Asymptotic Dynamics
of Nonlinear Schrödinger Equations:
Resonance-Dominated
and Dispersion-Dominated Solutions
TAI-PENG TSAI
AND
HORNG-TZER YAU
Courant Institute
Abstract
We consider a linear Schrödinger equation with a nonlinear perturbation in R3 .
Assume that the linear Hamiltonian has exactly two bound states and its eigenvalues satisfy some resonance condition. We prove that if the initial data is sufficiently small and is near a nonlinear ground state, then the solution approaches to
certain nonlinear ground state as the time tends to infinity. Furthermore, the difference between the wave function solving the nonlinear Schrödinger equation
and its asymptotic profile can have two different types of decay: The resonancedominated solutions decay as t −1/2 or the dispersion-dominated solutions decay
c 2002 John Wiley & Sons, Inc.
at least like t −3/2 . Contents
1. Introduction
154
2. Preliminaries
162
3. Main Oscillation Terms
171
4. Outline of Proofs and Basic Estimates
175
5. Estimates of the Dispersive Wave
179
6. Excited-State Equation and Normal Form
182
7. Change of the Mass of the Ground State
191
8. Existence of Dispersion-Dominated Solutions
201
9. Proof of Linear Estimates
205
Bibliography
215
Communications on Pure and Applied Mathematics, Vol. LV, 0153–0216 (2002)
c 2002 John Wiley & Sons, Inc.
154
T.-P. TSAI AND H.-T. YAU
1 Introduction
Consider the nonlinear Schrödinger equation
(1.1)
i∂t ψ = (−1 + V )ψ + λ|ψ|2 ψ ,
ψ(t = 0) = ψ0 ,
where V is a smooth localized potential, λ is an order-1 parameter, and ψ =
ψ(t, x) : R × R3 −→ C is a wave function. Let e0 < 0 be the ground state
energy to −1 + V , and denote H1 = −1 + V − e0 . The nonlinear bound states to
the Schrödinger equation (1.1) are solutions to the equation
(1.2)
(−1 + V )Q + λ|Q|2 Q = E Q .
They are critical points to the energy functional
Z
1
1
1
H[φ] =
|∇φ|2 + V |φ|2 + λ|φ|4 d x
2
2
4
2
subject to the constraint of fixed L norm. For each bound state Q = Q E , ψ(t) =
Qe−i Et , is a solution to the nonlinear Schrödinger equation. We may obtain a
family of such bound states by standard bifurcation theory: For each E sufficiently
close to e0 so that E − e0 and λ share the same sign, there is a unique small positive
solution Q = Q E to equation (1.2) that decays exponentially as x → ∞; see
Lemma 2.1. We call this family the nonlinear ground states and shall refer to it as
{Q E } E .
Let
(1.3)
HE = −1 + V − E + λQ 2E .
We have HE Q E = 0. Since Q E is small and E is close to e0 , the spectral properties
of HE are similar to those of H1 .
Suppose the initial data of the nonlinear Schrödinger equation ψ0 is near some
Q E . Under rather general conditions, the family of nonlinear ground states is stable
in the sense that if
inf ψ(t) − Q E ei2 2
2,E
L
is small for t = 0, it remains so for all t; see, for example, [13] for the case λ < 0.
See also [18, 19]. Let k·k L 2 denote a local L 2 norm; a precise choice will be made
loc
later on. One expects that this difference actually approaches zero when measured
by a local L 2 norm, i.e.,
(1.4)
lim inf ψ(t) − Q E ei2 2 = 0 .
t→∞ 2,E
L loc
If −1 + V has only one bound state, it is proven in [15] that the evolution will
eventually settle down to some ground state Q E∞ with E ∞ close to E. (See also
[9] for another proof using techniques from dynamical systems.)
Suppose now that −1 + V has multiple bound states, say, two bound states: a
ground state φ0 with eigenvalue e0 and an excited state φ1 with eigenvalue e1 , i.e.,
H1 φ1 = e01 φ1 where e01 = e1 −e0 > 0. The question is whether the evolution with
initial data ψ0 near some Q E will eventually settle down to some ground state Q E∞
DYNAMICS OF NLS
155
with E ∞ close to E. Furthermore, can we characterize the asymptotic evolution?
In this paper, we shall answer this question positively in the case of two bound
states and estimate precisely the rate of relaxation for a certain class of initial data.
We now state the main assumptions of this paper.
A SSUMPTION A0: −1 + V acting on L 2 (R3 ) has two simple eigenvalues
e0 < e1 < 0 with normalized eigenvectors φ0 and φ1 .
A SSUMPTION A1: Resonance condition. Let e01 = e1 − e0 be the spectral
gap of the ground state. We assume that 2e01 > |e0 | so that 2e01 is in the
continuum spectrum of H1 . Furthermore, for some constant γ0 > 0 and all
real s sufficiently small,
1
2
H1
2
(1.5)
lim φ0 φ1 , Im
P φ0 φ1 ≥ γ0 > 0 .
σ →0+
H1 − σ i − 2e01 − s c
We shall use 0i to replace σ i and the limit limσ →0+ later on.
A SSUMPTION A2: For λQ 2E sufficiently small, the bottom of the continuous
spectrum to −1 + V + λQ 2E , 0, is not a generalized eigenvalue; i.e., it
is not a resonance. Also, we assume that V satisfies the assumption in
Yajima [20] so that the W k, p estimates k ≤ 2 for the wave operator W H =
limt→∞ eit H eit (1+E) hold for k ≤ 2; i.e., there is a small σ > 0 such that
|∇ α V (x)| ≤ Chxi−5−σ
for |α| ≤ 2 .
Also, the functions (x · ∇)k V for k = 0, 1, 2, 3, are −1 bounded with a
−1 bound less than 1:
(x · ∇)k V φ ≤ σ0 k−1φk2 + C kφk2 , σ0 < 1 , k = 0, 1, 2, 3.
2
Assumption A2 contains some standard conditions to assure that most tools in
linear Schrödinger operators apply. These conditions are certainly not optimal. The
main assumption above is the condition 2e01 > |e0 | in assumption A1. The rest of
assumption A1 just consists of generic assumptions. This condition states that the
excited state energy is closer to the continuum spectrum than to the ground state
energy. It guarantees that twice the excited state energy of H1 (which one obtains
from taking the square of the excited state component) becomes a resonance in
the continuum spectrum (of H1 ). This resonance produces the main relaxation
mechanism. If this condition fails, the resonance occurs in higher-order terms and
a proof of relaxation will be much more complicated. Also, the rate of decay will
be different.
Define the notation
p
∼ min εn, n −1 t −1/2 ,
(1.6) hxi = 1 + x 2 , {t}ε = ε−2 n −2 + 20t , {t}−1/2
ε
where n = kψ0 k L 2 and 0 = O(n 2 ) is a positive constant to be specified later in
(6.3) of Lemma 6.1. For the moment, we remark that 0 is of order 2λ2 n 2 times
the quantity in (1.5). The subscript ε is a small parameter to be specified in Theorems 1.3 and 1.4. We shall often drop it in the proofs of Theorems 1.3 and 1.4. We
156
T.-P. TSAI AND H.-T. YAU
denote by L r2 the weighted L 2 spaces (r may be positive or negative),
(1.7)
L r2 (R3 ) ≡ φ ∈ L 2 (R3 ) : hxir φ ∈ L 2 (R3 ) .
Our space for initial data is
Y ≡ H 1 (R3 ) ∩ L r20 (R3 ) ,
(1.8)
r0 > 3 .
to denote
The parameter r0 > 3 is fixed, and we can choose,
We shall use
say, r0 = 4 for the rest of this paper.
Our first theorem states that if the initial data ψ0 is small in Y and the distance
between ψ0 and a nonlinear ground state Q ∗ is small, then the solution ψ(t) has to
settle down to some asymptotic nonlinear ground state as t → ∞. Furthermore,
the difference between ψ0 and the asymptotic nonlinear ground state at t = ∞ is
bounded above by the order O(t −1/2 ). Recall that λ = O(1) is fixed.
L 2loc
L 2−r0 .
T HEOREM 1.1 Assume that assumptions A0, A1, and A2 on V hold. Then there
are small universal positive constants ε0 and n 0 > 0 such that, for any nonlinear
ground state Q ∗ with mass n = kQ ∗ k L 2 < n 0 and any initial data ψ0 satisfying
kψ0 − ei20 Q ∗ kY ≤ ε02 n 2 for some 20 ∈ R, there exist an energy E ∞ and a function
2(t) such that kQ E∞ k L 2 − n = O(ε02 n), 2(t) = −E ∞ t + O(log t), and
ψ(t) − Q E ei2(t) 2 ≤ C(1 + t)−1/2 .
(1.9)
∞
L loc
To describe more detailed behavior of the solution ψ(t), we need various spectral properties of the linearized operator. All statements we make here will be
proven in Section 2. Let L be the operator obtained from linearizing the Schrödinger equation (1.1) around the trivial solution Qe−i Et (see Section 2), i.e.,
(1.10)
Lh = i −1 (−1 + V − E + λQ 2 )h + λQ 2 (h + h) .
Notice that L is not self-adjoint due to the conjugation. If we can decompose h
into its real and imaginary parts, the operator L can be written in the matrix forms
0
L−
L ←→
−L + 0
where
(1.11)
L − := −1 + V − E + λQ 2E = HE ,
L + = L − + 2λQ 2E .
Explicitly, we have
L( f + ig) = L − g − i L + f .
Notice that L is a small perturbation of the Hamiltonian H1 . Hence the spectral
properties of L are closely related to those of H1 . Since H1 has two eigenvalues, 0
and e01 , we expect L to have two eigenvalues as well.
From (1.2) we have L − Q E = 0. If we differentiate (1.2) with respect to E, we
have L + R E = Q E , where R E = ∂ E Q E . Let S E be the space spanned by i Q E and
its tangent R E , i.e.,
(1.12)
S = S E = spanR {R E , i Q E } .
DYNAMICS OF NLS
157
Clearly, S is the generalized eigenspace of L with eigenvalue 0.
L has a pair of eigenvalues ±iκ, where κ is a perturbation of e01 = e1 − e0 of
H1 . The corresponding generalized eigenfunctions, u and v, are perturbations of
the linear excited state φ1 . They are real-valued functions characterized by
(1.13)
L + u = κv ,
L − v = κu ,
(u, v) = 1 .
In particular, for all real numbers α and β, we have L2 (αu + βiv) = −κ 2 (αu +
βiv). Thus
(1.14)
Eκ (L) = spanR {u, iv}
is the generalized eigenspace with eigenvalues ±iκ.
Finally, the space of the continuous spectrum, Hc (L), is characterized by the
relation
(1.15)
Hc (L) = f + ig ∈ L 2 : f, g real, f ⊥ Q E , v; g ⊥ R E , u .
We have the following spectral decomposition result, to be proven in Section 2.2.
L EMMA 1.2 (Spectral Decomposition) The space of complex-valued L 2 functions
in R3 can be decomposed as the direct sum of S, Eκ (L) and Hc (L), i.e.,
(1.16)
L 2 (R3 ) = S ⊕ M = S ⊕ Eκ (L) ⊕ Hc (L) ,
M := Eκ (L) ⊕ Hc (L) .
The decomposition is not orthogonal; the three real subspaces S, Eκ (L), and
Hc (L) are invariant under L but not under multiplication by i. The space M can
also be characterized by
(1.17)
M = M E = f + ig ∈ L 2 : f, g real, f ⊥ Q E ; g ⊥ R E .
From this lemma, for a fixed energy E we can decompose a wave function via
(1.16). It is more convenient for our analysis to decompose the wave function into
the following form:
(1.18)
ψ(t, x) = [Q E (x) + a E (t)R E (x) + h E (t, x)] ei2(t)
with h E (t, x) ∈ M E . We shall prove in Lemma 2.2 the existence and uniqueness
of such a decomposition for ψ(t, x) near a ground state. If we allow the energy E
to vary, it is possible to choose E such that the component along the R E direction
vanishes; i.e., there exists E(t) such that
(1.19)
ψ(t, x) = [Q E(t) (x) + h(t, x)]ei2(t)
with h E (t, x) ∈ M E ; see Lemma 2.3. We can view this ground state Q E(t) as
the best approximation to the wave function ψ(t). This choice is different from
choosing the best approximation by minimizing the difference in the L 2 norm
inf E,2 kψ(t, ·) − Q E ei2 k L 2 .
Although (1.19) can be viewed as the best decomposition of a wave function,
it introduces a time-dependent linearized operator. So for analytical purposes, it is
more convenient to work with (1.18). Notice that Q E + a E (t)R E is the first-order
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T.-P. TSAI AND H.-T. YAU
approximation of Q (E+a E (t)) . So the decomposition (1.18) can be viewed as the
best time-independent decomposition. Since h E (t, x) ∈ M E , we can write it as
(1.20)
h E (t) = ζ E (t) + η E (t) ,
ζ E ∈ Eκ (L) ,
η E ∈ Hc (L) .
Theorem 1.1 gives an upper bound to the difference between the wave function
and the asymptotic ground state. In fact, we can estimate the component along the
excited states and the continuum spectrum more precisely in the following theorem:
T HEOREM 1.3 Suppose the assumptions of Theorem 1.1 hold. Let
1/2 (1.21)
ε ≡ n −1 kζ E0 ,0 k + kη E0 ,0 kY ≤ ε0 .
If we decompose ψ(t) as in (1.18) with Q = Q E∞ being the profile at time t = ∞
and h(t) = ζ (t) + η(t) as in (1.20), then we have
(1.22)
kζ (t)k L 2 ≤ C {t}−1/2 ,
kη(t)k L 4 ≤ C {t}−3/4+σ ,
|a(t)| ≤ C {t}−1 ,
kη(t)k L 2 ≤ C {t}−1 ,
loc
where {t} = {t}ε and σ = 0.01.
Theorem 1.3 provides rather precise upper bounds on the asymptotic evolution.
These bounds are optimal for a large class of initial data described by the next
theorem. In the following, for two functions f and g, we denote
(1.23)
f ≈g
if C1 kgk ≤ k f k ≤ C2 kgk
for some constants C1 , C2 > 0.
T HEOREM 1.4 (Resonance-Dominated Solutions) Assume that assumptions A0
through A2 on V hold. Suppose that the initial data ψ0 is decomposed as in (1.19)
with respect to the unique E 0 , i.e.,
ψ0 = [Q E0 + ζ E0 ,0 + η E0 ,0 ]ei2 E0 ,0 .
Write ζ E0 ,0 = z 0 u + i z 1 v and denote z E0 ,0 = z 0 + i z 1 . Suppose that these components satisfy
kz E0 ,0 k
kψ0 k = n < n 0 , 0 < ε :=
(1.24)
≤ ε0 , kη E0 ,0 kY ≤ ε2 n 2 ,
n
where n 0 and ε0 are the same constants as in Theorem 1.1. Then the conclusions
of Theorem 1.1 and Theorem 1.3 hold. In particular, there is a limit frequency E ∞
such that the estimates (1.22) hold for {t} = {t}ε . If ζ (t) = ζ E∞ (t) denotes the
excited state component with respect to the linearized operator with energy E ∞ ,
the lower bound holds as well, and we have
(1.25)
ζ E∞ (t) ≈ {t}−1/2
.
ε
Furthermore, for all λ > 0 or λ < 0, we have kQ E∞ k2 > kQ E0 k2 and
1
(1.26)
kQ E∞ k2L 2 = kQ E0 k2L 2 + kζ E0 ,0 k2L 2 + o(ε2 n 2 ) .
2
DYNAMICS OF NLS
159
Solutions satisfying (1.25) are called resonance-dominated solutions.
Roughly speaking, Theorem 1.4 states that, if the excited state component is
much bigger than the dispersive component, then the decay mechanism is dominated by resonance. Furthermore, (1.26) states that approximately half of the probability density of the excited state, |ζ E0 (x)|2 /2, is transferred to the ground state
independently of the sign of λ. Since the total probability is conserved, the other
half is transferred to the dispersive part.
We have decomposed the wave function according to the optimal energy E 0
(1.19). Alternatively, we can first fix an energy E ∗ and decompose the initial data
ψ0 as
(1.27)
ψ0 = [Q E∗ + a∗ R E∗ + ζ∗ + η∗ ]ei2∗ .
If the components satisfy
(1.28)
kψ0 k = n < n 0 ,
kζ∗ k = εn ,
kη∗ kY ≤ Cε2 n 2 ,
0 < ε ≤ ε0 ,
|a∗ | ≤ ε2 n 2 ,
then we can re-decompose ψ0 as in Theorem 1.4 with respect to E 0 satisfying the
estimate (1.24); see Lemma 2.3. This implies that the set of all such ψ0 contains
an open set with nonlinear ground states in its boundary. Therefore, the class of
resonance-dominated solutions is in a sense large.
The condition (1.24) is very subtle. It states that the excited-state component
is much bigger than the dispersive component with respect to a decomposition
according to L. Since L differs from HE by order n 2 , the condition (1.24) does not
hold for a decomposition of ψ0 with respect to the linear Hamiltonian HE .
Finally, the following existence result shows that the resonance-dominated solutions are not all solutions. We expect that the dispersion-dominated solutions
constructed by the following theorem are rare:
T HEOREM 1.5 (Dispersion-Dominated Solutions) Let assumptions A0 through A2
apply on V . Let Z = H 2 ∩W 2,1 (R3 ). For a given nonlinear ground state Q E∞ with
kQ E∞ k = n ≤ n 0 , let E = E ∞ and L = L E∞ . For any given ξ∞ ∈ Hc (L) ∩ Z with
sufficiently small Z norm, there exists a solution ψ(t) of (1.1) and a real function
θ (t) = O(t −1 ) for t > 0 so that
kψ(t) − ψas (t)k H 2 (R3 ) ≤ Ct −2 ,
where
ψas (t) = Q E e−i Et+iθ (t) + e−i Et et L ξ∞ .
In particular, ψ(t) − Q E∞ e−i Et+iθ (t) = O(t −3/2 ) in L 2loc .
Theorem 1.5 constructs dispersion-dominated solutions based on the operator
et L . The scattering property of this operator is very similar to the standard operator
ei1t . In particular, for χ∞ ∈ e
Z = H 2 (R3 ) ∩ W 2,1 (R3 , (1 + |x|4 )d x) sufficiently
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T.-P. TSAI AND H.-T. YAU
small in e
Z with χ̂∞ (0) = 0 and ∇ χ̂∞ (0) = 0, the same statement holds if we
replace ψas by
eas (t) = Q E e−i Et + ei1t χ∞ .
ψ
This will be shown in the proof of Theorem 1.5.
The resonance solutions were first observed by Buslaev and Perel0 man [3] for
some one-dimensional Schrödinger equation with two bound states and a higher
nonlinear term (we thank the referee for supplying us with this reference). In the
physical dimension d = 3, this type of solution was proven by Soffer and Weinstein
in an important paper [16], but for real solutions to the nonlinear Klein-Gordon
equation
(1.29)
∂t2 u + B 2 u = λu 3 ,
B 2 := (−1 + V + m 2 ) ,
where λ is a small nonzero number. Assume that B 2 has only one eigenvector (the
ground state) φ, B 2 φ = 2 φ, with the resonance condition  < m < 3 (and
some positivity assumption similar to that appearing in assumption A1). Rewrite
real solutions to equation (1.29) as u = aφ + η with
a(t) = Re A(t)eit ,
Re A0 (t)eit = 0 .
Then A and η satisfy the equations
(1.30)
(1.31)
1 −it
e
φ, λ(aφ + η)3
2i
2
2
(∂t + B )η = Pc λ(aφ + η)3 .
Ȧ =
Theorem 1.1 in [16] states that all solutions decay as
A(t) ∼ hti−1/4 ,
kη(t)k L ∞ ∼ hti−3/4 .
In particular, the ground state is unstable and will decay as a resonance with
rate t −1/4 .
We first remark that the proof in [16] has only established the upper bound
t −1/4 . Furthermore, a universal lower bound of the form t −1/4 is in fact incorrect.
From the previous work of [1, 5], it is clear that dispersion-dominated solutions
decaying much faster than t −1/4 exist. Similar to Theorems 1.4 and 1.5, we have
the following two cases:
(1) η(0) A(0): The dominant term on the right side of (1.30) is λa 3 φ 3 .
(2) η(0) A(0): The dominant term is λη3 .
In case 2 another type of solutions arises, namely, those with decay rate
A(t) ∼ hti−2 ,
kηk L ∞ ∼ hti−3/2 .
We shall sketch a construction of such solutions at the end of Section 8.
