:18)e µ C jk ∗ | x | (cid:19) | p | + K where K is compact.Next, for every k we insert another localization as follows1 | p | C ′ k (cid:18)e µ C jk ∗ | x | (cid:19) | p | = ( C jk + R \ C jk ) 1 | p | C ′ k (cid:18)e µ C jk ∗ | x | (cid:19) | p | ( C jk + R \ C jk ) . The operator R \ C jk | p | C ′ k se µ C jk ∗ | x | is compact, by Lemma 7. Indeed, we have C ′ k (cid:18)e µ C jk ∗ | x | (cid:19) = C ′ k (cid:18)e µ C jk ∗ C jk + C ′ k | x | (cid:19) ∈ L ( C ′ k )so that its square root is in L ( C ′ k ), and ˆ R \ C jk dx (1 + | x | ) < ∞ . This proves that1 | p | B R (cid:18)e µ ∗ | x | (cid:19) | p | = X k C jk | p | B R ∩ C ′ k (cid:18)e µ C jk ∗ | x | (cid:19) | p | C jk + K where K is compact. By Kato’s inequality (25) the first operator is boundedabove as follows: X k C jk | p | B R ∩ C ′ k (cid:18)e µ C jk ∗ | x | (cid:19) | p | C jk π X k : C k ∩ B R = ∅ C jk e µ ( C j k ) ε π X k : C k ∩ B R = ∅ C jk π ε. Here we have used that e µ ( C j ) ε for every j by our choice of r . Thereforeour initial operator is the norm-limit of a sequence of compact operatorsand it must be compact. (cid:3) Now we can provide the
Proof of Theorem 1.
Our goal is to show thatlim sup | s |→∞ (cid:13)(cid:13)(cid:13)(cid:13)q | V µ | D + is q | V µ | (cid:13)(cid:13)(cid:13)(cid:13) max R ∈ R | µ ( { R } ) | < , (26) where we recall that V µ := µ ∗ | x | − . We write µ in the form µ = ∞ X m =1 ν m δ R m + e µ where it is understood that the R m are all distinct and where max m | ν m | < e µ has no atom. Here we allow infinitelymany singularities for simplicity of notation but many of the ν m could vanish.We know that P ∞ m =1 | ν m | < ∞ .Similarly as in (24) we can first write µ = K X m =1 ν m δ R m + e µ B N ! + X m > K +1 ν m δ R m + e µ R \ B N := µ + µ . (27)Using Kato’s inequality (25) and the fact that | µ | ( R ) is small for K and N large enough, we see that it suffices to show the limit (26) for µ havingfinitely many atoms and for e µ of compact support, which we assume for therest of the proof. In this case we have gained that | V µ ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12) ˆ R dµ ( y ) | x − y | (cid:12)(cid:12)(cid:12)(cid:12) | µ | ( R ) | x | − N where supp( µ ) ⊂ B N . In particular, V µ is bounded at infinity.We then write the potential V µ in the form V µ = K X m =1 ν m B η ( R m ) | x − R m | + B R V e µ + K X m =1 ν m R \ B η ( R m ) | x − R m | + R \ B R V e µ where B η ( R m ) is the ball of radius η centered at R m . We choose η < min m = ℓ K | R m − R ℓ | / pP V j P p V j , this shows that | V µ | K X m =1 p | ν m | B η ( R m ) | x − R m | + B R q V | e µ | + vuut K X m =1 | ν m | η + r | e µ | ( R ) R − N . (28)Next we replace p | V µ | by the function on the right in (cid:13)(cid:13)(cid:13)(cid:13)q | V µ | D + is q | V µ | (cid:13)(cid:13)(cid:13)(cid:13) , expand everything and estimate all the terms separately.The dominant terms are the ones close to the singularities K X m =1 | ν m | B η ( R m ) | x − R m | D + is B η ( R m ) | x − R m | . IRAC-COULOMB OPERATORS WITH GENERAL CHARGE DISTRIBUTION I 17
It is time to recall that for Coulomb potentialsSp | x | α · p + β | x | ! = Sp ess | x | α · p + β | x | ! = Sp | x | α · p | x | ! = Sp ess | x | α · p | x | ! = [ − ,
1] (29)and that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | x | α · p + is | x | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | x | α · p + β + is | x | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 1 . (30)for all s ∈ R . See [29, 39, 23, 22, 1]. This implies immediately that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K X m =1 | ν m | B η ( R m ) | x − R m | D + is B η ( R m ) | x − R m | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) max m | ν m | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K X m =1 B η ( R m ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = max m | ν m | < . Note that the estimate does not depend on K but requires η to be smallenough to guarantee that the balls do not overlap. The choice of η dependson the smallest distance between the nuclei.All the other terms are going to be small for s large enough. First, usingthat | p | | D + is | = | p | p p + 1 + s p | s | we see that all the terms involving the constant potential vuut K X m =1 | ν m | η + r | e µ | ( R ) R − N in (28) have a norm of the order O ( | s | − / ). The terms involving B R p V e µ can all be written in the form q | V µ ′ | | p | | p | D + is | p | q | V e µ | B R for some measure µ ′ . They all tend to 0 in norm when s →
