PAIR EXCITATIONS AND THE MEAN FIELD APPROXIMATION OF INTERACTING BOSONS, I

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1 PAIR EXCITATIONS AND THE MEAN FIELD APPROXIMATION OF INTERACTING BOSONS, I M. GRILLAKIS AND M. MACHEDON Abstract. In our previous work [0],[] we introduced a correction to the mean field approximation of interacting Bosons. This correction describes the evolution of pairs of particles that leave the condensate and subsequently evolve on a background formed by the condensate. In [] we carried out the analysis assuming that the interactions are independent of the number of particles N. Here we consider the case of stronger interactions. We offer a new transparent derivation for the evolution of pair excitations. Indeed, we obtain a pair of linear equations describing their evolution. Furthermore, we obtain apriory estimates independent of the number of particles and use these to compare the exact with the approximate dynamics.. Introduction The purpose of our present work is to investigate certain aspects of the evolution of a large number of indistinguishable Quantum particles Bosons) under binary interactions. If we call ψt, x, x... x N ) the wavefunction describing the N particles with x j R 3 the coordinates for j =,... N, then ψ satisfies an evolution equation of the form /i) t ψ = H N V ) ψ ) where H is a sum of the from N j= x j. The term V models two body interactions of the following general type V := /) x j x k N 3β v N β x j x k ) ) ; 0 β ) where v C0 is non-negative, spherically symmetric, and decreasing. In the equation ) above we consider non-relativistic particles and set h = m = for simplicity. Here and for the rest of this paper we denote v N := N 3β v N β ). The fact that we consider Bosons means that the wavefunction is invariant under all permutations of the indices j =,... N and one would like to solve the evolution equation under some initial condition at, say t = 0, ψ0, x, x... x n ) :=

2 M. GRILLAKIS AND M. MACHEDON ψ 0 x, x... x N ). Presently we are interested in the evolution of factorized or approximately factorized) initial data i.e. we would like to consider special initial data of the form ψ 0 := N φ 0 x j ). 3) j= The evolution of ) with initial data 3) is quite complicated for N large and one would like to have an effective approximate description of the evolution. The motivation for this type of problem comes from Bose-Einstein condensation where one considers a large number of identical indistinguishable) particles in a trap. Einstein following ideas of Bose, observed that nonineracting particles in a box undergo a phase transition at a critical temperature proportional to density /3, so that below this temperature a macroscopic number of particles occupy the ground state, furthermore at zero temperature all particles condense to the ground state. It is natural and more realistic to consider interacting particles. Following ideas of Landau a heuristic theory based on the idea of the mean-field approximation was developed by Gross and Pitaevski [3], []. On a more fundamental level, the problem of a weakly interacting Quantum gas was taken up in the pioneering work of Lee, Huang and Yang as well as Dyson) [7], [0]. More recent theoretical developments are due to Lieb, Solovej, Yngvanson, Seiringer et. al. see [9] and references therein. In particular, Theorems 6., 7. in [9], as well as Appendix C in [3] strongly suggest that that the ground state is well approximated by a tensor product 3) where φ 0 describes a mean field approximation. Let us point out that we can in fact treat more general initial conditions corresponding to the Nth component of e NAφ) e Bk) 0, see section ). Experimental confirmation of Bose-Einstein condensates was finally achieved [], using alkali atoms. The reason for the use of alkali atoms is the fact that they contain a single valence electron in the outermost s-orbital for example 5-s for Rubidium). The other contributions to the total spin comes from the nuclear spin. If the nuclear isotope is one with odd number of protons and neutrons it will have a net half integer spin. For example Rubidium 87 has S = 3/ from the nucleus. The total spin takes the values S = or S =. If we prepare the sample so that only one of these states is present then this will be a gas of identical Bosons. If two different states are present then we should consider it as a mixture of two different gasses. Since alkali atoms are complicated composite particles their interactions are not known explicitly, which

3 PAIR EXCITATIONS,I 3 means that the potential v in ) is not explicitly known, moreover one can treat the atoms as Bosons only for a sufficiently dilute gas. At shorter distances the internal structure of the atoms should be taken into account. Here we consider a sufficiently dilute Boson gas and we make the reasonable assumption that the interactions are repulsive i.e. v 0 and that they are short range in the sense that vx)dx <. It is clear from the above comments that one should consider particles with spin. The present framework can be generalized in this case in a straightforward manner, however in the name of simplicity we forgo this generalization. Let us comment on the scaling present in the form of binary interactions. The parameter β describes the strength of particle interactions. It is a reasonable but not obvious) idea to assume that the evolution of ) is approximated by a tensor product i.e. N ψt) φt, x j ) 4) j= and the issues are, first to explain the nature of the approximation described in 4) and second to derive an evolution equation for φt, x) consistent with the dynamics of ). On the second question, the general idea is that the evolution of the mean field φt, x) satisfies an equation of the form i t φ = φ g β φ ) φ, 5) and the nonlinear term g β φ ) depends on β in the following manner, g 0 φ ) = dy { vx y) φt, y) } g β φ ) = vy)dy φt, x) ; 0 < β < g φ ) = 8πa φt, x), where a appearing in g is the scattering length corresponding to the potential v. In the case β = 0 one obtains a Hartree type evolution for the mean field and considerations similar to our present work where taken up in [0],[]. The case β = is probably the most interesting. In this case the scaling is critical in the sense that particles develop short scale correlations which in the limit N lead to the appearance of the scattering length in the equation. A heuristic argument for this is well known in the Physics community, however the explanation on how the scattering length emerges from the N body dynamics was recently given in the work of Erdös, Schlein and Yau [6],[5].

4 4 M. GRILLAKIS AND M. MACHEDON Our aim is to introduce pair excitations as a correction to the mean field approximation. This goal is achieved by introducing a kernel kt, x, y) which describes pair excitations and one would like to derive an evolution equation for k consistent with the N body dynamics, which means that we should be able to obtain estimates comparing the exact with the approximate dynamics. The general idea of the approximation can be described in the following manner. Two particles leave the condensate and form a pair v N x x )φx )φx ) which in turn drives the evolution of pair interactions. It turns out that a natural way to introduce pair excitations as a correction to the mean field is via a Fock space formalism which we will outline in the next section. Let us comment here on the nature of our approximation. The mean field approximation 4) is a simple description of the N-body wavefunction, however the nature of the approximation is quite involved and uses the BBGKY hierarchy and its limit as N as shown by Elgart, Erdös, Schlein and Yau [,, 3, 4, 5, 6]. See the approach of [5], [6], [5] based on space-time estimates. We also mention the related case of 3 body interactions [6], and switchable quadratic traps [7, 8]. Moreover the approximation does not track the exact dynamics, rather its true usefullness lies in the fact that it can approximately) track observables. In contrast our approximation is more complicated but it tracks the exact dynamics in Fock space norm. As a matter of fact a heuristic explanation of our approximation runs as follows: The N-body wave function consists of three parts, particles that live in the condensate, bound pairs and particles that decayed after forming pairs. Controlling the number of particles that formed pairs leads to another justification of the mean field approximation. We will not pursue this line of inquiry here, however the approximation can be readily used to estimate observables. There are two main points in our present work. First we have a new transparent derivation of the evolution equation of pair excitations, indeed we derive a new system of linear equations. Second we obtain apriory estimates for the pair excitations kernel which are independent of N and this, in turn, allows us to estimate the difference between the exact and approximate solutions provided that β is sufficiently small β < ). 6 Our work was inspired by [35] as well as [37]. Previous works directly related to the present are [8] and [4]. See also [] and [3]. See Theorem.3) below for the precise statement of our main result. The paper is organized as follows. In section we develop the Fock space formalism which is necessary for our computations and derive the evolution equations for the pair excitation kernel. In section

5 PAIR EXCITATIONS,I 5 and 3 we we derive the apriory estimates for the mean-field and for the pair excitation kernel. In section 4 we show how this information can be implemented in order to compare the exact solution to our approximation.. Fock space formalism and the new derivation In this section we introduce the Fock space formalism and the Hamiltonian evolution in symmetric Fock space. F is a Hilbert space consisting of vectors of the form ψ = ψ 0, ψ x ), ψ x, x ),... ) where ψ 0 C and ψ k are symmetric L functions. The norm of such a vector is, ψ = ψ ψ = ψ0 + ψn n= Thus F is a direct sum of sectors F n of the form, F = F n ; F n := L ) s R 3n n=0 with F 0 = C and L sr 3 ) denoting the subspace of symmetric functions. In the Fock space F we introduce creation and anihilation distribution valued operators denoted by a x and a x respectively which act on sectors F n and F n+ in the following manner, a xψ n ) := n L. n δx x j )ψ n x,..., x j, x j+,..., x n ) j= a x ψ n+ ) := n + ψ n+ [x], x,..., x n ) with [x] indicating that the variable x is frozen. In addition a x kills F 0 i.e. a x ψ 0 ) = 0. The vacuum state will play an important role later and we define it as follows 0 :=, 0, 0...) [ so that a x 0 = 0. One can easily check that ax, ay] = δx y) and since the creation and anihilation operators are distribution valued we can form operators that act on F by introducing a field, say φx), and form a φ := dx { } φx)ax and a φ := dx {φx)a x} where by convention we associate a with φ and a with φ. These operators are well defined, unbounded, on F provided that φ is square

6 6 M. GRILLAKIS AND M. MACHEDON integrable. The creation and anihilation operators provide a way to introduce coherent states in F in the following manner, first define Aφ) := dx { } φx)ax φx)a x and then introduce N-particle coherent states as ψφ) := e NAφ) 0. 6) It is easy to check that e n ) NAφ) 0 =... c n φx j )... j= with c n = e N N n /n! ) /. In particular, by Stirling s formula, the main term that we are interested in has the coefficient c N πn) /4 7) Thus a coherent state introduces a tensor product in the sector F N, hence we can use such states as a mean field approximation to the Hamiltonian evolution in Fock space, see 4). The Fock Hamiltonian acting on Fock space vectors) is where we set H := H N V where, 8a) H := dxdy { x δx y)a xa y } and 8b) V := dxdy { } v N x y)a xa ya x a x, 8c) v N x y) := N 3β v N β x y ), 9) and the evolution in Fock space is described by the equation, i t ψ = H ψ 0) which has the formal solution ψt) = e ith ψ0. ) Notice that H preserves the sectors F n and that H agrees with the classical Hamiltonian ) on F N. However in this framework we allow any number of particles to evolve and one is interested, in particular, in the evolution on the sector F N.

7 PAIR EXCITATIONS,I 7 Our goal is to approximate ψt) in ) and for this purpose we introduce two fields φt, x) and kt, x, y) and the associated operators, Aφ) := dx { } φt, x)ax φt, x)a x a) Bk) := dxdy { kt, x, y)ax a y kt, x, y)a xay}. b) The coherent initial data are introduced via ψ0 = e NAφ 0 ) 0 which means that the initial data are a tensor product on F N as desired, see 3). Our approximation scheme is ψ appr := e NAt) e Bt) e inχt) 0 3) with χt) a phase factor, and we plan to show that ψt) ψappr t). The issue for us is to determine the dynamics of the fields φ and k. It turns out the the evolution of k is described via a set of new fields, shk) := k + 3! k k k +..., chk) := δx y) +! k k +..., 4a) 4b) where indicates composition, namely k l stands for the product, k l)x x ) := dy {kx, y)ly, x )}. A crucial property of the above multiplication is that it is not commutative i.e. k l l k. In order to describe the evolution we need g N t, x, y) := x δx y) + v N φ )t, x)δx y) + v N x y)φt, x)φt, y) 5a) m N x, y) := v N x y)φx)φy). 5b) Using g N we can construct two operators as follows: For a function st, x, y) symmetric in x, y) and a function pt, x, y) conjugate symmetric in x, y) i.e. p T = p, we define Ss) := i s t + g T N s + s g N Wp) := i p t + [g T N, p]. 6a) 6b)

