Long–time dynamics of the

Schrodinger–Poisson–Slater system

Oscar Sanchez∗& Juan Soler †

March 11, 2004

Abstract

In this paper we analyze the asymptotic behaviour of solutions tothe Schrodinger–Poisson–Slater (SPS) system in the frame of semi-conductor modeling. Depending on the potential energy and on thephysical constants associated with the model, the repulsive SPS sys-tem develops stationary or periodic solutions. These solutions pre-serve the Lp(R3) norm or exhibit dispersion properties. In compari-son with the Schrodinger–Poisson (SP) system, only the last kind ofsolutions appear.

Keywords: Open quantum system, Schrodinger-Poisson system,dispersion, Xα-approach, asymptotic behaviour, stationary solutions.

1 Introduction

The aim of this paper is to analyze the asymptotic behaviour of solutionsto the Schrodinger–Poisson–Slater (SPS) system.

The SPS system can be written in terms of the wave function ψ : R3 ×[0, T [→ C and the charge density n(x, t) = |ψ(x, t)|2 as follows:

i~∂ψ

∂t= − ~2

2m∆xψ + V ψ − CSn

13ψ, lim|x|→∞ ψ(x, t) = 0, (1)

ψ(x, t = 0) = φ(x), (2)∆xV = −ε n, lim|x|→∞ V (x, t) = 0, (3)

∗Departamento de Matematica Aplicada, Facultad de Ciencias, Universidad deGranada, 18071 Granada, SPAIN. ossanche@ugr.es

†Departamento de Matematica Aplicada, Facultad de Ciencias, Universidad deGranada, 18071 Granada, SPAIN. jsoler@ugr.es

1

where ε = 1 (repulsive case) or ε = −1 (attractive case) depending onthe type of interaction considered. Here, ~ and m respectively hold forthe Planck constant and the particle mass. Also, CS stands for the Slaterconstant. The system coincides with the Schrodinger–Poisson (SP) systemwhen the contribution of the last term (the Slater term) is not considered.As usual, equations (1)–(3) are understood in a weak sense and (3) de-termines the self–consistent potential V originated by the charge of theparticles. This potential can be equivalently written in integral form as

V (x, t) =ε

4π

∫R3

|ψ(x′, t)|2

|x− x′|dx′.

The physical constants (~,m) that are usually involved in the formulationof the Schrodinger equation can be normalized to unity for the sake ofsimplicity. However, this normalization also modifies the Slater constant,whose value is relevant for the subsequent analysis.

The SPS system describes the evolution of an electron ensemble in asemiconductor crystal. The mixed-state SP system is commonly used tomodel semiconductor devices (see [23]). However, the repulsive effect ofthe Coulomb potential seems to be too strong when we compare the be-haviour of the solutions to the SP system to simulations of superlatticestructures (see [25, 27]). These phenomena are also observed in the con-text of attractive Coulomb potential. Some different approximations havebeen studied to overcome this problem, obtaining appropriate adaptationsof the Poisson potential. The Hartree–Fock model has been used to ana-lyze a wide variety of phenomena in Quantum-Chemistry and Solid StatePhysics (see [1, 16, 22]). The time–dependent Hartree–Fock equation hasbeen analyzed in [14] and [9]. One of the most interesting corrections tothe Poisson potential in the SP system is found by deriving nonlinear |ψ|αterms from the Fock potential via various limits, in particular the low den-sity limit [3], which gives α = 2/3. This kind of |ψ|α approximations tothe Fock term is usually called the Xα-approach. Another motivation forthis approximation in Quantum–Chemistry is the enormous quantity ofcalculations necessary to evaluate the Fock term, usually of order N4, Nbeing the number of particles. In this direction, the Xα approach to theFock correction (Dirac, Slater, . . . ) has been proved relevant in differentcontexts. These local approximations to nonlocal interaction terms giveexcellent results when studying stationary states, for example in Quan-tum Chemistry (see [7] and [17, 20, 4] for some derivations and analysis ofthese systems). Then, the calculations are reduced from N4 to N3, eventhere might be some place for improvements. However, there is no rigorousfoundation of the Xα-model in the time-dependent case. In this direction,following the classical ideas of the thermodynamical limit in Statistical Me-chanics (see [8]) some recent advances are being done from the continuum

2

and mean–field limit of the N–quantum–particle system by C. Bardos etal., see [2].

The aim of this work is to analyze the qualitative differences between theSchrodinger–Poisson–Slater (α = 2

3 ) evolution system and the Schrodinger–Poisson and Hartree–Fock systems. We are mainly interested in the Xα

case studied in semiconductor theory, that is α = 23 , which is derived

from the Fock term by means of a low density limit, see [3]. This |ψ|2/3ψcorrection is also known as the Dirac exchange term. Another interestingapproach comes from the limit of heavy atoms, i.e. the high charge ofnuclei limit; this leads to the Thomas-Fermi correction (α = 4/3) of thekinetic energy, see [17, 18]. In this paper we do not approximate the kineticenergy term (which is also called the von Weizsacker correction). However,the Thomas–Fermi term can be alternatively seen as a correction of theFock interaction, that always appears as a repulsive potential, see [18]. Aswe will point out later, these otherXα–approaches, useful in many scientificcontexts, can be treated in our mathematical framework.

One important feature of the SPS system is that its associated potentialenergy can reach negative values depending on the constants of the system(mass, initial energy or Slater constant). This fact implies some relevantproperties of the SPS system in the repulsive case: 1) the minimum of thetotal energy operator is negative for some choices of the physical constants;2) there are solutions (depending on the initial energy) that do not havedispersive character; 3) there are steady-state solutions, i.e. solutions withconstant density; 4) there are solutions, even with positive energy, whichpreserve the Lp norm and do not decay with the time evolution. Theseproperties show important qualitative differences between the SPS systemand the SP and Hartree–Fock systems, see [12, 6, 9, 14]. On the otherhand, the Xα–Slater–model appears as an appropriate correction to theself–consistent Coulomb potential in semiconductor heterostructures mo-deling, in the sense that it covers different phenomenologies observed in thiscontext. Some of our results hold true under the hypothesis of a relationbetween the value of the Slater constant, the mass and the energy of thesystem. However, the Slater constant is a characteristic of the componentmetals in the semiconductor device as it was pointed out in [11] wheninterpreting the exchange–correlation potential of Kohn–Sham type. Inthis way, our study covers the whole range of variation for these constants,and the relation between these constants appears in a natural way and is nota restriction from a physical point of view. As we have commented before,the main differences with respect to the SP system occur in the repulsivecase, where non dispersive effects, stationary and periodic solutions appear.However, the attractive case is also of interest in applications related toquantum gravity in the limit of very heavy particles (see for example [24]),thus we analyze both cases. We focus our study in the the single–state

3

case.In [3], the mixed–state case for the SPS system has been dealt with.

