Hardy inequalities for large fermionic systems

. Given 0 < s < d2 with s (cid:20) 1 , we are interested in the large N -behavior of the optimal constant (cid:20) N in the Hardy inequality P Nn D 1 . (cid:0) (cid:129) n / s (cid:21) (cid:20) N P n<m j X n (cid:0) X m j (cid:0) 2s , when restricted to antisymmetric functions. We show that N 1 (cid:0) 2sd (cid:20) N has a positive, finite limit given by a certain variational problem, thereby generalizing a result of Lieb and Yau related to the Chandrasekhar theory of gravitational collapse.


Introduction and main result
A prototypical form of Hardy's inequality states that when d ≥ 3 and u ∈ Ḣ1 (R d ), the homogeneous Sobolev space.This and other forms of Hardy's inequality are fundamental tools in many questions in PDE, harmonic analysis, spectral theory and mathematical physics.We refer to the survey paper by Davies [6] and the books of Maz'ya [35] and Opic and Kufner [36] for extensive results, as well as background and further references.
In [19], the second and third authors and their coauthors studied what they called many-particle Hardy inequalities.These are inequalities for functions defined on R dN with coordinates denoted by X = (X 1 , . . ., X N ) with X 1 , . . ., X N ∈ R d .Here N ≥ 2 can be interpreted as the number of (quantum) particles in R d and the X n , n = 1, . . ., N, as their positions.The Hardy weight takes the form 1≤n<m≤N |X n −X m | −2 .It is shown in [19] that for all u ∈ Ḣ1 (R dN ) with a certain explicit lower bound on the optimal constant β (N ) d that is positive, provided d ≥ 3 and N ≥ 2. What is of interest is the behavior of © 2024 by the authors.This paper may be reproduced, in its entirety, for non-commercial purposes.Partial support through US National Science Foundation grant DMS-1954995 (R.L.F.), as well as through the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through Germany's Excellence Strategy EXC-2111-390814868 (R.L.F.) is acknowledged.Add acknowledgements.
1 the optimal constant β (N ) d as N → ∞ for fixed d.This corresponds to the manyparticle limit, a classical topic in mathematical physics.The explicit lower bound for β (d) N obtained in [19] shows that lim inf N →∞ N −1 β (d) N > 0. It is also noted in [19] that lim sup N →∞ N −1 β (d) N < ∞.The methods of this paper allow us to prove that lim N →∞ N −1 β (d) N exists and give an explicit expression for it in terms of a variational problem for functions on R d ; see Subsection 2.7 for details.
Here sgn σ ∈ {+1, −1} denotes the sign of σ.The antisymmetry requirement appears naturally in physics in the description of fermions.(Note that we restrict ourselves here to scalar functions, corresponding to the spinless or spin-polarized situation, although in our results for dimension d ≥ 3 spin could be incorporated; however, for d = 1, 2, when |x| −2 is not locally integrable, the spin-polarization is crucial.) We denote by κ N the optimal constant in inequality (1) when restricted to antisymmetric functions, that is, As emphasized in [19], there are signifcant differences between the inequality on all functions in Ḣ1 (R dN ) and its restriction to antisymmetric ones.One important difference is that κ N , but, as we will see in the present paper, this is not optimal, at least when d ≥ 3.
In fact, our main result states that lim N exists as a positive and finite number when d ≥ 3, and gives an explicit expression for it in terms of a variational problem for functions on R d .This is the special case of a more general result, which concerns the inequality for all u ∈ Ḣs (R dN ).Here s is a real number satisfying 0 < s < d 2 and the operator (−∆ n ) s 2 acts on the n-th variable of X = (X 1 , . . ., X N ) by multiplication by |ξ n | s in Fourier space.The homogeneous Sobolev space Ḣs (R dN ) is the completion of C ∞ c (R dN ) with respect to the quadratic form on the left side of (2).It is relatively straightforward to see that inf Indeed, for each pair (n, m) with n = m we have, by the ordinary fractional Hardy inequality (see Lemma 9 below), Integrating this inequality with respect to the remaining variables and summing over n and m gives (3).As an aside, we mention that the optimal constant in (4) is known; see [17,20] and also [39,12].
Our interest is again in the sharp constant in (2) when restricted to antisymmetric functions, that is, in To state the limiting variational problem, we introduce for nonnegative, measurable functions The fact that τ d > 0 follows from the Hardy-Littlewood-Sobolev inequality [25, Theorem 4.3], together with Hölder's inequality.Finally, let The superscript TF stands for 'Thomas-Fermi' and it will become clear in the proof that this constant is related to the Thomas-Fermi approximation for the kinetic energy.
