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PAIR EXCITATIONS AND THE MEAN FIELD APPROXIMATION OF INTERACTING BOSONS, II M. GRILLAKIS AND M. MACHEDON
Abstract. We consider a large number of Bosons with interaction potential vN (x) = N 3β v(N β x). In our earlier papers [19]-[21] we considered a set of equations for the condensate φ and pair excitation function k and proved that they provide a Fock space approximation to the exact evolution of a coherent state for β < 31 . In [22] we introduced a coupled refinement of our original equations and conjectured that they provide a Fock space approximation in the range 0 ≤ β ≤ 1. In the current paper we prove that this is indeed the case for β < 23 , at least locally in time. In order to do that, we re-formulate the equations of [22] in a way reminiscent of BBGKY and apply harmonic analysis techniques in the spirit of those used by X. Chen and J. Holmer in [9] to prove the necessary estimates. In turn, these estimates provide bounds for the pair excitation function k. While our earlier papers provide background material, the methods of this paper paper are mostly new, and the presentation is self-contained.
1. Introduction The problem considered in this paper (as well as our earlier papers [19]- [22]) is the N -body linear Schr¨odinger equation ! N X 1∂ 1 X − ∆x + vN (xi − xj ) ψN (t, ·) = 0 (1) i ∂t j=1 j N i<j ψN (0, x1 , · · · , xN ) = φ0 (x1 )φ0 (x2 ) · · · φ0 (xN ) kψN (t, ·)kL2 (R3N ) = 1 where vN (x) := N 3β v(N β x) with 0 ≤ β ≤ 1, v ∈ S and v ≥ 0. The goal is to find a rigorous, simple approximation (in a suitable norm) to ψN which is consistent with ψapprox (t, x1 , · · · , xN ) ∼ eiχ(t) φ(t, x1 )φ(t, x2 ) · · · φ(t, xN )
(2)
as N → ∞, where φ (which represents the Bose-Einstein condensate) satisfies a non-linear Schr¨odinger equation. The problem becomes more difficult, interesting, and requires new ideas as β approaches 1, and this 1
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M. GRILLAKIS AND M. MACHEDON
explains why several authors, (including us) devoted several papers to their programs. We refer to [36] for extensive background on (static) Bose-Einstein condensation. During recent years, in a series of papers by Erd¨os and Yau [12], and Erd¨os, Schlein and Yau [13] to [15], it was proved γ1N (t, x, x0 ) → φ(t, x)φ(t, x0 )
(3)
in trace norm as N → ∞, and similarly for the higher order marginal density matrices γkN , where k is fixed. Recent simplifications and generalizations were given in [29], [27], [5], [8], [9], [10]. See also [17], [30] for a different approach. We also mention the approach based on the quantum de Finetti theorem in [6], as well as results in the case of negative interaction potentials in [11] and the different approach of Knowles and Pickl [30]. Another approach to this problem is based on Fock space techniques and the second quantization. In physics, it was pioneered in the papers by Bogoliubov [4], Lee, Huang and Yang [33] in the static case, and Wu [43] in the time dependent case. See also the more recent papers [37], [38]. In the rigorous mathematical literature devoted to the time evolution problem, it originates in the of work Hepp [23], Ginibre and Velo [18] and, after lying dormant for about 30 years, Rodnianski and Schlein [40], followed by [19]. Currently, it is an active field. Our project, initiated in collaboration with Margetis in[19], is to study a PDE describing additional second order corrections (given by a Bogoliubov transformation eB ) to the right hand side of the approximation (2). Mathematically, Boguliubov transformations are representations of a group isomorphic to a real symplectic group, corresponding to the Segal-Shale-Weil representation in infinite dimensions, due to Shale. Interestingly, the theories seem to have evolved independently in physics and pure mathematics. Several important recent papers also use coherent states and Bogoliubov transformations. These include [2] and [1]. In fact, Theorem 2.2 in [1] proves that the unitary operators of the type used in [23] and [18] can be obtained, abstractly, as Bogoliubov transformations. We mention in passing that our concrete1 in eB (to be discussed below) agrees with that Bogoliubov transformation (up to a phase), but only when applied to the vacuum. Roughly speaking, our eB is not the 1We
mean that B = B(k) where k satisfies a PDE in 6+1 dimensions and is, in principle, computable numerically .
PAIR EXCITATIONS, II
3
operator corresponding to the evolution of the quadratic Bogoliubov Hamiltonian, but rather diagonalizes it. In the current paper we initiate the analysis of solutions to a coupled system of PDEs for the condensate φ and pair excitation k, see(22), (23a)-(23b) below. It would be very interesting to obtain estimates including the case β = 1, as well as large time estimates for the solutions of these equations which are uniform in N , similar to those obtained in earlier works for the uncoupled equations ((14a)-(14c) below) which describe the case β < 1/3. We hope to address these question in future work. In this paper we only consider the case 1/3 < β < 2/3, locally in time. See also the paper by Lewin, Nam, and Schlein [34] for a Fock space type approach to an L2 (R3N ) estimate based on corrections to a pure tensor product (Hartree state) rather than a coherent state. That approach has been generalized very recently to the case β < 1/3 my Nam and Napiorkowski in [39], and their work uses the linear equations (14b), (14c) introduced in [21], as well as some of the estimates from that paper. Very recently, after completing this work, we have also learned of the related and important paper [3]. A brief comparison of the results of that paper with ours is included at the end next section.
2. Background and statement of the main result We start with a very brief review of symmetric Fock space. This is included for the convenience of the reader, and follows closely the exposition from our earlier papers. We refer the reader to [22] for more details and comments. The elements of F are vectors of the form ψ = ψ0 , ψ1 (x1 ) , ψ2 (x1 , x2 ) , . . .
where ψ0 ∈ C and ψk are symmetric L2 functions. The inner product is ∞ Z X
φ, ψ = φ0 ψ0 + φn ψn . n=1
Thus we use physicists’ convention of an inner product linear in the second variable. The creation and annihilation distribution valued operators denoted by a∗x and ax respectively which act on vectors of the
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M. GRILLAKIS AND M. MACHEDON
form (0, · · · , ψn−1 , 0, · · · ) and (0, · · · , ψn+1 , 0, · · · ) by n
1 X := √ δ(x − xj )ψn−1 (x1 , . . . , xj−1 , xj+1 , . . . , xn ) n j=1 √ ax (ψn+1 ) := n + 1ψn+1 ([x], x1 , . . . , xn )
a∗x (ψn−1 )
with [x] indicating that the variable x is frozen. The vacuum state is defined as follows: Ω := (1, 0, 0 . . .) One can easily check the canonical relations ax , a∗y = δ(x − y) and since the creation and annihilation operators are distribution valued we can form operators that act on F by introducing a field, say φ(x), and form Z Z ∗ ¯ ¯ a(φ) := dx φ(x)ax and a (φ) := dx {φ(x)a∗x } where by convention we associate a with φ¯ and a∗ with φ. Also define the skew-Hermitian operator Z ∗ ¯ A(φ) := dx φ(x)a (4) x − φ(x)ax and coherent states √
ψ := e−
N A(φ)
Ω.
(5)
It is easy to check that e
√ − N A(φ)
Ω=
. . . cn
n Y
! φ(xj ) . . .
2
with cn = e−N kφkL2 N n /n!
1/2
j=1
We also consider 1 B(k) := 2
Z
¯ x, y)ax ay − k(t, x, y)a∗ a∗ . dxdy k(t, x y
(6)
This particular construction and the corresponding unitary opera√ − N A −B tor M := e e were introduced (at least in the mathematics literature related to the problem under consideration) in [19]. The construction is in the spirit of Bogoliubov theory in physics, and the Segal-Shale-Weil representation in mathematics.
.
PAIR EXCITATIONS, II
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The Fock Hamiltonian is 1 H := H1 − V where, (7a) N Z (7b) H1 := dxdy {∆x δ(x − y)a∗x ay } and Z 1 V := dxdy vN (x − y)a∗x a∗y ax ax , (7c) 2 where vN (x) = N 3β v N β x . It is a diagonal operator on Fock space, and it acts as a differential operator in n variable Hn, P DE =
n X
∆xj −
j=1
1 X 3β N v N β (xj − xk ) N i<j
on the nth component of F. Notice this is the same as (1), except that the dimension n is decoupled from the parameter N . Our goal is to study the evolution of coherent initial conditions of the form ψexact = eitH e−
√
N A(φ0 )
Ω
(8)
In our earlier papers[19, 20, 21, 22] we considered an approximation of the form √ ψappr := e− N A(φ(t) e−B(k(t)) Ω (9) and derived suitable Schr¨odinger type equations equations for φ(t, x), k(t, x, y) so that ψexact (t) ≈ eiN χ(t) ψappr (t), with χ(t) a real phase factor, in order to find precise estimates in Fock space, see Theorem (2.1) below. Our strategy is to consider ψred = eB(t) e
√
√ N A(t) itH − N A(0)
e
e
Ω
and then compute a ”reduced Hamiltonian” Hred =
1 ∂t M∗ M + M∗ HM i
(10)
so that 1 ∂t ψred = Hred ψred . i To state the results of [19, 20, 21] we define the operator kernel
(11)
gN (t, x, y) := −∆x δ(x − y) + (vN ∗ |φ|2 )(t, x)δ(x − y) + vN (x − y)φ(t, x)φ(t, y)
(12)
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M. GRILLAKIS AND M. MACHEDON
and 1 T Sold (s) := st + gN ◦ s + s ◦ gN and i 1 T , p] Wold (p) := pt + [gN i mN (x, y) := −vN (x − y)φ(x)φ(y) , vN (x) = N 3β v(N β x)
(13)
The main result of [21] can be summarized as follows: Theorem 2.1. Let φ and k satisfy 1 ∂t φ − ∆φ + vN ∗ |φ|2 φ = 0 i Sold (sh(2k)) = mN ◦ ch(2k) + ch(2k) ◦ mN Wold ch(2k) = mN ◦ sh(2k) − sh(2k) ◦ mN .
(14a) (14b) (14c)
with prescribed initial conditions φ(0, ·) = φ0 , k(0, ·, ·) = 0. If φ, k satisfy the above equations, then there exists a real phase function χ such that 4
ψexact (t) − eiN χ(t) ψappr (t) ≤ C(1 + t) log (1 + t) . (15) F N (1−3β)/2 provided 0 < β < 31 . See [21] for the reasons behind these equations. We also mention a very recent simple derivation of these equations in [39]. This result was extended to the case β < 12 in [32], where it was also argued informally that the equations of [21] do not provide an approximation for β > 12 . In the hope of obtaining an approximation for all β ≤ 1, in [22] we introduced a coupled refinement of the system (14a), (14b), (14c). The coupled equations of [22] were introduced the following way: Since Hred is a fourth order polynomial in a and a∗ , Hred Ω = (X0 , X1 , X2 , X3 , X4 , 0, · · · ).
(16)
The new, coupled equations for φ and k that we introduce in [22] can be written abstractly as X1 = 0 and X2 = 0.
(17)
It was shown there that they are Euler-Lagrange equations for the Lagrangian density X0 . Remark 2.2. The static terms of X0 (t) (not involving time derivatives) also appear in the recent paper [2], but do not serve as a Lagrangian there.
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To write down the equations (17) explicitly in terms of φ and k, we introduce Definition 2.3. Define 1 Λ(t, x1 , x2 ) = sh(2k)(t, x1 , x2 ) + φ(t, x1 )φ(t, x2 ) 2N 1 Γ(t, x1 , x2 ) = sh(k) ◦ sh(k) (t, x1 , x2 ) + φ(t, x1 )φ(t, x2 ) N and the new operator kernel
(18) (19)
g˜N (t, x, y) := −∆x δ(x − y) + vN (x − y)(trΓ)(t, x)δ(x − y) + vN (x − y)Γ(t, x, y) where tr denotes trace density, and define 1 1 T T ˜ ˜ ◦ s + s ◦ g˜N and W(p) := pt + [˜ gN , p] S(s) := st + g˜N i i In this notation, the following is proved in [22] Theorem 2.4. The equation X1 = 0 is equivalent to Z 1 ∂t φ(t, x) − ∆φ + vN (x − y)Λ(t, x, y)φ(t, y)dy i 1 + (vN ∗ T r(sh(k) ◦ sh(k)))(t, x)φ(t, x) NZ 1 + vN (x − y)(sh(k) ◦ sh(k))(t, x, y)φ(t, y)dy = 0 N
(20)
(21)
(22)
Here T r(sh(k) ◦ sh(k))(t, x) = (sh(k) ◦ sh(k))(t, x, x) denotes the trace density. The equation X2 = 0 is equivalent to ˜ (sh(2k)) + (vΛ) ◦ ch(2k) + ch(2k) ◦ (vΛ) = 0 S (23a) ˜ ch(2k) + (vΛ) ◦ sh(2k) − sh(2k) ◦ (vΛ) = 0 W (23b) Notice the similarity with (14a), (14b), (14c). Since it is difficult to prove estimates for these equations directly, we will write them down in a different, equivalent form. The derivation will be self-contained, and in fact most of the rest of this paper is independent of our previous work [19]-[22]. We now state the main result of our current paper. Theorem 2.5. Let 13 < β < 32 , and let the interaction potential v ∈ S satisfy v ≥ 0 and |ˆ v | ≤ wˆ for some w ∈ S. Let φ, k be solutions to (22), (23a), (23b) with φ(0, ·) ∈ S, k(0, ·) = 0. Then there exists a real
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M. GRILLAKIS AND M. MACHEDON
function χ(t) = χN (t) such that for every > 0 there exists T0 > 0 and C such that √
kψexact − ψappr kF := keitH e− C ≤ 3β N 1− 2 −
N A(φ0 )
Ω − eiχ(t) e−
√
N A(φ(t)) −B(k(t))
e
ΩkF
uniformly for 0 ≤ t ≤ T0 . Remark 2.6. In the very recent and important paper [3], Bocatto, Cenatiempo and Schlein prove a result closely related to ours in the full range β < 1, and the estimate is global in time. However, there are substantial differences between their work and ours. Both papers approximate eitH ψinitial for some initial vector ψinitial . Our ψinitial is a √ − N A(φ0 ) coherent state e Ω, and the Bogoliubov term e−B(k) develops dynamically, while in [3] ψinitial must have e−B(k0 ) built into the initial conditions, where k0 is chosen to have a crucial explicit form. We√ mention that our approach could also accommodate ψinitial = e− N A(φ0 ) e−B(k0 ) Ω for k0 satisfying suitable estimates (see remark 3 in [19]), but we chose k0 = 0 for simplicity, since the proof would be the same. Also, the techniques used in [3] are quite different than ours, and the√ approximation in [3] is given (translating to our notation) by eiχ(t) e− N A(φ(t)) e−B(k(t)) U2,N (t)Ω where k(t) = k(t, x, y) is explicit (and related but different from our k(t)) and U2,N (t) is an evolution in Fock space with a quadratic generator (see the page preceding Theorem 1.1 in [3]). Given the complexity the evolution equation defining of U2,N (t), we believe there is still sufficient interest in having an approximation √ iχ(t) − N A(φ(t)) −B(k(t)) given by just e e e Ω where k satisfies a classical PDE in 6 + 1 variables.
3. The equations for Λ and Γ (self-contained derivation) As already mentioned, it seems difficult to obtain estimates (uniformly in N ) for φ and k directly from equations (23a), (23b). The equations seem linear, but the ”coefficients” vN Λ, vN Γ depend on sh(2k), ch(2k). We will proceed indirectly by deriving and studying equations for Λ and Γ.