8 8 M. GRILLAKIS AND M. MACHEDON The dynamics of the fields are determined via, i tφ φ + v N φ ) φ = 0 7a) S shk)) = m N chk) + chk) m N 7b) W chk) = m N shk) shk) m N. 7c) Recall we assume v C 0 is non-negative, spherically symmetric, and decreasing. Remark.. It is clear that chk) and shk) are not independent of each other, thus we can ignore the third equation, however in the form stated above the equations are readily amenable to the derivation of apriori estimates. The equation for φ is of Hartree type and its formal limit as N is NLS. The theorem concerning the evolution of the mean field φ and the pair excitation kernel k reads as follows. Theorem.. Suppose that 0 < β < in 9). Given initial data φ0, x) := φ 0 x) W k, k derivatives in L, with k sufficiently large) and k0, x, y) := 0 the system 7a)7b)7c) has global solutions in time which satisfy the apriori estimates, φt) H s R 3 ) C s φt) L R 3 ) + t φt) L R 3 ) C t 3/ shk)t) L R 6 ) + chk)t) δ L R 6 ) C log + t). 8a) 8b) 8c) The main difficulty in obtaining the estimates in the theorem above is the fact that v N defined in 9) has a formal limit v N x y) cδx y) which means that m N has a limit which is not square integrable, as a matter of fact it does not belong to any L p for p >. In view of the theorem above, we can compare the exact with the approximate evolutions and the result is the following theorem. Theorem.3. Suppose that ψt) is the solution of 0) with initial data ψ0 := e NAφ 0 ) 0 and ψappr t) is the approximation in 3) where the evolution of the fields φ and k is determined from theorem.). Under these conditions the following estimate holds, ψt) ψappr t) F C + t) log4 + t). 9) N 3β)/ provided 0 < β <. This is a meaningful approximation of the Nth 3 component of ψ provided 0 < β <, because of formula 7). A slightly 6

9 PAIR EXCITATIONS,I 9 more precise form of the estimate could be obtained by integrating the right hand side of the inequalities in Proposition 6.). Remark.4. The real phase factor χ is described via χt) := t { dt µ0 t )+ N µ t ) } where µ 0 t) = dxdy { v N x y) φt, x) φt, y) } 0) and µ is a complicated integral given in 73) Proof. Here is an outline of the proof of this theorem. In order to relate the exact with the approximate solution we introduce the reduced dynamics ψred t) := e Bt) e NAt) e ith e NA0) 0 ) i.e. we follow the exact dynamics for time t and then go back following the approximate evolution. Notice that ψ red 0) = 0 and if our approximation was following the exact evolution we would have that ψred t) = 0. Thus our goal is to estimate the deviation of the evolution from the vacuum state. It is straightforward to compute the evolution of ψred and it is i t ψ red = Hred ψ red ) where the self-adjoint) reduced Hamiltonian is, H red := t e B) e B i + e B t e ) ) NA e NA + e NA He NA e B. 3) i The main idea is that the evolution of the fields φ and k is chosen so that the reduced Hamiltonian looks like H red = Nµt) + dxdy {Lt, x, y)a xa y } N / Et) where Et) is an error term containing polynomials in a, a ) up to degree four, and L is some self-adjoint expression which is irrelevant for the rest of the argument. Next consider ψ ) t := e inχt) ψred 0 where χt) := µt )dt where we called µ := µ 0 + N µ. Thus i t H red + Nµt) e inχt) ψred = 0.

10 0 M. GRILLAKIS AND M. MACHEDON and therefore ) i t H red + Nµt) ψ = N / Et) 0 The equation above has a forcing term namely N / Et) 0 and a standard energy estimate together with the fact that e NA and e B are unitary, gives ψt) ψapprt) F = t { ψt) F N / dt Et ) } 0 F. 4) The proof will be complete if we estimate the right hand side in the above inequality. Notice that E 0 has entry only in Fock sectors Fj for j =,, 3, 4 and in order to estimate it we need the lemma below. Lemma.5. The error term is described as follows, E := e B P 3 + N / P 4 ) e B where P 3 and P 4 are cubic and quartic polynomials in a, a ) respectively. Moreover the following estimate holds if 0 β < 3, Et) 0 F CN 3β/ log 4 + t). 5) A more precise estimate is given in Proposition 6.). Remark.6. The polynomials P 3 and P 4 appearing in the error term are given by the expressions, P 3 := dxdy { v N x y) φy)a xa ya x + φy)a )} xa x a y P 4 :=/) dxdy { } v N x y)a xa ya x a y as we will see shortly. The rest of this section is devoted to the derivation of 7a)7b)7c). We have to compute H red above, see 3), and for this task there are two crucial ingredients. They are based on the formal identities below for any two operators, say A and H, e A He A n = ada H 6a) n! t e A) e A = n=0 n= 0 n ada At n! 6b)

11 PAIR EXCITATIONS,I [ ] where ad A H := A, H. They indicate that we have to compute repeated commutators of various operators. The series defining the exponentials in 6a), 6b) converge absolutely on the dense subset of vectors with finitely many nonzero entries provided that A = Aφ) is a polynomial of degree one with φ L or A = Bk) is second order with k L small. If B is skew-hermitian, e B extends as a unitary operator for all k L. This construction is closely related to the Segal-Shale- Weil representation, as explained in [4], [7], [36], and our appendix 7). This calculation was also used in our previous papers [0, ]. The first observation is the fact that since Aφ) is a degree one polynomial, if we denote by P n a homogeneous polynomial of degree n then commuting with A produces [ ] A, P n = Pn i.e. a homogeneous polynomial of degree n. This in turn implies that repeated commutators produce a finite series in 6a), 6b) which can be computed explicitly after some tedious but straightforward calculations. The second observation is that in 6a), 6b) when we replace A with B we obtain infinite series with a certain periodicity which allows for explicit summation. This can be expressed via a Lie algebra isomorphism. For symplectic matrices of the blocked form dx, y) lx, y) L := kx, y) dy, x) where d, k and l are kernels in L, and k and l are symmetric in x, y), we define the map from L to quadratic polynomials is a, a ) in the following manner, I L ) = { } dx, y) lx, y) a dxdy a x, a y x). 7) kx, y) dy, x) a y The crucial property of this map is the Lie algebra isomorphism [ IL ), IL ) ] = I [L, L ] ) 8) thus any computation that involves commutations can be performed in the realm of symplectic matrices and then transfered to polynomials in a, a ). In particular if we call IH) = H for a quadratic Hamiltonian and IK) = B then we have the two formulas below, e B He B = I e K He K) 9a) t e B) e B = I t e K )e K). 9b) Actually, to avoid the infinite trace in 9a), we write e B He B = H + [e B, H]e B = H + I [e K, H]e K)