In particular, the well-posedness and regularity of local–in–time and globalsolutions was analyzed, with L2 or H1 initial data. Also, the basic conser-vation laws and the minimal energy solutions in the attractive case werederived under a variational framework. A different approach for the single-state case can be seen in [5].

Most of these results are valid for other Xα–approaches. However,motivated by the applications in semiconductor theory, we focus our effortsin the Slater approach to the Fock term. We will comment along the paperon some extensions of the results to other Xα-terms or some combinationof them.

Let us summarize the main results and the techniques used in the paperin comparison with previous results. Section 2 is devoted to the minimiza-tion of the energy functional in the repulsive case. This allows us to deducethe existence of stationary solutions with negative energy as well as optimalbounds for the kinetic energy. To deal with this nonconvex minimizationproblem we can use different techniques introduced in [17, 19, 21]. Thisproblem was treated in [3] (in the attractive case) by using symmetric de-creasing rearrangement inequalities, but this tool seems to be fruitless inthe repulsive case. Some minimization problems related to the repulsivecase are studied in references [17] and in [22, 4] for small enough valuesof the mass upon using a perturbative argument. Alternatively, we pro-pose here a scaling argument which provides effective bounds on the mass.In the third section we analize the long time behaviour of SPS solutions.The balance between the Coulombian potential and the Slater correctionmakes the usual arguments (based on the pseudo–conformal law) power-less. In our analysis we combine this property, or the equivalent dispersionequation, with the Galilean invariance in order to conclude some Lp(R3)estimates. Also, from the dispersion equation (which relates the total en-ergy to the momentum and position dispersions) it can be deduced that thesolution is expansive in the sense that its second order moment increaseswith time. Finally, in Section 4 we analyze the asymptotic behaviour ofthe SPS solutions under attractive Coulomb forces. Actually, we prove theexistence of stationary solutions in the case of negative energy.

4

2 Minimum of the energy in the repulsivecase: stationary solutions, solutions pre-serving the Lp norm and evolution of thekinetic energy

The Slater term introduces some qualitative differences in the behaviourof the solutions to the SPS system when compared to solutions to theSP system. While the SP energy in the repulsive case is always positive,this can be negative when the Slater contribution is considered. The totalenergy operator associated with the solutions to the SPS system has thefollowing form:

E[ψ]=∫

R3

|∇ψ(x, t)|2

2+∫

R3

ε|ψ(x, t)|2|ψ(x′, t)|2

8π|x− x′|dx′− 3CS

4|ψ(x, t)| 83

dx. (4)

E is an invariant of motion (i.e., E is preserved along the time evolution)provided that ψ is such that E[ψ] is well–defined (see [3]). We refer tothe first term in the right–hand side of (4) as the kinetic energy EKIN (ψ),while the sum of the last two terms is the potential energy EPOT (ψ). In(4), the expression of the Coulombian potential has been identified as

12

∫R3V (x)n(x) dx =

ε

2

∫R3|∇V (x)|2 dx.

However, in the repulsive case ε = 1, we can prove that the potential energyis always negative for some choice of the Slater constant in terms of themass of the system. The following result corroborates this feature.

Lemma 2.1 If the L2(R3) norm of the initial data φ associated with theSPS system verifies

‖φ‖L2(R3) ≤(

3CS2C

) 34

,

where CS is the Slater constant and C is a positive constant determinedby

1C

= Inf

‖ψ(·, t)‖

83

L83 (R3)

‖∇V (ψ)(·, t)‖2L2(R3)

; ψ ∈ L 83 (R3) ∩ L2(R3), ‖ψ(·, t)‖L2 = 1

,

then the potential energy of the solutions is negative along the time evolu-tion.

5

Proof. This result is based on the following inequality, valid for all ψ ∈L2(R3) ∩ L 8

3 (R3):

‖∇V (ψ)(·, t)‖2L2(R3) ≤ C‖ψ(·, t)‖

43L2(R3)‖ψ(·, t)‖

83

L83 (R3)

. (5)

This inequality is a direct consequence of Holder and Hardy-Littlewood-Sobolev inequalities as well as of the interpolation inequalities for Lp spaces.

Applying (5) to solutions ψ of the SPS system and using that the L2–norm of the initial data is preserved along the time evolution, we concludethe proof by writing

EPOT (ψ)(t) ≤(C

2‖ψ(·, t)‖

43L2(R3) −

34CS

)‖ψ(·, t)‖

83

L83 (R3)

.

2

Remark. The inequality (5) as well as an upper bound for the sharpconstant C were found by Lieb and Oxford in [16]. In our context, thisbound takes the value C = 1.092

2π = 0.1737.

Furthermore, since the potential energy associated with the initial datacan be negative in the repulsive case we can find initial conditions for whichthe total energy is also negative, as proved in the following

Proposition 2.2 There exist initial data φ ∈ H1(R3) for which the totalenergy in the repulsive case is negative.

Proof. Let ψ ∈ H1(R3) such that the associated potential energy is nega-tive (this may happen by virtue of Lemma 2.1). Then, there is σ > 0 smallenough such that the total energy of ψσ(x) = σ

32ψ(σx),

E[ψσ] = σ2EKIN (ψ) + σEPOT (ψ), (6)

is nonpositive. Then, by choosing φ = ψσ as initial condition, the energyassociated with this problem is nonpositive. 2

Remark. The same thing occurs when other Xα terms are considered.The total energy functional also reaches negative values when couplings ofthe Coulombian potential with power nonlinearities |ψ|αψ α ∈ (0, 4/3] areconsidered. Combinations of some of these terms could be also possible intheir attractive or repulsive versions. However, some other kind of prob-lems appear in the minimization argument, as we will mention in the nextsubsection.