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Theorem 3.1. The equations of Theorem (2.4) are equivalent to Z 1 (24a) ∂t − ∆x1 φ(x1 ) = − dy {v(x1 − y)Γ(y, y)} φ(x1 ) i Z − dy v(x1 − y)φ(y) Γ(y, x1 ) − φ(y)φ(x1 ) + v(x1 − y)φ(y) Λ(x1 , y) − φ(x1 )φ(y) 1 1 ∂t − ∆x1 − ∆x2 + v(x1 − x2 ) Λ(x1 , x2 ) (24b) i N Z = − dy {v(x1 − y)Γ(y, y) + v(x2 − y)Γ(y, y)} Λ(x1 , x2 ) Z n o − dy v(x1 − y) + v(x2 − y) Λ(x1 , y)Γ(y, x2 ) + Γ(x1 , y)Λ(y, x2 ) + Z + 2 dy v(x1 − y) + v(x2 − y) |φ(y)|2 φ(x1 )φ(x2 ) 1 ∂t − ∆x1 + ∆x2 Γ(x1 , x2 ) (24c) i Z = − dy v(x1 − y) − v(x2 − y) Λ(x1 , y)Λ(y, x2 ) + Z o n − dy v(x1 − y) − v(x2 − y) Γ(x1 , y)Γ(y, x2 ) + Γ(y, y)Γ(x1 , x2 ) Z + 2 dy v(x1 − y) − v(x2 − y) |φ(y)|2 φ(x1 )φ(x2 ) See (31a)-(31c) for the conceptual meaning of these equations in terms of the density matrices L defined below. While it is easy to prove this by direct calculation, we proceed with a derivation which is independent of Theorem (2.4) of our previous paper [22]. As in our previous papers [19]-[22], we consider M := e−
√
N A −B
e
and we have the evolution
1 ∂t M = HM − MHred and of course, i 1 − ∂t M∗ = M∗ H − Hred M∗ . i The evolution equation for M above is obvious from (10). Take a monomial of the form: (Wick ordered) Pm,n = a∗y1 a∗y2 . . . a∗ym ax1 ax2 . . . axn
(25) (26) (27)
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M. GRILLAKIS AND M. MACHEDON
and define the L matrices as follows,
1 Lm,n (t, y1 , . . . , ym ; x1 , . . . , xn ) := (m+n)/2 ay1 · · · aym MΩ, ax1 · · · axn MΩ N
1 = (m+n)/2 Ω, M∗ Pm,n MΩ N The notation is chosen so that the second set of variables are un-barred. We will often skip the t dependence, since it is passive in the calculations that we have in mind. Fortunately we will only need L0,1 , L1,1 and L0,2 (which turn out to be φ, Γ and Λ) but the computation is quite general. To get started, we observe that from the evolution of the operator M we have, 1 ∗ hence ∂t M PM = Hred , M∗ PM + M∗ P, H M i
1 1 ∂t L = (n+m)/2 Ω, Hred , M∗ PM Ω + Ω, M∗ P, H MΩ . i N (28) At this point we record the following lemma Lemma 3.2. If Hred Ω = µ, 0, 0, X3 , X4 , 0 . . . and P := a
or
P := aa
P := a∗ a
or
then
Ω, Hred , M∗ PM Ω = 0,
(29)
leaving only the second term in (28) Proof. We use the following notation: c = ch(k), Z ax (c) := Z ∗ ax (c) := Z ax (u) :=
u = sh(k) a∗x (u)
dy {ay c(y, x)} , dy a∗y c(y, x) =
Z
Z :=
dy a∗y u(y, x)
dy c(x, y)a∗y
by symmetry
dy {u(x, y)ay }
by symmetry
Z dy {ay u(y, x)} =
We have the conjugation formulas (see also (76)) √ √ M∗ ax M = ax (c) + a∗x (u) + N φ(x) := bx + N φ(x) √ √ M∗ a∗x M = a∗x (c) + ax (u) + N φ(x) := b∗x + N φ(x)
(30a) (30b)
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which implies the transformation of the monomial, √ √ M∗ P(a∗ , a)M = P b∗ + N φ , b + N φ . Now if P = a then, using (17), √ √
Ω, Hred bx + N φ(x) Ω − Ω, bx + N φ(x) Hred Ω = 0 . The argument is similar if P = aa or P = a∗ a since only the entries in the zeroth slot survive. Based on this, we easily prove the following proposition. Proposition 3.3. Under the assumptions of Lemma (3.2), the following equations hold 1∂ − ∆x1 L0,1 (t, x1 ) (31a) i ∂t Z = − vN (x1 − x2 )L1,2 (t, x2 ; x1 , x2 )dx2 1∂ + ∆x1 − ∆y1 L1,1 (t, x1 ; y1 ) (31b) i ∂t Z Z = vN (x1 − x2 )L2,2 (t, x1 , x2 ; y1 , x2 )dx2 − vN (y1 − y2 )L2,2 (t, x1 , y2 ; y1 , y2 )dy2 1∂ 1 − ∆x1 − ∆x2 + vN (x1 − x2 ) L0,2 (t, x1 , x2 ) (31c) i ∂t N Z Z = − vN (x1 − y)L1,3 (t, y; x1 , x2 , y)dy − vN (x2 − y)L1,3 (t, y; x1 , x2 , y)dy The equation (31b) is one of the BBGKY equations, but in our case L2,2 can be expressed in terms of the earlier matrices, see Lemma (3.4). Proof. With P any monomial of degree one or two we know from Lemma (3.2) that
Ω, Hred , M∗ PM Ω = 0 and for any of the corresponding matrices L we arrive at the equation, 1
1 , α = 1/2 , 1 . ∂t L = α Ω, M∗ P, H MΩ i N We need to compute [P, H] and for this purpose recall our original Hamiltonian, Z H=
dxdy {∆x δ(x −
y)a∗x ay }
1 − 2N
Z
dxdy vN (x − y)a∗x a∗y ax ax
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M. GRILLAKIS AND M. MACHEDON
Below we list the commutators with each mononomial: Z 1 ax1 , H = ∆x1 ax1 − dy v(x1 − y)a∗y ay ax1 N 1 ax1 ax2 , H = ∆x1 ax1 ax2 + ∆x2 ax1 ax2 − v(x1 − x2 )ax1 ax2 N Z 1 − dz v(x1 − z) + v(x2 − z) a∗z az ax1 ax2 N ∗ ax1 ax2 , H = a∗x1 ∆x2 ax2 − (∆x1 ax1 )∗ ax2 Z 1 + dz v(x1 − z) − v(x2 − z) a∗x1 a∗z az ax2 N from which we can derive the corresponding evolution equations for the L matrices:
Z
1 1 ∂t − ∆x1 L0,1 (t, x1 ) = − 3/2 dyv(x1 − y) Ω, M∗ a∗y ay ax1 MΩ i N (32)
1 1 ∂t − ∆x1 − ∆x2 + v(x1 − x2 ) L0,2 (t, x1 , x2 ) = i N Z
1 − 2 dz v(x1 − z) + v(x2 − z) Ω, M∗ a∗z az ax1 ax2 MΩ (33) N
and finally 1 ∂t + ∆x1 − ∆x2 L1,1 (t, x1 , x2 ) = i Z
1 + 2 dz v(x1 − z) − v(x2 − z) Ω, M∗ a∗x1 a∗z az ax2 MΩ (34) N
which implies the statement of the proposition.
The proof of Theorem (3.1) is finished by computing the necessary matrices L. We need the following (writing throughout this proof [x]
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indicates freezing the variable x):
M∗ ax1 MΩ √ √ = bx1 + N φ(x1 ) Ω = N φ([x1 ]), u(y, [x1 ]), 0, 0 . . .
(35)
M∗ ax1 ax2 MΩ = √ √ bx1 + N φ(x1 ) bx2 + N φ(x2 ) Ω = f0 , f1 , f2 , 0, 0 . . . where the entries are : f0 ([x1 ], [x2 ]) = N φ([x1 ])φ([x2 ]) + u ◦ c))[x1 ], [x2 ]) = N Λ([x1 ], [x2 ]) (36) i √ h f1 (y, [x1 ], [x2 ]) = N φ([x1 ])u(y, [x2 ]) + φ([x2 ])u(y, [x1 ]) (37) h i 1 √ f2 (y1 , y2 , [x1 ], [x2 ]) = u(y1 , [x1 ])u(y2 , [x2 ]) + u(y2 , [x1 ])u(y1 , [x2 ]) 2 (38)
and similarly,
M∗ a∗x1 ax2 MΩ = √ √ b∗x1 + N φ(x1 ) bx2 + N φ(x2 ) Ω = g0 , g1 , g2 , 0, 0 . . . where the entries are : g0 [x1 ], [x2 ]) = N φ([x1 ])φ([x2 ]) + (u ◦ u)([x1 ], [x2 ]) = N Γ([x1 ], [x2 ]) (39) h i √ g1 (y, [x1 ], [x2 ]) = N φ([x1 ])u(y, [x2 ]) + c(y, [x1 ])φ([x2 ]) (40) i 1 h g2 (y1 , y2 , [x1 ], [x2 ]) = √ c(y1 , [x1 ])u(y2 , [x2 ]) + c(y2 , [x1 ])u(y1 , [x2 ]) 2 (41)
Based on this we easily compute
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M. GRILLAKIS AND M. MACHEDON
Lemma 3.4. The L matrices are given by √ 1
L0,1 (t, x1 ) = √ Ω, bx1 + N φ(x1 ) Ω = φ(x1 ) N √ √ 1
Ω, bx1 + N φ(x1 ) bx2 + N φ(x2 ) Ω L0,2 (t, x1 , x2 ) = N 1 = (u ◦ c)(x1 , x2 ) + φ(x1 )φ(x2 ) N 1 = ψ(t, x1 , x2 ) + φ(t, x1 )φ(t, x2 ) = Λ 2N where ψ = sh(2k) = 2u ◦ c 1 L1,1 (t, x1 ; x2 ) = u ◦ u)(x1 , x2 ) + φ(x1 )φ(x2 ) N 1 = ω(t, x1 , x2 ) + φ(t, x1 )φ(t, x2 ) = Γ 2N where ω := ch(2k) − 1 = 2u ◦ u , 1 1 ψ(x3 , x2 )φ(x1 ) + u ◦ u(x1 , x3 )φ(x2 ) L1,2 (x1 ; x2 , x3 ) = Γ(x1 , x2 )φ(x3 ) + 2N N Z 1 L2,2 (y1 , y2 ; x1 , x2 ) = Λ(y1 , y2 )Λ(x1 , x2 ) + 2 f 1 (y, y1 , y2 )f1 (y, x1 , x2 )dy N Z 1 + 2 f 2 (y, z, y1 , y2 )f2 (y, z, x1 , x2 )dydz N Z 1 L1,3 (y1 ; x1 , x2 , x3 ) = Γ(x1 , y1 )Λ(x2 , x3 ) + 2 g 1 (y, x1 , y1 )f1 (y, x2 , x3 )dy N Z 1 + 2 g 2 (y, z, x1 , y1 )f2 (y, z, x2 , x3 )dydz N where the integrals in the last two formulas can be trivially expressed in terms of ψ and ω, which in turn can be expressed in terms of φ, Λ and Γ. Proof. All calculations are straightforward. For instance, 1
1
L1,2 (x1 ; x2 , x3 ) = 3/2 Ω, M∗ a∗x1 ax2 ax3 MΩ = 3/2 Ω, M∗ a∗x1 ax2 MM∗ ax3 MΩ N N 1 ∗ ∗ = 3/2 M ax2 ax1 MΩ, M∗ ax3 MΩ N Z 1 = g 0 (x2 , x1 )φ(x3 ) + g 1 (y, x2 , x1 )u(y, x3 )dy N 1 1 = Γ(x1 , x2 )φ(x3 ) + u ◦ c(x3 , x2 )φ(x1 ) + u ◦ u(x1 , x3 )φ(x2 ) N N 1 1 = Γ(x1 , x2 )φ(x3 ) + sh(2k)(x3 , x2 )φ(x1 ) + u ◦ u(x1 , x3 )φ(x2 ) 2N N
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15
1 ∗ ∗ M a a MΩ, M a a MΩ y y x x 1 2 1 2 N2 Z 1 1 = 2 f 0 (y1 , y2 )f0 (x1 , x2 ) + 2 f 1 (y, y1 , y2 )f1 (y, x1 , x2 )dy N N Z 1 + 2 f 2 (y, z, y1 , y2 )f2 (y, z, x1 , x2 )dydz N
L2,2 (y1 , y2 ; x1 , x2 ) =
With these ingredients at hand we proceed to write down the evolution of L1 = φ : Z 1 ∂t − ∆x1 + dy {v(x1 − y)Γ(y, y)} φ(x1 ) = i Z 1 − dy v(x1 − y)φ(y)ω(y, x1 ) + v(x1 − y)φ(y)ψ(x1 , y) 2N and we can eliminate ψ and ω by the substitution, ω = 2N Γ − φ ⊗ φ ψ = 2N Λ − φ ⊗ φ so that we have a system involving only Λ, Γ and φ matrices. The evolution of Λ is given by the expression below: Z 1 1 ∂t − ∆x1 − ∆x2 + v(x1 − x2 ) + dy {(v(x1 − y) + v(x2 − y)) Γ(y, y)} Λ = i N Z n o 1 − dy v(x1 − y) + v(x2 − y) φ(y) ω(y, x1 )φ(x2 ) + ω(y, x2 )φ(x1 ) 2N Z n o 1 − dy v(x1 − y) + v(x2 − y) φ(y) ψ(x1 , y)φ(x2 ) + ψ(x2 , y)φ(x1 ) 2N Z n o 1 − dy v(x − y) + v(x − y) ψ(x , y)ω(y, x ) + ω(y, x )ψ(x , y) 1 2 1 2 1 2 4N 2 Finally the evolution of Γ is given by: Z 1 ∂t + ∆x1 − ∆x2 Γ = dy v(x1 − y) − v(x2 − y) Λ(x1 , y)Λ(y, x2 ) + i Z n o 1 dy v(x1 − y) − v(x2 − y) φ(y)ω(y, x2 )φ(x1 ) + φ(y)ω(x1 , y)φ(x2 ) + 2N Z n o 1 2 dy v(x1 − y) − v(x2 − y) |φ(y)| ω(x1 , x2 ) + ω(y, y)φ(x1 )φ(x2 ) + 2N Z n o 1 dy v(x − y) − v(x − y) ω(x , y)ω(y, x ) + ω(y, y)ω(x , x ) 1 2 1 2 1 2 4N 2
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M. GRILLAKIS AND M. MACHEDON
If we substitute ψ = 2N Λ − φ ⊗ φ and ω = 2N Γ − φ ⊗ φ we obtain the equations of Theorem (3.1). Remark 3.5. Using the ideas in this section one can easily
show that the expected number of particles for our approximation, MΩ,
N MΩ R ∗ (where N = ax ax dx), as well as the energy MΩ, HMΩ are constant in time. This provides an easier proof of some of the results of section 8 of [22]. 4. Estimates In this paper, S and S± stand for the pure differential operators, ˜ = S + potential terms, so that the operators of Theorem (2.4) are S ˜ = S± + potential terms W 1 S = ∂t − ∆ (in 6 + 1 or 3 + 1 dimensions, as will be clear from the context) i 1 S± = ∂t − ∆x + ∆y i and potential terms are given by composition with vN (x − y)Γ(x, y) and multiplication by vN ∗ T rΓ. The symbol of S is τ +|ξ|2 or τ +|ξ|2 +|η|2 , depending on dimensions, and the symbol of S± is τ + |ξ|2 − |η|2 . We will use the following norms: δ kf kX δ = k 1 + |τ + |ξ|2 | fˆ(τ, ξ)kL2 δ kf kXSδ = k 1 + |τ + |ξ|2 + |η|2 | fˆ(τ, ξ, η)kL2 2 2 δ ˆ f (τ, ξ, η)kL2 kf kXW δ = k 1 + |τ + |ξ| − |η| | and refer to [42], section 2.6 for their history and properties. Of special importance is the following result: (proposition 2.12 in [42]): if Su = f , δ > 0 and χ(t) is a fixed C0∞ , cut-off function, then kχ(t)ukX 21 +δ .χ,δ ku(0, ·)kL2 + kf kX − 21 +δ
(42)
For the rest of the paper, we will use the standard notation A .χ,δ B to mean ”there exists a constant C depending of χ and δ such that A ≤ CB. We will also use freely the general principle that if an Lp estimate holds for solutions to the homogeneous equation, it also holds in X 1/2+δ spaces, see Lemma 2.9 in [42]. To get started, fix w ∈ S such that |ˆ v | ≤ wˆ and fix > 0 depending 1 1 + on β < 2/3 so that < ∇x > 2 < ∇y > 2 + N1 wN (x − y), which is a
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17
function of x − y, satisfies 1
1 C wN kL6/5 (d(x−y) ≤ small power N N
1
k < ∇x > 2 + < ∇y > 2 +
The basic space-time collapsing estimates, in the spirit of [29], are: Lemma 4.1. If SΛ = 0 then 1/2
k|Λ(t, x, x)kL2 (dtdx) . k|∇|x−y Λ0 (x, y)kL2 (dxdy)
(43)
k|∇|1/2 x Λ(t, x, x)kL2 (dtdx)
(44)
.
k|∇|x1/2 |∇|y1/2 Λ0 (x, y)kL2 (dxdy)
As a consequence, if SΛ = F and δ > 0, then 1
sup k < ∇x > 2 + χ(t)Λ(t, x, x + z)kL2 (dtdx)
(45)
z 1
1
1 + 2
1 + 2
.δ k < ∇x > 2 + < ∇y > 2 + χ(t)ΛkX 1/2+δ S
.δ k < ∇x > + k < ∇x >
1 + 2
< ∇y >
< ∇y >
1 + 2
Λ0 (x, y)kL2 (dxdy)
F kX −1/2+δ S
If S± Γ = 0, then 1
k|∇x | 2 + Γ(t, x, x)kL2 (dtdx) + k|∇x |1/2 Γ(t, x, x)kL2 (dtdx) 1
1
. k < ∇x > 2 + < ∇y > 2 + Γ0 (x, y)kL2 (dxdy)
(46)
As a consequence, if S± Γ = F , then 1
sup k|∇x | 2 + χ(t)Γ(t, x, x + z)kL2 (dtdx) + sup k|∇x |1/2 χ(t)Γ(t, x, x + z)kL2 (dtdx) z
.,δ < ∇x >
z 1 + 2
< ∇y >
1 + 2
1 + 2
W
.,δ k < ∇ >x < ∇y > + k < ∇x >
1 + 2
< ∇y >
χ(t)ΓkX 1/2+δ
1 + 2
1 + 2
Γ0 (x, y)kL2 (dxdy)
F kX −1/2+δ
(47)
W
Remark 4.2. Notice that the estimates for Λ are different from those for Γ at low frequencies. We cannot estimate kΓ(t, x, x)kL2 (dtdx) . The ”lowest” derivative we can estimate in (46) is k|∇x |1/2 Γ(t, x, x)kL2 (dtdx) .
18
M. GRILLAKIS AND M. MACHEDON
e denote Proof. The same method, inspired by [28], works for both. Let Λ the space-time Fourier transform of Λ. For (43) ^ |Λ(t, x, x)(τ, ξ)|2 Z 1 . δ(τ − |ξ − η|2 − |ξ + η|2 ) dη |η| Z 1 \ 2 δ(τ − |ξ − η|2 − |ξ + η|2 )|∇x−y Λ0 (ξ − η, ξ + η)|2 dη In order to prove the estimate, we must show Z 1 sup δ(τ − |ξ|2 − |η|2 ) dη . 1 |η| τ,ξ which is obvious. For (44) 2 ^ |∇1/2 x Λ(t, x, x)(τ, ξ)| Z . δ(τ − |ξ − η|2 − |ξ + η|2 )
|ξ| dη |ξ − η||ξ + η| Z 1 1 \ δ(τ − |ξ − η|2 − |ξ + η|2 )|∇x2 ∇y2 Λ0 (ξ − η, ξ + η)|2 dη
In order to prove the estimate, we must show Z |ξ| sup δ(τ − |ξ|2 − |η|2 ) dη . 1 |ξ − η||ξ + η| τ,ξ Without loss of generality, consider the region |ξ − η| ≤ |ξ + η|. If |ξ| 1 . |η| and the integral can be evaluated |ξ − η| ∼ |ξ + η|, |ξ−η||ξ+η| in polar coordinates. If |ξ − η| 1+ < ξ + η >1+ τ,ξ To prove this, take ξ = (|ξ|, 0, 0). Then Z |ξ||ξ − η| dη δ(τ − ξ · η) < ξ − η >1+ < ξ + η >1+ Z Z |ξ| 1 = δ(τ − |ξ|η1 ) dη . dη2 dη3 . 1 1+ 2+ < ξ − η >< ξ + η > R2 < η2,3 > Next, we record some Strichartz type estimates Lemma 4.3. The following estimate holds keit(∆x ±∆y ) f kL2 (dt)L6 (dx)L2 (dy) . kf kL2 .
(48)
keit(∆x ±∆y ) f kL2 (dt)L∞ (dx)L2 (dy) . k < ∇x >1/2+ f kL2 .
(49)
and, as a consequence kF kL2 (dt)L6 (dx)L2 (dy) .δ kF kX 1/2+δ
(50)
where X can be either XS or XW . Proof. We argue as follows: kkeit(∆x ±∆y ) f kL2 (dy) kL2 (dt)L6 (dx) = kkeit∆x f kL2 (dy) kL2 (dt)L6 (dx) . kkeit∆x f kL2 (dt)L6 (dx) kL2 (dy) . kkeit∆x f kL2 (dt)L6 (dx) kL2 (dy) . kf kL2 (dxdy) 3 + 1 end-point Strichartz [25] The proof of (48) is similar, using Sobolev. This type of estimate has first appeared, we believe, in [7]. Versions of it were used in [9], [10], [32]. We will need the following refinements. Notice that for S we can choose x, y coordinates or x + y, x − y coordinates, but this is not possible for S± . Lemma 4.4. For each 0 < δ < 12 there exists 6/5+ > 6/5 a number which can be chosen arbitrarily close to 6/5 if δ is small such that the following estimate holds kΛk
−1 2 +δ
XS
.δ kΛkL2 L6/5+ (x−y)L2 (x+y)
(51) (52)
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M. GRILLAKIS AND M. MACHEDON
Proof. This is proved by interpolating the estimate dual to (50) kΛk
−1 2 −δ
XS
. kΛkL2 L6/5 (x−y)L2 (x+y)
(53)
= kΛkL2 L2 (x−y)L2 (x+y)
(54)
with kΛk
1+1 −2 2
XS
Also we will need the closely related Lemma 4.5. For each 0 < δ < 21 there exist numbers 2−, 6/5+ arbitrarily close to 2, 6/5 if δ is close to 0, so that kF kX −1/2+δ .δ kF kL2− (dt)L6/5+ (dx)L2 d(y)
(55)
W
Proof. We start with kF kL2+ (dt)L6− (dx)L2 (dy) . kF kX 1/2+δ (interpolate (50) with energy estimates) W
kF kX −1/2−δ . kF kL2− (dt)L6/5+ (dx)L2 (dy) (dual estimates) W
and interpolate with the trivial estimate kF kX −1/2+1/2 = kF kL2 (dt)L2 (dxdy) W
we get (55).