0. This is because BA s K tends to zero in norm when B is bounded, K is compact and A s → k A s k C uniformly in s . The operator | p | − / p | V e µ | B R iscompact by Lemma 8.We are left with the more complicated interaction terms between the balls B η ( R k ) | x − R k | D + is B η ( R m ) | x − R m | (31)with k = m . We have1 D + is = α · pp + 1 + s + βp + 1 + s − i sp + 1 + s
28 M.J. ESTEBAN, M. LEWIN, AND ´E. S´ER´E and the term involving β is easily controlled by | p | − (1 + s ) − / and Kato’sinequality. Like in Lemma 7, our idea to deal with the other two terms is touse pointwise kernel bounds. Note that operator bounds are not very usefulhere since we have different functions on both sides of ( D + is ) − . Recallalso that | A ( x, y ) | B ( x, y ) implies k A k k B k . The kernel of the firstoperator is α · pp + 1 + s ( x, y ) = i α · ( x − y )4 π | x − y | e −√ s | x − y | + i p s α · ( x − y )4 π | x − y | e −√ s | x − y | and it can be bounded by (cid:12)(cid:12)(cid:12)(cid:12) α · pp + 1 + s ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) e −√ s | x − y | π | x − y | + p s e −√ s | x − y | π | x − y | C | s | | x − y | . Similarly, we have (cid:12)(cid:12)(cid:12)(cid:12) isp + 1 + s ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) = | s | e −√ s | x − y | π | x − y | C | s | | x − y | . We obtain that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B η ( R k ) | x − R k | D + is B η ( R m ) | x − R m | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C | s | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B η ( R k ) | x − R k | | p | B η ( R m ) | x − R m | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + C | s | . (32)Using Lemma 7 the norm on the right is finite and we conclude that theinteractions between balls are a O ( | s | − / ). This concludes the proof of (26).By the result of Nenciu [29, Cor. 2.1] (see also Klaus [23]) based on theresolvent expansion (6), the estimate (26) proves that D − V µ has a uniqueself-adjoint extension whose domain is included in H / ( R ). In addition, D − V µ ( | V µ | n ) converges to D − V µ in the norm-resolvent sense when n → ∞ [25, 23]. That the essential spectrum is equal to ( −∞ , − ∪ [1 , ∞ ) isalso a consequence of the resolvent formula (6) as in [29, 25]. This is because1 D − z q | V µ | is compact for z ∈ C \ σ ( D ). This follows from the Hardy-Littlewood-Sobolev inequality and the fact that p | V µ | ∈ ( L − ε + L ε )( R ) whereas( α · p + β − z ) − ∈ L r ( R ) for all r >
3, hence belongs to ( L − ε ∩ L ε )( R ).It remains to prove the statement (8) about the domain. We call R , ..., R K all the points such that | µ ( { R k } ) | > / x ∈ R \ { R , ..., R K } , we have lim r → | µ | ( B r ( x )) < B r ( x ) is the ball of radius r centered at x . Let χ be a smoothfunction supported on B / (0) and let χ r ( x ) = χ (( x − x ) /r ). Every Ψ in IRAC-COULOMB OPERATORS WITH GENERAL CHARGE DISTRIBUTION I 19 the domain of D − µ ∗ | x | − satisfies D Ψ − V µ Ψ = Φ ∈ L ( R )in H − / , so that D ( χ r Ψ) − V µ χ r Ψ = χ r Φ − i ( α · ∇ χ r )Ψ ∈ L ( R ) . We decompose µ = µ B r + µ R \ B r and use that (cid:12)(cid:12)(cid:12) V µ R \ Br (cid:12)(cid:12)(cid:12) | µ | ( R ) r on B r/ .This gives (cid:0) D − V µ Br (cid:1) χ r Ψ ∈ L ( R ) . For µ ( B r ) < /
2, the operator D − V µ Br is self-adjoint on H ( R ) byHardy’s inequality. This proves, as stated, that χ r Ψ ∈ H ( R ). Using that µ ( R \ B R ) → R → ∞ we can prove in a similar manner that(1 − χ R )Ψ ∈ H ( R ). We obtain the claim (8) by covering R \ ∪ Kj =1 B r ( R j )with finitely many balls together with the complement of a large ball. Thisconcludes the proof of Theorem 1. (cid:3) Proof of Theorem 2