Proposition 2.2 allows to remark some important differences betweenthe asymptotic behaviour of solutions to our system and those to the SP

6

system. For the repulsive SP system it was proved (see [6],[12]) that theLp norms of the solutions tend asymptotically in time to zero for p ∈]2, 6].However, when we analyze the evolution of solutions to the repulsive SPSsystem whose initial data has negative energy, we observe that the L

83

norm of the wave function ψ cannot go to zero as t→∞. This is becausethe total energy of the system is preserved and the Slater term is the onlynonpositive contribution to the total energy.

One of the relevant points in the analysis of this problem is the existenceof a global minimum of the energy functional in H1(R3) under the constraint‖ψ‖L2(R3) = M . This problem has no solution for the repulsive SP systembecause the infimum of the energy is always 0, which is not a minimumexcept for the case M = 0. In the following section we prove the existenceof such a minimum for the SPS problem, for solutions with sufficientlysmall L2(R3) norm.

2.1 Minimum of energy in the repulsive case

In this section we study the following minimization problem associatedwith the total energy of the repulsive SPS system

IM = infE[ψ]; ψ ∈ H1(R3), ‖ψ‖L2(R3) = M

, (7)

where M > 0 and E[ψ] is defined by (4). The main result of this sec-tion claims that this functional reaches a minimum value, which allows usto deduce two interesting consequences. The first one is the existence ofstationary profiles, which are periodic-in-time solutions to the SPS systempreserving the density. We also note that this kind of solutions does notexist for the repulsive SP system. The second consequence is the derivationof optimal bounds for the kinetic energy of solutions for which the totalenergy is well-defined.

Let us prove the results that ensure the existence of a minimum of (7).Firstly we observe that the energy operator is bounded from below in termsof the problem (7). From the Gagliardo–Nirenberg inequality we get

‖ψ‖83

L83 (R3)

γ83 ‖ψ‖

53L2(R3)

≤ ‖∇ψ‖L2(R3), (8)

which holds for all ψ(·, t) ∈ H1(R3). Using (8) and the fact that in thiscase the Coulombian potential term is nonnegative we obtain

E[ψ] ≥

(∫R3 |ψ(x)| 83 dxγ

83M

53

)2

− 32CS

∫R3|ψ(x)| 83 dx. (9)

7

The right-hand side of (9) can be seen as a second order polynomial ax2+bxin∫

R3 |ψ|83 dx, where

a =(γ

83M

53

)−2

and b = −32CS .

Thus, we immediately conclude that the total energy is bounded from be-low. Furthermore, we can deduce the boundedness in H1(R3) of any mini-mizing sequence, which plays an important role in our argument to obtainthe minimum of the functional.

The technical difficulties arising in this nonconvex minimization prob-lem come from the invariance of the total energy functional by the non-compact group of translations. The possible loss of compactness due tothat invariance has to be detected by the techniques used in the proofs. Inthis way, two methods are proposed in the previous literature to analizethe class of problems (7): the concentration–compactness method [21] andthe method of the nonzero weak convergence after translations [19]. Infact, we can prove that every minimizing sequence is in essence relativelycompact provided that a certain sub-additivity property is strict. Thiscondition implies that a minimizing sequence is concentrated in a boundeddomain. Recently, this lack of compactness has been analyzed in [10] forthe Sobolev embedding. Since an important part of the intermediate stepsare common to both (concentration–compactness and nonzero weak con-vergence after translations in Sobolev sapces) techniques, we will commentthe application of them.

2.1.1 Concentration–compactnes argument

We can use the following formulation of the concentration–compactnessprinciple adapted to our situation.

Proposition 2.3 For every M > 0, the following inequality

IM ≤ Iα + IM−α , ∀α ∈ (0,M), (10)

holds. Furthermore, every minimizing sequence of (7) is relatively compactin H1(R3) (up to a translation) if and only if

IM < Iα + IM−α , ∀α ∈ (0,M). (11)

Proof. The proof is a consequence of Lemma III.1 and Lemma I.1 in [21].In order to make the paper self-consistent, we adapt these results to ournotation. The general framework for minimization problems proposed by

8

P. L. Lions allows us to establish the condition (10). Consider a minimiz-ing sequence un of (7). Since this sequence is bounded in H1(R3) with||un||2L2(R3) = M , then there exists a subsequence nk ∈ N for which eithercompactness or vanishing or dichotomy occurs (Lemma III.1 [21]). In or-der to prove compactness let us prove that vanishing and dichotomy cannotoccur. The strict sub–additivity condition (11) prevents the subsequencefrom dichotomy. This property is stated as follows: there exists α ∈]0,M [such that for every ε > 0, there exist k0 ≥ 1 and u1

k, u2k bounded in H1(R3)

satisfying‖unk

− (u1k + u2

k)‖Lp(R3) ≤ δp(ε) → 0, ε→ 0+ , 2 ≤ p < 6;∣∣∫R3 |u1

k|2 dx− α∣∣ ≤ ε,

∣∣∫R3 |u2

k|2 dx− (M − α)∣∣ ≤ ε;

dist(Suppu1k,Suppu2

k) →∞, k →∞;lim infk

∫R3|∇unk

|2 − |∇u1k|2 − |∇u2

k|2 dx ≥ 0;

for k ≥ k0. Indeed, if dichotomy occurs we easily deduce that

IM ≥ Iα + IM−α,

which yields a contradiction. On the other hand, if strict sub–additivitydoes not occur, then a minimizing sequence can be constructed withoutconvergent subsequences (see [21] for details). Vanishing occurs when

limk→∞

supy∈R3

∫y+BR

|unk(x)|2 dx = 0, ∀R <∞, BR = x ∈ R3, |x| < R.

It can be proved that the subsequence does not vanish as follows from thefact that IM < 0 and from the following result (Lemma I.1 [21] with q = 2,p = 2 and α = 8/3):

Lemma 2.4 Let 1 < p ≤ ∞ and 1 ≤ q < ∞ with q 6= 3p3−p if p < 3.

Assume that un is bounded in Lq(R3), ∇un is bounded in Lp(R3) and

supy∈R3

∫y+BR

|un|q dx→n

0 for some R > 0 .

Then, un →n

0 in Lα(R3) for α ∈ [q, 3p3−p ].

Hence, we have proved that any minimizing sequence satisfies the followingcompactness criterium: there exists yk ∈ R3 such that |unk

(·+yk)|2 is tight

∀ε > 0, ∃R <∞,

∫yk+BR

|unk(x)|2 ≥M − ε .