Finally, we have one more estimate along the same lines Lemma 4.6. If SΛ = 0 then 1
1
kΛkL2 (dt)L∞ (d(x−y))L2 (d(x+y)) . k < ∇x > 2 + < ∇y > 2 + Λ0 kL2 (dxdy) and, as a consequence, if SΛ = F then kχ(t)ΛkL2 (dt)L∞ (d(x−y))L2 (d(x+y)) 1
1
1
1
. k < ∇x > 2 + < ∇y > 2 + Λ0 kL2 (dxdy) + k < ∇x > 2 + < ∇y > 2 + F k
−1 2 +δ
XS
Proof. Writing (49) in x + y, x − y coordinates keit(∆x +∆y ) Λ0 kL2 (dt)L∞ (d(x−y))L2 (d(x+y)) 1
1
1
. k < ∇x−y > 2 + Λ0 kL2 (dxdy) . k < ∇x > 2 + < ∇y > 2 + Λ0 kL2 (dxdy)
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21
Note that here we are forced to use < ∇ > rather than ∇. With the help of Lemma (4.6) we can estimate solutions to 1 S + vN (x − y) Λ = F N Λ(0, ·) = Λ0 (·)
(56)
The next proposition is the key estimate of our paper. Proposition 4.7. If Λ satisfies (56), β < 2/3, and χ(t) is a smooth cut-off function which is 1 on [0, 1], δ is sufficiently small, and N ≥ N0 is sufficiently large, then 1
1
k < ∇x > 2 + < ∇y > 2 + χ(t)Λk 1
(57)
1 +δ
XS2
1
1
1
. k < ∇x > 2 + < ∇y > 2 + Λ0 (x, y)kL2 (dxdy) + k < ∇x > 2 + < ∇y > 2 + F k
1 +δ −2
XS
and thus, by Lemma (4.1), 1
sup k < ∇x > 2 + χ(t)Λ(t, x + z, x)kL2 (dtdx)
(58)
z 1
1
1
1
. k < ∇x > 2 + < ∇y > 2 + Λ0 (x, y)kL2 (dxdy) + k < ∇x > 2 + < ∇y > 2 + F k ∂ Similar estimates hold for ∂t , ∇x+y commute with the potential.
n
1 +δ −2
XS
(χ(t)Λ) because these derivatives
Proof. Recall we introduces a potential w ∈ S such that w ˆ ≥ |ˆ v |. For (57), we use another cut-off function χ1 (t) which is identically 1 on the support of χ and notice that the solution of (56) agrees with that of 1 vN χ1 (t)Λ = F N Λ(0, ·) = Λ0 (·) SΛ +
(59)
on the support of χ. So we work with the solution of (59). Also, for technical reasons, we define bχ1 (t)Λc to be the inverse Fourier transform of the absolute value of the (space-time) Fourier transform of χ1 (t)Λ. Then 1
1
k < ∇x > 2 + < ∇y > 2 + χ1 (t)Λk 12 +δ XS 1 1 1 + + . k < ∇x > 2 < ∇y > 2 vN χ1 (t)Λ k − 21 +δ XS N 1
1
(60) (61) 1
1
+ k < ∇x > 2 + < ∇y > 2 + Λ0 (x, y)kL2 (dxdy) + k < ∇x > 2 + < ∇y > 2 +δ F k
1 +δ −2
XS
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M. GRILLAKIS AND M. MACHEDON
We plan to absorb (61) in the LHS of (60). The reason we introduced w and bχ1 (t)Λc, which have non-negative Fourier transforms, is to have the following cheap substitute for the Leibnitz rule: 1 k < ∇x > < ∇y > χ1 (t)vN Λ k − 21 +δ XS N 1 1 1 . k < ∇x > 2 + < ∇y > 2 +δ wN bχ1 (t)Λck − 21 +δ XS N 1 1 1 + k < ∇x > 2 + wN < ∇y > 2 + bχ1 (t)Λck − 21 +δ XS N 1 1 1 + k < ∇y > 2 + wN < ∇x > 2 + bχ1 (t)Λck − 21 +δ XS N 1 1 1 + k wN < ∇x > 2 + < ∇y > 2 + bχ1 (t)Λck − 21 +δ XS N 1 + 2
1 + 2
1
(62a) (62b) (62c) (62d)
1
For the most singular term, (62a), < ∇x > 2 + < ∇y > 2 + N1 wN (x − y) is a function of x − y and we recall was chosen so that 1 1 k < ∇x > 2 + < ∇y > 2 + N1 wN kL6/5 ≤ N smallC power . At this stage we insist δ is so small that the corresponding number 6/5+ also satisfies 1 1 k < ∇x > 2 + < ∇y > 2 + N1 wN kL6/5+ ≤ N smallC power . We estimate, using (51) and Lemma (4.6) 1 1 1 + + wN bχ1 (t)ΛckX − 21 +δ k < ∇x > 2 < ∇y > 2 N 1 + 12 + 1 2 . k ∇x ∇y wN bχ1 (t)ΛckL2 (dt)L6/5+ (d(x−y))L2 (d(x+y)) N 1 + 1 + 1 wN kL6/5+ (d(x−y)) kbχ1 (t)ΛckL2 (dt)L∞ (d(x−y))L2 (d(x+y)) . k∇x2 ∇y2 N 1 1 1 . small power k < ∇x > 2 + < ∇y > 2 + bχ1 (t)ΛckX 21 +δ N 1 1 1 = small power k < ∇x > 2 + < ∇y > 2 + χ1 (t)ΛkX 21 +δ N
This can be absorbed in the LHS of (60). The other terms are easier. While we will use the X spaces as tools in our proofs, the actual norms in which we prove well-posedness are Lp norms defined for some
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23
0 < T ≤ 1. 1
NT (Λ) = sup k < ∇x > 2 + Λ(t, x + z, x)kL2 ([0,T ]×R3 ) z 1
1
+ k < ∇x > 2 + < ∇y > 2 + Λ(t, x, y)kL∞ ([0,T ]×L2 (R6 )) ˙ T (Γ) = sup k|∇x | 21 + Γ(t, x + z, x)kL2 ([0,T ]×R3 ) N z
+ sup k||∇x |1/2 Γ(t, x + z, x)kL2 ([0,T ]×R3 ) z 1
1
+ k < ∇x > 2 + < ∇y > 2 + Γ(t, x, y)kL∞ ([0,T ]×L2 (R6 )) 1
1
NT (φ) =k < ∇x > 2 + φkL∞ ([0,T ])L2 + k < ∇x > 2 + φkL2 ([0,T ])L6 Based on energy and Strichartz estimates, the collapsing estimates of Lemma (4.1) and general properties of X spaces which allow one to transfer estimates for solutions to the homogeneous equation to estimates in X 1/2+δ spaces we have 1
1
NT (Λ) . k < ∇x > 2 + < ∇y > 2 + Λk
1 +δ
XS2
˙ T (Γ) . k < ∇x > 12 + < ∇y > 12 + Γk N
1 +δ
2 XW
˙ T (φ ⊗ φ) . k < ∇x > 21 + φk2 NT (φ ⊗ φ) + N
1
X 2 +δ
Our basic estimates for the inhomogeneous linear equations are Proposition 4.8. If Γ and φ satisfy S± Γ = F Sφ = G 0 ≤ T ≤ 1, and 2− < 2, 6/5+ > 6/5 are fixed numbers which can be chosen close to 2 and 6/5, then ˙ T (Γ) N 1
1
1
1
. k < ∇x > 2 + < ∇y > 2 + Γ(0, x, y)kL2 (dxdy) + k < ∇x > 2 + < ∇y > 2 + F kL2− [0,T ]L6/5+ (dx)L2 (dy) NT (φ) 1
1
. k < ∇x > 2 + φ(0, ·)kL2 (dx) + k < ∇x > 2 + GkL2− [0,T ]L6/5+ (dx) ˙ T (φ ⊗ φ) . NT (φ)2 NT (φ ⊗ φ) + N Similar estimates hold when x and y are reversed.
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M. GRILLAKIS AND M. MACHEDON
Proof. We will prove this for the equation for Γ, the other one for φ being easier. The standard energy estimate and the collapsing estimate of lemma (4.1) prove that if S± Γ = 0, then ˙ T (Γ) . k < ∇x > 21 + < ∇y > 12 + Γ0 (x, y)kL2 (dxdy) N From here and general properties of X 1/2+δ spaces, we get, for any δ > 0, ˙ T (Γ) .δ k < ∇x > 12 + < ∇y > 21 + Γk 1/2+δ N X W
uniformly in T . Finally, to prove the stated estimate, let FT = F in [0, T ], and 0 otherwise. Let ΓT be the solution to S± ΓT = FT ΓT (0, ·) = Γ(0, ·) Then Γ = ΓT in [0, T ] and recalling 0 < T ≤ 1 and χ = 1 on [0, 1], ˙ T (Γ) = N ˙ T (χ(t)ΓT ) N 1
1
. k < ∇x > 2 + < ∇y > 2 + χ(t)ΓT k 1
1
1
1
1 +δ
XS2
. k < ∇x > 2 + < ∇y > 2 + Γ0 (x, y)kL2 (dxdy) + k < ∇x > 2 + < ∇y > 2 + FT k 1
1
1
1
1
1
1
1
−1 2 +δ
XS
(by (42))
. k < ∇x > 2 + < ∇y > 2 + Γ0 (x, y)kL2 (dxdy) + k < ∇x > 2 + < ∇y > 2 + FT kL2− (dt)L6/5+ (dx)L2 (dy) (by (55)) = k < ∇x > 2 + < ∇y > 2 + Γ0 (x, y)kL2 (dxdy) + k < ∇x > 2 + < ∇y > 2 + F kL2− [0,T ]L6/5+ (dx)L2 (dy) The same type of result holds for the equation for Λ. Proposition 4.9. There exist numbers 2− < 2, 6/5+ > 6/5, which can be chosen arbitrarily close to 2 and 6/5 such that, if Λ satisfies 1 S + vN (x − y) Λ = F (63) N Λ(0, ·) = Λ0 (·) β < 2/3, and 0 ≤ T ≤ 1, then
PAIR EXCITATIONS, II
NT (Λ)
25
(64)
. k < ∇x >
1 + 2
< ∇y >
1
1 + 2
Λ0 (x, y)kL2 (dxdy)
1
+ k < ∇x > 2 + < ∇y > 2 + F kL2− [0,T ]L6/5+ (dx)L2 (dy) ∂ n (∇x+y )m Λ: Similar estimates hold for ∂t n ∂ m NT (∇x+y ) Λ (65) ∂t n 1 1 ∂ m + + (∇x+y ) Λ(t, x, y) kL2 (dxdy) . k < ∇x > 2 < ∇y > 2 ∂t t=0 n 1 1 ∂ + k < ∇x > 2 + < ∇y > 2 + (∇x+y )m F kL2− [0,T ]L6/5+ (dx)L2 (dy) ∂t Proof. The proof is the same as the one of Proposition (4.8), except that, because of the potential vN we don’t get the estimate (42) from general principles but use Proposition (4.7). To prove (65), first differentiate the equation, noting that these derivatives commute with the potential, then apply the previous result. 5. The non-linear equations Now we come to our main PDE result Theorem 5.1. Let Λ, Γ and φ be solutions of (24b), (24c), (24a) with initial conditions φ0 , k0 ∈ S. There exists N0 such that for all N ≥ N0 , the following estimates hold: 1
1
NT (Λ) . k < ∇x > 2 + < ∇y > 2 + Λ(0, ·)kL2 smallpower 4 ˙ +T NT (Λ) NT (Γ) + NT (φ) ˙ T (Γ) . k < ∇x > 21 + < ∇y > 12 + Γ(0, ·)kL2 N smallpower 2 2 4 ˙ +T NT (Λ) + NT (Γ) + NT (φ) 1
NT (φ) . k < ∇x > 2 + φ(0, ·)kL2 smallpower 2 ˙ +T NT (Λ) + NT (Γ) + NT (φ) NT (φ)
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M. GRILLAKIS AND M. MACHEDON
So there exists T0 > 0 such that, if T ≤ T0 , 1
1
NT (Λ) . k < ∇x > 2 + < ∇y > 2 + Λ(0, ·)kL2 ˙ T (Γ) . k < ∇x > 21 + < ∇y > 12 + Γ(0, ·)kL2 N 1
NT (φ) . k < ∇x > 2 + φ(0, ·)kL2
Also, similar estimates hold for the derivatives which commute with the potential: n ∂ m ∇x+y Λ (66) NT ∂t n 1 1 ∂ + + m 2 2 . k < ∇x > < ∇y > ∇x+y Λ kL2 ∂t t=0 n ∂ ˙T N ∇m x+y Γ ∂t n 1 1 ∂ + + m ∇x+y Γ kL2 . k < ∇x > 2 < ∇y > 2 ∂t nt=0 n 1 ∂ ∂ NT ∇m . k < ∇x > 2 + ∇m x φ x φ|t=0 kL2 ∂t ∂t Proof. For equation (24a), which can be abbreviated as 1 ∂t φ − ∆φ i = −(vN Λ) ◦ φ − (vN Γ) ◦ φ − (vN ∗ T rΓ) · φ + 2(vN ∗ |φ|2 )φ := RHS(24a) we first apply Proposition (4.8): 1
1
NT (φ) . k < ∇x > 2 + φ(0, ·)kL2 + k < ∇x > 2 + RHS(24a)kL2− ([0,T ])L6/5+ (dx) 1
1
. k < ∇x > 2 + φ(0, ·)kL2 + T small power k < ∇x > 2 + RHS(24a)kL2 ([0,T ])L6/5+ (dx) 1 + small power 2 ˙ 2 . k < ∇x > φ(0, ·)kL2 + T NT (Λ) + NT (Γ) + NT (φ) NT (φ)
The last line follow from the classical fractional Leibnitz rule in Lp spaces due to Kato and Ponce, [24] . We only present a typical term
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in RHS(24a): k < ∇x > (vN ∗ T rΓ) · φ kL2 ([0,T ])L6/5+ (dx) 1 + . k(vN ∗ T rΓ) · φkL2 ([0,T ])L6/5+ (dx) + k|∇x | 2 (vN ∗ T rΓ) · φ kL2 ([0,T ])L6/5+ (dx) 1 + 2
. kT rΓkL2 ([0,T ])L3 (dx) kφkL∞ L2+ 1
1
+ k|∇x | 2 + T rΓkL2 ([0,T ])L2 (dx) kφkL∞ L3+ + kT rΓkL2 ([0,T ])L3+ (dx) k∇x | 2 + φkL∞ L2 ˙ T (Γ)NT (φ) .N Now we deal with the equation (24b), which can be abbreviated as 1 S + vN Λ N = −(vN Λ) ◦ Γ − Γ ◦ (vN Λ) − (vN ∗ T rΓ) (x) + (vN ∗ T rΓ) (y) Λ(x, y) − vN Γ ◦ Λ − Λ ◦ (vN Γ) + 2(vN ∗ |φ|2 )(x)φ(x)φ(y) + 2(vN ∗ |φ|2 )(y)φ(y)φ(x) := RHS(24b) Applying Proposition (4.9) to the equation (24b)
1
1
NT (Λ) . k < ∇x > 2 + < ∇y > 2 + Λ(0, ·)kL2 1
1
+ k < ∇x > 2 + < ∇y2
+
> RHS(24b)kL2− [0,T ]L6/5+ (dx)L2 (dy) 1
1
6
We would like to estimate < ∇x > 2 + < ∇y > 2 + RHS(24b) in L2 L 5 + L2 to gain a small power of T from Cauchy-Schwarz in time. Typical term in RHS (24b): 1 + 2
1 + 2
< ∇x > < ∇y > (vN Λ) ◦ Γ(x, y) Z 1 1 + = vN (z) < ∇x > 2 + < ∇y2 > (Λ(x, x − z)Γ(x − z, y)) dz
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M. GRILLAKIS AND M. MACHEDON
Applying the fractional Leibniz rule in Lp spaces, Z 1 1 + + 2 2 vN (z)k < ∇x > < ∇y > Λ(x, x − z)Γ(x − z, y) kL2 ([0,T ])L 56 + (dx)L2 (dy) dz Z 1 . vN (z)k < ∇x > 2 + Λ(x, x − z)kL2 ([0,T ])L2 (dx) × 1
k < ∇y2
+
> Γ(x − z, y)kL∞ ([0,T ])L3+ (dx)L2 (dy) dz
Z +
vN (z)kΛ(x, x − z)kL2 ([0,T ])L3+ (dx) × 1
1
k < ∇x > 2 + < ∇y2 ˙ T (Γ) . NT (Λ)N
+
> Γ(x − z, y)kL∞ ([0,T ])L2 (dx)L2 (dy) dz
All other terms are treated in a similar manner. In fact, all terms on the RHS(24b) and RHS(24c) are of the form (vN F ) ◦ G or (vN ∗ T rF )G where F , G can be Λ, Γ, φ ⊗ φ of φ ⊗ φ (or their complex conjugates) and NT (F ), NT (G) are estimated as above by Proposition (4.8) or Proposition (4.9). 6. Estimates for sh(2k) Recall the equation (23a), which can be written explicitly as S (sh(2k)) = −2vN Λ
(67)
− (vN Λ) ◦ p2 − p2 ◦ (vN Λ) − ((vN ∗ T rΓ) (x) + (vN ∗ T rΓ) (y)) sh(2k)(x, y) − (vN Γ) ◦ sh(2k) − sh(2k) ◦ (vN Γ) := RHS(67) Now that we control the quantities (66) we use proofs similar to those of section 4 of [21], or section 3 of [32] with −vN Λ playing the role of ˜ playing the role of S, at least locally in time. The crucial m and S ingredient is that sup kΛ(t, x + z, x)kL2 (dx) + sup k z
z
∂ Λ(t, x + z, x)kL2 (dx) ∂t
∂ + kΛkL∞ + k ΛkL∞ . 1 (68) ∂t The reader is warned, with apologies, that S in [21] is not the S of the current paper (which is a purely differential operator), but what was called Sold in the introduction. In this section, we will prove
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Theorem 6.1. Let sh(2k), ch(2k) satisfy the equations (23a), (23b). Then for every > 0, and T0 as in Theorem (5.1), k < Dx,y >1/2− sh(2k)(t, ·, ·)kL2 (dxdy) . 1
(69)
k < Dx,y >1/2+ sh(2k)(t, ·, ·)kL2 (dxdy) . N 2
(70)
sup ksh(2k)(t, x, ·)kL2 (dy) . N (0 ≤ t ≤ T0 )
(71)
x
Similar estimates hold for the higher time derivatives of sh(2k). Define ch(k) = δ(x − y) + p. The following is immediate, as in the proof of Corollary 4.2 in [21]: Corollary 6.2. Let > 0. The following estimates hold uniformly in 0 ≤ t ≤ T0 k < Dx,y >1/2− sh(k)(t, ·, ·)kL2 (dxdy) . 1 k < Dx,y >1/2+ sh(k)(t, ·, ·)kL2 (dxdy) . N 2 kp(t, ·, ·)kL2 (dxdy) . 1 sup ksh(k)(t, x, ·)kL2 (dy) . N x
sup kp(t, x, ·)kL2 (dy) . N x
with similar estimates on the higher time derivatives. We start with some preliminary lemmas. Lemma 6.3. (replacing Lemma 4.4 in [21].) Let s0a be the solution to Ss0a = −2vN (x − y)Λ s0a (0, x, y)
(72)
=0
Then k < Dx,y >1/2− s0a (t, ·, ·)kL2 (dxdy) . 1 k < Dx,y >1/2+ s0a (t, ·, ·)kL2 (dxdy) . N 2 (0 ≤ t ≤ T0 ) with similar estimates for higher time derivatives. This lemma is a particular case (with F = vN (x − y)Λ(t, x, y) of a more general result we will need later: Lemma 6.4. Let F be a function of 6+k+1 variables (k ≥ 0, x, y ∈ R3 , z ∈ Rk , t ∈ [0, 1]) and let Z t E(x, y, z, t) = ei(t−s)∆x,y,z F (s)ds 0
30
M. GRILLAKIS AND M. MACHEDON
Then k < Dx−y >1/2− E(t, ·)kL2 ∂ . sup kF (s, ·)kL2 (d(x+y))L2 (dz)L1 (d(x−y)) + k F (s, ·)kL2 (d(x+y))L2 (dz)L1 (d(x−y)) ∂s 0≤s≤t Proof. Change variables, so we work in x, y rather than x − y, x + y coordinates. In these coordinates, integrating by parts, Z t E(t, ·) = ei(t−s)∆x,y,z F (s, ·)ds (73) 0 Z t ∂ F (s, ·)ds (74) = ei(t−s)∆x,y,z ∆−1 x,y,z ∂s 0 −1 + eit∆x,y,z ∆−1 x,y,z F (0, ·) − ∆ F (t, ·) We are going to project in frequencies dual to x only. For the low frequency case we use (73): Z t 1/2− kP|ξ|≤1 < Dx > E(t, ·)kL2 = k ei(t−s)∆x,y,z P|ξ|≤1 < Dx >1/2− F (s, ·)dskL2 0 1/2−
≤ sup kP|ξ|≤1 < Dx >
F (s, ·)kL2 (dxdydz)
0≤s≤t
. sup kF (s, ·)kL2 (dydz)L1 (dx) 0≤s≤t
where we have used the fact that the (physical space) kernel corresponding to P|ξ|≤1 < Dx >1/2− is in L2 . For the high frequency case we use (74). We only write down one term, the boundary terms being similar: Z t ∂ 1/2− ei(t−s)∆x,y,z P|ξ|≥1 < Dx >1/2− ∆−1 F (s, ·)dskL2 kP|ξ|≥1 < Dx > E(t, ·)kL2 = k x,y,z ∂s 0 ∂ ≤ sup kP|ξ|≥1 < Dx >1/2− ∆−1 F (s, ·)dskL2 (dxdydz) x ∂s 0≤s≤t ∂ . sup k F (s, ·)kL2 (dydz)L1 (dx) 0≤s≤t ∂s where we have used that the kernel of P|ξ|≥1 < Dx >1/2− ∆−1 x in in 2 L. Proof. (of Lemma (6.3)). Applying the previous result and the estimates (66) we already know k < Dx−y >1/2− s0a (t, ·, ·)kL2 (dxdy) . 1
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with similar estimates for derivatives which commute with vN (x − y). Thus n ∂ k < Dx,y >1/2− s0a (t, ·, ·)kL2 (dxdy) .n 1 ∂t Using the equation (72), we also get n ∂ k ∆x,y s0a (t, ·, ·)kL2 (dxdy) . N 3β/2 ∂t so the second claim follows by interpolation.
