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RAIROMODÉLISATION MATHÉMATIQUE
ET ANALYSE NUMÉRIQUE
I. GASSER
R. ILLNER
P. A. MARKOWICH
C. SCHMEISERSemiclassical, t → ∞ asymptotics and dispersive effectsfor Hartree-Fock systemsRAIRO — Modélisation mathématique et analyse numérique, tome 32, no 6 (1998), p. 699-713.<http://www.numdam.org/item?id=M2AN_1998__32_6_699_0>
© SMAI, EDP Sciences, 1998, tous droits réservés.
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Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiques
http://www.numdam.org/
FT MATHEMATICAL MODELLING AND NUHERICAL ANALYSIS[MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE
(Vol 32, n° 6, 1998, p 699 à 713)
SEMICLASSICAL, f - *~ ASYMPTOTICS AND DISPERSIVE EFFECTS FOR HARTREE-FOCK SYSTEMSDedicated to Helmut Neunzert at the occasion of his 60th birthday
I. GASSER (*), R. ILLNER (**), P. A. MARKOWICH (*) and C. SCHMEISER (***)
Résumé — On analyse la limite semiclassique et Vasymptotique t —> °° pour des systèmes des équations de Schrodinger faiblement nonlinéaire en forme Hartree-Fock En utilisant des techniques de fonction de Wigner, on démontre que la limite semi-classique est représentéepar l'équation de Vlasov « s elf-consistent » En outre, on démontre des estimations du temps pour la densité et le potentiel électrique deHartree-Fock dans les norms if pour t —> » © Elsevier, Pans
Abstract — We analyze the semiclassical limit and the "t —> °° asymptotics" ofmildly nonhnear Schrodinger Systems of (self-consistent)Hartree-Fock form Using Wigner-functwn techniques we prove that the semiclassical limit is represented by the self-consistent Vlasovéquation Moreover we prove time decay for the position density and for the Hartree-potential in Lp norms as t —> » © Elsevier, Pans
1. INTRODUCTION
We consider Hartree-Fock Systems in Ud of the form
2ie -r- we, = — -=- -
2 X' V\(x, f) y/), x e W, t e U, l e N (Lia). 7 = 1
, leN (1.1b)
=| U(x-z)n(z,t)dz (l.ld)d
^ , 0 = 1 U(x-z)y/el(z,t)ye(z,t)dz. (1.1e)
Hère e > 0 dénotes the scaled Planck-constant, X]^ 0 the occupation number of the state y/ev n is the number
density of the considered particle System, V8H is the self-consistent Hartree potential (defined by the interaction
potential U = U(x)), V^ represents a given exterior potential and Vetj stands for the interaction of the l-th and
j-th state.
Manuscnpt received April 1, 1997(H) FB Mathematik MA 6-2, TU-Berlm, Stra(3e des 17 Juni 136, 10623 Berlin, Germany(xx) Department of Mathematics and Statistics, University of Victoria, RO Box 3045, Victoria, B.C. V8W 3P4, Canada.(***) Institut fur Angewandte und Numensche Mathematik, TU-Wien5 Wiedner Hauptstrape 8-10, 1040 Wien Austria
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700 I. GASSER, R. ILLNER, P. A. MARKOWICH, C. SCHMEISER
Hartree-Fock Systems are considered an accurate description of the quantum-mechanical évolution of a Fermionsystem, since their dérivation from many body phy sics takes into account the Pauli exclusion principle [ S ] , whichis not the case for Hartree Systems (obtained by setting V^:=0).
In this paper we consider two limits of Hartree-Fock Systems. The first one, analyzed in Section 2, is thesemiclassical limit s —» 0. We prove — under suitable assumptions on the data — that the Hartree-Fock exchangeterm does not give a contribution in the limit £ —» 0, i.e. the semiclassical limit of the Hartree-Fock system is —in a sense made précise in the next section — the selfconsistent Vlasov équation. The same result has alreadybeen shown for Hartree Systems [LPa, MM]. Clearly, this behaviour is physically plausible, since the Pauliprinciple is a purely quantum physical notion.
The second limit to be considered is the limit t —> oo in the purely répulsive case U ̂ 0. These results, whichimprove [DF] are contained in Section 3.
Section 4 is concerned with dispersive effects.