9

Setting un = un(·+yn), we can assume (up to a subsequence) that un → uweakly in H1(R3) and the compactness property implies∫

BR

|u|2 dx ≥M − ε.

Thus, un converges strongly in L2(R3) to u. By using the Gagliardo–Nirenberg inequality, un converges strongly to u in Lp(R3) for 2 ≤ p < 6.This fact allows to assure that u is a minimum of the problem IM asconsequence of the weak lower semi–continuity of the H1(R3) norm andthe convergence EPOT (un) → EPOT (u). Thus, a posteriori we deduce∫

R3|∇un|2 dx→

n

∫R3|∇u|2 dx,

showing the compactness in H1(R3). 2

To obtain the relative compactness of any minizing sequence (up totranslations) has been used the concentration–compactness argument thatcan be equivalently replaced by the arguments based on nonzero weakconvergence after translations [19]. The point which is common to bothtechniques is that hypothesis (11) is required.

2.1.2 Nonzero weak convergence after translations

In this approach, we apply the next two results to any minimizing sequenceunn∈N of (7) to guarantee the existence of a nonzero weak convergent sub-sequence in H1(R3) up to translations (see [19], Theorem 8.10 and exercise2.22, for more details):

Lemma 2.5 (Exercise 2.22 [19]) Suppose that 1 ≤ p < q < r ≤ ∞and that u is a function in Lp(Ω) ∩ Lr(Ω) with ‖u‖Lp(Ω) ≤ Cp < ∞,‖u‖Lr(Ω) ≤ Cr < ∞, and ‖u‖Lq(Ω) ≥ Cq > 0. Then, there are constantsε > 0 and M > 0, depending only on p, q, r, Cp, Cq, Cr, such thatMeas(x : |u(x)| > ε) > M .

Theorem 2.6 (Theorem 8.10 [19]) Let 1 < p <∞ and let unn∈N bea bounded sequence of functions in H1(R3). Suposse that for some ε > 0the set En := x : |un(x)| > ε satisfies Meas(En) > δ > 0 for some δand all n ∈ N. Then, there is a sequence of vectors yn ∈ R3 such that thetranslated sequence un(x) := u(x + yn) has a subsequence that convergesweakly in H1(R3) to a nonzero function.

Any function un verifies the hypothesis of Lemma 2.5 with p = 2,q = 8/3, p = 6, Cp = M and Cq = (−4IM/3CS)

83 , Cr being a constant

10

which comes from the boundedness of unn∈N in H1(R3). Then, the wholesequence satisfies the hypotheses of Theorem 2.6. As consequence, thereexist vectors yn such that a subsequence of un verifies

un := un(·+ yj) → u weakly in H1(R3), ‖u‖H1(R3) > 0 . (12)

In order to deduce that u is a minimizer of (7) we have to prove that

EPOT (u) ≤ lim inf EPOT (un) .

To this aim, it is enough to observe that no charge escapes to infinity,i.e. ‖u‖L2(R3) = M , because this would imply the convergence un → u inL2(R3) and EPOT (un) → EPOT (u). The inequality (11) plays a crucialrole at this point. If ‖u‖L2(R3) = α < M , then it can be proved that unis under the dichotomy hypothesis. Indeed, there exists R > 0 such that∫BR

|u|2 dx = α − ε/2, for ε > 0. On the other hand, let Rn be suchthat

∫BRn

|un|2 dx = α + ε/2. The sequence Rnn∈N → ∞ as n → ∞(otherwise, this would contradict (12)). We define u1

n := unχBRand u2

n :=unχR3−BRn

, where n ∈ N and χΩ denotes the characteristic function of theset Ω. Then, we have that un verifies

‖un − (u1n + u2

n)‖Lp(R3) ≤ δp(ε) → 0, ε→ 0+ , 2 ≤ p < 6 ;∣∣∫R3 |u1

n|2 dx− α∣∣ ≤ ε,

∣∣∫R3 |u2

n|2 dx− (M − α)∣∣ ≤ ε;

dist(Supp u1n,Supp u2

n) = Rn −R→∞, n→∞;lim infk

∫R3|∇un|2 − |∇u1

n|2 − |∇u2n|2 dx ≥ 0;

for n ≥ n0. The incompatibility between dichotomy and (11) allows toconclude that ‖u‖L2(R3) = M as well as the minimizing charater of u. Thisconcludes the proof with the technique of nonzero weak convergence aftertranslations in Sobolev spaces.

Before deriving the inequality (11) in the SPS context, let us introducesome notations. Let a, b, c be positive constants and consider the operatorsTKIN , TPOT , T, K : H1(R3) −→ R defined by

TKIN (ψ) = a

∫R3|∇ψ(x)|2 dx ,

TPOT (ψ) =∫

R3

b

∫R3

|ψ(x)|2|ψ(x′)|2

|x− x′|dx′ − c |ψ(x)| 83

dx ,

T (ψ) = TKIN (ψ) + TPOT (ψ) , K(ψ) = −14

(TPOT (ψ))2

TKIN (ψ).

Then, we have the following

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Lemma 2.7 The minimization problems associated with the operators Tand K over the set

BM = ψ ∈ H1(R3); ‖ψ‖L2(R3) = M, TPOT (ψ) < 0

are equivalent in the following sense

TM ≡ infT [ψ] ;ψ ∈ BM = infK[ψ] ;ψ ∈ BM ≡ KM .

In addition, if ψ is a function in which T achieves its minimum, then itis also the minimum for the functional K. On the other hand, if ψ isa function in which K achieves its minimum, then the function ψσ is aminimum for T , where ψσ(x) = σ

23ψ(σx) and σ = −TP OT (ψ)

2TKIN (ψ) .

Proof. We first remark that the scaling ψσ(x) = σ32ψ(σx), σ > 0, pre-

serves the properties of BM . Then, the result can be easily deduced byoptimizing for every ψ ∈ BM the value of the parameter σ for which thetotal energy reaches the minimum over the uniparametric family of func-tions ψσ ; σ ∈ R+. 2

As a particular case, we obtain a minimization problem equivalent to(7). Denoting AM =

ψ ∈ H1(R3) ; ‖ψ‖L2(R3) = M, EPOT [ψ] < 0

, we

have

IM = infE[ψ] ; ψ ∈ H1(R3), ‖ψ‖L2(R3) = M

= inf E[ψ]; ψ ∈ AM ,

which shows that our problem is equivalent to

inf−1

4(EPOT (ψ))2

EKIN (ψ); ψ ∈ AM

. (13)

Also, note that the set AM is nonempty for any value of M , see [22, 4].Furthermore we have −EPOT (ψM ) = 2EKIN (ψM ), where ψM denotes

the minimizer ofIM . Consequently,

E(ψM ) =12EPOT (ψM ) = −EKIN (ψM ). (14)

Using Lemma 2.7 we can prove the following result, which provides thestrict sub–additivity property (11).

