Optimal transport with Coulomb cost and the semiclassical limit of density functional theory

Optimal transport with Coulomb cost and the semiclassical limit of density functional theory

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OPTIMAL TRANSPORT WITH COULOMB COST AND THE SEMICLASSICAL LIMIT

OF DENSITY FUNCTIONAL THEORY

by Ugo Bindini & Luigi De Pascale

Abstract. — We present some progress in the direction of determining the semiclassical limit of the Levy-Lieb or Hohenberg-Kohn universal functional in density functional theory for Coulomb systems. In particular we give a proof of the fact that for Bosonic systems with an arbitrary number of particles the limit is the multimarginal optimal transport problem with Coulomb cost and that the same holds for Fermionic systems with two or three particles. Comparisons with previous results are reported. The approach is based on some techniques from the optimal transportation theory.

Résumé(Transport optimal avec coût coulombien et limite semi-classique de la théorie de la fonctionnelle de la densité)

Nous présentons des progrès récents en vue de la détermination de la limite semi-classique de la fonctionnelle universelle de Levy-Lieb ou Hohenberg-Kohn en théorie de la fonctionnelle de la densité pour des systèmes coulombiens. Nous donnons en particulier une preuve du fait que, pour des systèmes de bosons avec un nombre arbitraire de particules, la limite est le problème de transport optimal multi-marginal à coût coulombien, de même que pour les systèmes de fermions à deux ou trois particules. Nous établissons des comparaisons avec des résultats antérieurs. Nous nous appuyons sur certaines techniques de la théorie du transport optimal.

Contents

1. Introduction and preliminary results. . . 910

2. Tools: Γ-convergence and multimarginal optimal transport . . . 912

3. AΓ-convergence result and proofs of the main theorems. . . 914

References. . . 933

Mathematical subject classification (2010). — 49J45, 49N15, 49K30.

Keywords. — Density functional theory, multimarginal optimal transportation, Monge-Kantorovich problem, duality theory, Coulomb cost.

The research of the second author is part of the project 2010A2TFX2Calcolo delle Variazionifunded by the Italian Ministry of Research and is partially financed by the“Fondi di ricerca di ateneo”of the University of Pisa and by the INDAM-GNAMPA..

1. Introduction and preliminary results

The ground state of a system ofN electrons interacting through Coulomb forces between each others and withM nuclei is described by one of the following minimal values (ground values)

(1.1) E_~^S = min

S T_~(ψ) +V_ee(ψ)−V_ne(ψ), or

(1.2) E_~^A= min

A T_~(ψ) +V_ee(ψ)−V_ne(ψ), and by the corresponding minimizers (ground states).

The minimization domains are the following sets of wave functions S=n

ψ∈H¹((R^d×Z2)^N)

R|ψ|²dz= 1, for all permutationsσofN points ψ((x_σ(1), α_σ(1)), . . . ,(x_σ(N), α_σ(N))) =ψ((x₁, α₁), . . . ,(x_N, α_N))o

,

A=n

ψ∈H¹((R^d×Z2)^N)

R |ψ|²dz= 1, for all permutationsσofN points ψ((xσ(1), ασ(1)), . . . ,(xσ(N), ασ(N))) = sgn(σ)ψ((x1, α1), . . . ,(xN, αN))o

, where we adopted the common notation

Z

f(z)dz:=

Z

f(z1, . . . , zN)dz1· · ·dzN

:= X

α₁,...,α_N=0,1

Z

R^d

f(x1, α1;. . .;xN, αN)dx1· · ·dxN.

The three terms in the functionals are: the kinetic energy T_~(ψ) = ~²

2m Z

|∇ψ|²(z)dz, the electron-electron interaction energy

Vee(ψ) =

Z X

16i<j6N

|ψ|²(z)

|xi−xj|dz

and the nuclei-electrons interaction energy Vne(ψ) =

Z X

16i<j6N M

X

k=1

|ψ|²(z)Zk

|x_i−N_k| dz.

The values E_~^S and E_~^A represents, respectively, the ground value for a Bosonic and Fermionic system of particles with N electrons and M nuclei. The Nk are the positions of the nuclei, while the ordered pair(x_i, α_i)is the position-spin of the i-th electron. In the usual Born interpretation,|ψ|² is the probability distribution of the N electrons and, according with the indistinguishability principle, it is invariant with respect to permutations of theN (xi, αi)variables.

Computing the ground values above amounts to solving a Schrödinger equation in R^dN and the numerical cost scales exponentially with N. The Density Functional Theory (DFT from now on) is an alternative introduced in the late sixties by Ho- henberg, Kohn and Sham. However the desire to describe the system in term of a different variable is much older and we may consider the Thomas-Fermi model as a precursor of this theory.

We associate to every wave function ψ a probability density on R^d defined as follows:⁽¹⁾

ρψ(x) := X

α1,...,αN=0,1

Z

R^d(N−1)

|ψ|²(x, α1;. . .;xN, αN)dx2· · ·dxN.

The map which associatesρtoψwill be denoted byψ↓ρψandρwill be called single electron density or electronic density. If we wish to describe the system in terms ofρ, the problems above should be reformulated as a minimization of suitable energies with respect to ρ which, no matter how big N is, is always a probability measure inR^d. The exact image of the mapψ↓ρψ was characterized by Lieb [13] who proved that the image of bothS andAis

H =

ρ|06ρ, R

ρ= 1, √

ρ∈H¹(R^d) .