Step 1. Proof of the estimate (10) on k · k V µ . The upper bound in (10)follows immediately from the fact that (1 + V µ ) − µ be a finite non-negative measure on R and V µ := µ ∗ | · | − . Then, by Hardy’s inequality | x | − − ∆) D ) ,we have (cid:13)(cid:13)(cid:13)(cid:13) V µ D (cid:13)(cid:13)(cid:13)(cid:13) µ ( R ) . We also have (cid:13)(cid:13)(cid:13)(cid:13) | p | ( β + 1) 1 D (cid:13)(cid:13)(cid:13)(cid:13) sup p ∈ R | p | p | p | √ . By the Rellich-Kato theorem, this proves that the operator D − V µ µ ( R ) − | p | β + 14is self-adjoint on H ( R ) and that 0 is not in its spectrum, with a universalestimate on the gap around the origin. For the same reason, the operator D − tV µ µ ( R ) − | p | β + 14has a gap around the origin at least as big as when t = 1, for all t ∈ [0 , | p | ( β + 1) / − − tV µ / (8 µ ( R )) −
1. From the min-max principle and a continuation argument in t from [8], the fact that 0 is never in the spectrum is equivalentto saying that ˆ R | σ · ∇ ϕ ( x ) | V µ ( x )8 µ ( R ) dx > µ ( R ) ˆ R V µ ( x ) | ϕ ( x ) | dx + 12 h ϕ, | p | ϕ i − k ϕ k L ( R ) for all ϕ ∈ H ( R , C ) and all non-negative finite measure µ over R . Drop-ping the potential term and using the inequality11 + a > , b ) (cid:0) ab (cid:1) gives ˆ R | σ · ∇ ϕ ( x ) | V µ ( x ) dx > h ϕ, | p | ϕ i − k ϕ k L ( R ) (cid:0) , µ ( R ) (cid:1) , ∀ ϕ ∈ H ( R , C ) . (33)Denoting M = 2 max (cid:0) , µ ( R ) (cid:1) >
2, we have M ˆ R | σ · ∇ ϕ ( x ) | V µ ( x ) dx > h ϕ, | p | ϕ i−k ϕ k L ( R ) > h ϕ, | p | ϕ i− ( M − k ϕ k L ( R ) so that ˆ R | σ · ∇ ϕ ( x ) | V µ ( x ) dx + k ϕ k L ( R ) > h ϕ, | p | ϕ i + k ϕ k L ( R ) M which is the left side of (10). By definition of V µ as the closure of H ( R , C )(or, equivalently, C ∞ c ( R , C )) we get the same inequality on V µ . The exactconstant in this inequality is not important, but it is crucial that it onlydepends on µ through its mass µ ( R ). Step 2. V µ coincides with the maximal space. Next we prove the formula (11)which states that V µ coincides with the maximal space on which one can givea meaning to the associated norm (9). We start by proving the followinglemma. Lemma 9 (Regularity of (1+ V µ ) α ) . Let µ be a non-negative Radon measureover R and V µ := µ ∗ | · | − . Then ∇ (1 + V µ ) α ∈ L ( R ) for all α < / and we have ˆ R |∇ (1 + V µ ) α | C α µ ( R ) (34) for a constant C α depending only on α . When α = 0 we have the sameestimate with (1 + V µ ) α replaced by log(1 + V µ ) .Proof. We write the proof for a non-negative µ ∈ C ∞ c ( R ). The generalresult follows from an approximation argument. Let Ω := { V µ < } andΩ i := { i − V µ < i } . Then we have (with (1+ V µ ) α replaced by log(1+ V µ ) IRAC-COULOMB OPERATORS WITH GENERAL CHARGE DISTRIBUTION I 21 when α = 0) ˆ R |∇ (1 + V µ ) α | = α ˆ R |∇ V µ | (1 + V µ ) − α = α ∞ X i =0 ˆ Ω i |∇ V µ | (1 + V µ ) − α α ˆ Ω |∇ V µ | + α ∞ X i =1 i − ) − α ˆ Ω i |∇ V µ | . Since − ∆ V µ = 4 πµ we have for all i > ˆ Ω i |∇ V µ | = ˆ R ∇ V µ · ∇ v i = 4 π ˆ Ω i v i d µ π i µ ( R )where v i := ( ( V µ >
1) + V µ ( V µ <
1) for i = 0,2 i − ( V µ i − ) + 2 i ( V µ > i ) + V µ (2 i − < V µ < i ) for i > ˆ R |∇ (1 + V µ ) α | πα ∞ X i =1 i (1 + 2 i − ) − α ! µ ( R ) , where the series is finite since α < / (cid:3) The lemma says that for ϕ ∈ L ( R ), (1 + V ) − / σ · ∇ ϕ makes sense asa distribution. It is then equivalent to require the existence of g ∈ L suchthat σ · ∇ ϕ = (1 + V µ ) / g or to ask that the distribution (1 + V µ ) − / σ · ∇ ϕ belongs to L . In the following we freely use any of the two formulations.Next we turn to the proof that any function ϕ such that g := (1+ V µ ) − / σ ·∇ ϕ ∈ L ( R , C ) can be approximated by a sequence ϕ n in C ∞ c ( R , C ) forthe norm k · k V µ , that is, such that ϕ n → ϕ in L ( R , C ) and (1 + V µ ) − / σ ·∇ ϕ n → g in L ( R , C ).First we truncate the sequence in space. We define ϕ n ( x ) := ϕ ( x ) χ ( x/n )where χ ∈ C ∞ c is such that χ (0) = 1. We have of course ϕ n → ϕ in L ( R , C ). In addition, we have in the sense of distributions, σ · ∇ ϕ n = χ ( · /n ) σ · ∇ ϕ + ϕ ( σ · ∇ χ )( · /n ) n = (1 + V µ ) χ ( · /n ) g + ϕ ( σ · ∇ χ )( · /n ) n (1 + V µ ) ! where the function in parenthesis has a compact support and convergesstrongly to g in L ( R ). This proves that functions of compact support aredense. In the following we assume, without loss of generality, that ϕ and g both have a compact support.Next we approximate ϕ by a sequence in H by arguing as in [13]. Let u ∈ ˙ H ( R ) such that ϕ = − iσ · ∇ u . Then we have in the sense of distributions − iσ · ∇ ϕ = − ∆ u = (1 + V µ ) g. We have u = 14 π (cid:16) (1 + V µ ) g (cid:17) ∗ | x | , ϕ = i π (cid:16) (1 + V µ ) g (cid:17) ∗ σ · x | x | . Next we define u ε = 14 π (cid:16) (1 + V µ ) ( V µ ε − ) g (cid:17) ∗ | x | , ϕ ε = − iσ · ∇ u ε which satisfies − iσ · ∇ ϕ ε = − ∆ u ε = (1 + V µ ) g ε = i π (cid:16) (1 + V µ ) g ε (cid:17) ∗ σ · x | x | where g ε = g ( V µ ε − ). Since (1 + V µ ) ( V µ ε − ) g ∈ L ( R ) and g hascompact support, we have (1 + V µ ) ( V µ ε − ) g ∈ L / ( R ). From theHardy-Littlewood-Sobolev inequality, this shows that ϕ ε ∈ L ( R ). Fromthe definition we also have σ · ∇ ϕ ε ∈ L ( R ), hence ∇ ϕ ε ∈ L ( R ) and ϕ ε ∈ H ( R ). From the dominated convergence theorem, we have g ε → g in L ( R ) and we now have to show that ϕ ε → ϕ in L as well. We have,again by the Hardy-Littlewood-Sobolev inequality, ˆ R | ϕ − ϕ ε | C (cid:13)(cid:13)(cid:13) (1 + V µ ) ( V µ > ε − ) g (cid:13)(cid:13)(cid:13) L / ( R ) = C k g k L ( B ) (cid:18) ˆ B (1 + V µ ) ( V µ > ε − ) (cid:19) where supp( g ) ⊂ B . The right side tends to zero when ε → ϕ ε → ϕ for the norm k · k V µ . The density of C ∞ c is then proved usingthe fact that k · k V µ is dominated by the H norm and this concludes theproof of (11), hence of Theorem 2. (cid:3) Proof of Theorems 3 and 4
We consider the quadratic form q λ ( ϕ ) := ˆ R | σ · ∇ ϕ | λ + V µ dx + ˆ R (1 − λ − V µ ) | ϕ | (35)defined (first) on H ( R d ) and show that it is coercive for the norm of V µ , afteradding C k ϕ k L ( R ) for an appropriate constant C . For this the estimate (10)on the first term is not enough because we also have to control the negativeCoulomb part involving V µ . Our proof will be based on Theorem 1, wherewe have shown that (cid:13)(cid:13)(cid:13)(cid:13)p V µ D + is p V µ (cid:13)(cid:13)(cid:13)(cid:13) < s large enough. We will explain here how this can be used to get someinformation on the quadratic form q λ in (18). More precisely, we will use asimilar argument as in the previous section and show thatmax Sp p V µ D + C − ε | p | ( β + 1) p V µ ! < C large enough and ε small enough. Recall that C ( β + 1) / C , that is, to push the upper IRAC-COULOMB OPERATORS WITH GENERAL CHARGE DISTRIBUTION I 23 part of the essential spectrum. Like in the previous section, the estimate (37)implies that q ( ϕ ) > ε (cid:13)(cid:13)(cid:13) | p | ϕ (cid:13)(cid:13)(cid:13) L − C k ϕ k L (38)for all ϕ ∈ H ( R ). This is the relation between the method of Nenciu etal [29] based on estimates for the operator K λ , and the method initiated byEsteban-Loss [15, 16] based on the quadratic form q λ . To our knowledge,this is the first time that such a link is established. Step 1. Proof of (37) . The proof of (37) is based on the following lemma.
Lemma 10 (Relating resolvents) . For every ε and C > , we havethe operator bound D + C − ε | p | ( β + 1) (cid:18) α · p + β + i √ C + 1 α · p + β − i √ C (cid:19) + 8 ε (1 + C ) | p | . (39) Proof.
We start with the case ε = 0. Note that we have D + C β + 1) = α · p + (cid:18) C (cid:19) β + C ± s | p | + (cid:18) C (cid:19) + C . The upper function is clearly bounded from below by 1 + C whereas thelower function is bounded above by −
1. The gap is ( − , C ). For large p ,the new operator behaves like D . This allows us to compute the resolvent,which we express in the form1 D + C ( β + 1) = α · p + (cid:0) C (cid:1) β − C | p | + (cid:0) C (cid:1) − C = α · p | p | + 1 + C + β | p | + 1 + C + C β − | p | + 1 + C . (40)Inserting α · p | p | + C + 1 = 12 (cid:18) α · p + β + i √ C + 1 α · p + β − i √ C (cid:19) − β | p | + 1 + C we obtain the relation1 D + C ( β + 1)= 12 (cid:18) α · p + β + i √ C + 1 α · p + β − i √ C (cid:19) + C β − | p | + 1 + C .