2. THE SEMICLASSICAL LIMIT
We define the density matrix p£ in the usual way
/(r,M) = i^;(r ,0^(^) , r9seUd (2.1)
and dérive the Heisenberg formulation of the Hartree-Fock system:
(VeH(r,t)-Vls
H(s,t))pe
(U(r-z)-U(s-z))p\r,z>t)pXz,s,t)dz (2.2a)
n\x,t)=p\x,x,t) (2.2b)
-I.
40,0= U(x-z)n(z,t)dz (2.2c)Jud
z
p£(r,s, t=0) = ̂ X](p](r) (p](s) = \ p](r> s) . (2.2d)
The Wigner transform of the density matrix is the Fourier transform of the function
pEl x + 2J?'x~2ï1*t) w ^ r e s P e c t t o */> ^-e-
we(x, v, t) := ^1 pel x + 2 tf* x~~ ô 7» f ) eWV n drj (2.3)
{cf. [GMMP], [LPa], [W], where the Fourier transform is defined by
)ew xdx. (2.4){Infini
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It is the solution of the Wigner-Hartree-Fock équation, obtained from (2.2) by an easy calculation [M]:
xwe + e\VE1 wE + 6\Ve
H] we + £2e[_we] = 0 ,
VH(x,t)=\ U(x-z)n\z,t)dz,
Jut
n\x,t)=\ w\x,v,t)dv,
(2.5a)
(2.5b)
(2.5c)
(2.5d)
For a given potential V = V( x ) the pseudo-differential operator ö£ [ V] is defined by
where >v dénotes the inverse Fourier transform of w-w(x, v) with respect to v:
W(JC, ^ ) = vw(x ïv)é~ iV '7 dv .Ju-
r3e is the (quadratically) nonlinear operator
The following estimate is basic for carrying out the limit e —» 0 + in the Hartree-Fock system.
LEMMA 2.1: Let we L2(Udxxüj), ç>s S(Ud
x X
f Ö6C dv
and
A\cp)
on Ud. Then
wffA y/( ri ) := sup | <p( x, n ) |.
vol. 32, n" 6, 1998
(2.6a)
(2.6b)
(2.7)
(2.8a)
(2.8b)
702 I. CASSER, R. ILLNER, P. A. MARKOWICH, C. SCHMEISER
Proof: With the substitution s = -z + %> r = z + % w e obtain
Qe[w] (j>dxdv = i\ [ed~l(U(sr) - U(es))] w(jt + | J , r ) W(JC - | r, s) <p(x9 r + s) dr ds dx
We estimate
tpdxdv
( |w(jc, drds
X M \w(x,s)\2dx\ (j\wUr)\2dx) drds
and thus
dx dv « 2 l\U(er)\fdr\\w{x,.1/2
f |M>(X, \r)\2dxdrds
The assertion of the Lemma now follows immediately. DThe subséquent Lemma is conceraed with a priori conserved quantities of the Hartree-Fock system:
LEMMA 2.2: Let U(x) = U(-x) on Ud hold. Then
n(x,t)dx=\ nI(x)dx, Vf G M ( charge conservation) , (2.9)
where nj(x) :=2
Ee(t)=E£(0), VfeR, (2.10)
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where
U(x-z)n(xft)nXz,t)dxdz
U(x-z) \pË(x,z,t)\2dxâz
(energy conservation).
Proof: (2.9) is obtained by multiplying the Hartree-Fock équation (1.1a) by X]y/], integrating by parts, takingimaginary parts and summing over /.