The following is our main result.
Remarks 2. (a) This result in the special case s = 1 2 , d = 3 is due to Lieb and Yau [32], following earlier work by Lieb and Thirring [31].While our overall strategy is similar to theirs, there are some significant differences, which we explain below.(b) Our proof of the asymptotics (5) comes with remainder bounds.We show that ); see (6) and ( 13).(c) We believe that Theorem 1 remains valid without the extra assumption s ≤ 1.This would probably require significant additional effort at various places and, since our main interest is the case s = 1, we decided to impose this simplifying assumption.(d) Theorem 1 extends to the case where spin is taken into account, except that the limiting expression in ( 5) is multiplied by a power of the number of spin states.We refer to [27] for an explanation of this terminology and to [32] for proofs where spin is taken into account.(e) Finding the asymptotic behavior of κ Let us give some background on Theorem 1 and explain some aspects of its proof.The basic idea is that it is a combined semiclassical and mean-field limit.Such a limit is behind what is called Thomas-Fermi approximation for Coulomb systems and has first been made rigorous by Lieb and Simon in [27].Parts of this proof were simplified through the use of coherent states [38,24] and the Lieb-Thirring inequality [30,24], and these will also play an important role for us.For a recent study of this combined semiclassical and mean-field limit for quite general systems we refer to [13].
One difficulty that we face here, compared to the analysis of nonrelativistic Coulomb systems [28] or the systems in [13], is that the kinetic energy and the potential energy scale in the same way, so that there is no natural scale.This problem was overcome by Lieb and Yau [32], following earlier work of Lieb and Thirring [31], in their rigorous derivation of Chandrasekhar theory of gravitational collapse of stars.In important ingredient in the proofs of [31,32] and also in the more recent [13], is the Lévy-Leblond method.This method will also play a crucial role in our proof.It consists in dividing the N particles into two groups, treating one part as 'electrons' and the other part as 'nuclei'.The electrons repel each other, and similarly the nuclei, while electrons and nuclei attract each other.The construction involves a further, free parameter that corresponds to the quotient between the charges of the electrons and the nuclei.At the end one averages over all such partitions.
There is one important structural property, however, that Lieb and Yau can take advantage of and we cannot.They deal with the case s = 1 2 , d = 3, where the interaction potential |x| −1 is, up to a constant, the fundamental solution of the Laplacian and the corresponding (sub/super)harmonicity properties enter into the proof of [32, Lemma 1].The same phenomenon occurs, for instance, for s = 1, d = 4, or in general for s = d−2 2 , d ≥ 3, but in the general case the interaction |x| −2s is not harmonic outside of the origin.Therefore a substantial part of our effort goes into proving bounds for systems interacting through Riesz potentials |x| −2s for general exponents 0 < 2s < d; see Propositions 3 and 11.In both cases our proof relies on the Fefferman-de la Llave decomposition of this interaction potential.As a curiosity, we mention that also the Cwikel-Lieb-Rozenblum inequality, and therefore the Lieb-Thirring inequality, which is another important ingredient in our proof, can be established using the Fefferman-de la Llave decomposition [9].For more on this decomposition, see also [16].
Neither the Lévy-Leblond method nor the Fefferman-de la Llave decomposition seem to work for d ≤ 2s and this case remains open (except for s = d = 1).In Appendix B we give a suggestion of what might be the relevant mechanism in the borderline case d = 2s.
Finally we mention that the results in this paper, with the exception of those in Section 2, are contained in a preprint with the same title, dated October 30, 2006, that was circulated among colleagues.The present paper corrects some minor mistakes therein and adds a proof of the sharp asymptotic lower bound.
It is our pleasure to dedicate this paper to Brian Davies in admiration of his many profound contributions to spectral theory and mathematical physics and, in particular, to the topic of Hardy inequalities.Happy birthday, Brian!