Proposition 2.8 For all CS > 0 and M > 0 such that

M <

(7CS10C

) 34

, (15)

the sub–additivity condition (11) holds. Here, CS denotes the Slater con-stant and C is the sharp constant in (5).

12

Proof. Assume that M satisfies (15). Using the scaling

ψ(x) −→M4ψ(M2x),

the set AM can be seen, for each M ∈ R+, as a transformation of the set

B′M := ψ ∈ H1(R3) ; ‖ψ‖L2(R3) = 1, EMPOT (ψ) < 0,

where

EMPOT (ψ) =M6

8π

∫R3

∫R3

|ψ(x)|2|ψ(y)|2

|x− y|dxdy − 3CSM

143

4

∫R3|ψ(x)| 83 dx .

In the same way, (7) can be rewritten as

IM = infEMKIN (ψ) + EMPOT (ψ) ; ψ ∈ B′M

,

where EMKIN (ψ) = M6

2

∫R3 |∇ψ(x)|2 dx. Since EMPOT (ψ) < 0 by (15) and the

proof of Lemma 2.1, we can take B′M = ψ ∈ H1(R3) ; ‖ψ‖L2(R3) = 1.Under this assumption, our minimization problem reads

IM= M143 −p inf

M 43+p

2

∫R3|∇ψ(x)|2 dx+

M43+p

8π

∫R3

∫R3

|ψ(x)|2|ψ(y)|2

|x− y|dxdy

−3CSMp

4

∫R3|ψ(x)| 83 dx ; ψ ∈ H1(R3), ‖ψ‖L2(R3) = 1

,

where p is a positive parameter to be precised. Then, we can apply Lemma2.7 to show that this problem is equivalent to

IM=M143 −p inf

−

(M

23+ p

2∫

R3 |∇V (ψ)|2dxdy − 3CS

2 Mp2−

23∫

R3 |ψ(x)| 83 dx)2

8∫

R3 |∇ψ(x)|2dx

; ψ ∈ H1(R3), ‖ψ‖L2(R3) = 1def= M

143 −pIM1 .

Now (11) can be written as

M143 −p IM1 < α

143 −p Iα1 + (M − α)

143 −p IM−α

1 , ∀α ∈ (0,M). (16)

This inequality is based on the bound

Mk > αk + (M − α)k, ∀α ∈ (0,M), M ∈ R+, ∀k > 1 .

We easily deduce

M143 −p IM1 < α

143 −p IM1 + (M − α)

143 −p IM1 , ∀α ∈ (0,M),

13

for some p ∈ ( 43 ,

113 ). To get (16) it is enough to show that for all η ∈ (0,M),

IM1 ≤ Iη1 holds. This is true according to the nonincreasing character ofthe function

fψ : η −→ −18

(η

23+ p

2∫

R3 |∇V (ψ)|2 dx− 3CS

2 ηp2−

23∫

R3 |ψ|83 dx

)2∫R3 |∇ψ|2 dx

,

for η ∈ (0,M) and p ∈ ( 43 ,

113 ), independently of ψ. Indeed, given M there

exists p ∈ ( 43 ,

113 ) such that

dfψdη

= −14

1∫R3 |∇ψ|2 dx

(η

23+ p

2

∫R3|∇V (ψ)|2 dx− 3CS

2η

p2−

23

∫R3|ψ| 83 dx

)×((

23

+p

2

)η

p2−

13

∫R3|∇V (ψ)|2 dxdy −

(p

2− 2

3

)3CS2η

p2−

53

∫R3|ψ| 83 dx

)is nonpositive for every ψ ∈ BM , where we have used (5). The optimalbound is obtained as p approaches 11

3 . Finally, this allows to establish theinequality IM1 ≤ Iη1 , which concludes the proof. 2

Remark. It is not clear to the authors if the constant in (15) is or notoptimal. Some idea about its optimality could open the discussion on thenonexistence of minimizers when (11) in Proposition 2.3 is violated.

Remark. Note that the Thomas-Fermi correction usually appears withpositive sign (see [18]), which can be seen as a repulsive contribution tothe potential. Then, the addition of this kind of correction simplifies theminimizing argument because combining the repulsive Thomas–Fermi withthe attractive Slater correction allows to convexify the functional, see [17].

Now, a simple application of Propositions 2.3 and 2.8 yields the exis-tence of a minimum, since every minimizing sequence is bounded in H1(R3)and relatively compact (up to a translation). Furthermore, by standard ar-guments (see [15]) the regularity of the minimum can be deduced.

Theorem 2.9 Under the hypothesis of Proposition 2.8, there exists a min-imizer ψM ∈ C∞(R3) of (7) which satisfies the following Euler-Lagrangeequation associated with the total energy functional E[ψ]:

−12∆ψM (x) +

14π

∫R3

|ψM (x′)|2ψM (x)|x− x′|

dx′ − CS |ψM |23ψM (x)

= βψM (x) (17)

in a distributional sense, for some β < 0.

The following paragraph is devoted to show some consequences of thisresult.

14

2.1.3 Stationary solutions and solutions preserving the Lp normin the repulsive case with negative energy

From Theorem 2.9 we can deduce the existence of standing waves ψ(x, t) =e−iβtψ(x) as solutions of the SPS system in the repulsive case. Actually,these are time-periodic solutions which preserve the density. For this kindof solutions, the repulsive SPS system is reduced to the time-independentSchrodinger equation

βψ = −12∆ψ + V ψ − CSn

13ψ, lim

|x|→∞ψ = 0, (18)

coupled to the Poisson equation

∆V = |ψ|2, lim|x|→∞

V = 0. (19)

The system (18)-(19) can be written as an Euler-Lagrange equation as-sociated with (7) (cf. (17)). Then, Theorem 2.9 implies the existence ofsolutions ψM . Since these functions minimize the total energy operator,(14) holds.

Let us also note that this kind of solutions do not exist for the SPsystem in the repulsive case, where every solution is dispersive.

