Commun. Math. Phys. 225, 223 – 274 (2002)
Communications in
Mathematical
Physics
© Springer-Verlag 2002
On the Point-Particle (Newtonian) Limit
of the Non-Linear Hartree Equation
Jürg Fröhlich1 , Tai-Peng Tsai2 , Horng-Tzer Yau2,∗
1 Theoretical Physics, ETH-Hönggerberg, 8093 Zürich, Switzerland. E-mail: juerg@itp.phys.ethz.ch
2 Courant Institute, New York University, New York, NY 10012, USA.
E-mail: ttsai@cims.nyu.edu; yau@cims.nyu.edu
Received: 30 June 2000 / Accepted: 25 June 2001
Abstract: We consider the nonlinear Hartree equation describing the dynamics of
weakly interacting non-relativistic Bosons. We show that a nonlinear Møller wave operator describing the scattering of a soliton and a wave can be defined. We also consider the
dynamics of a soliton in a slowly varying background potential W (εx). We prove that
the soliton decomposes into a soliton plus a scattering wave (radiation) up to times of
order ε −1 . To leading order, the center of the soliton follows the trajectory of a classical
particle in the potential W (εx).
1. Introduction and Summary of Main Results
The problem of identifying classical regimes of quantum mechanics is a long standing
problem of quantum theory. For simple systems it was first studied by Schrödinger in
1926; see [1]. In this paper, we explore a classical regime for a class of systems of
identical, non-relativistic bosons, e.g., bosonic atoms such as 7 Li, with very weak twobody interactions described by a potential −κ of van der Waals or Newtonian type
satisfying certain regularity properties described below. These bosons move under the
influence of an external potential λV , where V is a smooth, positive function on physical
space R3 and λ ≥ 0. The potential λV describes e.g. a trap confining the bosons.
Let κ denote the strength of the two-body interaction between two bosons as compared
to their average kinetic energy, (e.g. in the sense that is small as compared to the kinetic
energy operator of two bosons, in the sense of Kato and Rellich,
[2]).
We are interested in
understanding the dynamics of a “condensate” of N = O κ −1 bosons in the “meanfield regime”, where κ is very small. By a “condensate” we mean a state of the system
with the property that all except for o(N ) bosons are in the same one-particle state
described by a wave function ψ(x), x ∈ R3 . N -particle states of this kind are also called
coherent states.
∗ Work partially supported by National Science Foundation Grant no. DMS-9703752 and DMS-0072098
224
J. Fröhlich, T.-P. Tsai, H.-T. Yau
Let ψ0 = ψ0 (x), x ∈ R3 , denote the initial one-particle wave function of a coherent
state of the system at time t = 0. In the mean-field limit,
κ → 0, N → ∞,
with κ · N =: ν = const.,
(1.1)
the quantum-mechanical time evolution of a condensate of bosons has the property that
it maps the initial coherent state with a one-particle wave function ψ0 to a coherent state
at a later time t with a one-particle wave function ψt . As proven by K. Hepp [3] (see
also [4] for some refinements and extensions), the one-particle wave function ψt of the
condensate turns out to be a solution of the (non-linear) Hartree equation, Eq. (1.2)
below.
If the two-body interactions are dominantly attractive, as for 7 Li atoms, and, given
κ, the number of bosons is large enough (i.e., N > Ncrit. (κ), or ν > νcrit. ), the system has bound states. In other words, the bosons may condense into a tightly bound,
spatially sharply localized cluster. In the mean-field regime, such bound states appear
to be (weakly) well approximated by coherent states with a one-particle wave function
corresponding to a non-trivial local minimum of the Hartree energy functional.
Turning on a very slowly varying external potential,
λV (x) := W (εx),
(1.2)
where W is a smooth, positive function, and ε is much smaller than the diameter of a
bound state of N bosons when λ = 0, one expects that the position, r(t) ∈ R3 , of the
center of mass of that bound state closely follows a solution of Newton’s equations of
motion,
ṙ(t) = v(t), v̇(t) = −ε (∇W ) (εr (t)) ,
(1.3)
−1 for times t with |t| < O ε .
It is in this precise sense that the quantum system of bosons described above approaches a classical regime in the mean-field limit.
For attractive two-body interactions, the Hartree equation describing the dynamics of
a condensate (coherent state) in the mean-field limit has a self-focussing non-linearity.
As a consequence, it has non-trivial “solitary wave solutions” looking like approximate
δ-functions, for ν sufficiently large. These solitary wave solutions are precisely the oneparticle wave functions of coherent bound states in the mean-field limit.
Our main objective in this paper is to study slow motion of solitons of the Hartree
equation. We propose to show that, under the influence of a slowly varying external
potential W (εx), the center of mass position, r(t), of a solitary-wave solution of the
self-focussing Hartree equation remains close to asolution
of Newton’s equations of
motion stated above, for all times t with |t| < O ε −1 . (We do, however, not prove
rigorous results on the precise way in which a system of identical bosons approaches its
mean-field limit; but see [3–5].)
Our main results on the self-focussing Hartree equations have been announced in [6],
where the reader can find additional background material and motivation coming from
physics.
In order to be able to describe our main results concisely, we introduce some notation
and recall some known results on the Hartree equation.
Let H 1 (Rn ) denote the Sobolev space,
(1.4)
H 1 (Rn ) = ψ(x), x ∈ Rn ∇ψ2 + ψ2 < ∞ ,
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
225
where ψ denotes a measurable complex function on Rn , ∇ψ denotes its gradient, and
(·)2 denotes the L2 -norm. We study properties of solutions of the Hartree equation
1
i∂t ψt = − ψt + λV ψt − ν ∗ |ψt |2 ψt .
(1.5)
2
In Eq. (1.5),
ψt (x) = ψ(x, t), x ∈ Rn , t ∈ R,
is a time (t)-dependent, complex-valued scalar function on physical space Rn belonging to the Sobolev space H 1 (Rn ), for each time t; denotes the scalar Laplacian, λV (x), λ ∈ R, is an external potential, with V a smooth, bounded, positive
function on Rn , and −(x) is a radially symmetric two-body potential, with ∈
Lp (Rn , d n x) + L∞ (Rn ), p ≥ n2 ; furthermore ∗ denotes convolution. We shall use the
following standard notation:
For an arbitrary measurable function ψ on Rn ,
ψ := ψ(x) d n x,
(1.6)
Rn
ψp :=
p
1/p
|ψ|
(1.7)
is the norm on the space Lp = Lp (Rn , d n x), 1 ≤ p < ∞,
ψH 1 := ∇ψ2 + ψ2
is the norm on H 1 = H 1 (Rn ), and
(ψ ∗ χ )(x) :=
ψ(x − y)χ (y)d n y
(1.8)
(1.9)
Rn
denotes the convolution of ψ with another such function χ .
There are two important functionals on Sobolev space H 1 which are conserved under
the flow ψ := ψ0 → ψt , ψ ∈ H 1 , determined by the Hartree equation (1.5). The first
one is the L2 -norm of ψ
N ψ̄, ψ := |ψ|2 = ψ22
(1.10)
and the second one is the Hamilton (or energy) functional
1
λ
2
H ψ̄, ψ :=
|∇ψ| +
V |ψ|2
4
2
1 −
∗ |ψ|2 |ψ|2 .
4
(1.11)
We note that if is a non-negative function belonging to Lp + L∞ , p ≥ n2 , then,
for an arbitrary δ > 0, there exists a finite constant C(δ) such that
2
0≤
∗ |ψ|2 |ψ|2 ≤ δN ψ̄, ψ ∇ψ22 + C(δ) N ψ̄, ψ ,
(1.12)
226
J. Fröhlich, T.-P. Tsai, H.-T. Yau
see e.g. [7]. Thus, for an arbitrary, but fixed value of N (ψ̄, ψ), and for arbitrary λ, |λ| <
∞, the Hamilton functional H(ψ̄, ψ) is bounded from below.
Under the assumptions that λV (x) has a minimum at x = x∗ , |x∗ | < ∞, that
(x) ≥ 0 and that the value, N, of the functional N (ψ̄, ψ) is large enough, one can
show (see Sect. 3) that the Hamilton functional H(ψ̄, ψ) restricted to the sphere
SN := ψ ψ ∈ H 1 , N ψ̄, ψ = N
(1.13)
in Sobolev space reaches its minimum on a positive function QN ∈ SN concentrated
near x∗ and decaying exponentially fast in |x|, as |x| → ∞ . This result still holds
when
λ = 0 (i.e., for a vanishing external potential); but if QN is a minimizer of H S then
N
so is QN,a , where QN,a (x) := QN (x − a), for arbitrary a ∈ RN . This is a consequence
of the translation invariance
of H, for λ = 0.
A minimizer, QN of H S is a solution of the non-linear eigenvalue equation
N
−
1
Q + λV Q − ∗ Q2 Q = EQ,
2
for some real number E, with
(1.14)
N Q̄, Q = N .
Then
ψ(x, t) = QN (x)e−iEt
is a stationary solution of the Hartree equation (1.5). Multiplying Eq. (1.14) by Q := QN
and integrating, we find that
1
λ
1
E=
(1.15)
V Q2N −
∗ Q2N Q2N .
(∇QN )2 +
2N
N
N
One should notice that EN is not the value of the energy functional H(ψ̄, ψ) S evalN
uated on the minimizer ψ = QN , because one is minimizing H(ψ̄, ψ) in the presence
of a constraint, namely N (ψ̄, ψ) = N .
(0)
Let QN be a minimizer of the Hamilton functional H(ψ̄, ψ) S , with λ = 0,
N
(0)
centered at x = 0; (QN is known to exist and to be non-trivial, for N large enough).
We set λ = 1 and choose
V (x) ≡ V (ε) (x) := W (εx),
(1.16)
where W is a fixed, smooth, bounded, positive function on Rn , and ε > 0 is a parameter.
Our main concern, in this paper, is to construct local (in time t) solutions of the Hartree
equation (1.5), with λ = 1 and V = V (ε) as in (1.16), of the form
(0)
(1.17)
ψ(x, t) = QN (x − r(t)) + hε (x − r(t), t) eiθ(x,t) ,
(0)
where hε is a small, dispersive correction to the solitary wave described by QN (x −
r(t))eiθ(x,t) , with
hε (·, t)H 1 0(ε3/2 ),
(1.18)
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
227
θ (x, t) is a time-dependent phase,
θ (x, t) = v(t)·x − Et + ϑ0 (t),
(1.19)
where v(t) = dr(t)
is the velocity of the solitary wave, and ϑ0 (t) is independent of
dt
x, for all times t with |t| 0(ε −1 ), and provided the soliton trajectory (r(t), v(t))
solves appropriate equations of motion. It will be shown that (r(t), v(t)) must solve the
Newtonian equations of motion
ṙ(t) = v(t),
v̇(t) = −ε(∇W )(εr(t)) + a(t),
(1.20)
where a(t) is a “friction force”, with
|a(t)| 0(ε 2 ),
(1.21)
for |t| 0(ε−1 ). The friction force a(t) will be determined more precisely in Sect. 3.
Neglecting the friction force a(t), Eqs. (1.20) are Newton’s equations of motion for
a point particle of mass N moving in an external acceleration field of strength ε with
potential V (ε) . Thus, for the velocity v(t) of this particle to deviate substantially from
the initial condition v(0) = v0 , the time t must be 0(ε −1 ). For times t, with |t| 0(ε −1 ),
the friction force a(t) has a negligibly small effect, for small ε.
A solution of the Hartree equation (1.5) of the form (1.17), with properties (1.18)
through (1.21), for times t with |t| 0(ε −1 ), describes the motion of an extended
particle in a shallow potential well V (ε) interacting weakly with a dispersive medium
of infinitely many degrees of freedom with which it can exchange mass and energy.
The point-particle limit in which Newton’s laws of motion become exact is the limit
ε → 0. For ε > 0, the interactions between the extended particle and the dispersive
medium can lead to phenomena such as mass accretion, loss of mass and energy from
the particle into dispersive waves, and friction, for times t large on a scale of ε −1 . The
intuitive picture is one of a bound cluster of “dust” describing an extended particle,
which exhibits Newtonian motion with friction. The friction is caused by the loss of
some “dust” originally bound to the particle. This loss of “dust” is only observed when
the motion of the particle is not inertial (i.e., accelerated or decelerated) and is described
by dispersive waves satisfying a wave equation which is essentially the linearization of
(0)
Eq. (1.5) around a solitary wave described by QN(t) (x − r(t))eiθ(x,t) . For very large
times, the trajectory of the extended particle is expected either to approach an inertial
motion diverging to spatial infinity (if W (x) → const, as |x| → ∞ and if the initial
mass and velocity of the particle were large enough), or to approach a local minimum of
W where the particle will come to rest. This dissipative behavior of the particle motion is
an example of the general phenomenon of “dissipation through radiation”. Some simple
results on the large-time asymptotics of solutions of the Hartree equation (1.5) (existence
of wave operators) are proven in Sect. 4. But it is fair to say that we do not yet have a
good mathematical understanding of large-time behavior of solutions of Eq. (1.5). For
some earlier results on scattering for the Hartree and nonlinear Schrödinger equation,
see, e.g., [7,8] and references given there.
Our analysis of solutions of the Hartree equation (1.5) of the form described in (1.17),
with properties (1.18) through (1.21), is based on a key assumption, which is, implicitly,
228
J. Fröhlich, T.-P. Tsai, H.-T. Yau
an assumption on the two-body potential − that will not be made explicit in this paper:
Let
(f, g) := f¯g
denote the usual scalar product on L2 , and let H denote the Hessian of the Hamilton
(0)
denote the
functional H(ψ̄, ψ), with λ = 0, at ψ = QN . Furthermore, let Hreal
restriction of H on real-valued functions, and extend it to a complex-linear operator. It
is given by an unbounded, selfadjoint operator on L2
will be shown in Sect. 3 that Hreal
defining a quadratic form on H 1 which is bounded from below. It is not hard to see that
(0)
Q, (Hreal
− E)Q = ε0 (Q, Q) < 0, for Q := QN ,
(1.22)
where E = EN and
2
ε0 = −
N
∗ Q2 Q 2 .
(1.23)
−E has only one negative eigenvalue. Since H is translation-invariant,
Actually, Hreal
it follows that ∇Q := {∂1 Q, . . . , ∂n Q}, ∂j := ∂x∂ j , j = 1, . . . , n, are n non-vanishing,
− E orthogonal to Q, i.e.,
linearly independent zero-modes for Hreal
− E)∂j Q = 0, and ∂j Q, Q = 0,
(1.24)
(Hreal
− E.
for all j = 1, . . . , n. Thus 0 is an at least n-fold degenerate eigenvalue of Hreal
Since Q is a minimizer of H(ψ̄, ψ)S , there is no spectrum of Hreal − E in the interval
N
− E in the interval
(ε0 , 0). Furthermore, it is easy to see that the spectrum of Hreal
[0, −E), where
1
1
2
2
E=
(1.25)
∗ Q Q2 < 0
(∇Q) −
N 2
is pure-point, while, on the half-line [−E, ∞), it is continuous. Thus, there is a gap,
− E in [0, ∞); see Sect. 3 for
ε2 > 0, between 0 and the rest of the spectrum of Hreal
details.
−E is precisely
Our key assumption is that the multiplicity of the eigenvalue 0 of Hreal
equal to n. This implies that
h, (Hreal
− E)h ≥ ε2 (h, h), ε2 > 0,
(1.26)
for all functions h ∈ H 1 with h ⊥ {Q, ∇Q} in the L2 -scalar product (·, ·).
We are now prepared to summarize the contents of this paper and to state our main
results in the form of theorems.
In Sect. 2, we recall the Hamiltonian nature of the Hartree equation (1.5) on the phase
space H 1 . We exhibit continuous symmetries of the Hamilton functional that give rise
to Eq. (1.5) and derive the corresponding conservation laws. We show that the Hartree
equation can also be viewed as the Euler–Lagrange equation derived from an action
functional. The Lagrangian formulation of the non-linear Hartree equation is useful to
study the formal point-particle limit (the ε → 0 limit in (1.16) through (1.21)). This limit
is discussed, in general terms but without mathematical proofs, in Sect. 2, using ideas
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
229
similar to those in [9] in an analysis of vortex motion in the Ginzburg-Landau equation,
which is based on an effective-action formalism. We also discuss some expected features
of the non-linear Hartree dynamics in the large-time limit.