(2.10) is the result of a somewhat more tedious calculation based on multiplying (1.1a) by X] -jr y/p taking realparts and summing over L Details can be found in [CG] (at least for the Coulomb interaction potentialU(x)=T^ronU3). D
EE(t) is the total energy, which by effect of Lemma 2.2 is constant in time. We remark that, by a well knowncalculation (see, e.g. [MM], [LPa]) the kinetic energy can be written as
z Jndï=i| f Q (2.11)
The folio wing Lemma pro vides an a priori L2 -estimate for the Wigner function ws :
LEMMA 2.3: Assume that the initial states {ç?/}"=1 are an orthonormal system in L (U.x) and thatU(x) = U(-x) on Md. Then
Proof: Similarly to the proof of (2.9) we show that initially orthogonal states (pev (p\ remain orthogonal for all
time under the Hartree-Fock évolution. The result then follows directly from the formulas (2.3), (2.1). DAlso, we remark that the local conservation law
^ e = 0 (2.13)
holds, where the current density f can be calculated from the Wigner function in the usual way
Je(x, t) = vw£(x, v, t) dv (2.14)
(see [MM, LPa]). We now make the following assumptions on the data:(Al) (i) Ve e (0, e0], l e N : X]> 0 ;
(ii) Vee (0, e0] : {ç^i i s a n O N S i n ̂ 2 /
(iii) 3 O 0 :
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704 I. GASSER, R. ILLNER, R A. MARKOWICH, C. SCHMEISER
forée (0,e0].On the external potential we assume
(A2) VEe Hll0C(Ud) ; 3Ve M : VE(x) ^ V°n Ud,
and on the interaction potential:(A3) (i) U(x) = U(-x) on Ud
(ii)
' 2<5<oo if d = 29
Ue Lr(Rd) + Ls'"(Rd) with < | ^ s < ~ > i f d = 3 ,
C/e C&(R) and U(0) = 0 if rf= 1 ;
(iii) Vf/ G L̂ rfT8~( Ud) + L*°°( Ud) wi th 2 / ++
88 < q < 2.
For the définition of the 'weak Lp -spaces' Lp' we refer to [RS, page 30]. (Al) implies a uniform bound forne LT(nt;L\n*)) for the initial kinetic energy and (with (A3) (i)) on w3 e L°°(Ut ; L
2(Udx Ud)).In order to carry out the Hmit s —> 0 in the Wigner-Hartree-Fock System we proceed as in [LPa] to establish
uniform a priori bounds. We start with the initial energy:
PROPOSITION 2.1: EE(0) ^ C.
From now on we dénote by C generic, not necessarily equal constants which are independent ofs e (0 f e 0 ] .
Proof: The following estimate can be found in [LPa] (cf. the Theorem in the Appendix)
(2.15)
where Co is also independent of w£ and
0= 4
d + 4'
d + 4 ,
Evaluating at t = 0 gives a uniform bound for nr G Ld + 2(Ux). The generalized Young inequah'ty [RS, page 32]
then yields (together with (A3) (ii)) a uniform bound for |C/ (x-z) | n}(x) n)(z) dzdx. Since (by theSchwartz inequality)
ne(x,t)n\z9t)& \pe(x9z9t)\2 (2.16)
obtain a uniform bound for | U(x- z)| \p)(x,z)\2 dzdx and the assertion of Proposition 2.1 follows,J ni x M*
wex
Next we dérive a uniform bound for the total kinetic energy:
PROPOSITION 2.2:
=\ teUt.
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Proof: From (2.10), (2.11) we obtain
E^t) £(f = O ) - V n(t)dx
- i f U+(x-z) (n\x, t) n\z, t) - \p\x, z, t)\2) dz dx
+ \ f t IT(x-z) (n(x, t) n\z, t) - \p\x, z, t)\2) dzdx
and (2.15) gives
EL0) ^ c + \ \ „ d \U{x-z)\ n{x,t)n\z,t)dzdx.
For d = 1 the assertion follows. For d > 1 we again apply the generalized Young inequality
f d\U(x-z)\n(x,t)n(z,t)dzdx^ C( 1 + \\n(0]\2LP,
1 2with 2 = - + - where s is of (A3) (ii). By interpolation we have
and (2.15) gives
and 2( 1 - 0, ) ( 1 - 0) = / ï " < 1 by (A3) (ü). Doo d + 4 d
We thus obtain a uniform bound for n G L°°( M.t ; Ld+2( Mx) ) from (2.15) and uniform bounds for
fand
inL"(R,) follow.Finally we need
f \U(x-z)\ \pe(x,z,t)\2dzdx
LEMMA 2.4: Let (A3) (i), (Ui) hold and assume that we G LT{ Ut ; L
2( M? x IR )̂ ) uniformly as e —> 0.
vol. 32, n° 6, 1998
706 I CASSER, R ILLNER, P A. MARKOWICH, C SCHMEISER
Proof: The assertion for d=\ follows immediately from Lemma2.1. Thus, we assume d>\ for thefollowing.