Let us now introduce some other solutions which preserve the Lp norm.

Proposition 2.10 There exist solutions of the SPS system with negativepotential energy and constant Lp norm along the time evolution.

Proof. The proof is based on the Galilean invariance of the system, see[3]. In fact, this property guarantees that if ψ(x, t) is a solution to theSPS system with initial data ψ0, then the solution corresponding to initialdata ψN (x, 0) = eiNxψ0(x), with N ∈ R3, is ψN (x, t) = eiNx−itN

2ψ(x −

2tN, t). Now, using the minimal energy solution we can construct thesolution e−iβteiNx−it

N22 ψM (x − tN), which has initial data eiNxψM (x).

This solution preserves the Lp norm, has negative potential energy and itstotal energy is

E(e−iβteiNx−itN22 ψM (x− tN)) =

12N2‖ψM‖L2(R3) + IM ,

which obviously exceeds the minimal energy. A similar idea has been usedin [13]. 2

15

2.1.4 Optimal kinetic energy bounds

Minimizing the total energy functional implies, by Lemma 2.7, the mini-mization of the associated functional

T (ψ) = −14

(EPOT (ψ))2

EKIN (ψ).

In the next result we use this fact to deduce optimal bounds for the kineticenergy of a solution, depending on the initial total energy and the minimumof the energy functional.

Proposition 2.11 The kinetic energy associated with a solution of therepulsive SPS system in H1(R3), EKIN , ranges between the optimal values

E±KIN = −2IM

(1− E0

2IM±√

1− E0

IM

), (20)

where E0 is the initial energy and IM is the infimum of the total energyover the set ψ ∈ H1(R3) ; ‖ψ‖L2(R3) = M. Here, M is assumed to satisfy(15).

Proof. As before, this is a direct consequence of the equivalence betweenthe energy minimization problem and (13). Since ψM minimizes (7), wehave

−14

(EPOT (ψ))2

EKIN (ψ)≥ −1

4(EPOT (ψM ))2

EKIN (ψM )= −1

44 I2

M

−IM= IM ,

for all ψ ∈ H1(R3) such that ‖ψ‖L2(R3) = M . Then, given ψ(·, t) ∈ H1(R3)a solution of the SPS system we find

EPOT (ψ) ≥ −2√−IM

√EKIN (ψ), ∀t ≥ 0.

This yields a relation between the kinetic and the total energy:

E0(ψ) ≥ −2√−IM

√EKIN (ψ) + EKIN (ψ) ∀t ≥ 0,

or, using that the potential energy is negative,

E2KIN + (4IM − 2E0)EKIN + E2

0 ≤ 0.

This concludes the proof. 2

16

3 Asymptotic behaviour in the repulsive case

In this section we study the time evolution of solutions to the SPS system.The standard arguments used to obtain various bounds on the Lp(R3)norms of solutions to nonlinear Schrodinger equations are fruitless in ourcase. This is due to the fact that the sign of the potential energy dependson the balance between the Coulombian potential and the Slater correction.Then, we have to combine these arguments with some other techniques tofind the Lp(R3) bounds.

By using similar arguments to those of [26] we can derive an equationwhich models the dispersion of solutions to the SPS system. Let us define

(∆x)2def= 〈x2〉(t)− 〈x〉2(t) and (∆p)2

def= 〈p2〉(t)− 〈p〉2(t),

where 〈x〉(t) denotes the expected value for the position operator

〈x〉(t) =∫

R3ψ∗(x, t)xψ(x, t) dx ,

while 〈p〉(t) is the expected value for the linear momentum operator

〈p〉(t) =1i

∫R3ψ∗(x, t)∇ψ(x, t) dx, (21)

which is preserved along the time evolution (see [3]). In terms of theseoperators we can prove the following result.

Theorem 3.1 The position and momentum dispersions for a solution ψ(x, t)of the SPS system with initial data in Σ = u ∈ H2(R3) ;xu ∈ L2(R3)satisfy the following equation

d2

dt2(∆x)2(t) = 2

(E(t)− 1

2〈p〉2(t)

)+ (∆p)2(t),

or equivalently

d2

dt2〈x2〉 = 2

(12〈p2〉+ E(t)

), (22)

where E(t) denotes the total energy.

By simple computations starting from the dispersion equation (22) wecan deduce the pseudo-conformal law verified by a SPS solution ψ(t, x) :

d

dt

(‖(x+ it∇)ψ‖2

L2(R3) + t2∫

R3V n dx− 3

2CSt

2

∫R3|ψ| 83 dx

)=

t

∫R3V ndx− 3

2CSt

∫R3|ψ| 83 dx. (23)

17

This shows an alternative derivation for the pseudo-conformal law widelystudied in the literature ([5], [12]).

Equation (22) allows to deduce (for positive energies) some importantconsequences about the long time behaviour of the solutions. The firstone is that the solutions tend to expand unboundedly when the energy ispositive. The second consequence is a decay bound for the potential energy.

Proposition 3.2 Let φ ∈ Σ the initial data of the SPS system such thatE(φ) > 0. Then, the system expands unboundedly for large times and theposition dispersion 〈x2〉(t) grows like O(t2).

Proof. To deduce this result we consider again the dispersion equation(22), rewritten as

12d2

dt2〈x2〉 = EKIN + E(t) = 2E(t)− EPOT . (24)

Since ‖φ‖L2 = M and E(φ) ≡ E are time invariant, we can bound theright-hand side of (24) by using (20) and obtain

E + E−KIN ≤ 12d2

dt2〈x2〉 ≤ E + E+

KIN .

By using the lower bound of the Slater potential, we also find

E <12d2

dt2〈x2〉 ≤ 2E + C(E,M).

If E is positive, then the upper and lower bounds are also positive. Thisallows to deduce the result by integrating twice in time. 2

As an immediate consequence we can deduce lower bounds for the Lp

norm of the solutions. These lower bounds are either positive constantsor coincide with the usual decay rates of the free Schrodinger equation,depending on a relation between the total energy, the mass and the linearmomentum (21). For simplicity we shall denote M [ψ] ≡< p > (0) =1i

∫R3 ψ

∗(x)∇ψ(x).