Notice that all solutions in [16] decay as a function of t. Therefore, we can view
[16] as a study of asymptotic dynamics around a vacuum. Although most works
concerning asymptotic dynamics of nonlinear evolution equations have been concentrated on cases with vacuum as the unique profile at t = ∞, more interesting
DYNAMICS OF NLS
161
and relevant cases are asymptotic dynamics around solitons (such as the Hartree
equations [5]; see also [2, 3]). The soliton dynamics have extra complications
involving translational invariance. The current setting of nonlinear Schrödinger
equations with local potentials eliminates the translational invariance and constitutes a useful intermediate step. This greatly simplifies the analysis but preserves a
key difficulty that we now explain.
Recall that we need to approximate the wave function ψ(t) by nonlinear ground
states for all t. Since we aim to show that the error between them decay like t −1/2 ,
we have to track the nonlinear ground states with accuracy at least like t −1/2 . Although the nonlinear ground states approximating the wave function ψ(t) can in
principle be defined, say, via equation (1.4) or (1.6), neither characterization is
useful unless we know the wave functions ψ(t) precisely. Furthermore, even assuming we can track the approximate ground states reasonably well, the linearized
evolution around these approximate ground states will be based on time-dependent,
non-self-adjoint operators Lt . At this point we would like to mention the approach
of [15] based on perturbation around the unitary evolution eit H1 , where H1 is the
original self-adjoint Hamiltonian. While we do not know whether this approach
can be extended to the current setting by adding the ideas of [16] (it was announced
in [16] that its method can be extended to (1.1) as well), such an approach can be
difficult to extend to the Hartree or other equations with nonvanishing solitons. The
main reason is that these dynamics are not perturbations of linear dynamics.
We believe that perturbation around the profile at t = ∞ is a more natural
setup. In this approach, at least we do not have to worry about the time dependence
of approximate ground states in the beginning. But the linearized operator L =
L∞ is still non-self-adjoint, and it does not commute with the multiplication by
i. So calculations and estimates based on L are rather complicated. Our idea
is to map this operator to a self-adjoint operator by a bounded transformation in
Sobolev spaces. This map in a sense brings the problem back to the self-adjoint
case for various calculations and estimates. Another important input we used is the
existence and boundedness of the wave operator for L. The boundedness of the
wave operator in Sobolev spaces for the standard one-body Schrödinger operator
is a classical theorem of Yajima [20]. In the current setting it was recently proved
by Cuccagna [4]. See Section 2 for more details.
The next step is to identify and calculate the leading oscillatory terms of the
nonlinear systems involving the bound-states components and the continuum-spectrum components. The leading-order terms, however, depend on the relative sizes
of these components, and thus we have two different asymptotic behaviors: The
resonance-dominated solutions and the dispersion-dominated solutions. Finally,
we represent the continuum spectrum component in terms of the bound-states components, and this leads to a system of ordinary differential-integral equations for
the bound-states components. This system can be put into a normal form, and the
size of the excited-state component can be seen to decay as t −1/2 . Notice that the
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T.-P. TSAI AND H.-T. YAU
phase and the size of the excited-state component decay differently. It is thus important to isolate the contribution of the phase in the system. Finally, we estimate
the error terms using estimates of the linearized operators. These estimates are
based on standard methods from scattering theory [6, 14, 16] and estimates on the
wave operators [4, 20].
2 Preliminaries
We first fix our notation. Let H k denote the Sobolev spaces W k,2 (R3 ). The
weighted Sobolev space L r2 (R3 ) is defined in (1.7). The inner product ( , ) is
Z
(2.1)
( f, g) =
f¯g d 3 x .
R3
For two functions f and g, we denote
f = O(g)
if k f k ≤ C kgk for some constant C > 0 .
We will denote, as in (1.23),
f ≈g
if C1 kgk ≤ k f k ≤ C2 kgk for some constants C1 , C2 > 0 .
For a double index α = (α0 , α1 ), α0 and α1 nonnegative integers, we denote
(2.2)
z α = z α0 z̄ α1 ,
(32)
|α| = α0 + α1 ,
[α] = −α0 + α1 .
= z z̄ . We will define z = µ−1 p where µ = eiκt . Hence
For example, z
z α = µ[α] p α . In what follows |α| = 2 and |β| = 3.
3 2
2.1 Nonlinear Ground States Family
We establish here the existence and basic properties of the nonlinear groundstates family mentioned in Section 1. Our statement and proof are also valid for
nonlinear excited states.
L EMMA 2.1 Suppose that −1 + V satisfies assumptions A0 and A2. Then there is
an n 0 sufficiently small such that for E between e0 and e0 +λn 20 there is a nonlinear
ground state {Q E } E solving (1.2). The nonlinear ground state Q E is real, local,
and smooth, λ−1 (E − e0 ) > 0, and
Z
−1/2
Q E = nφ0 + O(n 3 ) , n ≈ C[λ−1 (E − e0 )]1/2 , C =
φ04 d x
.
Moreover, we have R E = ∂ E Q E = O(n −2 )Q E + O(n) = O(n −1 ) and ∂ E2 Q E =
O(n −3 ). If we define c1 ≡ (Q, R)−1 , then c1 = O(1) and λc1 > 0.
P ROOF : Suppose that Q = Q E is a nonlinear ground state satisfying
(−1 + V − e0 )Q + λQ 3 = E 0 Q
where E 0 = E − e0 is small. Write Q = nφ0 + h with real h ⊥ φ0 . Then h satisfies
(2.3)
(−1 + V − e0 − E 0 )h + λ(nφ0 + h)3 = E 0 nφ0 .
DYNAMICS OF NLS
163
Taking the inner product with φ0 , we see that it is necessary that λ and E 0 be of the
same sign. For E 0 sufficiently small, since the spectral gap of the operator −1 +
V − e0 is of order 1, the same conclusion holds for the operator −1 + V − e0 − E 0 .
Thus we can check directly from (2.3) that
(2.4)
kQ E k := n 2 ∼ λ−1 E 0 ,
khk2 = O(λn 3 ) .
The existence of solutions h and E 0 can be established from the implicit function
theorem, or the solutions can simply be obtained by a contraction mapping argument.
By differentiating the equation of Q E with respect to E, we have
(2.5)
L + RE = Q E ,
L + := −1 + V − E + 3λQ 2 .
Denote by |0i the ground state to L + with L 2 norm 1 and eigenvalue ≈ λn 2 . We
have
Q E = O(n)|0i + O(λn 3 ) .
Hence (using the spectral gap)
n
|0i + O(λn 3 )
R = (L + ) [O(n)|0i + O(λn )] = O
λn 2
−1
(2.6)
3
= O(λ−1 n −2 )Q + O(n) .
Hence c1 = O(1). Since the sign of the ground-state energy to L + is the same as
λ, we have λ(Q, R) > 0. By differentiating the equation, we can prove ∂ E2 Q E =
O(n −3 ) in a similar way.
Suppose Q E = O(n) and a is a small parameter with |a| n. By Taylor
expansion and ∂ E2 Q E = O(n −3 ), we have
(2.7)
Q E+a = Q E + a R E + O(n −3 a 2 ) = Q E + O(n −1 a) .
Since (a R E )2 = O(n −2 a 2 ), from (2.7) we also have
(2.8)
kQ E+a k2 = kQ E k2 + 2a(Q E , R E ) + O(n −2 a 2 ) .
We will need the following renormalization lemmas. We will formulate them
in terms of nonlinear ground states, although they also hold for nonlinear excited
states. Let Y1 be either Y , defined in (1.8), or L 2−r0 , defined in (1.7) with r0 > 3.
Recall from Lemma 1.2 that h ∈ M if and only if Re h ⊥ Q and Im h ⊥ R.
L EMMA 2.2 Let ψ ∈ Y1 be given with kψkY1 = n, 0 < n ≤ n 0 . Suppose that for
some nonlinear ground state Q E ei2 we have kψ − Q E ei2 kY1 ≤ Cτ n, 0 ≤ τ ≤ ε0 .
Then there are unique small a, θ, and h such that
(2.9)
ψ = [Q E + a R E + h] ei(2+θ ) ,
Furthermore, kQ E kY1 >
9
n,
10
h ∈ ME .
a = O(τ n 2 ), θ = O(τ ), and h = O(τ n).
164
T.-P. TSAI AND H.-T. YAU
P ROOF : By considering ψe−i2 instead of ψ, we may assume 2 = 0. Write
ψ = Q + k with a small k = O(τ n). We want to solve small a and θ such that
h = ψe−iθ − Q E − a R E ∈ M E , i.e.,
0
(Q, Re(Q + k)e−iθ − Q − a R)
=
,
0
(R, Im[(Q + k)e−iθ ])
with the left side of order [ τ τn 2 ] if (a, θ ) = (0, 0). We compute the derivative of
this vector with respect to a and θ at (a, θ ) = (0, 0) and obtain the matrix
−(Q, R)
(Q, Im k)
−(Q, R)
O(τ n 2 )
,
=
0
(R, − Re(Q + k))
0
−(R, Q) + O(τ )
which is invertible. Here we have used Q = O(n) and R = O(n −1 ). Since
(Q, R) = O(1), by the implicit function theorem we can solve a = O(τ n 2 ) and
θ = O(τ ) satisfying the equation. Hence h = (Q E + k)e−iθ − Q E − a R E =
O(τ n).
The next lemma shows that the best decomposition as defined by (1.19) can
always be achieved for a small vector near a nonlinear ground state.
L EMMA 2.3 Let ψ ∈ Y1 with kψkY1 = n bounded by a small number n 0 , i.e.,
0 < n ≤ n 0 . If ψ is near a nonlinear ground state Q E1 ei21 in the sense that
kψ − Q E1 ei21 kY1 ≤ Cτ n for some τ with 0 ≤ τ ≤ ε0 and 0 small. Then there is
a unique E near E 1 such that the component along the R E direction as defined by
(2.9) vanishes; i.e., there is unique small θ and h such that
ψ = [Q E + h] ei(21 +θ ) ,
h ∈ ME .
Moreover, E − E 1 = O(τ n ), θ = O(τ ), and h = O(τ n).
If ψ is given explicitly as ψ = [Q E1 + a1 R E1 + h 1 ]ei21 with h 1 ∈ M E1 and
a1 = O(τ 2 n 2 ), we have
5
(2.10)
|E − E 1 | ≤ |a1 | = O(τ 2 n 2 ) , θ = O(τ 3 ) , h − h 1 = O(τ 2 n) .
4
2
2
P ROOF : As in the proof of the previous lemma, we may assume 21 = 0. Write
ψ = Q E1 + k1 with h 1 ∈ M E1 and k1 = a1 R E1 + h 1 = O(τ n) small. Let
E = E 1 + γ . We want to find small γ and θ such that h = ψe−iθ − Q E ∈ M E ,
that is,
(Q E1 +γ , Re(Q E1 + k1 )e−iθ − Q E1 +γ )
0
(2.11)
=
.
0
(R E1 +γ , Im[(Q E1 + k1 )e−iθ ])
When (γ , θ ) = (0, 0), the left side equals
2
(Q E1 , a1 R E1 )
τn
, which is of order
.
0
0
The derivative of this vector with respect to γ and θ at (γ , θ) = (0, 0) is given by
DYNAMICS OF NLS
165
(Q E1 , Im k1 )
(R E1 , −Q E1 + Re k1 )
=
(∂ E R E | E=E1 , Im k1 ) (R E1 , − Re(Q E1 + k1 ))
−(Q, R)
O(τ n 2 )
,
O(τ n −2 ) −(R, Q) + O(τ )
which is invertible. Here we have used Q = O(n), R = O(n −1 ), and ∂ E R E =
O(n −3 ). By the implicit function theorem we can solve γ = O(τ n 2 ) and θ =
O(τ 2 ) satisfying the equation. Hence h = (Q E1 + k1 )e−iθ − Q E1 +γ = O(τ n).
Finally, when a1 = O(τ 2 n 2 ), the vector on the left side of (2.11) at (γ , θ) =
(0, 0) is of order
2 2
τ n
.
0
Hence we can solve γ = O(τ 2 n 2 ) and θ = O(τ 3 ). We also have h − h 1 =
(Q E1 + a1 R E1 + h 1 )e−iθ − h 1 − Q E1 +γ = O(nθ) + O(n −1 a1 ) = O(τ 2 n). The last
statement is proven.
The next lemma provides estimates on the components in (1.18) when the reference nonlinear ground state varies.
L EMMA 2.4 Let ψ = [Q E1 + a1 R E1 + h 1 ]ei21 with kQ E1 kY1 = n ≤ n 1 , kh 1 kY1 =
ρ ≤ ε1 n, and |a1 | ≤ Cρ 2 . Suppose E 2 = E 1 + γ with |γ | ≤ Cρ 2 . By Lemma 2.2
we can rewrite ψ uniquely with respect to E 2 as
(2.12)
ψ = Q E2 + a2 R E2 + h 2 ei22 ,
where h 2 ∈ M E2 , a2 , and θ = 22 − 21 are small. We have the estimates
θ = O(n −3 ργ ) ,
E 1 + a1 − E 2 − a2 = O(n −1 ργ ) ,
(2.13)
h 1 − h 2 = O(n −2 ργ ) .
Notice that n −3 ργ ≤ Cε13 is small.
P ROOF : Note |γ | ≤ Cρ 2 ≤ ε12 n 2 . Using (2.7) we have
Q E1 − Q E2 = O(n −1 γ ) ,
(2.14)
R E1 − R E2 = O(n −3 γ ) ,
λQ 2E1 − λQ 2E2 = O(γ ) .
Since
e−i21 ψ − Q E2 = [Q E1 − Q E2 ] + a1 R E1 + h 1
= O(n −1 γ ) + O(ρ 2 n −1 ) + O(ρ) = O(ρ) ≤ Cε1 n ,
Lemma 2.2 is applicable with τ = ρ/n, and we have (2.12) with
a2 = O(τ n 2 ) = O(nρ) ,
h 2 = O(τ n) = O(ρ) ,
θ = O(τ 2 ) = O(n −2 ρ 2 ) .
We have
(2.15)
[Q E1 + a1 R E1 + h 1 ]e−iθ = Q E2 + a2 R E2 + h 2 .
166
T.-P. TSAI AND H.-T. YAU
Denote by (RS) the right side and by (LS) the left side. Taking the imaginary part
and then taking the inner product with R E2 , we have
(R E2 , Im(RS)) = (R E2 , Im h 2 ) = 0 ,
since h ∈ M E2 , and
(R E2 , Im(LS)) = (R E2 , Q E1 + a1 R E1 + Re h 1 )(− sin θ) + (R E2 , Im h 1 ) cos θ .
Since (R E2 , Q E1 + a1 R E1 + Re h 1 ) ≈ 1 and
(R E2 , Im h 1 ) = (R E1 + O(n −3 γ ), Im h 1 ) = 0 + O(n −3 ργ ) ,
we conclude that sin θ +O(n −3 ργ ) = 0. Since θ is small, we have θ = O(n −3 ργ ).
Write eiθ = 1 + O(θ ) in (2.15). Since
[Q E1 + a R E1 + h 1 ]O(θ) = O(n)O(n −3 ργ ) ,
equation (2.15) gives
(2.16)
Q E1 + a1 R E1 − Q E2 − a2 R E2 = O(n −2 ργ ) + h 2 − h 1 .
Taking the real part and then taking the inner product with Q E2 , we get from the
right side O(n · n −2 ργ ) + 0 + (Q E1 + O(n −1 γ ), h 1 ) = O(n −1 ργ ). On the other
hand, by (2.7)
Q E 1 + a1 R E 1 − Q E 2 − a2 R E 2
= Q E2 − γ R E2 + O(γ 2 n −3 ) + a1 R E2 + O(γ n −3 ) − Q E2 − a2 R E2
= (−γ + a1 − a2 )R E2 + O(n −3 ρ 2 γ ) ,
where we have used E 1 = E 2 − γ . Hence we conclude
(E 1 − E 2 + a1 − a2 )(Q E2 , R E2 ) + O(n −3 ρ 2 γ ) = O(n −1 ργ ) .
Since (Q E2 , R E2 ) = O(1), we have E 1 − E 2 + a1 − a2 = O(n −1 ργ ). From (2.16),
we thus have
h 2 − h 1 = O(n −1 ργ )R E2 + O(n −3 ρ 2 γ ) − O(n −2 ργ ) = O(n −2 ργ ) .
2.2 Linearized Operator: Spectral Analysis
We now study the spectral properties of L. In this section, we identify C with
R and the multiplication by i becomes
0 −1
J=
.
1
0
2
Recall the definition of M and the decomposition L 2 (R3 ) = S ⊕ M. This decomposition is nonorthogonal and presents some problems in analysis. Let X be the
space orthogonal to Q,
⊥
Q
2
2
3
X = 5(L ) = φ ∈ L (R ) : φ ⊥ Q ; X ←→
,
Q⊥
DYNAMICS OF NLS
167
where 5 is the orthogonal projection that eliminates the Q direction:
(Q, h)
Q.
(Q, Q)
We claim that there is a “nice” operator U from M to X and a self-adjoint operator
A that is a perturbation of H1 such that
(2.17)
L = −U −1 J AU .
5h = h −
M
Let PM be the projection from L 2 onto M according to the decomposition
L (R3 ) = S ⊕ M. It has the matrix form
2
⊥
L
P1 0
Q
PM :
−→
, PM =
,
0 P2
R⊥
L2
2
where the projections are given by
(2.18)
P1 : L 2 −→ Q ⊥ ,
P1 = Id − c1 |RihQ| ,
P2 : L 2 −→ R ⊥ ,
P2 = Id − c1 |QihR| .
c1 = (Q, R)−1 ,
Clearly P1 R = 0 and P2 Q = 0. One can easily check that
I 0
I 0
: X −→ M .
(2.19)
RM X ≡
: M −→ X , R X M ≡
0 5
0 P2
We now define
(2.20)
1/2 1/2
1/2
A := (H 2 + H 1/2 2λQ 2 H 1/2 )
= H L + H 1/2
,
H = L− .
A is a self-adjoint operator acting on L 2 (R3 ), with Q as an eigenvector with eigenvalue 0. We shall often view A as an operator restricted to its invariant subspace
X . Define
1/2 −1/2
A H
0
(2.21)
U0 : X −→ X , U0 ≡
,
0
A−1/2 H 1/2
and let
(2.22)
U = U0 R M X : M −→ X ,
U −1 = R X M U0−1 : X −→ M .
Notice that H −1/2 is defined only on Q ⊥ . We can easily check that U −1U | M = Id M
and UU −1 | X = Id X . Moreover, we have
0
H 1/2 A−1/2 A3/2 H −1/2
0
−1
−(U J )(AU ) M =
0
A1/2 H 1/2
−P2 H −1/2 A1/2
0
0
H
=
.
−P2 5L + 0
Since P2 5L + = L + when acting on R ⊥ , we have proven (2.17).
If we define
1
(2.23)
U± = 5A1/2 H −1/2 P1 ± 5A−1/2 H 1/2 5 ,
2
168
T.-P. TSAI AND H.-T. YAU
then (U± )∗ = 12 (P2 H −1/2 A1/2 5 ± 5H 1/2 A−1/2 5) and
(2.24)
U = U+ + CU− ,
U −1 = (U+ )∗ − C(U− )∗ ,
where C = [ 10 −10 ] is the conjugation operator in L 2 (R3 ). Although U+ and U− are
not self-adjoint operators, they commute with the multiplication operator i. Hence,
as operators in L 2 (R3 ),
(2.25)
U i = (U+ + CU− )i = i(U+ − CU− ) .
Since the mass of Q E is small, A is a self-adjoint perturbation of H1 = −1 +
V −e0 . As H1 has two simple eigenvalues, so does A. The ground state of A is just
Q (which with the normalized ground state equals φ0A = Q/ kQk2 ) with energy
0. Standard perturbation theory implies that A has an excited state Aφ1A = κφ1A
such that κ = e10 + O(n 2 ) and φ1A = φ1 + O(n 2 ). The spectral decomposition
Lemma 1.1 is a corollary of (2.17) and the spectral properties of A. For example,
define
A
u
−1 φ1
.
=U
v
φ1A
We can check easily that u and v are the generalized eigenvectors of (1.13). We
summarize the results here in the following lemma:
L EMMA 2.5 Let X be the orthogonal complement of Q in L 2 . Let A be defined by (2.20). Let U and U −1 be defined by (2.22). Then (2.17) holds, i.e.,
L| M = −U −1 J AU . If we denote by PκA and PcA the orthogonal projections onto
the eigenspace and continuous spectrum space of A, we have
(2.26)
U PM = U ,
U PκL = PκA U ,
A
U PL
c = Pc U .
Furthermore, we have U = U+ + U− C and U −1 = U+∗ − U−∗ C, with U+ and U−
commuting with i.
2.3 The Equations
In this subsection we derive the equations for the components of the solution
ψ(t) of the Schrödiger equation (1.1) according to the decomposition (1.18).