Lower bound
Our goal in this section is to prove the lower bound in Theorem 1.That is, we will show lim inf . More precisely, we will prove the following quantitative version of it, As explained in the introduction, we mostly follow the method in [32], but an important new ingredient, which replaces their [32, Equation (2.21)], is the electrostatic inequality in Proposition 3.
2.1.An electrostatic inequality.For a probability measure µ on R dM we denote by ρ µ the nonnegative measure on R d obtained by summing the marginals of µ.That is, for any bounded continuous function f on R d we have The definition of D λ is extended to nonnegative measures on R d in a natural way, namely, by where the implied constant depends only on d and λ.
The following proof uses some ideas from that of [7,Corollary 1].
Proof.According to the Fefferman-de la Llave formula [7], we have for all y, y ′ ∈ R d with a constant depending only on d and λ.This implies that where we have introduced, for any ball B, Note that K B is a nonnegative integer.We claim that for any n ≥ 0 and any K ′ ∈ N 0 , Indeed, this is true when K ′ = 0, and when K ′ ≥ 1, we write the left side as and bound Thus, we have shown that Indeed, when the left side does not vanish, we have |y − a| < r and there is a k ∈ {1, . . ., K} such that By performing first the a and then the r integration, we obtain for each This implies the claimed inequality.

Lieb-Thirring inequality. Associated to a normalized function
and therefore we can consider the measure dρ µ on R d as in the previous subsection.In the present case, this measure turns out to be absolutely continuous and we denote its density by ρ ψ .Explicitly, This density appears in the following famous Lieb-Thirring inequality.
Lemma 4. Let d ≥ 1 and s > 0. Then for any M ∈ N and any antisymmetric and where the implied constant depends only on d and s.
For s = 1, this inequality is due to Lieb and Thirring [30].Their original proof generalizes readily to the full regime s > 0; see also [27,Chapter 4], as well as [11,Theorem 4.60 and Section 7.4] and [8].For the currently best known values of the constants, see [10].

Coherent states.
The following lemma is a rigorous version of the Thomas-Fermi approximation for the kinetic energy.It is proved with the help of coherent states. Here dx denotes the Fourier transform of g.Bounds of the same type as in the lemma appear in [24, Eqs.(5.14)- (5.22)] in the special case s = 1 and d = 3; a general version is formulated in [32,Lemma B.4].Because of a subtlety in the application of that lemma, we sketch the proof.
Proof.We will show that for any where (Compared to [25,Section 12.7] and other presentations, we find it convenient to use y − x instead of x − y in the definition of v.) Once ( 8) is shown, the inequality in the lemma follows as in [32,Lemma B.3].
To prove (8), we observe that We multiply this identity by |η| 2s and integrate with respect to η.
In case s ≤ and argue similarly as for s ≤ 1 2 to obtain the claimed inequality (8).The second inequality in (9) follows from the subadditivity of t → t s , and to prove the first inequality we write attains its maximum at t = 0.
We expect a similar bound as in Lemma 5 told for s > 1 as well, but the structure of the remainder term will probably more complicated.
2.4.Summary so far.Let us combine the bounds from this section.
Proof.Let ψ ∈ H s (R dM ) be antisymmetric and L 2 -normalized.We recall that according to Proposition 3 we have The second term on the right side appears in the claimed error bound.To bound the first term, let g ∈ H s (R d ) be L 2 -normalized.Using the definition of τ d,s , Young's convolution inequality and Lemma 5 we find .
We now assume that g ) and a parameter ℓ > 0 to be chosen.We have This implies the claimed bound 2.5.Domination of the nearest neighbor attraction.For X = (X 1 , . . ., X N ) and n ∈ {1, . . ., N}, let Then for any antisymmetric u ∈ Ḣs (R dN ) we have with an implicit constant that only depends on d and s.This bound appears as [33,Theorem 5] in the cases d = 3 and s ∈ { 1  2 , 1}, but the proof readily generalizes to the stated parameter regime and is omitted.We also mention an alternative proof in [7, Corollary 2], which is based on a Fefferman-de la Llave type formula for the Ḣs (R d )-seminorm and which generalizes to the regime s < 1.
Probably Proposition 7 remains valid for 1 < s < d 2 , but this would require an argument and for the sake of brevity we do not consider this case.The IMS localization formula in [34,Lemma 14] might be useful.
2.6.Proof of the lower bound in Theorem 1.We turn to the proof of ( 6), for which we use the Levy-Leblond method [22], similarly as in [31,32].Given N ≥ 3 we choose an integer M ∈ {1, . . ., N − 2} and a real number Z > 0. We set K := N − M and consider partitions π = (π 1 , π 2 ) of {1, . . ., N} into two disjoint sets π 1 and π 2 with M and K elements, respectively.We have Let u ∈ Ḣs (R dN ) be antisymmetric.Our goal is to bound the Hardy quotient for u.By density we may assume that u ∈ H s (R dN ) and by homogeneity we may assume that u is L 2 -normalized.We integrate the left side of (11) against |u(X)| 2 .Correspondingly, on the right side we obtain a sum over partitions and we bound the integral for each fixed such partition P .We first carry out the integral over the variables in π 1 .Denoting these variables as (Y 1 , . . ., Y M ) and the variables in π 2 as (R 1 , . . ., R K ) we infer from Corollary 6 that Here dπ 1 (X) denotes integration with respect to the variables X m with m ∈ π 1 and, for m ∈ π 1 , we have Inserting this into the above bound and carrying out the integration over the variables in π 2 , we obtain According to (11), summing this bound over π gives Using Proposition 7, the right side can be bounded by with Our goal is to choose the parameters M and Z (depending on N) in such a way that C → 1 as N → ∞.We choose Z = M/K and obtain With the choice This completes the proof of (6).
Remark 8.Under the additional assumption d > 4s one can prove (6) (with a worse remainder bound) without using Proposition 7. Indeed, inserting the bound from Lemma 9 below into the bound in Corollary 6, we can drop the last term there at the expense of replacing the factor in front of the first term by with an implicit constant depending only on d and s.
The following proof has some similarities with [32,Lemma B.1].