12 M. GRILLAKIS AND M. MACHEDON As a matter of fact if we define the following quardatic expressions, D xy := a x a y ; D xy := a xa y Q xy := a xa y ; Q xy := a x a y then we can write, IL) = dxdy { } dx, y)d xy + dy, x)d xy + kx, y)q xy lx, y)q xy. Remark.7. Notice that D xy = D yx + δx y) thus we can write IL) = dxdy { } dx, y)d yx + dy, x)d xy + kx, y)q xy lx, y)q xy dx{dx, x)}. In our present formalism if we define the matrix 0 k K =, k 0 then we have that IK) = B, see the expression in b). The exponential of K can be computed, chk) shk) e K = where, shk) chk) shk) := k + 3! k k k +..., chk) := δx y) +! k k +..., and indicates composition. For completeness and for the convenience of the reader we include in the appendix the derivation of 8), see also [0], [7]. Let us now proceed with the calculations. First look at the expression inside the parentheses in the reduced Hamiltonian 3). It is straightforward after repeated but finite) commutations with A to come up with the expression below see section 3 of [0]), t e ) NA e NA + e NA He NA i = Nµ 0 + N / P + P N / P 3 N P 4 30) where P n indicate polynomials of degree n to be given explicitly below. The first term µ 0 is a scalar which can be absorbed in the evolution as an extra phase factor. It is given by the commutators, A, t A i[ ] + A, [A, H ] [ ] [A, [ A, [A, [A, V]] ]] 4!

13 PAIR EXCITATIONS,I 3 which reduce to the expression below, { } µ 0 := dx φ φt i φφ ) t φ dxdy { v N x y) φx) φy) }. 3) The first degree polynomial P arise from the commutators, i ta + [ ] [ ] A, H A, [A, [A, V]] 3! and it can be expressed as follows, P = dx { ht, x)a x + ht, } x)a x 3) where h := /i) t φ + φ v N φ ) φ. The second degree polynomial consists of the terms H [ ] A, [A, V] and can expressed P = + dxdy { g N t, x, y)d yx g N t, y, x)d x,y } dxdy { m N t, x, y)q xy + m N t, x, y)q x,y } 33) where g N and m N are given by, see 5a), 5b) g N t, x, y) := x δx y) + v N φ )t, x)δx y) + v N x y)φt, x)φt, y) m N x, y) := v N x y)φx)φy). It is clear that g N and m N above depend on the number of particles N. Subsequently, for simplicity, we will suppres this subscript and recall it only when it is relevant in an argument. Let us define the two operators below H G := dxdy { g N t, x, y)d yx g N t, y, x)d x,y } 34a) } M := = I 0 m m 0 dxdy { m N t, x, y)q xy + m N t, x, y)q x,y ) 34b) so that we can write P = H G + M. The relevance of this splitting will become clear shortly. The third and fourth degree polynomiasl

14 4 M. GRILLAKIS AND M. MACHEDON arise from the commutators [ A, V ] and V respectively and are given below P 3 := dxdy { v N x y) φy)a xa ya x + φy)a )} xa x a y 35a) P 4 := /) dxdy { } v N x y)a xa ya x a y. 35b) The mean field approximation emerges from the first degree polynomial P. Since µ 0 can be absorbed into the evolution it is reasonable to pick the field φ so that hφ) = 0. This leads to the evolution i tφ φ + v N φ ) φ = 0 36) which is of Hartree type. The formal limit of the equation above is the cubic NLS where the constant in front of the nonlinear term is the integral of the potential v. If φ satisfies 36) then µ 0 reduces to µ 0 = dxdy { v N x y) φt, x) φt, y) }. 37) Now we can compute the reduced Hamiltonian in 3) using the splitting in 34a), 34b). First let us first give a name to E := e B P 3 + N / P 4 ) e B 38) which will be treated later as an error term. Now we can write, see 3), H red = i t e B) e B + H G + [e B, H G ]e B + e B IM)e B + Nµ 0 e B N / P 3 + N P 4 e B + Nµ 0 = H G + I /i) t e K) e K + [e K, G]e K + e K Me K) N / E + Nµ 0 = H G + IR) N / E + Nµ 0, 39) where R is defined to be the expression, R := /i) t e K) e K + [e K, G]e K + e K Me K.

15 PAIR EXCITATIONS,I 5 For the convenience of the reader, let us recall our set up, 0 k chk) shk) K := and e K = k 0 shk) chk) shk) := k + 3! k k k +..., chk) := δx y) +! k k +..., gt, x, y) := x δx y) + v N φ )t, x)δx y) + v N x y)φt, x)φt, y), mx, y) := v N x y)φx)φy) where v N x) = N 3β vn β x) g 0 0 m G := 0 g T and M := m 0 I 0 N u := this corresponds to the Number operator) 0 I Ss) := i s t + g T s + s g and Wp) := i p t + [g T, p]. Thus S describes a Shrödinger type evolution, while W is a Wigner type operator. These operators will emerge shortly. Recall the formula 39) that we derived earlier for the reduced Hamiltonian H red = H G + IR) N / E, where H G has only a a terms which annihilate the vacuum) and R can be computed explicitly. In fact we have, R = chk)t shk) t i shk) t chk) t [chk), g] sh m + shk) g + g T shk) ch m chk) shk) shk) chk) ) ) shk) g T g shk) + ch m [chk), g T ] + sh m matrix product, where kernel products mean compositions) The condition that we would like to impose is that R is block diagonal so that IR) contains only terms of the form aa and a a so that, apart from a trace when we commute a with a, we obtain an operator which annihilates the vacuum state. The remaining trace can be absorbed in

16 6 M. GRILLAKIS AND M. MACHEDON the evolution as a phase factor. Thus our requirement is i t ek e K + [e K, G]e K + e K Me K is block diagonal. 40) We proceed to show this equivalent to equations 7b), 7c). Let us make the elementary observations t ek e K = t I ek t e K [e K, G]e K = e K Ge K G so removing the part of 40) that is diagonal already we have the equivalent formulation of 40) e K ) i t + G + M e K is block diagonal. 4) Now we make the observation that a matrix is block-diagonal if and only if it commutes with the number operator matrix N u, as well as for arbitrary matrices A and B) we have [e K Ae K, B] = 0 if and only if [A, e K Be K ] = 0, so our equation 4) reads, [ i t + G + M ) ], e K N u e K = 0. 4) A direct calculation gives chk) shk) e K N u e K = shk) chk) after which is is straightforward to compute [ ] i t + G, e K N u e K = W chk) S shk)) and simlarly, [ M, e K N u e K] m shk) + shk) m = m chk) chk) m S shk)) W chk) ) m chk) chk) m m shk) + shk) m Finally combining the two formulas above we obtain, see 4), the linear pair of equations below S shk)) = m chk) + chk) m 43a) W chk) = m shk) shk) m. 43b) This completes the derivation of the evolution equations for the pair excitations and the mean field, namely 43a), 43b), together with 36)

