ESTIMATES FOR THE OPTIMAL CONSTANTS IN MULTIPOLAR HARDY INEQUALITIES FOR SCHRÖDINGER AND DIRAC OPERATORS

By expanding squares, we prove several Hardy inequalities with two critical singularities and constants which explicitly depend upon the distance between the two singularities. These inequalities involve the L2 norm. Such results are generalized to an arbitrary number of singularities and compared with standard results given by the IMS method. The generalized version of Hardy inequalities with several singularities is equivalent to some spectral information on a Schrodinger operator involving a potential with several inverse square singularities. We also give a generalized Hardy inequality for Dirac operators in the case of a potential having several singularities of Coulomb type, which are critical for Dirac operators.


Introduction
For N ≥ 3, the simplest form of Hardy's inequality is easily obtained by the "expansion of the square" method as follows: for any function u ∈ H 1 (R N ), which shows for α = (N − 2)/2 that, for all u ∈ H 1 (R N ), and it is well known that the constant (N − 2) 2 /4 is optimal. With two singularities located at ±y ∈ R N , from the above inequality we get without effort that |x| 2 dx ≥ 0 where u ± (·) = u(· ± y). For a given function u with compact support and d := |y| large enough, it is however clear that the constant (N − 2) 2 /8 can be replaced by (N − 2) 2 /4. To improve upon (N − 2) 2 /8 for general functions and in presence of two singularities, one has to break the scaling invariance by introducing a new scale. This can be done by adding a lower order term. One of the goals in this paper is to obtain estimates for the best constant λ = λ(µ, d), that is, the smallest positive constant, in the inequality for any µ ∈ (0, (N − 2) 2 /4] and any y ∈ R N with d := |y| > 0.
The above result can of course be reinterpreted as a lower estimate on the spectrum of the Schrödinger operator −∆ − µ 1 |x − y| 2 + 1 |x + y| 2 since for a fixed µ ∈ (0, (N − 2) 2 /4), −λ is the bottom of the spectrum of this operator. Such an operator is indeed semi-bounded from below, as we shall see below. By a simple rescaling argument, it is easy to see that λ(µ, d)/d 2 is independent of d. A similar fact also holds in the case of more than two singularities. In such a case, after division by the correct scaling parameter, the best constant does not depend on the distances between the singularities anymore, but still depends on the relative position of each of them, i.e., on the geometric pattern of the singularity points.
It is certainly out of the scope of this introduction to present all various forms of Hardy's inequality. One can for instance refer to [45] for an up-to-date introduction. Multipolar potentials have recently been studied in [28], in a different context. Huge efforts have been done over the past years to improve on Hardy's inequality, but mostly in the case of a single singularity: see [22,3,4,15,23]. Also see [21,20,19] and references therein for some existence results of ground states for linear and nonlinear Schrödinger equations with multipolar inverse square singular potentials, and [19] for questions related to the self-adjointness of the operators. Among other results, it is proved in these papers that the operator −∆ − M k=1 µ k |x − y k | −2 is positive if M k=1 µ + k < (N − 2) 2 /4, where {y k : k = 1, 2 , . . . M } is any set of disjoint poles. Reciprocally, if M k=1 µ + k > (N − 2) 2 /4, there exists a configuration of poles such that −∆ − M k=1 µ k |x − y k | −2 is not positive. Moreover, there exists a configuration of poles such that this operator is positive if and only if µ + k < (N − 2) 2 /4 for all k and M k=1 µ + k < (N − 2) 2 /4. Felli, Marchini and Terracini also consider in [21] modified potentials which are sums of inverse square potentials restricted to compact supports around a given number of singularities. In this case, they discuss again the positivity of the operator and, in some cases, also give estimates for the lowest eigenvalue of the corresponding Schödinger operator.