Corollary 3.3 Let ψ be a SPS solution with initial data φ ∈ Σ such that

E[φ] <12|M [φ]|2

‖φ‖L2(R3). (25)

Then, there exist positive constants C, C ′ and C ′′ depending on ‖φ‖L2(R3),E[φ], |M [φ]|2 and p such that

‖ψ(t)‖Lp(R3) ≥ C, EPOT [ψ] ≤ −C ′ , ∀t ≥ 0, p ∈ [83, 6]. (26)

18

In the case

E[φ] ≥ 12|M [φ]|2

‖φ‖L2(R3), (27)

the following lower bound

‖ψ(·, t)‖Lp(R3) ≥C ′′

t3p−62p

, ∀t > ξ > 0, p ∈ [2, 6], (28)

holds.

Proof. To show the relevance of (25) and (27) we shall use again theGalilean invariance of the system. The solutions ψN associated with aninitial condition φN = eiNxφ0(x) have the same Lp(R3) norm and thesame potential energy for every N and time t, while the total energy is

E[φN ] =12N2‖φN‖L2(R3) +NM [φ] + E[φ0] .

It is a simple matter to observe that for every φ0 the Galilean invariancegives a parametric family of initial data φN for which the time evolutionof the Lp(R3) norm and of the potential energy are the same. The analysisof a particular member of the family of Galilean transformations allows todeduce the behaviour of the Lp(R3) norm and of the potential energy forthe whole family.

By a simple optimization argument one can easily check that (25) im-plies the existence of initial data with negative energy in the family, whilethe energy of the initial data is nonnegative if (27) holds.

Under hypothesis (27), the initial data φ can be assumed to have pos-itive energy ( if not, φ can be replaced by φN ′ belonging to the class ofGalilean transforms of φ with positive energy). We have

‖ψ(·, t)‖2L2(R3) =

∫|x|≤R

|ψ(x, t)|2dx+∫|x|≥R

|ψ(x, t)|2dx ,

≤ CR3p−6

p ‖ψ(x, t)‖2Lp(R3) +

1R2

〈x2〉.

By optimizing over R we obtain

‖ψ(·, t)‖2L2(R3) ≤ C

(‖ψ(·, t)‖Lp(R3)

) 4p5p−6 〈x2〉

3p−65p−6 .

This concludes (28) by using Proposition 3.2 and the positivity of the totalenergy.

On the other hand, if φ fulfills (25), then we can choose a Galileantranslation φN ′ whose total energy is negative. In this case, we find

−‖ψN ′(t)‖83

L83 (R3)

< E[φN ′ ] < 0 , ∀t ≥ 0 .

19

We conclude (28) by using the Holder inequality, mass preservation andthe invariance of the Lp norm of the solutions under Galilean translations.2

The next result provides a rate–of–decay estimate for the potential en-ergy. However, the potential energy may be negative as shown before. Forinstance, from (5) we know that the potential energy is always nonpositivein the repulsive case.

Proposition 3.4 Let φ ∈ Σ the initial data of the SPS system. Then, thepotential energy associated with the solution ψ(x, t) satisfies the inequality

EPOT (ψ)(t) ≤ Cξt, ∀ t ≥ ξ > 0, (29)

where Cξ is a positive constant depending on ξ.

Proof. Integrating the pseudo–conformal law from ξ to t we find

‖(x+ it∇)ψ(·, t)‖2L2(R3) + t2

∫R3|∇V (x, t)|2 dx− 3

2CSt

2

∫R3|ψ(x, t)| 83 dx

= C +∫ t

ξ

(s

∫R3V (x, s)n(x, s)dx− 3

2CSs

∫R3|ψ(x, s)| 83 dx

)ds, (30)

where

C= ‖(x+ iξ∇)ψ(·, ξ)‖2L2(R3) + ξ2

∫R3|∇V (x, ξ)|2dx

−32ξ2CS

∫R3|ψ(x, ξ)| 83 dx (31)

and ξ ≥ 0. Notice that this constant can be chosen positive if ξ is smallenough because the right-hand side in (31) goes to ‖xφ‖2

L2(R3) as t → 0.

Let g(t) = t2∫

R3 V ndx− 32CSt

2∫

R3 |ψ|83 dx. Then, from (30) we deduce

g(t) ≤ C +∫ t

ξ

g(s)s

ds.

Now Gronwall’s lemma yields

g(t) = t2∫

R3V ndx− 3

2CSt

2

∫R3|ψ| 83 dx ≤ Ct

ξ≡ Cξt , ∀t ≥ ξ,

and we are done with the proof. 2

20

Consider the function

fψ(t)=‖(x+it∇)ψ(·, t)‖2L2(R3)+t

2

∫R3|∇V (x, t)|2 dx−3

2CSt

2

∫R3|ψ(x, t)| 83 dx.

From (29) we get

fψ(t) ≤ C +∫ t

ξ

Cξs

sds ≤ Cξt.

The evolution of fψ (more precisely, the evolution of its sign) implies quali-tative differences in the behaviour of the associated solution. The followingresult provides a decay estimate for the potential energy in the attractivecase or a weak decay property for some Lp,q–norms of the wave functions.

Corollary 3.5 If there exists t0 ∈ R+ such that fψ(t0) < 0, then fψ(t) < 0for all t ≥ t0. Furthermore,

2EPOT (ψ) ≤(fψ(t0)t0

)1t< 0, ∀t ≥ t0.

Otherwise we have∫ ∞

ξ

‖ψ(s)‖4p

3(p−2)

Lp(R3)ds ≤ C, ∀p ∈ (2, 6],

where C is a positive constant depending on p, ‖φ‖L2 , ‖xφ‖L2 and ξ.

Proof. The first part of the Corollary is deduced by using similar argu-ments to those of Proposition 3.4, when taking ξ = t0.

Setting ψg(x, t) := exp(− ix2

2t )ψ(x, t) we have

it∇ψg(x, t) = exp(− ix2

2t

)(x + it∇)ψ, (32)

which implies

fψ(t) = t2‖∇ψg‖2L2(R3) + t2

∫R3V (x, t)n(x, t) dx− 3

2CSt

2

∫R3|ψ(x, t)| 83 dx.

In the case fψ > 0 we have

‖∇ψg‖2L2(R3) +

∫R3V (x, t)n(x, t) dx− 3

2CS

∫R3|ψ(x, t)| 83 dx > 0. (33)

On the other hand, we can rewrite (23) in the following form

d

dt

(t‖∇ψg(·, t)‖2

L2(R3) + 2tEPOT (ψ)(t))

= −‖∇ψg(·, t)‖2L2(R3).