Our first main result is proven in Sect. 3.
(0)
Theorem 1.1. Suppose that assumption (1.26) holds for all minimizers Q = QN , with
N in an open neighborhood of some N0 > 0. We also assume that
(x) is radial,
W 2,1 (R3 )∩W 2,∞ (R3 ) ≤ C
(1.27)
for some constant C . Then there is a positive constant C0 such that, for an arbitrary
T < ∞, there is an ε0 > 0 with the property that, for any 0 < ε ≤ ε0 and any initial
condition of the form
(1.28)
ψ(x, 0) = ψ0 (x) = Q (x − r0 ) + hε,0 (x) eiv0 x ,
with Q = QN0 and hε,0 H 1 ≤ C0 ε 3/2 , the Hartree equation, Eq. (1.5), with λ = 1
and V (x) = W (εx) as in (1.16), has a solution of the form (1.17), for all times t with
|t| < T ε−1 , with the following properties:
1. The phase θ (x, t) is as in (1.19);
2. the trajectory (r(t), v(t)) of the extended-particle solution (1.17) is a solution of the
equations of motion (1.20) with initial conditions r(t) = r0 , v(t) = v0 , for a friction
force a(t) bounded by
|a(t)| ≤ C1 ε 2 ;
3. the dispersive correction hε satisfies
hε (·, t)H 1 ≤ C2 ε 3/2 ,
for some finite constants C1 , C2 depending on T .
This result makes the point-particle limit (ε → 0) of the Hartree equation (1.5) precise
for initial conditions describing a single extended particle (solitary wave) moving in a
shallow potential well, W (εx), and perturbed by a small amount of radiation (described
by hε ). It is a special case of the more general situation considered in Sect. 3. A more
detailed discussion and the proof of Theorem 1.1 form the contents of Sect. 3.
The results just described raise the issue of asymptotic properties of the dynamics
determined by the Hartree equation, as time t tends to ±∞. In Sect. 4, we establish a result
on the scattering of small-amplitude waves off a single solitary wave. For simplicity,
we suppose that physical space is three-dimensional, n = 3, (but our methods can be
applied whenever n ≥ 3), we set λ = 0, and we choose to be a non-negative, bounded
function of rapid decrease, as |x| → ∞. We consider an “asymptotic profile” described
by
ψas (x, t) = Q (x − r0 − v0 t) e
i x·v0 − 21 v02 +E t
+ has (x, t),
(1.29)
where has is a solution of the free-particle Schrödinger equation
1
i∂t has (x, t) = − has (x, t),
2
(1.30)
230
J. Fröhlich, T.-P. Tsai, H.-T. Yau
with initial condition has (x, 0)
to and being sufficiently small
=: has,0 (x) belonging
in the space H 4 (R3 ) ∩ W 3,1 R3 , 1 + |x|2 d 3 x and such that the Fourier transform,
(0)
ĥas,0 (k), vanishes at k = v0 . In (1.29), Q = QN0 ∈ SN0 is a solution of Eq. (1.14), with
(0)
λ = 0, and it is assumed that inequality (1.26) is satisfied for Q = QN ∈ SN , for all N
in a small neighborhood of N0 > 0.
Theorem 1.2. For an asymptotic profile ψas (x, t) as described in (1.29), (1.30), and
under the hypotheses stated above, there are solutions, ψ± (x, t), of the Hartree equation
(1.5) (for λ = 0) such that
ψ± (x, t) −→ ψas (x, t), as t → ±∞,
(1.31)
in H 2 (R3 ). Their difference is of order O(t −1 ).
Thus the non-linear Møller wave maps /± : ψas −→ ψ± exist as symplectic maps
on asymptotic profiles of the form (1.29), (1.30). We emphasize that the effect of the
scattering wave on the location and the phase of the soliton has to be tracked precisely
for all time. The stability of the soliton is quite simple and can be obtained purely
from energy consideration. A review can be found in Sect. 3 (see also Weinstein [14]).
Therefore, the key points of Theorem 1.2 are its two precise assertions: 1. The location of
the soliton is almost “linear.” 2. The scattering wave behaves like an ordinary dispersive
wave, (described by has (x, t)), plus a small correction. The condition on the Fourier
transform of has,0 is a technical one and we expect to remove it later on. Our result
constitutes the first step toward scattering theory.
The proof of Theorem 1.2 is the contents of the final section, Sect. 4, of this paper.
2. The Hartree Equation as a Hamiltonian System with Infinitely Many Degrees
of Freedom, and Its Point-Particle Limit
In the introduction, we have described results indicating how the Hartree equation (1.2)
captures the dynamics of a system of very many non-relativistic bosons with very weak
two-body interactions in a condensate state. This regime has been called the “mean-field
limit”. Actually, the mean-field limit is equivalent, mathematically, to the classical limit
in which the value of Planck’s constant, h̄, is sent to 0. We are accustomed to expect
(actually in general erroneously) that the unitary dynamics of a quantum-mechanical
system reduces to the Hamiltonian dynamics of a corresponding classical system, in the
classical limit. In the examples studied in this paper, this expectation is justified.
2.1. The Hamiltonian nature of the Hartree equation. The phase space, 0, for the
Hartree equation (1.5) is the Sobolev (energy) space H 1 (Rn ) defined in (1.4). We use
ψ(x) and its complex conjugate ψ̄(x), x ∈ Rn , as complex coordinates for 0. The
symplectic 2-form on 0 is given by 2i dψ ∧ d ψ̄. It leads to the following Poisson
brackets:
{ψ(x), ψ(y)} = ψ̄(x), ψ̄(y) = 0,
ψ(x), ψ̄(y) = 2iδ(x − y) .
(2.1)
(2.2)
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
231
The Hamilton functional, H(ψ̄, ψ), leading to the Hartree equation (1.5) is given by
1 H(ψ̄, ψ) =
(2.3)
|∇ψ|2 + 2λV |ψ|2 − ∗ |ψ|2 |ψ|2 .
4
For ∈ Lp + L∞ , p ≥ n2 , H is well defined on 0 and bounded below on the spheres
SN = ψ ψ ∈ 0, N ψ̄, ψ = N < ∞ ,
(2.4)
where
N ψ̄, ψ =
|ψ|2 ;
see inequality (1.12). Hamilton’s equations of motion for ψ are given by
ψ̇t (x) = H ψ̄t , ψt , ψt (x)
1
=i
ψt (x) − λV (x)ψt (x)
2
+ ∗ |ψt |2 (x)ψt (x) ,
(2.5)
(2.6)
which is precisely the Hartree equation (1.5).
From (2.3) we infer the following symmetries and corresponding conservation laws.
(1) Gauge invariance of the first kind. The phase transformations
ψ(x) → eiθ ψ(x), ψ̄(x) → e−iθ ψ̄(x)
(2.7)
leave H(ψ̄, ψ) invariant. These transformations describe the symplectic flow generated
by the Hamiltonian vector field corresponding to the function 21 N (ψ̄, ψ). Since they
are a symmetry of H(ψ̄, ψ), it follows that
{H, N } = 0,
(2.8)
and hence N is conserved, and the spheres SN defined in (2.4) are invariant under the
time evolution ψ → ψt described by (2.6).
(2) Galilei invariance, for λ = 0. We shall assume henceforth that is rotationinvariant. If the external potential λV vanishes then arbitrary Galilei transformations
are symmetries of H.
Space translations, x → x + a, are represented on 0 by
ψ(x) → ψa (x) := ψ(x − a), a ∈ Rn ,
and are generated by the momentum functional
i
P(ψ̄, ψ) :=
ψ̄∇ψ .
2
(2.9)
They clearly leave H(ψ̄, ψ) invariant, hence P is conserved under the time evolution
and
{H, P} = 0 .
(2.10)
232
J. Fröhlich, T.-P. Tsai, H.-T. Yau
Rotations, Rab , in the (ab)-plane of Rn , 1 ≤ a < b ≤ n, are represented on 0 by
−1
x .
ψ(x) → ψRab (x) := ψ Rab
They are generated by the angular momentum functionals
Lab (ψ̄, ψ) :=
i
2
ψ̄ x a ∂b − x b ∂a ψ,
(2.11)
with ∂b = ∂/∂x b . Since has been assumed to be rotation-invariant, rotations leave
H(ψ̄, ψ) invariant, hence the functionals Lab are conserved under the time evolution
and Poisson-commute with H, for all (ab).
Finally, boosts (velocity transformations), x → x − vt, v ∈ Rn , t denotes time, are
represented on time-dependent trajectories, ψt (x), in 0 by
ψt (x) → ψt (v; x) := ψt (x − vt)e
2
i v·x− v2 t
.
(2.12)
They do not leave H invariant, but one easily checks that if ψt (x) is a solution of
Hamilton’s equations of motion (2.6) then so is ψt (v; x), for arbitrary v ∈ Rn . The
conserved quantity corresponding to (2.12) is given by
Mv ψ̄t , ψt :=
ψ̄t v· (x + it∇) ψt .
(2.13)
It follows that the “centre of mass motion” of a solution ψt of (2.6) is inertial.
We conclude this section by noting that, as usual, Hamilton’s equations of motion
(2.6) can also be viewed as Euler–Lagrange equations derived from an action principle.
The action functional is defined on a space of continuously differentiable (in time)
trajectories in phase space 0. It is given by
t2
S ψ̄, ψ :=
t1
i
dt
ψ̄t ψ̇t − H ψ̄t , ψt .
2
(2.14)
The Hartree equation (2.6) is obtained from the action functional S(ψ̄, ψ) by variation
with respect to ψ̄, i.e., it is equivalent to the equation
δS ψ̄, ψ δ ψ̄t (x) = 0,
(2.15)
under the boundary conditions that
δψti (x) = 0, i = 1, 2 .
(2.16)
Global existence and uniqueness of solutions of the equations of motion (2.6), for
∈ Lp + L∞ , p ≥ n2 , is proven in [7] and refs. given there.
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
233
2.2. Stationary solutions of the Hartree equations, for fixed values of N , P and Lab . In
this section, we consider stationary solutions of the non-linear Hartree equations (2.6),
assuming that λV = 0 and that is rotation-invariant. Since the L2 -norm N (ψ̄, ψ), the
momentum functional P(ψ̄, ψ) and the angular momentum functionals Lab (ψ̄, ψ) are
conserved, we may put them to fixed values, N, P and Lab , respectively. In order to find
stationary solutions of (2.6), with N (ψ̄, ψ) = N, P(ψ̄, ψ) = π and Lab (ψ̄, ψ) = λab ,
we may look for critical points of the generalized energy functional
E N − N ψ̄, ψ
E ψ̄, ψ; E, P , Lab := H ψ̄, ψ +
2
ab + P · π − P ψ̄, ψ +
L λab − Lab ψ̄, ψ ,
(2.17)
a<b
where E, P and Lab are Lagrange multipliers. By varying E ψ̄, ψ; E, P , Lab with
respect to ψ̄, ψ, E, P and Lab , we find the equations
−
1
ψ − ∗ |ψ|2 ψ − Eψ
2
− iP ·∇ψ − i
Lab x a ∂b − x n ∂a ψ = 0,
(2.18)
a<b
(variation with respect to ψ̄), and
N ψ̄, ψ = N (variation with respect to E),
P ψ̄, ψ = π (variation with respect to P ),
(2.19)
(2.20)
and
Lab ψ̄, ψ = λab (variation with respect to Lab ),
(2.21)
1≤a<b≤n.
Not much is known about the general solution of Eqs. (2.18) through (2.21). But, for
the purposes of this paper, the following solutions are particularly important: We look
(0)
for a rotation-invariant absolute minimum, QN , of the Hamilton functional H(ψ̄, ψ)
restricted to the sphere SN , which has zero momentum. Equations (2.18) through (2.21)
then simplify to
−
1
ψ − ∗ |ψ|2 ψ = Eψ,
2
N ψ̄, ψ = N,
(2.22)
(2.23)
(0)
and the solution, ψ = QN , must satisfy
and
i (0)
(0)
(0)
(0)
QN , ∇QN = 0
P Q̄N , QN =
2
(2.24)
(0)
x a ∂b − x b ∂a QN = 0, for all a < b .
(2.25)
234
J. Fröhlich, T.-P. Tsai, H.-T. Yau
Equation (3.22) is identical to Eq. (1.14), for λV = 0, and E is given by
1
E=
2N
1
(0) 2
(0)2
(0)2
QN ,
∇QN
∗ QN
−
N
(2.26)
(0)
see Eq. (1.15), which is strictly negative, for a non-trivial minimizer QN .
Lemma 2.1. For a positive, rotation-invariant potential ∈ Lp + L∞ , p ≥ n2 , with
(x) → 0, as |x| → ∞, there exists a constant N∗ = N∗ (), with 0 ≤ N∗ < ∞ , such
(0)
(0)
(0)
that, for N > N∗ , (2.23) has a non-trivial solution ψ = QN , with N Q̄N , QN = N ,
(0)
corresponding to a local minimum of H ψ̄, ψ S . The phase of QN can be chosen
N
(0)
such that QN > 0. The non-linear eigenvalue E is given by (2.26) and is strictly
(0)
negative, for N > N∗ . The function QN (x) is smooth and decays exponentially, as
√
|x| → ∞, with decay rate −E.
Remarks. (i) From the theory of quantum-mechanical bound states we infer that, in
n = 1, 2 dimensions, N∗ = 0, while, for n ≥ 3, N∗ is strictly positive if is integrable,
but vanishes for potentials of very long range, such as the Coulomb potential; see [10].
(0)
(ii) Given a solution, QN , of (3.22), the function
(0)
ψt (v; x) := QN (x − r − vt)e
i v·x− 21 v 2 +E t
(2.27)
solves the Hartree equation (2.6), with λV = 0, for arbitrary r ∈ Rn and v ∈ Rn . This
follows from the Galilei invariance of the theory. For ψt as in (2.27),
P ψ̄t , ψt = N v .
(2.28)
Equation (2.6) also has wave-like solutions with P = 0, (e.g. ψt (x) =
ψ0 exp i (k · x − E (k, ψ0 ) t), which has infinite energy and momentum). It would be
of interest to also study square-integrable, stationary rotating soliton solutions of (2.6)
with Lab = 0.
(iii) It is straightforward to extend Lemma 2.1 to systems where λV = 0. Such
generalizations are
of particular interest when V has symmetries. Then minimizers,
(0)
QN , of H(ψ̄, ψ)S tend to break the symmetries of λV if N is large enough.
N
(0)
(iv) Let H denote the Hessian of H(ψ̄, ψ) at ψ = QN , (λ = 0). In our proofs
of Theorems 1.1 and 1.2 (see Sects. 3 and 4), we shall always assume that assumption
(1.26) holds, for all N in an open neighborhood of some N0 > N∗ .
Since the proof of Lemma 2.1 is standard, it is omitted. The interesting analytical
issues arise in the problems described in Remarks (iii) and (iv). They deserve further
study.
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
235
2.3. A heuristic discussion of the point-particle limit of the Hartree equation. In this
section we start from the results reviewed in the last section (see Lemma 2.1) to study
the point-particle (Newtonian) limit of the Hartree equation. In this limit the Hartree
equation reduces to the Newtonian mechanics of point-particles interacting through
two-body potential forces. We use ideas closely related to those proposed in [9] in an
analysis of vortex motion in the plane, as described by the Ginzburg–Landau equations.
Let λV and be as in Eqs. (1.5), (2.6). We set λ = 1 and consider a family of
external potentials of the form
V (x) ≡ V (ε) (x) := W (εx),
(2.29)
where W is some smooth, positive function on Rn , and ε > 0 is a parameter. Furthermore,
the two-body potential, −, is chosen to be
(x) = s (x) + 6 (εx),
(2.30)
where s (x) is a rotation-invariant, smooth function decaying rapidly in ρ := |x|, as
ρ → ∞, and with the properties that
ds (ρ)
< 0, for ρ > 0,
dρ
(2.31)
and that the key gap assumption (1.26) stated in Sect. 1 holds for = s . The perturbing
potential 6 is rotation-invariant and smooth and may be of long range, e.g.