Fix an energy E, the corresponding ground state Q = Q E , and its tangent
R = R E = ∂ E Q E . Recall the decomposition (1.18) of the wave function ψ(t, x).
We let
(2.27)
2(t) = θ(t) − Et .
Denote by h a = a(t)R E (x) + h(t, x). Substituting the above ansatz into (1.1), we
have the equations
∂t h a = Lh a + i −1 (F + θ̇ (Q + h a )) ,
F = λQ(2|h a |2 + h a2 ) + λ|h a |2 h a
= λQ(2|h|2 + h 2 ) + 2λQ Ra(2h + h̄) + 3λQ R 2 a 2 + λ(a R + h)2 (a R + h̄) ,
DYNAMICS OF NLS
169
where L is already defined in (1.10). By (1.2) and (2.5), we have (L − ∂t )a R =
−ia Q − ȧ R. Hence
∂t h = Lh + i −1 (F + θ̇(Q + a R + h)) − ia Q − ȧ R .
Since h(t) ∈ M and M is an invariant subspace of L, we have i −1 (F + θ̇(Q +a R +
h)) − ia Q − ȧ R ∈ M. From the characterization M = [ f + ig : f ⊥ Q, g ⊥ R],
the functions a(t) and θ (t) satisfy
(Q, Im(F + θ̇ h) − ȧ R) = 0 ,
(R, Re(F + θ̇(Q + a R + h)) + a Q) = 0 .
The equation on M is
(2.28)
∂t h = Lh + PM Fall ,
Fall = i −1 (F + θ̇(a R + h)) .
Recall the representation of L in terms of A. The previous equation can be rewritten
as
∂t (U h) = −i A(U h) + U PM Fall .
(2.29)
We can write U h =
of A. Then we have
zφ1A + w
where w denotes the part on the continuum spectrum
ż = −iκz + (φ1A , U PM Fall ) ,
ẇ = −i Aw + PcA U PM Fall .
A
A
A
Since U PL
c = Pc U , we have Pc U PM = Pc U . Using the definitions of u and v,
we have
(φ1A , U PM Fall ) = (v, Re Fall ) + i(u, Im Fall ) = (u + , Fall ) − (u − , F all )
where u ± = 12 (u ± v). If we define p(t) = eiκt z(t), we have
e−iκt ṗ(t) = i −1 {(u + , F) + (u − , F) + [(u + , h) + (u − , h) + (u, R)a]θ̇} .
Summarizing, we have the decomposition
ψ = (Q + a R + h)e−i Et+iθ ,
U h = zφ1A + w ,
z = e−iκt p , µ = eiκt ,
and the equation

−1

ȧ = (c1 Q, Im(F + θ̇ h)) , c1 = (Q, R) ,
(2.30)
∂t w = −i Aw + PcA U i −1 (F + θ̇(a R + h)) ,

 −iκt
ṗ = (u + , F) + (u − , F) + [(u + , h) + (u − , h) + (u, R)a]θ̇ ,
ie
where
(2.31)
F = λQ(2|h|2 + h 2 ) + 2λQ Ra(2h + h̄) + 3λQ R 2 a 2
+ λ(a R + h)2 (a R + h̄) ,
(2.32)
θ̇ = − [a + (c1 R, Re F)] · [1 + a(c1 R, R) + (c1 R, Re h)]−1 .
This is a system of equations involving a, z, and η only. It is the system that we
shall solve in the rest of this paper. Note that θ does not appear explicitly in the
system (2.30). We see θ̇ only on the right side. Although θ will appear when we
170
T.-P. TSAI AND H.-T. YAU
integrate the main term of η, it appears only in the form eiθ ; hence we do not need
estimates for θ.
We also have the following decomposition of h according to (1.16):
η = U −1 w ∈ Hc (L) ,
(2.33)
h = ζ +η,
(2.34)
ζ = U −1 zφ1A = Re(z)u + Im(z)iv = zu + + z̄u − ∈ Eκ (L) .
The equation for η can be obtained from that of w or more directly from projecting
the original equation of h onto the continuum spectrum of L:
(2.35)
−1
∂t η = Lη + PL
c i (F + θ̇(a R + h)) .
2.4 Linear Estimates
Here we collect a few estimates on the operators U , L, and A. They will be
proven at the end of this paper.
We denote the wave operators for L (respectively, A and H ) by WL (respectively, W A and W H ). They are defined by
WL = lim e−t L e−it H∗ ,
(2.36)
t→∞
W H = lim e
t→∞
W A = lim eit A e−it H∗ ,
t→∞
it H −it H∗
e
,
where
(2.37)
H∗ = −1 − E .
Note that if W A exists, we have the intertwining property that f (A)Pc (A) =
W A f (H∗ )W A∗ for suitable functions f . We also have a similar property for L.
L EMMA 2.6 For k = 0, 1, 2, there is a positive constant C1 so that
(2.38)
C1−1 kφk H k ≤ ke−it A φk H k ≤ C1 kφk H k
for all φ ∈ X ∩ H k and all t ∈ R.
L EMMA 2.7 (Decay Estimates for e−it A ) For q ∈ [2, ∞] and q 0 = q/(q − 1),
(2.39)
−it A A
−3 12 − q1
e
kφk L q 0 .
Pc 5φ L q ≤ C|t|
For smooth local functions φ and sufficiently large r1 , we have
−r −it A
1
A
−r
−9/8
1 hxi 1 e
(2.40)
P
5hxi
φ
c
2 ≤ Chti
(A − 0i − 2κ)l
L
where l = 1, 2 and 0i means σ i with limσ →0+ outside of the bracket.
Estimate (2.39) for A = −1 +√V was proven in [8] using estimates from [7]
and [10]. Estimate (2.40) for A = H was proven by Soffer and Weinstein [16].
DYNAMICS OF NLS
L EMMA 2.8
(2.41)
φ0 φ12 , Im
171
1
A
2
P 5φ0 φ1 =
A − 0i − 2κ c
1
φ0 φ12 , Im
PcH1 φ0 φ12 + O(n 2 ) > 0 .
H1 − 0i − 2κ
This lemma is a perturbation result. Notice that κ = e01 + O(n 2 ); hence the
second term is greater than γ0 by assumption A1.
(i) The operators U0 and U0−1 are bounded operaL EMMA 2.9 (Operator U )
tors in W k, p ∩ X for k ≤ 2, 1 ≤ p < ∞, and in L r2 ∩ X for |r | ≤ r0 . (L r2 is the
weighted L 2 space defined in (1.7).) Hence U : M → X and U −1 : X → M are
bounded in W k, p and L r2 norms.
(ii) The commutator [U, i] is a local operator in the sense that
(2.42)
k[U, i]φk L 8/7 ∩L 4/3 ≤ C kφk L 4 .
L EMMA 2.10 (Wave Operators) The wave operators WL and W A defined by (2.36)
exist and satisfy W k, p estimates for k ≤ 2, 1 ≤ p < ∞ (similar estimates hold for
their adjoints):
WL PL k, p k, p ≤ C , W A P A 5 k, p k, p ≤ C .
c (W ,W )
c
(W ,W )
The statement on WL was proven in [4], following the proof of [20]. Hence we
need only to prove the statement on W A .
3 Main Oscillation Terms
We now identify the main oscillation terms in equation (2.30). Most quantities
treated in this paper, such as F, a, z, and η, are strongly oscillatory. Hence it is
necessary to identify their oscillatory parts before we can estimate. We shall use the
complex amplitude of the excited state, z, as the reference. Recall z(t) = e−iκt p(t).
We will show that p(t) ∼ t −1/2 and its oscillation (phase speed) is much smaller
than κ. The change of mass on the direction of the nonlinear ground state is given
by a. We will also show that a = O(z 2 ) and the order of the dispersive wave η(t)
is also of order O(z 2 ). Assuming these orders, the second-order term in F is given
explicitly by
(3.1)
F (2) = λQ(2|ζ |2 + ζ 2 ) = z 2 φ(20) + z z̄φ(11) + z̄ 2 φ(02)
where ζ = zu + + z̄u − and
φ(20) = λQ(u 2+ + 2u + u − ) = λQφ12 + O(n 3 ) ,
(3.2)
φ(11) = 2λQ(u 2+ + u 2− + u + u − ) = 2λQφ12 + O(n 3 ) ,
φ(02) = λQ(u 2− + 2u + u − ) = O(n 3 ) .
172
T.-P. TSAI AND H.-T. YAU
We shall write F (2) = z α φα , where α is a double indices (i j) with i + j = 2 and
i, j ≥ 0. The repeated indices mean summation. We shall use β later on to denote
double indices summing to 3 and γ summing to 4.
3.1 Leading Oscillatory Terms in a
We first identify the main oscillation terms of a(t). Recall from (2.30) that
ȧ = (c1 Q, Im F+θ̇ h) and c1 = (Q, R)−1 . We shall impose the boundary condition
of a at t = T . This is in fact a condition imposed on the choice of E T . Hence we
use the following equivalent integral equation:
Z t
(3.3)
a(t) = a(T ) +
(c1 Q, Im F + θ̇ h)(s)ds .
T
(2)
The main term of Im F + θ̇ h is ImR F = Im λQζ 2 , and thus the leading oscillation
t
term of a(t) is from the integral T A(2) ds with
A(2) = (c1 Q, λQ Im ζ 2 ) .
Since ζ = zu + + z̄u − , we have Im ζ = Im z(u + − u − ) and Im ζ 2 = (Im z 2 )(u 2+ −
u 2− ); therefore, we have
A(2) = C1 Im z 2 ,
C1 = (c1 Q, λQ(u 2+ − u 2− )) .
Write
z 2 (s) = e−2iκs p 2 =
(3.4)
1 d −2iκs 2
)p .
(e
−2iκ ds
We can integrate A(2) by parts to get
Z t
Z t
Z t
(2)
2
2 t
−2iκs
(3.5)
A ds = Im C1
z ds = 2a20 Re [z ]T −
e
2 p ṗ ds ,
T
T
T
where
C1
1
c1 Q, λQ(u 2+ − u 2− ) = O(n 2 )
=
4κ
4κ
and the last integral is a higher-order term. Let us denote
(3.6)
a20 =
a (2) (t) = a20 (z 2 + z̄ 2 )(t) ,
(3.7)
which is the main oscillatory term in a. We denote the rest of a by b, i.e.,
a(t) = a (2) (t) + b(t) ,
(3.8)
and we have
b(t) = a(T ) − a (2) (T )
Z t
(3.9)
+
(c1 Q, Im[F − F (2) + θ̇ h]) − 4a20 Re e−2iκs p ṗ (s)ds .
T
It is easy to see that a, b, ȧ = O(z 2 ), but ḃ = O(z 3 ). In other words, a (2) is the
part of a with strong oscillation and b is the part of a that has slower oscillation.
The use of b is convenient when we work on normal forms of p and a later. (We
DYNAMICS OF NLS
173
will integrate by parts terms involving b, and in some sense replace terms involving
b by terms involving ḃ, which is smaller. If we work with a instead, then we replace
a by ȧ, which is of the same order and we need one more step.) In fact, in principle
one should treat the normal forms of p and a together, since they correspond to the
excited states and the ground state. See [17] for more elaboration on this point.
It should be noted that, although b(t) is not oscillatory, it is in fact larger than
a (2) (t), and hence is the main term of a(t). Another point to make here is that the
introduction of b(t) is for computational convenience. The true variable we work
with is still a(t). In particular, the induction assumption we make later in Section 5
is on a, not b.
Notice that when integrating θ̇, we take Re F instead of Im F. Hence the term
C|z|2 survives and cannot be integrated; thus θ(t) = O(log t), although a(t) =
O(t −1 ).
3.2 Leading Oscillatory Terms in η
We now identify the main oscillation term in η. From the basic equation (2.30),
we rewrite the w equation as
∂t w = −i Aw − i θ̇ w + PcA U i −1 [F + θ̇(a R + ζ )] − PcA [U, i]θ̇η .
where we have used the commutator [U, i] to interchange U and i so as to produce
the term i θ̇ w. This term is a global linear term in η and cannot be treated as an
error term (however, [U, i]θ̇η is an error term). We can eliminate it by rewriting
the last equation in terms of η̃ = eiθ w = eiθ U η, i.e.,
∂t η̃ = −i Aη̃ + eiθ PcA U i −1 [F + θ̇(a R + ζ )] − eiθ PcA [U, i]θ̇η ,
or in integral form,
(3.10)
η̃(t) = e−i At η̃0 +
Z
t
e−i A(t−s) PcA Fη (s)ds ,
0
where
(3.11)
Fη = eiθ U i −1 [F + θ̇(a R + ζ )] − eiθ [U, i]θ̇η .
Since U and U −1 are bounded in Sobolev spaces by Lemma 2.9 and
(3.12)
η(t) = U −1 e−iθ (t) η̃(t) ,
for the purpose of estimation we can treat η and η̃ as the same.
Let Fη,2 be the second-order term of Fη , and let Fη,3 denote the rest, i.e.,
Fη = Fη,2 + Fη,3 ,
(3.13)
Fη,2 = eiθ U i −1 F (2) = eiθ U i −1 z α φα ,
Fη,3 = eiθ U i −1 (F − F (2) ) + θ̇(a R + ζ ) − eiθ [U, i]θ̇η .
174
T.-P. TSAI AND H.-T. YAU
Since U = U+ + CU− with U+ and U− commuting with i, see (2.24), we can write
Fη,2 as
Fη,2 = eiθ (U+ + CU− )i −1 z α φα
(3.14)
8(20)
(3.15)
= i −1 eiθ (U+ − CU− )z α φα = i −1 eiθ z α 8α ,
= U+ φ(20) − U− φ(02) , 8(02) = U+ φ(02) − U− φ(20) ,
8(11) = (U+ − U− )φ(11) .
Replacing Fη in (3.10) by Fη,2 and integrating by parts, we have
Z t
Z t
e−i A(t−s) PcA Fη,2 ds =
e−it A eis(A−0i+[α]κ) (eiθ p α )(s) i −1 PcA 8α ds
0
0
ηα + (∗) ,
= η̃(2) (t) − e−it A (eiθ z α )(0)e
(3.16)
where
(3.17)
η̃(2) (t) = eiθ(t) z α (t)e
ηα ,
e
ηα =
−1
P A 8α ,
A − 0i + [α]κ c
and (∗) denotes the remainder term from integration by parts,
Z t
d
e−i(t−s)A eisκ[α] (eiθ p α )e
ηα ds .
(∗) = −
ds
0
In the integration we have inserted the regularizing factor eis(−0i) with the sign of
0i chosen so that e−it Ae
ηα decays as t → +∞. See Lemma 2.7.
We shall see that η̃(2) (t) is the main oscillation term in η̃(t). We denote the rest
by η̃(3) (t):
η̃ = η̃(2) + η̃(3) .
(3.18)
Notice that this is not a decomposition in L 2 : Both η̃(2) and η̃(2) are not in L 2 .
Nevertheless, this decomposition is useful for studying the local behavior of η̃.
Also note that, although η̃α 6∈ L 2 but is still “orthogonal” to the eigenvector of A,
(φ1A , η̃α ) = 0.
From (3.10) we obtain the equation for η̃(3) :
(3.19)
η̃(3) (t) = e−i At η̃0 − e−i At (eiθ z α )(0)e
ηα
Z t
−i A(t−s)
isκ[α] d
iθ α
−
e
ηα ds
(e p )e
e
ds
0
Z t
e−i A(t−s) PcA Fη,3 − eiθ U i −1 η2 η̄ ds
+
0
Z t
e−i A(t−s) PcA eiθ U i −1 η2 η̄ ds
+
0
≡
η̃1(3)
+ η̃2(3) + η̃3(3) + η̃4(3) + η̃5(3) .
DYNAMICS OF NLS
175
We treat η̃5(3) separately because η2 η̄ is a nonlocal term. Recall that η =
U −1 e−iθ η̃. We have the similar decomposition for η,
(3.20)
η(t) = η(2) (t) + η(3) (t) ,
η(3) = U −1 e−iθ η̃(3) ,
η(2) = U −1 e−iθ η̃(2) = U −1 z αe
ηα ,
−1 −iθ (3)
η(3)
e η̃ j .
j =U
If we view b and η(2) as order z 2 , and η(3) as order z 3 , we can now decompose
F into
(3.21)
e(3) + F (4)
F = F (2) + F (3) + F
where
F (2) = λQ(2|ζ |2 + ζ 2 ) ,
F (3) = 2λQ (ζ + ζ̄ )η(2) + ζ η̄(2) + λ|ζ |2 ζ + 2λQ Ra (2) (2ζ + ζ̄ ) ,
e(3) = 2λQ Rb(2ζ + ζ̄ ) = 2λQ R(a − a (2) )(2ζ + ζ̄ ) ,
(3.22) F
F (4) = 2λQ (ζ + ζ̄ )η(3) + ζ η̄(3) + λQ 2|η|2 + η2 + 2λa Q R(2η + η̄)
+ 3λa 2 Q R 2 + λ |k|2 k − |ζ |2 ζ .
e(3) of terms of order
In view of (4.5), F (2) consists of terms of order z 2 ; F (3) and F
3
(4)
(3)
e
z ; and F of higher-order terms. We separate the F term since it depends on b
and requires different methods to estimate.
From (2.32), we have
θ̇ = −[a + (c1 R, Re F)] · [1 + a(c1 R, R) + (c1 R, Re h)]−1
= −[a + (c1 R, Re F)] · [1 − (c1 R, Re h)] + O(z 4 + a 2 + η2 )
(3.23)
where
= −[a + (c1 R, Re F (2) )] + Fθ,3 + Fθ,4
e(3) + a + (c1 R, Re F (2) ) (c1 R, Re ζ )
Fθ,3 = − c1 R, Re F (3) + F
and Fθ,4 = O(z 4 + a 2 + η2 ).
4 Outline of Proofs and Basic Estimates
We are now ready to outline the proofs for Theorems 1.1 through 1.4.
4.1 Initial Data and Basic Quantities
Under the assumptions of either Theorem 1.1 or Theorem 1.3, we can rewrite
the initial data ψ0 using Lemma 2.3 as
ψ0 = [Q E0 + h E0 ,0 ]ei2 E0 ,0 ,
h E0 ,0 ∈ M E0 ,
for some E 0 , h E0 ,0 , and 2 E0 ,0 , with
n = kψ0 kY ,
kQ E0 k ≥
9
n
10
,
kh E0 ,0 kY ≤ Cε0 n .
176
T.-P. TSAI AND H.-T. YAU
We can assume that 2 E0 ,0 = 0 without loss of generality. We now write h E0 ,0 =
ζ E0 ,0 + η E0 ,0 according to the spectral decomposition (1.16) in M E0 . Recall the
definition of ε in (1.21). We have
(4.1)
kζ E0 ,0 k ≤ Cεn ,
kη E0 ,0 kY ≤ Cε2 n 2 ,
ε ≤ ε0 .
For the rest of this paper, we shall take as either given by (1.21) or (1.24). The
choice will not be important, and, whenever a specific choice is needed, we shall
remark upon it.
For all t ≥ 0, we have kψ(t)k L 2 = n. If we write ψ(t) = xφ0 + yφ1 + ξ ,
ξ ∈ Hc (−1 + V ), then we have |x|2 + |y|2 + kξ k2L 2 = n 2 . Since the spectral
projections of −1 + V and L differ by an order of O(λQ 2 ) = O(n 2 ), we have
kQ E k = |x| + O(n 3 ), kζ k L 2 = |y| + O(n 3 ), and kηk L 2 = kξ k L 2 + O(n 3 ). In
particular,
(4.2)
kQ E k ≤ 98 n ,
kζ (t)k L 2 ≤ 98 n ,
kη(t)k L 2 ≤ 98 n .
We now give a list of the expected sizes of frequently used quantities. Let
1
(4.3)
0 ≡ Im 2820 ,
PcA 820 ≥ λ2 n 2 γ0 ,
A − 0i − 2κ
where 820 = O(n) is defined as in (3.15), and γ0 is the constant from assumption A1. The last statement 0 ≥ λ2 n 2 γ0 will be proven in Lemma 6.1. Recall
that
{t} ≡ ε−2 n −2 + 0t, {t}−1/2 ∼ min εn, n −1 t −1/2 .
(4.4)
{t}−1/2 will be the typical size of |z(t)|. The following is a table of order in t for
functions:
|z(t)| = O(t −1/2 ) , |a(t)| = O(t −1 ) ,
(4.5)
kη(t)kloc = O(t −1 ) ,
|θ(t)| = O(log t) .
The following is a table of order in n for constants:
(4.6)
Q = O(n) ,
c1 = O(1) ,
R = O(n −1 ) ,
u + = O(1) ,
u − = O(n 2 ) .
The first three estimates in the last equation are due to Lemma 2.1. The last one
is because the differences between L + , L − , and H1 = −1 + V − e0 are of order
O(λn 2 ), and hence so are the differences between φ1 , u, and v.