Our purpose in the case of the Schrödinger operators is to give an as good as possible estimate for the lower bound of the spectrum of the operator −∆− M k=1 µ |x−y k | 2 , µ ∈ (0, (N − 2) 2 /4], M ≥ 2, y 1 , . . . y M ∈ R N . We will use two different methods, the so-called IMS method, see Section 1, and the "expansion of the squares" method, see Section 2, which has already been introduced above in the case of a single singularity.
From a mathematical point of view, inverse square potentials are interesting because of their criticality. There are various motivations for applications in physics, see for instance a list of topics in [21,19], and related nonlinear problems in PDEs. On the other hand, the 1/|x| 2 term appears in the linearization of critical nonlinear PDEs, see for instance [5,43,7], and plays a crucial role for understanding the asymptotic behavior of branches of solutions in some bifurcation problems.
The second topic of this paper is concerned with Hardy inequalities for Dirac equations in the framework of relativistic quantum mechanics. In such a setting, the Coulomb potential is critical. Hardy inequalities have been studied long ago for simplified relativistic equations coupled with multipolar potentials of quantum chemistry for instance in [9,34], motivated by the question of the stability of matter. A key estimate in such a case is a Hardy inequality which was derived by Kato in [30]: in dimension N = 3. A refined relativistic operator was introduced by Brown and Ravenhall and optimal corresponding inequalities have been derived in [6,24]. Also see [26,2] for some consequences. In the case of the Dirac operator coupled to Coulomb interactions, Hardy type inequalities are much more recent, see [13,12]. For a pure Coulomb singularity these inequalities read as follows. For all ν ∈ (0, 1], for all φ ∈ L 2 (R 3 , C 2 ), with S(ν) : Here some of the terms can be infinite and the matrices σ i , i = 1, 2, 3 are the so-called Pauli matrices. See Section 3 for more precisions and explanations on these matrices and the above inequality. In particular, exactly as in the case of the Schrödinger operator, the constant S(ν) is the smallest eigenvalue of H − ν/|x|, where H is the Dirac operator and ν ∈ (0, 1). We postpone the precise explanation of the relation of the optimal value of S(ν) and the spectrum of H to Section 3, as well as some comments on the self-adjointness of the operators. For a general introduction to the Dirac equation we refer to [44]. Scaling properties explain why inverse square potentials and Coulomb potentials are critical respectively for the Schrödinger and the Dirac operators. It is clear that 1/|x| 2 has the same scaling law as −∆. As for Dirac's operator, the principal part of H scales like 1/|x|, which can also be seen in the above inequality at the level of the operator φ → −( σ · ∇)(|x| σ · ∇ φ). Some further details will be given at the beginning of Section 3. More strikingly, one can also prove that the standard Hardy inequality for the Schrödinger operator is a consequence of the Hardy-like inequality for the Dirac operator coupled to one Coulomb singularity. For the convenience of the reader, we reproduce here some computations which already appeared in [12]. By taking the limit as ν → 1, we get If we replace φ(·) by ε −1 φ(ε −1 ·) and take the limit as ε → 0, we obtain By taking φ = (f, 0) with f purely real, we end up with as shown by considering u = |x| f . We observe that (N − 2) 2 /4 = 1/4 if N = 3, so that we recover the optimal constant, and even corrective terms at all orders. See [22,12] for more details and further references.
A major difference between Dirac and Schrödinger operators coupled respectively to Coulomb type interactions and to multiple inverse square singular potentials lies in the structure of the continuous spectrum. While the Schrödinger operators are semi-bounded from below, thus allowing only for some positive continuous spectrum, Dirac operators have continuous spectrum everywhere except in a gap. There is therefore a natural limitation of the coupling constant, called the coupling constant threshold, which has been studied in [32] for smooth potentials and in [11] in the case of one Coulomb singularity and a constant magnetic field.