6 (ρ) ∼ ρ 2−n , as ρ → ∞,
(2.32)
for n ≥ 3, which is the behavior of the Coulomb and of Newton’s gravitational potential.
For simplicity, we assume that |d6 (ρ) dρ is uniformly bounded in ρ.
We pick k positive integers N1 , . . . , Nk , with Nj > N∗ (s ), for all j . For λV = 0
and N > N∗ (s ), we define
δN :=
N −1
(0)
d n xQN (x)2 x 2 ,
(2.33)
(0)
where QN is a rotation-invariant minimizer of the functional H(ψ̄, ψ)S , as described
N
in Lemma 2.1.
We consider an initial condition, ψ0 (x), for the Hartree equation (2.6) describing
(0)
a configuration of k far-separated “solitons”, QNj (x − rj ), rj ∈ Rn , j = 1, . . . , k,
(perturbed by a small-amplitude wave), with the following properties: Each soliton
(0)
QNj (x) is a rotation-invariant solution of Eq. (3.22), with = s and N = Nj ,
minimizing H(ψ̄, ψ) (for λ = 0, = s ). Furthermore
SN
max
j =1,... ,k
δNj
min
1≤i<j ≤k
ri − rj where ε is the parameter introduced in (2.29), (2.30).
≤ ε,
(2.34)
236
J. Fröhlich, T.-P. Tsai, H.-T. Yau
Our goal is to construct a solution, ψt , of the Hartree equation (2.6) of the form
ψt (x) =
k
j =1
(0) QNj (t) x − rj (t) eiθj (x,t) + hε (x, t),
(2.35)
where rj (0) = rj , as in (2.34), and ṙj (0) = vj ∈ Rn , j = 1, . . . , k, with the following
properties: There is a positive constant T such that, for all times t with |t| < Tε ,
(a)
(b)
hε (·, t) ∼ o(ε),
for an appropriately chosen norm (·) ,
θj (x, t) = ṙj (t)· x − rj (t) + ϑj (t),
where ϑj (t) is independent of x, and
Ṅj (t) = o(ε).
(c)
The trajectories r1 (t), . . . , rk (t) and the phases ϑ1 (t), . . . , ϑk (t) will turn out to satisfy
equations of motion which can be derived from the Hartree equation. In this section we
do not present a mathematical proof of the claim that solutions of the Hartree equation
(2.6) of the form (2.35) with properties (a)–(c) exist; (but see Sect. 3). We merely verify
that a function ψt (x) of the form (2.35) with properties (a)–(c) approaches a critical
point of the action functional S(ψ̄, ψ) introduced in (2.14), as ε → 0, provided the
trajectories rj (t) satisfy certain Newtonian equations of motion and the phases ϑj (t)
are suitably chosen (j = 1, . . . , k). Since critical points of S(ψ̄, ψ) satisfy the Hartree
equation (2.6), this makes it plausible that solutions of (2.6) of the form (2.35) with
properties (a) – (c) exist. This claim is proven in Sect. 3 for k = 1.
Our heuristic analysis is based on the following simple facts:
(1) For
i = j,
(0)
(0) d n xQNi (x − ri ) QNj x − rj → 0,
exponentially fast, as |ri − rj | = 0(ε−1 ) → ∞. This follows from Lemma 2.1.
T
(0)
(2) QNi (t) , hε (·, t) = o(ε), for |t| ≤ ,
ε
as ε → 0, for all i = 1, . . . , k; see (2.35) and property (a).
(0)
(0)
(3) QNi , ∇QNi = 0, for all i,
by translation invariance (see Eq. (1.24)).
(4) For y := x − ri (t),
(0)
2
d n y QNi (t) (y) y = 0, for all i,
by rotation invariance.
(0)
(0)
(5) Ṅi (t) = 2 QNi (t) , QṄ (t) , for all i,
i
(0)
(0)
because Ni = QNi , QNi .
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
Using that
∂ (0) QNj (t) x − rj (t) eiθj (x,t)
∂t
(0) (0) = QṄ (t) x − rj (t) − ṙj (t)·∇QNj (t) x − rj (t)
j
(0) + i θ̇j (x, t)QNj (t) x − rj (t) eiθj (x,t) ,
with
237
(2.36)
θ̇j (x, t) = r̈j (t) x − rj (t) − ṙj (t)2 + ϑ̇j (t),
(2.37)
∇θj (x, t) = ṙj (t),
(2.38)
and
we find that, for ψt (x) as in (2.35), the action functional S(ψ̄, ψ) introduced in (2.14),
with − Tε ≤ t1 < t2 ≤ Tε , is given by
k (0) 2 i
dt
Ṅj − QNj x − rj r̈j · x − rj
2
j =1
t1
Nj 2
1 (0) 2
2
∇QNj −
ṙ
+ Nj ṙj − Nj ϑ̇j −
2
2 j
(0) 2 (0) 2
1
∗ QNj QNj − Nj W εrj +
2
1 +
Ni Nj 6 ε ri − rj + sε ,
2
1
S ψ̄, ψ =
2
t2
(2.39)
i:i=j
where sε is an error term ∼ o(ε). In the first term on the R.S. of (2.39) we have used
(5), the second term proportional to r̈j vanishes by (4), in the third and fourth term we
have used (2.37), in the sixth term we have used (2.38), and various cross terms vanish
because of (3) or only contribute to the error term because of (1) and (2). We have also
used that
(0) 2
d n xW (εx)QNj x − rj = Nj W εrj + o(ε) ;
and that, for i = j ,
(0)
(0) 2
2
d n x d n y QNi (x − ri ) (x − y) QNj y − rj = Ni Nj 6 ri − rj + o(ε),
by (4) and because s (x) decays rapidly in |x|. Thus
1
S ψ̄, ψ =
SNewton rj , Nj j =1,... ,k
2
t2 k i
1
(0)
(0)
dt
Ṅj − Nj ϑ̇j − 2H QNj , QNj + sε ,
+
2
2
t1
j =1
(2.40)
238
J. Fröhlich, T.-P. Tsai, H.-T. Yau
where
SNewton
rj , Nj
t2
dt
=
j =1,... ,k
k Nj 2
ṙj − Nj W εrj
2
j =1
t1
1 Ni Nj 6 ε ri − rj
+
2
(2.41)
i:i=j
is the usual Hamiltonian action for k point particles with masses N1 , . . . , Nk in an
external acceleration field
potential
W (ε·) and interacting through two-body forces
with
with potential Ni Nj 6 ε ri − rj .
In order to guarantee that the ansatz (2.35) yields a solution of the Hartree equation (2.6) with properties (a), (b) and (c), we must require that the variation of the
action S ψ̄, ψ calculated in (2.40), (2.41) with respect to the variational parameters rj , Nj , ϑj , j = 1, . . . , k, and hε vanish! To write down the variational equations,
we observe that the second term on the R.S. of (2.40) isindependent
of r1 , . . . , rk , except
for the error term sε , which is o(ε). Thus, varying S ψ̄, ψ with respect to r1 , . . . , rk
yields Newton’s equations of motion
r̈j = − ε (∇W ) εrj
ε +
Ni (∇6 ) ε rj − ri + aj ,
(2.42)
2
i:i=j
where aj comes from the error term sε , and |aj (t)| ∼ o(ε), for |t| ≤
Variation with respect to N1 , . . . , Nk yields the equations
ϑ̇j =
T
ε
; j = 1, . . . , k.
1 2
Ni 6 ε ri − rj
ṙj − W εrj +
2
i:i=j
∂
(0)
(0)
H QNj , QNj + o(ε) .
−
∂Nj
(2.43)
It is easy to see that
1
∂
(0)
(0)
(0) 2
∇QNj
H QNj , QNj =
∂Nj
2Nj
1
(0)2
(0)2
∗ QNj QNj
−
Nj
= Ej − Nj 6 (0) + o(ε),
see Eq. (2.26). Hence, for |t| ≤
T
ε
(2.44)
,
k
ϑ̇j =
1 2
ṙj − W εrj +
Ni 6 ε ri − rj − Ej + o(ε) .
2
i=1
(2.45)
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
239
Variation with respect to ϑ1 , . . . , ϑk yields the equations
Ṅj = o(ε),
(2.46)
(approximate conservation of masses of particles), and, finally, variation with respect to
hε yields an equation of motion of the form
k ∂
(2.47)
hε (x, t) = X hε , rj , Nj , ϑj j =1 (x, t),
∂t
with X ∼ o(ε), for |t| ≤ Tε , where (·) is an appropriately chosen norm.
At a heuristic level, eqs. (2.42) and (2.46) show very clearly that the limit ε → 0
corresponds to the point-particle limit in which the masses, N1 , . . . , Nk , of the particles
(“solitons”) are constant and their trajectories are solutions of Newton’s equations of
motion, on time scales of 0(ε−1 ).
It is interesting and useful to work out explicit expressions for all the terms of o(ε) in
Eqs. (2.42), (2.45), (2.46) and (2.47), in order to understand more about the corrections
to the Newtonian point-particle limit and to get a handle on phenomena like radiation
loss and dissipation through emission of small-amplitude dispersive radiation. But, since
our discussion in this section is at a formal level, let’s not! In the special case where
k = 1, the terms of size o(ε) are analyzed in Sect. 3.
The analysis of the correction term sε in expression (2.40) for the action functional
and of the properties of solutions of Eq. (2.47) is crucial in attempting to understand the
long-time behavior of solutions of the Hartree equation (2.6). In the introduction, we have
drawn attention to results of Soffer and Weinstein [8], see also [12], concerning “nonlinear Rayleigh scattering” for small-amplitude solutions of the non-linear Schrödingeror Hartree equations with a suitable external potential λV . One would like to extend their
results in the direction of a theory of non-linear resonances (metastable states) and gain
understanding of the phenomenon of “approach to a groundstate”. Of particular interest
are situations where the Hamilton functional H(ψ̄, ψ), see (2.3), restricted to a sphere
SN in phase space has several distinct local minima, for N large enough. This happens
when λV has several minima separated by large barriers and − is the potential of an
attractive force. One would then like to understand the shape of the “basins of attraction”
in phase space of the local minima of H(ψ̄, ψ)S : The forward (backward) basin of
N
attraction of a family of local minima of H(ψ̄, ψ)S parametrized by N consists of all
N
initial conditions in phase space which approach an element of this family plus dispersive
radiation decaying to 0 at the free dispersion rate, as t → +∞ (t → −∞). This is the
phenomenon of “approach to a groundstate”.
More ambitiously, one might try to construct a “centre manifold” of asymptotically
attracting configurations of solitons to which solutions of the Hartree equation with
initial conditions sufficiently close to the centre manifold converge locally in space, as
|t| → ∞. See [12] for some preliminary results.
Let us consider an example: We choose an initial condition for the Hartree equation
describing two far-separated solitons at positions r1 , r2 and with initial velocities v1 , v2 .
We suppose that λV = 0 and that −6 is purely attractive and of short range. The
“masses” N1 , N2 of the solitons and the initial conditions r1 , v1 and r2 , v2 are chosen
such that the two solitons form a bound state, i.e., that
N1 2 N2 2
v +
v − N1 N2 6 (ε (r1 − r2 )) < 0 .
2 1
2 2
(2.48)
240
J. Fröhlich, T.-P. Tsai, H.-T. Yau
One would then like to calculate the power, PR (t), of emission of dispersive radiation
through a sphere of radius R " max (|r1 |, |r2 |) . Moreover, one would like to show
that, as t → ±∞, a typical configuration of two solitons satisfying (2.48) collapse to
a single soliton moving through space at a constant velocity. This phenomenon would
describe the “radiative collapse of a binary system”.
More generally, it would be interesting to understand how, at intermediate times,
small inhomogeneities in the initial conditions for solutions of the Hartree equation
grow to form a structure of rotating bodies (solitons) perturbed by outgoing, dispersive
radiation, before it eventually approaches a number of far separated solitons escaping
from each other. [In studying such problems, one finds out that the Hartree equation
not only “knows” about Newton’s equations of motion, it also “knows” about the Euler
equations for the motion of rigid bodies.]
The problems described here are problems on the scattering theory for the Hartree
equation. If − is attractive, i.e., for a self-focussing non-linearity, scattering theory is
bound to be very subtle, involving infinitely many “scattering channels”, and is beyond
the reach of our methods; (see, however, Sect. 4 for some preliminary results, and [7]
for the case where − is repulsive).
3. Proof of Theorem 1.1
In this section, we prove the first main result (Theorem 1.1) of this paper.
3.1. Stability of soliton solutions of Hartree equations. We first review the stability of
the soliton solutions to the Hartree equation without external potential, i.e., for λ = 0.
The equation is
i∂t ψ = 2
1
∂H
= − 9ψ − ( ∗ |ψ|2 )ψ,
2
∂ ψ̄
(3.1)
where ∂∂H
(H = H(λ=0) , see (1.11)) is the first variation of the energy functional w.r.t.
ψ̄
ψ̄. Recall that Q is a minimizer of H under the constraint N (ψ̄, ψ) := ψ2 = N , for
some N fixed, and thus Q satisfies the equation
1
− 9Q − ( ∗ |Q|2 )Q = EQ,
2
(3.2)
for some non-linear eigenvalue (Lagrange multiplier) E. Suppose the function ψ can be
written in the form ψ = (Q + h)e−iEt . Then the linearized equation satisfied by h takes
the form
i∂t h = Lh,
(3.3)
1
Lh = − 9h − Eh − ( ∗ Q2 )h − Q( ∗ (Q(h + h̄))).
2
(3.4)
where
Due to the appearance of h̄ on the right side of (3.4), L is not a complex-linear operator.
It is therefore convenient to separate the last equation into real and imaginary parts
Lh = L+ A + iL− B,
h = A + iB,
(A and B real),
(3.5)
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
241
where
1
L− = − 9 − E − ∗ Q2 ,
2
L+ = L− − 2Q[ ∗ (Q·)],
In matrix form,
∂
∂t
(L+ A = L− A − 2Q[ ∗ (QA)]).
A
0 L−
A
A
=
=: L
.
B
−L+ 0
B
B
(3.6)
(L is the matrix form of −iL; it determines a linear Hamiltonian vector field.)
The operators L− and L+ also appear naturally in the second variation of the energy
functional H. Writing ψ = u + iv, we have by explicit computation
∂H ∂H H(Q + h) = H(Q) + dx
A + dx
B
∂u Q
∂v Q
1
∂ 2 H ∂ 2 H +
dxA
A + 2 dxA
B
2
∂u∂u Q
∂u∂v Q
∂ 2 H + dxB
B + O(h3 )
∂v∂v Q
where
∂H =
∂u Q
∂H
∂u
ψ=ψ̄=Q
.
Notice that H has no cross terms in u and v, except in the nonlinear term depending
only on |ψ|2 . Since Q is real, we have that
∂H =0.
∂v Q
Thus the first order term is just
∂H ∂H dx
A = E dxQA,
A = 2 dx
∂u Q
∂ ψ̄ Q
where we have used Eq. (3.2). Similarly,
∂ 2 H = 0,
∂u∂v Q
and the second order term is just
1
∂ 2 H ∂ 2 H dxA
A + dxB
B
2
∂u∂u Q
∂v∂v Q
=
dxAL+ A + dxBL− B + E dx(A2 + B 2 ) .
− E = L .) We have thus proved that
(Observe that Hreal
+
H(Q + h) = H(Q) + E[(Q, A) + h2 ] + Re(Lh, h) + O(h3 ),
(3.7)
242
J. Fröhlich, T.-P. Tsai, H.-T. Yau
where (f, g) =
f¯gdx is the standard L2 scalar product and
Re(Lh, h) = dxAL+ A + dxBL− B.
Let Qε ≡ QN+ε be the (real) minimizer centered at the origin, with Qε 2 = N + ε.