It remains to bound the weak
This gives the claimed bound.

2.7.
The case without antisymmetry.In this subsection we explain how the proof of ( 13) can be modified to give a lower bound on the optimal constant β (d,s) N in (2).We denote .
It is not difficult to show that ω d,s > 0 when 0 < s < d 2 and that there is an optimizer ρ * ; see, e.g., [32,Theorem 4] in the case s = 1 2 , d = 3.Then one can show that by taking u(X) = N n=1 ρ * (X n ).For s = 1, d ≥ 3 this argument appears in [19,Theorem 2.3].We now state the lower bound corresponding to (12).
Proof.We proceed from inequality (10), which did not use the antisymmetry of ψ.
Using the definition of ω d,s and Lemma 9 we can bound the right side of ( 10) by The second inequality here is the Hoffmann-Ostenhof inequality [18] for s = 1 and its generalization to s < 1 by Conlon [5].Following the Lévy-Leblond method we deduce from this bound that We choose again Z = M/K and obtain Choosing M = 1 we arrive at the claimed bound.

Upper bound
Our goal in this section is to prove the upper bound in Theorem 1.That is, we shall show lim sup More precisely, we will prove the following quantitative version of it, For the proof we follow rather closely the method in [31, Section 3].One new ingredient is an exchange inequality, which appears in Proposition 11.