Our purpose in the case of Dirac operators is to give estimates of the lowest energy level for a Dirac-Coulomb operator in terms of the interdistance between the M singularities, or equivalently to determine estimates of the coupling threshold. Such a coupling threshold can be seen from two points of view: (i) Either we fix the configuration of the singularities and look for estimates of the largest coupling constant ν for which the lowest eigenvalue is above −1, the upper bound of the negative continuous spectrum. (ii) Or we fix the coupling constant ν to some value in (0, 1) and look for estimates that guarantee that the lowest eigenvalue of the operator is above −1. This determines a minimal distance between singularities if νM > 1. Both approaches are equivalent and amount to finding estimates for the optimal constant in Hardy type inequalities for Dirac operators with multiple Coulomb singularities.
In the Schrödinger and in the Dirac case, we investigate the asymptotics d → +∞ when the singularities are asymptotically far from each other, and also the limit d → 0 corresponding to all singularities merged in a single point (further assumptions are however needed for the Dirac operator).
From a physical point of view, the multipolar Dirac-Coulomb operator describes the state of one charged particle in a molecule with M fixed nuclei, in the Born-Oppenheimer approximation. In atomic units, ν = α Z, Z is the nuclear charge number and α is the Sommerfeld fine-structure constant, whose value is 1/137.037 . . . Hence the condition ν < 1 means that the nuclear charge cannot be larger than 1/α. Finding the lowest energy level in the gap of the Dirac-Coulomb operator is a basic question for studying stability, computing energy levels and getting estimates for related nonlinear models.
To our knowledge, when there is more than one singularity, no explicit estimates for the lowest eigenvalue of Dirac operators have been derived yet. In the case of a crystal, which is a slightly different setting, Hardy inequalities for Dirac type operators are currently under investigation, see [8]. Estimates could be deduced from Kato's inequality or from the inequality stated in [6,24] for the Brown-Ravenhall Hamiltonian and their extensions to the multipolar case, but they would anyway not cover the whole range of the coupling constant, as we do here.
In [31], Klaus studied conditions under which the Dirac operator coupled to a potential with several Coulomb singularities is self-adjoint. The case of two singularities when they are far apart from each other has been studied in [25]. Also see [10] for a more general approach of double well Hamiltonians, and [46] for a semi-classical analysis in the case of potentials with multiple wells.
The paper is organized as follows. In Section 1 we use the well-known IMS method to derive some lower estimates on the lowest eigenvalue of the operator −∆ − M k=1 µ |x−y k | 2 , µ ∈ (0, (N − 2) 2 /4], M ≥ 2 and y 1 , . . . y M ∈ R N . This method consists in localizing the wave functions around the singularities by using a well chosen partition of R N . In some cases the geometric pattern defined by the singularity centers allows for better estimates than the general ones.
In Section 2 we expand some well chosen squared quantities and integrate by parts to prove another type of estimates, which in some cases improve those obtained by the IMS method. The idea here is to show that for some function Q(x, y 1 , . . . y M ), the operator −∆ − M k=1 µ |x−y k | 2 + Q is nonnegative. The constant λ is then defined as the infimum of Q. Of course, this procedure cannot provide the best constant λ, since it is based on a pointwise estimate of Q.
Section 3 deals with Hardy-type inequalities for Dirac operator with Coulomb singularities. Due to the non homogeneity of the Dirac operator, the best constant heavily depends on the interdistance between the singularities, and not only on the geometric pattern defined by them. In this case we are only able to use the IMS method, which gives good results only for large interdistances, that is, for very distant singularities. For nearby singularities, we introduce a slightly modified version of the Hardy inequality.
When dealing with the Schrödinger operator, we always consider dimensions N ≥ 3. The Dirac case is studied only in the physical space with N = 3. Also note that in the case of Dirac operators, the essential spectrum is not bounded from below, and so, when we speak of the lowest eigenvalue, we always mean the lowest eigenvalue in the gap of the essential spectrum. All estimates can be worked out for complex valued functions and spinors, but since we do not consider evolution equations, results will be stated for real valued functions whenever possible, and without further notice. For simplicity, we have assumed that coupling constants are the same for all singularities. The extension to the case when they differ can be worked out by the same methods, but is left to the reader.