Let hε = Qε − Q. Then
ε = Q + hε 2 − Q2 = 2 Qhε + h2ε = 2 Qhε + O(ε 2 ).
We define E(N ) as the minimal energy subject to the constraint ψ2 = N :
E(N ) =
inf
ψ2 =N
H(ψ).
The last two equations and (3.7) then yield the standard relation
∂E(N )
= E/2.
∂N
For an arbitrary h with Reh ⊥ Q, Eq. (3.7) yields
dxAL+ A + dxBL− B = H(Q + h) − H(Q) − Eh2 + O(h3 ).
(3.8)
(3.9)
Since H(Q + h) ≥ E(Q + h2 ) = E(N + h2 ), (because Reh ⊥ Q, Q + h2 =
Q2 + h2 ), we obtain from Eq. (3.8)
H(Q + h) − H(Q) − Eh2 ≥ O(h3 ).
This proves that
dxAL+ A ≥ 0,
dxBL− B ≥ 0,
for all A ⊥ Q and arbitrary B. Thus L− ≥ 0, and L+ has at most one negative eigenvalue.
From the explicit form of L− and L+ we conclude that
L− Q = 0,
L+ ∇Q = 0,
L− (xQ) = −∇Q.
(3.10)
Since Q is positive and L− ≥ 0, its null space is the span of Q, i.e.,
L− ≥ 0,
N (L− ) = spanR {Q}.
From the explicit form of L+ we have that
(Q, L+ Q) =: ε0 · (Q, Q) < 0,
where
ε0 = −2(N (Q))
−1
(3.11)
Q2 ∗ Q2 < 0.
Thus L+ has exactly one negative eigenvalue. The continuous spectra of L− and L+ can
easily be shown to be the half-line [−E, ∞). Since L+ ∇Q = 0, 0 is an at least n-fold
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
243
degenerate eigenvalue of L+ . A key assumption in our analysis is that the whole null
− E is spanned by ∇Q, i.e.,
space of L+ = Hreal
N (L+ ) = spanR {∇Q}.
(3.12)
Since the continuous spectrum of L− and of L+ is the half-line [−E, ∞), 0 is an isolated
point. Hence there is a positive number δ such that
(h, L+ h) ≥ δ(h, h),
if h is orthogonal to the span of ∇Q and to the ground state of L+ . In particular, the
number of eigenvalues strictly below δ is exactly n + 1. We have proved the following
lemma.
Lemma 3.1. Assume that (3.12) holds. Then the null spaces of L− and L+ are given
by N (L− ) = spanR {Q}, N (L+ ) = spanR {∇Q}. Furthermore, there is a constant
ε2 > 0 such that (a) (g, L− g) ≥ ε2 (g, g) if g ⊥ Q. (b) (f, L+ f ) ≥ ε2 (f, f ) if
f ⊥ spanR {Q, ∇Q}.
If we assume that
Q + h2 = Q2
the term with the factor E in (3.7) vanishes, because
2(Q, A) = −h2 ,
and we have that
H(Q + h) = H(Q) +
dxAL+ A +
dxBL− B + O(h3 ).
(3.13)
Thus if h = A + iB, with
A ⊥ spanR {∇Q},
B ⊥ Q,
Q + h2 = Q2 ,
(3.14)
then we can write A = A1 +cQ, with (A1 , Q) = 0, for some c of order h2 , (c(Q, Q) =
(A, Q) = −(h, h)/2). Since (Q, ∇Q) = 0, we have that (∇Q, A1 ) = 0, provided that
(A, ∇Q) = 0. Therefore, under assumption (3.14), we can rewrite (3.7) as
H(Q + h) − H(Q) = (A, L+ A) + (B, L− B) + O(h3 )
(3.15)
3
= (A1 , L+ A1 ) + (B, L− B) + O(h ).
(3.16)
Since A1 ⊥ spanR {Q, ∇Q}, we can apply Lemma 3.1 to conclude that (A1 , L+ A1 ) +
(B, L− B) ≥ ε2 (A1 2 + B2 ). Since the difference between A1 2 and A2 is of
higher order, we obtain
H(Q + h) − H(Q) ≥ ε2 h2 + O(h3 ).
(3.17)
The last equation implies the global (modulational) stability of the soliton solution
under small perturbations. To see this, suppose the initial data is of the form Q + h0 ,
with Q + h0 2 = Q2 ; (the last condition always holds, since we can choose a Q
with the mass of the initial value). At a later time t, we can find r and θ such that
ψt (x − r)e−iθ = h(x) + Q(x)
with the mass of the correction, h2 , minimized. One can easily check that h satisfies
condition (3.14). By inequality (3.17), h2 is bounded from above by the left hand side,
which is conserved under the time evolution.
244
J. Fröhlich, T.-P. Tsai, H.-T. Yau
3.2. Dynamical linearization of the Hartree equation around solitons. We now return
to the Hartree equation (1.5) with external potential λV (x) = W (εx). Since our time
scale is of order t ∼ ε −1 , the change in the external potential during the evolution on
this time scale may not be small. Thus the argument in the last section no longer applies.
We shall show that, nevertheless, the soliton solution is stable on this time scale, and we
shall track the motion of the soliton precisely.
The Hartree equation (1.5) is
1
i∂t ψ = − 9ψ + W (εx)ψ − ( ∗ |ψ|2 )ψ =: H (ψ)ψ.
2
(3.18)
The solutions we are interested in are of the form
ψ(x, t) = [Q (x − r(t)) + hε (x − r(t), t)] eiθ(x,t) ,
(3.19)
for ε > 0 small enough, where Q(x) = Q(ε=0) (x) is a minimizer of the energy functional
H, and hε (x − r(t), t) is a small correction term which tends to 0, as ε → 0; θ(x, t) is
a time-dependent phase of the form
θ (x, t) = v(t) · (x − r(t)) − Et + θ1 (t).
Also, we expect that, to leading order, the velocity v(t) and the location r(t) of the
soliton are given by
ṙ(t) = v(t) ,
v̇(t) = −ε (∇W ) (εr(t)) .
For the time being, there is no canonical way to determine corrections to these equations,
as the decomposition (3.19) is not unique. We require v, r, and θ1 to obey the following
equations:
ṙ(t) = v(t),
v̇(t) = −ε(∇W )(εr(t)) + a(t),
θ˙1 (t) = 1 v 2 (t) − W (εr(t)) + ω(t),
2
where the (vector) acceleration correction a(t) and the (scalar) “angular velocity” correction ω(t) are of higher order in ε and will be used for fine adjustment, later on. Their
initial values will be discussed in Subsect. 4.5.1, when we adjust the initial datum hε,0 .
We now derive the equations for a, ω and h. Let ξ(x, t) = Q(y)eiθ , y = x − r(t).
By explicit computation,
1
−1
−1
ξ {i∂t − H (ξ )} ξ = ξ
i∂t ξ + 9ξ − W (εx) + ∗ |ξ |2
2
∇Q
1 2
˙
= −θ1 (t) − ∂t [v(t)(x − r(t)] − 2 v (t) + i
(v(t) − ṙ(t))
Q
9Q
− W (εx) + E +
+ ∗ |ξ |2 .
2Q
We expand the potential W around the point r(t):
W (εx) = W (εr(t)) + ∇W (εr(t))ε(x − r(t)) + /0 (x, t),
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
245
where the remainder /0 (x, t) is real and, by the mean value theorem,
|/0 (x, t)| ≤ Cε2 |y|2 ,
(3.20)
where C = C(W ) depends on W . Recalling the equation for r, v and (3.2), we then
have
ξ −1 {i∂t − H (ξ )} ξ = −/ξ,
(3.21)
where
/ = −W (εr) + W (εx) + v̇y + ω = /0 (x, t) + a(t)y + ω(t).
(3.22)
Next, we consider h(y, t) = hε (x − r(t), t). Substituting ψ = (Q + h)eiθ into
Eq. (3.18) and canceling eiθ we get
i {(Q + h)(i∂t θ) − ṙ · ∇(Q + h) + ∂t h}
1
1
= − 9(Q + h) − iv · ∇(Q + h) + v 2 (Q + h)
2
2
+ W (εx)(Q + h) − ( ∗ |Q + h|2 )(Q + h),
where Q and h are taken at (y, t) = (x − r(t), t), that is, Q = Q(x − r(t)) = Q(y)
and h = h(x − r(t), t) = h(y, t). Using ṙ(t) = v(t), Eq. (3.2), and
1 2
1
∂t θ = v̇ · x + − v − v̇r − W (εr) + ω(t) − E = −W (εx) + / − v 2 − E,
2
2
we obtain
1
i∂t h = − 9h − Eh + /(Q + h) − ( ∗ |Q + h|2 )(Q + h) − ( ∗ Q2 )Q .
2
(3.23)
Treating /h as an error term, we can rewrite this equation as
∂t h = −iLh + G,
(3.24)
where the operator L is given by (3.4), and the nonlinear part is
G = − i/(Q + h) − iF (h),
with F (h) = − ( ∗ |h|2 )(Q + h) + ( ∗ [Q(h + h̄)])h .
(3.25)
In matrix form,
∂
∂t
A
0 L−
A
ReG
=
+
.
B
B
ImG
−L+ 0
(3.26)
We observe that, except for /0 which is part of / (and thus appears in G), all quantities
in this system are evaluated at (y, t).
246
J. Fröhlich, T.-P. Tsai, H.-T. Yau
3.3. Properties of the linearized flow. We have shown that the linear part in the dynamical linearization of the nonlinear Hartree equation results in the standard linear
evolution (3.3) with matrix form given in (3.6). We notice that L+ and L− , the real and
imaginary part of L, can be reinterpreted as complex-linear operators which turn out to
be self-adjoint in the usual L2 space. The operator
0 −L+
0 L−
,
with L∗ =
L=
−L+ 0
L− 0
acting on H 1 × H 1 is, however, not symmetric. Although our functions A and B are
real, we shall view L− and L+ as self-adjoint operators on the Sobolev space H 1 of
complex-valued functions. The operator L is, however, not self-adjoint and thus does not
have a spectral decomposition. A standard procedure is to decompose the space H 1 ×H 1
into a direct sum of its generalized null space,
S := Ng (L) = {v : Ln v = 0 for some n} ,
and the orthogonal compliment of the generalized null space of its adjoint, i.e., the space
M = Ng (L∗ )⊥ . It is simple to check that both spaces, S = Ng (L) and M = Ng (L∗ )⊥ ,
are invariant under L. Note that the decomposition H 1 × H 1 = M ⊕ S is, however, not
an orthogonal decomposition.
Following M. I. Weinstein [13], we want to establish the following picture:
1. H 1 × H 1 = M ⊕ S.
2. The H 1 × H 1 -norm on M remains uniformly bounded under the linearized flow for
all time.
3. The dynamics on S can be computed explicitly.
We use PM and PS to denote (non-orthogonal) projections with respect to the decomposition M ⊕ S. We first establish some spectral properties of L+ and L− .
3.3.1. Generalized null space. We first determine the generalized null space S = Ng (L).
We recall Lemma 3.1 and the equations
L− Q = 0,
L+ ∇Q = 0,
L− xQ = −∇Q.
Since Q ⊥ spanR {∇Q} = N (L+ ) and L+ is self-adjoint, there exists a solution, 01 ,
of the equation L+ 01 = Q. We may assume 01 ⊥ ∇Q by subtracting its projection on
the ∇Q-direction. If 01 ⊥ Q, then (01 , Q) = (01 , L+ 01 ) > ε1 (01 , 01 ), by Lemma 3.1.
This contradiction shows (01 , Q) = 0. Now we let 0 = 01 +b∇Q with b = 2(01 , xQ).
Then (0, Q) = (01 , Q) = 0, and (0, xQ) = 0. To summarize, we have found a 0 such
that
L+ 0 = Q,
(0, xQ) = 0,
(0, Q) = 0.
(3.27)
We require (0, xQ) = 0, in order to construct a dual basis on S in Proposition 3.2 below.
To determine
the generalized null space, we need to solve all solutions of the equation
Ln ( uv ) = 00 for some n. If n = 2k is even, it is equivalent to (L− L+ )k u = 0 and
(L+ L− )k v = 0. If n = 2k + 1 is odd, it is equivalent to L+ (L− L+ )k u = 0 and
k = 0. We have solved the solutions for the case n = 1 above: It is the span
L−(L+
L− ) v∇Q
0
of Q
and 0 .
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
247
We next consider the case n = 2. The null space of L+ L− is
N (L+ L− ) = L− −1 N (L+ ) = N (L− ) ⊕ spanR {xQ} = spanR {Q, xQ}.
(3.28)
Similarly,
N (L− L+ ) = N (L+ ) ⊕ spanR {0} = spanR {∇Q, 0}.
(3.29)
For the case n = 3, we have
N (L− L+ L− ) = L− −1 N (L− L+ ) = L− −1 spanR {∇Q, 0} .
Since N (L− ) = spanR {Q} and (Q, 0) = 0, 0 is not in the range of L− . Thus
L− −1 spanR {∇Q, 0} = L− −1 spanR {∇Q} = N (L− L+ ).
This proves that N (L− L + L− ) = N (L+ L− ). Similarly, N (L + L
− L+ ) = N (L− L+ ).
Therefore, if Ln ( uv ) = 00 for some n ≥ 2, then L2 ( uv ) = 00 . Thus we have found
a basis for Ng (L). We also have similar statements for Ng (L∗ ). Summarizing, we have
proved
Proposition 3.2.
0 ∇Q 0 0 , xQ , 0 },
Q ,
0
0 xQ
0 Q
∗
Ng (L ) = spanR { 0 , 0 , ∇Q , 0 }.
S = Ng (L) = spanR {
(3.30)
Notice that these vectors are dual bases and we have ordered them correspondingly.
In particular, for an arbitrary function g we have
0
PS (g) = κ1 (Img, 0) Q
+ κ2 (Reg, xQ) ∇Q
0
0 + κ1 (Reg, Q) 00 ,
+ κ2 (Img, ∇Q) xQ
where κ1 = 1/(Q, 0) and κ2 = 1/(xj Q, ∂j Q) = −2. Also note that we have
0 0
0
0 0
L Q
= 0 , L ∇Q
= 0 , L yQ
= − ∇Q
, L 00 = − Q
.
0
0
(3.31)
Let g(t) be a solution to the linear evolution (3.3) and denote the projection onto S
by
PS g(t) = α(t)
0
Q
+ β(t)
∇Q 0
+ γ (t)
0
xQ
+ δ(t)
0
0
.
Then by (3.31) the equations for the coefficients (α(t), β(t), γ (t), δ(t)) (note β(t) and
γ (t) are vector functions) are given by
0
Q : α̇ = −δ,
∇Q : β̇ = −γ ,
0
0 yQ : γ̇ = 0,
0
δ̇ = 0.
0 :
248
J. Fröhlich, T.-P. Tsai, H.-T. Yau
3.3.2. The flow on M. We have decomposed the space H 1 × H 1 into a direct sum of
the generalized null space S = Ng (L) and M = Ng (L∗ )⊥ . The generalized null spaces
for L and L∗ are given by Proposition 3.2. Thus, M is the space
M = {( uv ) : u ⊥ spanR {Q, xQ}, v ⊥ spanR {∇Q, 0}}.
Since all functions in the space S = Ng (L) and M ⊥ = Ng (L∗ ) are smooth, the projections PS and PM are bounded in any H k space.
Our first aim is to prove
Lemma 3.3 (H 1 -norm on M).
1. If g ∈ M, then Re(Lg, g) is non-negative and comparable to g2H 1 .
2. If g(t) = e−itL φ and 0 = φ ∈ M, then g(t)H 1 is uniformly bounded below and
above.
To prove this lemma, we first show that, for all vectors ( uv ) ∈ M,
C −1 u2L2 ≤ (u, L+ u),
C −1 v2L2 ≤ (v, L− v) ,
(3.32)
for some constant C, as follows from Lemma 3.1. In fact, it is sufficient to assume that
u ⊥ spanR {Q, xQ} and v ⊥ spanR {0}. (As will become clear, we only use (0, Q) = 0
and (xQ, ∇Q) = 0 in this argument.)