Next, we include the potential terms: Lemma 6.5. Let sa be the solution to ˜ a = −2vN Λ Ss
(75)
sa (0, x, y) = 0 Then n ∂ s0a (t, ·, ·)kH 1/2− (dxdy) .n 1 (0 ≤ t ≤ T0 ) k ∂t n ∂ k s0a (t, ·, ·)kH 1/2+2 (dxdy) .n N (0 ≤ t ≤ T0 ) ∂t ˜ Proof. Let V be the ”potential” part of S:
V (u) = ((vN ∗ T rΓ) (x) + (vN ∗ T rΓ) (y)) u + (vN Γ) ◦ u + u ◦ (vN Γ) ∂ n Form the estimates (66) for Γ we see that ∂t Γ . 1, thus V and ∂ n [ ∂t , V ] have bounded operator norm (on L2 ). Write sa = s0a + s1a (s0a as in the previous lemma) so that s1a satisfies the equation ˜ 1 = −V (s0 ) Ss a a n n n n ∂ ∂ ∂ ∂ 1 0 0 ˜ ˜ 1 S sa = −V sa − [ , V ]sa − [ , S]s a ∂t ∂t ∂t ∂t Using energy estimates and induction on n we see n ∂ k s1a kL2 . 1 ∂t which, in turn, implies (using the equation) n−1 ∂ k ∆s1a kL2 . 1 ∂t uniformly for 0 ≤ t ≤ 1. The result follows from the previous lemma.
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M. GRILLAKIS AND M. MACHEDON
Finally, we can prove Theorem (6.1). Proof. (of theorem (6.1)) Write sh(2k) := s2 = sa +se where sa satisfies (75) and ch(2k) = δ(x − y) + p2 . In analogy with (64a), (64b) of [21], they satisfy ˜ e ) = −(vN Λ) ◦ p2 − p2 ◦ (vN Λ) S(s ˜ 2 ) = −(vN Λ) ◦ sa + sa ◦ (vN Λ) W(p −(vN Λ) ◦ se + se ◦ (vN Λ) := M − (vN Λ) ◦ se + se ◦ (vN Λ) Using the result of Lemma (6.5), as well as estimates (66) for Λ we see that k
∂ ∂t
n M kL2 (dxdy) . 1 (0 ≤ t ≤ T0 )
Differentiating the equation with respect to t and using energy estimates as in [32], we see k
∂ ∂t
n
se kL2 + k
∂ ∂t
n−1 ∆se kL2 . 1
Finally, to prove (71) we will use the L2 estimates that have been proved already (69) and (70). We go back to the equation (67), and write sh(2k) = s2a + s3a the following way: Let c = 2δ(x − y − z) |x − y − z| cχ(x − y − z) ∆x,y = 2δ(x − y − z) + χ(x ˜ − y − z) |x − y − z| ∆x,y
where χ, χ˜ are C0∞ functions, χ = 1 and χ˜ = 0 in a neighborhood of 0. Next, compare sh(2k) with the ”ansatz” s2a given by s2a
Z =
cχ(x − y − z) vN (z)Λ |x − y − z|
x+y+z x+y−z , 2 2
dz
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so that (splitting ∆x,y = ∆x+y + ∆x−y , Sx,y = Sx+y − ∆x−y , where, of course, ∆x+y = 21 (∇x + ∇y ) · (∇x + ∇y ) ) Sx,y s2a Z x+y+z x+y−z = (−2δ(x − y − z) − χ(x ˜ − y − z)) vN (z)Λ , dz 2 2 Z cχ(x − y − z) x+y+z x+y−z − vN (z)Sx+y Λ , dz |x − y − z| 2 2 = −2vN (x − y)Λ(x, y) Z x+y+z x+y−z − χ(x ˜ − y − z)vN (z)Λ , dz 2 2 Z cχ(x − y − z) x+y+z x+y−z vN (z)Sx+y Λ , dz − |x − y − z| 2 2 with initial conditions s2a (0, x, y) Using (68), it is easy to see that s2a satisfies the estimates in the statement of the theorem. In fact, supx ks2a (t, x, ·)kL2 (dy) . 1. To estimate the difference, S sh(2k) − s2a Z x+y+z x+y−z = χ(x ˜ − y − z)vN (z)Λ , dz 2 2 Z cχ(x − y − z) x+y+z x+y−z + vN (z)Sx+y Λ , dz |x − y − z| 2 2 − (vN Λ) ◦ p2 − p2 ◦ (vN Λ) − ((vN ∗ T rΓ) (x) + (vN ∗ T rΓ) (y)) sh(2k)(x, y) − (vN Γ) ◦ sh(2k) − sh(2k) ◦ (vN Γ) := RHS with initial conditions s2a (0, x, y) Using (66), we see that kRHSkL2 (dxdy) + k|Dx |1/2+ RHSkL2 (dxdy) . N 2 uniformly in 0 ≤ t ≤ T0 , with similar estimates for the time derivatives, and the same holds for s2a (0, x, y). The Strichartz estimate (49), Duhamel’s formula and Sobolev in time show that sup ksh(2k) − s2a kL2 (dy) . N 2 x
uniformly in [0, T0 ].
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M. GRILLAKIS AND M. MACHEDON
7. The reduced Hamiltonian Recall (10) for the definition of the reduced Hamiltonian. Hred was computed, for instance, in Section 5 of [21]. We will write it in a different way, using Wick’s theorem. Recall the conjugation formulas Z B −B ch(k)(y, x)ay + sh(k)(y, x)a∗y dy (76) e ax e =
eB a∗x e−B
= a(ch(k)(x, ·)) + a∗ (sh(k)(x, ·)) := bx Z = sh(k)(y, x)ay + ch(k)(y, x)a∗y dy = a(sh(k)(x, ·)) + a∗ (ch(k)(x, ·)) := b∗x .
The reduced Hamiltonian is Hred = N µ0 (t) (77a) Z ¯ + N 1/2 dx h(φ(t, x))b∗x + h(φ(t, x))bx (77b) Z 1 ∂ B(k(t)) −B(k(t) e e + dxb∗x ∆bx (77c) + i ∂t Z 1 dxdyvN (x − y) φ(x)φ(y)bx by + φ(x)φ(y)b∗x b∗y + 2φ(x)φ(y)b∗x by − 2 (77d) Z − dx(vN ∗ |φ|2 )b∗x bx (77e) Z 1 −√ dx1 dx2 vN (x1 − x2 ) φ(x2 )b∗x1 bx1 bx2 + φ(x2 )b∗x1 b∗x2 bx1 N (77f) Z 1 dx1 dx2 vN (x1 − x2 )b∗x1 b∗x2 bx1 bx2 . − (77g) 2N The functions µ0 (t) and h(φ(t, x)) appearing in (77a),(77b) are given below, Z Z 2 1 1 ¯ ¯ t − ∇φ − φφt − φφ dxdy vN (x − y)|φ(x)|2 |φ(y)|2 µ0 := dx 2i 2 1 h(φ(t, x)) := − ∂t φ + ∆φ − vN ∗ |φ|2 φ . i We re-arrange the terms in Hred using Wick’s theorem . Define the contraction of A(f ) := a(f¯1 ) + a∗ (fR2 ) and A(g) := a(¯ g1 ) + a∗ (g2 ) to be C(A(f ), A(g)) = [a(f¯1 ), a∗ (g2 )] = f¯1 g2 , and define the normal order
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Nor A(f )A(g)A(h)A(k) to be A(f )A(g)A(h)A(k) expanded and rearranged so all starred terms have been moved to the left, as if they commuted with unstarred terms. Wick’s theorem, which can easily be proved by induction, says, in particular, that A(g)A(h)A(k) = Nor A(g)A(h)A(k) + C(A(g), A(h))A(k) + C(A(g), A(k))A(h) + C(A(h), A(k))A(g) A(f )A(g)A(h)A(k) = Nor A(f )A(g)A(h)A(k) + C(A(f ), A(g))A(h)A(k) + C(A(f ), A(h))A(g)A(k) + · · · (6 terms) = Nor A(f )A(g)A(h)A(k) + C(A(f ), A(g))A(h)A(k) + C(A(f ), A(h))A(g)A(k) + · · · (6 ordered terms with 1 contraction) + C(A(f ), A(g))C(A(h), A(k)) + C(A(f ), A(h))C(A(g), A(k)) + · · · (6 ordered terms with 2 contractions) Applying Wick’s theorem to Hred (putting the quartic and cubic terms in normal order, but not the quadratics) we get Hred = N µ0 (t) Z o n ¯ 1/2 ∗ ˜ ˜ +N dx h(φ(t, x))bx + h(φ(t, x))bx Z 1 ∂ B(k(t)) −B(k(t) + e e + dxb∗x ∆bx (78a) i ∂t Z 1 dxdyvN (x − y) Λ(x, y)bx by + Λ(x, y)b∗x b∗y + 2Γ(x, y)b∗x by − 2 (78b) Z − dx(vN ∗ T rΓ)b∗x bx (78c) Z 1 dx1 dx2 Nor vN (x1 − x2 ) φ(x2 )b∗x1 bx1 bx2 + φ(x2 )b∗x1 b∗x2 bx1 −√ NZ 1 − dx1 dx2 Nor vN (x1 − x2 )b∗x1 b∗x2 bx1 bx2 . 2N
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M. GRILLAKIS AND M. MACHEDON
˜ is the modified Hartree operator (22). Let us remark that, in where h complete analogy to (67c) in [21], the quadratic terms (78a)+(78b)+ (78c) can be written concisely and explicitly as ! w˜ T f˜ (78a) + (78b) + (78c) = HG˜ − I −f˜ −w˜ where ˜ ˜ − sh(k) ◦ (vN Λ) ◦ sh(k) f˜ := S(sh(k)) + ch(k) ◦ (vN Λ) ◦ ch(k) − W(ch(k)) ˜ ˜ w˜ := W(ch(k)) − sh(k) ◦ (vN Λ) ◦ ch(k) − S(sh(k)) + ch(k) ◦ (vN Λ) ◦ sh(k) Z Hg˜ = g˜(x, y)a∗x ay dxdy
Also, I is the Lie algebra isomorphism used in our previous papers [19]-[22] (see for instance formula (27) in [21]) is ! Z n o T ˜ 1 w˜ f I =− dxdy w(y, ˜ x)ax a∗y + w(x, ˜ y)a∗x ay − f˜(x, y)a∗x a∗y − f˜(x, y)ax ay 2 −f˜ −w˜ ˜ 2 is a multiple of f˜, thus f˜ = 0 if our equations are In this notation, X satisfied. If we also put the quadratics in normal order, the above formula becomes Hred = X0 (t)+ Z o n ¯ 1/2 ∗ ˜ ˜ N dx h(φ(t, x))bx + h(φ(t, x))bx ! w˜ T f˜ + Hg˜ + Nor I −f˜ −w˜ Z 1 −√ dx1 dx2 vN (x1 − x2 ) φ(x2 )b∗x1 bx1 bx2 + φ(x2 )b∗x1 b∗x2 bx1 N ! Z 1 − . dx1 dx2 vN (x1 − x2 )b∗x1 b∗x2 bx1 bx2 2N where X0 is written down explicitly in Section 6 on [22]. In conclusion, ˜ if φ and k satisfy our equation X1 = 0, X2 = 0, then h(φ(t, x)) = 0,
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f˜ = 0 and P := Hred − H − X0 (79) T w˜ 0 = Hg˜ − H1 + Nor I 0 −w˜ Z 1 +Nor − √ dx1 dx2 vN (x1 − x2 ) φ(x2 )b∗x1 bx1 bx2 + φ(x2 )b∗x1 b∗x2 bx1 2 N Z 1 ∗ ∗ − dx1 dx2 vN (x1 − x2 )bx1 bx2 bx1 bx2 2N Z 1 + dx1 dx2 vN (x1 − x2 )a∗x1 a∗x2 ax1 ax2 . 2N Remark that the third and fourth term are − √1N Nor eB [A, V]e−B and − N1 Nor eB Ve−B (see Section 5 of [21]).
8. Estimates for the error term In this section we choose < 1 − 3β and apply the estimates of 2 Corollary (6.2) (where, in order to simplify notation, the in that corollary is taken to be the current divided by 10.) We proceed as in our previous papers [19]-[21], using the identity R
R
ψexact (t) − ei X0 (t)dt ψappr (t) = e−i X0 (t)dt ψred − Ω F F
and estimate the right hand side term using the equation
R 1∂ −i X0 (t)dt ˜ − Hred + X0 e ψred − Ω = (0, 0, 0, X3 , X4 , 0, · · · ) := X i ∂t
Denote E = e−i
R
X0 (t)dt
ψred − Ω, so that
1∂ ˜ − Hred + X0 E = (0, 0, 0, X3 , X4 , 0, · · · ) := X i ∂t E(0, ·) = 0
(80)
The terms X0 , X3 and X4 were computed in Section 5 of [21], see formulas (74c) and (72c). We only need X3 and X4 here, and they are
38
M. GRILLAKIS AND M. MACHEDON
(modulo symmetrization and normalization) Z 1 X3 (y1 , y2 , y3 ) = √ ch(y1 , x1 ) ch(x2 , y2 )vN (x1 − x2 )φ(x2 )sh(y3 , x1 )dx1 dx2 N 1 = √ (vN (y1 − y2 )sh(y3 , y1 )sh(y2 , y4 ) + LOT ) Z N 1 X4 (y1 , y2 , y3 , y4 ) = ch(y1 , x1 ) ch(x2 , y2 )vN (x1 − x2 )sh(y3 , x1 )sh(x2 , y4 )dx1 dx2 N 1 = vN (y1 − y2 ) (sh(y3 , y1 )sh(y2 , y4 ) + LOT ) N where the lower order terms LOT come from the p component of ch(k) = δ(x − y) + p. The main result of this section, which completes the proof of Theorem (2.5), is Theorem 8.1. Let 0 < < 1 −
3β , 2
and let E satisfy (80). Then
kEkF . N
3β −1+ 2
uniformly for 0 ≤ t ≤ T0 . Proof. This is proved by splitting 1∂ SF := − Hred + X0 i ∂t := SD − P where 1∂ −H SD = Z i ∂t 1 H = a∗x ∆ax − V 2N Thus H is the original (unconjugated) Fock space Hamiltonian (7a)(7c), and P accounts for the rest of the terms: P = Hred − H − X0 Recall H acts on Fock space as 1 X 3β ∆− N v N β (xj − xk ) N j
Abstract. We consider a large number of Bosons with interaction potential vN (x) = N 3β v(N β x). In our earlier papers [19]-[21] we considered a set of equations for the condensate φ and pair excitation function k and proved that they provide a Fock space approximation to the exact evolution of a coherent state for β < 31 . In [22] we introduced a coupled refinement of our original equations and conjectured that they provide a Fock space approximation in the range 0 ≤ β ≤ 1. In the current paper we prove that this is indeed the case for β < 23 , at least locally in time. In order to do that, we re-formulate the equations of [22] in a way reminiscent of BBGKY and apply harmonic analysis techniques in the spirit of those used by X. Chen and J. Holmer in [9] to prove the necessary estimates. In turn, these estimates provide bounds for the pair excitation function k. While our earlier papers provide background material, the methods of this paper paper are mostly new, and the presentation is self-contained.