Hardy inequalities for the Schrödinger operator and the IMS method
The results of this section rely on the so-called "IMS" (for Ismagilov, Morgan, Morgan-Simon, Sigal, see [35,41]) truncation method. Our goal is to obtain estimates for the lowest eigenvalue of |x−y k | 2 , M ≥ 2 and y 1 , . . . y M are M points of R N . Notice that the Hamiltonian is essentially self-adjoint if µ ≤ (N − 2) 2 /4 − 1, otherwise one has to use the corresponding Friedrichs extension. See [29,19] for more details. Define d by where Ω k := Int (supp(J k )), k = 1 , . . . M, then we obtain an explicit formula for the sum of the gradients: In the sequel we will always use partitions of unity that satisfy (1). Note that to avoid a singularity for the gradient of J M+1 at the points where 1 − J 2 k = 0, from (d) we shall assume that the additional constraint . . M and for some F ∈ L ∞ (R N ). The functions sinus and cosinus, for example, satisfy this last requirement.
Partitions of unity can be used to localize the Schrödinger operator. Here is a statement taken from [41], with its proof.
k=1 be a partition of unity satisfying (1). For any u ∈ H 1 (R N ) and any Proof. On the one hand, On the other hand, Combined with Property (a), this proves the result.
Thanks to Lemma 2 and Property (d), we can write Here we successively used Properties (b) and (d) and |∇(J M+1 u)| 2 ≥ 0. Using the fact that Ω j ∩ Ω k = ∅ for any j, k = 1 , . . . M, j = k, then the above inequality can be rewritten as Consider now the case M = 2, y 1 = y, y 2 = −y and assume that 0 < d ≤ |y|.
for any x ∈ R N , 0 < d ≤ |y|, such that, for any µ > 0, there exists a constant K 2 ∈ [0, π 2 ) for which, almost everywhere for all x ∈ Ω := supp(J 1 ) ∪ supp(J 2 ), we have In the proof of this result, we use first a partition of unity defined as follows. Let Proof. The lower bound K 2 ≥ 0 immediately follows by evaluating the l.h.s. at x = 0. As for the upper bound, consider the above partition of unity. Thus defined, {J 1 , J 2 , J 3 } is a partition of unity of R N satisfying (1). Let θ := |y|/d ≥ 1 and Ω + := {x ∈ Ω , x · y > 0}. By Property (d), using the symmetry with respect to the hyperplane {x ∈ R N : x · y = 0}, we get The choice (2) is not optimal and can be improved by taking and g as the solution to the ODE with g(1/2) = 1, and by adjusting A > 0 such that g(1) = 0 and g(t) > 0 for any t ∈ (1/2, 1). Hence we can slightly improve the value of the constant K 2 , but this value now depends on µ.
Consider the case M = 2, V = V 2 . With the notations of Lemma 3, we observe that using the Hardy inequality for one singularity, for µ ≤ (N − 2) 2 /4. As a consequence, we obtain Theorem 1 is a generalization of this inequality to M ≥ 3.
Proof of Theorem 1. Consider a partition of unity k=1 Ω k . For every k = 1 , . . . M, the Hardy inequality for one singularity and the estimate |x − y l | ≥ d on Ω k for l = k, give and thus provide a lower bound for Q[J k u]: For every k = 1, 2 , . . ., M, we can apply Lemma 3 on Ω k with (y k , y l ) = (−y, y) up to a change of coordinates, for some y l = y k , and for all j = k, l, bound |x − y j | −2 by 1/d 2 . Hence we get Collecting the terms, that is ( from which the result follows. Remarks. There are many possibilities for improving the result of Theorem 1 for M ≥ 2 and µ < (N − 2) 2 /4.