Since β D + C ( β + 1) (cid:18) α · p + β + i √ C + 1 α · p + β − i √ C (cid:19) . (41) Next we consider the case 0 < ε D + C − ε | p | β + 1) = α · p + (cid:18) C − ε | p | (cid:19) β + C − ε | p | ± s | p | + (cid:18) C − ε | p | (cid:19) + C − ε | p | ± s | p | + 1 + C − ε | p | + (cid:18) C − ε | p | (cid:19) + C − ε | p | . Noticing that | p | + 1 + C − ε | p | > | p | + 1 + C − | p | > | p | C > . we see that the two eigenvalues do not approach the origin, hence the oper-ator is invertible. We can next estimate the difference by (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D + C − ε | p | ( β + 1) − D + C ( β + 1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ε | p | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D + C − ε | p | ( β + 1) ( β + 1) 1 D + C ( β + 1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε | p | (cid:12)(cid:12)(cid:12) D + C − ε | p | ( β + 1) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) D + C ( β + 1) (cid:12)(cid:12) . (42)From (40) we have1 (cid:12)(cid:12) D + C ( β + 1) (cid:12)(cid:12) | p | + 1 + C | p | + 1 + C √ C | p | √ C | p | . On the other hand, using (40) with C replaced by C − ε | p | we obtain1 (cid:12)(cid:12)(cid:12) D + C − ε | p | ( β + 1) (cid:12)(cid:12)(cid:12) | p | + 1 + C + ε | p || p | + 1 + C − ε | p | | p | + 1 + C | p | + + C √ C | p | √ C | p | . Inserting this bound in (42) gives the claimed inequality. The constants arenot at all optimal and they are only displayed for concreteness. (cid:3)
Using Lemma 10 and Kato’s inequality, we obtainmax Sp p V µ D + C − ε | p | ( β + 1) p V µ ! πε (1 + C ) µ ( R )+ 12 (cid:18)(cid:13)(cid:13)(cid:13)(cid:13)p V µ α · p + β + i √ C p V µ (cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)p V µ α · p + β − i √ C p V µ (cid:13)(cid:13)(cid:13)(cid:13)(cid:19) . We have shown in the proof of Theorem 1 that the two operator norms areless than 1 for C large enough. Taking ε small enough then concludes ourproof of (37). IRAC-COULOMB OPERATORS WITH GENERAL CHARGE DISTRIBUTION I 25
Step 2. Proof of (38) and equivalence of quadratic forms.
Let us truncate V µ into W n = V µ ( V µ n ) and notice that, by Step 1,max Sp p W n D + C − ε | p | ( β + 1) p W n ! max Sp p V µ D + C − ε | p | ( β + 1) p V µ ! < W n V µ pointwise, for C large enough and ε small enough. Forthe bounded potential W n , the min-max formula and the Birman-Schwingerprinciple are well known. The previous condition implies that D − tW n + C − ε | p | β + 1)has no eigenvalue in ( − ,
0) for every t ∈ [0 , t from [8], this is equivalent to saying that q ,W n ( ϕ ) > − C k ϕ k L + ε (cid:13)(cid:13)(cid:13) | p | ϕ (cid:13)(cid:13)(cid:13) L for all ϕ ∈ H ( R , C ), where of course q ,W n denotes the quadratic formwith V µ replaced by W n . Passing to the limit n → ∞ we obtain (38). Thisis not yet enough to show that q is equivalent to the norm of V µ . Now usingKato’s inequality, for η < q ( ϕ ) + C k ϕ k L > η ˆ R | σ · ∇ ϕ | µ ∗ | x | − dx − η ˆ R V µ | ϕ | + (1 − η ) ε (cid:13)(cid:13)(cid:13) | p | ϕ (cid:13)(cid:13)(cid:13) > η ˆ R | σ · ∇ ϕ | µ ∗ | x | − dx + (cid:16) (1 − η ) ε − π ηµ ( R ) (cid:17) (cid:13)(cid:13)(cid:13) | p | ϕ (cid:13)(cid:13)(cid:13) . After taking η < εε + π µ ( R )we see that the quadratic form q + C is equivalent to the norm of the space V µ . This quadratic form is thus closable on H ( R , C ) and its closure isequivalent to the norm of V µ . Step 3. Domain and min-max principle.