1. Introduction The problem considered in this paper (as well as our earlier papers [19]- [22]) is the N -body linear Schr¨odinger equation ! N X 1∂ 1 X − ∆x + vN (xi − xj ) ψN (t, ·) = 0 (1) i ∂t j=1 j N i<j ψN (0, x1 , · · · , xN ) = φ0 (x1 )φ0 (x2 ) · · · φ0 (xN ) kψN (t, ·)kL2 (R3N ) = 1 where vN (x) := N 3β v(N β x) with 0 ≤ β ≤ 1, v ∈ S and v ≥ 0. The goal is to find a rigorous, simple approximation (in a suitable norm) to ψN which is consistent with ψapprox (t, x1 , · · · , xN ) ∼ eiχ(t) φ(t, x1 )φ(t, x2 ) · · · φ(t, xN )
(2)
as N → ∞, where φ (which represents the Bose-Einstein condensate) satisfies a non-linear Schr¨odinger equation. The problem becomes more difficult, interesting, and requires new ideas as β approaches 1, and this 1
2
M. GRILLAKIS AND M. MACHEDON
explains why several authors, (including us) devoted several papers to their programs. We refer to [36] for extensive background on (static) Bose-Einstein condensation. During recent years, in a series of papers by Erd¨os and Yau [12], and Erd¨os, Schlein and Yau [13] to [15], it was proved γ1N (t, x, x0 ) → φ(t, x)φ(t, x0 )
(3)
in trace norm as N → ∞, and similarly for the higher order marginal density matrices γkN , where k is fixed. Recent simplifications and generalizations were given in [29], [27], [5], [8], [9], [10]. See also [17], [30] for a different approach. We also mention the approach based on the quantum de Finetti theorem in [6], as well as results in the case of negative interaction potentials in [11] and the different approach of Knowles and Pickl [30]. Another approach to this problem is based on Fock space techniques and the second quantization. In physics, it was pioneered in the papers by Bogoliubov [4], Lee, Huang and Yang [33] in the static case, and Wu [43] in the time dependent case. See also the more recent papers [37], [38]. In the rigorous mathematical literature devoted to the time evolution problem, it originates in the of work Hepp [23], Ginibre and Velo [18] and, after lying dormant for about 30 years, Rodnianski and Schlein [40], followed by [19]. Currently, it is an active field. Our project, initiated in collaboration with Margetis in[19], is to study a PDE describing additional second order corrections (given by a Bogoliubov transformation eB ) to the right hand side of the approximation (2). Mathematically, Boguliubov transformations are representations of a group isomorphic to a real symplectic group, corresponding to the Segal-Shale-Weil representation in infinite dimensions, due to Shale. Interestingly, the theories seem to have evolved independently in physics and pure mathematics. Several important recent papers also use coherent states and Bogoliubov transformations. These include [2] and [1]. In fact, Theorem 2.2 in [1] proves that the unitary operators of the type used in [23] and [18] can be obtained, abstractly, as Bogoliubov transformations. We mention in passing that our concrete1 in eB (to be discussed below) agrees with that Bogoliubov transformation (up to a phase), but only when applied to the vacuum. Roughly speaking, our eB is not the 1We
mean that B = B(k) where k satisfies a PDE in 6+1 dimensions and is, in principle, computable numerically .
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3
operator corresponding to the evolution of the quadratic Bogoliubov Hamiltonian, but rather diagonalizes it. In the current paper we initiate the analysis of solutions to a coupled system of PDEs for the condensate φ and pair excitation k, see(22), (23a)-(23b) below. It would be very interesting to obtain estimates including the case β = 1, as well as large time estimates for the solutions of these equations which are uniform in N , similar to those obtained in earlier works for the uncoupled equations ((14a)-(14c) below) which describe the case β < 1/3. We hope to address these question in future work. In this paper we only consider the case 1/3 < β < 2/3, locally in time. See also the paper by Lewin, Nam, and Schlein [34] for a Fock space type approach to an L2 (R3N ) estimate based on corrections to a pure tensor product (Hartree state) rather than a coherent state. That approach has been generalized very recently to the case β < 1/3 my Nam and Napiorkowski in [39], and their work uses the linear equations (14b), (14c) introduced in [21], as well as some of the estimates from that paper. Very recently, after completing this work, we have also learned of the related and important paper [3]. A brief comparison of the results of that paper with ours is included at the end next section.
2. Background and statement of the main result We start with a very brief review of symmetric Fock space. This is included for the convenience of the reader, and follows closely the exposition from our earlier papers. We refer the reader to [22] for more details and comments. The elements of F are vectors of the form ψ = ψ0 , ψ1 (x1 ) , ψ2 (x1 , x2 ) , . . .
where ψ0 ∈ C and ψk are symmetric L2 functions. The inner product is ∞ Z X
φ, ψ = φ0 ψ0 + φn ψn . n=1
Thus we use physicists’ convention of an inner product linear in the second variable. The creation and annihilation distribution valued operators denoted by a∗x and ax respectively which act on vectors of the
4
M. GRILLAKIS AND M. MACHEDON
form (0, · · · , ψn−1 , 0, · · · ) and (0, · · · , ψn+1 , 0, · · · ) by n
1 X := √ δ(x − xj )ψn−1 (x1 , . . . , xj−1 , xj+1 , . . . , xn ) n j=1 √ ax (ψn+1 ) := n + 1ψn+1 ([x], x1 , . . . , xn )
a∗x (ψn−1 )
with [x] indicating that the variable x is frozen. The vacuum state is defined as follows: Ω := (1, 0, 0 . . .) One can easily check the canonical relations ax , a∗y = δ(x − y) and since the creation and annihilation operators are distribution valued we can form operators that act on F by introducing a field, say φ(x), and form Z Z ∗ ¯ ¯ a(φ) := dx φ(x)ax and a (φ) := dx {φ(x)a∗x } where by convention we associate a with φ¯ and a∗ with φ. Also define the skew-Hermitian operator Z ∗ ¯ A(φ) := dx φ(x)a (4) x − φ(x)ax and coherent states √
ψ := e−
N A(φ)
Ω.
(5)
It is easy to check that e
√ − N A(φ)
Ω=
. . . cn
n Y
! φ(xj ) . . .
2
with cn = e−N kφkL2 N n /n!
1/2
j=1
We also consider 1 B(k) := 2
Z
¯ x, y)ax ay − k(t, x, y)a∗ a∗ . dxdy k(t, x y
(6)
This particular construction and the corresponding unitary opera√ − N A −B tor M := e e were introduced (at least in the mathematics literature related to the problem under consideration) in [19]. The construction is in the spirit of Bogoliubov theory in physics, and the Segal-Shale-Weil representation in mathematics.
.
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The Fock Hamiltonian is 1 H := H1 − V where, (7a) N Z (7b) H1 := dxdy {∆x δ(x − y)a∗x ay } and Z 1 V := dxdy vN (x − y)a∗x a∗y ax ax , (7c) 2 where vN (x) = N 3β v N β x . It is a diagonal operator on Fock space, and it acts as a differential operator in n variable Hn, P DE =
n X
∆xj −
j=1
1 X 3β N v N β (xj − xk ) N i<j
on the nth component of F. Notice this is the same as (1), except that the dimension n is decoupled from the parameter N . Our goal is to study the evolution of coherent initial conditions of the form ψexact = eitH e−
√
N A(φ0 )
Ω
(8)
In our earlier papers[19, 20, 21, 22] we considered an approximation of the form √ ψappr := e− N A(φ(t) e−B(k(t)) Ω (9) and derived suitable Schr¨odinger type equations equations for φ(t, x), k(t, x, y) so that ψexact (t) ≈ eiN χ(t) ψappr (t), with χ(t) a real phase factor, in order to find precise estimates in Fock space, see Theorem (2.1) below. Our strategy is to consider ψred = eB(t) e
√
√ N A(t) itH − N A(0)
e
e
Ω
and then compute a ”reduced Hamiltonian” Hred =
1 ∂t M∗ M + M∗ HM i
(10)
so that 1 ∂t ψred = Hred ψred . i To state the results of [19, 20, 21] we define the operator kernel
(11)
gN (t, x, y) := −∆x δ(x − y) + (vN ∗ |φ|2 )(t, x)δ(x − y) + vN (x − y)φ(t, x)φ(t, y)
(12)
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M. GRILLAKIS AND M. MACHEDON
and 1 T Sold (s) := st + gN ◦ s + s ◦ gN and i 1 T , p] Wold (p) := pt + [gN i mN (x, y) := −vN (x − y)φ(x)φ(y) , vN (x) = N 3β v(N β x)
(13)
The main result of [21] can be summarized as follows: Theorem 2.1. Let φ and k satisfy 1 ∂t φ − ∆φ + vN ∗ |φ|2 φ = 0 i Sold (sh(2k)) = mN ◦ ch(2k) + ch(2k) ◦ mN Wold ch(2k) = mN ◦ sh(2k) − sh(2k) ◦ mN .
(14a) (14b) (14c)
with prescribed initial conditions φ(0, ·) = φ0 , k(0, ·, ·) = 0. If φ, k satisfy the above equations, then there exists a real phase function χ such that 4
ψexact (t) − eiN χ(t) ψappr (t) ≤ C(1 + t) log (1 + t) . (15) F N (1−3β)/2 provided 0 < β < 31 . See [21] for the reasons behind these equations. We also mention a very recent simple derivation of these equations in [39]. This result was extended to the case β < 12 in [32], where it was also argued informally that the equations of [21] do not provide an approximation for β > 12 . In the hope of obtaining an approximation for all β ≤ 1, in [22] we introduced a coupled refinement of the system (14a), (14b), (14c). The coupled equations of [22] were introduced the following way: Since Hred is a fourth order polynomial in a and a∗ , Hred Ω = (X0 , X1 , X2 , X3 , X4 , 0, · · · ).
(16)
The new, coupled equations for φ and k that we introduce in [22] can be written abstractly as X1 = 0 and X2 = 0.
(17)
It was shown there that they are Euler-Lagrange equations for the Lagrangian density X0 . Remark 2.2. The static terms of X0 (t) (not involving time derivatives) also appear in the recent paper [2], but do not serve as a Lagrangian there.
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To write down the equations (17) explicitly in terms of φ and k, we introduce Definition 2.3. Define 1 Λ(t, x1 , x2 ) = sh(2k)(t, x1 , x2 ) + φ(t, x1 )φ(t, x2 ) 2N 1 Γ(t, x1 , x2 ) = sh(k) ◦ sh(k) (t, x1 , x2 ) + φ(t, x1 )φ(t, x2 ) N and the new operator kernel
(18) (19)
g˜N (t, x, y) := −∆x δ(x − y) + vN (x − y)(trΓ)(t, x)δ(x − y) + vN (x − y)Γ(t, x, y) where tr denotes trace density, and define 1 1 T T ˜ ˜ ◦ s + s ◦ g˜N and W(p) := pt + [˜ gN , p] S(s) := st + g˜N i i In this notation, the following is proved in [22] Theorem 2.4. The equation X1 = 0 is equivalent to Z 1 ∂t φ(t, x) − ∆φ + vN (x − y)Λ(t, x, y)φ(t, y)dy i 1 + (vN ∗ T r(sh(k) ◦ sh(k)))(t, x)φ(t, x) NZ 1 + vN (x − y)(sh(k) ◦ sh(k))(t, x, y)φ(t, y)dy = 0 N
(20)
(21)
(22)
Here T r(sh(k) ◦ sh(k))(t, x) = (sh(k) ◦ sh(k))(t, x, x) denotes the trace density. The equation X2 = 0 is equivalent to ˜ (sh(2k)) + (vΛ) ◦ ch(2k) + ch(2k) ◦ (vΛ) = 0 S (23a) ˜ ch(2k) + (vΛ) ◦ sh(2k) − sh(2k) ◦ (vΛ) = 0 W (23b) Notice the similarity with (14a), (14b), (14c). Since it is difficult to prove estimates for these equations directly, we will write them down in a different, equivalent form. The derivation will be self-contained, and in fact most of the rest of this paper is independent of our previous work [19]-[22]. We now state the main result of our current paper. Theorem 2.5. Let 13 < β < 32 , and let the interaction potential v ∈ S satisfy v ≥ 0 and |ˆ v | ≤ wˆ for some w ∈ S. Let φ, k be solutions to (22), (23a), (23b) with φ(0, ·) ∈ S, k(0, ·) = 0. Then there exists a real
8
M. GRILLAKIS AND M. MACHEDON
function χ(t) = χN (t) such that for every > 0 there exists T0 > 0 and C such that √
kψexact − ψappr kF := keitH e− C ≤ 3β N 1− 2 −
N A(φ0 )
Ω − eiχ(t) e−
√
N A(φ(t)) −B(k(t))
e
ΩkF
uniformly for 0 ≤ t ≤ T0 . Remark 2.6. In the very recent and important paper [3], Bocatto, Cenatiempo and Schlein prove a result closely related to ours in the full range β < 1, and the estimate is global in time. However, there are substantial differences between their work and ours. Both papers approximate eitH ψinitial for some initial vector ψinitial . Our ψinitial is a √ − N A(φ0 ) coherent state e Ω, and the Bogoliubov term e−B(k) develops dynamically, while in [3] ψinitial must have e−B(k0 ) built into the initial conditions, where k0 is chosen to have a crucial explicit form. We√ mention that our approach could also accommodate ψinitial = e− N A(φ0 ) e−B(k0 ) Ω for k0 satisfying suitable estimates (see remark 3 in [19]), but we chose k0 = 0 for simplicity, since the proof would be the same. Also, the techniques used in [3] are quite different than ours, and the√ approximation in [3] is given (translating to our notation) by eiχ(t) e− N A(φ(t)) e−B(k(t)) U2,N (t)Ω where k(t) = k(t, x, y) is explicit (and related but different from our k(t)) and U2,N (t) is an evolution in Fock space with a quadratic generator (see the page preceding Theorem 1.1 in [3]). Given the complexity the evolution equation defining of U2,N (t), we believe there is still sufficient interest in having an approximation √ iχ(t) − N A(φ(t)) −B(k(t)) given by just e e e Ω where k satisfies a classical PDE in 6 + 1 variables.
3. The equations for Λ and Γ (self-contained derivation) As already mentioned, it seems difficult to obtain estimates (uniformly in N ) for φ and k directly from equations (23a), (23b). The equations seem linear, but the ”coefficients” vN Λ, vN Γ depend on sh(2k), ch(2k). We will proceed indirectly by deriving and studying equations for Λ and Γ.