(1) One can notice that when applying the Hardy inequality for one singularity, we can write using the optimal constant in the inequality. Consequently, On the other hand, we know that Collecting these estimates, withμ := 2 µ − (N − 2) 2 /4 < µ, this proves that The estimate |x − y l | ≥ d on Ω k for l = k is certainly extremely rough for large values of M, due to volume filling effects. (3) Other partitions of unity can be considered. For a given set of poles y 1 , y 2 , . . . y M ∈ R N , define for instance the corresponding Voronoi cells Γ k by and let d k := dist(y k , ∂Γ k ). Then the truncation functions J k can be defined such that We now illustrate points (3) and (4) in the simple case M = 2, y 1 = y and y 2 = −y. Consider the partition of the unity {J k } 3 k=1 such that Proposition 4. Let M = 2 and assume that µ ∈ (0, (N − 2) 2 /4]. Then for any u ∈ H 1 (R N ), Proof. Up to a translation, we can work with y 1 = y and y 2 = −y, that is with the potential where I Ω is the characteristic function of the set Ω = Ω 1 ∪ Ω 2 . Our aim is to estimate Q By Hardy's inequality, we get Further, for any x ∈ supp(J 3 ), Hence we obtain Applying again the Hardy's inequality, it results that and the proof is complete.

"Expansion of the square" and Hardy inequalities for the Schrödinger operator
The estimates of Theorem 1 are not very good when M µ is close to (N − 2) 2 /4. The goal of this section is to remedy to this problem by the "expansion of the square" method, already used in the introduction in the case of a single singularity. We start with an elementary computation in the case of two singularities, see Lemma 5, (see Lemma 8 in case M ≥ 2). Then we show how the optimal constant in the multipolar Hardy inequality can be estimated. The result is twofold: we establish some qualitative properties of the best constant, see Lemmas 6 and 9, and then explicitly estimate it in Theorems 7 and 10. This is done first in the case of two singularities, and then extended to M ≥ 2 singularities.
Lemma 5. For any u ∈ H 1 (R N ), for any y ∈ R N , When y = 0 we recover the standard Hardy inequality with one singularity.
Proof. Assume that u ∈ D(R N ) and α > 0. We compute where we have used an integration by parts. From the parallelogram law, |x − y| 2 + |x + y| 2 = 2 |x| 2 + 2 |y| 2 , we get We also have with equality if and only if α = (N − 2)/4. By choosing this value and writing we get the result. By density, we extend the inequality to any u ∈ H 1 (R N ).
With y = 0, the inequality in Lemma 5 covers the optimal case with only one singularity. For a given u, the optimal case is also recovered in the limit d = |y| → +∞. The factor |y| 2 |x| 2 +|y| 2 indeed converges to 1 for any x ∈ R N as d → +∞. Hence, replacing u by u(· − y), we recover The inequality in Lemma 5 is therefore an optimal interpolation inequality, with a weight, which interpolates between the case y = 0 and the limit d → +∞. It is however not very useful in the sense that for a fixed value of |y|, one cannot obtain improved values (uniformly with respect to u) of the constant (N − 2) 2 /8, as a simple scaling argument shows.
To do better than this we look for estimates of the positive constant L(µ, N ) such that the following inequality holds for any u ∈ H 1 (R N ) and any y ∈ R N , d := |y| > 0. Let V 2 (x) := 1 |x−y| 2 + 1 |x+y| 2 and define Proof. As a function of µ, L(µ, N ) is nondecreasing by its definition, using the fact that V 2 is positive. (i) is a consequence of Lemma 5. By Theorem 1, L(µ, N ) takes finite nonnegative values in case (ii). Let us prove that L(µ, N ) has to be positive. If it were not the case for some µ ∈ ((N − 2) 2 /8 < µ ≤ (N − 2) 2 /4), by applying the inequality to u ε (x) = u(x/ε), we would be able to write Letting ε → 0, this would prove that Hardy's inequality with only one singularity holds for some µ > (N − 2) 2 /4, a contradiction with the fact that (N − 2) 2 /4 is optimal.