4.2 Outlines of the Proofs for Theorems 1.1 and 1.3
We now estimate solutions to the equations (2.30) with the decompositions into
main oscillatory and higher-order terms in Section 3. We first fix a time T and let
E = E(T ) be the best approximation at the time T so that (1.19) holds at time
T , i.e.,
(4.7)
ψ(T, x) = [Q E (x) + h E (T, x)] ei2(T ) .
DYNAMICS OF NLS
177
We can now decompose the wave function for all time with respect to this ground
state as in (1.18), namely,
ψ(t, x) = Q E(T ) (x) + a E(T ) (t)R E(T ) (x) + h E(T ) (t, x) ei2(t) ,
where h E(T ) (t, x) = ζ E(T ) (t) + η E(T ) (t) ∈ M E(T ) . To simplify the notation, we
write a E(T ) = aT , etc. By assumption, we have aT (T ) = 0. We now wish to
estimate aT , ζT , and ηT for all time t ≤ T . The following propositions are stated
with respect to a decomposition of a fixed nonlinear ground state profile E.
We first choose a suitable norm. We need to control the excited-state component
z and a local norm of η. We also need a global norm of η to control the nonlocal term η3 . Recall that we can decompose η into a sum of η(2) + η(3) with η(3)
decomposed further into a sum of five terms; see (3.20). Since η(2) , η1(3) , and η2(3)
are explicit, we need only to control
(3)
η3−5
(4.8)
≡
5
X
(3)
(3)
(3)
η(3)
j = η3 + η4 + η5 ,
j=3
and we define
M(T ) := sup
0≤t≤T
(4.9)
n
{t}1/2 |z(t)| + {t}3/4−σ kη(t)k L 4 + n σ {t} kη(t)k L 2
(3)
+ εσ (εn)−3/4 {t}9/8 η3−5
(t) L 2
o
loc
loc
where 0 = O(n ) is defined in (4.3) and B22 =
+ O(n 3 ) will be defined
explicitly later in (7.22). For the time being, we need only know that B is a constant
with the size given above; hence D = c1 + O(n).
If we assume that a(t) is bounded by t −1 up to some time T , we have the
following control of M(t).
1
c0
2 1
2
P ROPOSITION 4.1 Suppose z(t) and η(t) are solutions to the equations (2.30) for
0 ≤ t ≤ T . (T can be finite or infinite.) Assume
(4.10)
|a(t)| ≤ D {t}−1 ,
0≤t ≤T.
Then we have
M(t) ≤ 2 for all t ≤ T .
Moreover, if we further assume |z 0 | = εn > 0 and kη0 kY ≤ ε2 n 2 , then |z(t)| ≥
c {t}−1/2 .
Recall b(t) := a(t) − a (2) (t). Let ST (t) = b(t) − b(T ) so that
t
(4.11)
a(t) = a(T ) + a20 (z 2 + z̄ 2 ) T + ST (t) .
Assuming that a(t) is bounded by t −1 , we can control ST (t) by the following
proposition:
178
T.-P. TSAI AND H.-T. YAU
P ROPOSITION 4.2 Suppose that M(t) ≤ 2 and |a(t)| ≤ D {t}−1 for 0 ≤ t ≤ T .
Recall D = 2B22 / 0 = c1 + O(n). Then we have
D −1
D
{t} , |a(t)| ≤ |a(T )| + {t}−1 .
(4.12)
|ST (t)| ≤
2
2
P ROOF OF T HEOREMS 1.1 AND 1.3: Assuming the previous two propositions,
we now prove that
(4.13)
sup |aT (t)| {t} < D
t≤T
for all time, where aT is the component with respect to the best approximation
at the time T defined in (1.19). From (4.1), the relation (4.13) holds for T = 0.
Suppose it holds for time T = T1 . From the continuity of the Schrödinger equation,
there exists an δ > 0 (depending on T1 ) such that |E(T )− E(T1 )| ≤ exp[−T1 ](D −
supt≤T1 |aT1 (t)| {t}) for all |T − T1 | < δ. Thus the bound (4.13) holds for all
|T − T1 | < δ. This proves that the set of such T is open. Now suppose that T1
is the first time that (4.13) is violated. Clearly, we have supt≤T1 |aT1 (t)| {t} ≤ D.
From Propositions 4.1 and 4.2, formula (4.13) holds for T1 .
Once we have proven (4.13), the conclusions of Propositions 4.1 and 4.2 hold,
and they imply both Theorem 1.1 and Theorem 1.3.
We remark that the main reason that the previous continuity argument works
is due to the explicit identification of the leading terms in a, η, and z. Since these
leading terms are essentially computed without using assumptions on the decay of
a, the bound on a, i.e., (4.13), is used only to control error terms. We shall prove
Theorem 1.4 in Section 7, where we obtain more detailed information about the
wave function.
The proof of Proposition 4.1 will be given in the next two sections and is followed by a proof of Proposition 4.2. Once again, our strategy of proof is to show
that M(0) ≤ 32 and that M(t) ≤ 32 if M(t) ≤ 2. By continuity of M(t), this
would imply that M(t) ≤ 32 for all t ≤ T . So for the rest of this and the next
sections, we shall freely use that M(t) ≤ 2 and |a(t)| ≤ D {t}−1 . It will follow
from Lemmas 5.2 and 5.3 that we can bound
{t}3/4−σ kη(t)k L 4 + εσ (εn)−3/4 {t}9/8 η(3) (t) 2
3−5
L loc
by 14 . It will follow from Lemma 6.2 that
{t}1/2 |z(t)| ≤ (1 + 2σ )
for some small σ . We have thus M(T ) ≤ 32 and conclude Proposition 4.1.
We note that, under the assumptions (4.10) and M(T ) ≤ 2, we have, from the
equations (2.31) for F, (2.32) for θ̇, (2.30) for ṗ, and (3.9) for b,
kFk L 2 ≤ Cn {t}−1 , F − F (2) L 2 ≤ C {t}−3/2 ,
loc
loc
(4.14)
|θ̇| ≤ C {t}−1 , | ṗ| ≤ Cn {t}−1 , |ḃ| ≤ Cn {t}−3/2 .
DYNAMICS OF NLS
179
It follows from (4.14) that
|θ(t)| ≤ C log {t} .
(4.15)
Note that, although θ̇ and ȧ are of the same order, θ̇ is not oscillatory and we cannot
expect an estimate better than (4.15). This estimate will not be used in the rigorous
proof, but it provides an idea about its size.
5 Estimates of the Dispersive Wave
For the dispersive part η, our main interest is its local decay estimate. Due to
the presence of the nonlocal term η3 in F, we will also need a global estimate on
η. Our goal is to prove, for σ = 1/100,
(3)
kη(t)k L 4 ≤ 18 {t}−3/4+σ , η3−5
(t) L 2 ≤ Cε−σ (εn)3/4 {t}−9/8 .
loc
We will need the following calculus lemma:
L EMMA 5.1 Let 0 < d < 1 < m, and {t} ≡ ε−2 + 20t.
Z t
(5.1)
|t − s|−d {s}−m ds ≤ Cε2m−2 n 2m+2d−4 {t}−d ,
0
Z t
(5.2)
|t − s|−d {s}−1 ds ≤ Cn 2d−2 {t}−d log(1 + ε2 n 4 t) .
0
If, instead, d ≥ m > 1,
Z
t
(5.3)
ht − si−d {s}−m ds ≤ C {t}−m .
0
P ROOF : Denote the first integral by (I). If t ≤ ε −2 n −4 , then {t} ∼ ε−2 n −2 and
Z t
(I) ∼
|t − s|−d (εn)2m ds . (εn)2m (ε−2 n −4 )1−d
0
= ε2m−2 n 2m+2d−4 (εn)2d ∼ ε2m−2 n 2m+2d−4 {t}−d .
If t ≥ ε−2 n −4 , then {t} ∼ 20t and
Z t/2
Z
−d
−m
(I) ≤
Ct {s} ds +
0
≤
Ct
t
C|t − s|−d C {t}−m ds
t/2
−d
(ε n )
0
2 4 m−1
+ Ct 1−d {t}−m ≤ Cε2m−2 n 2m+2d−4 {t}−d .
Estimate (5.2) is an obvious modification of (5.1). For the last estimate (5.3),
denote the second integral by (II). If t ≤ ε−2 n −4 , then {t} ∼ ε−2 n −2 and
Z t
ht − si−d (εn)2m ds ≤ (εn)2m · C ∼ C {t}−m .
(II) ≤ C
0
180
T.-P. TSAI AND H.-T. YAU
If t ≥ ε−2 n −4 , then {t} ∼ 0hti and
Z t
ht − si−d 0 −m hsi−m ds ≤ C0 −m hti−m ∼ C {t}−m .
(II) ∼
0
We conclude the lemma.
5.1 Estimates in L 4
L EMMA 5.2 Suppose that η̃ is given by equation (3.10) and recall η = U −1 e−iθ η̃.
Assuming the estimate (4.10) on a(t) and M(T ) ≤ 2, we have
kη(t)k L 4 ≤
1
8
{t}−3/4+σ ,
where σ = 1/100 if ε is sufficiently small.
P ROOF : From the defining equation (3.10) of η̃, we have
Z t
−3/4
ke
η0 kY hti
+
C|t − s|−3/4 kFη (s)k4/3 ds ,
η(t)k L 4 ≤ C ke
0
where Fη = eiθ U i −1 [F + θ̇ (a R + ζ )] − eiθ [U, i]θ̇η. By Lemma 2.9,
iθ
e [U, i]θ̇η(s) 4/3 ≤ C|θ̇ | kη(s)k4 .
L
Therefore we have
kFη (s)k4/3 ≤ C|θ̇| |a| + kζ (s)k4/3 + kη(s)k4 + kF(s)k L 4/3 .
From the Hölder inequality and the definition of kF(s)k L 4/3 , we can bound
kF(s)k L 4/3 by
kF(s)k L 4/3 ≤ C khk2L 4 + |a| khk L 4 + |a|2 + kh 3 k L 4/3 + |a|3 .
Since kh 3 k L 4/3 = khk3L 4 and kh(s)k L 4 ≤ kζ (s)k L 4 + kη(s)k L 4 , we have from the
assumption MT (t) ≤ 2 that
kh(s)k L 4 ≤ C{s}−1/2 + C{s}−3/4 log{s} ≤ C{s}−1/2 .
Therefore we have kF(s)k L 4/3 ≤ Cn{s}−1 . Similarly, we have |θ̇(s)| ≤ C{s}−1 .
1
{t}−3/4+σ , we conSince C kη̃0 kY hti−3/4+σ ≤ C kη̃0 kY ε−3/2+2σ {t}−3/4+σ ≤ 16
clude
Z t
−3/4+σ
1
kη(t)k L 4 ≤ 16 {t}
+
|t − s|−3/4 Cn{s}−1 ds
0
≤
1
16
{t}−3/4+σ + Cnn −1/2 {t}−3/4 log[(εn)2 {t}] ≤
Here we have used Lemma 5.1 for the integration.
1
8
{t}−3/4+σ .
DYNAMICS OF NLS
181
5.2 Local Decay
Recall that η(3) = U −1 e−iθ η̃(3) and η̃(3) satisfies the equation (3.19). We want to
show that η̃(3) is smaller than η̃(2) locally. Recall that the local L 2 norm is defined
in (1.7) with r0 the exponent in Lemma 2.7.
L EMMA 5.3 Assuming the estimate (4.10) on a(t) and M(T ) ≤ 2, we have
(3) η 2 ≤ Cε2 n 2 hti−9/8 , η(3) 2 ≤ C(εn)3/4 {t}−9/8 .
(5.4)
1−2 L
3−5 L
loc
loc
−1
Hence we have kηk L 2 ≤ C {t} , and for a local function φ we have
loc
(φ, |η|2 + |η|3 ) ≤ C hxi2r0 φ ∞ kηk2 2 + kηk 2 kηk2 4
L loc
L
L
L loc
(5.5)
≤ C hxi2r0 φ L ∞ {t}−2 .
P ROOF : Since U is bounded, it is sufficient to prove the corresponding esti(3)
mates for each η̃(3)
j , j = 1, 2, . . . , 5, defined in the decomposition (3.19) for η̃ .
We first estimate η̃1(3) ,
(3) η̃ 2 ≤ Ckη0 kY hti−9/8 ≤ Cε2 n 2 hti−9/8 .
1
L
loc
η̃2(3)
η̃3(3) ,
For
and
the two terms involving ηα (ηα 6∈ L 2 ) from the definition and the
linear estimates in Lemma 2.7, we have
(3) η̃ 2 ≤ (εn)2 nhti−9/8 ,
2
L loc
Z t
(3) η̃ 2 ≤ C
ht − si−9/8 n 2 {s}−3/2 ds ≤ n 2 (εn)3/4 {t}−9/8 .
3
L
loc
0
Here we have used e
ηα = O(n), | ṗ| ≤ Cn {t}−1 , and |θ̇| ≤ C {t}−1 in (4.14) and
(5.3). Note that, when t is of order 1, η̃1(3) and η̃2(3) are of order η0 and ε2 n 3 , which
are larger than C(εn)3/4 {1}−9/8 = O((εn)3 ).
Rt
We treat η̃4(3) and η̃5(3) together. For η̃4(3) = 0 e−i A(t−s) PcA {Fη,3 −eiθ U i −1 η2 η̄}ds,
we can rewrite the integrand using the definition of Fη,3 from (3.13),
Fη,3 − eiθ U i −1 η2 η̄ = eiθ U i −1 (F − F (2) − η2 η̄) + θ̇(a R + ζ ) − eiθ [U, i]θ̇η ,
which consists of only local terms. Since MT (t) ≤ 2, we have
Fη,3 − eiθ U i −1 η2 η̄ 4/3 8/7 ≤ C{s}−3/2 .
L
∩L
Here we have used Lemma 2.9 to estimate eiθ [U, i]θ̇η. Note that, in terms like
Qζ η, η is estimated by kηk L 2 ≤ Cn −σ {s}−1 .
loc
From the Hölder inequality, the L 2 bound (4.2), and the global estimate of η in
Lemma 5.2, we have
5/2
1/2
5/2
≤ C{s}−7/4 ,
kη3 (s)k L 8/7 ≤ kηk2 kηk4 ≤ Cn 1/2 {s}−3/4+σ
3
kη3 (s)k L 4/3 ≤ kηk34 ≤ C {s}−3/4+σ ≤ C{s}−7/4 .
182
T.-P. TSAI AND H.-T. YAU
Rt
Thus we have η̃4(3) + η̃5(3) = 0 e−i A(t−s) PcA Fη,3 (s)ds, with
Fη,3 (s) 4/3 8/7 ≤ C{s}−3/2 .
L ∩L
We can bound the L 2loc norm by either the L 4 or L 8 norm. If t > 1, we bound
the L 2loc norm by L 8 in 0 ≤ s ≤ t − 1, and by L 4 in t − 1 ≤ s ≤ t. By Lemma 2.7,
Z t−1
(3)
1
η̃ + η̃(3) 2 ≤ C
kFη,3 (s)k L 8/7 ds
4
5
L loc
|t − s|9/8
0
Z t
1
+C
kFη,3 (s)k L 4/3 ds
|t
−
s|3/4
t−1
Z t
C ht − si−9/8 {s}−3/2 ds + C {t}−3/2
≤C
0
≤ C(εn)3/4 {t}−9/8 + C(εn)3/4 {t}−9/8 .
In the last line we have used (5.3). If t ≤ 1, we can use the second integral to
estimate; hence we get the desired estimate for kη̃4(3) + η̃5(3) k L 2 .
loc
6 Excited-State Equation and Normal Form
6.1 Excited-State Equation
Let us return to the equation for ṗ from (2.30),
(6.1)
ṗ = −ieiκt (u + , F) + (u − , F) + (u + , h) + (u − , h) + (u, R)a θ̇ .
In view of the decompositions (3.22) of F and (3.23) of θ̇, we can expand the
right-hand side of equation (6.1) into terms in order of z:
(6.2)
ṗ = eiκt cα z α + dβ z β + d1 bz + d2 bz̄ + P (4) .
Here α and β are summing over all |α| = 2 and |β| = 3. The coefficients are
computed and their properties summarized in the following lemma:
L EMMA 6.1 We can rewrite equation (6.1) of ṗ into the form (6.2). The coefficients
d1 and d2 and all cα are purely imaginary. P (4) denotes higher-order terms and
will be defined in (6.6). Moreover, Re d21 = −0, with
1
A
(6.3)
0 = 2820 , Im
P 820 ≥ λ2 n 2 γ0 > 0 ,
A − 0i − 2κ c
where γ0 is the constant in assumption A1, and 820 = 5λQφ12 + O(n 3 ) is defined
in (3.15). We also have cα = O(n) and dβ , d1 , d2 = O(1).
P ROOF : There are two parts in equation (6.1): The part with F and the part
with θ̇. We first consider the second part. Recall from (3.23),
θ̇ = −a − (c1 R, Re F (2) ) + O(z 3 ) ,
DYNAMICS OF NLS
183
where O(z 3 ) denotes terms of the order z 3 +az+zη plus higher-order terms. Hence
− ieiκt {(u + , h) + (u − , h̄) + (u, R)a}θ̇
= −ieiκt (u + , ζ ) + (u − , ζ̄ ) + O(z 2 ) · − a − (c1 R, Re F (2) ) + O(z 3 )
= −ieiκt (u + , ζ ) + (u − , ζ̄ )
· − a20 (z 2 + z̄ 2 ) − b − (c1 R, Re F (2) ) + O(z 4 ) .
The leading terms are of the form z β , bz, and bz̄. These terms are of order O(z 3 ).
Since the coefficients in each bracket are real, the coefficients of the leading terms,
that is, their contributions to dβ , d1 , and d2 , are all purely imaginary due to the i
factor in front.
Now we look at the first part of the equation (6.1). The contribution to cα z α is
from F (2) = z α φα ,
−i {(u + , z α φα ) + (u − , z̄ α φα )} .
Clearly all coefficients of z α are purely imaginary. Since there is no contribution
from the second part to cα z α , we know that the cα are purely imaginary. We also
have cα = O(φα ) = O(n).
e(3) :
The contribution from the first part to d1 bz + d2 bz̄ is from F
−i (u + , 2λQ Rb(2ζ + ζ̄ ) + (u − , 2λQ Rb(ζ + 2ζ̄ )
with ζ = zu + + z̄u − . Hence all coefficients of bz and bz̄ are purely imaginary.
Together with the analysis of the second part of (6.1), we know d1 and d2 are purely
imaginary.
The contribution from the first part to dβ z β is from F (3) :
−i (u + , F (3) ) + (u − , F̄ (3) ) .
Among all dβ , we are only interested in the real part of d21 , which does not come
from the second part as we already showed. A coefficient in F (3) has to have an
imaginary part in order to have a contribution to Re d21 . The only such source
is z 2 η̃20 in η(2) , which lies in the first group of terms in F (3) . Let us call these
terms F1(3) :
F1(3) = 2λQ (ζ + ζ̄ )η(2) + ζ η̄(2) .
Recall that η(2) = U −1 z α η̃α . Denote η0 = z α η̃α and recall (2.24) that U −1 =
(U+ )∗ − (U− )∗ C. Hence η(2) = [(U+ )∗ − (U− )∗ C]η0 = (U+ )∗ η0 − (U− )∗ η̄0 , and
u + , F1(3) + u − , F̄1(3)
= 2λQ, (u + ζ + u ζ̄ )η(2) + 2λQ, (uζ + u − ζ̄ )η̄(2)
= 2λQ, (u + ζ + u ζ̄ )[(U+ )∗ η0 − (U− )∗ η̄0 ]
+ 2λQ, (uζ + u − ζ̄ )[(U+ )∗ η̄0 − (U− )∗ η0 ]
184
T.-P. TSAI AND H.-T. YAU
Z
=
U+ 2λQ(u + ζ + u ζ̄ ) − U− 2λQ(uζ + u − ζ̄ ) η0 d x
Z
+
−U− 2λQ(u + ζ + u ζ̄ ) + U+ 2λQ(uζ + u − ζ̄ ) η̄0 d x .
We want to collect terms of the form C z 2 z̄ with Re C 6= 0. However, the only term
from η̄0 with a resonance coefficient is z 2 η̃20 , which is of the form z̄ 2 , with two
bars. Hence the last integral does not contain z 2 z̄ and is irrelevant. From the first
integral, we want to choose z 2 from η0 , i.e., z 2 η̃20 , and choose z̄ from ζ or ζ̄ . The
terms with z̄ are
U+ 2λQ(u + z̄u − + u z̄u + ) − U− 2λQ(u z̄u − + u − z̄u + ) = 2(U+ φ20 − U− φ02 )z̄
= 2820 z̄ .