Forρ∈Hwe introduce the Levy-Lieb functional [11, 13] also known as Hohenberg- Kohn functional⁽²⁾

F_~^S(ρ) = inf

ψ∈S, ψ↓ρ{T_~(ψ) +Vee(ψ)}, (1.3)

F_~^A(ρ) = inf

ψ∈A, ψ↓ρ{T_~(ψ) +V_ee(ψ)}.

(1.4)

Then the problems above can be reformulated as follows:

E_~^S = min

ρ∈H F_~^S(ρ) +N Z

v(x)ρ(x)dx, and

E_~^A= min

ρ∈H F_~^A(ρ) +N Z

v(x)ρ(x)dx, where we denoted

v(x) =−

M

X

k=1

Z_k

|x−Nk|.

This paper deals with the semiclassical limit of the two functionals F_~^S(ρ) and F_~^A(ρ)and more precisely it concerns the relations of the multimarginal optimal trans- port theory with these semiclassical limits. The ties between the DFT for Coulomb systems and optimal transport appeared first in [5, 7] and by now have revealed to

(1)In the usual definition the integral in the definition ofρψis also multiplied by a factorN, but here we prefer to deal with probability measures.

(2)These functionals are sometimes denoted byFHK(ψ). For the sake of a lighter notation, here we will not report theHK.

be a precious tool for the understanding. To detail better the results, let us shortly introduce the multimarginal optimal transport problem of interest here.

For everyρ∈H consider the set

Π(ρ) :={P∈ P(R^dN)|πⁱ_]P =ρ, i= 1, . . . , N},

and then the multimarginal optimal transport problem with Coulomb cost

(1.5) C(ρ) := min

P∈Π(ρ)

Z X

16i<j6N

1

|xi−xj|dP.

Duality for (1.5) and some properties of the functional C(ρ) have been studied in [10, 4]. The structure of1-dimensional minimizers has been investigated in [6].

The results on optimal transportation needed in this paper will be reported in the next section.

In this paper we will prove the following results.

Theorem1.1. — For all ρ∈H andd, N ∈N, F_~^S(ρ)^~→0−→C(ρ).

Theorem1.2. — For all ρ∈H,d= 1,2,3,4,N = 2,3, F_~^A(ρ)^~→0−→C(ρ).

The same problem was previously studied in [7]. It is easy to see that the proof presented in that paper adapts to prove Theorem 1.1 for any N. In this case our contribution consists in a more direct use of optimal transport techniques with the consequent simplifications. The convergence 1.2 was proved in [7] for N = 2. Here we extend the result toN = 3 and we are able to enlarge the class of approximating antisymmetric wave functions also forN = 2.

Remark1.3. — Although the usual description is limited to the physical dimension d= 3, here we explored also other dimensions in the hope to shed some light on the problems which are still open.

Since the functionals appearing in Theorems 1.1 and 1.2 above are all expressed as minimal values, the natural tool to deal with their convergence is theΓ-convergence which also we shortly introduce in the next section.

Addendum. — After this paper was accepted, two new papers appeared [12, 8]. Both papers contain a proof of the convergence for generalN. The approach is different.

2. Tools:Γ-convergence and multimarginal optimal transport 2.1. Definition ofΓ-convergence and basic results. — A crucial tool that we will use throughout this paper isΓ-convergence. All the details can be found, for instance, in Braides’ book [3] or in the classical book by Dal Maso [9]. In what follows,(X, d) is a metric space or a topological space equipped with a convergence.

Definition2.1. — Let(Fn)n be a sequence of functionsX 7→R. We say that(Fn)n

Γ-converges to F and we write F_n−→^Γ

n F if for anyx∈X we have – for any sequence(x_n)_n ofX converging tox

lim inf

n F_n(x_n)>F(x) (Γ-liminf inequality);

– there exists a sequence(x_n)_n converging toxand such that lim sup

n

Fn(xn)6F(x) (Γ-limsup inequality).

This definition is actually equivalent to the following equalities for anyx∈X:

F(x) = infn lim inf

n Fn(xn) :xn−→xo

= infn lim sup

n

Fn(xn)|xn −→xo . The function x 7→ inf

lim inf

n Fn(xn) : xn → x is called Γ-liminf of the sequence (Fn)n and the other one its Γ-limsup. A useful result is the following (which for instance implies that a constant sequence of functions does notΓ-converge to itself in general).

Proposition2.2. — TheΓ-liminf and theΓ-limsup of a sequence of functions(F_n)_n are both lower semi-continuous on X.

The main interest ofΓ-convergence resides in its consequences in terms of conver- gence of minima:

Theorem 2.3. — Let (Fn)n be a sequence of functions X → R and assume that F_n−→^Γ

n F. Assume moreover that there exists a compact and non-empty subsetKofX such that

∀n∈N, inf

X Fn= inf

K Fn

(we say that(F_n)_n is equi-mildly coercive on X). Then F admits a minimum on X and the sequence (inf_XF_n)_n converges to minF. Moreover, if (x_n)_n is a sequence of X such that

limn F_n(x_n) = lim

n (inf

X F_n)

and if (x_φ(n))n is a subsequence of(xn)n having a limitx, then F(x) = infXF. 2.2. Multimarginal optimal transportation and composition of optimal transport plans. — In this subsection we present some basic results about multimarginal opti- mal transportation with Coulomb cost.

An element P of Π(ρ)is commonly called a transport plan for ρ. If ρ ∈ P(R^d), define

µ_ρ(t) = sup

x∈R^d

ρ(B(x, t)).

Definition2.4. — Theconcentration ofρis µρ= lim

t→0µρ(t).