For the v-part, if (v, Q) = 0, the claim follows from Lemma 3.1. Hence we may
assume tv = Q + w for some t = 0, w ⊥ Q. By assumption 0 = (0, tv) = (0, Q + w),
hence |(0, Q)| = |(0, w)| ≤ 02 w2 . Therefore we have w2 ≥ c3 > 0 and
ε2 c32
ε2 w22
(w, L− w)
(v, L− v)
=
≥
≥
.
(v, v)
Q22 + w22
Q22 + w22
Q22 + c32
For the u-part, if (u, ∇Q) = 0, the claim follows from Lemma 3.1. Hence we
may assume u = b∇Q + w for some vector b = 0 and some w ⊥ Q, ∇Q. By
assumption, 0 = b(xQ, u) = (bxQ, b∇Q + w) = C|b|2 + (bxQ, w), with C = 0.
Hence w2 > C|b| and
ε2 w22
(w, L+ w)
(u, L+ u)
=
≥
≥ Cε2 ,
2
(u, u)
Cb2 + w2
Cb2 + w22
by a similar estimate. Hence (3.32) is proved.
Now, since ∇u2 is bounded by (u, L+ u) and u2 , (and hence by (u, L+ u), see
(3.32)), we can replace the norm on the left hand side of (3.32) by the H1 -norm; (here
the H1 -norm is the sum of the L2 -norm plus the L2 -norm of the derivative). Therefore
we have proved that, for ( uv ) ∈ M,
C −1 (u, u)H 1 ≤ (u, L+ u) ≤ C(u, u)H 1 ,
C
−1
(3.33)
(v, v)H 1 ≤ (v, L− v) ≤ C(v, v)H 1 .
The upper bounds on (u, L+ u) and (v, L− v) are obvious. Hence the first part of the
lemma is proved.
The second part follows from the first part and the next lemma, which states that the
quantity (u, L+ u) + (v, L− v), which is equivalent to the H 1 -norm on M, is actually
conserved by the linear flow (3.3). We note that
(u, L+ u) + (v, L− v) = Re(Lg, g) = Im(Lf, g).
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
249
Lemma 3.4. Recall that L = −iL, and iL = Li, (see (4.5)).
1. Re(Lf, g) = Im(Lf, g) = Im(Lg, f ) = −Im(f, Lg).
2. If g(t) = e−itL φ, then Im(Lk g, g) is constant for any integer k ≥ 0.
3. For any g(t) with ∂t g = Lg + G, one has
d
Im(Lg, g) = 2Im(Lg, G).
dt
Proof. All these assertions can be checked by simple computations. We only prove the
last one in the following.
d
Im(Lg, g) = Im(L2 g + LG, g) + Im(Lg, Lg + G)
dt
= Im(LG, g) + Im(Lg, G) = 2Im(Lg, G).
'
&
The following two lemmas will be used to prove inequality (3.61) below. (Note H k
denotes the Sobolev space W k,2 .)
Lemma 3.5. (a) For any m ≥ 1, e−itL is a bounded map from M ∩ H m into itself.
Explicitly, for any φ ∈ M ∩ H m ,
−itL φ m ≤ Cm φH m .
e
H
(b) L = −iL, restricted to M, has an inverse which is bounded from M ∩ L2 to M ∩ H 2 .
Proof. Proof of (a): Let g(t) = e−itL . The case m = 1 is Lemma 3.3, part 2. If m ≥ 3
is odd, we have that
g(t)2H m ≤ C Im(Lm g, g) + C g(t)2H m−2
≤ C Im(Lm φ, φ) + C φ2 m−2 ≤ C φ2 m .
H
H
The second inequality uses Lemma 3.4, part 2. (Note: If m is even, Im(Lm g, g) = 0,
and the first inequality fails.) The general case follows from interpolation.
Proof of (b): For ( uv ) ∈ M we seek xy ∈ M such that L xy = ( uv ), i.e., L− y = u
and L+ x = −v. Notice that u ⊥ Q and v ⊥ ∇Q, and the null spaces of the self-adjoint
operators L− and L+ are spanned by Q and ∇Q respectively. Since 0 is an isolated
−1
eigenvalue of L− and L+ , it follows that L−1
− and L+ are bounded operators on the
orthogonal complements of the null spaces. This proves that L has a bounded inverse
on M ∩ L2 .
To prove the bound, write w = ( uv ) ∈ M ∩ H 2 . By (3.33)
1
u22 + C L+ u22 .
2
Hence uW 1,2 ≤ C L+ u2 . Similarly vW 1,2 ≤ C L− v2 . Furthermore, write
L+ = − 21 9 + V . (The explicit form of V can easily be read from the definition of
L+ .)
u2W 1,2 ≤ C(u, L+ u) ≤
9u2 = 2 L+ u − V u2 ≤ 2 L+ u2 + C(V ) u2 ≤ C L+ u2 .
Hence uW 2,2 ≤ C L+ u2 . Similarly vW 2,2 ≤ C L− v2 . We conclude that
wW 2,2 ≤ C Lw2 . The lemma follows by a duality argument. &
'
250
J. Fröhlich, T.-P. Tsai, H.-T. Yau
Let
Xk = H k ∩ L2 (1 + |y|2k )dy
(3.34)
denote the subspace of H k of functions with prescribed decay at infinity.
Lemma 3.6 (Finite propagation speed). For any integer k ≥ 0, for any real m ≥ 1,
and for φ ∈ M ∩ Xk ∩ H k+m ,
m tL y e φ Hk
≤ C y m φ X + C(1 + |t|m ) φH k+m .
k
(3.35)
The constant C depends on k and m.
Remark. For the free Schrödinger equation, one need not assume that φ ∈ M, since
Lemma 3.5 (a) always holds.
Proof. Let α be any multi-index with |α| = k. Let g(t) = e−itL φ and v(t) = ∇yα g(t).
We have
∂t v = Lv + Ig,
with I = [∇ α , L].
Hence,
d
dt
y
2m
|v| = 2Re y 2m v̄vt
≤ C y 2m−1 |v||∇v|dy + C v22 + C y 2m |v||Ig|.
2
Since I is a localized operator involving derivatives only up to (k − 1)st order, (I
vanishes for k = 0), the last term is bounded by v2 gH k−1 ≤ C φ2H k . Hence, by
interpolation,
m−1 d m 2 y v ≤ C y m v · ∇v + C φ2H k
dt
2 y
2
2
m 2(1−1/2m)
1/m
≤ C y v 2
· vH m + C φ2H k .
Let f (t) = y m v22 and N = C φ2H k+m . By Hölder’s inequality,
f ≤ f 1−1/2m N 1/2m + N ≤
f
+ C(1 + t)2m−1 N,
1+t
hence f (t) ≤ Cf (0) + C(1 + t)2m N , which proves the claim.
We will need the case k = 1 when we prove Lemma 3.8.
'
&
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
251
3.4. The fine adjustment. We first recall the conclusion of dynamical linearization. We
decompose the function ψ into the sum
ψ(x, t) = [Q (x − r(t)) + hε (x − r(t), t)] eiθ(x,t) ,
where θ (x, t) is a time-dependent phase of the form
θ (x, t) = v(t) · (x − r(t)) − Et + θ1 (t),
with
ṙ(t) = v(t),
v̇(t) = −∇W (εr(t))ε + a(t),
1
θ˙1 (t) = v 2 (t) − W (εr(t)) + ω(t).
2
Here the (vector) acceleration correction a(t) and the (scalar) angular velocity correction
ω(t) are of smaller orders, and we shall determine their values in this subsection. The
main correction h satisfies the equation
∂
h = Lh + G,
∂t
h(0) = hε,0 ,
(3.36)
with
G = −i/(Q + h) − iF (h),
/ = /0 + ay + ω,
/0 = W (εx) − W (εr) − εy∇W (εr),
F = −( ∗ |h|2 )(Q + h) − ( ∗ [Q(h + h̄)])h.
We decompose h(t) into a sum of its components in S and M: h(t) = hS (t) + hM (t).
The component in S is a sum of the basis vectors (3.30)
0
0 + β(t) ∇Q
+ γ (t) yQ
+ δ(t) 00 .
hS (t) = α(t) Q
0
We now consider projections of Eq. (3.36) onto S and M. Taking inner products with
the dual basis, (see Proposition 3.2), we obtain the equations on S:
0
(3.37)
Q : α̇ = −δ +κ1 (ImG, 0),
∇Q (3.38)
: β̇ = −γ +κ2 (ReG, yQ),
0
0 (3.39)
κ2 (ImG, ∇Q),
yQ : γ̇ =
0
(3.40)
δ̇ =
κ1 (ReG, Q).
0 :
The equation on M is
∂
hM = LhM + PM G.
∂t
(3.41)
Notice that /0 = W (εx) − W (εr) − εy∇W (εr) is determined by r(t), which solves
r̈ = −ε∇W (εr) + a.
252
J. Fröhlich, T.-P. Tsai, H.-T. Yau
This system is not closed, yet, since we still need to determine a and ω, which are
used for the fine adjustment. Observe that a and ω appear explicitly in the equation on
S only through ImG, that is, ayQ and ωQ. These two terms appear in (3.37) and (3.39),
the equations for α and γ . Our strategy is to choose a and ω so that α̇ = 0 and γ̇ = 0.
Then hS (t) has at most linear growth.
It is important to understand the orders of these quantities. Assume that h ≤ o(ε).
Since the force G contains an external input /0 Q ∼ ε 2 , G is of the form O(h2 ) + ε 2 .
The equation for hM , i.e., (3.41), is thus of the form
f ≤ f 2 + c2 ε 2 ,
c>1,
(3.42)
(and we have assumed that we can take care of the linear part). The solutions of this
equation can blow up at t = (cε)−1 . Explicitly, if f (0) = 0 then
f (t) = cε tan(cεt).
A more careful examination shows that, due to a cancellation property when integrating in time (which is due to an oscillatory behaviour in time), one can show that
h(t) ∼ ε3/2 .
Based on this observation, we will prove that
a(t) ∼ ε2 ,
ω(t) ∼ ε2 ,
PS (h) ∼ ε3 ,
PM (h) ∼ ε3/2 .
(3.43)
In the following subsections, we will prove the existence of h(y, t) by proving a
priori bounds and using its local existence. It is also possible to prove existence by a
contraction mapping argument, as we will do in Sect. 4 for the wave operator.
3.5. Initial value and equations on S.
3.5.1. Initial value. Recall that the initial datum is given by
ψ0 (x) = Q (x) + hε,0 (x) eiv0 x .
The coordinates of the initial value hε,0 in the S direction can be calculated:
0
Q : α(0) = κ1 (Imhε,0 , 0),
∇Q : β(0) = κ2 (Rehε,0 , yQ),
0
0 yQ : γ (0) = κ2 (Imhε,0 , ∇Q),
0
δ(0) = κ1 (Rehε,0 , Q).
0 :
By our assumption on hε,0 , these initial values are of order ε 3/2 , which is too large for our
purpose. They can be made smaller by introducing suitable normalization conventions.
We first replace Q by Q∗ = Qψ0 2 , with Q∗ L2 = ψ0 L2 . We then define h1 by
the equation
ψ0 (x) = Q (x) + hε,0 (x) eiv0 x = Q∗ (x) + h1 (x) eiv0 x .
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
253
From the assumption Q∗ 2 = ψ0 2 , we have
2(Q∗ , Reh1 ) = −h1 2 .
Next, we want to choose r ∗ , v ∗ , θ ∗ , and write
∗
∗
∗
ψ0 (x) = Q∗ x − r ∗ + h∗ x − r ∗ eiv (x−r )+iθ ,
(3.44)
so that PS h∗ is essentially zero. Notice that h∗ is determined once we have chosen r ∗ , v ∗
and θ ∗ : As a function of y = x − r ∗ ,
∗
∗
∗
h∗ (y) = Q∗ y + r ∗ + h1 y + r ∗ ei[v0 (y+r )−v y−θ ] − Q∗ (y) .
The leading term of h∗ is given by (we will choose r ∗ ∼ 0, v ∗ ∼ v0 and θ ∗ ∼ v0 r ∗ )
h∗ (y) ∼ h1 (y) + Q∗ (y) i(v0 − v ∗ )y + i(v0 r ∗ − θ ∗ ) + r ∗ · ∇Q∗ (y) .
We can now calculate the initial value of h∗ along the S direction (w.r.t. Q∗ ) as before.
The conclusion is
0
∗
∗
+κ1 (v0 r ∗ − θ ∗ ) Q∗ , 0 ∗ ,
Q : α ∼ κ1 Imh1 , 0
∇Q : β ∗ ∼ κ2 Reh1 , yQ∗ +κ2 k rk∗ ∇k Q∗ , yQ∗ ,
0
0 ∗
∗
∗
∗
∗
yQ : γ ∼ κ2 Imh1 , ∇Q +κ2 (v0 − v ) Q , ∇Q ,
0
δ ∗ ∼ κ1 Reh1 , Q∗ .
0 :
Since the initial value hε,0 is of order ε 3/2 , h1 is of the same order and we can choose
v0 r ∗ − θ ∗ , r ∗ and v0 − v ∗ of order ε3/2 such that α ∗ , β ∗ and γ ∗ vanish to leading order.
It is easy to check that the next order is bounded by ε3 . Furthermore, δ ∗ is of order ε 3
as well, thanks to the relation 2(Q∗ , Reh1 ) = −h1 2 .
In the remaining part of this section, we will prove Theorem 1.1 with ψ0 of the
form (3.44), and PS h∗ ∼ ε3 . The initial values of r(0), v(0), and θ(0) are defined
correspondingly. Notice that, by the assumption of the Theorem, (3.12) is satisfied by
Q∗ . After this case is proved, the statement in the theorem, with Q = QN0 , can be
obtained by defining h as
h(y, t) = ψ(x, t)e−iθ − QN0 (y)
= (ψe−iθ − Q∗ ) + (Q∗ − QN0 )
= h∗ (y, t) + (Q∗ − QN0 ) = O(ε3/2 ).
From now on, we may and will drop the superscript ∗ and assume
PS hε,0 ≤ Cc0 ε 3 ,
(3.45)
where ε is sufficiently small: ε ≤ ε−1 , with ε−1 and C depending only on the initial
setting (H, N0 , Q,...) but not on W or T . Equation (3.45) will be used in (3.52) below.
We note that the smallness of c0 is only used to find a suitable h∗ (0). It is no longer
needed in the future and hence c0 is independent of T and W . Also note that we may
assume hε,0 ≤ c0 ε 1+σ for σ ∈ (0, 1/2]. Then we replace ε 3/2 , inthe above
argument, by
ε1+σ , and we get a similar conclusion, with (3.45) replaced by PS hε,0 ≤ Cc0 ε 2+2σ .
254
J. Fröhlich, T.-P. Tsai, H.-T. Yau
3.5.2. Equations on S. From now on, C denotes a constant which may depend on the
quantities (, Q...), but not on W or T .
Recall that we want to set α̇ = 0 and γ̇ = 0 in (3.37) and (3.39), which yield equations
for a and ω. From the definition of G and the inner product relations (xQ, 0) = 0 and
κ1 = 1/(Q, 0), we have
κ1 (ImG, 0) = −ω − κ1 (G2 , 0),
where
G2 = /0 Q + /Reh + ReF (h).
(3.46)
Similarly, from (Q, ∇Q) = 0 and κ2 = 1/(xj Q, ∂j Q), we have
κ2 (ImG, ∇Q) = −a − κ2 (G2 , ∇Q).
Therefore, in order to have α̇ = 0 and γ̇ = 0, it suffices to set
ω = −δ − κ1 (G2 , 0),
a = −κ2 (G2 , ∇Q).
(3.47)
With this choice of a and ω, we have α(t) = α(0), γ (t) = γ (0); β(t) and δ(t) are
defined by solving the ODEs (3.38) and (3.40), i.e.,
t
δ(t) =
κ1 (ReG(s), Q)ds + δ(0),
(3.48)
0
t
β(t) =
κ2 (ReG(s), yQ)ds − γ (0)t + β(0).