Now we can apply the resultsof [15, 16, 31] to the operator D + C ( β + 1) / − V µ and we obtain a uniqueself-adjoint extension, distinguished from the property that its domain isincluded in V µ × L ( R , C ). This relies on the fact that the multiplicationoperator − V µ is essentially self-adjoint on C ∞ c ( R , C ) [31]. The domainis as described in the statement of the theorem. Of course, the same holdsfor D − V µ since C ( β + 1) / V µ × V µ . Let Ψ = ( ϕ, χ ) ∈V µ × L ( R , C ) be in the domain. Then we have ( (1 − V µ ) ϕ + σ · pχ = f ∈ L ( R , C ) , − (1 + V µ ) χ + σ · pϕ = g ∈ L ( R , C ) . where the terms on the left side are interpreted as distributions. Since ϕ ∈ V µ ⊂ H / ( R , C ), we have (1 + V µ ) / ϕ ∈ L ( R , C ), by Kato’sinequality. But the first equation can then be written in the form σ · pχ = ( V µ + 1) ( V µ + 1) ϕ + f − ϕ ( V µ + 1) ! where the function in parenthesis belongs to L ( R , C ). By the character-ization (11) of V µ , this gives immediately that χ ∈ V µ . Therefore we haveshown that D ( D − V µ ) ⊂ V µ × V µ ⊂ H / ( R , C ) . By uniqueness in H / we conclude that this extension must be the sameas the one from Theorem 1. The Birman-Schwinger principle was shownin [29, 23]. This concludes the proof of Theorem 3.The validity of the min-max formulas was shown for one-center Diracoperators in [8] in the domain of the distinguished extension and then in H / ( R ) in [27, 28]. This was extended to all spaces between C ∞ c ( R , C )and H / ( R , C ) in [13]. Here we can follow the same approach. Actually,the abstract result from [31] gives immediately the min-max methods in H / ( R , C ) and the density of C ∞ c ( R , C ) in V µ allows to conclude thatthe formula must hold in all spaces in between, following the argumentof [13]. In particular, the numbers λ ( k ) F are independent of the chosen space F . One notable difference is that in those works it is often assumed that λ (1) > − λ ( k ) > − k > Step 4. Proof of λ ( k ) ր . Let us first prove that for any positive integer k , λ ( k ) < µ = 0. For every k we choose a k -dimensional subspace ofradial functions in C ∞ c ( R , C ), denoted by W k . Let U R ( f ) = R − / f ( · /R )be the unitary operator which dilates the function by a factor R . Introduce W k,R := U R W k . Then for every normalized ϕ R = U R ϕ ∈ W k,R , we have q λ ( ϕ R ) = 1 R ˆ R | σ · ∇ ϕ ( x ) | λ + V µ ( Rx ) dx + ˆ R (1 − λ − V µ ( Rx )) | ϕ ( x ) | dx Decomposing µ = µ B η + µ B cη with a large but fixed η , we have (cid:12)(cid:12)(cid:12)(cid:12) ˆ R V µ ( Rx ) | ϕ ( x ) | dx − ˆ R V µ Bη ( Rx ) | ϕ ( x ) | dx (cid:12)(cid:12)(cid:12)(cid:12) µ ( R \ B η ) π R k ϕ k H ( R ) . On the other hand, after passing to Fourier variables and using | [ µ B η ( k ) − [ µ B η (0) | Cη | k | , we find (cid:12)(cid:12)(cid:12)(cid:12) ˆ R V µ Bη ( Rx ) | ϕ ( x ) | dx − µ ( B η ) R ˆ R | ϕ ( x ) | | x | dx (cid:12)(cid:12)(cid:12)(cid:12) CηR k ϕ k H ( R ) . In our finite-dimensional space, all the norms are equivalent, hence we obtain ˆ R V µ ( Rx ) | ϕ ( x ) | dx > (cid:18) c µ ( B η ) − Cµ ( R \ B η ) R − CηR (cid:19) k ϕ k L ( R ) for some c > W k . Choosing η large enough and λ = 1 − ε/R with ε > q − ε/R ( ϕ R ) < W k,R for R IRAC-COULOMB OPERATORS WITH GENERAL CHARGE DISTRIBUTION I 27 large enough. The min-max formula (14) can be reformulated in terms ofthe quadratic form q λ as [8, 31] λ ( k ) = inf { λ : ∃ W ⊂ V µ , dim( W ) = k : q λ ( ϕ ) , ∀ ϕ ∈ W } (44)Using the characterization on the first line, this proves that λ ( k ) − ε/R ,as we wanted.Next we prove that λ ( k ) → k → ∞ . Note that k λ ( k ) isnon-decreasing and < ess ( D − V µ ) = ( −∞ , − ∪ [1 ∪ ∞ )by Theorem 1. From this we conclude that if we have λ ( k ) > − k , then λ ( k ) is an eigenvalue of D − V µ and it can only converge to 1.Let us argue by contradiction and assume that λ ( k ) = − k > W k ⊂ V µ of dimension dim( W k ) = k and ε k → + such that q − ε k is negative on W k . By monotonicity with respect to λ , we conclude that q is also negative on W k . After extraction this provides a sequence ϕ n ∈ V µ such that k ϕ n k L = 1, ϕ n ⇀ q ( ϕ n ) <
0. By (38) we knowthat q ( ϕ n ) > ε (cid:13)(cid:13)(cid:13) | p | ϕ n (cid:13)(cid:13)(cid:13) L ( R ) − C λ k ϕ n k L ( R ) and this proves that the sequence { ϕ n } is bounded in H / ( R ). UsingKato’s inequality for the negative term involving V µ in q and again thefact that q ( ϕ n ) <