Note that ρ∈H impliesµρ = 0, since √

ρ∈ H¹ impliesρ∈ L¹. It is commonly assumed that, if µρ < 1/N, then C(ρ) < +∞. Here we provide a simple argument adapted to the particular caseρ∈H:

Proposition2.5. — If p>2d/(2d−1) andρ∈L¹(R^d)∩L^p(R^d), thenC(ρ)<+∞.

Proof. — Consider the transport planP(x1, . . . , xN) =ρ(x1)· · ·ρ(xN); then, C(P) =

Z

c(X)dP(X) = X

16i<j6N

Z

R^d×R^d

ρ(x_i)ρ(x_j)

|xi−xj| dxidxj6C(d, p)kρk²_p, where the last inequality follows from the Hardy-Littlewood-Sobolev and Hölder in-

equalities.

Definition2.6. — Ifα >0, let Dα=

X ∈R^{N d}| ∃i6=j with |xi−xj|< α be an open strip around

D₀=

X ∈R^{N d}| ∃i6=j withx_i=x_j , which is the set where the cost functionc is singular.

Note that if a transport planP has the property thatP(D_α) = 0for someα >0 thenC(P)is finite. The converse is not true in general, i.e., there may well be transport plans of finite cost whose supports have distance 0 from the setD0.

However, this is not the case ifP is optimal, as recently proved in [4]:

Theorem2.7. — Let ρ∈ P(R^d)withµρ<1/N(N−1)², with C(ρ)<+∞, and let β be such that

µρ(β)< 1 N(N−1)².

If P ∈Π(ρ)is an optimal plan for the problem (1.5), then P |_Dα= 0 for everyα such that

α < 2β N²(N−1).

3. AΓ-convergence result and proofs of the main theorems

It is useful to reformulate the different variational problems so that they have a common domain. For every ψ ∈ S or ψ ∈ A there is a natural transport plan Pψ∈Π(ρψ)defined by

Pψ= X

α1,...,αN

|ψ|²(x1, α1;. . .;xN, αN).

Then for everyρ∈H defineF_~^S, F_~^A: Π(ρ)→R⁺∪ {+∞} as follows:

F_~^S(P) =

(T_~(ψ) +Vee(ψ) ifP =Pψ for someψ∈ S,

+∞ otherwise,

and

F_~^A(P) =

(T_~(ψ) +V_ee(ψ) ifP =P_ψ for some ψ∈ A,

+∞ otherwise.

It follows that

(3.1) F_~^S(ρ) = min

Π(ρ)F_~^S(P), and

(3.2) F_~^A(ρ) = min

Π(ρ)

F_~^A(P).

Concerning the optimal transport problem we only need to incorporate the symmetry constraint in the transport functional. For every σ∈S^N permutation of {1, . . . , N}

and everyP ∈ P(R^{N d})we considerσ]P the image measure via σof the measureP, where with a little abuse of notations we have denoted by

σ(x1, . . . , xN) := (xσ(1), . . . , xσ(N)).

For everyP ∈Π(ρ)we can consider the measure Pe:= 1

N! X

σ∈S^N

σ]P.

We have thatPe∈Π(ρ)and, since the cost is also permutation invariant the transport cost ofPe is the same as the cost ofP. We say that P is symmetric if σ_]P =P for every σ∈S^N. Then, for example, the measurePe above is symmetric. Define, then Definition3.1. — C_S(P) =

(R c dP ifP is symmetric, +∞ otherwise.

We will omit theS and only useC(P)if symmetry is not required.

By the previous discussion

(3.3) C(ρ) = min

Π(ρ)C_S(P).

Then the common domain of minimization of F^S

~, F^A

~ and C_S is Π(ρ) which we consider embedded in the space of probability measures P equipped with the tight convergence.

Definition 3.2. — A generalized sequence of wave functions {ψ_~} converges to a transport planP ifP_ψ

~ * P. We will prove

Theorem3.3. — For every ρ∈H the functionalsF_~^S are mildly equicoercive and F_~^S −→^Γ C_S,

with respect to the tight convergence of measures.

Theorem3.4. — For everyρ∈H the functionalsF_~^Aare mildly equicoercive and for d= 2,3,4 andN = 2,3,

F_~^A−→^Γ C_S, with respect to the tight convergence of measures.

The proofs of Theorems 3.3 and 3.4 above coincide for a large part and will con- stitute the rest of this section.

3.1. Equicoerciveness

Lemma3.5. — Π(ρ)is compact with respect to the weak convergence.

Proof. — First we prove thatΠ(ρ) is tight. In fact, forε > 0 letK ⊆R^d compact such that

Z

K^c

ρ(x)dx6ε.

Observe that K^Nc

= K^c×R^d(N−1)

∪ R^d×K^c×R^d(N−2)

∪ · · · ∪ R^d(N−1)×K^c , and for everyP ∈Π(ρ)

Z

(K^N)^c

dP(X)6N ε

andK^N is compact. By Prokhorov’s Theorem we deduce thatΠ(ρ)is relatively com- pact. However, if P_n * P and P_n ∈ Π(ρ), for every function φ(x_j) ∈ C_b(R^{N d}) depending only on thej-th variable one has

Z

R^{N d}

φ(xj)dPn(X) = Z

R^d

φ(x)ρ(x)dx, which implies

Z

R^d

φ(x)ρ(x)dx= Z

R^{N d}

φ(x_j)dP(X) = Z

R^d

φ(x)dπ^j_#(P)(x).

Sinceφwas arbitrary,π^j_#(P)(x) =ρ(x), and henceP ∈Π(ρ).