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Theorem 3.1. The equations of Theorem (2.4) are equivalent to Z 1 (24a) ∂t − ∆x1 φ(x1 ) = − dy {v(x1 − y)Γ(y, y)} φ(x1 ) i Z − dy v(x1 − y)φ(y) Γ(y, x1 ) − φ(y)φ(x1 ) + v(x1 − y)φ(y) Λ(x1 , y) − φ(x1 )φ(y) 1 1 ∂t − ∆x1 − ∆x2 + v(x1 − x2 ) Λ(x1 , x2 ) (24b) i N Z = − dy {v(x1 − y)Γ(y, y) + v(x2 − y)Γ(y, y)} Λ(x1 , x2 ) Z n o − dy v(x1 − y) + v(x2 − y) Λ(x1 , y)Γ(y, x2 ) + Γ(x1 , y)Λ(y, x2 ) + Z + 2 dy v(x1 − y) + v(x2 − y) |φ(y)|2 φ(x1 )φ(x2 ) 1 ∂t − ∆x1 + ∆x2 Γ(x1 , x2 ) (24c) i Z = − dy v(x1 − y) − v(x2 − y) Λ(x1 , y)Λ(y, x2 ) + Z o n − dy v(x1 − y) − v(x2 − y) Γ(x1 , y)Γ(y, x2 ) + Γ(y, y)Γ(x1 , x2 ) Z + 2 dy v(x1 − y) − v(x2 − y) |φ(y)|2 φ(x1 )φ(x2 ) See (31a)-(31c) for the conceptual meaning of these equations in terms of the density matrices L defined below. While it is easy to prove this by direct calculation, we proceed with a derivation which is independent of Theorem (2.4) of our previous paper [22]. As in our previous papers [19]-[22], we consider M := e−
√
N A −B
e
and we have the evolution
1 ∂t M = HM − MHred and of course, i 1 − ∂t M∗ = M∗ H − Hred M∗ . i The evolution equation for M above is obvious from (10). Take a monomial of the form: (Wick ordered) Pm,n = a∗y1 a∗y2 . . . a∗ym ax1 ax2 . . . axn
(25) (26) (27)
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M. GRILLAKIS AND M. MACHEDON
and define the L matrices as follows,
1 Lm,n (t, y1 , . . . , ym ; x1 , . . . , xn ) := (m+n)/2 ay1 · · · aym MΩ, ax1 · · · axn MΩ N
1 = (m+n)/2 Ω, M∗ Pm,n MΩ N The notation is chosen so that the second set of variables are un-barred. We will often skip the t dependence, since it is passive in the calculations that we have in mind. Fortunately we will only need L0,1 , L1,1 and L0,2 (which turn out to be φ, Γ and Λ) but the computation is quite general. To get started, we observe that from the evolution of the operator M we have, 1 ∗ hence ∂t M PM = Hred , M∗ PM + M∗ P, H M i
1 1 ∂t L = (n+m)/2 Ω, Hred , M∗ PM Ω + Ω, M∗ P, H MΩ . i N (28) At this point we record the following lemma Lemma 3.2. If Hred Ω = µ, 0, 0, X3 , X4 , 0 . . . and P := a
or
P := aa
P := a∗ a
or
then
Ω, Hred , M∗ PM Ω = 0,
(29)
leaving only the second term in (28) Proof. We use the following notation: c = ch(k), Z ax (c) := Z ∗ ax (c) := Z ax (u) :=
u = sh(k) a∗x (u)
dy {ay c(y, x)} , dy a∗y c(y, x) =
Z
Z :=
dy a∗y u(y, x)
dy c(x, y)a∗y
by symmetry
dy {u(x, y)ay }
by symmetry
Z dy {ay u(y, x)} =
We have the conjugation formulas (see also (76)) √ √ M∗ ax M = ax (c) + a∗x (u) + N φ(x) := bx + N φ(x) √ √ M∗ a∗x M = a∗x (c) + ax (u) + N φ(x) := b∗x + N φ(x)
(30a) (30b)
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which implies the transformation of the monomial, √ √ M∗ P(a∗ , a)M = P b∗ + N φ , b + N φ . Now if P = a then, using (17), √ √
Ω, Hred bx + N φ(x) Ω − Ω, bx + N φ(x) Hred Ω = 0 . The argument is similar if P = aa or P = a∗ a since only the entries in the zeroth slot survive. Based on this, we easily prove the following proposition. Proposition 3.3. Under the assumptions of Lemma (3.2), the following equations hold 1∂ − ∆x1 L0,1 (t, x1 ) (31a) i ∂t Z = − vN (x1 − x2 )L1,2 (t, x2 ; x1 , x2 )dx2 1∂ + ∆x1 − ∆y1 L1,1 (t, x1 ; y1 ) (31b) i ∂t Z Z = vN (x1 − x2 )L2,2 (t, x1 , x2 ; y1 , x2 )dx2 − vN (y1 − y2 )L2,2 (t, x1 , y2 ; y1 , y2 )dy2 1∂ 1 − ∆x1 − ∆x2 + vN (x1 − x2 ) L0,2 (t, x1 , x2 ) (31c) i ∂t N Z Z = − vN (x1 − y)L1,3 (t, y; x1 , x2 , y)dy − vN (x2 − y)L1,3 (t, y; x1 , x2 , y)dy The equation (31b) is one of the BBGKY equations, but in our case L2,2 can be expressed in terms of the earlier matrices, see Lemma (3.4). Proof. With P any monomial of degree one or two we know from Lemma (3.2) that
Ω, Hred , M∗ PM Ω = 0 and for any of the corresponding matrices L we arrive at the equation, 1
1 , α = 1/2 , 1 . ∂t L = α Ω, M∗ P, H MΩ i N We need to compute [P, H] and for this purpose recall our original Hamiltonian, Z H=
dxdy {∆x δ(x −
y)a∗x ay }
1 − 2N
Z
dxdy vN (x − y)a∗x a∗y ax ax
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M. GRILLAKIS AND M. MACHEDON
Below we list the commutators with each mononomial: Z 1 ax1 , H = ∆x1 ax1 − dy v(x1 − y)a∗y ay ax1 N 1 ax1 ax2 , H = ∆x1 ax1 ax2 + ∆x2 ax1 ax2 − v(x1 − x2 )ax1 ax2 N Z 1 − dz v(x1 − z) + v(x2 − z) a∗z az ax1 ax2 N ∗ ax1 ax2 , H = a∗x1 ∆x2 ax2 − (∆x1 ax1 )∗ ax2 Z 1 + dz v(x1 − z) − v(x2 − z) a∗x1 a∗z az ax2 N from which we can derive the corresponding evolution equations for the L matrices:
Z
1 1 ∂t − ∆x1 L0,1 (t, x1 ) = − 3/2 dyv(x1 − y) Ω, M∗ a∗y ay ax1 MΩ i N (32)
1 1 ∂t − ∆x1 − ∆x2 + v(x1 − x2 ) L0,2 (t, x1 , x2 ) = i N Z
1 − 2 dz v(x1 − z) + v(x2 − z) Ω, M∗ a∗z az ax1 ax2 MΩ (33) N
and finally 1 ∂t + ∆x1 − ∆x2 L1,1 (t, x1 , x2 ) = i Z
1 + 2 dz v(x1 − z) − v(x2 − z) Ω, M∗ a∗x1 a∗z az ax2 MΩ (34) N
which implies the statement of the proposition.
The proof of Theorem (3.1) is finished by computing the necessary matrices L. We need the following (writing throughout this proof [x]
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indicates freezing the variable x):
M∗ ax1 MΩ √ √ = bx1 + N φ(x1 ) Ω = N φ([x1 ]), u(y, [x1 ]), 0, 0 . . .
(35)
M∗ ax1 ax2 MΩ = √ √ bx1 + N φ(x1 ) bx2 + N φ(x2 ) Ω = f0 , f1 , f2 , 0, 0 . . . where the entries are : f0 ([x1 ], [x2 ]) = N φ([x1 ])φ([x2 ]) + u ◦ c))[x1 ], [x2 ]) = N Λ([x1 ], [x2 ]) (36) i √ h f1 (y, [x1 ], [x2 ]) = N φ([x1 ])u(y, [x2 ]) + φ([x2 ])u(y, [x1 ]) (37) h i 1 √ f2 (y1 , y2 , [x1 ], [x2 ]) = u(y1 , [x1 ])u(y2 , [x2 ]) + u(y2 , [x1 ])u(y1 , [x2 ]) 2 (38)
and similarly,
M∗ a∗x1 ax2 MΩ = √ √ b∗x1 + N φ(x1 ) bx2 + N φ(x2 ) Ω = g0 , g1 , g2 , 0, 0 . . . where the entries are : g0 [x1 ], [x2 ]) = N φ([x1 ])φ([x2 ]) + (u ◦ u)([x1 ], [x2 ]) = N Γ([x1 ], [x2 ]) (39) h i √ g1 (y, [x1 ], [x2 ]) = N φ([x1 ])u(y, [x2 ]) + c(y, [x1 ])φ([x2 ]) (40) i 1 h g2 (y1 , y2 , [x1 ], [x2 ]) = √ c(y1 , [x1 ])u(y2 , [x2 ]) + c(y2 , [x1 ])u(y1 , [x2 ]) 2 (41)
Based on this we easily compute
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M. GRILLAKIS AND M. MACHEDON
Lemma 3.4. The L matrices are given by √ 1
L0,1 (t, x1 ) = √ Ω, bx1 + N φ(x1 ) Ω = φ(x1 ) N √ √ 1
Ω, bx1 + N φ(x1 ) bx2 + N φ(x2 ) Ω L0,2 (t, x1 , x2 ) = N 1 = (u ◦ c)(x1 , x2 ) + φ(x1 )φ(x2 ) N 1 = ψ(t, x1 , x2 ) + φ(t, x1 )φ(t, x2 ) = Λ 2N where ψ = sh(2k) = 2u ◦ c 1 L1,1 (t, x1 ; x2 ) = u ◦ u)(x1 , x2 ) + φ(x1 )φ(x2 ) N 1 = ω(t, x1 , x2 ) + φ(t, x1 )φ(t, x2 ) = Γ 2N where ω := ch(2k) − 1 = 2u ◦ u , 1 1 ψ(x3 , x2 )φ(x1 ) + u ◦ u(x1 , x3 )φ(x2 ) L1,2 (x1 ; x2 , x3 ) = Γ(x1 , x2 )φ(x3 ) + 2N N Z 1 L2,2 (y1 , y2 ; x1 , x2 ) = Λ(y1 , y2 )Λ(x1 , x2 ) + 2 f 1 (y, y1 , y2 )f1 (y, x1 , x2 )dy N Z 1 + 2 f 2 (y, z, y1 , y2 )f2 (y, z, x1 , x2 )dydz N Z 1 L1,3 (y1 ; x1 , x2 , x3 ) = Γ(x1 , y1 )Λ(x2 , x3 ) + 2 g 1 (y, x1 , y1 )f1 (y, x2 , x3 )dy N Z 1 + 2 g 2 (y, z, x1 , y1 )f2 (y, z, x2 , x3 )dydz N where the integrals in the last two formulas can be trivially expressed in terms of ψ and ω, which in turn can be expressed in terms of φ, Λ and Γ. Proof. All calculations are straightforward. For instance, 1
1
L1,2 (x1 ; x2 , x3 ) = 3/2 Ω, M∗ a∗x1 ax2 ax3 MΩ = 3/2 Ω, M∗ a∗x1 ax2 MM∗ ax3 MΩ N N 1 ∗ ∗ = 3/2 M ax2 ax1 MΩ, M∗ ax3 MΩ N Z 1 = g 0 (x2 , x1 )φ(x3 ) + g 1 (y, x2 , x1 )u(y, x3 )dy N 1 1 = Γ(x1 , x2 )φ(x3 ) + u ◦ c(x3 , x2 )φ(x1 ) + u ◦ u(x1 , x3 )φ(x2 ) N N 1 1 = Γ(x1 , x2 )φ(x3 ) + sh(2k)(x3 , x2 )φ(x1 ) + u ◦ u(x1 , x3 )φ(x2 ) 2N N
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15
1 ∗ ∗ M a a MΩ, M a a MΩ y y x x 1 2 1 2 N2 Z 1 1 = 2 f 0 (y1 , y2 )f0 (x1 , x2 ) + 2 f 1 (y, y1 , y2 )f1 (y, x1 , x2 )dy N N Z 1 + 2 f 2 (y, z, y1 , y2 )f2 (y, z, x1 , x2 )dydz N
L2,2 (y1 , y2 ; x1 , x2 ) =
With these ingredients at hand we proceed to write down the evolution of L1 = φ : Z 1 ∂t − ∆x1 + dy {v(x1 − y)Γ(y, y)} φ(x1 ) = i Z 1 − dy v(x1 − y)φ(y)ω(y, x1 ) + v(x1 − y)φ(y)ψ(x1 , y) 2N and we can eliminate ψ and ω by the substitution, ω = 2N Γ − φ ⊗ φ ψ = 2N Λ − φ ⊗ φ so that we have a system involving only Λ, Γ and φ matrices. The evolution of Λ is given by the expression below: Z 1 1 ∂t − ∆x1 − ∆x2 + v(x1 − x2 ) + dy {(v(x1 − y) + v(x2 − y)) Γ(y, y)} Λ = i N Z n o 1 − dy v(x1 − y) + v(x2 − y) φ(y) ω(y, x1 )φ(x2 ) + ω(y, x2 )φ(x1 ) 2N Z n o 1 − dy v(x1 − y) + v(x2 − y) φ(y) ψ(x1 , y)φ(x2 ) + ψ(x2 , y)φ(x1 ) 2N Z n o 1 − dy v(x − y) + v(x − y) ψ(x , y)ω(y, x ) + ω(y, x )ψ(x , y) 1 2 1 2 1 2 4N 2 Finally the evolution of Γ is given by: Z 1 ∂t + ∆x1 − ∆x2 Γ = dy v(x1 − y) − v(x2 − y) Λ(x1 , y)Λ(y, x2 ) + i Z n o 1 dy v(x1 − y) − v(x2 − y) φ(y)ω(y, x2 )φ(x1 ) + φ(y)ω(x1 , y)φ(x2 ) + 2N Z n o 1 2 dy v(x1 − y) − v(x2 − y) |φ(y)| ω(x1 , x2 ) + ω(y, y)φ(x1 )φ(x2 ) + 2N Z n o 1 dy v(x − y) − v(x − y) ω(x , y)ω(y, x ) + ω(y, y)ω(x , x ) 1 2 1 2 1 2 4N 2
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M. GRILLAKIS AND M. MACHEDON
If we substitute ψ = 2N Λ − φ ⊗ φ and ω = 2N Γ − φ ⊗ φ we obtain the equations of Theorem (3.1). Remark 3.5. Using the ideas in this section one can easily
show that the expected number of particles for our approximation, MΩ,
N MΩ R ∗ (where N = ax ax dx), as well as the energy MΩ, HMΩ are constant in time. This provides an easier proof of some of the results of section 8 of [22]. 4. Estimates In this paper, S and S± stand for the pure differential operators, ˜ = S + potential terms, so that the operators of Theorem (2.4) are S ˜ = S± + potential terms W 1 S = ∂t − ∆ (in 6 + 1 or 3 + 1 dimensions, as will be clear from the context) i 1 S± = ∂t − ∆x + ∆y i and potential terms are given by composition with vN (x − y)Γ(x, y) and multiplication by vN ∗ T rΓ. The symbol of S is τ +|ξ|2 or τ +|ξ|2 +|η|2 , depending on dimensions, and the symbol of S± is τ + |ξ|2 − |η|2 . We will use the following norms: δ kf kX δ = k 1 + |τ + |ξ|2 | fˆ(τ, ξ)kL2 δ kf kXSδ = k 1 + |τ + |ξ|2 + |η|2 | fˆ(τ, ξ, η)kL2 2 2 δ ˆ f (τ, ξ, η)kL2 kf kXW δ = k 1 + |τ + |ξ| − |η| | and refer to [42], section 2.6 for their history and properties. Of special importance is the following result: (proposition 2.12 in [42]): if Su = f , δ > 0 and χ(t) is a fixed C0∞ , cut-off function, then kχ(t)ukX 21 +δ .χ,δ ku(0, ·)kL2 + kf kX − 21 +δ
(42)
For the rest of the paper, we will use the standard notation A .χ,δ B to mean ”there exists a constant C depending of χ and δ such that A ≤ CB. We will also use freely the general principle that if an Lp estimate holds for solutions to the homogeneous equation, it also holds in X 1/2+δ spaces, see Lemma 2.9 in [42]. To get started, fix w ∈ S such that |ˆ v | ≤ wˆ and fix > 0 depending 1 1 + on β < 2/3 so that < ∇x > 2 < ∇y > 2 + N1 wN (x − y), which is a
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17
function of x − y, satisfies 1
1 C wN kL6/5 (d(x−y) ≤ small power N N
1
k < ∇x > 2 + < ∇y > 2 +
The basic space-time collapsing estimates, in the spirit of [29], are: Lemma 4.1. If SΛ = 0 then 1/2
k|Λ(t, x, x)kL2 (dtdx) . k|∇|x−y Λ0 (x, y)kL2 (dxdy)
(43)
k|∇|1/2 x Λ(t, x, x)kL2 (dtdx)
(44)
.
k|∇|x1/2 |∇|y1/2 Λ0 (x, y)kL2 (dxdy)
As a consequence, if SΛ = F and δ > 0, then 1
sup k < ∇x > 2 + χ(t)Λ(t, x, x + z)kL2 (dtdx)
(45)
z 1
1
1 + 2
1 + 2
.δ k < ∇x > 2 + < ∇y > 2 + χ(t)ΛkX 1/2+δ S
.δ k < ∇x > + k < ∇x >
1 + 2
< ∇y >
< ∇y >
1 + 2
Λ0 (x, y)kL2 (dxdy)
F kX −1/2+δ S
If S± Γ = 0, then 1
k|∇x | 2 + Γ(t, x, x)kL2 (dtdx) + k|∇x |1/2 Γ(t, x, x)kL2 (dtdx) 1
1
. k < ∇x > 2 + < ∇y > 2 + Γ0 (x, y)kL2 (dxdy)
(46)
As a consequence, if S± Γ = F , then 1
sup k|∇x | 2 + χ(t)Γ(t, x, x + z)kL2 (dtdx) + sup k|∇x |1/2 χ(t)Γ(t, x, x + z)kL2 (dtdx) z
.,δ < ∇x >
z 1 + 2
< ∇y >
1 + 2
1 + 2
W
.,δ k < ∇ >x < ∇y > + k < ∇x >
1 + 2
< ∇y >
χ(t)ΓkX 1/2+δ
1 + 2
1 + 2
Γ0 (x, y)kL2 (dxdy)
F kX −1/2+δ
(47)
W
Remark 4.2. Notice that the estimates for Λ are different from those for Γ at low frequencies. We cannot estimate kΓ(t, x, x)kL2 (dtdx) . The ”lowest” derivative we can estimate in (46) is k|∇x |1/2 Γ(t, x, x)kL2 (dtdx) .
18
M. GRILLAKIS AND M. MACHEDON
e denote Proof. The same method, inspired by [28], works for both. Let Λ the space-time Fourier transform of Λ. For (43) ^ |Λ(t, x, x)(τ, ξ)|2 Z 1 . δ(τ − |ξ − η|2 − |ξ + η|2 ) dη |η| Z 1 \ 2 δ(τ − |ξ − η|2 − |ξ + η|2 )|∇x−y Λ0 (ξ − η, ξ + η)|2 dη In order to prove the estimate, we must show Z 1 sup δ(τ − |ξ|2 − |η|2 ) dη . 1 |η| τ,ξ which is obvious. For (44) 2 ^ |∇1/2 x Λ(t, x, x)(τ, ξ)| Z . δ(τ − |ξ − η|2 − |ξ + η|2 )
|ξ| dη |ξ − η||ξ + η| Z 1 1 \ δ(τ − |ξ − η|2 − |ξ + η|2 )|∇x2 ∇y2 Λ0 (ξ − η, ξ + η)|2 dη
In order to prove the estimate, we must show Z |ξ| sup δ(τ − |ξ|2 − |η|2 ) dη . 1 |ξ − η||ξ + η| τ,ξ Without loss of generality, consider the region |ξ − η| ≤ |ξ + η|. If |ξ| 1 . |η| and the integral can be evaluated |ξ − η| ∼ |ξ + η|, |ξ−η||ξ+η| in polar coordinates. If |ξ − η| 1+ < ξ + η >1+ τ,ξ To prove this, take ξ = (|ξ|, 0, 0). Then Z |ξ||ξ − η| dη δ(τ − ξ · η) < ξ − η >1+ < ξ + η >1+ Z Z |ξ| 1 = δ(τ − |ξ|η1 ) dη . dη2 dη3 . 1 1+ 2+ < ξ − η >< ξ + η > R2 < η2,3 > Next, we record some Strichartz type estimates Lemma 4.3. The following estimate holds keit(∆x ±∆y ) f kL2 (dt)L6 (dx)L2 (dy) . kf kL2 .
(48)
keit(∆x ±∆y ) f kL2 (dt)L∞ (dx)L2 (dy) . k < ∇x >1/2+ f kL2 .
(49)
and, as a consequence kF kL2 (dt)L6 (dx)L2 (dy) .δ kF kX 1/2+δ
(50)
where X can be either XS or XW . Proof. We argue as follows: kkeit(∆x ±∆y ) f kL2 (dy) kL2 (dt)L6 (dx) = kkeit∆x f kL2 (dy) kL2 (dt)L6 (dx) . kkeit∆x f kL2 (dt)L6 (dx) kL2 (dy) . kkeit∆x f kL2 (dt)L6 (dx) kL2 (dy) . kf kL2 (dxdy) 3 + 1 end-point Strichartz [25] The proof of (48) is similar, using Sobolev. This type of estimate has first appeared, we believe, in [7]. Versions of it were used in [9], [10], [32]. We will need the following refinements. Notice that for S we can choose x, y coordinates or x + y, x − y coordinates, but this is not possible for S± . Lemma 4.4. For each 0 < δ < 12 there exists 6/5+ > 6/5 a number which can be chosen arbitrarily close to 6/5 if δ is small such that the following estimate holds kΛk
−1 2 +δ
XS
.δ kΛkL2 L6/5+ (x−y)L2 (x+y)
(51) (52)
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M. GRILLAKIS AND M. MACHEDON
Proof. This is proved by interpolating the estimate dual to (50) kΛk
−1 2 −δ
XS
. kΛkL2 L6/5 (x−y)L2 (x+y)
(53)
= kΛkL2 L2 (x−y)L2 (x+y)
(54)
with kΛk
1+1 −2 2
XS
Also we will need the closely related Lemma 4.5. For each 0 < δ < 21 there exist numbers 2−, 6/5+ arbitrarily close to 2, 6/5 if δ is close to 0, so that kF kX −1/2+δ .δ kF kL2− (dt)L6/5+ (dx)L2 d(y)
(55)
W
Proof. We start with kF kL2+ (dt)L6− (dx)L2 (dy) . kF kX 1/2+δ (interpolate (50) with energy estimates) W
kF kX −1/2−δ . kF kL2− (dt)L6/5+ (dx)L2 (dy) (dual estimates) W
and interpolate with the trivial estimate kF kX −1/2+1/2 = kF kL2 (dt)L2 (dxdy) W
we get (55).