Concerning (iii), if the inequality were true for some µ > (N − 2) 2 /4, we could also consider u λ (x) := u(λ (x − y)). Taking the limit λ → ∞, this would prove that Hardy's inequality holds with constant µ, again a contradiction with the optimality on (N − 2) 2 /4. The independence of L(µ, N ) in terms of d is also a consequence of the scaling properties of the inequality. By considering u(·/d), the problem can indeed be reduced to the case d = 1 without loss of generality.
The "expansion of the square" method goes beyond these estimates. We shall establish in Theorem 7 an explicit expression of a nondecreasing function µ → K(µ, N ) such that Assume that u ∈ D(R N ), y ∈ R N . In (4), we want to estimate pointwise the last term by a combination of the bipolar potential and a constant. For this purpose, we choose any β ∈ (0, 1) and look for an optimal pointwise upper estimate of Let r := |x|, |y| = d and cos θ = x · y/(r d). We can rewrite F as and since |x − y| 2 |x + y| 2 = (r 2 + d 2 ) 2 − 4 r 2 d 2 cos 2 θ is nonnegative, we get F (x) = 2 (1 − β) r 2 − (1 + β) d 2 (r 2 + d 2 ) 2 − 4 r 2 d 2 cos 2 θ and F achieves its maximum for cos θ = 1: under the condition we see that f realizes its maximum on 1+β 1−β d, ∞ at r = 1+3 β 1−β d and get This shows the pointwise inequality Recall that V 2 (x) = 1 |x−y| 2 + 1 |x+y| 2 and inject the above estimate in We thus get From the conditions µ > 0, β ∈ (0, 1) we deduce that In terms of α and µ, we have Recall that t, a and β are related by 1 = t a − a 2 (1 + β) , β ∈ (0, 1) , which means that at t fixed, a has to be taken such that (1 + β) a 2 − t a + 1 = 0 .
Solving the equation, we get a = a ± β = t ± t 2 − 4 (1 + β) 2 (1 + β) for some β ∈ (0, 1). The admissible domain D for a is therefore given by Notice that on such a domain, 2 a 2 − t a + 1 > 0. The function achieves its infimum on D at a = 1 4 t ± √ t 2 − 8 , that is for β = 1. This is admissible only for with h(a) := 4 a 3 − 6 t a 2 + (t 2 + 6) a − t, the sign of g ′ is therefore the same as the one of −h on D. For any given t > 2, h changes sign at three different valuesā 1 (t) <ā 2 (t) <ā 3 (t). For t > 2, the curves t →ā 1 (t) and t →ā 3 (t) do not intersect D, while the curve t →ā 2 (t) intersects D for 2 < t < 2 √ 2. The minimum of g on D is therefore achieved for a =ā 2 (t), 2 < t ≤ 2 √ 2, and is 0 for t ≥ 2 √ 2. The curve t →ā 2 (t) is explicitly given for t > 2 bȳ Notice that the curves t →ā 1 (t) and t →ā 3 (t) can also be computed Define the function κ by We have therefore proved the following result.
Theorem 7. With the above notations, let K(µ, N ) : holds for any u ∈ H 1 (R N ) and any y ∈ R N , d := |y| > 0.  Figure 1. Curves t → a ± 0 (t), a ± 1 (t),ā 1 (t),ā 2 (t),ā 3 (t). Notice that the estimate of Theorem 7 is optimal as long as one uses only a local estimate of the potential. Next we prove an inequality in the case of a potential with M ≥ 2 singularities. As a first step, we state the M -poles version of Lemma 5.
Lemma 8. For any u ∈ H 1 (R N ) and any set of poles y 1 , y 2 , . . . y M ∈ R N , M ≥ 2, When y k = y l for all k, l, one recovers the standard Hardy inequality with one singularity.