3.2. Γ-lim inf inequality

Proposition3.6. — Let ρ∈H and letP a probability measure on R^{N d}. If{ψ_~}_~⊆ S(ρ)(orA(ρ)) is a generalized sequence such thatPψ_~* P. Then

(i) P is symmetric, (ii) lim inf_~→0F^S

~(P_ψ

~)>C_S(P).

Proof. — It is easy to see that invariance with respect to permutations is a closed con- dition in the space of probability measures. For the inequality:lim inf_~→0F_~^S(Pψ_~) = lim inf_~→0T_~(ψ_~) +Vee(ψ_~)>lim inf_~→0Vee(ψ_~)>CS(P).

3.3. An approximation procedure forP ∈Π(ρ)

Proposition3.7. — Let ρ∈H, andP ∈Π(ρ)be a transport plan such that

(3.4) P|D_α = 0

for someα >0. Then there exists a family of plans{Pε}ε>0 such that:

(i) for every ε > 0, Pε ∈ Π(ρ) and is absolutely continuous with respect to the Lebesgue measure, with density given byϕ²_ε(X), whereϕ_ε is a suitableH¹ function;

(ii) Pε* P asε→0;

(iii) lim sup_ε→0C(Pε)6C(P);

(iv) the kinetic energy ofϕε is explicitly controlled:

Z

|∇ϕ_ε(X)|² dX 6N k√

ρk²_H1+ K 4ε²

for a suitable constant K >0.

The proof is made of several steps. We start regularizing by convolution. The plans we obtain are regular but do not have the same marginals asP. We use the standard mollifiersη:R^d→Ras

η(u) =

(ke^{−1/(1−|u|2)} |u|<1 0 |u|>1,

where k=k(d)>0 is a suitable constant, which depends only on the dimensiond, such thatR

R^dη(u)du= 1. Set also

ηε(u) = 1

ε^dη(u/ε).

The next Lemma is well known and it is a useful tool to estimate someL² norms which will appear later.

Lemma3.8. — There exists a constant K=K(d)>0, depending only on the dimen- siond, such that

Z

B(0,ε)

|∇ηε(u)|²

ηε(u) du=K ε². Proof. — Point out that

∇η(u) =







−2ku

(1− |u|²)²e^{−1/(1−|u|2)} |u|<1

0 |u|>1

and

∇η_ε(u) = 1

ε^d+1∇η(u/ε).

Now Z

B(0,ε)

|∇ηε(u)|² η_ε(u) du=

Z

B(0,ε)

1 ε^d+2

|∇η(u/ε)|²

η(u/ε) du= 1 ε²

Z

B(0,1)

|∇η(v)|² η(v) dv

= 1 ε²

Z

B(0,1)

4k|v|²

(1− |v|²)⁴e^{−1/(1−|v|2)}dv=K(d)

ε² .

Now we define the function

Hε(Y −X) =

N

Y

i=1

ηε(yi−xi).

and use it as mollifier to regularize the transport planP defining Pe_ε(Y) =

Z

H_ε(Y −X)dP(X).

Note that the marginals of Peε(3)are different fromρ, but may be written explicitly:

ρε(y) = Z

R^(N−1)d

Peε(y, y2, . . . , yN)dy2· · ·dyN

= Z

R^(N−1)d

Z

ηε(y−x1)ηε(y2−x2)· · ·ηε(yN −xN)dP(X)dy2· · ·dyN

= Z

η_ε(y−x₁)dP(X) = Z

R^d

ρ(x)η_ε(y−x)dx= (ρ∗η_ε)(y).

Lemma3.9. — Let α be as in the statement of Proposition 3.7 and let Y ∈R^{N d} be such that |y_i−y_j|< α/2 for somei6=j, andε < α/4, then Pe_ε(Y) = 0.

Proof. — Note that

Peε(Y) = Z

Hε(Y −X)

N

Y

j=1

χ_B(yj,ε)(xj)dP(X).

IfY andεare as in the statement, andxi∈B(yi, ε), xj ∈B(yj, ε), then

|xi−xj|6|xi−yi|+|yi−yj|+|yj−xj|6α.

The thesis follows from (3.4).

Define now

ϕe_ε(Y) = q

Pe_ε(Y).

Lemma3.10. — For every ε >0,ϕeε∈H¹(R^{N d}).

(3)As usual we mean the marginals ofPeε(Y)dY.

Proof. — Fixε >0. ClearlyϕeεisL², since Z

ϕe²_ε(Y)dY = Z

Pe_ε(Y)dY = Z Z

H_ε(Y −X)dP(X)dY

= Z Z

H_ε(Y −X)dY dP(X) = Z

dP(X) = 1.

Now we estimate|∇ϕeε|². Using the Cauchy-Schwarz inequality, ∇Peε(Y)

6 Z

|∇Hε(Y −X)|dP(X)

6 s

Z |∇Hε(Y −X)|² H_ε(Y −X) dP(X)

s Z

H_ε(Y −X)dP(X)

= s

Z |∇Hε(Y −X)|² H_ε(Y −X) dP(X)

q Peε(Y),

where the first integral is extended to the set where the integrand is defined, namely

|xj−yj|< ε∀j. Therefore, with the same convention, Z

|∇ϕeε(Y)|²dY = 1 4

Z |∇Peε(Y)|²

Peε(Y) dY 61 4

Z Z |∇Hε(Y −X)|²

H_ε(Y −X) dP(X)dY

= 1 4

N

X

j=1

Z Z |∇jHε(Y −X)|²

H_ε(Y −X) dY dP(X)

6 1 4

N

X

j=1

Z Z |∇ηε(y_j−x_j)|²

ηε(yj−xj) dyjdP(X)

= N 4

Z |∇ηε(u)|²

η_ε(u) du= KN

4ε² .