Here we have used (3.2) and (3.15); hence
(6.4) Re d21 = Re(−i) (2820 , η20 ) = Im 2820 ,
−1
A
P 820 = −0 .
A − 0i − 2κ c
To show 0 ≥ λ2 n 2 γ0 , it suffices to show that
(6.5)
820 = 5λQφ12 + O(n 3 ) .
Note that 820 = U+ φ20 − U− φ02 = U+ λQφ12 + O(n 3 ) by (3.15) and (3.2). Now
U+ = 12 (5A1/2 H −1/2 P1 +5A−1/2 H 1/2 5) is defined in (2.23). Since A1/2 H −1/2 =
1+ O(n 2 ) and A−1/2 H 1/2 = 1+ O(n 2 ) by (9.7) and (9.8), we have U+ = 12 (5P1 +
5) + O(n 2 ). By (2.18), P1 = 5 + O(n 2 ). Therefore we have (6.5). It will follow
from the induction assumption that Q = mφ0 + O(n 3 ) with m ≥ 45 n. Then we
have, by Lemma 2.8,
1
2 2
2
A
2
0 = 2λ m Im 5φ0 φ1 ,
P 5φ0 φ1 + O(n 4 )
A − 0i − 2κ c
2
≥ 2λ2 45 n γ0 + O(n 4 ) .
Collecting terms of order O(z 4 + a 2 + η2 ), we have
(4)
P (4) = −i (u + , F (4) ) + (u − , F ) + [(u + , η) + (u − , η) + (u, R)a]θ̇
− i (u + , ζ ) + (u − , ζ ) (Fθ,3 + Fθ,4 ) .
(6.6)
6.2 Normal Form
L EMMA 6.2 We can rewrite equation (6.2) of ṗ into a normal form:
(6.7)
q̇ = δ21 |q|2 q + d1 bq + g ,
where q is a perturbation of p given in the proof. The coefficient δ21 satisfies
(6.8)
Re δ21 = Re d21 = −0 .
DYNAMICS OF NLS
185
If we assume estimate (4.10) on a(t) and M(T ) ≤ 2, then the error term g(t), to
be given by (6.22), satisfies the bound
(6.9)
|g(t)| ≤ Cε 3/4−σ n 7/4 {t}−13/8 .
Furthermore, there is a small positive constant σ such that |q(t)| and |z(t)| are
bounded by
(6.10)
|q(t)| ≤ (1 + σ ) {t}−1/2 ,
|z(t)| ≤ (1 + 2σ ) {t}−1/2 .
If we have |z(0)| = εn + o(εn), we also have lower bound
|q(t)| ≥ (1 − σ ) {t}−1/2 ,
|z(t)| ≥ (1 − 2σ ) {t}−1/2 .
P ROOF : From (6.2) we have
ṗ = µ cα z α + dβ z β + d1 bz + d2 bz̄ + P (4) ,
µ = eiκt ,
and we want to obtain the normal form (6.7). We will repeatedly use the following
formula:
µm p α
d µm p α
m α
−
(6.11)
µ p =
f α (z)
dt iκm
iκm
where, if α = (α0 α1 ), |α| = α0 + α1 = 2, 3, 4, . . . ,
(6.12)
f α (z) = (α0 + α1 C)( p −1 ṗ)
= (α0 + α1 C)z −1 cα z α + dβ z β + d1 bz + d2 bz̄ + P (4) ,
and C denotes the conjugation operator. The formula is equivalent to integration
by parts.
We first remove cα z α . Let
cα
p1 = p −
µz α .
iκ(1 + [α])
Since [α] is even, 1 + [α] 6= 0. By (6.11)
cα
ṗ1 = ṗ − cα µz α −
µz α f α (z) .
iκ(1 + [α])
Decomposing f α (z) into two parts, we can write
cα
µz α (α0 + α1 C)z −1 cα̃ z α̃ ,
dβ+ z β = −
iκ(1 + [α])
cα
g1 = −
µz α (α0 + α1 C)z −1 dβ z β + d1 bz + d2 bz̄ + P (4) ,
iκ(1 + [α])
and we get
ṗ1 = δβ µz β + d1 µbz + d2 µbz̄ + µP (4) + g1
with δβ = dβ + dβ+ . Since cα are purely imaginary, hence so are dβ+ , and we have
the relation
(6.13)
Re δβ = Re dβ .
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T.-P. TSAI AND H.-T. YAU
Next we remove d2 µbz̄. Let
p2 = p1 −
µd2 bz̄
.
2iκ
We have
ṗ2 = ṗ1 − µd2 bz̄ −
µ2 d2
˙ = µδβ z β + d1 µbz + (µP (4) + g1 + g2 ) ,
(ḃ p̄ + b p̄)
2iκ
where
µ2 d2
˙ .
(ḃ p̄ + b p̄)
2iκ
Now we deal with δβ z β terms. Let
X
δβ µz β
p3 = p2 −
.
iκ(1
+
[β])
β6 =(21)
g2 = −
Note 1 + [β] 6= 0 for β 6= (21). We have
ṗ3 = ṗ2 − µδβ z β + g3 = δ21 µz 2 z̄ + µd1 bz + (µP (4) + g1 + g2 + g3 )
with
g3 = −
X
δβ µ1+[β] d β
(p ) .
iκ(1 + [β]) dt
β6 =(21)
Notice that the equation for p3 has the desired form (6.7), and that the error
term (µP (4) + g1 + g2 + g3 ) has the desired decay in t when t is large. However,
certain terms in the error term are not small compared to 0z 3 when t is of order 1,
and hence need to be treated. (Recall 0 = − Re δ21 .) We will also integrate these
terms and include them in the perturbation of p.
(3)
(3)
from P (4) , where η1,2
= η1(3) + η2(3) . Since
One such term is of the form Qζ η1,2
the size of η1(3) and η2(3) are comparable to η(2) when the time t is of order 1, and 0z 3
(3)
comes from terms of the form Qζ η(2) , we need to treat terms of the form Qζ η1,2
.
We will integrate these terms and include them in the perturbation of p. Recall
−1 −iθ (3)
e e
η j , j = 1, 2, and
η(3)
j =U
η0 − (eiθ z α )(0)e
e
η1(3) + e
η2(3) = e−i At e
ηα
where e
η0 = eiθ(0)U η0 , and e
ηα , |α| = 2, are defined as in (3.17). Denote χ =
(3)
(3)
(3)
e0 = eiθ (0) [U η0 − z α (0)e
ηα ] ∈ Hc (A) for the computation
η1,2 = η1 + η2 and χ
below. We have
(6.14)
χ (t) = η1(3) + η2(3) (t) = U −1 e−iθ (t) e−i At χ
e0 .
Recall (6.1). The only source of Qζ χ is from F (4) , where we have a term 2λQ((ζ +
ζ̄ )χ + ζ χ). This is the same source for the resonance term Qζ η(2) . Hence terms
of the form Qζ χ in µP (4) are exactly
Pzη(3) ≡ −iµ u + , 2λQ((ζ + ζ̄ )χ + ζ χ̄) − iµC u − , 2λQ((ζ + ζ̄ )χ + ζ χ̄) .
1,2
DYNAMICS OF NLS
187
Clearly these terms can be summed in the form
Pzη(3) = µ(zφ + z̄φ, χ ) + Cµ−1 (zφ + z̄φ, χ ) .
(6.15)
1,2
Here each φ = O(n) stands for a different local smooth function. Recall U −1 =
U++ − U−∗ C from (2.24); hence
U −1 (z f + z̄g) = z(U+∗ f − U−∗ ḡ) + z̄(U+∗ g − U−∗ f¯)
(cf. (7.14)). Together with (6.14) we have
Z
pφ1 + µ2 p̄φ2 e−iθ e−i At χ
e0 d x
Pzη(3) =
1,2
Z
(6.16)
e0 d x
+C
µ−2 pφ3 + p̄φ4 e−iθ e−i At χ
for some local functions φ1 , φ2 , φ3 , φ4 = O(n). Recall µ(s) = eiκs . Define
Z tZ
f 1 (t) =
φ1 e−i As χ
e0 d x ds ,
∞
Z tZ
f 2 (t) =
φ2 ei2κs e−i(A−0i)s χ
e0 d x ds ,
∞
Z tZ
φ3 e−i2κs e−i As χ
f 3 (t) =
e0 d x ds ,
∞
Z tZ
f 4 (t) =
e0 d x ds .
φ4 e−i As χ
∞
η20 =
These functions in t depend on the initial data only. Recall χ
e0 contains e
−1 A
−(A − 0i − 2κ) Pc 820 , which is not a local term. It is clear by Lemma 2.7 that
Z t
| f j (t)| ≤
Cnhsi−9/8 ke
χ0 k ds ≤ Cε 3/2 n 2 hti−1/8 .
∞
However, we claim we have
| f j (t)| ≤ Cε3/2 n 2 hti−9/8 ,
(6.17)
j = 1, 2, 3, 4 .
In fact, (6.17) is clear for f j (t), j 6= 2, since we can integrate them explicitly. For
example,
Z
f 1 (t) =
(−i A)−1 (PcA φ1 )e−i At χ
e0 d x = O(hti−9/8 ) ,
by Lemma 2.7. Here we have used that A−1 is bounded in L r2 ∩ Hc (A). The
e0 is
problem for the term f 2 (t) is that the factor ei2κs e−i As gives resonance while χ
not a local term. However, the main term in f 2 that concerns us is
Z tZ
Z
i2κs −i(A−0i)s
φ2 e e
e
η20 d x ds = −i φ2 (A − 2κ − 0i)−2 PcA 820 d x ,
∞
which is still of order O(hti−9/8 ) by Lemma 2.7. Thus we also have (6.17) for f 2 .
188
T.-P. TSAI AND H.-T. YAU
We can now integrate Pzη(3) :
1,2
Pzη(3) =
(6.18)
1,2
d
( p4 ) + g4 ,
dt
where p4 is of similar form to Pzη(3) ,
1,2
f 2 + C pe−iθ f 3 + p̄e−iθ f 4 ,
(6.19)
d
d
d
d
−iθ
−iθ
−iθ
−iθ
( pe ) f 3 + ( p̄e ) f 4 .
g4 = ( pe ) f 1 + ( p̄e ) f 2 + C
dt
dt
dt
dt
By (6.17),
p4 = pe
(6.20)
−iθ
f 1 + p̄e
−iθ
| p4 | ≤ Cε3/2 n 2 hti−9/8 |z| ,
|g4 | ≤ Cε3/2 n 2 hti−9/8 (|z θ̇| + | ṗ|) .
(3)
.
We have integrated terms of the order Qzη1−2
Other terms of concern for t = O(1) are related to a R and η from P (4) . (We
are not worried about terms from g1 , g2 , and g3 , since the coefficients of these
remainder terms of integration by parts are multiplied by O(n); see (4.14).) These
terms are of the form
Pa 2 ,aη = µ (φ, z 2 a R) + (φ, Q(a R)2 ) + (φ, Qa Rη) + (φ, za Rη) ,
where φ denotes some local functions. Since a = O({t}−1 ), R = O(n −1 ), and
η = O({t}−1 ) locally, the largest terms here are of order n −1 {t}−2 , which is larger
than 0z 3 when t is small. However, terms linear in η can be integrated as A(zη)
in the next section. The first two terms, µz 2 a R and µQ(a R)2 , are multiplied
by µ = eiκt ; hence they are oscillatory and can be integrated as cα z α terms and
as A(zb) in the next section. (Note a = a20 (z 2 + z̄ 2 ) + b and all terms of the
form µz γ bm with |γ | + 2m = 4 are oscillatory; i.e., they have a nonzero phase.)
Since the computation is the same as for A(zη) , cα z α , and A(zb) , we shall skip the
computation and just give the result:
d
Pa 2 ,aη = ( p5 ) + g5 ,
dt
where p5 is of similar form to Pa 2 ,aη and
| p5 | ≤ Cn −1 {t}−2 ,
(6.21)
|g5 | ≤ C {t}−5/2 .
Now we let q = p3 − p4 − p5 . We have
q̇ = δ21 µz 2 z̄ + µd1 bz + µP (4) + g1 + g2 + g3 − Pzη(3) + g4 − Pa 2 ,aη + g5
1,2
2
(4)
= δ21 | p| p + d1 bp + µP − Pzη(3) − Pa 2 ,aη + g1 + g2 + g3 + g4 + g5
1,2
= δ21 |q|2 q + d1 bq + g ,
where
(6.22)
g=
µP (4) − Pzη(3) − Pa 2 ,aη + g1 + · · · + g5
1,2
+ δ21 (| p| p − |q|2 q) + d1 b( p − q) .
2
DYNAMICS OF NLS
189
Hence we have arrived at the normal form. (Note that, although we kept ṗ and ḃ
in the definition of g2 , it should be replaced by their corresponding equations (6.2)
and (7.2)). Also, note we have
X
cα
δβ µz β
µd2 bz̄
(6.23)
q = p−
µz α −
−
− p4 .
iκ(1 + [α])
2iκ
iκ(1 + [β])
β6 =(21)
Finally, from the explicit form of g(t), it is bounded by terms of the form
(3)
nz 4 + nb2 + (nφ, η2 ) + nzη3−5
.
Assuming the bounds on a(t) and M(t), we get the bound (6.9) for g(t):
|g(t)| ≤ Cε 3/4−σ n 7/4 {t}−13/8 .
To conclude this lemma, it remains to prove the estimate on q, which follows
from the next lemma.
Our main restriction on the size of η0 comes from the term (nφ, (η1(3) )2 ) in g(t).
Since we want it to be smaller than 0|z|3 , we need η0 ε3/2 n 2 for t = O(1). We
assume η0 ≤ ε2 n 2 for simplicity.
6.3 Decay Estimates
In this subsection we present a calculus lemma that deals with the decay of q(t).
We will write
(6.24)
q(t) = ρ(t)eiω(t) ,
where ρ = |q| and ω is the phase of q. Recall
{t} = ε−2 n −2 + 20t ,
{t} ∼ max{ε −2 n −2 , t} .
Before we proceed with the proof, we give some simple facts of an ordinary
differential equation.
Example. Consider real functions r (t) > 0 that solve
(6.25)
ṙ (t) = −r (t)3 − ε(1 + t)−3
for t > 0, where ε > 0 is small. We have the following facts:
(1) All solutions r (t) satisfy
r (t) ≤ (C + 2t)−1/2
with C = r (0)−2 .
(2) There is a number r1 > 0 such that if r (0) > r1 , then r (t) ∼ (C + 2t)−1/2 .
(3) There is a unique global solution r0 (t) of (6.25) such that
r0 (t) ∼ t −2
as t → ∞ .
(4) If r (0) < r0 (0), then r (t) = 0 in finite time.
(5) If r (0) > r0 (0), then
Z ∞
r (s)2 ds = ∞ .
0
190
T.-P. TSAI AND H.-T. YAU
Thus the behavior of r (t) depends on which term on the right side of (6.25) dominates.
L EMMA 6.3 Recall that n 0 , ε0 , n, and ε given in Theorems 1.1 through 1.4 satisfy
0 < n ≤ n 0 and 0 < ε ≤ ε0 . Let 0 ≈ n 2 and σ = 0.01. Suppose a positive
function ρ(t) satisfies
ρ̇ = −0ρ 3 + e
g (t) ,
(6.26)
where the error term
|e
g (t)| ≤ Cε 3/4−σ n 7/4 {t}−13/8 ,
{t} = {t}ε = ε−2 n −2 + 20t .
(i) Suppose ρ0 = εn. Then there is a constant m = m(ε0 ) > 1 such that
m −1 {t}−1/2 ≤ ρ(t) ≤ m {t}−1/2 .
Moreover, m(ε0 ) → 1+ as ε0 → 0+ .
(ii) Suppose that ρ(0) ≤ εn. Then we have
ρ(t) ≤ m {t}−1/2 .
The above example shows that we cannot expect a lower bound for ρ(t) under
the assumption of part (ii), when ρ(0) is too small compared with the error term.
P ROOF : We first prove part (i). Let ρ+ = m {t}−1/2 and ρ− = m −1 {t}−1/2 ,
with m > 1 to be determined. We have ρ+ (0) > ρ(0) > ρ− (0). Moreover,
ρ̇+ = −0m −2 ρ+3 = −0ρ+3 + 0(1 − m −2 )ρ+3 ≥ −0ρ+3 + e
g
if
0(1 − m −2 )m 3 {t}−3/2 ≥ Cε3/4−σ n 7/4 {t}−13/8 .
Also,
ρ̇− = −0m 2 ρ−3 = −0ρ−3 − 0(m 2 − 1)ρ−3 ≤ −0ρ−3 + e
g
if
0(m 2 − 1)m −3 {t}−3/2 ≥ Cε3/4−σ n 7/4 {t}−13/8 .
Since {t}−3/2 ≥ (εn)1/4 {t}−13/8 , both inequalities hold if
0(1 − m −2 )m 3 , 0(m 2 − 1)m −3 ≥ Cε1−σ n 2 .
Since 0 ≈ n 2 and ε ≤ ε0 , the above is true for m > 1 arbitrarily close to 1 by
choosing ε0 sufficiently small. By comparison, we have ρ− (t) < ρ(t) < ρ+ (t) for
all t.
The upper bound in part (ii) follows from the same proof by comparing ρ(t)
with ρ+ (t).
DYNAMICS OF NLS
191
7 Change of the Mass of the Ground State
Recall that a(t) satisfies the integral equation (3.3), and we have derived the
main oscillatory terms of a(t) in (3.8): a(t) = a (2) (t)+b(t) and a (2) = a20 (z 2 + z̄ 2 ),
with b(t) given by (3.9); that is,
Z t
a(t) = a(T ) +
(7.1)
(c1 Q, Im(F + θ̇ h))ds
T
Z t
(2)
(2)
= a(T ) + a (t) − a (T ) +
ḃ(s)ds ,
T
where c1 = (Q, R)
(7.2)
−1
and
ḃ = c1 Q, Im[F − F (2) + θ̇ h] − 4a20 Re e−2iκs p ṗ .
(2)
(3)
2
2
Note that, after
P the3substitution η = η + 4η and2 a = a20 (z + z̄ ) + b, ḃ is of
the form Re {C z + C zb + (zφ, η) + C z + C z b} plus error. We will perform
several integrations by parts to arrive at the form
Z t
−1
a(t) = O(t ) +
O(t −2 )ds = O(t −1 )
T
−1
and obtain the estimate |a(t)| ≤ O(t ).
We first decompose the integrand of a(t) in (7.1) according to order in z; see
table (4.5). Recall η = η(2) + η(3) and a = a20 (z 2 + z̄ 2 ) + b. Also, recall from
(3.23) that
(7.3) θ̇ = − a (2) + b + (c1 R, Re F (2) ) + Fθ,3 + Fθ,4
e(3) + a (2) + b + (c1 R, Re F (2) ) (c1 R, Re ζ ) ,
Fθ,3 = − c1 R, Re F (3) + F
and Fθ,4 = O(z 4 + a 2 + η2 ). Thus we have
(c1 Q, Im θ̇ h) = θ̇ c1 Q, Im ζ + η(2) + η(3)
(7.4)
= − a (2) + b + (c1 R, Re F (2) ) (c1 Q, v) Im z
+ Fθ,3 (c1 Q, v) Im z − a (2) + b + (c1 R, Re F (2) ) c1 Q, Im η(2)
(7.5)
+ Fθ,4 (c1 Q, v) Im z + Fθ,3 + Fθ,4 c1 Q, Im η(3)
(7.6)
+ θ̇ c1 Q, Im η(3) .
Here the second line (7.4) is of order O(z 2 ) and the third line (7.5) of order O(z 3 ),
and the last line (7.6) contains higher-order terms. Together with the decompositions (3.22) of F we can decompose the integrand of a(t) in (7.1) according to
order in z,
(c1 Q, Im(F + θ̇ h)) = A(2) + A(3) + A(4) + A(5) ,
A(3) = [A(zb) + A(zη) + A(z ) ] ,
3
192
T.-P. TSAI AND H.-T. YAU
where A(2) consists of O(z 2 ) terms, A(3) of O(z 3 ) terms, A(4) of O(z 4 ) terms, and
A(5) of higher-order terms. They are given explicitly by
A(2) = (c1 Q, Im λQζ 2 ) ,
A(zb) = (c1 Q, Im 2λQ Rbζ ) − b(c1 Q, v) Im z ,
A(zη) = (c1 Q, Im 2λQζ η) ,
3
A(z ) = c1 Q, Im λ|ζ |2 ζ + 2λQ Ra (2) ζ
− a (2) + (c1 R, Re F (2) ) (c1 Q, v) Im z ,
A(4) + A(5) = c1 Q, Im λQη2 + 2λQ Raη + λ |a R + ζ |2 (a R + ζ ) − |ζ |2 ζ
+ (7.5) + (7.6) .