(3.49)
0
qLet
Cw = 1 + W W 3,∞ .
Then |/0 (x, t)| ≤ CCw
ε 2 |y|2 ,
(cf. (3.20)). Define
ζ (t) := |a(t)| + |ω(t)| + ε1/2 h(t)H 1 + Cw ε 2 .
(We would like to have that ζ (t) = O(ε 2 ) for 0 ≤ t ≤ T ε −1 .) In the following we work
in the time range [0, t1 ] where
ζ (t) ≤ C∗ ε 2 ,
with ε ≤ ε0 ≤ (C∗ + T + 100)−2
(3.50)
holds. Here C∗ > Cw is a (large) constant to be determined later. Equation (3.50) is
true for t = 0 if C∗ is sufficiently large with respect to c0 . Moreover, if ζ (s) < C∗ ε 2
for some s < T ε−1 , then (3.50) holds for a small time interval [s, s + δs] by a local
wellposedness result. Our goal is to show that Eq. (3.50) holds for 0 ≤ t ≤ T ε−1 , by
requiring ε0 sufficiently small. Our strategy is to show that, indeed, ζ (t) ≤ 21 C∗ ε 2 if
(3.50) holds. A local wellposedness result then guarantees that (3.50) holds for the whole
time range.
The quantities ω and a are defined in terms of G2 in (3.47), and recall the definition
of G2 , Eq. (3.46). Note that /0 Q is the leading term in G2 . In their definitions, the main
term comes from /0 Q, and we have
|(/0 Q, 0)| + |(/0 Q, ∇Q)| ≤ CCw ε 2 .
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
255
Also
|(/(t)Reh(t), 0)| + |(/(t)Reh(t), ∇Q)| ≤ C(Cw ε 2 + |a(t)| + |ω(t)|) h2
≤ Cε −1/2 ζ (t)2 .
From the assumption (1.27) on , we have for a general φ ∈ H 1 ,
∗ |φ|2 φ 1 ≤ ∗ |φ|2 ∞ · φH 1 + (∇) ∗ |φ|2 H
L∞
L
From the Young inequality, we have
∗ |φ|2 ∞ ≤ L∞ |φ|2 L
L1
· φL2 .
= L∞ φ2L2 .
Similarly, we can bound the term with replaced by ∇. Thus we have proved that
F (φ)H 1 ≤ C φ2H 1 + C φ3H 1
for some constant depending on . Hence we can bound (F (h(t)), 0) and (F (h(t)), ∇Q)
by
|(F (h(t)), 0)| + |(F (h(t)), ∇Q)| ≤ C h2H 1 + C h3H 1 ≤ Cε −1 ζ (t)2 .
Under assumption (3.50), we have thus proved that
|ω(t)| + |a(t)| ≤ CCw ε 2 + Cε −1 ζ (t)2 .
(3.51)
To estimate β and δ, we note that
ReG = (/0 + ay + ω)Imh + ImF (h).
Since we are only interested in the inner products of ReG with Q or yQ, and Q has
exponential decay, we can treat y to be of order one. Thus we have the bound
t
dsε −1 ζ (s)2 ,
(3.52)
|β(t)| + |δ(t)| ≤ Cc0 (T + 1)ε 3 +
0
where we have used (3.45) and εt ≤ T .
3.6. Modified linear operator on M. It is important to observe that / is not bounded.
In fact,
(3.53)
/ =W (εx) − W (εr) − εy∇W (εr) + ay + ω = O ε 2 (y 2 + 1)
= O (1 + ε|y|) ,
(3.54)
depending on whether we use Taylor expansion. In either case / is not bounded. This
makes the term −i/h in the nonlinear term G hard to control, although the term −i/Q
stays fine since Q is localized. By a finite propagation speed estimate we will see that
/ is of order 1. However, −i/h still cannot be considered an error term. To overcome
this difficulty, we will include this term in the linear operator.
256
J. Fröhlich, T.-P. Tsai, H.-T. Yau
Explicitly, Eq. (3.41) for hM can be rewritten as
∂t hM = (L + PM 1i /)hM + PM G,
= −i/(Q + hS ) − iF (h).
G
(3.55)
Hence we must consider the solution propagator P(s, t) which solves the following
problem: If u(t) = P(s, t)φ, then u is a solution of the equation
∂t u(t) = (L + PM 1i /)u(t),
u(s) = φ.
We note that the operator L + PM 1i / leaves M and S invariant; but we will primarily
consider P(s, t) on M.
Now the equation for hM can be written as
t
hM (t) =
+ P(0, t)hM (0).
P(s, t)PM G(s)ds
(3.56)
0
into the sum of a main part, φ(s), and a remainder, PM G3 (s),
We decompose PM G
where
φ(s) = PM (−i/(s)Q) = PM (−i/0 (s)Q),
G3 = −i/hS − iF (h).
The following lemma provides a basic estimate on the propagator P(s, t).
Lemma 3.7. Assume (3.50) is true for 0 ≤ t ≤ T ε−1 . For φ ∈ M,
P(s, t)φH 1 ≤ C10 φH 1
for 0 ≤ s ≤ t ≤ T ε−1 , where C10 = eCCw T is independent of ε.
We shall prove this lemma in the next subsection. Assuming this lemma and recalling
that G3 (s) is of order h2 + h3 , we can bound the contribution of G3 (s) to hM by
t
t
P(s, t)PM G3 (s)ds ≤ CC10
ε −1 ζ (s)2 ds.
H1
0
0
The key observation is the following lemma, which takes into account cancellations
in the time integration.
Lemma 3.8 (Cancellation). Assume (3.50) is true for 0 ≤ t ≤ T ε −1 . Let φ ∈ M ∩ X3 .
For 1 ( t ≤ T ε−1 , we have that
t
P(s, t)φds ≤ C12 t 1/2 φX
3
H1
0
for a constant C12 = C12 (W, T ) independent of ε. Furthermore, for φ(t) : [0, T ε −1 ] →
M ∩ X3 ,
t
P(s, t)φ(t)ds ≤ C12 t 1/2 sup φ(s)X + C12 t sup φ(s) − φ(σ )H 1 .
3
0
H1
s
|s−σ |≤t 1/2
(3.57)
The space X3 has been defined in (3.34).
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
257
We also claim the following bounds on the main term φ(s) = PM (−i/0 (s)Q),
φ(s)X3 ≤ CCw ε 2 ,
φ(s) − φ(σ )H 1 ≤ CCw ε 3 |s − σ |.
(3.58)
We will prove the lemma and the claim in next subsection.
Assuming Lemma 3.8 and the claim, we get
t
P(s, t)PM (−i/0 (s)Q)ds H1
0
≤ C12 t 1/2 CCw ε 2 + C12 tCCw ε 3 t 1/2 ≤ C13 ε 3/2 ,
where C13 = CC12 Cw (T + 1)3/2 . Hence, by (3.56), hM (t) is bounded by
hM (t)H 1 ≤ C13 ε 3/2 + CC10
t
ε −1 ζ (s)2 ds + Cc0 ε 3/2 .
(3.59)
0
Recall that ζ (t) = |a(t)| + |ω(t)| + ε1/2 h(t)H 1 . Then we can combine all these
bounds, (3.51), (3.52), and (3.59), to obtain the following estimate:
ζ (t) ≤ C(Cw + c0 (1 + ε 1/2 T ) + C13 )ε 2 + Cε −1 ζ 2 (t) + CC10
t
ε −1 ζ (s)2 ds
0
≤ Cε2 (Cw + c0 (1 + ε 1/2 T ) + C13 + C10 C∗2 ε(1 + T ))
≤ Cε2 C14 ,
where C14 = Cw + 2c0 + C13 + 1,
if we require ε1/2 T ≤ 1 and C10 C∗2 ε(1 + T ) < 1, in addition to assumption (3.50). We
now choose
C∗ = 2CC14
and then ε0 such that
ε0 ≤ (C∗ + 100)−2 ,
1/2
ε0 T ≤ 1,
C10 C∗2 ε0 (1 + T ) < 1.
With these choices, we have proven that
ζ (t) ≤
1
C∗ ε 2
2
(3.60)
under assumption (3.50) that ζ (t) ≤ C∗ ε 2 . Suppose that [0, t1 ] is the maximal time
interval such that (3.50) holds and t1 < T ε−1 . Then the equality must hold at t = t1 by
local existence and continuity, and ζ (t) must be slightly less than C∗ ε 2 for some t < t1 .
This is a contradiction to (3.60) and hence (3.50) holds true for all t ≤ T ε−1 .
258
J. Fröhlich, T.-P. Tsai, H.-T. Yau
3.7. Proofs of lemmas.
3.7.1. Proof of Lemma 3.7. Here we prove that the flow given by P(s, t) is bounded in
M:
Proof. Let u(t) = P(s, t)φ ∈ M, and
f (t) = Im(Lu(t), u(t)) ≥ 0.
Recall the second assertion of Lemma 3.4: It implies that fˆ(t) = Im(Lg(t), g(t)),
with g(t) := et L φ, is constant. We propose that f (t) does not grow in t very fast, for
s, t ∈ (0, T ε −1 ). More precisely, we will prove that
d
f (t) ≤ Cεf (t),
dt
which implies f (t) ≤ Cf (s), and hence Lemma 4.7 follows.
We recall the third assertion of Lemma 3.4. In our case, ∂t u = Lu + PM 1i /u, hence
d
f (t)/2 = Im(Lu, PM 1i /u) = Im(Lu, −i/u) − Im(Lu, PS (−i/u))
dt
1
= Im
∇ ū(∇/)u + O(ε 2 u22 ) − Im(Lu, PS (−i/u)) .
2
j
(e , −i/u)ej =
If {ej } and {ej } denote dual bases of S, then PS (−i/u) =
(i/ej , u)ej . Hence PS (−i/u)H 1 ≤ Cε2 uL2 , and
Im(Lu, PS (−i/u)) ≤ C uH 1 · PS (−i/u)H 1 ≤ Cε2 u2H 1 .
Since ∇/∞ = ε∇W
(εx) − ε∇W (εr) + a∞ ≤ 2Cw ε + C∗ ε 2 , (with no y depen
dence), the term Im 21 ∇ ū(∇/)u dominates, and
d
f (t) ≤ Im ∇ ū(∇/)u + Cε 2 u2H 1 ≤ (2Cw ε + Cζ (t)) u2H 1 ≤ CCw εf.
dt
The last inequality follows from (3.50) and Lemma 3.3. Hence
f (t) ≤ eCCw εt f (0) ≤ eCCw T f (0)
for t ≤ T ε−1 .
'
&
3.7.2. Proof of Lemma 3.8. Next we prove the key cancellation lemma. The cancellation
is due to oscillatory behavior in time. We first prove a variant of Lemma 3.8 for the
original flow et L , which will help us to visualize the oscillation. Then we will prove a
weaker result for the modified flow in Lemma 3.8.
d
ρ(t) = O(ε) in H 1 . (One such
Suppose ρ(t) ∈ M satisfies ρ(t) = O(1) and dt
−2
example is ε /0 (t)Q.) Then there is a C > 0 such that
t
e(t−s)L ρ(s)ds ≤ C(1 + εt).
(3.61)
0
H1
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
259
By Lemma 3.5, L−1 is defined on M and commutes with e(t−s)L . Thus
t
e
(t−s)L
d (t−s)L −1
L ρ(s)ds
−e
0 ds
t t
d
(t−s)L −1
= −e
L ρ(s) +
e(t−s)L L−1 ρ(s)ds
0
ds
0
t
= O(1) +
e(t−s)L O(ε)ds
ρ(s)ds =
0
t
0
= O(1) + O(εt),
in H 1 .
Here we have used Lemma 3.3. (Notice the analogy with the integration of eit , which
does not increase the order of eit because of its oscillation.)
Now we prove the lemma.
Proof. Choose τ ∼ t 1/2 " 1. We have
t
P(s, t)φds =
0
j
=
(j +1)τ
jτ
P(s, t)φds
P((j + 1)τ, t)
j
(j +1)τ
jτ
P(s, (j + 1)τ )φds.
We write each summand as
(j +1)τ
P(s, (j + 1)τ )φds ≡ (I)
jτ
=
(j +1)τ
jτ
+
e((j +1)τ −s)L φds
(j +1)τ
jτ
(j +1)τ
s
P(σ, (j + 1)τ )PM 1i /(σ )e(σ −s)L φdσ ds
≡ (II) + (III).
We have
(II)H 1 ≤ C φH 1 (1 + ετ ) ≤ C φH 1
by (3.61) and (3.50). For (III), since φ is localized, we expect it is not affected much by
the large potential in PM 1i /(σ ) for large y. To prove this, we use the finite propagation
speed estimate (3.35): For s ∈ (0, τ ),
PM 1i /(·)es L φ 1 ≤ C Cw ε 2 y 2 + C∗ ε 2 (1 + |y|) es L φ 1
H
H
2 2 2
≤ CCw ε (1 + y )φ 1 + (1 + s ) φH 3
H
2
+ CC∗ ε (1 + |y|)φH 1 + (1 + s) φH 2 .
260
J. Fröhlich, T.-P. Tsai, H.-T. Yau
Hence
IIIH 1 ≤ CC10 ε 2 τ 2 (Cw + C∗ ) (1 + y 2 )φ H1
≤ CC10 Cw φX3 ,
+ (Cw τ 2 + C∗ τ ) φH 3
τ 2ε
since
< 2 and ε1/2 C∗ ≤ 1, see (3.50).
Therefore (I )H 1 ≤ C11 φX3 with C11 = C + CC10 Cw and
t
P(s, t)φds ≤
C10 C11 φX3 ≤ CC10 C11 t 1/2 φX3 .
H1
0
j
Next, for a suitably localized function φ(t) ∈ M ∩ X3 ,
t
P(s, t)φ(t)ds 1
0
H
(j +1)τ
=
P((j + 1)τ, t)
P(s, (j + 1)τ ) φ(j τ ) + φ(s) − φ(j τ ) ds jτ
j
1
H
2
≤
C10 C11 φ(j τ )X3 +
C10
τ sup φ(s) − φ(σ )H 1
j
≤ C12 t
|s−σ |≤τ
j
1/2
sup φ(s)X3 + C12 t
s
sup
|s−σ |≤t 1/2
2 = C(1 + C eCCw T )2 .
with C12 = CC11
w
φ(s) − φ(σ )H 1
'
&
This estimate is mainly used for φ(s) = PM (−i/0 (s)Q).
3.7.3. Proof of claim (3.58). We rewrite /0 in the form
/0 (x, t) = W (εr + εy) − W (εr) − ∇W (εr) · εy
1
{∇W (εr + uεy) · εy} du − ∇W (εr) · εy
=
0
1 1
0
0
=
∇ 2 W (εr + vuεy) : εy ⊗ uεy dvdu.
From the first line we have that ∇ 3 /0 ∞ ≤ ε3 ∇ 3 W ∞ . From the third line we obtain
/0 e−ν|y| ≤ ε2 ∇ 2 W . Hence, for φ(s) = PM (−i/0 (s)Q), we have that
∞
∞
3 φ(s)X3 ≤ C ∇ /0 + C /0 e−ν|y| ≤ C W W 3,∞ ε 2 ,
∞
∞
e−ν|y|
where the factor
is due to the exponential decay of Q. Furthermore, since
2
∇ W (εr(s) + vuεy) − ∇ 2 W (εr(σ ) + vuεy) ≤ sup |∇ 3 W | · ε|r(s) − r(σ )|
and |r(s) − r(σ )| ≤ CC∗ |s − σ |, (note |v(t)| ≤ CC∗ ), we conclude that
φ(s) − φ(σ )L2 ≤ C ∇ 3 W C∗ ε 3 |s − σ |.
By rewriting ∇/0 (x, t) =
for ∇[φ(s) − φ(σ )]L2 .
1
0
∞
∇ 2 W (εr(t) + uεy) · ε 2 y du, we get the same bound
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
261
4. Møller Wave Operator
In this section we prove Theorem 1.2. We assume for simplicity that the space dimension
n = 3. All arguments can be modified easily to n > 3.