0, we finally obtain that the sequence ( ϕ n ) is boundedin V µ . Next we pick a localization function χ R ( x ) = χ ( x/R ) where χ ∈ C ∞ c ( R , [0 , χ ≡ B and χ ≡ R \ B and let η R := q − χ R .We use the pointwise IMS formula for the Pauli operator which states that X k | σ · ∇ ( J k ϕ ) | = X k X i,j =1 h ∂ i ( J k ϕ ) , σ i σ j ∂ j ( J k ϕ ) i C = | σ · ∇ ϕ | + X k X i,j =1 h ϕ, σ i σ j ϕ i C ∂ i J k ∂ j J k + 2 ℜ X k X i,j =1 h ∂ i ϕ, σ i σ j ϕ i C J k ∂ j J k = | σ · ∇ ϕ | + | ϕ | X k |∇ J k | (45)for any real partition of unity P k J k = 1. We have used that σ i σ j + σ j σ i = 0for i = j in the second term of the second equality and that 2 P k J k ∂ j J k = ∂ j P k J k = 0 for the last term. We obtain q ( ϕ n ) = ˆ R | σ · ∇ ( χ R ϕ n ) | V µ − ˆ R V µ χ R | ϕ n | + 1+ ˆ R | σ · ∇ ( η R ϕ n ) | V µ − ˆ R V µ η R | ϕ n | − ˆ R |∇ χ R | + |∇ η R | V µ | ϕ n | . For the first two terms involving χ R we use that q is bounded from belowby (38), which yields ˆ R | σ · ∇ ( χ R ϕ n ) | V µ − ˆ R V µ χ R | ϕ n | > − C ˆ R χ R | ϕ n | . Hence we obtain q ( ϕ n ) > − C ˆ R χ R | ϕ n | − ˆ R V µ η R | ϕ n | − CR . (46)We will prove that the negative terms on the right side are all small, whichgives q ( ϕ n ) >
0, a contradiction. We start with the second negative term.We decompose µ = µχ R/ + µη R/ and remark that V µχ R/ η R C/R whereas ˆ R V µη R/ η R | ϕ n | π µ ( R \ B R/ ) k η R ϕ n k H / Cµ ( R \ B R/ )by Kato’s inequality. Hence ˆ R V µ η R | ϕ n | C (cid:18) R + µ ( R \ B R/ ) (cid:19) . We may therefore choose R large enough such that ˆ R V µ η R | ϕ n | + CR ϕ n ⇀ H / , we have for thisfixed R lim n →∞ ˆ R χ R | ϕ n | = 0and this proves, as we claimed, that q ( ϕ n ) > Step 5. Proof that λ (1) > under Tix’s condition (16) . We claim that forevery finite positive measure µ k K k µ ( R ) π + π K := p V µ D p V µ . (47)Then, the statement that λ (1) > − , K we introduce the free Diracspectral projections P ± = R ± ( D ) and write − p V µ P − √ − ∆ p V µ p V µ D p V µ p V µ P +0 √ − ∆ p V µ . We obtain (cid:13)(cid:13)(cid:13)(cid:13)p V µ D p V µ (cid:13)(cid:13)(cid:13)(cid:13) max τ ∈{±} (cid:13)(cid:13)(cid:13)(cid:13)p V µ P τ √ − ∆ p V µ (cid:13)(cid:13)(cid:13)(cid:13) . IRAC-COULOMB OPERATORS WITH GENERAL CHARGE DISTRIBUTION I 29
By charge-conjugation the two norms on the right are equal in the maximum.One can bound them by (cid:13)(cid:13)(cid:13)(cid:13)p V µ P +0 √ − ∆ p V µ (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) P +0 √ − ∆ V µ P +0 √ − ∆ (cid:13)(cid:13)(cid:13)(cid:13) µ ( R ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P +0 p p + 1 1 | x | P +0 p p + 1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = µ ( R ) π + π . The inequality is by convexity in µ and translation-invariance of P +0 (1 − ∆) − . The last equality is due to Tix [36]. (cid:3) Proof of Theorem 6
We start by providing the
Proof of Lemma 5.
We assume M >
2, otherwise the result has already beenproved before in [8] with d = 0. We follow arguments from [3] and introducea smooth partition of unity P M +1 k =1 J k = 1 so that J k ≡ B ( R k , d k / J k ≡ B ( R k , d k /
4) for k = 1 , ..., M , where d k = min ℓ = k | R k − R ℓ | isthe distance from the other nuclei. Setting d := min k = ℓ | R k − R ℓ | the smallest distance between the nuclei, we may assume that M +1 X j =1 k∇ J k k ∞ ≤ κd (48)for a universal constant κ . By the IMS formula (45) we have ˆ R | σ · ∇ ϕ | λ − V dx = M +1 X k =1 ˆ R | σ · ∇ ( J k ϕ ) | λ − V − ˆ R M +1 X k =1 |∇ J k | | ϕ | λ − V dx > M +1 X k =1 ˆ R | σ · ∇ ( J k ϕ ) | λ − V − κd (1 + λ ) ˆ R | ϕ | dx . We can write, similarly as in [13, Sec. 1.4], ˆ R | σ · ∇ ( J k ϕ ) | λ − V dx = (1 − κ k ) ˆ R | σ · ∇ ( J k ϕ ) | λ − V dx + κ k ˆ R | σ · ∇ ( J k ϕ ) | λ − V dx , and use the following Hardy-type inequality which was proved in [8, 7]: ∀ a > , ˆ R | σ · ∇ ϕ ( x ) | a + 1 / | x | dx + ˆ R (cid:18) a − | x | (cid:19) | ϕ ( x ) | dx > . (49)Using our assumption (5) and the fact that V ≥ − κ k | x − R k | − ( M − νd on the support of J k for k = 1 , ..., M , with ¯ ν := max( κ m ) we obtain κ k ˆ R | σ · ∇ ( J k ϕ ) | − V + λ dx > κ k ˆ R | σ · ∇ ( J k ϕ ) | ν k | x − R k | − + ( M − νd + λ dx > ˆ R (cid:18) κ k | x − R k | − − λ − ( M − νd (cid:19) | J k ϕ | dx > − ˆ R V | J k ϕ | dx − (cid:18) λ + 2( M − νd (cid:19) ˆ R | J k ϕ | dx (50)for k = 1 , . . . , M . For k = M + 1 we use that ˆ R V | J M +1 ϕ | dx ≥ − M ¯ νd ˆ R | J M +1 ϕ | dx. Since − < λ < M − > M for M >
2, we obtain ˆ R | σ · ∇ ϕ | λ − V dx + ˆ R (1 − λ + V ) | ϕ | dx > (1 − ¯ ν ) M X k =1 ˆ R | σ · ∇ ( J k ϕ ) | − V dx + ˆ R | σ · ∇ ( J M +1 ϕ ) | − V dx − (cid:18) λ + 2( M − νd + Cd (1 + λ ) (cid:19) ˆ R | ϕ | dx > (1 − ¯ ν ) ˆ R | σ · ∇ ϕ | − V dx − (cid:18) λ + 2( M − νd + Cd (1 + λ ) (cid:19) ˆ R | ϕ | dx. (51) (cid:3) We are now ready to provide the
Proof of Theorem 6.