(7.7)
A
(4)
A(5)
= (7.5) + c1 Q, Im λQ(η(2) )2 + 2λQ Raη(2)
+ λ 2|ζ |2 (a R + η(2) ) + ζ 2 (a R + η(2) ) ,
= c1 Q, Im λQ[2η(2) η(3) + (η(3) )2 ] + 2λQ Raη(3)
+ c1 Q, Im λ 2|ζ |2 η(3) + ζ 2 η(3) + `2 ζ̄ + 2|`|2 ζ + `2 `¯ + (7.6) ,
` = aR + η .
Since ζ = zu + + z̄u − , we have
Im ζ = Im z(u + − u − )
and
Im ζ 2 = (Im z 2 )(u 2+ − u 2− ) .
We now proceed to integrate them term by term. We have already integrated
A(2) in Section 3.1. Write
(7.8)
e−iκt ṗ = cα z α + dβ z β + d1 bz + d2 bz̄ + P (4) := cα z α + P (3,4) ,
where P (3,4) is defined by the last equality. We can rewrite our result in Section 3.1 as
Z t
Z t
(2)
(2)
(2)
(7.9)
A ds = a (t) − a (T ) +
A2,3 + A2,4 + A2,5 ds ,
T
T
a (2) = a20 (z 2 + z̄ 2 ) , a20 = (4κ)−1 c1 Q, λQ(u 2+ − u 2− ) = O(n 2 ) ,
(7.10)
A2,3 = −4a20 Re zcα z α , A2,4 = −4a20 Re dβ zz β + d1 bz 2 ,
A2,5 = −4a20 Re z P (4) .
Note here we have used Re zd2 bz̄ = 0 from Lemma 6.1.
We can also rewrite (7.2), the equation for b = a − a (2) , as
(7.11)
3
ḃ = A(zb) + A(zη) + A(z ) + A2,3 + A(4) + A2,4 + A(5) + A2,5 .
DYNAMICS OF NLS
193
7.1 Integration of O(z 3 ) Terms
There are three terms of order O(z 3 ): A(zb) , A(zη) , and A(z ) (we will absorb
3
A2,3 into A(z ) ). We first integrate A(zb) . From its explicit form we have A(zb) =
C2 b Im z for a real constant C2 = O(n). Hence, using the integration-by-parts
formula (6.11) and the decompositions (7.8) and (7.11),
Z t
Z t
Z t
1
(zb)
−iκs d
A ds =
C2 b Im z ds = C2 Im
e
bz −
(bp) ds
−iκ
dt
T
T
T
Z t
C2
=
z ḃ + b(cα z α + P (3,4) )ds
2 Re zb −
2κ
T
Z t
A zb,4 + A zb,5 ds ,
= czb b(z + z̄) +
3
T
where P (3,4) is defined in (7.8) and
C2
= O(n) ,
2κ
3
= −czb (z + z̄) A(zb) + A(zη) + A(z ) + A2,3 − 2czb b Re cα z α ,
czb =
(7.12)
A zb,4
A zb,5 = −czb (z + z̄) A(4) + A2,4 + A(5) + A2,5 − 2czb b Re P (3,4) .
We now integrate A(zη) . Recall from (2.24) that U = U+ + U− C and U −1 =
U+∗ − U−∗ C. We will also use the following formulae:
Z
Z
Re d x f (Cg) = Re d x(C f )g ,
Z
Z
(7.13)
Im d x f (Cg) = − Im d x(C f )g ,
(7.14)
U (z f + z̄g) = z(U+ f + U− ḡ) + z̄(U+ g + U− f¯) .
Recall η = U −1 e−iθ η̃. Denote η0 = e−iθ η̃ and hence η = U −1 η0 . We have
from (7.13) and (7.14) the identity
A(zη) = (c1 Q, Im 2λQζ η)
Z
= Im d x2c1 λQ 2 ζ η
Z
= Im d x2c1 λQ 2 (zu + + z̄u − )(U+∗ − U−∗ C)η0
Z
= Im d x (U+ + U− C)[z(2c1 λQ 2 u + ) + z̄(2c1 λQ 2 u − )] η0
Z
= Im d x(zφ3 + z̄φ4 )η0 ,
194
T.-P. TSAI AND H.-T. YAU
where
φ3 = U+ (2c1 λQ 2 u + ) + U− (2c1 λQ 2 u − ) ,
φ4 = U+ (2c1 λQ 2 u − ) + U− (2c1 λQ 2 u + ) .
We now rewrite
(7.15)
η0 (s) = e−iθ η̃ = e−iθ e−i As f (s) ,
Z s
i As
f (s) = e η̃ = η̃0 +
eiτ A PcA Fη (τ )dτ .
0
The reason we work with f instead of η̃ is that those terms of the same order (z 2 )
in ∂s f (s) = eis A PcA Fη (s) are explicit. We now have
Z t
Z t
A(zη) ds = Im (φ3 , zη0 ) + (φ4 , z̄η0 )ds
T
T
Z t
φ3 , e−is(A+κ) pe−iθ f ds
= Im
T
Z t
+ Im
φ4 , e−is(A−κ) p̄e−iθ f ds .
T
Note f (s) ∈ Hc (A). We first compute the first integral, which is equal to
t
1
−is(A+κ)
−iθ
pe f
= Im φ3 ,
e
−i(A + κ)
T
Z t
1
d
−is(A+κ)
−iθ
− Im
e
pe f ds
φ3 ,
−i(A + κ)
ds
T
t
1
A
−iθ
= Re
P 5φ3 , ze η̃
A+κ c
T
Z t
1
A
−iκs
0
0
−iθ A
− Re
ṗη − i θ̇ zη + ze Pc Fη ds
P 5φ3 , e
A+κ c
T
Z t
t
φ5 , e−iκs ṗη0 − i θ̇ zη0 + ze−iθ PcA Fη ds
= Re φ5 , zη0 T − Re
T
where
1
P A 5φ3 .
A+κ c
We are careful in adding PcA 5 so that φ5 makes sense. We can do so since f (s) ∈
Hc (A). Similarly, the second integral is equal to
Z t
t
= Re φ6 , z̄η0 T − Re
φ6 , eiκs ṗη0 − i θ̇ z̄η0 + z̄e−iθ PcA Fη ds
φ5 =
T
with
φ6 =
1
P A 5φ4 .
A−κ c
DYNAMICS OF NLS
195
Rt
Using η0 = U η, we can rewrite the leading terms of T A(zη) ds in the form
Z
Re(φ5 , zη0 ) + Re(φ6 , z̄η0 ) = Re d x(zφ5 + z̄φ6 )(U+ + U− C)η
Z
= Re d x[(U+∗ + U−∗ C)(zφ5 + z̄φ6 )]η
Z
= Re d x(zφ8 + z̄φ7 )η
= Re(zφ7 + z̄φ8 , η) ,
where we have used (7.13) and (7.14) again, with the convention (2.1) and
φ8 = U+∗ φ5 + U−∗ φ6 ,
φ7 = U+∗ φ6 + U−∗ φ5 .
Tracking our definition, we have φi = O(n 2 ), i = 3, 4, . . . , 8.
The remaining integral has the integrand
= − Re φ5 , e−iκs ṗη0 − i θ̇η0 + ze−iθ PcA Fη
− Re φ6 , eiκs ṗη0 − i θ̇ z̄η0 + z̄e−iθ PcA Fη
Z
= − Re d x e−iκs ṗφ5 + eiκs ṗφ6 U η
+ (zφ5 + z̄φ6 ) −i θ̇U η + e−iθ PcA Fη .
Using (3.11) we have
−i θ̇U η + e−iθ PcA Fη = PcA U i −1 [F + θ̇(a R + ζ + η)] .
Thus the integrand of the remaining integral can be written as A zη,3 + A zη,4 + A zη,5 ,
and we have
Z t
Z t
t
(zη)
(7.16)
A ds = [Re(zφ7 + z̄φ8 , η)]T +
A zη,3 + A zη,4 + A zη,5 ds ,
T
Z
A zη,3 = − Re
Z
T
d x(zφ5 + z̄φ6 )PcA U i −1 z α φα
d x cα z α φ5 + cα z α φ6 U η(2)
e(3)
+ (zφ5 + z̄φ6 )PcA U i −1 F (3) + F
(2)
− a + b + c1 R, Re F (2) ζ
Z
= − Re d x cα z α φ5 + cα z α φ6 U η(3) + P (3,4) φ5 + P (3,4) φ6 U η
A zη,4 = − Re
(7.17)
A zη,5
+ (zφ5 + z̄φ6 )PcA U i −1 F (4) + θ̇(a R + η) + (Fθ,3 + Fθ,4 )ζ ,
where P (3,4) stands for the higher-order terms in e−iαs ṗ (7.8), and we have used
the definitions of F (3.22) and θ (7.3). We observe that the only appearance of θ
196
T.-P. TSAI AND H.-T. YAU
is as the exponent of eiθ . Also, A zη,3 is a sum of monomials of the form cz β with
|β| = 3. There is no a or η in A zη,3 .
Summarizing our effort, we have obtained
t
a(t) = a(T ) + a20 z 2 + z̄ 2 + czb b(z + z̄) + Re(zφ7 + z̄φ8 , η) T
Z t
(z 3 )
(7.18)
A + A2,3 + A(zη,3) + A(4) + A2,4 + A(zb,4) + A(zη,4)
+
T
+ A(5) + A2,5 + A(zb,5) + A(zη,5) ds .
All terms in [A(z ) + A2,3 + A(zη,3) ] are of the form cz β , |β| = 3. We define Aβ so
that
3
Aβ z β = A(z ) + A2,3 + A(zη,3)
3
= c1 Q, Im λ|ζ |2 ζ + 2λQ Ra (2) ζ − a (2) + c1 R, Re F (2) (c1 Q, v) Im z
Z
− 4a20 Re zcα z α − Re d x(zφ5 + z̄φ6 )PcA U i −1 z α φα .
From this explicit form, we have Aβ = O(n). By integration by parts, we have
Z t
Z t
Aβ
Aβ z β ds = aβ z β −
aβ z β f β (z) ds , aβ :=
= O(n) ,
i[β]κ
T
T
Z t
β
= aβ z +
(7.19)
A3,4 + A3,5 ds
T
where f β is defined in (6.12) and
X
A3,4 =
(7.20)
−aβ z β (β0 + β1 C)z −1 cα z α
β=(β0 ,β1 ),β0 +β1 =3
A3,5 =
X
−aβ z β (β0 + β1 C)z −1 P (3,4) .
β=(β0 ,β1 ),β0 +β1 =3
Notice that [β] 6= 0 and aβ = aβ̄ .
Substituting (7.19) into (7.18), we obtain
t
a(t) = a(T ) + a20 z 2 + z̄ 2 + czb b(z + z̄) + Re(zφ7 + z̄φ8 , η) + aβ z β T
Z t
(4)
+
(7.21)
A + A2,4 + A(zb,4) + A(zη,4) + A3,4
T
+ A(5) + A2,5 + A(zb,5) + A(zη,5) + A3,5 ds .
We have finished the integration of all terms of order O(z 3 ) in equation (7.1) for
a(t). Thus there are no order-O(z 3 ) terms in the integrand in (7.21). The order z 4
terms are collected in first group of integrands in (7.21), and the higher-order terms
in the second group.
DYNAMICS OF NLS
197
In principle, order-z 4 terms have the following forms: z 4 , b2 , η2 , z 2 b, z 2 η, bη,
and zη(3) . Closer examination shows that b2 , b|z|2 , or zη(3) never occur. (Recall that zη(3) is part of zη, which we already treated when we integrated A(zη) .)
Furthermore, the terms involving η can be removed by making the substitution
η = η(2) + η(3) . A precise statement on these integrands is given in the following
lemma. It is crucial that no term involving b2 appear inR the integrand. Otherwise,
t
it will result in an inequality of the form b(t) ≤ C/t + b(s)2 ds, which does not
guarantee b(t) ≤ C/t.
L EMMA 7.1 The integrands of order O(z 4 ) in (7.21) can be summed into the form
A(4) + A2,4 + A(zb,4) + A(zη,4) + A3,4 =
B22 |z|4 + Re A40 z 4 + A31 z 3 z̄ + Ab2 bz 2 .
There are no terms of the form b2 , b|z|2 , or zη(3) . Moreover, we have
(7.22)
B22 = 12 c1 0 + O(n 3 ) ,
A40 , A31 = O(n) ,
Ab2 = O(n) .
P ROOF : Recall the definitions (7.7), (7.10), (7.12), (7.17), and (7.20):
A(4) = c1 Q, Im λQ(η(2) )2 + 2λQ Raη(2)
+ λ 2|ζ |2 (a R + η(2) ) + ζ 2 (a R + η(2) )
+ Fθ,3 (c1 Q, v) Im z − a (2) + b + c1 R, Re F (2) (c1 Q, Im η(2) )
A2,4 = −4a20 Re dβ zz β + d1 bz 2
3
A(zb,4) = −czb (z + z̄) A(zb) + A(zη) + A(z ) + A2,3 − 2czb b Re cα z α
Z
A(zη,4) = − Re d x cα z α φ5 + cα z α φ6 U η(2)
e(3) − a (2) + b + c1 R, Re F (2) ζ
+ (zφ5 + z̄φ6 )PcA U i −1 F (3) + F
X
A3,4 =
−aβ z β (β0 + β1 C)z −1 cα z α .
β=(β0 ,β1 ),β0 +β1 =3
The first part of the lemma is obtained by direct inspection. Note we can deal with
A zη,4 in the same manner as we dealt with A(zη) . The orders of the coefficients are
also obtained by direct check with the following table in mind:
Q = n,
R = n −1 ,
a20 = n ,
2
c1 = 1 ,
aβ = n ,
ζ ∼ z + n 2 z̄ ,
cα = n ,
czb = n ,
a R ∼ n −1 ρ 2 ,
dβ , d1 , d2 = 1 ,
φ7 , φ 8 = n 2
η ∼ nρ 2 .
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T.-P. TSAI AND H.-T. YAU
We note that all C|z|4 terms with C = O(n 2 ) but one are killed by the Im operator.
The only surviving term is the following resonance term of A(4) :
c1 Q, Im λζ 2 η(2) =
X
O(n 2 )z γ .
c1 Q, Im λz 2 u 2+ z 2 η20 + O(n 3 )|z|4 +
|γ |=4,γ 6 =(22)
The first term on the right side is equal to 12 c1 0|z|4 + O(n 3 ), and this is the main
term in B22 . This proves Lemma 7.1.
7.2 Integration of Higher-Order Terms
Next we proceed to integrate out those oscillatory terms in the integrand of
(7.21) for a(t). The first group consists of terms of the form z γ , |γ | = 4.
Z t
Re
A40 z 4 + A31 z 3 z̄ + Ab2 bz 2 ds
T
A40 z 4
A31 z 3 z̄
= Re
+
+
−4iκ
−2iκ
Z t
A40 z 4
− Re
f 40 (z) +
T −4iκ
where
d
(bp 2 )
ds
t
Ab2 bz 2
−2iκ T
A31 z 3 z̄
Ab2 µ−2 d
f 31 (z) +
(bp 2 )ds
−2iκ
−2iκ ds
= ḃ p 2 + 2bp ṗ = O(z 5 ). Let
A40
A31
Ab2
= O(n) , a31 =
= O(n) , ab2 =
= O(n) ,
−4iκ
−2iκ
−2iκ
d
A4,5 = − Re a40 z 4 f 40 (z) + a31 z 3 z̄ f 31 (z) + ab2 (bp 2 ) .
ds
Then we have the identity
Z t
Re
A40 z 4 + A31 z 3 z̄ + Ab2 bz 2 ds =
T
Z t
4
3
2 t
Re a40 z + a31 z z̄ + ab2 bz T +
A4,5 ds .
a40 =
T
After this integration, the integrand in (7.21) becomes
A(5) + A2,5 + A(za,5) + A(zη,5) + A3,4 + A4,5 .
Since |A4,5 | is of order z 5 , the integrands are of order z 5 or higher. More precisely,
they are of the orders
(3)
(3)
+ z 2 η3−5
+ ··· .
n z 5 + z(a R)2 + z(a R)η + (a R)2 η + zη2 + z 2 η1−2
(The first n is from the c1 Q in equation (2.30) for ȧ.) These terms are of order
t −2−1/8 and thus can be considered as error terms when t large. Unfortunately, we
need to control their behavior even for t of order 1. In this region, their orders in
terms of n are crucial.
DYNAMICS OF NLS
199
We first note that nz 5 is bounded by εn 2 |z|4 |B22 ||z|4 . Since kηk L 2
−1
C {t} , the term nzη is also much smaller than |B22 ||z| . The term
2
4
loc
(3)
nz 2 η3−5
≤
is
−2−1/8
|B22 ||z|4 . Thus the main trouble lies in terms
bounded by ε−σ n(εn)3/4 {t}
of order
(3)
nz(a R)2 + nz(a R)η + n(a R)2 η + nz 2 η1−2
.
These terms have faster time-decay order than |B22 ||z|4 , namely, t −2−1/8 , but they
are larger than |B22 ||z|4 when time t = O(1). They are oscillatory terms of order
z 5 that we will integrate. (In contrast, terms of order z 2k may be nonoscillatory.)
Since the integration procedures for these terms are the same as those for O(z 3 )
terms, we only sketch the main steps:
(1)
(2)
(3)
(4)
(5)
Replace all a in the above group by a20 (z 2 + z̄ 2 ) + b.
3
All terms z γ with |γ | = 5 are oscillatory and can be integrated as A(z ) .
nz 3 b R 2 and nzb2 R 2 are also oscillatory and can be integrated as A(zb) .
All terms linear in η—z 3 η, zbη, and b2 η—can be integrated as A(zη) .
(3)
can be integrated as Pzη(3) , defined in (6.15). Specifically, we have
z 2 η1−2
1−2
terms of the form
X
d
(3) A z 2 η(3) =
Re nφ , z α η1−2
az 2 η(3) + A z 2 η(3) ,5
=
1−2
1−2
1−2
dt
|α|=2
where φ = O(1) are some complex local functions and az 2 η(3) and A z 2 η(3) ,5
1−2
1−2
are defined similarly to p4 and g4 in (6.19) with the decay
a 2 (3) ≤ C {t}−1 ε2 n 3 hti−9/8 ,
z η1−2
A 2 (3) ≤ C {t}−3/2 ε2 n 3 hti−9/8 ≤ Cε|B22 | {t}−2 .
z η ,5
1−2
Note that, as in p4 and g4 , we have terms of the form (φ, (A − 2κ −
0i)−2 PcA 8)nz 2 in az 2 η(3) , which still has the claimed decay by Lemma 2.7.
1−2
To summarize, we have obtained
t
a(t) = a(T ) + a20 z 2 + z̄ 2 + czb b(z + z̄) + Re(zφ7 + z̄φ8 , η) + aβ z β T
t
+ Re a40 z 4 + a31 z 3 z̄ + ab2 bz 2 + az 2 η0 T
(7.23)
(3) t
+ Re C z 5 + C z 3 b + C zb2 + C z 3 η + C zbη + Cb2 η + C z 2 η1−2
T
Z t
+
B22 |z|4 + B5 ds ,
T
1
c0
2 1
+ O(n 3 ), the terms in the third line denote various terms of
where B22 =
similar form (e.g., C z 5 means C z γ with |γ | = 5), and
(7.24)
|B5 (t)| ≤ C(ε + n)σ n 7/4 {t}−2−1/8 0 {t}−2 ,
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T.-P. TSAI AND H.-T. YAU
assuming the bound (4.10) on a(t) and M(T ) ≤ 2.
Recall b(t) := a(t) − a (2) (t) and ST (4.11) is defined by
t
a(t) = a(T ) + a20 z 2 + z̄ 2 T + ST (a, z, η, θ )(t) .
From (7.23), ST can be viewed as a function of a, z, η, and θ and explicitly given by
ST (a, z, η, θ)(t)
t
:= czb b(z + z̄) + Re(zφ7 + z̄φ8 , η) + aβ z β T
t
+ Re a40 z 4 + a31 z 3 z̄ + ab2 bz 2 + az 2 η0 T
(7.25)
(3) t
+ Re C z 5 + C z 3 b + C zb2 + C z 3 η + C zbη + Cb2 η + C z 2 η1−2
T
Z t
+
B22 |z|4 + B5 ds .
T
The right-hand side depends on b, η(3) , etc., which are not explicitly given as variables in ST . But all these variables can be traced back to the basic variables a, z,
η, and θ, e.g., b = a − a20 (z 2 + z̄ 2 ).
Making the same assumptions as in Proposition 4.1, we have
B22 −1
D
{t} ≤ o(1) +
{t}−1 .