In the next step we introduce a natural technique to get back the original marginals ρ without losing too much regularity. This technique is original and different from the one presented in [7]. We point out that, in a different context, this construction may well be generalized to a plan with different marginalsρ¹, . . . , ρ^N.

The construction fits in the general scheme of composition of transport plans as presented in [1].

Forx, y∈R^d define

γε(x, y) := ρ(x)ηε(y−x) ρε(y)

with the convention that it is zero ifρε(y) = 0. The two variables functionρ(x)ηε(y−x) is the key point of the construction, since it has the following property that links the different marginals:

Z

R^d

ρ(x)ηε(y−x)dx=ρε(y), Z

R^d

ρ(x)ηε(y−x)dy=ρ(x).

To simplify the notation we set mε(X, Y) =

N

Y

i=1

γε(xi, yi), m^j_ε(X, Y) =

N

Y

i=1i6=j

γε(xi, yi).

Note that

Z

R^d

γε(x, y)dx=χρ_ε>0(y).

Now set

Γε(X, Y) :=ϕe²_ε(Y)mε(X, Y) and observe that

Z

Γε(X, Y)dX =ϕe²_ε(Y)

N

Y

i=1

χρ_ε>0(yi) =ϕe²_ε(Y) =Peε(Y), where we used the following remark, which will be implicit from now on:

Remark3.11. — IfY is such thatρ_ε(y_i) = 0, then 0 =ρε(yi) =

Z

R^(N−1)d

ϕeε(y1, y2, . . . , yn)²dy1· · ·dyci· · ·dyN

and henceϕeε(Y) = 0.

Define

Pε(X) = Z

Γε(X, Y)dY and calculate the marginals ofPεto get

Z

R^(N−1)d

Pε(X)dx2· · ·dxN = Z

R^(N−1)d

Z

Γε(X, Y)dY dx2· · ·dxN

= Z

ϕe²_ε(Y)γ_ε(x₁, y₁)dY

= Z

R^d

ρε(y1)γε(x1, y1)dy1=ρ(x1)

and similarly also the otherN−1marginals are equal toρ.

Lemma3.12. — Let α be as in the statement of Proposition 3.4. If |xi−xj|< α/4 for somei6=j, andε < α/8, thenPε(X) = 0.

Proof. — FixX and εas in the statement, and supposemε(X, Y)>0. Then neces- sarily|yi−xi|< εand|yj−xj|< ε, so that

|yi−yj|6|yi−xi|+|xi−xj|+|xj−yj|6α/2.

From Lemma 3.9 it follows thatϕe²_ε(Y) = 0.

We now define the functionϕε(X) =p

Pε(X), and proceed to estimate its kinetic energy.

Estimate for the kinetic energy. — Calculate first the gradient with respect toxj ofPε:

∇jPε(X) =∇j

Z

ϕe²_ε(Y)m^j_ε(X, Y)ρ(xj)ηε(yj−xj) ρ_ε(y_j) dY

= Z

ϕe²_ε(Y)m^j_ε(X, Y)∇ρ(xj)ηε(yj−xj) ρε(yj) dY

− Z

ϕe²_ε(Y)m^j_ε(X, Y)ρ(xj)∇ηε(yj−xj) ρε(yj) dY

=A(X) +B(X).

We define for simplicity J(X) =

Z

ϕe²_ε(Y)m^j_ε(X, Y)ηε(yj−xj) ρ_ε(y_j) dY so that, for example,

Pε(X) =ρ(xj)J(X), A(X) =∇ρ(xj)J(X).

Now

∇jϕε(X) =∇j

pPε(X) = ∇jPε(X) 2p

P_ε(X) = 1 2p

P_ε(X)[A(X) +B(X)]

and we estimate the square of the L²norm of every term.

Z |A(X)|² P_ε(X) dX=

Z |∇ρ(xj)|²

ρ(x_j) J(X)dX

=

Z Z |∇ρ(xj)|²

ρ(x_j) ϕe²_ε(Y)m^j_ε(X, Y)ηε(yj−xj) ρ_ε(y_j) dY dX

= Z

R^d

Z |∇ρ(xj)|²

ρ(xj) ϕe²_ε(Y)ηε(yj−xj) ρε(yj) dY

dxj

= Z

R^d

Z

R^d

|∇ρ(xj)|²

ρ(xj) ηε(yj−xj)dyjdxj = Z

R^d

|∇ρ(xj)|²

ρ(xj) dxj64k√ ρk²_H1.

By Cauchy-Schwarz inequality,

|B(X)|²6ρ(xj)J(X)· Z

ϕeε(Y)²m^j_ε(X, Y)ρ(xj)|∇ηε(yj−xj)|² ρε(yj)ηε(yj−xj) dY.

(Here the integral is extended to the region whereηε(yj−xj)>0.) Therefore, with the same convention,

Z |B(X)|² P_ε(X) dX 6

Z Z

ϕe_ε(Y)²m^j_ε(X, Y)ρ(xj)|∇ηε(yj−xj)|² ρ_ε(y_j)η_ε(y_j−x_j) dY dX

= Z

R^d

Z

ϕeε(Y)²ρ(xj)|∇ηε(yj−xj)|² ρ_ε(y_j)η_ε(y_j−x_j) dY dxj

= Z

R^d

Z

R^d

ρ(x_j)|∇ηε(y_j−x_j)|²

ηε(yj−xj) dy_jdx_j= Z

R^d

|∇ηε(y)|²

ηε(y) dy= K ε².