17 PAIR EXCITATIONS,I 7 describe the evolution of φ and k and are the equations in 7a), 7b) and 7c). In particular, we have proved that in that if φ, k satisfy these equations, then the energy estimate 4) holds. 3. Estimates for the solution to the Hartree equation This section adapts classical results for NLS due to Lin and Strauss [30], Ginibre and Velo [9], Bourgain [3], as well as Colliander, Keel, Staffilani, Takaoka and Tao [4] to the Hartree equation. Assume i t φ φ + v N φ ) φ = 0 44) φ0, ) = φ 0. where v C 0 is non-negative, spherically symmetric, and decreasing. We recall the relevant conserved quantities, following the notation []: ρ := /) φ ; p j := /i) φ j φ φ j φ ) ; p 0 = /i) φ t φ φ t φ ) ; σ jk := j φ k φ + k φ j φ ; λ := Iφ t φ) + φ + v φ ) φ = φ v φ ) φ ) ; e : = φ + v φ ) φ. The associated conservation laws are t ρ j p j = 0, t p j k { σ k j σ 0j = j φ t φ + t φ j φ 45a) δ k j λ } + l j = 0, 45b) t e j σ j 0 + l 0 = 0. 45c) These laws express the conservation of mass, momentum and energy, respectively, where the vector l j, l 0 ) is l j := v N ρ)ρ j v N ρ j )ρ), l 0 := v N ρ)ρ 0 v N ρ 0 )ρ). In the case of NLS, v N = δ and l j, l 0 are 0. We adapt the well-known method of interaction Morawetz estimates, due to Colliander, Keel, Staffilani, Takaoka and Tao, outlined in [4]. Start with Qt) = j p j t, x)ρt, y) + ρt, x) j p j t, x) ) x y dxdy.

18 8 M. GRILLAKIS AND M. MACHEDON Using 45) we get Qt) = + j p j t, x) k p k t, y) x y dxdy j k { σ k j t, x) δ k j λt, x) } l j t, x) ) ρt, y) { + ρt, x) j k σ k j t, y) δj k λt, y) } l j t, y) )) ) x y dxdy λt, x)ρt, y) ρt, x)λt, y)) x y dxdy main term) j l j t, x))ρt, y) + ρt, x) j l j t, y))) x y dxdy error term.) We have used the fact which we recall for the reader s convenience see [4]), that j k a) x y) p j t, x)p k t, y) + σj k t, x)ρt, y) + σj k t, y)ρt, x) ) φx)φj = j k a) x y) y) + φ j x)φy) ) φx)φ j y) + φ j x)φy) ) ) + φx)φ j y) φ j x)φy)) φx)φ j y) φ j x)φy)) 0 where ax) = x. It is easy to check main term) c φt, ) 4 L 4 + vn ρ)t, x)ρt, x)ρt, y) + v N ρ)t, y)ρt, y)ρt, x) ) x y dxdy with c > 0

19 We proceed to analyze the error term: PAIR EXCITATIONS,I 9 error term = j l j t, x))ρt, y) + ρt, x) j l j t, y))) x y dxdy = j l j t, x))ρt, y) x y dxdy x y)j = l j t, x)ρt, y) x y dxdy vn = 4 ρ)t, x)ρ j t, x) v N ρ j )t, x)ρt, x) ) x y)j ρt, y) x y dxdy = 4 v N x z) ρt, z)ρ j t, x) ρ j t, z)ρt, x) ) x y)j ρt, y) = 8 4 = 4 v N x z ) x z)j x z v N x z)ρt, z)ρt, x)ρt, y) x,j x y) j x z) v N x j x y) j z ) x z x y x y x y dxdydz y)j ρt, z)ρt, x)ρt, y)x x y dxdydz ) dxdydz + z x)j z x z y) j z y ) ρt, z)ρt, x)ρt, y) vn ρ)t, x)ρt, x)ρt, y) + v N ρ)t, y)ρt, y)ρt, x) ) x y dxdy The next-to-last line is 0 because of the assumption v N 0 and the elementary trigonometric inequality x z) j x y) j z x)j + x z x y z x = cosθ ) + cosθ ) 0 z y) j z y The last line is negative, but cancels part of the main term. Thus main term) + error term) c φ 4 L 4 Since Qt) is bounded uniformly in time by φ 0 4 H, we have shown the following proposition. Proposition 3.. Let φ be a solution to the Hartree equation 44). There exists C depending only on φ 0 H such that φ L 4 [0, ) R 3 ) C and, as an immediate consequence of conservation of energy, φ L 8 [0, )L 4 R 3 ) C. 46)

20 0 M. GRILLAKIS AND M. MACHEDON Remark 3.. It was shown by Bourgain in [3] that if φ is a solution to cubic NLS, then there exists C s depending only on φ 0 H s such that φt, ) H s C s t. Using the above Morawetz estimate which was not yet discovered when Bourgain did this work), we can easily prove the same for our Hartree equation. Proposition 3.3. Let φ be a solution to the equation 44). exists C s depending only on φ 0 H s such that such that uniformly in time. φt, ) H s C s Proof. Split [0, ) into finitely many intervals I k where φ L 8 I k )L 4 R 3 ) ɛ where ɛ is to be prescribed later. Differentiating 44) There i t Ds φ D s φ = D s vn φ )φ ) 47) where D s vn φ )φ ) = v N φ ) D s φ + similar and easier terms. For the first interval, I, we get, using the L 8/3 L 4 Strichartz estimate, D s φ L 8/3 I )L 4 R 3 ) C φ 0 H s + C v N φ ) D s φ L 8/5I ) L 4/3 R 3 ) C φ 0 H s + C φ L 8 I )L 4 R 3 ) Ds φ L 8/3 I )L 4 R 3 ). At this stage, we pick ɛ so that C ɛ to conclude D s φ L 8/3 I )L 4 R 3 ) C φ 0 H s. In turn, this allows us to control the inhomogeneity of 47) v N φ ) D s φ L 8/5 I )L 4/3 R 3 ) φ L 8 I )L 4 R 3 ) Ds φ L 8/3 I )L 4 R 3 ) and therefore C φ 0 3 H s φt, ) H s φ 0 H s + C φ L 8 I )L 4 R 3 ) Ds φ L 8/3 I )L 4 R 3 ) C φ 0 3 H s for all t I. Repeating the process finitely many times, we are done.