Proof. The proof is similar to the one of Lemma 5. Assume that u ∈ D(R N ) and α > 0. We compute where we have applied an integration by parts. To rewrite the mixed term we use the identity and note that M j,k=1 As a consequence, The proof of this lemma is straighforward and follows the same lines as the one of Lemma 6. The best constant L := sup Let us consider the square expansion (5) and look for an optimal pointwise upper estimate of If we notice that x − y k = (x − (y j + y k )/2) − (y k − (y j + y k )/2) = (x − (y j + y k )/2) − y with y := (y k − y j )/2 and x − y j = (x − (y j + y k )/2) − (y j − (y j + y k )/2) = (x − (y j + y k )/2) + y, up to a translation of (y j + y k )/2, what we have to estimate is and exactly as in the proof of Lemma 6, we get Returning to the original coordinates we find From the previous estimate and (5) we obtain Now, as in the case with 2 poles, we set (N − 2) α − α 2 (1 + β M ) =: µ to get Recall that t, a, M and β are related by 1 = t a − a 2 (1 + β (M − 1)) , β ∈ (0, 1) , which means that at t fixed, a has to be taken such that i.e., for some β ∈ (0, 1), As in the case of 2 poles we investigate the function g M (a) = (t a−M a 2 −1) 2 t a−a 2 −1 and see that its infimum is equal to 0 for t ≥ 2 √ M , that is, for µ ∈ 0, (N − 2) 2 /(4 M ) , while for µ ∈ (N − 2) 2 /(4 M ), (N − 2) 2 /4 , the infimum of g M in its interval of definition is equal to the value of g M at the second root a 2,M =ā 2,M (t) of the function The result of this computation can be summarized into the following statement.
holds for any u ∈ H 1 (R N ) and any y 1 , . . . y M in R N . Here I n the identity matrix in (C 3 ) n and σ 1 = 0 1 1 0 ,

Hardy inequalities for the Dirac operator
If W is a bounded function which tends to 0 at infinity, one can easily prove that the operator H − W with domain H 1 (R 3 ) is self-adjoint. If W has singularities, as it is the case for instance if W = ν W M , one is interested in defining self-adjoint extensions of (H − W ) ↾ C ∞ 0 (R 3 ). The method used to do this depends on the singularity. Let us for instance consider Coulomb potentials ν W 1 (x) = ν/|x|, ν > 0. Then for ν ∈ (0, π/2] one can use the pseudo-Friedrich extension method to define an extension which satisfies This result is obtained by using Kato's inequality : |H| ≥ 2 π |x| . Actually one can prove that H ν,W1 is essentially self-adjoint if ν < √ 3/2 ( [40]). When the singularities are stronger, that is for ν ≥ √ 3/2, other methods need to be used. Various works have dealt with this issue, and it appears that for potentials W which have a singularity at the origin, the condition is sufficient to define a distinguished extension of (H − W ) ↾ C ∞ 0 (R 3 ). This has been done by various methods : see [47,48,49] in the case of semibounded potentials W and [39] without this assumption. The extension T is then defined as T := T * ↾D(T * ) ∩ D(|x| −1/2 ). On the other hand, under Assumption (6), Nenciu proved in [38] the existence of a unique extension T with domain contained in H 1/2 (R 3 , C 4 ). Finally, in the case of semibounded potentials satisfying (6), Klaus and Wüst proved in [33] that the aforementioned methods lead to the same self-adjoint extension.
As in the case of the Schrödinger operator, Hardy inequalities for Dirac operators provide us with lower estimates for some eigenvalues. If ψ = φ χ is an eigenfunction of H ν,W with eigenvalue λ, the eigenvalue equation can be rewritten as where K := −i σ · ∇. We can eliminate the lower component χ. The equation for φ is and so, φ appears as a critical point of φ → R 3 | σ·∇ φ| 2 , with critical value 0. Such a functional is monotone decreasing as a function of λ. Hence if Λ = λ 1 is the smallest eigenvalue of H ν,W in (−1, 1), then holds with Λ = λ 1 . Reciprocally, for a large class of potentials W, any eigenvalue of H ν,W in (−1, 1) can be characterized as a min-max of the Rayleigh quotient (H ν,W ψ, ψ) / ψ 2 is decomposed into an upper component φ and a lower one, χ. Under appropriate conditions, see [13,14], . The minimization step is then reduced to establish that This completes the proof, at least at a formal level, that finding the lowest eigenvalue of H ν,W in the gap (−1, 1) is equivalent to getting the best constant in Inequality (7). It was proved in [13,14] that for a large class of potentials with at most one singularity, the first eigenvalue of H − W in the spectral gap (−1, 1) is the largest constant Λ for which (7) holds. For instance, in the case of the radial Coulomb potential, −ν/|x|, for all ν ∈ (0, 1), where S(ν) := √ 1 − ν 2 is the best possible constant. Passing to the limit ν → 1 − , Inequality (8) also holds for ν = 1.