Finally, we have one more estimate along the same lines Lemma 4.6. If SΛ = 0 then 1
1
kΛkL2 (dt)L∞ (d(x−y))L2 (d(x+y)) . k < ∇x > 2 + < ∇y > 2 + Λ0 kL2 (dxdy) and, as a consequence, if SΛ = F then kχ(t)ΛkL2 (dt)L∞ (d(x−y))L2 (d(x+y)) 1
1
1
1
. k < ∇x > 2 + < ∇y > 2 + Λ0 kL2 (dxdy) + k < ∇x > 2 + < ∇y > 2 + F k
−1 2 +δ
XS
Proof. Writing (49) in x + y, x − y coordinates keit(∆x +∆y ) Λ0 kL2 (dt)L∞ (d(x−y))L2 (d(x+y)) 1
1
1
. k < ∇x−y > 2 + Λ0 kL2 (dxdy) . k < ∇x > 2 + < ∇y > 2 + Λ0 kL2 (dxdy)
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21
Note that here we are forced to use < ∇ > rather than ∇. With the help of Lemma (4.6) we can estimate solutions to 1 S + vN (x − y) Λ = F N Λ(0, ·) = Λ0 (·)
(56)
The next proposition is the key estimate of our paper. Proposition 4.7. If Λ satisfies (56), β < 2/3, and χ(t) is a smooth cut-off function which is 1 on [0, 1], δ is sufficiently small, and N ≥ N0 is sufficiently large, then 1
1
k < ∇x > 2 + < ∇y > 2 + χ(t)Λk 1
(57)
1 +δ
XS2
1
1
1
. k < ∇x > 2 + < ∇y > 2 + Λ0 (x, y)kL2 (dxdy) + k < ∇x > 2 + < ∇y > 2 + F k
1 +δ −2
XS
and thus, by Lemma (4.1), 1
sup k < ∇x > 2 + χ(t)Λ(t, x + z, x)kL2 (dtdx)
(58)
z 1
1
1
1
. k < ∇x > 2 + < ∇y > 2 + Λ0 (x, y)kL2 (dxdy) + k < ∇x > 2 + < ∇y > 2 + F k ∂ Similar estimates hold for ∂t , ∇x+y commute with the potential.
n
1 +δ −2
XS
(χ(t)Λ) because these derivatives
Proof. Recall we introduces a potential w ∈ S such that w ˆ ≥ |ˆ v |. For (57), we use another cut-off function χ1 (t) which is identically 1 on the support of χ and notice that the solution of (56) agrees with that of 1 vN χ1 (t)Λ = F N Λ(0, ·) = Λ0 (·) SΛ +
(59)
on the support of χ. So we work with the solution of (59). Also, for technical reasons, we define bχ1 (t)Λc to be the inverse Fourier transform of the absolute value of the (space-time) Fourier transform of χ1 (t)Λ. Then 1
1
k < ∇x > 2 + < ∇y > 2 + χ1 (t)Λk 12 +δ XS 1 1 1 + + . k < ∇x > 2 < ∇y > 2 vN χ1 (t)Λ k − 21 +δ XS N 1
1
(60) (61) 1
1
+ k < ∇x > 2 + < ∇y > 2 + Λ0 (x, y)kL2 (dxdy) + k < ∇x > 2 + < ∇y > 2 +δ F k
1 +δ −2
XS
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M. GRILLAKIS AND M. MACHEDON
We plan to absorb (61) in the LHS of (60). The reason we introduced w and bχ1 (t)Λc, which have non-negative Fourier transforms, is to have the following cheap substitute for the Leibnitz rule: 1 k < ∇x > < ∇y > χ1 (t)vN Λ k − 21 +δ XS N 1 1 1 . k < ∇x > 2 + < ∇y > 2 +δ wN bχ1 (t)Λck − 21 +δ XS N 1 1 1 + k < ∇x > 2 + wN < ∇y > 2 + bχ1 (t)Λck − 21 +δ XS N 1 1 1 + k < ∇y > 2 + wN < ∇x > 2 + bχ1 (t)Λck − 21 +δ XS N 1 1 1 + k wN < ∇x > 2 + < ∇y > 2 + bχ1 (t)Λck − 21 +δ XS N 1 + 2
1 + 2
1
(62a) (62b) (62c) (62d)
1
For the most singular term, (62a), < ∇x > 2 + < ∇y > 2 + N1 wN (x − y) is a function of x − y and we recall was chosen so that 1 1 k < ∇x > 2 + < ∇y > 2 + N1 wN kL6/5 ≤ N smallC power . At this stage we insist δ is so small that the corresponding number 6/5+ also satisfies 1 1 k < ∇x > 2 + < ∇y > 2 + N1 wN kL6/5+ ≤ N smallC power . We estimate, using (51) and Lemma (4.6) 1 1 1 + + wN bχ1 (t)ΛckX − 21 +δ k < ∇x > 2 < ∇y > 2 N 1 + 12 + 1 2 . k ∇x ∇y wN bχ1 (t)ΛckL2 (dt)L6/5+ (d(x−y))L2 (d(x+y)) N 1 + 1 + 1 wN kL6/5+ (d(x−y)) kbχ1 (t)ΛckL2 (dt)L∞ (d(x−y))L2 (d(x+y)) . k∇x2 ∇y2 N 1 1 1 . small power k < ∇x > 2 + < ∇y > 2 + bχ1 (t)ΛckX 21 +δ N 1 1 1 = small power k < ∇x > 2 + < ∇y > 2 + χ1 (t)ΛkX 21 +δ N
This can be absorbed in the LHS of (60). The other terms are easier. While we will use the X spaces as tools in our proofs, the actual norms in which we prove well-posedness are Lp norms defined for some
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23
0 < T ≤ 1. 1
NT (Λ) = sup k < ∇x > 2 + Λ(t, x + z, x)kL2 ([0,T ]×R3 ) z 1
1
+ k < ∇x > 2 + < ∇y > 2 + Λ(t, x, y)kL∞ ([0,T ]×L2 (R6 )) ˙ T (Γ) = sup k|∇x | 21 + Γ(t, x + z, x)kL2 ([0,T ]×R3 ) N z
+ sup k||∇x |1/2 Γ(t, x + z, x)kL2 ([0,T ]×R3 ) z 1
1
+ k < ∇x > 2 + < ∇y > 2 + Γ(t, x, y)kL∞ ([0,T ]×L2 (R6 )) 1
1
NT (φ) =k < ∇x > 2 + φkL∞ ([0,T ])L2 + k < ∇x > 2 + φkL2 ([0,T ])L6 Based on energy and Strichartz estimates, the collapsing estimates of Lemma (4.1) and general properties of X spaces which allow one to transfer estimates for solutions to the homogeneous equation to estimates in X 1/2+δ spaces we have 1
1
NT (Λ) . k < ∇x > 2 + < ∇y > 2 + Λk
1 +δ
XS2
˙ T (Γ) . k < ∇x > 12 + < ∇y > 12 + Γk N
1 +δ
2 XW
˙ T (φ ⊗ φ) . k < ∇x > 21 + φk2 NT (φ ⊗ φ) + N
1
X 2 +δ
Our basic estimates for the inhomogeneous linear equations are Proposition 4.8. If Γ and φ satisfy S± Γ = F Sφ = G 0 ≤ T ≤ 1, and 2− < 2, 6/5+ > 6/5 are fixed numbers which can be chosen close to 2 and 6/5, then ˙ T (Γ) N 1
1
1
1
. k < ∇x > 2 + < ∇y > 2 + Γ(0, x, y)kL2 (dxdy) + k < ∇x > 2 + < ∇y > 2 + F kL2− [0,T ]L6/5+ (dx)L2 (dy) NT (φ) 1
1
. k < ∇x > 2 + φ(0, ·)kL2 (dx) + k < ∇x > 2 + GkL2− [0,T ]L6/5+ (dx) ˙ T (φ ⊗ φ) . NT (φ)2 NT (φ ⊗ φ) + N Similar estimates hold when x and y are reversed.
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M. GRILLAKIS AND M. MACHEDON
Proof. We will prove this for the equation for Γ, the other one for φ being easier. The standard energy estimate and the collapsing estimate of lemma (4.1) prove that if S± Γ = 0, then ˙ T (Γ) . k < ∇x > 21 + < ∇y > 12 + Γ0 (x, y)kL2 (dxdy) N From here and general properties of X 1/2+δ spaces, we get, for any δ > 0, ˙ T (Γ) .δ k < ∇x > 12 + < ∇y > 21 + Γk 1/2+δ N X W
uniformly in T . Finally, to prove the stated estimate, let FT = F in [0, T ], and 0 otherwise. Let ΓT be the solution to S± ΓT = FT ΓT (0, ·) = Γ(0, ·) Then Γ = ΓT in [0, T ] and recalling 0 < T ≤ 1 and χ = 1 on [0, 1], ˙ T (Γ) = N ˙ T (χ(t)ΓT ) N 1
1
. k < ∇x > 2 + < ∇y > 2 + χ(t)ΓT k 1
1
1
1
1 +δ
XS2
. k < ∇x > 2 + < ∇y > 2 + Γ0 (x, y)kL2 (dxdy) + k < ∇x > 2 + < ∇y > 2 + FT k 1
1
1
1
1
1
1
1
−1 2 +δ
XS
(by (42))
. k < ∇x > 2 + < ∇y > 2 + Γ0 (x, y)kL2 (dxdy) + k < ∇x > 2 + < ∇y > 2 + FT kL2− (dt)L6/5+ (dx)L2 (dy) (by (55)) = k < ∇x > 2 + < ∇y > 2 + Γ0 (x, y)kL2 (dxdy) + k < ∇x > 2 + < ∇y > 2 + F kL2− [0,T ]L6/5+ (dx)L2 (dy) The same type of result holds for the equation for Λ. Proposition 4.9. There exist numbers 2− < 2, 6/5+ > 6/5, which can be chosen arbitrarily close to 2 and 6/5 such that, if Λ satisfies 1 S + vN (x − y) Λ = F (63) N Λ(0, ·) = Λ0 (·) β < 2/3, and 0 ≤ T ≤ 1, then
PAIR EXCITATIONS, II
NT (Λ)
25
(64)
. k < ∇x >
1 + 2
< ∇y >
1
1 + 2
Λ0 (x, y)kL2 (dxdy)
1
+ k < ∇x > 2 + < ∇y > 2 + F kL2− [0,T ]L6/5+ (dx)L2 (dy) ∂ n (∇x+y )m Λ: Similar estimates hold for ∂t n ∂ m NT (∇x+y ) Λ (65) ∂t n 1 1 ∂ m + + (∇x+y ) Λ(t, x, y) kL2 (dxdy) . k < ∇x > 2 < ∇y > 2 ∂t t=0 n 1 1 ∂ + k < ∇x > 2 + < ∇y > 2 + (∇x+y )m F kL2− [0,T ]L6/5+ (dx)L2 (dy) ∂t Proof. The proof is the same as the one of Proposition (4.8), except that, because of the potential vN we don’t get the estimate (42) from general principles but use Proposition (4.7). To prove (65), first differentiate the equation, noting that these derivatives commute with the potential, then apply the previous result. 5. The non-linear equations Now we come to our main PDE result Theorem 5.1. Let Λ, Γ and φ be solutions of (24b), (24c), (24a) with initial conditions φ0 , k0 ∈ S. There exists N0 such that for all N ≥ N0 , the following estimates hold: 1
1
NT (Λ) . k < ∇x > 2 + < ∇y > 2 + Λ(0, ·)kL2 smallpower 4 ˙ +T NT (Λ) NT (Γ) + NT (φ) ˙ T (Γ) . k < ∇x > 21 + < ∇y > 12 + Γ(0, ·)kL2 N smallpower 2 2 4 ˙ +T NT (Λ) + NT (Γ) + NT (φ) 1
NT (φ) . k < ∇x > 2 + φ(0, ·)kL2 smallpower 2 ˙ +T NT (Λ) + NT (Γ) + NT (φ) NT (φ)
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M. GRILLAKIS AND M. MACHEDON
So there exists T0 > 0 such that, if T ≤ T0 , 1
1
NT (Λ) . k < ∇x > 2 + < ∇y > 2 + Λ(0, ·)kL2 ˙ T (Γ) . k < ∇x > 21 + < ∇y > 12 + Γ(0, ·)kL2 N 1
NT (φ) . k < ∇x > 2 + φ(0, ·)kL2
Also, similar estimates hold for the derivatives which commute with the potential: n ∂ m ∇x+y Λ (66) NT ∂t n 1 1 ∂ + + m 2 2 . k < ∇x > < ∇y > ∇x+y Λ kL2 ∂t t=0 n ∂ ˙T N ∇m x+y Γ ∂t n 1 1 ∂ + + m ∇x+y Γ kL2 . k < ∇x > 2 < ∇y > 2 ∂t nt=0 n 1 ∂ ∂ NT ∇m . k < ∇x > 2 + ∇m x φ x φ|t=0 kL2 ∂t ∂t Proof. For equation (24a), which can be abbreviated as 1 ∂t φ − ∆φ i = −(vN Λ) ◦ φ − (vN Γ) ◦ φ − (vN ∗ T rΓ) · φ + 2(vN ∗ |φ|2 )φ := RHS(24a) we first apply Proposition (4.8): 1
1
NT (φ) . k < ∇x > 2 + φ(0, ·)kL2 + k < ∇x > 2 + RHS(24a)kL2− ([0,T ])L6/5+ (dx) 1
1
. k < ∇x > 2 + φ(0, ·)kL2 + T small power k < ∇x > 2 + RHS(24a)kL2 ([0,T ])L6/5+ (dx) 1 + small power 2 ˙ 2 . k < ∇x > φ(0, ·)kL2 + T NT (Λ) + NT (Γ) + NT (φ) NT (φ)
The last line follow from the classical fractional Leibnitz rule in Lp spaces due to Kato and Ponce, [24] . We only present a typical term
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in RHS(24a): k < ∇x > (vN ∗ T rΓ) · φ kL2 ([0,T ])L6/5+ (dx) 1 + . k(vN ∗ T rΓ) · φkL2 ([0,T ])L6/5+ (dx) + k|∇x | 2 (vN ∗ T rΓ) · φ kL2 ([0,T ])L6/5+ (dx) 1 + 2
. kT rΓkL2 ([0,T ])L3 (dx) kφkL∞ L2+ 1
1
+ k|∇x | 2 + T rΓkL2 ([0,T ])L2 (dx) kφkL∞ L3+ + kT rΓkL2 ([0,T ])L3+ (dx) k∇x | 2 + φkL∞ L2 ˙ T (Γ)NT (φ) .N Now we deal with the equation (24b), which can be abbreviated as 1 S + vN Λ N = −(vN Λ) ◦ Γ − Γ ◦ (vN Λ) − (vN ∗ T rΓ) (x) + (vN ∗ T rΓ) (y) Λ(x, y) − vN Γ ◦ Λ − Λ ◦ (vN Γ) + 2(vN ∗ |φ|2 )(x)φ(x)φ(y) + 2(vN ∗ |φ|2 )(y)φ(y)φ(x) := RHS(24b) Applying Proposition (4.9) to the equation (24b)
1
1
NT (Λ) . k < ∇x > 2 + < ∇y > 2 + Λ(0, ·)kL2 1
1
+ k < ∇x > 2 + < ∇y2
+
> RHS(24b)kL2− [0,T ]L6/5+ (dx)L2 (dy) 1
1
6
We would like to estimate < ∇x > 2 + < ∇y > 2 + RHS(24b) in L2 L 5 + L2 to gain a small power of T from Cauchy-Schwarz in time. Typical term in RHS (24b): 1 + 2
1 + 2
< ∇x > < ∇y > (vN Λ) ◦ Γ(x, y) Z 1 1 + = vN (z) < ∇x > 2 + < ∇y2 > (Λ(x, x − z)Γ(x − z, y)) dz
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M. GRILLAKIS AND M. MACHEDON
Applying the fractional Leibniz rule in Lp spaces, Z 1 1 + + 2 2 vN (z)k < ∇x > < ∇y > Λ(x, x − z)Γ(x − z, y) kL2 ([0,T ])L 56 + (dx)L2 (dy) dz Z 1 . vN (z)k < ∇x > 2 + Λ(x, x − z)kL2 ([0,T ])L2 (dx) × 1
k < ∇y2
+
> Γ(x − z, y)kL∞ ([0,T ])L3+ (dx)L2 (dy) dz
Z +
vN (z)kΛ(x, x − z)kL2 ([0,T ])L3+ (dx) × 1
1
k < ∇x > 2 + < ∇y2 ˙ T (Γ) . NT (Λ)N
+
> Γ(x − z, y)kL∞ ([0,T ])L2 (dx)L2 (dy) dz
All other terms are treated in a similar manner. In fact, all terms on the RHS(24b) and RHS(24c) are of the form (vN F ) ◦ G or (vN ∗ T rF )G where F , G can be Λ, Γ, φ ⊗ φ of φ ⊗ φ (or their complex conjugates) and NT (F ), NT (G) are estimated as above by Proposition (4.8) or Proposition (4.9). 6. Estimates for sh(2k) Recall the equation (23a), which can be written explicitly as S (sh(2k)) = −2vN Λ
(67)
− (vN Λ) ◦ p2 − p2 ◦ (vN Λ) − ((vN ∗ T rΓ) (x) + (vN ∗ T rΓ) (y)) sh(2k)(x, y) − (vN Γ) ◦ sh(2k) − sh(2k) ◦ (vN Γ) := RHS(67) Now that we control the quantities (66) we use proofs similar to those of section 4 of [21], or section 3 of [32] with −vN Λ playing the role of ˜ playing the role of S, at least locally in time. The crucial m and S ingredient is that sup kΛ(t, x + z, x)kL2 (dx) + sup k z
z
∂ Λ(t, x + z, x)kL2 (dx) ∂t
∂ + kΛkL∞ + k ΛkL∞ . 1 (68) ∂t The reader is warned, with apologies, that S in [21] is not the S of the current paper (which is a purely differential operator), but what was called Sold in the introduction. In this section, we will prove
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Theorem 6.1. Let sh(2k), ch(2k) satisfy the equations (23a), (23b). Then for every > 0, and T0 as in Theorem (5.1), k < Dx,y >1/2− sh(2k)(t, ·, ·)kL2 (dxdy) . 1
(69)
k < Dx,y >1/2+ sh(2k)(t, ·, ·)kL2 (dxdy) . N 2
(70)
sup ksh(2k)(t, x, ·)kL2 (dy) . N (0 ≤ t ≤ T0 )
(71)
x
Similar estimates hold for the higher time derivatives of sh(2k). Define ch(k) = δ(x − y) + p. The following is immediate, as in the proof of Corollary 4.2 in [21]: Corollary 6.2. Let > 0. The following estimates hold uniformly in 0 ≤ t ≤ T0 k < Dx,y >1/2− sh(k)(t, ·, ·)kL2 (dxdy) . 1 k < Dx,y >1/2+ sh(k)(t, ·, ·)kL2 (dxdy) . N 2 kp(t, ·, ·)kL2 (dxdy) . 1 sup ksh(k)(t, x, ·)kL2 (dy) . N x
sup kp(t, x, ·)kL2 (dy) . N x
with similar estimates on the higher time derivatives. We start with some preliminary lemmas. Lemma 6.3. (replacing Lemma 4.4 in [21].) Let s0a be the solution to Ss0a = −2vN (x − y)Λ s0a (0, x, y)
(72)
=0
Then k < Dx,y >1/2− s0a (t, ·, ·)kL2 (dxdy) . 1 k < Dx,y >1/2+ s0a (t, ·, ·)kL2 (dxdy) . N 2 (0 ≤ t ≤ T0 ) with similar estimates for higher time derivatives. This lemma is a particular case (with F = vN (x − y)Λ(t, x, y) of a more general result we will need later: Lemma 6.4. Let F be a function of 6+k+1 variables (k ≥ 0, x, y ∈ R3 , z ∈ Rk , t ∈ [0, 1]) and let Z t E(x, y, z, t) = ei(t−s)∆x,y,z F (s)ds 0
30
M. GRILLAKIS AND M. MACHEDON
Then k < Dx−y >1/2− E(t, ·)kL2 ∂ . sup kF (s, ·)kL2 (d(x+y))L2 (dz)L1 (d(x−y)) + k F (s, ·)kL2 (d(x+y))L2 (dz)L1 (d(x−y)) ∂s 0≤s≤t Proof. Change variables, so we work in x, y rather than x − y, x + y coordinates. In these coordinates, integrating by parts, Z t E(t, ·) = ei(t−s)∆x,y,z F (s, ·)ds (73) 0 Z t ∂ F (s, ·)ds (74) = ei(t−s)∆x,y,z ∆−1 x,y,z ∂s 0 −1 + eit∆x,y,z ∆−1 x,y,z F (0, ·) − ∆ F (t, ·) We are going to project in frequencies dual to x only. For the low frequency case we use (73): Z t 1/2− kP|ξ|≤1 < Dx > E(t, ·)kL2 = k ei(t−s)∆x,y,z P|ξ|≤1 < Dx >1/2− F (s, ·)dskL2 0 1/2−
≤ sup kP|ξ|≤1 < Dx >
F (s, ·)kL2 (dxdydz)
0≤s≤t
. sup kF (s, ·)kL2 (dydz)L1 (dx) 0≤s≤t
where we have used the fact that the (physical space) kernel corresponding to P|ξ|≤1 < Dx >1/2− is in L2 . For the high frequency case we use (74). We only write down one term, the boundary terms being similar: Z t ∂ 1/2− ei(t−s)∆x,y,z P|ξ|≥1 < Dx >1/2− ∆−1 F (s, ·)dskL2 kP|ξ|≥1 < Dx > E(t, ·)kL2 = k x,y,z ∂s 0 ∂ ≤ sup kP|ξ|≥1 < Dx >1/2− ∆−1 F (s, ·)dskL2 (dxdydz) x ∂s 0≤s≤t ∂ . sup k F (s, ·)kL2 (dydz)L1 (dx) 0≤s≤t ∂s where we have used that the kernel of P|ξ|≥1 < Dx >1/2− ∆−1 x in in 2 L. Proof. (of Lemma (6.3)). Applying the previous result and the estimates (66) we already know k < Dx−y >1/2− s0a (t, ·, ·)kL2 (dxdy) . 1
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with similar estimates for derivatives which commute with vN (x − y). Thus n ∂ k < Dx,y >1/2− s0a (t, ·, ·)kL2 (dxdy) .n 1 ∂t Using the equation (72), we also get n ∂ k ∆x,y s0a (t, ·, ·)kL2 (dxdy) . N 3β/2 ∂t so the second claim follows by interpolation.