21 PAIR EXCITATIONS,I If we assume the data φ 0 and sufficiently many derivatives are not only in L but also in L, we can also get decay. Corollary 3.4. Let φ be a solution to 44). There exists C depending only on φ 0 W k, for k sufficiently large such that φt, ) L C t 3 48a) and also t φt, ) L C t 3 48b) Proof. The proof follows the outline of [30], except that we have two modern ingredients which were not available to Lin and Strauss in 977: φ C s R 3+ ) C s s N φ L 4 R 3+ ) C This implies that φt, ) L R 3 ) 0 as t. Indeed, φ ) L 4 [n,n+] R 3 ) φ L R 3+ ) φ L 4 [n,n+] R 3 ) 0 This implies φ L p [n,n+] R 3 ) 0 for any fixed 4 < p <. Repeating the process one more time implies φt, ) L R 3 ) 0. We solve 44) by Duhamel s formula and use the standard L L decay estimate for the linear equation. We use the following estimate: e it s) v φ )φs) ) L 49) C t s 3/ v φ )φs) L C t s 3/ φs, ) L We would also like to estimate e it s) v φ )φs)) L by e it s) v φ )φs)) L 3. This is a false end-point, but becomes true if one replaces 3 by 3 + ɛ. To keep numbers easy, skip the ɛ and notice first that e it s) v φ )φs) ) C L 3 t s v / φ )φs) L 3/ 50) C t s φ C / L φ L 6 φs, ) /3 t s / L φs, ) 4/3 4 L C φs, ) 4/3 t s / L

22 M. GRILLAKIS AND M. MACHEDON Now, using 3 + ɛ rather than 3 leads to an estimate of the form e it s) v φ )φs) ) L C t s /+ɛ φs, ) 4/3 ɛ L 5) Combining 49) and 5) we get: There exists a kernel L [0, )) and δ > 0 such that Putting all together e it s) v φ )φs) ) L kt s) φs, ) +δ L 5) φt, ) L C t 3/ φ 0 L + t/ 0 C t t s φs, ) 3/ L ds + kt s) φs, ) +δ L ds t/ Denoting Mt) = sup 0<s<t + s 3/ ) φs, ) L, We have, for t >, Mt) C φ 0 L + C + t 3/ ) t/ 0 Ms)ds + C sup us, ) δ + s 3/ L ) Mt) t/<s<t The last term can be absorbed in Mt), and the result follows by Gronwall s inequality. Now that we know that φs, ) L C, it is very +s 3/ easy to estimate t φ. We use 49) and 5) with 3 + ɛ replacing 3), as well as the fact that all norms α φs, ) L p C α,p uniformly in s, for all pk). This is a consequence of Proposition 3.3) boundedness of the H s norms). t φt, ) L C t tφ 3/ 0 L + C t + C 3/ + C t t C t + C 3/ + C t t C t + C 3/ + C t t t 0 t + t s 3/ s 0 e it s) s v φ )φs) ) L v φ )φs) ) L ds + t s /+ɛ s v φ )φs) ) ds L 3/ ɛ t 0 + t s 3/ s v φ )φs) ) L ds + t s /+ɛ s v φ )φs) ) ds L 3/ ɛ t 0 + t s 3/ φs) L ds + t s /+ɛ φs) L ds If we estimate φs) L using 48a), we are done.

23 PAIR EXCITATIONS,I 3 By interpolating with the L uniform bound we get the next Corollary. Corollary 3.5. Let φ be a solution to 44). There exists C depending only on φ 0 W k, for k sufficiently large such that φt, ) L 3 + t φt, ) L 3 C + t. 4. Estimates for the pair excitations Define chk) := δ + p, shk) := s, and also chk) := δ + p, shk) := s so that, see 43a), 43b) become S s ) = m + m p + p m W p ) = m s s m s 0, ) = p 0, ) = 0 The goal of this section is to prove the following theorem. 53a) 53b) Theorem 4.. Assume φ 0 W k, for k sufficiently large. The following estimates hold: s t, ) L R 6 ) + p t, ) L R 6 ) C log + t) 54) where C depends on φ0, ) W k, for some finite k. A similar result holds for the higher time derivatives, but we will not use it or prove it. An immediate corollary is of the above theorem is, Corollary 4.. The following estimates hold: s t, ) L R 6 ) C log + t) p t, ) L R 6 ) C log + t) p x, x) dx C log + t). Proof. of corollary 4.)) Since shk) = shk) chk), we get s t, ) L R 6 ) s t, ) L R 6 ) chk) operator s t, ) L R 6 ).

24 4 M. GRILLAKIS AND M. MACHEDON We also have p x, x) 0, p p x, x) 0, so taking traces in the relation gives the other estimates. p p + p = s s Before starting the proof of Theorem 4.), we need some preliminary lemmas. Lemma 4.3. Recall mt, x, y) = v N x y)φt, x)φt, y). Then there exists C such that mn t, ξ, η) ξ + η ) dξdη C φt, ) 4 L 55a) 3 and also t m N t, ξ, η) ξ + η ) dξdη φt, ) L 3 tφt, ) L. 55b) 3 Similar estimates hold for higher time derivatives. Proof. Write v N x y)φt, x)φt, y) = δx y z)v N z)φt, x)φt, y)dz. The Fourier transform of δx y z)φt, x)φt, y) is easily computed to be e iz η φφz t, ξ + η) where we denote φ z x) = φx z). Thus m N t, ξ, η) = v N z)e iz η φφz t, ξ + η)dz v N L v N z) φφ z t, ξ + η) dz Thus, after a change of variables, the left hand side of 55a) is dominated by v N z) φφ z t, ξ) ξ + η ) dξdηdz C v N z) φφ z t, ξ) dξdz ξ C v N z) φ 4 L 3dz = C φ 4 L. 3