To prove ( i ), let us fix some R , ..., R M all distinctfrom each other and let d := min j = k | R j − R k | >
0. Let χ ∈ C ∞ c ( B d/ )be a function such that χ B d/ ≡ χ m := χ ( x − R m ) thefunction centered at the m -th nucleus. Let finally η := 1 − P Mm =1 χ m bethe function which localizes outside of the M nuclei. Next we consider somenew positions R ′ , ..., R ′ M ∈ R such that | R m − R ′ m | ε d/
10 and definethe following deformation of space
T x = M X m =1 χ m ( x )( x + R ′ m − R m ) + η ( x ) x = x + M X m =1 χ m ( x )( R ′ m − R m )which sends each nucleus R m onto R ′ m and does not move the points locatedat a distance > d/ | T x − x | ε and | DT ( x ) − | Cε (cid:18) d δ ( x ) d (cid:19) , δ ( x ) := min m | x − R m | hence T is a C ∞ –diffeomorphism for ε small enough. For a function ϕ ∈ C ∞ c ( R , C ) we define ϕ T ( x ) := ϕ ( T − x ). Denoting by q λ and q ′ λ the qua-dratic forms corresponding respectively to the nuclear positions R m and R ′ m IRAC-COULOMB OPERATORS WITH GENERAL CHARGE DISTRIBUTION I 31 we find after a change of variable in the integrals q λ ( ϕ T ) q ′ λ ( ϕ ) + Cε ˆ d δ d |∇ ϕ | + | ϕ | q ′ λ − Cε ( ϕ )for ε small enough and λ far enough from −
1. By (44) this proves that λ D − M X m =1 κ m | x − R εm | ! ≤ λ D − M X m =1 κ m | x − R ′ m | ! + C ε for ε small enough. We get the reverse inequality by using the inversetransformation.To prove ( ii ), we assume for simplicity of notation that max κ m = κ M .We then use the pointwise bound M X m =1 κ m | x − R m | > κ M | x − R M | and the monotonicity of λ (1) with respect to the potential to deduce that λ D − M X m =1 κ m | x − R m | ! ≤ λ (cid:18) D − κ M | x − R M | (cid:19) = q − κ M . The reverse inequality in the limit | R j − R k | → ∞ is proved by localizingexactly as in [3, Cor. 4.7].Finally, we discuss the proof of ( iii ). After a translation we may assumethat R = 0 and that R m → m = 2 , ..., M . We denote by q R,λ and q ,λ the quadratic forms associated to the potentials V R = P Mm =1 κ m | x − R m | − and V = P Mm =1 κ m | x | − , respectively. We denote by λ ( R ) and λ (0) the corresponding min-max values. From Theorem 4 it is known that λ ( R ) > R whenever M X m =1 κ m π/ /π , as was assumed in the statement. By a continuation principle as in [8] weknow that the first min-max value λ (1) is non-negative and coincides withthe first eigenvalue. Moreover, we recall from (47) that k K R, k π/ /π M X m =1 κ m < K R, := p V R D p V R → p V D p V := K , strongly. Then we have (1 − K R, ) − → (1 − K , ) − strongly. Since( D ) − √ V R converges in norm, this proves using (6) that ( D − V R ) − → ( D − V ) − in norm, hence that the eigenvalues converge. Note thatthe other parts of the statement also follow from this argument, when P Mm =1 κ m < π/ /π ) − . (cid:3) References [1]
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CEREMADE, CNRS, Universit´e Paris-Dauphine, PSL Research University,Place de Lattre de Tassigny, 75016 Paris, France
E-mail address : [email protected] CEREMADE, CNRS, Universit´e Paris-Dauphine, PSL Research University,Place de Lattre de Tassigny, 75016 Paris, France
E-mail address : [email protected] CEREMADE, Universit´e Paris-Dauphine, PSL Research University, CNRS,Place de Lattre de Tassigny, 75016 Paris, France
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