Moreover,

Z A(X)·B(X)

4Pε(X) dX=−1 4

Z Z ∇ρ(xj)· ∇ηε(xj−yj)

ρε(yj) ϕe²_ε(Y)m^j_ε(X, Y)dY dX

=−1 4

Z Z

R^d×R^d

∇ρ(xj)· ∇ηε(xj−yj)dxjdyj

=−1 4

Z

R^d

∇ρ(x)dx

· Z

R^d

∇ηε(z)dz

and the second factor is equal to 0, as is easy to see integrating in spherical coordinates.

Thus Z

|∇jϕε(X)|² dX6 1 4

Z |A(X)|²+|B(X)|²

Pε(X) dX6k√

ρk²_H1+ K 4ε², and by summation overj

Z

|∇ϕ_ε(X)|² dX6N k√

ρk²_H1+ K 4ε²

.

Here we prove (ii). First we give the following lemma, which specifies that, for ε→0, the mass ofPeεis concentrated near the mass of P.

Lemma3.13. — SupposeR, δ >0 are such that Z

|X|>R

dP(X)6δ;

then, ifε√

N < R,

Z

|Y|>2R

Peε(Y)dY 6δ.

Proof Z

Y >2R

Peε(Y)dY = Z Z

|Y|>2R

Hε(Y −X)dP(X)dY

= Z Z

|Y|>2R∩|X|>R

H_ε(Y −X)dP(X)dY

6 Z Z

|X|>R

Hε(Y −X)dY dP(X) = Z

|X|>R

dP(X)6δ

since, where |X| 6R, one has |Y −X|>R > ε√

N, and hence there existsi such

that|xi−y_i|> ε.

Next, to prove thatP_ε* P, we interpolate byPe_εin between.

Lemma3.14. — Peε* P.

Proof. — Letφ∈Cb(R^{N d});

Z

φ(Y)dPeε(Y)− Z

φ(X)dP(X)

=

Z Z

[φ(Y)−φ(X)]Hε(Y −X)dP(X)dY 6

Z Z

|φ(Y)−φ(X)|Hε(Y −X)dP(X)dY.

Given δ > 0, let R be such that the hypothesis of Lemma 3.13 holds. We divide R^{N d}×R^{N d} in three disjoint regions:

E₁={|X|> R}, E₂={|X|6R,|Y|62R}, E₃={|X|6R,|Y|>2R}. As before, ifε√

N < R, onE3 one hasHε(X−Y)≡0.

Z Z

E1

|φ(Y)−φ(X)|Hε(Y −X)dP(X)dY 62kφk_∞ Z Z

E1

Hε(Y −X)dP(X)dY

62δkφk_∞.

On the other hand, E₂ is compact; take ε₀ such that |X−Y| 6 ε₀ implies

|φ(X)−φ(Y)|6δ. Ifε√

N 6ε0 we get Z Z

E2

|φ(Y)−φ(X)|H_ε(Y −X)dP(X)dY 6δ Z Z

E2

H_ε(Y −X)dP(X)dY

6δ Z Z

Hε(Y −X)dP(X)dY =δ.

Lemma3.15. — Pε* P.

Proof. — Letφ∈Cb(R^{N d}). Using the fact thatPeε * P (Lemma 3.14), it is left to estimate

Z

φ(X)P_ε(X)dX− Z

φ(Y)Pe_ε(Y)dY

=

Z Z

[φ(X)−φ(Y)]Γ_ε(X, Y)dX dY 6

Z Z

|φ(X)−φ(Y)|Γε(X, Y)dX dY.

As in the proof of Lemma 3.14, given δ >0 let Rbe such that the hypothesis of Lemma 3.13 holds. We divide R^{N d}×R^{N d}in three disjoint regions:

E₁={|Y|>2R}, E₂={|Y|62R,|X|63R}, E₃={|Y|62R,|X|>3R}. Ifε√

N < R, as before, onE3the integral is zero sinceΓε(X, Y)≡0there. Thanks to Lemma 3.13,

Z Z

E₁

|φ(X)−φ(Y)|Γε(X, Y)dX dY 62kφk∞

Z Z

E₁

Γε(X, Y)dX dY

= 2kφk_∞ Z

{|Y|>2R}

Peε(Y)dY 62δkφk_∞.

Exactly as before, using that E2 is compact and φ is absolutely continuous the

thesis follows.

It is left to prove (iii). The cost function c is not continuous neither bounded.

However, recall Lemma 3.12, it is bounded on the complement ofD_α/4. With this in mind, consider the functionv:R^2d→Rdefined as

v(x, y) =





 1

|x−y| if |x−y|>α/4, 4/α elsewhere.

and set

c(X) = X

16i<j6N

v(xi, xj).

Clearlyc(X)6c(X), and cis continuous (sum of continuous functions), bounded by ^N₂

4/α; moreover, thanks to the property (3.4) and Lemma 3.12, Z

c(X)dP(X) = Z

c(X)dP(X), Z

c(X)Pε(X)dX = Z

c(X)Pε(X)dX.

We can conclude the estimate as follows:

lim sup

ε→0

C(Pε) = lim sup

ε→0

Z

c(X)Pε(X)dX = Z

c(X)dP(X) = Z

c(X)dP(X).

Proposition 3.7 can be extended to all plans. Since we will make use of Theorem 2.7 we limit ourself to the case of symmetric transport plans. The same proof for non symmetric plans requires a version of Theorem 2.7 for plans with different marginals.