In the main argument of this section, we assume v0 = 0 and work with the profile
ξ∞ = has,0 , with ξ̂∞ (0) = 0. At the end of this section we deal with general v0 by
applying a Galilei transform. In either case, we have has,0 (x) = ξ∞ (x)eiv0 ·x , and
ĥas,0 (v0 ) = ξ̂∞ (0) = 0.
4.1. Dynamical linearization. We recall the Hartree equation
1
i∂t ψ = − 9ψ − ( ∗ |ψ|2 )ψ,
2
and the equation for the ground state Q,
1
− 9Q − ( ∗ Q2 )Q = EQ.
2
We consider solutions of the Hartree equation of the form
ψ = (Q(y) + h(y, t))eiθ(y,t) ,
where
y = x − r(t),
ṙ(t) = v(t),
θ (y, t) = v(t)y − Et + θ1 (t),
v̇(t) = a(t),
∞
1 2
θ1 (t) = −
v
+
ω
ds .
2
t
The argument here is the same as that in Subsect. 4.2, with W ≡ 0. We obtain the
equation for h:
∂t h = Lh + G(h),
where the linear part
1 1
−29 − E + A ,
i
A(h) = −( ∗ Q2 )h − Q( ∗ (Q(h + h̄))),
L=
and the nonlinear part
1
{/(Q + h) + F (h)} ,
/ = ω + ay,
i
F (h) = −( ∗ |h|2 )(Q + h) − ( ∗ [Q(h + h̄)])h.
G=
We take projections of Eq. (4.1) onto S and M. The equations on S are
0
Q : α̇ = −δ +κ1 (ImG, 0),
∇Q : β̇ = −γ +κ2 (ReG, yQ),
0
0 κ2 (ImG, ∇Q),
yQ : γ̇ =
0
δ̇ =
κ1 (ReG, Q).
0 :
(4.1)
262
J. Fröhlich, T.-P. Tsai, H.-T. Yau
(See Proposition 4.2.) The equation on M is
∂t hM = LhM + PM G(h).
Next we consider the wave operator. Given a profile ξ∞ at t = ∞, we hope to find a
function h(y, t) such that
h(y, t) − et L0 ξ∞ → 0,
as t → ∞, in a sense to be made more precise. Here
L0 = −i − 21 9 − E
so that L = L0 − iA. Our strategy is to write
h(·, t) = ξ(·, t) + g(·, t),
where ξ(t) is the main term, which satisfies a linear equation and has the desired profile
explicitly; g(t) is an error and converges to zero, as t → ∞, in a suitable sense.
In view of the equation for h, we would like ξ to satisfy the linear equation
∂t ξ = Lξ + PM J ξ
ξ(t) ∈ M,
(4.2)
with ξ(t) → et L0 ξ∞ , as t → ∞. The operator J is a modification of the multiplication
operator −i/ and is to be defined later in (4.9).
Define the propagator P(s, t) such that u(t) := P(s, t)φ solves the equation
∂t u = Lu + PM J u,
u(s) = φ ∈ M.
Clearly, P(s, t) leaves the space M invariant so that u ∈ M. Note that t < s in this
section, cf. Sect. 3. We define ξ to be given by
∞
1
PM (4.3)
ξ(t) = PM et L0 ξ∞ −
P(s, t)PM [ A + PM J (s)]es L0 ξ∞ ds.
i
t
We have that ξ ∈ M, by definition, and that ξ satisfies (4.2) (differentiate (4.3) and use
that [L, PM ] = 0!). We shall prove later on that
ξ(t) → et L0 ξ∞
in L2 ,
as t → ∞,
(4.4)
under the assumption
ξ̂∞ (0) = 0.
(4.5)
The potential / = ω + ay is unbounded and complicates the analysis. One may
prove certain finite propagation speed estimates, so that y is effectively cut off, as in
Sect. 3. Alternatively, we can modify the form of ψ so that the unbounded potential is
cut off. We shall follow the second option in this section. Specifically, we would like h
not to “see” the fast phase change vy in θ when y is large. Let χ (·) be a smooth cutoff
function with χ (x) = 1, for |x| ≤ 1, and χ (x) = 0, for |x| ≥ 2. We consider ψ of the
form:
ψ = Q(y)eiθ + h(y, t)ei(χvy−Et+θ1 ) = Q + µ−1 h eiθ ,
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
263
where θ = vy − Et + θ1 , µ = exp(i(1 − χ )vy), y = x − r(t) and χ = χ (C∗ y/t),
(C∗ > 0 is a constant to be chosen later). Then µ−1 h satisfies (4.1)
∂t (µ−1 h) = L(µ−1 h) − i/(Q + µ−1 h) − iF (µ−1 h).
(4.6)
Now µ∂t (µ−1 h) = ∂t h + h∂t (−i(1 − χ )vy) and ∂t (−i(1 − χ )vy) =: −i(ay + J (1) ),
where
J (1) = −χ ay − (1 − χ )v 2 + (∇χ )(vt + y)t −2 vy .
Also µL(µ−1 h) = Lh + µ[L, µ−1 ]h. Explicit computation gives
µ∇µ−1 = i −(1 − χ )v + (∇χ )t −1 vy = iJ (2) ,
µ9µ−1 = −(J (2) )2 + i∇ · J (2) ,
∇ · J (2) = 2(∇χ ) · t −1 v + (9χ )t −2 vy.
Recall that L = −i(−9/2 − E + A). Thus,
iµ
[9, µ−1 ]h − iµ[A, µ−1 ]h
2
i (2) 2 ∇ · J (2)
= − (J ) −
h − J (2) · ∇h + J (3) h,
2
2
µ[L, µ−1 ]h =
where
J (3) h := −iµ[A, µ−1 ]h = iµQ ∗ [Q(µ−1 h + µh̄)]. − iQ ∗ [Q(h + h̄)]
This yields the following equation for h:
∂t h = Lh + J h − i/µQ − iµF (µ−1 h),
(4.7)
where
J h = −iω + iJ
(1)
i
∇ · J (2)
− (J (2) )2 −
2
2
h − J (2) · ∇h + J (3) h.
Notice that J depends on ω, a and v with v̇(t) = a(t). Throughout the rest of this section
we assume that there is a constant C∗ such that
t 3 |a(t)| + t 2 |ω(t)| ≤ C∗ .
(4.8)
We shall prove later on that this assumption holds. Under this assumption, one finds that
J (1) ∞ ≤ O(t −2 ),
J (2) ∞ ≤ O(t −2 ),
∇ · J (2) ∞ ≤ O(t −3 ),
J (3) ∞ ≤ O(e−t ).´
We write
J = Ja + Jb · ∇ + Jc ,
(4.9)
264
J. Fröhlich, T.-P. Tsai, H.-T. Yau
with
Ja = i[−ω − χ ay + (∇χ )t −2 (vy)y],
Jb = −J (2) = − −(1 − χ )v + (∇χ )t −1 vy ,
i
∇ · J (2)
Jc = −i(1 − χ )v 2 + i(∇χ )vt −1 (vy) − (J (2) )2 −
+ J (3) .
2
2
Note that Jb is real. Furthermore, the only appearance of µ in J is in J (3) , which is
exponentially small. Assuming the bound (4.8) on a and ω, we can check the following
bounds on J :
Ja ∞ + Jb ∞ ≤ O(t −2 ),
Jc ∞ ≤ O(t −3 ).
(4.10)
Once J (t) is defined, so is ξ(t) by (4.2). We can now use (4.7) and (4.2) to obtain an
equation for g := h − ξ :
∂t g = (L + J )g + PS J ξ − i/µQ − iµF (µ−1 (ξ + g)).
(4.11)
−1
G(1)
µ := J gS − i/(µ − 1)Q − iµF (µ (ξ + g)).
(4.12)
Let
(1)
Since −i/Q ∈ S, we have that PM G = PM J gM + PM Gµ , and the equation for g on
M is
gM (t) = −
t
∞
P(s, t)PM G(1)
µ ds,
(4.13)
Let
−1
G(2)
µ := J g + PS J ξ − i/(µ − 1)Q − iµF (µ (ξ + g)).
(4.14)
(2)
Then PS G = −i/Q + PS Gµ , and the equations on S are
0
Q
∇Q 0
0
yQ
0
0
:
α̇ = −δ − ω +κ1 (ImG(2)
µ , 0),
:
β̇ = −γ
:
γ̇ =
:
δ̇ =
+κ2 (ReG(2)
µ , yQ),
−a+κ2 (ImG(2)
µ , ∇Q),
κ1 (ReG(2)
µ , Q).
Here we have used that κ1 (−/Q, 0) = −ω, κ2 (−/Q, ∇Q) = −a.
(4.15)
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
265
4.2. Bounds on the Propagator P(s, t). The following lemma shows that P(s, t) conserves the H 1 -norm in M.
Lemma 4.1. Assume the bound (4.8). Then P(s, t) is bounded in M ∩ H k , k = 1, 2, 3.
More precisely, there is a constant C such that for any C∗ (the bound in (4.8)), any
T ≥ 1, and any φ ∈ M ∩ H k , we have
P(s, t)φ H k ≤ eCC∗ /T φH k , k = 1, 2, 3,
provided that s, t ≥ T . (The larger T is, the better the estimate.)
Proof. We first consider the case k = 1. Assume u(t) ∈ M, ∂t u(t) = Lu + PM J (t)u,
u(s) = φ. Let f (t) = Im(Lu, u) ≥ 0. Then
d
f (t) = Im(Lu, PM J u) = Im(Lu, J u) − Im(Lu, PS J u).
2dt
Here we have used Lemma 4.3. Note |(Lu, PS J u)| ≤ CC∗ t −2 u2L2 and Im(Lu, J u) =
CRe(9u, Jb · ∇u) + O(t −2 ) u2H 1 , also, (recall Jb is real)
2Re(9u, Jb · ∇u) = − Jb · ∇|∇u|2 − 2Re (∇ ū · ∇)Jb · ∇u
2
= (∇ · Jb )|∇u| − 2Re (∇ ū · ∇)Jb · ∇u ≤ CC∗ t −3 u2H 1 .
(4.16)
Hence we have
d
f (t) ≤ CC∗ t −2 u2 1 ≤ CC∗ t −2 f (t).
dt
H
(4.17)
Hence we get
t
[ln f ]t ≤ −CC∗ t −1 ≤ CC∗ T −1 .
s
s
In particular,
f (t) f (s)
−1
,
≤ eCC∗ T .
f (s) f (t)
Now we consider the case k = 3. The case k = 2 follows by interpolation. Let u(t)
be as above and w = Lu ∈ M. We have
∂t w = Lw + LPM J u = Lw + J w + [L, J ]u − LPS J u.
This time we let f3 (t) = Im(Lw, w) and have
d
f3 (t) = Im(Lw, J w + [L, J ]u − LPS J u).
2dt
We have |(Lw, LPS J u)| ≤ CC∗ t −2 w2 u2 , and we already showed
|Im(Lw, J w)| ≤ CC∗ t −2 w2H 1 when we considered f (t), see especially (4.16). Finally
|Im(Lw, [L, J ]u)| = |Im(Lw, −i(∇Jb ) · ∇u + O(t −2 )u)|
≤ CC∗ t −3 wH 1 uH 2 + CC∗ t −2 wH 1 uH 1 ,
266
J. Fröhlich, T.-P. Tsai, H.-T. Yau
by integration by parts. Since w2H 1 is comparable with f3 , we conclude
d
f3 (t) ≤ CC∗ t −2 f3 (t) + f3 (t) u(t)H 1 ≤ CC∗ t −2 [f3 (t) + f (t)] .
dt
Together with (4.17), we see (f + f3 ) satisfies the same inequality in (4.17), and hence
the same bound. Since (f + f3 ) ∼ u(t)2H 3 , the lemma is proved. &
'
Remark. Due to the spatial cut-off in our Eq. (4.7), we do not need to prove a finite
speed estimate for P, (as we did in Lemma 4.6 for P), in order to prove the above lemma.
4.3. Estimates of ξ . We now estimate ξ precisely. Recall (4.2) and (4.3), the equations
∞
of ξ . Our goal is to estimate the term − t P(s, t)PM ( 1i A + PM J (s))es L0 ξ∞ ds.
We need the following standard results on the free evolution.
Lemma 4.2 (Decay of eit9/2 ). Let k > 0 be a positive integer and assume ∇pm ξ̂∞ (0) = 0
for all non-negative integers m ≤ 2k − 2, then
C
n it9/2
ξ∞ (x)
≤ d/2+k (1 + |y|2k )|∇yn ξ∞ (y)|dy,
(4.18)
∇x e
x=O(1)
t
for any integer n ≥ 0.
Proof. We first consider the case n = 0. Write r =
(e
it9/2
ξ∞ )(x) =
1
e
i|x−y|2
2t . We
have
i|x−y|2
2t
ξ∞ (y)dy
(2πit)d/2
1
1 2
1
k−1
k
=
1 + r + r + ··· +
+ O(r ) ξ∞ (y)dy.
r
2
(k − 1)!
(2πit)d/2
Therefore, the conclusion of the lemma holds if
|x − y|2l ξ∞ (y)dy = 0 for all x, for all l < k,
which is true under the assumption of the lemma. For general n, we take the derivative
m n
n
first and then proceed as above. Note ∇pm (∇
x ξ∞ )(0) = ∇p (p ξ̂∞ )(0) = 0 for all
m ≤ 2k − 2. &
'
We now use that P(s, t) is bounded in H1 (Lemma 4.1) to have
∞
∞
1 s L0
1 s L0 P(s, t)PM Ae ξ∞ ds ≤
PM i Ae ξ∞ 1 ds .
i
t
t
H1
H
From Lemma 4.2 with k = 1, the last term is bounded by
∞
∞
s L0 ξ
ds
≤
s −5/2 ds ≤ Ct −3/2 .
e
∞ 1,∞
t
W
(y∼1)
t
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
267
Notice that this is the only place we use assumption (4.5). Now we recall J = Ja + Jb ·
∇ + Jc . Since Jc (s)∞ ≤ s −3 , we have
∞
∞
s L0
−
J
(s)e
ξ
ds
≤
Cs −3 dsξ∞ H 1 ≤ Ct −2 ξ∞ H 1 .
P
(s,
t)P
M c
∞ t
H1
t
We now expand P(s, t) once more to get
∞
P(s, t)PM (Ja + Jb · ∇)(s)es L0 ξ∞ ds = ξJ + ξAJ + ξJ J ,
−
t
where
∞
e(t−s)L0 PM (Ja + Jb · ∇)(s)es L0 ξ∞ ds,
ξJ = −PM
t
∞ s
1
P(σ, t)PM Ae(σ −s)L0 dσ PM (Ja + Jb · ∇)(s)es L0 ξ∞ ds,
ξAJ =
i
t
t
∞ s
P(σ, t)PM J (s)e(σ −s)L0 dσ PM (Ja + Jb · ∇)(s)es L0 ξ∞ ds.
ξJ J =
t
t
Recall from J.-L. Journe, A. Soffer and C. D. Sogge [15],
C V̂ 1
is0 H0 is1 H0 Ve
.
1 ∞ ≤
e
(L ,L )
(s0 + s1 )d/2
(4.19)
Suppose that we can neglect the second projection PM in the definition of ξJ . Since
Jb ∇es L0 = Jb es L0 ∇, and we can write Ja = −iω + Ja2 , Jb = v + Jb2 , where Ja2 and
Jb2 have compact supports and Ja2 (s)L1 (p) + Jb2 (s)L1 (p) = O(s −2 ), the L∞ -norm
of the integrand of ξJ is bounded by Ct −3/2 s −2 . Integrating in s we get
ξJ (t)L∞ ≤ Ct −3/2 ξ∞ W 1,1 .
To handle the PM , we simply use that PM = 1 − PS . Since PS is a projection onto local
smooth functions, the same proof applies. We shall not repeat the argument to handle
the projection PM later on.
We can also bound ξJ (t) in the H 1 norm by brutal force as we deal with Jc :
ξJ (t)H 1 ≤ Ct −1 ξ∞ H 2 ,
since ξJ involves only free evolution.