In this section we show that inequalities like (7) can be proved for the multipolar potentials ν W M and provide us with lower estimates for the eigenvalues of H − ν W M in the interval (−1, 1). We shall use the IMS method to localize the integrals and to reduce the problem to locally radial potentials.
We now decompose the term | σ · ∇φ| 2 by using the partition of unity defined in Section 1 and take advantage of the fact that the function J is scalar and takes real values.
k=1 be any partition of unity such that Int (supp(J k )) ∩ Int (supp(J l )) = ∅ if Proof. We first denote by J an arbitrary element of the partition of unity. By applying the definition of | σ · ∇ (Jφ)| 2 and grouping the terms with J 2 , 3 a=1 |∂ a J| 2 and J ∂ a J, we obtain the following: where a, b = a b and Re(z) denotes the real part of the complex number z. To obtain the result we have to sum over all the indexes k = 1 , . . . M + 1 and remember that M+1 k=1 J k ∂ a J k = 0 for a = 1, 2, 3.
The next result is a local estimate on domains where the potential W admits a radial dominant potential.
We can then apply Lemma 12 to R 3 | σ·∇ (J k φ)| 2 1+λ+ν WM dx with for k = 1 , . . . M, and for k = M + 1. From this and Lemma 11, it follows that Property (d) of the partition of unity, see Section 1, gives To get λ * , we now choose the largest possible λ such that that is the largest root of (13) (1 + b) λ 2 + a + (2 + a) b λ + (1 + a) b + a + π 2 d 2 − 1 = 0 . The value d * (ν) is the minimal d for which the discriminant of the second order equation (13) is nonnegative. For such a d, one can check that λ * > −1.
In the case M = 2, a better result can be achieved by considering the partition of unity {J k } 3 k=1 defined by (3), i.e. such that Proof. From Property (d) of Section 1 we derive Moreover, we have the estimate The first inequality holds because |x + y| ≥ d on supp(J 1 ) = {x ∈ R 3 : |x − y| ≤ |x + y|}. The second inequality is similar. The last one can be obtained as in the proof of Proposition 4. The proof is then the same as the one of Theorem 13, with K 2 (λ, ν, d, 2) replaced by K 2 (λ, ν, d, 1).
In the case M > 2, if one wants to improve on Theorem 13, one has to make a geometric assumption, which is always satisfied for M = 2. Proof. One proceeds as in the proof of Theorem 13, with K 2 replaced by K 2 (λ, ν, d, M − 1) = K 1 (λ, ν, d, M ).
Theorem 13 has several consequences. Assume first that b M = M .
3.4. Further estimates. One can actually use Lemma 12 to obtain slightly different inequalities which are not Hardy inequalities for Dirac operators as (7). The difference lies in the fact that the coefficient in front of the L 2 term is taken bigger than in (7). Such inequalities are valid for all d > 0 and are useful to obtain asymptotics both in the cases d → +∞ and d → 0, for ν small enough.
Putting the above estimates together we complete the proof.
We are able to recover the Hardy-like inequality for the Dirac operator with a radial Coulomb potential (8) by taking the limit as d → 0 only under the following technical assumption: lim d→0 1 d 2 |(x·y)|≤d 2 /2 |φ| 2 dx = 0 .