The extension is contained, for example, in [2].

Proposition3.16. — Let P ∈Π(ρ) a symmetric plan (not necessarily satisfying the property (3.4)). Then there exists a family of plans{P_ε}_ε>0 such that:

(i) for every ε > 0, P_ε ∈ Π(ρ) and is absolutely continuous with respect to the Lebesgue measure, with density given byϕ²_ε(X), whereϕ_ε is a suitableH¹ function;

(ii) Pε* P asε→0;

(iii) lim sup_ε→0CS(Pε)6CS(P);

(iv) the kinetic energy ofϕ_ε is explicitly controlled:

Z

|∇ϕε(X)|² dX 6N k√

ρk²_H1+ K 4ε²

for a suitable constant K >0.

Proof. — The proof of (i), (ii) and (iv) holds in general since it does not require (3.4).

And, in fact, carefully following the constructions in Propositions 3.7 one may observe that ifPis permutations invariant then the approximatingPεhave the same property.

Then we only need to prove (iii) and so if CS(P) = ∞ there is nothing to prove.

SupposeC_S(P) =K <∞. Letr >0be a parameter, and split P =Qr+P|D_r.

Letσr be the marginals ofQr, andρer those ofP|D_r; clearlyσr+ρer=ρ. Sinceρ∈ L¹∩L^d/(d−2)by Sobolev embedding, andρer6ρpointwise, we haveρer∈L¹∩L^d/(d−2)

Althoughρerneeds not to be a probability measure onR^d, we can suppose there exists λr>0 such that

Z

R^d

ρer(x)dx= 1 λr

<1,

otherwiseP|D_r = 0and we get the result directly by Proposition 3.7. Let nowPerbe a symmetric optimal transport plan inΠ(λ_rρe_r), and define

Pr=Qr+Pe_r λr

,

which lies inΠ(ρ). On the one hand we have the following Lemma3.17. — Pr* P.

Proof. — Recall thatC(P)is finite to get

K=C(P)>C(P|D_r)> 1 rP(Dr), and a fortiori forPe_r/λ_r due to the optimality. Hence

r→0limP(Dr) = lim

r→0

Pe_r(D_r) λr

= 0.

Takef ∈Cb(R^{N d}), and estimate

Z

f(X)dPr(X)− Z

f(X)dP(X)

= Z

f(X)dP|D_r(X)− 1 λr

Z

f(X)dPer(X) 6kfk∞

P(Dr) +Pe_r(D_r) λr

−→0 as r−→0.

On the other hand,

C_S(P_r) =C_S(Q_r) +C_S(Per)

λ_r 6C_S(Q_r) +C_S(P|_Dr) =C_S(P), thus

lim sup

r→0

CS(Pr)6CS(P).

Thanks to Proposition 2.5 C_S(Pe_r) is finite, and by Theorem 2.7 there exists α=α(r)>0such thatPe_ris supported outsideD_α.⁽⁴⁾Recall now Proposition 3.7 to findϕε,r weakly converging toPr asε→0, with

Z

|∇ϕ_ε,r(X)|² dX6N k√

ρk²_H1+ K 4ε²

and

lim sup

ε→0

C_S(|ϕε,r|²) =C_S(Pr).

It suffices now to take{ϕr,r}_r>0to conclude. In fact, givenδ >0, letR be such that Z

{|X|>R}

dP_r(X)6δ.

(4)Observe thatα(r)may be chose decreasing asr→0, as follows from Theorem 2.7.

Note thatRmay be chose independent fromr, since the marginals ofPrare all equal toρ, and we may chooseK⊆R^dcompact such that

Z

K

ρ(x)dx6 δ N, andR sufficiently large such thatK^N ⊆B(0, R)_RN d.

Now to prove weak convergence takeφ∈Cb(R^{N d})and proceed as in the previous paragraph to estimate

Z

φ(X)ϕ²_r,r(X)dX− Z

φ(X)dP_r(X) , using in addition Lemma 3.17 to estimate

Z

φ(X)dPr(X)− Z

φ(X)dP(X)

.

3.4. Constructing wave-functions. — A wave-function depends on N space-spin variables. In the previous subsections we worked mainly in R^{N d}, since we were con- sidering transport plans in Π(ρ). To introduce the spin we will separate the spin dependence as follows: for every s binary string of length N we consider the func- tionψ_s(x₁, . . . , x_N) =ψ(x₁, s₁, . . . , x_N, s_N), then we describe ψas a2^N-dimensional vector

ψ(z1, . . . , zN) = (ψs(X))_s∈S. As an example, ifN = 2we would have

ψ(z₁, z₂) =









ψ00(x1, x2) ψ01(x1, x2) ψ₁₀(x₁, x₂) ψ₁₁(x₁, x₂)







 ,

and forN = 3a wave-function would be represented as

ψ(z1, z2, z3) =



























ψ₀₀₀(x₁, x₂, x₃) ψ001(x1, x2, x3) ψ010(x1, x2, x3) ψ₀₁₁(x₁, x₂, x₃) ψ100(x1, x2, x3) ψ101(x1, x2, x3) ψ110(x1, x2, x3) ψ₁₁₁(x₁, x₂, x₃)

























 .

Note that now the density|ψ(X)|² is simply the square of the Euclidean norm of the vectorψ, and the same holds for∇ψ, once we set

∇ψ(z1, . . . , zN) = (∇ψs(X))_s∈S.