At first we observe that the L°°( Ud) part of U(x) gives an O(ed~1 ) contribution to A£(<P ) defined in (2.8) (b)and consequently by (2.8) (a) its contribution to £2Ê[v/] in S" is of the same order. Therefore, to complete theproof, it suffices to assume U e Z/'°°([Rrf) with s as of (A3) (ii).
We dénote Zs(x) := \ed"1 U(ex)\2 and estimate the convolution in (2.8b) using the generahzed Younginequality:
for y/ e Ll(Ud ) r\L°°(Ud), where 1 < q < <*>, 0<S<- and pà-—T~,—^r—r. Keeping q fixed and taking
ô to zero gives
II v * z 8 n j . . .
Since
we conclude the assertion with s = 2q, DThe existence of a unique solution of the Hartree-Fock problem (or, equivalently, the Wigner-Hartree-Fock
System) for e > 0 can easily be shown by generalizing the methods of [CG]. Details are left to the reader. Thelimit e —> 0 can now be carried out by applying the methods that lead to Theorem IV.5 in [LPa].
THEOREM 2.1: Let (Al), (A2), (A3) hold. Then, for every séquence e —> 0 there exists a subsequence (denotedby the same symbol) such that
w)-> w°j^ 0 in L2( Udx x Ud
v ) weakly , (2.17a)
w£ -> w° ^ 0 in L~( Rt ; L2( Ud X Ud
v ) ) weak- * , (2.17b)
ne ->n°=\w° dv in L°°( Ut ; Ld7i( Rd ) ) weak- * , (2.17c)
f _> 7° = f Ï;W° di? |W L°°( Kr ; L ^ ( Ud ) ) wcofe- * , (2.17d)
V Z C/(JC-Z)«°(Z,O& w L°°(nt;L2(nd))weak-*, (2.17e)
( w , n , £"° = VV^) are weak solutions of the self consistent Vlasov équation:
w°t+v- Vxw°-VxV°H • Vvw° = Q inUdxUdxUt (2.18a)
w°(t = 0)=w°I. (2.18b)
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SEMICLASSICAL, t -» <~ ASYMPTOTICS AND DISPERSIVE EFFECTS FOR HARTREE-FOCK SYSTEMS 707
Remark 2.1: The important case of the Coulomb interaction in 3 dimensions
C/( J C)=-i-9 xG U3 (2.19)
is contained in the assumptions. We then have q = j and s = 3 in (A3). The limiting problem (2.18), (2.17c),(2.17e) is the Vlasov-Poisson équation [LPe], [LPa].
Remark 2.2: The case of the Poisson interaction U(x) = |x| in 1 dimension is not included because of (A3)(iii), which was imposed in order to be able to treat the (relatively uninteresting) 1-dimensional case analogouslyto the case d > 1. However, the assumption U e Cb( IR ) can easily be replaced by at most polynomial growthat oo and continuity at 0.
Remark 23: Both the attractive case ( U ^ 0) and the répulsive case ( U 2= 0) are covered by Theorem 2.1.
Remark 2.4: The cases of e-independent occupation probabilities X\ and of finitely many states (i.e. X] = 0 for/ > N) is not included because of (Al) (iii). While it can be dealt with rather easily in the Hartree case with asmooth interaction potential U (cf. [LPa]), it créâtes difficulties for the Hartree-Fock problem when the completesemiclassical information is sought. Then the Schrödinger problem (1.1) has to be dealt with as a fully coupledsystem of TV équations and methods as presented in [GMMP] have to be applied (passage to the semiclassical limitin the Wignermatrix of the Schrödinger system). Serious mathematical difficulties then occur at points in(x, ?)-space where the spectral décomposition of the Hartree interaction potential matrix V]. dégénérâtes. To ourknowledge, this problem has not been solved yet.
However, the semiclassical limit of ws(t) (and consequently of ne(t) and Je(t)) can still be computed.Therefore, assume that t / e C (IR) for some 0 < / ? ^ 1. A simple modification of the proof of Lemma 2.1shows that (2.8a) also holds with
Thus, by the regularity of U and since V£/(0) = 0 we obtain
Ae(<p) = O(ed+fi) inS(MdxUd). (2.20)
Instead of the uniform bound on — 2 (X\)2 in (iii) assume now that 2 i^])2 is bounded uniformly in £ (e.g.8 i = i i = i , v
finitely many states only). Lemma 2.3 then implies ||w£(t) \\L\udxUd) = ol — ) and Lemma 2.1 (with (2.20)
instead of (2.8b)) gives
The other terms in (2.5a) and (2.5c) can be taken to the limit as in [LP]. Thus, Theorem 2.1 also applies for smoothinteraction potential (instead of (A3)) without thprocess (2.17) have to be changed accordingly.interaction potential (instead of (A3)) without the uniform L2 -bound on ws, however the topologies for the limit
3. ASYMPTOTIC BEHAVÏOUR AS t -> <*> IN THE REPULSIVE CASE
In this section we investigate the time decay properties of the Hartree-Fock-System (2.5). We shall assumevanishing external potential VE = 0.