21

Integrating between ξ > 0 and t > ξ yields

t‖∇ψg(·, t)‖2L2(R3) + 2tEPOT (ψ)(t) =

fψ(ξ)ξ

−∫ t

ξ

‖∇ψg(·, t)‖2L2(R3). (34)

Now, the left-hand side of (34) can be estimated by using (33), which gives∫ t

ξ

‖∇ψg(·, t)‖2L2(R3) ≤

fψ(ξ)ξ

.

The proof concludes by noting that the Lp norm of ψ and ψg coincide, thenwe can apply the Gagliardo-Nirenberg inequality to ψg. 2

Let us now prove some decay properties of the solutions in the case ofnonnegative potential energy.

Proposition 3.6 Let φ ∈ Σ the initial data of the SPS system and let ψbe the corresponding solution. If the potential energy associated with ψis nonnegative along the time evolution, then there exist constants C > 0which depend on ‖φ‖L2(R3) and ‖xφ‖L2(R3) such that

(i) ∀ |t| ≥ 1, ∀ p ∈ [2, 6] , ‖ψ(·, t)‖Lp(R3) ≤ C

|t|32 ( 1

2−1p

),

(ii) ∀ |t| ≥ 1, ∀ p ∈ [1, 3] , ‖n(·, t)‖Lp(R3) ≤ C

|t|32 (1− 1

p),

(iii) ∀ |t| ≥ 1, ∀ p ∈ ]3,∞[ , ‖V (·, t)‖Lp(R3) ≤ C

t( 12−

32p

),

(iv) ∀ |t| ≥ 1, ∀ p ∈ ] 32 ,∞[ , ‖∇V (·, t)‖Lp(R3) ≤ C

t(1− 3

2p).

Proof. The proof follows the steps of Proposition 3.4 and the argumentsgiven in [6], [12].

Using (32) the pseudo–conformal law can be written as

t2‖∇ψg(·, t)‖2L2(R3) = C +

∫ t

ξ

(s

∫R3|∇V (x, s)|2dx− 3

2CSs

∫R3|ψ(s)| 83 dx

)ds

−t2∫

R3|∇V (x, t)|2n(x, t)dx+

32CSt

2

∫R3|ψ(x, t)| 83 dx.

Then, applying (29) and taking into account the nonnegativity of the po-tential energy we find

‖∇ψg(·, t)‖2L2(R3) ≤

C ′

tξ,

22

for all t ≥ ξ, where C ′ = C ′(C, ξ) > 0. Now, the Gagliardo–Nirenberginequality (applied to ψg) allows to get (i) for p ∈ [2, 6] and a = 3

(12 −

1p

):

‖ψ(·, t)‖Lp(R3) = ‖ψg(·, t)‖Lp(R3) ≤ γ(p)‖∇ψg(·, t)‖aL2(R3)‖ψg(·, t)‖1−aL2(R3)

≤ γ(p)‖∇ψg(·, t)‖aL2(R3)‖φ‖1−aL2(R3) ≤

C ′

t32 ( 1

2−1p ).

(ii) is a consequence of ‖n(·, t)‖Lp(R3) = ‖ψ(·, t)‖2L2p , while (iii) can be

deduced from the Hardy–Littlewood-Sobolev inequality and (ii):

‖V (·, t)‖Lp(R3) ≤ C ′∥∥∥∥1r∗ n(·, t)

∥∥∥∥Lp(R3)

≤ C ′‖n(·, t)‖Lq(R3)

≤ C ′1

t32 (1− 1

q )≤ C ′

1

t(12−

32p )

,

where 1q = 1

p + 23 and q ∈]1, 3[. The proof of (iv) is analogous to that of

(iii). 2

4 Minimization of the energy in the attrac-tive case

The aim of this section is to give some results concerning the asymptoticbehaviour in time of solutions to the SPS system under the assumption ofattractive interactions. In this case the energy functional reads

E[ψ]=∫

R3

12|∇ψ(x, t)|2−

∫R3

|ψ(x, t)|2|ψ(x′, t)|2

8π|x− x′|dx′− 3

4CS |ψ(x, t)| 83

dx. (35)

Using the same arguments developed before to bound the energy in therepulsive case and the inequality (5), it can be shown that this functionalhas a lower bound over the set ψ ∈ H1(R3) ; ‖ψ‖L2(R3) = M. In [3] it wasproved the existence of a minimizer ψM of the energy functional (35) inH1(R3) under the constraint ‖ψ‖L2(R3) = M, M ∈ R+. Furthermore, thisminimum was found to be spherically symmetric. The proof given abovecan be also adapted to this case, therefore it might give an alternative wayto obtain the existence of a minimum. In this case the restriction on theL2–norm is not necessary because the potential energy is always negative.

Theorem 4.1 For all M > 0 there exists a minimizer ψM ∈ C∞(R3) ofthe problem

minE[ψ];ψ ∈ H1(R3), ‖ψ‖L2(R3) = M,

23

where E[ψ] denotes the energy functional (35). Also, psiM satisfies theEuler-Lagrange equation

−12∆ψM − 1

4π

∫R3

|ψM (x′, t)|2

|x− x′|dx′ ψM − CS |ψM |

23ψM = βψM

in a distributional sense, for some β < 0.

As an immediate consequence we get the existence of stationary waves ofthe form ψ(x, t) = e−iβtψM (x) and we can construct solutions of the sametype than in Proposition 2.10 satisfying (14). Also, from the minimizationof the total energy operator we can deduce the same bound for the kineticenergy as in (20).

The dispersion properties (in the positive energy case) as well as thedispersion and pseudo–conformal laws are also valid in this case. However,since the potential energy is always negative in the attractive case, thedecay properties of the solutions are no longer verified.

It is also possible to study the asymptotic behaviour of the solutionsat t = 0. Actually, this analysis is a straightforward adaptation of thetechniques developed in [6] and shall be omitted here.

Acknowledgements

The authors would like to thank very much Isabelle Catto, Claude LeBris,Jose L. Lopez, Reinhard Illner and Luis Vega for fruitful discussions andremarks that have improved the final version of this paper. Also, they thankthe referees for some interesting comments. This research was partiallysupported by the EU financed network IHP-HPRN-CT-2002-00282 and byMCYT (Spain), Proyectos PB98-1281 and BFM2002–00831.

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