Next, we include the potential terms: Lemma 6.5. Let sa be the solution to ˜ a = −2vN Λ Ss
(75)
sa (0, x, y) = 0 Then n ∂ s0a (t, ·, ·)kH 1/2− (dxdy) .n 1 (0 ≤ t ≤ T0 ) k ∂t n ∂ k s0a (t, ·, ·)kH 1/2+2 (dxdy) .n N (0 ≤ t ≤ T0 ) ∂t ˜ Proof. Let V be the ”potential” part of S:
V (u) = ((vN ∗ T rΓ) (x) + (vN ∗ T rΓ) (y)) u + (vN Γ) ◦ u + u ◦ (vN Γ) ∂ n Form the estimates (66) for Γ we see that ∂t Γ . 1, thus V and ∂ n [ ∂t , V ] have bounded operator norm (on L2 ). Write sa = s0a + s1a (s0a as in the previous lemma) so that s1a satisfies the equation ˜ 1 = −V (s0 ) Ss a a n n n n ∂ ∂ ∂ ∂ 1 0 0 ˜ ˜ 1 S sa = −V sa − [ , V ]sa − [ , S]s a ∂t ∂t ∂t ∂t Using energy estimates and induction on n we see n ∂ k s1a kL2 . 1 ∂t which, in turn, implies (using the equation) n−1 ∂ k ∆s1a kL2 . 1 ∂t uniformly for 0 ≤ t ≤ 1. The result follows from the previous lemma.
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M. GRILLAKIS AND M. MACHEDON
Finally, we can prove Theorem (6.1). Proof. (of theorem (6.1)) Write sh(2k) := s2 = sa +se where sa satisfies (75) and ch(2k) = δ(x − y) + p2 . In analogy with (64a), (64b) of [21], they satisfy ˜ e ) = −(vN Λ) ◦ p2 − p2 ◦ (vN Λ) S(s ˜ 2 ) = −(vN Λ) ◦ sa + sa ◦ (vN Λ) W(p −(vN Λ) ◦ se + se ◦ (vN Λ) := M − (vN Λ) ◦ se + se ◦ (vN Λ) Using the result of Lemma (6.5), as well as estimates (66) for Λ we see that k
∂ ∂t
n M kL2 (dxdy) . 1 (0 ≤ t ≤ T0 )
Differentiating the equation with respect to t and using energy estimates as in [32], we see k
∂ ∂t
n
se kL2 + k
∂ ∂t
n−1 ∆se kL2 . 1
Finally, to prove (71) we will use the L2 estimates that have been proved already (69) and (70). We go back to the equation (67), and write sh(2k) = s2a + s3a the following way: Let c = 2δ(x − y − z) |x − y − z| cχ(x − y − z) ∆x,y = 2δ(x − y − z) + χ(x ˜ − y − z) |x − y − z| ∆x,y
where χ, χ˜ are C0∞ functions, χ = 1 and χ˜ = 0 in a neighborhood of 0. Next, compare sh(2k) with the ”ansatz” s2a given by s2a
Z =
cχ(x − y − z) vN (z)Λ |x − y − z|
x+y+z x+y−z , 2 2
dz
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so that (splitting ∆x,y = ∆x+y + ∆x−y , Sx,y = Sx+y − ∆x−y , where, of course, ∆x+y = 21 (∇x + ∇y ) · (∇x + ∇y ) ) Sx,y s2a Z x+y+z x+y−z = (−2δ(x − y − z) − χ(x ˜ − y − z)) vN (z)Λ , dz 2 2 Z cχ(x − y − z) x+y+z x+y−z − vN (z)Sx+y Λ , dz |x − y − z| 2 2 = −2vN (x − y)Λ(x, y) Z x+y+z x+y−z − χ(x ˜ − y − z)vN (z)Λ , dz 2 2 Z cχ(x − y − z) x+y+z x+y−z vN (z)Sx+y Λ , dz − |x − y − z| 2 2 with initial conditions s2a (0, x, y) Using (68), it is easy to see that s2a satisfies the estimates in the statement of the theorem. In fact, supx ks2a (t, x, ·)kL2 (dy) . 1. To estimate the difference, S sh(2k) − s2a Z x+y+z x+y−z = χ(x ˜ − y − z)vN (z)Λ , dz 2 2 Z cχ(x − y − z) x+y+z x+y−z + vN (z)Sx+y Λ , dz |x − y − z| 2 2 − (vN Λ) ◦ p2 − p2 ◦ (vN Λ) − ((vN ∗ T rΓ) (x) + (vN ∗ T rΓ) (y)) sh(2k)(x, y) − (vN Γ) ◦ sh(2k) − sh(2k) ◦ (vN Γ) := RHS with initial conditions s2a (0, x, y) Using (66), we see that kRHSkL2 (dxdy) + k|Dx |1/2+ RHSkL2 (dxdy) . N 2 uniformly in 0 ≤ t ≤ T0 , with similar estimates for the time derivatives, and the same holds for s2a (0, x, y). The Strichartz estimate (49), Duhamel’s formula and Sobolev in time show that sup ksh(2k) − s2a kL2 (dy) . N 2 x
uniformly in [0, T0 ].
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M. GRILLAKIS AND M. MACHEDON
7. The reduced Hamiltonian Recall (10) for the definition of the reduced Hamiltonian. Hred was computed, for instance, in Section 5 of [21]. We will write it in a different way, using Wick’s theorem. Recall the conjugation formulas Z B −B ch(k)(y, x)ay + sh(k)(y, x)a∗y dy (76) e ax e =
eB a∗x e−B
= a(ch(k)(x, ·)) + a∗ (sh(k)(x, ·)) := bx Z = sh(k)(y, x)ay + ch(k)(y, x)a∗y dy = a(sh(k)(x, ·)) + a∗ (ch(k)(x, ·)) := b∗x .
The reduced Hamiltonian is Hred = N µ0 (t) (77a) Z ¯ + N 1/2 dx h(φ(t, x))b∗x + h(φ(t, x))bx (77b) Z 1 ∂ B(k(t)) −B(k(t) e e + dxb∗x ∆bx (77c) + i ∂t Z 1 dxdyvN (x − y) φ(x)φ(y)bx by + φ(x)φ(y)b∗x b∗y + 2φ(x)φ(y)b∗x by − 2 (77d) Z − dx(vN ∗ |φ|2 )b∗x bx (77e) Z 1 −√ dx1 dx2 vN (x1 − x2 ) φ(x2 )b∗x1 bx1 bx2 + φ(x2 )b∗x1 b∗x2 bx1 N (77f) Z 1 dx1 dx2 vN (x1 − x2 )b∗x1 b∗x2 bx1 bx2 . − (77g) 2N The functions µ0 (t) and h(φ(t, x)) appearing in (77a),(77b) are given below, Z Z 2 1 1 ¯ ¯ t − ∇φ − φφt − φφ dxdy vN (x − y)|φ(x)|2 |φ(y)|2 µ0 := dx 2i 2 1 h(φ(t, x)) := − ∂t φ + ∆φ − vN ∗ |φ|2 φ . i We re-arrange the terms in Hred using Wick’s theorem . Define the contraction of A(f ) := a(f¯1 ) + a∗ (fR2 ) and A(g) := a(¯ g1 ) + a∗ (g2 ) to be C(A(f ), A(g)) = [a(f¯1 ), a∗ (g2 )] = f¯1 g2 , and define the normal order
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Nor A(f )A(g)A(h)A(k) to be A(f )A(g)A(h)A(k) expanded and rearranged so all starred terms have been moved to the left, as if they commuted with unstarred terms. Wick’s theorem, which can easily be proved by induction, says, in particular, that A(g)A(h)A(k) = Nor A(g)A(h)A(k) + C(A(g), A(h))A(k) + C(A(g), A(k))A(h) + C(A(h), A(k))A(g) A(f )A(g)A(h)A(k) = Nor A(f )A(g)A(h)A(k) + C(A(f ), A(g))A(h)A(k) + C(A(f ), A(h))A(g)A(k) + · · · (6 terms) = Nor A(f )A(g)A(h)A(k) + C(A(f ), A(g))A(h)A(k) + C(A(f ), A(h))A(g)A(k) + · · · (6 ordered terms with 1 contraction) + C(A(f ), A(g))C(A(h), A(k)) + C(A(f ), A(h))C(A(g), A(k)) + · · · (6 ordered terms with 2 contractions) Applying Wick’s theorem to Hred (putting the quartic and cubic terms in normal order, but not the quadratics) we get Hred = N µ0 (t) Z o n ¯ 1/2 ∗ ˜ ˜ +N dx h(φ(t, x))bx + h(φ(t, x))bx Z 1 ∂ B(k(t)) −B(k(t) + e e + dxb∗x ∆bx (78a) i ∂t Z 1 dxdyvN (x − y) Λ(x, y)bx by + Λ(x, y)b∗x b∗y + 2Γ(x, y)b∗x by − 2 (78b) Z − dx(vN ∗ T rΓ)b∗x bx (78c) Z 1 dx1 dx2 Nor vN (x1 − x2 ) φ(x2 )b∗x1 bx1 bx2 + φ(x2 )b∗x1 b∗x2 bx1 −√ NZ 1 − dx1 dx2 Nor vN (x1 − x2 )b∗x1 b∗x2 bx1 bx2 . 2N
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M. GRILLAKIS AND M. MACHEDON
˜ is the modified Hartree operator (22). Let us remark that, in where h complete analogy to (67c) in [21], the quadratic terms (78a)+(78b)+ (78c) can be written concisely and explicitly as ! w˜ T f˜ (78a) + (78b) + (78c) = HG˜ − I −f˜ −w˜ where ˜ ˜ − sh(k) ◦ (vN Λ) ◦ sh(k) f˜ := S(sh(k)) + ch(k) ◦ (vN Λ) ◦ ch(k) − W(ch(k)) ˜ ˜ w˜ := W(ch(k)) − sh(k) ◦ (vN Λ) ◦ ch(k) − S(sh(k)) + ch(k) ◦ (vN Λ) ◦ sh(k) Z Hg˜ = g˜(x, y)a∗x ay dxdy
Also, I is the Lie algebra isomorphism used in our previous papers [19]-[22] (see for instance formula (27) in [21]) is ! Z n o T ˜ 1 w˜ f I =− dxdy w(y, ˜ x)ax a∗y + w(x, ˜ y)a∗x ay − f˜(x, y)a∗x a∗y − f˜(x, y)ax ay 2 −f˜ −w˜ ˜ 2 is a multiple of f˜, thus f˜ = 0 if our equations are In this notation, X satisfied. If we also put the quadratics in normal order, the above formula becomes Hred = X0 (t)+ Z o n ¯ 1/2 ∗ ˜ ˜ N dx h(φ(t, x))bx + h(φ(t, x))bx ! w˜ T f˜ + Hg˜ + Nor I −f˜ −w˜ Z 1 −√ dx1 dx2 vN (x1 − x2 ) φ(x2 )b∗x1 bx1 bx2 + φ(x2 )b∗x1 b∗x2 bx1 N ! Z 1 − . dx1 dx2 vN (x1 − x2 )b∗x1 b∗x2 bx1 bx2 2N where X0 is written down explicitly in Section 6 on [22]. In conclusion, ˜ if φ and k satisfy our equation X1 = 0, X2 = 0, then h(φ(t, x)) = 0,
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f˜ = 0 and P := Hred − H − X0 (79) T w˜ 0 = Hg˜ − H1 + Nor I 0 −w˜ Z 1 +Nor − √ dx1 dx2 vN (x1 − x2 ) φ(x2 )b∗x1 bx1 bx2 + φ(x2 )b∗x1 b∗x2 bx1 2 N Z 1 ∗ ∗ − dx1 dx2 vN (x1 − x2 )bx1 bx2 bx1 bx2 2N Z 1 + dx1 dx2 vN (x1 − x2 )a∗x1 a∗x2 ax1 ax2 . 2N Remark that the third and fourth term are − √1N Nor eB [A, V]e−B and − N1 Nor eB Ve−B (see Section 5 of [21]).
8. Estimates for the error term In this section we choose < 1 − 3β and apply the estimates of 2 Corollary (6.2) (where, in order to simplify notation, the in that corollary is taken to be the current divided by 10.) We proceed as in our previous papers [19]-[21], using the identity R
R
ψexact (t) − ei X0 (t)dt ψappr (t) = e−i X0 (t)dt ψred − Ω F F
and estimate the right hand side term using the equation
R 1∂ −i X0 (t)dt ˜ − Hred + X0 e ψred − Ω = (0, 0, 0, X3 , X4 , 0, · · · ) := X i ∂t
Denote E = e−i
R
X0 (t)dt
ψred − Ω, so that
1∂ ˜ − Hred + X0 E = (0, 0, 0, X3 , X4 , 0, · · · ) := X i ∂t E(0, ·) = 0
(80)
The terms X0 , X3 and X4 were computed in Section 5 of [21], see formulas (74c) and (72c). We only need X3 and X4 here, and they are
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M. GRILLAKIS AND M. MACHEDON
(modulo symmetrization and normalization) Z 1 X3 (y1 , y2 , y3 ) = √ ch(y1 , x1 ) ch(x2 , y2 )vN (x1 − x2 )φ(x2 )sh(y3 , x1 )dx1 dx2 N 1 = √ (vN (y1 − y2 )sh(y3 , y1 )sh(y2 , y4 ) + LOT ) Z N 1 X4 (y1 , y2 , y3 , y4 ) = ch(y1 , x1 ) ch(x2 , y2 )vN (x1 − x2 )sh(y3 , x1 )sh(x2 , y4 )dx1 dx2 N 1 = vN (y1 − y2 ) (sh(y3 , y1 )sh(y2 , y4 ) + LOT ) N where the lower order terms LOT come from the p component of ch(k) = δ(x − y) + p. The main result of this section, which completes the proof of Theorem (2.5), is Theorem 8.1. Let 0 < < 1 −
3β , 2
and let E satisfy (80). Then
kEkF . N
3β −1+ 2
uniformly for 0 ≤ t ≤ T0 . Proof. This is proved by splitting 1∂ SF := − Hred + X0 i ∂t := SD − P where 1∂ −H SD = Z i ∂t 1 H = a∗x ∆ax − V 2N Thus H is the original (unconjugated) Fock space Hamiltonian (7a)(7c), and P accounts for the rest of the terms: P = Hred − H − X0 Recall H acts on Fock space as 1 X 3β ∆− N v N β (xj − xk ) N j