vol. 32, n° 6, 1998
708 I. GASSER, R. ILLNER, P. A. MARKOWICH, C. SCHMEISER
Also, we assume that a global unique strong solution of the Hartree-Fock-System exists. The assumptions ofthe previous section are sufficient for this; we remark that the (Al) (iii) can be weakened. In addition we imposethe following assumptions on the interaction potential:(A3) (iv) U=U0(\x\)&Q.
(v) ^ ( r ) ^ - ^ £ / 0 ( r ) , r > 0 , a > 0 .Also, for the sake of clarity óf the présentation we consider the case d ^ 2.
Note that results along the lines of the ones presented below entirely based on the Schrödinger formalismrestricted to the 3d Coulomb case and finitely many coupled states can be found in [DF, P]. Decay results for theHartree case with Coulomb interaction can be found in [ELZ].
We state
LEMMA 3.1: The following relation holds:
( l -hf 2 ~ a ) , a < 2
with c independent of t.
Proof: Using the équation (2.5a) we obtain
vt\2w\t)dxdv4 | f \x-dtiudjud
= - f f \x-vt\2{Ge[V£H] we
= 2t\ f x-v{eë[VeH]we + Q
~t2\ f \v\2{&s[VBH]w£
e] \dxdv
}dxdv.
An easy but tedious calculation gives
2t\ f x-v{€>£[VeH]we + Qe[we] }dxdv =
JRÎJMÎ
tl f z<VU(z){nE(x-z)nE(x)-\p£(x-z,x)\2}dzdx.Jud
xJudv
Now, combining the energy conservation (2.10) with VE = 0 and (2.11) gives
it\\\ \ \v\2™edxdv+\\ f U(x-z){ne(x)nXz)~\pe(x,z)\2}dxdz]=0.
With the relation
hél f v2wdxdv+\\ f \v\2{e£[V£H]
zatJudJui zJudJui(3.2)
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obtained by multiplying the équation (2.5a) by ^ |i;|2 and integrating over IRf x R j , we conclude
-t2] f \v\2{0e[VeH]w£ + Q£[we] }dxdv
= -t2i\ f U(x-z){nXx)n\z)-\pe(x9z)\2}dxdz.ai 0UdJ Ud
Therefore
Adt
• vt\2 w8 dx dv + g(t)f i , -
f (x-z)VU(x-z){nE(x)nE(z)-\p\x,z)\2}dxdz +ud
xJmdv l
= t2ï f U(x-z){nXx)ne(z)-\p\x,z)\2}dxdz
with
holds. Using the assumptions on the interaction potential we have
x- VxU+aU=r- U\r) + odJ ̂ 0.
We obtain
1 (2-Adt f f \x-
JudJuivt\2w£dxdv
o,
It is (again an easy but tedious calculation)
O I I ^ R - ) = f f \x- vt\2w£dxdv.
Therefore we can apply Gronwall's lemma to (3.3) and arrive at (3.1).We need also
LEMMA 3.2: = 0 and 2 ^ll-xp* llL2(Rf) < o ° - ^ estimate
holds for the wavefunctions corresponding to the solution of problem (2.5).
Proof: It is
IJCI n( jc,
(3.3)
D
vol. 32, n° 6, 1998
710 I. GASSER, R. HXNER, P. A. MARKOWICH, C. SCHMEISER
and
-™ \x\2n(x,t)dx = 2 X'Je(x,t)dxdtJudx Jud
=5 2 \x\2n\x,t)dx}
VL^ 2 VEE(O) - / I \x\2 ne(x, t) dx .
Then the result follows applying the Gronwall's lemma. D
This result is needed in order to apply the well known.