We now use that P(s, t) is bounded in H1 to have
∞ s
(σ −s)L0
ξAJ (t)H 1 ≤
PM (Ja + Jb · ∇)(s)es L0 ξ∞ Ae
t
H1
t
From the definition of A, we have
(σ −s)L0
PM (Ja + Jb · ∇)(s)es L0 ξ∞ 1
Ae
H
(σ −s)L0
≤ e
PM (Ja + Jb · ∇)(s)es L0 ξ∞ ∞
L
(σ −s)L0
s L0
+ ∇e
PM (Ja + Jb · ∇)(s)e ξ∞ L∞
.
dσ ds.
268
J. Fröhlich, T.-P. Tsai, H.-T. Yau
Again we use (4.19) to have
(σ −s)L0
PM (Ja + Jb · ∇)(s)es L0 ξ∞ e
L∞
≤ σ −3/2 s −2 ξ∞ W 1,1 .
Since ∇ and e(σ −s)L0 commute, we can bound the term with ∇e(σ −s)L0 in the same way
−2
by also using ∇
y Ja2 L1 (p) + ∇
y Jb2 L1 (p) ≤ O(s ). We conclude that
∞ s
ξAJ (t)H 1 ≤
σ −3/2 s −2 dσ ds ξ∞ W 2,1 ≤ t −3/2 ξ∞ W 2,1 .
(4.20)
t
t
Finally, we can bound ξJ J (t) by
∞ s
ξJ J (t)H 1 ≤
σ −2 s −2 dσ ds ξ∞ H 3 ≤ t −2 ξ∞ H 3 .
t
t
(4.21)
Let ξ(t) = ξ (0) (t) + ξ (1) (t) + ξ (2) (t), where ξ (0) (t) = PM et L0 ξ∞ , ξ (1) = ξJ and ξ (2) (t)
denotes the rest. Then we have proved that
(0) ξ (t) ∞ + ξ (1) (t) ∞ ≤ Ct −3/2 ,
L
L
(4.22)
(1) −1
ξ (t) 1 ≤ Ct , ξ (2) (t) 1 ≤ Ct −3/2 ,
H
H
with the constants depending on ξ∞ . In fact, tracking the proof we see that, since ∇
commutes with es L0 , we actually have
(0) ξ (t) 2,∞ + ξ (1) (t) 2,∞ ≤ Ct −3/2 ,
W
W
(4.23)
(1) −1
ξ (t) 2 ≤ Ct , ξ (2) (t) 2 ≤ Ct −3/2 .
H
H
Of course we need to use a stronger norm for ξ∞ . The following norm is sufficient:
ξ∞ H 4 + ξ∞ W 3,1 + ξ∞ W 2,1 ((1+x 2 )dx) ≤ C −1 C∗ ,
(4.24)
where C∗ is a small constant to be chosen in the next subsection.
4.4. Existence of g. In this section we construct the solution via a contraction mapping
argument. After defining the map in Step 1, we show the following bounds in Step 2:
t 2 |ω(t)| + t 3 |a(t)| + t 2 g(t)H 2 < C∗ ,
(t > T )
(4.25)
provided that ξ∞ ≤ C −1 C∗ with C∗ > 0 sufficiently small (see (4.24)) and T sufficiently large. Finally in Step 3 we show that the contraction mapping converges in the
norm
t 2 |ω(t)| + t 3 |a(t)| + t 2 g(t)H 1
in the ball t 2 |ω(t)| + t 3 |a(t)| + t 2 g(t)H 2 < C∗ . Notice that we use the H 1 norm
for g(t) in the contraction, which is weaker than the H 2 norm appearing in (4.25). Our
approach is certainly not the shortest. Once a certain apriori bound is established, we
can follow standard existence construction by taking weak limits. This will avoid the
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
269
proof of the contraction completely. Our approach however provides more information
to the scattering operator.
STEP 1
We first define the map
(ω, a, g) −→ (ω , a , g )
(4.26)
with the convention gS = PS g and gM = PM g , and so on. Recall that J (t) and ξ(t),
defined by (4.9) and (4.3) respectively, depend on ω and a. To solve the equation on
the S (4.15), we first solve β and δ from (4.15). Since we plan to solve the equation by
iteration, we define (we think γ = 0)
∞
κ1 (ReG(2)
(4.27)
δ (t) = −
µ (s), Q)ds ,
t
∞
κ2 (ReG(2)
β (t) = −
µ (s), yQ)ds .
t
Instead of solving the equation for α and γ , we choose ω and a such that α̇ = γ̇ = 0.
Therefore, we define ω , a to be
ω = −δ + κ1 (ImG(2)
µ , 0),
a =
(4.28)
κ2 (ImG(2)
µ , ∇Q).
With this choice, the component of g in the S direction is simply
gS (t) = β (t) ∇Q
+ δ (t) 00 .
0
Finally, the component on the M direction is given by
∞
P(s, t)PM G(1)
gM (t) = −
µ ds,
t
(4.29)
(1)
P(s, t) depends on a and ω, so is
where Gµ is defined in (4.12). Note the definition of µ. Our next step is to prove this map is bounded in a certain norm.
STEP 2
Suppose that ξ∞ ≤ C −1 C∗ (see (4.24)) and
t 2 |ω(t)| + t 3 |a(t)| + t 2 g(t)H 2 < C∗ .
(4.30)
We will prove the following bound:
t 2 |ω (t)| + t 3 |a (t)| + t 2 g (t)
H2
< C∗ /2
(4.31)
provided that C∗ is sufficiently small. The last statement seems to be contradictory as
the norm is getting smaller after each iteration and we can drive the constant to zero.
But this is impossible as the constant on the estimate of ξ∞ remains unchanged. Indeed,
the right hand side of the last bound depends mainly on the constant appearing in the
estimate of ξ∞ , i.e., in the inequality ξ∞ ≤ C −1 C∗ .
−2
Since a(t) satisfies (4.30)
= v̇, v = ṙ,
and a−1
we have |v(t)| ≤ CC∗ t and |r(t)| ≤
−1
CC∗ t . We now estimate µF (µ (ξ + g)) H 2 . By definition,
µF (µ−1 h) = − ∗ |h|2 (µQ + h) − 2 ∗ [QRe(µ−1 h)] h.
270
J. Fröhlich, T.-P. Tsai, H.-T. Yau
Recall the decomposition and the estimate for ξ (4.23) from Subsect. 5.3. Write h =
ξ + g = ξ (0) + ξ (1) + (ξ (2) + g). Because of the bound (1.27) on ,
∗ |h|2 (µQ + h) 2
H
2
≤ ∗ |h| 2,∞ · µQ + hH 2
W
(0)
≤ C ∗ |ξ + ξ (1) |2 2,∞ + C ∗ |ξ (2) + g|2 2,∞
W
W
2
2
(0)
(2)
(1) ≤ C W 2,1 · ξ + ξ 2,∞ + C W 2,∞ · ξ + g 2
W
≤ CC∗2 t −3 .
H
Since ∗ [QRe(µ−1 h)] h is a local term by the presence of Q, using the bound (1.27)
on we have
∗ [QRe(µ−1 h)] h 2 ≤ C h2H 2 (y∼1) ≤ CC∗2 t −3 .
H
We conclude that
µF (µ−1 (ξ + g))
H2
≤ CC∗2 t −3 .
From the bound of J (4.10) and the assumption on the norm of g (4.25), we have
J gS (t)H 2 ≤ CC∗2 t −2−2 . For any f ∈ S, we also have
|(f, J gM )| ≤ CC∗ t −2 f H1 gM L2 ≤ CC∗2 t −4 f H1 .
Also, |(f, PS J ξ )| ≤ CC∗2 t −2−3/2 . Finally −i/(µ − 1)Q is exponentially small in t.
(2)
Hence we conclude that |(f, Gµ )| ≤ CC∗2 t −3 f . Thus
1
C∗ t −2 ,
8
1
|a (t)| ≤ C∗ t −3 ,
8
|β (t)| + |δ (t)| ≤
1
|ω (t)| ≤ C∗ t −2 ,
8
1
gS (t) 2 ≤ C∗ t −2 ,
H
8
provided that C∗ is sufficiently small. One can also easily check that
∞
∞
1
(1) CC∗2 s −3 ds ≤ C∗ t −2 .
Gµ (s) 2 ds ≤
gM (t) 2 ≤
H
H
8
t
t
The claim (4.31) is proved.
STEP 3 Given two data (ω1 , a1 , g1 ) and (ω2 , a2 , g2 ) we denote by δ their differences:
δω = ω1 − ω2 , δa = a1 − a2 , δg = g1 − g2 , δg = g1 − g2 , and so on. We also let
δ 0 = sup t 2 |δω(t)| + t 3 |δa(t)| + t 2 δg(t)H 1 .
(4.32)
t
Note: different a(t) gives different µ, (µ = ei(1−χ)vy ), but χ is the same. Also, from the
−2
definition of J , we have δJa (t)∞ + δJb (t)∞ + δJ
(t)∞
c
≤ Cδ 0 t .
2
3
2
Our goal is to estimate t |δω (t)|+t |δa (t)|+t δg (t) H 1 . Recall the definition
of ω , a and g from (4.27), (4.28) and (4.29). In order to estimate the difference of
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
271
(2)
(2)
(2)
ω , a from two initial data, we need to control the difference of δGµ := Gµ,1 −Gµ,2 ,
(2)
where Gµ,k := Jk gk − iµk F (µ−1
k (ξk + gk )), k = 1, 2. Here µk , ξk and Jk denote
the corresponding µ, ξ and J , k = 1, 2 and thus ∂t ξk = (L + PM Jk )ξk . We shall
−1
first estimate δξ , then δF = µ1 F (µ−1
1 (ξ1 + g1 )) − µ2 F (µ2 (ξ2 + g2 )) and finally
δ(J g) := J1 g1 − J2 g2 and δ(J ξ ) := J1 ξ1 − J2 ξ2 .
From the equation of ξ , δξ satisfies
∂t δξ = (L + PM J1 )δξ + PM (δJ )ξ2 .
Since δξ(t) → 0 in H 1 as t → ∞, (see (4.23)), we have
∞
P1 (s, t)PM (δJ (s))ξ2 (s)ds
δξ = −
t
in H 1 . We now derive a bound on δξ . The last term can be decomposed into two parts
A + B with
∞
P1 (s, t)PM (δJ (s))PM es L0 ξ∞ ds,
A := −
t
∞
P1 (s, t)PM (δJ (s)) ξ2 (s) − PM es L0 ξ∞ ds.
B := −
t
Since δJ (s)∞ ≤ Cδ 0 s −2 and ξ2 (s)−es L0 ξ2 H 2 ≤ Cs −1 from (4.23), we can bound
B by
BH 1 ≤
Cδ 0 s −2 C∗ s −1 ds = CC∗ δ 0 t −2 .
We can bound A exactly as in Subsect. 4.5. In other words, it can be written as a sum of
three terms satisfying (4.23). More precisely, A = A(0) + A(1) + A(2) and
(0) A (t) 2,∞ + A(1) (t) 2,∞ ≤ Cδ 0 t −3/2 ,
W
(1) A (t)
H2
W
≤ Cδ 0 t −1 ,
(2) A (t)
H2
≤ Cδ 0 t −3/2 .
(In fact, A(0) = 0.) Notice that the constants on the right hand side now have a δ 0
factor. In particular, we can write δξ = (δξ )a + (δξ )b with (δξ )a = A(0) + A(1) and
(δξ )b = A(2) + B such that
(δξ )a (t)W 1,∞ ≤ Cδ 0 t −3/2 ,
(δξ )b (t)H 1 ≤ Cδ 0 t −3/2 .
(4.33)
From the definition of δF , we can bound δF in terms of δξ and δg. (Note that
(µ1 − µ2 )Q is exponentially small in t.) The previous bound on δξ and the bound (4.32)
on δg thus yields that
δF H 1 ≤ CC∗ δ 0 t −3 .
Also, δ(J g) = (δJ )g1 + J2 (δg). Thus, for any f ∈ S with f H 1 ≤ 1 we have
|(f, δ[J g])| ≤ CC∗ δ 0 t −4 .
272
J. Fröhlich, T.-P. Tsai, H.-T. Yau
Similarly,
|(f, δ[PS J ξ ])| ≤ CC∗ δ 0 t −7/2 .
Finally, δ(−i/(µ − 1)Q) ≤ CC∗ t −2 e−Ct . We conclude for any f ∈ S with f H 1 ≤ 1
that
−3
|(f, δG(2)
µ )| ≤ CC∗ δ 0 t .
Simple calculations then show that
1 −3
δ0 t ,
8
1
|δω (t)| ≤ δ 0 t −2 ,
8
1
δgS (t) 1 ≤ δ 0 t −2 ,
H
8
|δa (t)| ≤
provided that C∗ is sufficiently small.
Finally, the equation of gM (4.29) can be written explicitly as
∂t gM = LgM + PM J (gM + gS ) − i/(µ − 1)Q − iµF (µ−1 (g + ξ )) .
Hence for δgM = g1,M − g2,M we have
∂t δgM = (L + PM J1 )δgM + PM (−δJ )g2,M + δ(J gS ) − iδ(/(µ − 1))Q − iδF .
Since (δgM )(t) → 0 as t → ∞ in H 1 , we can put it in integral form:
(δgM )(t) =
∞
−
P1 (s, t)PM (−δJ )g2,M + δ(J gS ) − iδ(/(µ − 1))Q − iδF ds.
t
(4.34)
Therefore, we can bound the H1 norm of δgM by
∞
≤
C
δgM 1
(−δJ )g2,M + δ(J gS ) − iδ(/(µ − 1))Q − iδF H
Since
t
(δJ )g2,M H1
≤ Cδ 0 s −2 g2,M H2
H1
ds .
,
(that is why we needed to prove a stronger bound for g in Step 2), together with previous
bounds on δ(J gS ), −iδ(/(µ − 1))Q and iδF , we can bound the integrand by C∗ δ 0 s −3 .
Thus we have
1
δgM 1 ≤ δ 0 t −2
H
8
provided that C∗ is sufficiently small.
Point-Particle (Newtonian) Limit of Non-Linear Hartree Equation
Conclusion:
273
For the case v0 = 0, we have proved that
t 2 |δω (t)| + t 3 |δa (t)| + t 2 δg (t)
H1
≤ δ 0 /2
under the assumptions (4.24), (4.30) and (4.32). Thus the map (4.26) is a contraction.
Since (4.30) holds for a nonempty set of functions (including zero), we obtain a solution
(ω, a, g), together with ξ . Furthermore, we have proved that
t 2 |ω(t)| + t 3 |a(t)| + t 2 g(t)H 2 ≤ C∗
∞
for t greater than an aboulute constant T . Hence v(t) = − t a(s)ds = O(t −2 ).
Similarly r(t) = O(t −1 ) and θ0 (t) = O(t −1 ). Also recall y = x − r(t) and has,0 = ξ∞ .
Therefore, by Taylor expansion,
ψ(x, t) − ψas (x, t)
= Q(y)ei(vy−Et+θ0 ) + h(y, t)ei(χvy−Et+θ0 ) − Q(x)e−iEt + eit9/2 ξ∞ (x)
= O(t −1 )
in H 2 .
Note that our result is true for t > T . However, if we replace all previous estimates of
the form t −m by (t + T )−m , our contraction argument still holds. Hence Theorem 1.2
is proved for the case v0 = 0. To conclude Theorem 1.2 for general v0 , we apply the
following Galilei transform (boost):
ψ(x, t) −→ ψ(x − v0 t, t)ei(v0 ·x− 2 v0 t) .
1 2
(Recall has,0 (x) = ξ∞ (x)eiv0 ·x and ĥas,0 (v0 ) = ξ̂∞ (0) = 0.) Also, for general r0
we apply a translation, which does not require a change of assumption. The proof is
complete.
Acknowledgements. We thank J. Bourgain, I.M. Sigal and T. Spencer for some useful discussions and comments. T. P. Tsai and H. T. Yau would like to acknowledge the support of the Center for Theoretical Sciences
at Taiwan where part of this work was done.
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Communicated by A. Kupiainen