(7.26) |ST (a, z, η, θ)(t)| ≤ C1 (D) {t}−3/2 +
20
4
We also have a(t) = a(T )+[a20 (z 2 + z̄ 2 )]tT + S(a)(t) = a(T )+[o(1)+ D/4] {t}−1 .
We conclude with the following lemma:
L EMMA 7.2 Suppose that M(t) ≤ 2 and |a(t)| ≤ D {t}−1 for 0 ≤ t ≤ T . Recall
D = 2B22 / 0 = c1 + O(n). Then we have
(7.27)
|ST (a, z, η, θ)(t)| ≤
D −1
{t} ,
2
|a(t)| ≤ |a(T )| +
D −1
{t} .
2
This lemma gives Proposition 4.2 and concludes the proof of Theorems 1.1 and 1.3.
Since |E 0 − E j+1 | ≤ |E 0 − E j |+|γ | ≤ 98 Dε2 n 2 +|γ | ≤ 2Dε2 n 2 , by Lemma 2.4
again with E 1 standing for E 0 and E 2 standing for E j+1 ,
(7.28)
|E 0 − E j+1 − a j+1 (0)| ≤ Cn −1 (εn)(2Dε2 n 2 ) = Cε3 n 2 .
The constant C here is independent of j since we use a bound for |E 0 − E j+1 | that
is independent of j. Since |a j+1 (0)| ≤ 12 Dε2 n 2 , we have
(7.29)
|E 0 − E j+1 | ≤ Dε2 n 2 .
From this, we also get kψ0 − Q j+1 ei2j+1 (0) kY ≤ 2εn. We have thus proven the
induction hypothesis (I j+1 ) for t = Tj+1 . The induction proof is complete.
DYNAMICS OF NLS
201
P ROOF OF T HEOREM 1.4: Suppose the assumption of Theorem 1.4 holds. The
existence of E ∞ and the upper bound (1.22) follows from the same proof for Theorem 1.3. Recall from (1.24) that
|z E0 ,0 | = εn ,
kη E0 ,0 kY ≤ Cε2 n 2 ,
a E0 ,0 = 0 .
Let γ = E 0 − E ∞ be the energy shift. The energy shift can be bounded by |γ | ≤
Dε2 n 2 (2.10). Thus with respect to E ∞ Lemma 2.4 guarantees
|z E∞ (0)| = εn + O(ε 2 n) ,
kη E∞ (0)kY ≤ Cε2 n 2 ,
a E∞ (0) ≤ Cε 2 n 2 .
The second part of Proposition 4.1 thus provides the lower bound |z ∞ (t)| ≥ (1 −
2σ ) {t}−1/2 for all times t.
We now prove (1.26), the last statement of Theorem 1.4. Since |γ | ≤ Dε 2 n 2 ,
by (2.8) we have
(7.30)
kQ E0 k2 = kQ E∞ k2 + 2γ (Q ∞ , R∞ ) + O(n −2 γ 2 ) .
We can also estimate the energy shift γ = a∞ (0) + O(ε3 n 2 ) by Lemma 2.4 with
E 1 standing for E 0 and E 2 standing for E ∞ . Thus we have
Z 0
Z 0
4
9/4
a∞ (0) =
B22 |z| ds + O (εn)
B22 |q|4 ds + O (εn)9/4
=
∞
∞
where q = q∞ (t). Recall B22 =
+ O(n ) and c1 = (Q ∞ , R∞ )−1 . On the
other hand, we have dtd |q|2 = −20|q| + {t}−17/8 ; hence
Z ∞
2
|q(0)| =
−20|q|4 ds + O((εn)9/4 ) .
1
c0
2 1
4
3
0
From these formulae for a∞ (0) and q(0) we have
4a∞ (0)(Q ∞ , R∞ ) + |q(0)|2 = O (εn)9/4 .
Recall that the eigenfunctions for L satisfy u, v = φ1 + O(n 2 ). From the definition of z, we have |z(0)|2 = kζ (0)k2L 2 · [1 + O(n 2 )]. Since |q(0)|2 = |z(0)|2 +
O((εn)9/4 ), we conclude 2γ (Q ∞ , R∞ ) = − 12 kζ (0)k2L 2 + O(ε2 n 2 (ε1/4 + n 1/4 ))
and
(7.31)
kQ E0 k2 = kQ E∞ k2 − 12 kζ (0)k2L 2 + O ε2 n 2 (ε1/4 + n 1/4 ) .
Therefore (1.26) and thus Theorem 1.4 are proven.
8 Existence of Dispersion-Dominated Solutions
In this section we prove Theorem 1.5. For a given profile ξ∞ , we will construct
a solution of the form (1.18),
ψ = [Q(x) + a(t)R(x) + h(t, x)]ei[−Et+θ (t)] ,
so that ψ(t) − ψas (t) = O(t −2 ). Here a(t), θ(t) ∈ R and h(t) ∈ M.
By (2.29) we have
∂t U h = −i AU h − i θ̇U h + U i −1 (F + θ̇a R) − [U, i]θ̇ h .
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T.-P. TSAI AND H.-T. YAU
(Recall U PM = U .) We single out −i θ̇U h since it is a global linear term in h. Let
e
h = eiθ U h; we have
h = −i Ae
h + eiθ U i −1 (F + θ̇a R) − [U, i]θ̇ h .
(8.1)
∂t e
Let
e
h(t) = e−i At U ξ∞ + g(t) ,
(8.2)
h.
h = U −1 e−iθ e
Here g(t) ∈ X is the error term. Note that g(t) consists of both excited state and
subspace of continuity components. From (8.1) we have
Z t
(8.3)
g(t) =
e−i A(t−s) eiθ U i −1 (F + θ̇a R) − [U, i]θ̇ h ds .
∞
We will solve {a, θ, g} satisfying (2.30), (2.32), and (8.3), respectively, with e
h and
h defined in (8.2).
Note F = F(a R + h). The main term in F is
ξ(t) = U −1 e−iθ (t) e−i At U ξ∞ .
F0 = λQ 2|ξ |2 + ξ 2 + λ|ξ |2 ξ ,
By assumption ξ∞ is small in the space Z = H 2 ∩ W 2,1 (R3 ). Since U is bounded
in W k, p , kU ξ∞ k H 2 ∩W 2,1 (R3 ) ≤ ε for some ε sufficiently small. From Lemmas 2.6
and 2.7, we have the bounds kξ(t)k H 2 ≤ C1 ε and kξ(t)kW 2,∞ ≤ C1 ε|t|−3/2 . Hence
the term λ|ξ |2 ξ in F0 is bounded by k|ξ |2 ξ(t)k H 2 ≤ Ckξ(t)k3H 2 ≤ Cε3 . Moreover,
if t > 1,
2
|ξ | ξ(t) 2 ≤ C ξ(t)2 ξ(t) ≤ Cε3 t −3 ,
L
2
∞
2
∇ (|ξ |2 ξ )(t) 2 ≤ C ξ(t)2 ∇ 2 ξ(t) + C ξ(t) ∇ξ(t)2 ≤ Cε3 t −3 .
L
2
∞
∞
4
We conclude that
(8.4)
2
|ξ | ξ(t)
H2
≤ Cε3 hti−3 .
We also have the following estimate: For a function φ ∈ L 1 ∩ L 2 , we have
(φ, |ξ |2 ξ ) ≤ min kφk L 1 · k|ξ |2 ξ k L ∞ , kφk L 2 · k|ξ |2 ξ k L 2
(8.5)
≤ C kφk L 1 ∩L 2 ε3 hti−9/2 .
The other term λQ(2|ξ |2 + ξ 2 ) in F0 is actually larger than λ|ξ |2 ξ locally and can
be estimated similarly. We conclude
kF0 (t)k H 2 ≤ Cε2 hti−3 .
If kg(t)k H 2 ≤ Cε2 hti−2 , one can prove, for example,
|ξ + g|2 (ξ + g)(t) 2 ≤ Cε3 hti−3 , kQξ g(t)k H 2 ≤ Cε3 hti−7/2 .
H
From these estimates on F0 , the main term of F, the following bounds follow
from definitions (8.3), (2.30), and (2.32) of g, a, and θ̇, respectively,
(8.6)
g(t) . t −2 ,
a . t −2 ,
θ̇ . t −2 ,
θ . t −1 .
DYNAMICS OF NLS
203
One verifies with these orders that the main term of F is indeed F0 . Although
θ (t) = O(t −1 ), θ only appears in the form eiθ and does not change the estimates.
We now proceed to construct a solution. For convenience, we introduce a new
variable ω = θ̇ . Let A be the space
A = (ω, θ, a, g) : [0, ∞) → R × R × R × H 2 , |ω(t)| ≤ ε2−2σ hti−2 ,
|θ (t)| ≤ ε2−3σ hti−1 , |a(t)| ≤ ε2−σ hti−2 , kg(t)k H 2 ≤ ε2−σ hti−2 ,
where σ = 1/100 is small. We define a Cauchy sequence on the space A by
iterating the following map: (cf. (2.30), (2.32), and (8.3))
ω4 (t) := − [a + (c1 R, Re F)] · [1 + a(c1 R, R) + (c1 R, Re h)]−1 ,
Z t
4
ω ds ,
θ (t) :=
∞
Z t
a 4 (t) :=
(c1 Q, Im(F + ωh)) ds ,
∞
Z t
4
g (t) :=
e−i A(t−s) e−i A(t−s) eiθ U i −1 (F + ωa R) − [U, i]ωh ds ,
∞
where F = F(a R + h) and h(t) = U −1 e−iθ (e−i At U ξ∞ + g(t)). Our initial data
are
ω(t) ≡ 0, θ (t) ≡ 0 , a(t) ≡ 0 , g(t) ≡ 0 .
Given (ω, θ, a, g) ∈ A, using this assumption and (8.4) and (8.5), we have
k|g|2 gk H 2 ≤ kgk3H 2 ≤ Cε6−3σ hti−6 ,
kFk H 2 ≤ Cε2 hti−3 ,
|(φ, F)| ≤ C kφk L 1 ∩L 2 Cε2 hti−3 ,
|ω4 (t)| ≤ Cε2−σ hti−2 ≤ ε2−2σ hti−2 ,
Z t
4
ε2−2σ hsi−2 ds ≤ ε2−3σ hti−1 ,
|θ (t)| ≤
∞
Z t
|a 4 (t)| ≤
Cε2 hsi−3 ds ≤ ε2−σ hti−2 ,
∞
Z t
4
kg (t)k H 2 ≤
Cε2 hsi−3 ds ≤ ε2−σ hti−2 ,
∞
provided that ε is small enough. In the last line we have used Lemma 2.6 on the
boundedness of e−it A on X ∩ H 2 . We have also used the estimate
k[U, i]hk H 2 ≤ C kξ kW 2,∞ + C kgk H 2 .
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T.-P. TSAI AND H.-T. YAU
The estimate of ξ is similar to (2.42) in Lemma 2.9, which in some sense says [U, i]
is a local operator. Although it is not stated in Lemma 2.9, the proof of Lemma 2.9
in Section 9 gives the estimate. We have shown that (ω4 , θ 4 , a 4 , g 4 ) ∈ A, that is,
our mapping maps A into itself.
Next we show that the mapping is a contraction. Given (ω1 , θ1 , a1 , g1 ) and
(ω2 , θ2 , a2 , g2 ) ∈ A, we denote
δ0 = sup 8hti2 |δω(t)| + hti|δθ (t)| + 27 hti2 |δa(t)| + hti2 kδg(t)k H 2 ;
0≤t<∞
we know δ0 ≤ Cε2−3σ . Note that F0 is cancelled in δ F and that δ(eiθ ) = O(δθ).
We have (the norms of F, g, and h are taken in H 2 )
kδ(|g|2 g)k H 2 ≤ C kgk2 kδgk ≤ Cε4−2σ δ0 hti−6 ,
|(φ, δh)| ≤ Cφ εhti−3/2 |δθ| + Cφ kδgk ≤ Cφ δ0 hti−2 ,
kδ Fk H 2 ≤ C ka R + hk · (εhti−3/2 |δθ| + kδgk + |δa|) ≤ Cεδ0 hti−7/2 ,
5
1
|δω4 (t)| ≤ |δa| + C kδ Fk + Cεhti−3/2 |(R, δh)| ≤ δ0 hti−2 ,
4
64
Z t
1
|δθ 4 (t)| ≤
|δω|ds ≤ δ0 hti−1 ,
8
∞
Z t
4
kδ Fk + ε2−2σ hsi−2 (|δω| + |(Q, δh)|)ds ≤ 2−10 δ0 hti−2 ,
|δa (t)| ≤ C
∞
Z t
4
kδg (t)k H 2 ≤ C
ε2 hsi−3 |δθ| + kδ Fk ,
∞
1
+ ε2−2σ hsi−2 (|δω| + |δa| + δ0 hsi−2 )ds ≤ δ0 hti−2 .
8
Here we have used Lemma 2.6 and the localness of [U, i] again. Therefore we
have
sup 8hti2 |δω4 (t)| + hti|δθ 4 (t)| + 27 hti2 |δa 4 (t)| + hti2 kδg 4 (t)k H 2
0≤t<∞
1
≤ δ0 ,
2
and thus our map is a contraction. We conclude that we do have solutions e
h with
−i At
the main profile e
U ξ∞ .
Recall ψas (t) = Q E e−i Et+iθ (t) + e−i Et et L ξ∞ . By (8.2) we have
ψ(t) = [Q + a R + U −1 e−iθ (e−i At U ξ∞ + g)]e−i Et+iθ (t)
= ψas (t) + [a R + U −1 e−iθ g]e−i Et+iθ (t) + e−i Et+iθ [U −1 , e−iθ ]e−i At U ξ∞ .
DYNAMICS OF NLS
205
Since a(t), kg(t)k H 2 = O(t −2 ), and by the localness of [U −1 , i]
[U −1 , e−iθ ]e−i At U ξ∞ = − sin θ[U, i]e−i At U ξ∞ = O(t −1 ) · O(t −3/2 ) ,
we have ψ(t) − ψas (t) = O(t −2 ) in H 2 . Hence Theorem 1.5 is proved.
Finally, we address the remark after Theorem 1.5. Rewrite L = i(1 + E) + V1
where V1 = i −1 (V + 2λQ 2 + λQ 2 C) is a local operator. One may replace the
ξ (t) with
definition of ξ(t) by ξ(t) = U −1 e−iθ (t) Ue
Z t
i(1+E)t
i(1+E)s
e
(8.7)
ξ (t) = PL
χ∞ +
e(t−s)L PL
χ∞ ds ,
ce
c V1 e
∞
and proceed as in the previous proof to construct a solution h = U −1 e−iθ (Ue
ξ + g).
The assumption χ̂∞ (0) = 0 and ∇ χ̂∞ (0) = 0 ensures that ei(1+E)t χ∞ has a local
ξ (t)
decay of order O(t −7/2 ); see [5, lemma 5.2]. Hence the difference between e
and ei(1+E)t χ∞ is of order O(t −5/2 ) in H 2 . Thus globally F0 (t) = O(t −3 ) and
locally F0 (t) = O(t −5 ). Hence a, θ̇ = O(t −4 ) and θ(t) = O(t −3 ). Therefore
in H 2
ψ(t) = [Q + U −1 e−iθ (t) Ue
ξ (t)]e−i Et+iθ (t) + O(t −2 )
= Qe−i Et + ei1t χ∞ + O(t −2 ) .
Here we use again that ei1t χ∞ has a fast local decay and that [U −1 , i] is a local
operator. Note that (8.7) is in fact an expansion of et L WL χ∞ by the Duhamel’s
formula, where WL is the wave operator of L defined in (2.36). See [5, section 5]
for a similar argument.
Remark (Remark on Dispersion-Dominated Solutions to Klein-Gordon Equations).
We now sketch a construction for dispersion-dominated solutions to Klein-Gordon
equations. We follow the notation in the introduction. For a specified profile η± ,
let u = ξ + g where ξ(t) = ei Bt η+ + e−i Bt η− and g denotes the rest. Then we have
(∂t2 + B 2 )ξ = 0 ,
Hence g(t) satisfies
g(t) =
Z
t
∞
(∂t2 + B 2 )g = λ(ξ + g)3 .
ei B(t−s) − e−i B(t−s)
1
λ(ξ + g)3 ds .
2i B
Since theR main source term is bounded by kξ 3 k2 ≤ kξ k2∞ kξ k2 ≤ Ct −3 , we have
t
kgk2 ≤ ∞ Cs −3 ds ≤ Ct −2 . Then we proceed as in Section 8 to construct one
such solution by a contraction mapping argument.
9 Proof of Linear Estimates
We now proceed to prove the lemmas in Section 2.4. To simplify the presentation, we will assume λ > 0. The proof for the case λ < 0 is exactly the same.
Recall that X is the space of all functions in L 2 (R3 ) that are orthogonal to Q,
and 5 is the orthogonal projection from L 2 (R3 ) onto X . In what follows we will
only consider the restrictions of H and A on X . Hence we often omit the projection
206
T.-P. TSAI AND H.-T. YAU
5 in the definition of A. (It should be noted, however, H∗ acts on L 2 (R3 ).) In the
rest of this section, when we write L 2 , W k, p , or L r2 , we often mean their intersection
with X : L 2 ∩ X , W k, p ∩ X , or L r2 ∩ X .
We recall assumption A2 on V . We assume that 0 is neither an eigenvalue nor a
resonance for −1 + V . We also assume that V satisfies the assumption in Yajima
[20] so that the W k, p estimates for k ≤ 2 for the wave operator W H holds: For a
small σ > 0,
|∇ α V (x)| ≤ Chxi−5−σ for |α| ≤ 2 .
Also, the functions (x · ∇)k V , for k = 0, 1, 2, 3, are −1 bounded with an −1
bound less than 1:
(9.1)
k(x · ∇)k V φk2 ≤ σ0 k−1φk2 + C kφk2 ,
σ0 < 1 , k = 0, 1, 2, 3.
By the assumption, the following operators are bounded in L 2 :
(9.2)
H∗−1/2 (x · ∇)k V H∗−1/2 ,
(x · ∇)k V H∗−1 ,
H∗−1 (x · ∇)k V ,
for k = 0, 1, 2, 3.
Since Q is the ground state of H with V satisfying the previous assumptions, Q
is a smooth function with exponential decay at infinity. Hence the above statements
on V also hold for Q and Q 2 . Since V + λQ 2 and V have the same properties, in
what follows we will replace V + λQ 2 in H by V and write H = H∗ + V to make
the presentation simpler. So it should be kept in mind that the potential V in this
section is in fact V + λQ 2 .
For two operators S and T with bounded inverses, S is said to be T -bounded
if ST −1 is a bounded operator. If both S and T are self-adjoint, this implies T −1 S
is also bounded. A deeper result says S 1/2 is T 1/2 -bounded; see [11, theorem
X.18]. We say S and T are mutually bounded if both ST −1 and T S −1 are bounded
operators. This is the case if k(S − T )T −1 k(L 2 ,L 2 ) = θ < 1 for some θ. (It
implies immediately that kST −1 k < 2. Since kT φk ≤ kSφk + k(T − S)φk ≤
kSφk + θ kT φk, we have kT φk ≤ C kSφk, which implies T is S-bounded.)
L EMMA 9.1 For each k = 12 , 1, 32 , 2, 3, the operators H∗k , H k , and Ak are mutually
bounded.
P ROOF : That H∗k and H k are mutually bounded follows from our assumption
on V by standard argument. To show H k and Ak are mutually bounded, it suffices to prove the cases k = 2 and k = 3 by the previous remark. We first show
k(A2 − H 2 )H −2 k < 1, which implies the case k = 2.
2
(A − H 2 )H −2 = H 1/2 λQ 2 H 1/2 H −2 ≤ H 1/2 λQ 2 H −1 ≤ H∗1/2 λQ 2 H∗−1 ≤ 1/2 .
The last inequality can be obtained by writing
H∗1/2 Q 2 H∗−1 = H∗−1/2 Q 2 + H∗1/2 [Q 2 , H∗−1 ]
= H∗−1/2 Q 2 + H∗−1/2 [Q 2 , H∗ ]H∗−1 ,
DYNAMICS OF NLS
207
and noting [Q 2 , H∗ ] = 1Q 2 + 2∇ Q 2 · ∇.
To prove the case k = 3, it suffices to prove A6 ≤ C H 6 and H 6 ≤ C A6 . Note
that
( f A6 f ) = ( f A2 A2 A2 f ) ≤ ( f A2 H 2 A2 f ) .