25 PAIR EXCITATIONS,I 5 We have used the fact that φφz t, ξ) dξ C Dx / φφ z ) L ξ C φφ z L 3/ C φ 4 L 3. The proof of 55b) is similar. by Hardy-Littlewood-Sobolev) Lemma 4.4. Let s 0 a be the solution to i t R 6 s 0 at, x, y) = mt, x, y) 56) Then s 0 a0, x, y) = 0. s 0 at, ) L R 6 ) t C φ0, ) L3 + φt, ) L3+ C log + t) where C depends only on φ 0 W k,. Proof. Solving 56) by Duhamel s formula we get s 0 at, ) L R 6 ) t 0 ) φs, ) L 3 s φs, ) L 3ds C e is ξ + η ) ˆms, ξ, η)ds L R 6 ) 0 ) ˆm0, ξ, η) C ξ + η ˆmt, ξ, η) L R 6 ) + ξ + η L R 6 ) + t 0 57) e is ξ + η ) ˆms, ξ, η) s ξ + η ds L R 6 ). 58) The terms 57) are estimated directly by Lemma 4.3). For the last term 58) move the norm inside the integral and use Lemma 4.3) as well as Corollary 3.5). Lemma 4.5. Let s a be the solution to Then where C depends on φ 0 W k,. Ss a ) = mt, x, y) 59) s a 0, x, y) = 0. s a t, ) L R 6 ) C log + t)

26 6 M. GRILLAKIS AND M. MACHEDON Proof. Let V be the potential part of S, so that S = i t R 6 + V. The operator V is bounded from L to L, with norm bounded by φt, ) L C by Corollary 3.4)). Explicitly, +t 3 V u)t, x, y) = v N φ )t, x) + v N φ )t, y) ) ut, x, y) + v N x z)φt, x)φt, z)uz, y)dz + ux, z)v N z y)φt, z)φt, y)dz. Write s a = s 0 a + s a where s 0 a is as in the previous lemma, i t s0 at, x, y) R 6s 0 at, x, y) = m. Then s a satisfies the equation Ss a) = V s 0 at, ) ). 60) Both s 0 a and s a are zero initially, and the estimate is already known for s 0 a.. We apply energy estimates to 60). Recall Ws a s a) = Ss a) s a s a Ss a) = V s 0 at, ) ) s a + s a V s 0 at, )). Taking traces, we get t s a L R 6 ) V s 0 at, ) ) L R 6 ) s a L R 6 ) C + t 3 s0 at, ) L R 6 ) s a L R 6 ) C log + t) s + t at, ) 3 L R 6 ). Integrating, we get the estimate which implies the claim. s at, ) L R 6 ) C We are ready for the proof of Theorem 4.). Proof. Write s = s a +s e where S s a ) = m, as in the previous lemma. The kernels s e and p satisfy the following less singular system: S s e ) = m p + p m W p ) = m s a s a m ) + m s e s e m. 6a) 6b)

27 Using lemma 4.5) to estimate s a t, ) L we have since PAIR EXCITATIONS,I 7 Mt, ) = m s a s a m S s e ) = m p + p m and defining W p ) = M + m s e s e m where Using the general formulas Mt, ) L R 6 ) m s L C φ L s L C log + t) + t 3 Ws s) = Ss) s s Ss) C + t 3 s L. Wp p) = Wp) p + p Wp) 6a) 6b) and taking traces we get ) s e t, ) L t R 6 ) + p t, ) L R 6 ) C p t, ) + t 3 L R 6 ) s e t, ) L R 6 ) + C Mt, ) L R 6 ) p t, ) L R 6 ) which leads to the desired estimate. Explicitly, define then E t) = s e t, ) L R 6 ) + p t, ) L R 6 ) t Et) C + t Et) + Mt, ) 3 L thus Et) is uniformly bounded. Also, p L stays uniformly bounded and the logarithmic growth of s L can be traced back to s 0 a L from Lemma 4.4). 5. List of all the error terms Our purpose in the present section is to compute explicitly all the error terms and show how they can be estimated. Fortunately there are only a few terms for which one should be carefull, the rest being easier to handle. Let us recall here briefly our basic strategy, which is to define ψred := e Bkt)) e NAφt)) e ith e NAφ0)) 0

28 8 M. GRILLAKIS AND M. MACHEDON and compute H red such that /i) t ψred = Hred ψred. H red = e Bkt)) e Bkt)) i t ) + e Bkt)) i t e NAφt)) e NAφt)) + e NAφt)) He NAφt)) e Bkt)) = Nµ 0 t) 63a) + N / e Bkt)) dx { hφt, x))a x + hφt, x)) } e Bkt)) 63b) + e Bkt)) e Bkt) + e Bkt)) H )e i t [[A, [A, V]]] Bkt)) 63c) ) N / e [A, Bkt)) V] + N / V e Bkt)). 63d) The function µ 0 t) and hφt, x)) appearing in 63a),63b) are given below, { } µ 0 := dx φ φt i φφ ) t φ dxdy { v N x y) φx) φy) } hφt, x)) := i tφ + φ v N φ ) φ. As we know, 63b) is set to be zero due to the Hartree equation for φ, and 63c) which contains terms of order O) becomes block diagonal by an appropriate choice for the evolution of k. We can compute the O) term in 63c), w T f 63c) = H G I f w where f := Sshk)) chk) m ) chk) Wchk)) + shk) m ) shk) w := Wchk)) + shk) m ) chk) Sshk)) chk) m ) shk). The evolution implies that f = 0 thus w T 0 63c) = H G I. 0 w Multiplying f on the right with shk) and using the identities chk) shk) = shk) chk) and shk) shk) = chk)) we discover f shk) + w chk) = Wchk)) + shk) m.