Let us take now a fermionic (i.e., antisymmetric) wave-function ψ, and consider a spin states= (s1, . . . , sN). Ifi < j are such thatsi =sj, considerσ= (i j)∈SN to

get

ψs(x1, . . . , xN) = sgn(σ)ψ_σ(s)(x_σ(1), . . . , x_σ(N))

=−ψs(x1, . . . , xj, . . . , xi, . . . , xN).

Hence we get the following

Remark3.18. — Ifψis fermionic andsis a spin state,ψ_sisseparatelyantisymmetric with respect to the spatial variables such thatsj = 0, and with respect to the spatial variables such thatsj = 1.

Consider now two spin statess ands⁰ with the same number of ones and zeroes.

Thenψsandψs⁰ are related: takingσ∈SN such that σ(s) =s⁰, we get ψs⁰(x_σ(1), . . . , x_σ(N)) = sgn(σ)ψs(x1, . . . , xN).

These observations will be used in the following.

3.5. Fermionic wave-functions with given density. — Suppose we have α > 0 and a symmetric functionψ(x₁, . . . , x_N)>0ofH¹ with the property that

(3.5) ψ(X) = 0 if|x_i−x_j|< α for somei6=j.

We wonder if there exists a fermionic wave-functionψsuch that X

s∈S

|ψ(x1, s1, . . . , xN, sN)|²=ψ²(X),

and

kψkH¹ 6CkψkH¹

for a suitable constantC. In fact, we managed to prove the following

Proposition3.19. — ForN = 2,3,d= 3,4, givenψ∈H¹ symmetric, withψ|D_α = 0 for someα >0, there existsψ fermionic such that

X

s∈S

|ψ_s(X)|²=ψ²(X) and X

s∈S

|∇ψ_s(X)|²6|∇ψ(X)|²+ C

α²ψ²(X).

In [7], the followingψ is given as a wave-function:

ψ₀₀= 0, ψ₀₁=ψ(x1, x2), ψ₁₀=−ψ(x1, x2), ψ₁₁= 0,

which is in fact fermionic with bounded kinetic energy. Note, however, that this con- struction cannot work for a larger number of particles. Indeed, if a binary stringshas lengthN >3, then there are at least two ones, or two zeros, on placesi6=j – thus the corresponding function ψ_s must change sign for a suitable flip of the variables (namely, x_i 7→ x_j, x_j 7→ x_i). We will exhibit a wave-function ψ (with square den- sityψ²) such thatψ₁₀=ψ₀₁= 0. This forces ψ₀₀andψ₁₁ to be different from zero and antisymmetric. We remark also that, for this kind of construction, the condition (3.5) is “morally necessary”.

3.6. Construction forN = 2,d= 3. — The variableX will be expanded asX = (x, y) = (x1, x2, x3;y1, y2, y3). Setr=α/√

3, and takeξ=−¹₂+i

√3

2 a primitive cubic root of 1. The key point is to choose two auxiliary C^∞ functions a, b: R→R such that

(i) a²+b²= 1;

(ii) b is symmetric,b(t) = 0 if|t|>r;

(iii) ais antisymmetric, a(t) =−1 ift6−r, a(t) = 1ift>r;

(iv) |a⁰|,|b⁰|6k/r.

Note that the constantk >1may be chosen arbitrarily close to 1. To shorten the notation we seta_j =a(x_j−y_j),b_j =b(x_j−y_j)forj= 1,2,3. Now define

g₁(x, y) = 1

√3(a₁+b₁a₂+b₁b₂a₃), g_ξ(x, y) =

√2

√3(a₁+ξb₁a₂+ξ²b₁b₂a₃).

By direct computation one sees that|g1|²+|gξ|²=a²₁+b²₁a²₂+b²₁b²₂a²₃, since clearly a1+ξb1a2+ξ²b1b2a3

=

a1+ξb1a2+ξ²b1b2a3

. Next we define the wave-function ψ₀₀(x, y) =g1(x, y)ψ(x, y), ψ₀₁(x, y) = 0, ψ₁₀(x, y) = 0, ψ₁₁(x, y) =gξ(x, y)ψ(x, y).

The following equality is crucial in the construction (3.6) ψ²(x, y)b²₁b²₂a²₃=ψ²(x, y)b²₁b²₂.

This holds because where |a3|² = 1, i.e., where |x3−y3| >r, the equality holds. It also holds where b1 = 0 or b2 = 0, i.e., where |x1−y1| > r or |x2−y2| > r. The region where|x_j−y_j|6rfor every j= 1,2,3is left, but there it holds

|x−y|= q

|x₁−y₁|²+|x₂−y₂|²+|x₃−y₃|²6r√ 3 =α, and hence the equality holds becauseψ²(x, y) = 0.

Now one can compute

|ψ₀₀|²+|ψ₁₁|²=

|g1(x, y)|²+|gξ(x, y)|²

ψ²(x, y)

= a²₁+b²₁a²₂+b²₁b²₂a²₃

ψ²(x, y)

= a²₁+b²₁a²₂+b²₁b²₂

ψ²(x, y)

= a²₁+b²₁

ψ²(x, y) =ψ²(x, y).

Next come the estimates for the derivatives. Since

∇ψ₀₀(x, y) =ψ(x, y)∇g1(x, y) +g₁(x, y)∇ψ(x, y)

∇ψ₁₁(x, y) =ψ(x, y)∇gξ(x, y) +gξ(x, y)∇ψ(x, y), it follows

|∇ψ(x, y)|²=|∇ψ(x, y)|²+

|∇g1(x, y)|²+|∇gξ(x, y)|²

ψ²(x, y)

+ψ(x, y)∇ψ(x, y)·v(x, y),