Since A2 = H 2 + H 1/2 λQ 2 H 1/2 , we have
( f A2 H 2 A2 f ) ≤
C( f H 2 H 2 H 2 f ) + C( f (H 1/2 λQ 2 H 1/2 )H 2 (H 1/2 λQ 2 H 1/2 ) f )
where the cross terms are estimated by the Schwarz inequality. To show that the
last term is bounded by C( f H 6 f ), we shall show that H 3/2 Q 2 H −5/2 is bounded
in X . Rewrite
H 3/2 Q 2 H −5/2 = H 3/2 H −2 Q 2 H −1/2 + H 3/2 [Q 2 , H −2 ]H −1/2
= H −1/2 Q 2 H −1/2 + H −1/2 [Q 2 , H 2 ]H −5/2 .
P
Since [Q 2 , H 2 ] is of the form |α|≤3 G α (x)∇ α , the operators on the right side of
the equation are bounded in X . This shows A6 ≤ C H 6 . That H 6 ≤ C A6 is proven
similarly.
Recall the standard formula
Z ∞
−σ
(9.3)
T =
0
1 ds
,
s + T sσ
0 < σ < 1.
The operator T in the above formula will be A2 or H . Hence we also need to
Hm
estimate operators of the form s+H
2 . Clearly, for s ≥ 0,
−1/2 H2 H
k, p k, p ≤ C .
(9.4)
s + H 2 k, p k, p ≤ 1 ,
(W ,W )
(W ,W )
L EMMA 9.2 Let s ≥ 0. The operator H/(s + H 2 ) is bounded in W k, p ∩ X with
H −1/2
.
(9.5)
s + H 2 k, p k, p ≤ Chsi
(W ,W )
Also, for k = ±1, ±2, ±3,
k 1
−k −1
hxi
hxi
2 2 ≤ Chsi ,
s+H
L
,L
(
)
(9.6)
k H
−1/2
hxi
hxi−k .
2 2 ≤ Chsi
s + H2
( L ,L )
P ROOF : We can rewrite
1
1
H
=
√ +
√ .
s + H2
H + si
H − si
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T.-P. TSAI AND H.-T. YAU
2
Therefore, to prove statements
√ for H/(s + H ), it suffices to prove the corresponding statements for 1/(H ± si). We first prove (9.5) for k = 0. Let κ1 denote the
eigenvalue of the excited state of H , and let P1 denote the projection onto the
corresponding eigenspace. We can write
1
1
1
√ =
√ P1 + W H 2
√ W ∗ PH ,
H + si κ1 + si
p − E + si H c
X
where p = −i∇ and W H is the wave operator of H . Note E < 0. Since W H
and W H∗ are bounded
in W k, p for sufficiently nice V [20], it is sufficient√
to prove
√
2
k, p
2
that 1/( p − E ± si) are bounded in W . However, 1/( p − E ± si) are
convolution operators with explicit Green functions:
C −|x|(−E±√si)1/2
.
e
|x|
√
Since |e−|x|(−E± si) | ≤ e−c|x|hsi , the L 1 norms of the Green functions are
bounded by hsi−1/2 . By Young’s inequality we have
1
−1/2
,
p 2 − E ± √si p p ≤ Chsi
1/4
1/2
(L ,L )
which proves (9.5) for k = 0. For k ≥ 1 and for φ ∈ W k, p , we have
k/2 H
k/2 H
H
k/2 H
φ
∼ H
φ = H φ
2
2
2
s+H
s+H
s+H
p
W k, p
Lp
L
−1/2 k/2 −1/2
kφkW k, p .
≤ Chsi
H φ p ∼ hsi
L
This proves (9.5) for k ≥ 1.
For (9.6), we prove the second part. The proof for the first part is similar. For
k > 0, since
√
1
1
1
k
hxi ,
=
√
√ [hxik , H + si]
√
H + si
H + si
H + si
and
[hxik , H +
we have
√
si] = 2∇ ∗ (∇hxik ) − (1hxik ) ,
1
−k hxik ,
hxi √
H + si
k−1
1
1
∗
−k+1 ≤C
√ (∇ + 1) · hxi
√ hxi
H + si
H + si
≤ Chsi−1/2
by induction in k. We have the same estimate for [hxik , H −1√si ], and hence (9.6)
holds for positive k. The proof for the case k < 0 is similar.
DYNAMICS OF NLS
209
Recall L r2 is the weighted L 2 space with norm k f k L r2 = khxir f k L 2 .
L EMMA 9.3 The operators H 1/2 A−1/2 , A−1/2 H 1/2 , H −1/2 A1/2 , and A1/2 H −1/2 are
bounded operators in W k, p ∩ X and L r2 ∩ X .
P ROOF : By (9.3) we can write
Z ∞
1
ds
1/2 −1/2
1/2
H A
=H
2
1/2
2
1/2
s + H + H λQ H s 1/4
Z0 ∞ 1
1
= H 1/2
+
H 1/2 λQ
2
2
s
+
H
s
+
H
0
j
∞ X
H
ds
1
1/2
Q
Q
H
λQ
2
2 s 1/4
s
+
H
s
+
H
j=0
j
Z ∞
∞ X
H
H
H
−1/2 ds
=1+
λQ
Q Q
H
.
λQ
s + H2
s + H2
s + H2
s 1/4
0
j=0
H
−1/2
by Lemma 9.2, we have
Since k s+H
2 k ≤ hsi
Z
1/2 −1/2 H A
(W k, p ,W k, p ) ≤ 1 + C
∞
0
hsi−1/2 n
∞
X
ds
(n 2 hsi−1/2 ) j nhsi−1/2 1/4
s
j=0
≤ 1 + Cn 2 .
Similarly,
−1/2 1/2 A
H 2
(W k, p ,W k, p ) ≤ 1 + Cn .
Also, using (9.6), for r ≤ 3 we have
1/2 −1/2 H A
2 2 + A−1/2 H 1/2 2 2 ≤ 1 + Cn 2 .
( L ,L )
( L ,L )
r
r
r
r
The above proves that H 1/2 A−1/2 and A−1/2 H 1/2 are bounded in W k, p and L r2 .
Indeed, we have proven
(9.7)
3 1/2 −1/2
hxi (H A
− 1)hxi3 ( L 2 ,L 2 )
+ hxi3 (A−1/2 H 1/2 − 1)hxi3 ( L 2 ,L 2 ) ≤ Cn 2 .
We now consider H −1/2 A1/2 and A1/2 H −1/2 . Since
Z ∞
1
ds
1/2
2 −3/2
2
A =A A
=A
,
2
s + A s 3/4
0
210
T.-P. TSAI AND H.-T. YAU
we have
H
−1/2
A
1/2
Z
∞
1
ds
λQ H )
2 + H 1/2 λQ 2 H 1/2 s 3/4
s
+
H
0
Z ∞
1
ds
= (H 3/2 + λQ 2 H 1/2 )
2
1/2
2
1/2
s + H + H λQ H s 3/4
0
=H
−1/2
(H + H
2
1/2
2
1/2
= I1 + λQ 2 I2 .
The main term is I1 . The term I2 is similar to H 1/2 A−1/2 , and its integrand has a
better decay in s for large s. Hence
2 λQ I2 ≤ Cλ kI2 k ≤ Cn 2 .
For the main term I1 ,
Z ∞
I1 = 1 +
0
j
∞ X
H
H2
ds
H
Q
Q
Q Q
H −1/2 3/4 .
2
2
2
s+H
s+H
s+H
s
j=0
Hence
Z
∞
kI1 k ≤ 1 + C
n
Z
0
∞
≤1+C
0
∞
X
ds
(n 2 hsi−1/2 ) j nhsi−1/2 3/4
s
j=0
n 2 hsi−1/2
ds
≤ 1 + Cn 2 .
s 3/4
Here the norms are taken in (W k, p , W k, p ) and (L r2 , L r2 ). Hence we have proven
Lemma 9.3.
In fact, the last part of the above proof also shows
(9.8) hxi3 (H −1/2 A1/2 − 1)hxi3 ( L 2 ,L 2 )
+ hxi3 (A1/2 H −1/2 − 1)hxi3 ( L 2 ,L 2 ) ≤ Cn 2 .
P ROOF OF L EMMA 2.9: The above lemma proves that U0 and U0−1 are bounded
in W k, p and L r2 . Moreover, (9.7) and (9.8) mean that U0 −1 and U0−1 −1 are “local”
operators. In particular, they imply k[U0 , i] φk L 8/7 ∩L 4/3 ≤ C kφk L 4 . Since
P1 0
e
e
U = U 5 where 5 =
0 5
is a bounded projection, U is also bounded. Similarly U −1 is also bounded. Moreover, we have
0
1
e , i] + [U0 , i]5
e = U0 (P1 − 5)
e,
[U, i] = U0 [5
+ [U0 , i]5
1 0
where by (2.18) P1 − 5 = −c1 |5Ri hQ| is a local operator; hence [U, i] satisfies
the last estimate in Lemma 2.9. Lemma 2.9 is thus proven.
DYNAMICS OF NLS
211
P ROOF OF L EMMA 2.10: We need only to prove the statement on W A . Note
that the estimates (9.7) and (9.8) also show, for any φ ∈ L 2 ,
(U ±1 − 1)e−it H∗ φ → 0
(9.9)
in L 2
as t → ∞ .
Notice
W A = lim eit A e−it H∗ = lim U et LU −1 e−it H∗
t→∞
t→∞
= lim U et L e−it H∗ + lim U et L (U −1 − 1)e−it H∗ .
t→∞
By (9.9) we have
t→∞
W A = lim U et L e−it H∗ = U WL .
t→∞
The boundedness of W A follows from that of U and WL . This proves Lemma 2.10.
P ROOF OF L EMMA 2.7: Since
e−it A PcA φ = W A e−it H∗ W A∗ PcA φ ,
the estimate (2.39) follows from the usual (L p , L q ) estimate for e−it H∗ and the
boundedness of W A and PcA in L p spaces. To prove (2.40), either we prove the
boundedness of W A in weighted spaces L r2 , or we use the Mourre estimate. We
will follow the second approach and the argument in [16].
Let a = 2κ. We consider intervals 1 = (a − r, a + r ). Let g1 (t) = g0 ((t −
a)/r ), where g0 is a fixed smooth function with support in (−2, 2) and g0 (t) = 1
for |t| < 1. We will consider g1 (A) with r small enough. Let D = x p + px,
p = −i∇, and the commutators
ad0D (A) = A ,
k
adk+1
D (A) = [ad D (A), D] .
We need to prove the following lemma:
L EMMA 9.4 For 1 small enough, the Mourre estimate
g1 (A)[i A, D]g1 (A) ≥ θg1 (A)2
holds for some θ > 0. Also, g1 (A) adkD (A)g1 (A) are bounded operators in L 2 for
k = 0, 1, 2, 3.
We will use the following lemma:
L EMMA 9.5 The operators
H −3 D k H m/2 hxi−3
and
hxi−3 H m/2 D k H −3
are bounded in L 2 for k, m = 0, 1, 2, 3.
P ROOF : This is standard and we only sketch the proof. If m is even, we can
compute the commutator [D k , H m/2 ] explicitly and estimate
H −3 D k H m/2 hxi−3 = H −3 H m/2 D k hxi−3 + H −3 [D k , H m/2 ]hxi−3 .
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T.-P. TSAI AND H.-T. YAU
If m is odd, we write
H
−3
k
D H
m/2
hxi
−3
Z
=
∞
H −3 D k H (m+1)/2
0
1
ds
hxi−3 √
s+H
s
and proceed as in the case when m is even, by using (9.6). Here we have used
formula (9.3).
P ROOF OF L EMMA 9.4: Let G = A − H and write A = H + G. Since
[i A, D] = [H∗ + V + G, i D] = −1 + [V + G, i D]
= A − V − G + [V + G, i D]
and g1 (A)Ag1 (A) ≥ 2θg1 (A)2 for some 2θ > 0, it suffices to show that, for
M = −V + [V, i D], −G, and [G, D], the operators
g1 (A)Mg1 (A) = (g1 (A)H∗2 )(H∗−2 M H∗−2 ) (H∗2 g1 (A))
are bounded by g1 (A)2 , and the bound goes to zero when the interval 1 shrinks to
zero. Since both
g1 (A)H∗2 = (g1 (A)A2 )(A−2 H∗2 )
and
H∗2 g1 (A) = (H∗2 A−2 )(A2 g1 (A))
are bounded and converge to zero weakly when 1 shrinks to zero, this will be true
if one can show that H∗−2 M H∗−2 is compact. The case M = −V + [V, i D] is
standard and follows from our assumption, so we only consider H∗−2 G H∗−2 and
H∗−2 [G, D]H∗−2 .
We proceed to find an explicit form of G. By (9.3) with T = A2 , σ = 12 , we
write
Z ∞
1
ds
−1
A =
√
2
1/2
2
1/2
s + H + H λQ H
s
0
j
Z ∞
∞ X
1
H
1
1
ds
1/2
+
H λQ
Q Q H 1/2
=
λQ
√
2
2
2
2
s+H
s+H
s+H
s+H
s
0
j=0
= H −1 + H −1/2 hxi−3 J0 hxi−3 H −1/2
where
Z
J0 =
∞
0
j
∞ X
H
H
H
ds
hxi
λQ
Q Q
hxi3 √ .
λQ
2
2
s + H2
s
+
H
s
+
H
s
j=0
3
By Lemma 9.2,
Z
kJ0 k(L 2 ,L 2 ) ≤
0
∞
hsi−1/2 · n 2 · hsi−1/2 s −1/2 ds ≤ Cn 2 .
DYNAMICS OF NLS
213
Hence
A = A2 A−1
(9.10)
= (H 2 + H 1/2 λQ 2 H 1/2 ) H −1 + H −1/2 hxi−3 J0 hxi−3 H −1/2
= H +G,
G = H 1/2 λQ 2 H −1/2 + H 3/2 hxi−3 J0 hxi−3 H −1/2
+ H 1/2 λQ 2 hxi−3 J0 hxi−3 H −1/2 .
Since H −1/2 hxi−1 and hxi−1 H −1/2 are compact, from (9.10) H∗−2 G H∗−2 is compact. We can also write
H∗−2 G D H∗−2 = H∗−2 G H 1/2 hxi · hxi−1 H −1/2 D H∗−2 .
The second operator is bounded by Lemma 9.5. The first is compact since its terms
are of the form H −m · hxi−k · (bounded operator). Similarly, H∗−2 DG H∗−2 is also
compact. Hence we conclude the Mourre estimate.
To show that g1 (A) adkD (A)g1 (A) are bounded for k = 0, 1, 2, 3, we rewrite
g1 (A) adkD (A)g1 (A) =
(g1 (A)A3 )(A−3 H 3 )(H −3 adkD (A)H −3 )(H 3 A−3 )(A3 g1 (A)) .
We need only to show that H −3 adkD (A)H −3 are bounded since the other terms
are bounded by Lemma 9.1. Recall A = H + G. It is standard to prove that
H −3 adkD (H )H −3 is bounded. For H −3 adkD (G)H −3 , since it is a sum of terms of
the form
H −3 D k G D m H −3 , k + m ≤ 3 ,
it suffices to show that these terms are bounded. By the explicit form (9.10) of G
and Lemma 9.5, they are indeed bounded. For example,
H −3 D 2 H 3/2 hxi−3 J0 hxi−3 H −1/2 D 1 H −3 =
−3 2 3/2 −3 −3 −1/2 1 −3 H D H hxi
D H
J0 hxi H
,
a product of three bounded operators. We conclude that g1 (A) adkD (A)g1 (A) are
bounded for k = 0, 1, 2, 3.
With Lemma 9.4 (cf. the remark in [16, p. 27]), the minimal velocity estimate
in [6] and theorem 2.4 of [14] implies
F(D ≤ θt/2)e−i At g1 (A) hDi−3/2 2 2 ≤ Chti−5/4 .
(L ,L )
The same argument in [16] then gives the desired decay estimate (2.40).
P ROOF OF L EMMA 2.8: Let ψn = PcA 5φ0 φ12 and ψ0 = PcH1 φ0 φ12 . Recall H1 =
−1 + V − e0 . We have ψn = ψ0 + O(n 2 ). We write ψn = ψ0 + bφ1 + η, where
214
T.-P. TSAI AND H.-T. YAU
η ∈ Hc (H1 ) and b, η = O(n 2 ). Rewrite
1
ψn
ψn , Im
A − 0i − 2κ
Z ∞
= Im i
ψn , e−it (A−0i−2κ) ψn dt
0
(9.11)
Z
= Im i
∞
ψn , e−it (H1 −0i−2κ) ψn dt
0
(9.12)
Z
+ Im i
0
∞
Z
t
ψn , e−i(t−s)(A−0i−2κ) (λQ 2 + G)e−is(H1 −0i−2κ) ψn ds dt .
0
The main term lies in (9.11). It is
Z ∞
−it (H1 −0i−2κ)
Im i
ψ0 dt = ψ0 , Im
ψ0 , e
0
1
ψ0
H1 − 0i − 2κ
,
which is the desired main term in Lemma 2.8.
We want to show that the rest of (9.11) and (9.12) are integrable and of order
O(n 2 ). Recall that we write ψn = ψ0 + bφ1 + η. For the term η in ψn , by the decay
estimate we have
(9.13) ψn , e−it (H1 −0i−2κ) η ≤ Chti−3/2 kψn k L 1 ∩L 2 kηk L 1 ∩L 2 ≤ Chti−3/2 n 2 ;
hence this term is integrable. Also, since H1 φ1 = e01 φ1 ,
ψn , e−it (H1 −0i−2κ) bφ1 = ψn , e−it (e01 −2κ−0i) bφ1 ,
so we can integrate this oscillation term explicitly. (The boundary term at t =
∞ vanishes due to the decay of e−it (−0i) .) We conclude that the rest of (9.11) is
integrable and of order O(n 2 ).
For (9.12), it suffices to show its integrability since λQ 2 + G gives the order
O(n 2 ). Rewrite the last ψn in (9.12) as bφ1 + PcH1 ψn . For the part containing bφ1 ,
we have
ψn , e−i(t−s)(A−0i−2κ) (λQ 2 + G)e−is(H1 −0i−2κ) bφ1 =
ψn , e−it (A−0i−2κ) eis(A−e01 ) (λQ 2 + G)bφ1 .
Integration in s gives
(A − e01 )−1 ψn , e−it (A−0i−2κ) (λQ 2 + G)bφ1 .
Since e01 lies outside the continuous spectrum of A, the last expression is integrable
in t following the same argument as (9.13). For the part containing PcH1 ψn , since
(λQ 2 + G) is a “local” operator in the sense that it sends L ∞ functions to L 1 , we
DYNAMICS OF NLS
215
have
ψn , e−i(t−s)(A−0i−2κ) (λQ 2 + G)e−is(H1 −0i−2κ) P H1 ψn ≤
c
Cn 2 ht − si−3/2 hsi−3/2 kψn k2L 1 ∩L 2 ,
which can be integrated in s and t. Hence we have proven Lemma 2.8.
P ROOF OF L EMMA 2.6: Given φ ∈ H k ∩ X , k = 0, 1, 2, let u(t) = e−it A φ ∈
X . We have dtd (u(t), Am u(t)) = 0; hence (u(t), Am u(t)) is a conserved quantity.
When k = 0, this implies that the L 2 norm is conserved. For k = 2, we have
(u, A2 u) = u, (H 2 + H 1/2 λQ 2 H 1/2 )u
= (u, H 2 u) + O(n 2 kuk2H 1 ) ≤ C kuk2H 2 .
On the other hand, since kuk2H 1 ≤ C kuk2H 2 + C kuk2L 2 and (u, u) ≤ C(u, A2 u)
due to the spectral gap of A, we have
C −1 kuk2H 2 ≤ (u, H 2 u) = (1 − O(n 2 ))−1 (u, A2 u) + O(n 2 ) kuk2L 2
≤ C(u, A2 u) .
Thus we have (u(t), A2 u(t)) ≈ ku(t)k2H 2 in the sense of (1.23), and we have
ku(t)k2H 2 ≈ (u(t), A2 u(t)) = (u(0), A2 u(0)) ≈ ku(0)k2H 2 .
The case k = 2 is thus proven. The case k = 1 can be obtained by interpolation.
Acknowledgments. It is with great pleasure that we thank M. I. Weinstein
for explaining the beautiful ideas behind his work with A. Soffer [16]. We also
thank S. Cuccagna for his very helpful comments and, in particular, for pointing
out a gap in the proof of Theorem 1.5 in the manuscript. The work of H.-T. Yau
was partially supported by NSF Grant DMS-0072098. Part of this work was done
when both authors were visiting the Center for Theoretical Sciences, Taiwan, in
summer 2000.
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TAI -P ENG T SAI
Institute for Advanced Study
Einstein Drive
Princeton, NJ 08540
E-mail: ttsai@ias.edu
Received November 2000.
Revised July 2001.
H ORNG -T ZER YAU
Courant Institute
251 Mercer Street
New York, NY 10012
E-mail: yau@cims.nyu.edu