LEMMA 3.3: Let ue Hl(Ud) such that xu e L2(Ud). Then
\\ue\\LP^C(P,d) \\G(-t)ue\\l* Kx+ieNx)ue\\^af-X8a-1
where G(t) is the unitary group generated by the homogeneous linear Schrödinger équation,
2 =g p sg ̂ 2 and a is
The proof of this lemma can be found in [ILZ] or [GV].At this point we can state our decay result.
THEOREM 3.1: Under the assumptions of this section the following decay estimâtes hold:
(i) II n
c(te) 2U a\ a<2
I c(te)~ZKi~a) a ^ 2
(iü) 11^11^)= j (t x-a(i-fl)' ^ 9I c(t£) , a < 2
wïï / i 1—a = « ( l — — ) , — = — + ^-, 1 + — = — + ^r aw6? c independent of s. It is2 \ q / s q 2 r q d
1 ^ q ^ -3 ^, 1 ^ 51 < ̂ —5 T and
V
Proof: Following [ILZ] and using the Lemmas 3.1-3.3 we estimate
\-a
t m = 1
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SEMICLASSICAL, t -> °° ASYMPTOTICS AND DISPERSIVE EFFECTS FOR HARTREE-FOCK SYSTEMS
and the decay result of the density follows. The decay resuit for the current is obtained using
711
^ n^ V^IIL^, * s K
Vil "'(O II *.(,*) ,
since e|| V^ | | L 2 ( R ^ is uniformly bounded by the energy conservation (2.10) (with VE = 0). The estimate (iii)follows using the Sobolev inequality
Judv
(y)h(y)dy f f "'WJuilui \x- y\
with - + 1 + ̂ = 2. Therefore,/ q d
and the assertion follows. D
4. A DISPERSIVE IDENTITY
Let xo€ Rd fixed with rf>l, setv(x-x0)
• a 1/a gives the identity:
= 0 or ô~\ and a > 0. Then multiplying (2.5a) by
ir, iuiiuw(x,v, dvdxdt
^ f f (x-xo)VU(x-z) 7
i—-^ K—^-{\p\x,Ztt)\2~n\x,t)n\z,t))dzdxdt
(x-x0)—^•(Js(x,T1)-J
£(x,T2))dx (4.1)
for all - oo < Tx < T2 < °o. Intégral identities of this type were obtained in [LPel] for the free transport équationand in [P] for the Vlasov-Poisson and Wigner-Poisson Systems.
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712 I- GASSER, R. ILLNER, P. A. MARKOWICH, C. SCHMEISER
A lengthy calculation shows that the first term on the left hand side of (4.1) is nonnegative. For example inthe case d = 3 and ô = 0 it is equal to
'zi^r[j.(g^f- i(^^T')i>+8-'^')'^' <«>i=i JT,\_JU3\ \X x o i |JC — JCO| /
(see [LPel] also for the other cases). Assume now that VE = 0 (no exterior field) and that the interaction potentialis radial U=UQ(\x\) with U'o(r)^O. Then, an easy calculation using pB{x, z, t) = pe(z, x, t) and (2.16)shows that also the third term in (4.1) is nonnegative. Thus, the identity (4.1) gives the bound for the first termon its left hand side:
UA^ii^j+iirc^iifcRï)-
Energy conservation shows that \\Je(t) \\L\ui) *s uniformly bounded in e and t. Thus, we conclude for d = 3 andail x0 G U3 :
^g^nj t f(j z^- i ( i"^Lv^*o i>*+ ' 'r.^' )*< c- c ( ' '- j t o» <4-3>
(just as for the free Schrôdinger équation). Similar estimâtes can be obtained for dimensions différent from 3.Other applications of the dispersive identity (4.1) are also possible.
ACKNOWLEDGEMENTS
This research was supported by grant Nr. A7487 of the Natural Science and Engineering Research Council ofCanada. The first and the third author were supported by grant Nr. MA 1662/1-1 and 1-2 of the german 'DeutscheForschungsgemeinschafV. The last author was supported by grant Nr. P11308-MAT of the austrian 'Found zurFôrderung der Wissenschaft und Forschung'. We would like to thank Dr. Raimund Wegener who alerted us to theproblems posed by the Hartree Fock Systems. I. Gasser, P. A. Markowich and C. Schmeiser would also like toacknowledge the hospitality of the University of Victoria, where this research was started.
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