3.1.
A bound on the indirect part of the Riesz energy.Here we return to the setting of Subsection 2.1 and consider probability measures µ on R dN and their marginals ρ µ .
Proposition 11.Let d ≥ 1 and 0 < λ < d.Then for any N ∈ N and for any nonnegative Borel probability measure with an implicit constant depending only on d and λ.
This bound for λ = 1 and d = 3 is due to [23] with an improved constant in [26].Here we adapt the proof strategy from [29], which does not use (sub/super)harmonicity properties of the interaction potential.
Proof.The Fefferman-de la Llave formula (7) where, for any ball B, n B is defined as in the proof of Proposition 3 and where We will derive two different lower bounds on the integrand The first one is the trivial bound To derive the second lower bound, we estimate, with an arbitrary 0 Thus, with n B as in the proof of Proposition 3. Choosing ρ = ρ µ we have ρ B = n B and therefore Combining ( 14) and ( 15) we find We now bound n B from above using the maximal function ρ * µ of ρ µ .By its definition (see, e.g., [15, Definition 2.1.1])we obtain The assertion now follows from the boundedness of the maximal function on 3.2.Relaxation to density matrices.We use the result from the previous subsection to make the next step towards (13), namely by proving an upper bound in terms of density matrices.
We recall that a nonnegative trace class operator γ on L 2 (R d ) has a well defined density . (In the case of a non-simple eigenvalue λ i one can convince oneself easily that this is independent of the choice of the eigenfunction ψ i .) We claim that for any operator γ on L 2 (R d ) satisfying we have Here, as usual, we write Tr(−∆) s γ instead of Tr(−∆) Of course the bound is only meaningful if the latter quantity is finite.
Given Proposition 11 the proof of this assertion is relatively standard (see, e.g., [31,Section 3]), but we include some details for the sake of completeness.First, there exists a nonnegative operator Γ on the antisymmetric subspace of L 2 (R dN ) satisfying where Tr N −1 denotes the partial trace with respect to N − 1 variables.This is due to [4]; see also [27,Theorem 3.2].It follows that with orthonormal antisymmetric functions u i ∈ L 2 (R dN ) and nonnegative numbers p i we obtain Therefore, the definition of κ (d,s) N , applied to u i , yields the inequality We now apply Proposition 11 to the measure which, by the above properties, is indeed a probability measure.Moreover, using the partial trace relation between Γ and γ, we find ρ µ = ρ γ .Thus, the claimed inequality (17) follows from Proposition 11.

Construction of
, where ω d is the volume of the unit ball in R d .It is easy to see that this operator satisfies (16).Indeed, the bound γ ≥ 0 follows immediately by estimating ½(|ξ| 2s < cρ(x) 2s d ) ≥ 0 and the bound γ ≤ 1 follow by estimating ½(|ξ| 2s < cρ(x) 2s d ) ≤ 1 and using Plancherel and the normalization of g.To prove Tr γ = N we integrate the kernel on the diagonal, using the choice of c and, again, the normalization of g.In this connection we also note that the density of γ is Assuming that | g| is even, we claim that This is shown in the special case d = 3, s = 1 2 in [31, Section 3] (see also [25,Theorem 12.10]).The proof generalizes to the general case, the underlying estimates being the same as in the proof of Lemma 5.
If we insert these facts into (17), we obtain By the normalization of g and Minkowski's inequality, we have Moreover, as in the proof of ( 6), Young's convolution inequality shows that .
To summarize, we have Similarly as in the proof of the lower bound we now assume that g ) and a parameter ℓ > 0 to be chosen.We consider G as fixed and obtain, as before Thus, 3.4.The semiclassical problem.The following result states that the variational problem defining τ d,s has an optimizer.This result is not strictly necessary for our proof of the upper bound in Theorem 1, but it is readily available and makes the proof more transparent.
Lemma 12. Let d ≥ 1 and 0 < s < d 2 .Then there is a 0 In the special case s = 1 2 , d = 3 this appears in [26,Appendix A].The proof in the general case is exactly the same.
For the sake of completeness we mention that the uniqueness (up to translations, dilations and multiplication by a constant) of ρ * has been studied in [32], as well as in the recent papers [2,3].
3.5.Proof of the upper bound in Theorem 1.Let ρ * be the optimizer from Lemma 12.After a dilation and a multiplication by a constant we may assume that We then apply the construction outlined in this section with the choice ρ = Nρ * .Inequality (18) turns into Here, as before, δ R (y We will estimate the sum of the negative eigenvalues of the (one-particle) operator Then, for all K ≥ 2, R ∈ R dK and λ > 0, with an implicit constant that only depends on d and s.
Proof.By the Lieb-Thirring inequality (see, e.g., [11,Theorem 4.60]) we have To estimate the latter integral we write V R (y The first integral is easily found to equal a constant times |R k − R l | −2s .To estimate the second integral we note that {y : Extending the domain of integration we find This proves the assertion.

Now everything is in place for the
Proof of Proposition 13.In view of (3), it suffices to prove the bound for sufficiently large N.In fact, we will prove the bound for N ≥ N(τ ) for some N(τ ) to be determined later.
For given N ≥ 3, τ > 0 and κ > 0 we choose an integer M ∈ {1, . . ., N − 2} and parameters λ, α > 0. Setting K : Here the sum runs over all partitions π = (π 1 , π 2 ) of {1, . . ., N} into two disjoint sets π 1 , π 2 of sizes M and K, respectively, and for any such partition the operator h π is defined by In order that ( 21) be an identity we require It suffices to prove that for κ ≤ C(τ )N −1+ 2s d one has h π ≥ 0 for all partitions π as above.We denote the variables in π 1 by Y = (Y 1 , . . ., Y M ) and those in π 2 by R = (R 1 , . . ., R K ).Then one has the estimates These two estimates lead to the lower bound The right side is an operator in L 2 (R dN ), but there is no kinetic energy associated with the R variables.Hence if we define for fixed R ∈ R dK an operator h R in the antisymmetric subspace of L 2 (R dM ) by the expression on the right side of (24), then one has the estimate Further, since h R acts on antisymmetric functions one has and hence by Lemma 14 inf for all x ∈ R 2 .
A computation [14] (see also [11,Lemma 7.21]), based on the Fourier transform of the characteristic function of an interval, shows that the ϕ p are orthonormal in L 2 (R 2 ).We define u as their Slater determinant, where p 1 , . . ., p N is an enumeration of N .We have [14] (see also [11,Lemma 7.21]) in the asymptotic regime that we are considering.
Our task is to bound from below where Note that the integrand on the right side of (30) is nonnegative.Consequently, we obtain a lower bound by restricting it to for a certain constant C, independent of L and µ, and to be chosen below.Note that for Therefore the ρ-part of the integral on the right side of (30), restricted to Ω, is bounded from below by where Comparing this with (29) we arrive at the constant 4 on the right side of (28).
Thus it remains to prove that the γ-part of the right side of (30), restricted to Ω, is negligible for some (and, in fact, any) choice of C. Before giving a complete proof, let us explain the heuristics.The nonrigorous step is that we approximate From this we arrive at the expectation that for x, x ′ ∈ Q L−ℓ , at least on average, one has Accepting this bound, we obtain by straightforward estimates Recalling that our lower bound on the ρ-term in (30) is of size µ 2 L 2 ln(µL 2 ), we see that the γ-term is indeed negligible.
We now present a rigorous proof that the γ-term is neglibigle.We will not be able to prove the bound in (31), but we will be able to prove that Accepting this bound and combining it with the trivial bound |γ u (x, x ′ )| 2 ≤ ρ u (x)ρ u (x ′ ), we obtain The latter follows by summing a trigonometric series.Since the sum over p 2 j ≤ µ contains µ If |r| ≤ L 2 , then dist(r j , LZ) = |r j | for j = 1, 2.Moreover, max j r 2 j ≥ 1 2 |r| 2 .Therefore the right side is µ 1 2 L 2 |r| −1 .This, applied to r = x − x ′ , yields (32) and concludes the proof of the proposition.
Remarks 16.(a) In physics, the vanishing of ρ u (x) ρ u (x ′ ) − |γ u (x, x ′ )| 2 near x = x ′ is called the exchange hole.The intuition is that it is this exchange hole that leads to the Hardy inequality for (spin-polarized) fermions in two dimensions.This hole, which is of size µ − 1 2 in the above example, mitigates the logarithmic divergence of the integral |x − x ′ | −2 and leads to the logarithmic behavior of the constant κ

N
> 0 for all d ≥ 1 and N ≥ 2, while β (d) N = 0 for d = 1, 2 and N ≥ 2; see [19, Remarks 2.2(i) and Theorem 2.8].Remarkably, for d = 1 the explicit value of the sharp constant κ (1) N is known, κ (1) N = 1 2 for all N ≥ 2 [19, Theorem 2.5].For d ≥ 2, as far as we know, only the lower bound κ (d) N ≥ d 2 /N is known.This displays the same N −1 behavior as β (d) (d,s) N in the case d = 2s is an open problem.In Appendix B we discuss a conjecture of what might be the right order and prove the corresponding upper bound.
It is essential for the validity of κ