29 PAIR EXCITATIONS,I 9 Using the formula see [], lemma 3., stated in slightly different notation) we get Wchk)) chk) + chk) Wchk)) = chk) m shk) shk) m chk) wy, x) = W chk) ) chk) ) + shk) m chk) = chk) ) ) m shk) + shk) m chk) + [ Wchk)), chk) ) ]. If we write wy, x) for the above we have the quadratic term and a trace 63c) =H G + dxdy {wx, y)d yx + wy, x)d xy } 64) + Tr chk) ) m shk) + shk) m chk) ) ) ). The error terms, which are order ON / ) or higher, are all in the third line 63d) i.e. what we called E, Recall that [A, V] = V = E = e B [A, V] + N / V ) e B. dx dx { vn x x ) φx )a x Q x x + φx )Q x x a x )} dx dx { vn x x )Q x x Q x x } 65a) 65b) and the transformation of a, a ) is chk)y, e B a x e B = x)ay + shk)y, x)ay) dy := bx 66a) e B a xe B = shk)y, x)a y + chk)y, x)a y dy := b x. 66b) One can see that [b x, b y] = δx y). For the rest of this section, the argument of the trigonometric functions is always k, thus sh abbreviates shk), p stands for p, etc. Keeping in mind that chy, x) =

30 30 M. GRILLAKIS AND M. MACHEDON chx, y) and shy, x) = shx, y), shy, x) = shx, y) we compute e B a x a x e B = e B Q x x e B = dy dy { shy, x ) chx, y )D y y + chy, x )shx, y )D y y } + dy dy { chy, x ) chx, y )Q y y + shy, x )shx, y )Q y y } + sh ch ) x, x ) 67a) e B a x a x e B = e B Q x x e B = dy 3 dy 4 { chy3, x )shx, y 4 )D y4 y 3 + shy 3, x )chx, y 4 )D y3 y 4 } + dy 3 dy 4 { chy3, x )chx, y 4 )Q y3 y 4 + shy 3, x )shx, y 4 )Q y 3 y 4 } + ch sh ) x, x ). 67b) Let us look first at the all the terms that come from the quartic term V. Since e B Ve B 0 we are interested only in terms that do not anihilate the vacuum. This implies that we keep Q y 3 y 4 from 67b) with everything in 67a) as well as Q y y from 67a) with the last term in 67b) and the product of the two last terms in 67a), 67b). Below is the list of all the terms in N eb Ve B which do not annihilate 0 ) in raw form, N dx dx dy dy dy 3 dy 4 { shy, x ) chx, y )v N x x )shy 3, x )shx, y 4 )D y y Q y 3 y a) chy, x )shx, y )v N x x )shy 3, x )shx, y 4 )D y y Q y 3 y b) chy, x ) chx, y )v N x x )shy 3, x )shx, y 4 )Q y y Q y 3 y c) } shy, x )shx, y )v N x x )shy 3, x )shx, y 4 )Q y y Q y 3 y 4 68d) + { dx dx dy dy N sh ch x, x )v N x x )shy, x )shx, y )Q y y + 68e) chy, x ) chx, y )v N x x ) ch sh ) } x, x )Q y y 68f) + dx dx N sh ch x, x )v N x x ) ch sh ) x, x ). 68g)

31 PAIR EXCITATIONS,I 3 Next let us look at the terms that come from the cubic term, namely the expression e B [A, V]e B 0. They come in two sets. First from the term b x b x b x ) there are three terms, two coming from the product of Q from 67b) with 66b) and one coming from a in 66b) with the constant term in 67b). The are listed below N / dx dx dy dy dy 3 { shy, x )v N x x ) φx )shy, x )shx, y 3 ) a y Q y y a) chy, x )v N x x ) φx )shy, x )shx, y 3 ) a y Q y y 3 }+ 69b) N / dx dx dy chy, x )v N x x ) φx ) ch sh ) x, x ) a y 69c) The second set comes from b x b x )b x and gives five terms, namely all terms appearing in 67a) multiplied with a in 66b). They are listed below, N / dx dx dy dy dy 3 { shy, x ) chx, y )v N x x )φx )shy 3, x )D y y a y 3 + chy, x )shx, y )v N x x )φx )shy 3, x ) D y y a y 3 + chy, x ) chx, y )v N x x )φx )shy 3, x )Q y y a y a) 70b) 70c) shy, x )shx, y )v N x x )φx )shy 3, x ) Q y y a y 3 }+ 70d) N / dx dx dy sh ch ) x, x )v N x x )φx )shy, x )a y 70e) Some of the terms can be reduced to lower order by commuting a with a whenever they appear together in a product. The term in 70d) applied to 0 gives zero. Irreducible terms are those appearing in 68c) which is fourth order, as well as the terms appearing in 69b) and 70c) which are cubic. The quartic irreducible term in 68c) can

32 3 M. GRILLAKIS AND M. MACHEDON be writen, if we write chx, y) = δx y) + px, y) first, as follows, { N /) v N y y )shy 3, y )shy, y 4 ) + 7a) dx { py, x)v N y x)shx, y 4 )} shy 3, y )+ 7b) dx { py, x)v N x y )shy 3, x)} shy, y 4 )+ 7c) dx dx { py, x )px, y )v N x x )shy 3, x )shx, y 4 ) }}. 7d) The rest of the quartic terms can be reduced to either quadratic or zero order terms. For example if we look at 68b) we can write, D y y Q y 3 y 4 = δy y 4 )Q y y 4 + δy y 4 )Q y y 3 modulo terms which annihilate 0 ) and similarly for 68a). Below is a list of all quadatic terms, /N) dx dx { chy, x )shx, y ) sh sh ) x, x )v N x x )+ chy, x )shx, y ) sh sh ) x, x )v N x x )+ chy, x )shx, y ) sh sh ) x, x )v N x x )+ chy, x )shx, y ) sh sh ) x, x )v N x x )+ 7a) 7b) 7c) 7d) shy, x )shx, y ) sh ch ) x, x )v N x x )+ 7e) chy, x ) chx, y ) ch sh ) } x, x )v N x x ). 7f) In addition 68d) and 68g) together with the trace in 64) will supply a zero order term, µ = chk) 4 Tr ) ) m shk) + shk) m chk) + dx dx {sh sh)x, x )v N x x )sh sh)x, x )+ N sh sh)x, x )v N x x )sh sh)x, x )+ sh ch ) x, x )v N x x ) ch sh ) x, x )}. 73) This term can be absorbed as a phase factor in the evolution. The cubic irreducible terms are those appearing in 69b),70c). Writing

33 PAIR EXCITATIONS,I 33 chy, x) = δy x) + py, x) we can express them in the manner below { N / v N y y )φy )shy 3, y ) + 74a) dx { v N y x) φx)shx, y 3 ) } shy, y )+ 74b) dx { py, x)v N x y )shy 3, x)} φy ) + 74c) dx { py, x)v N y x)φx)} shy 3, y ) + 74d) dx dx { py, x )v N x x ) φx )shy, x )shx, y 3 ) } + 74e) } dx dx { py, x )px, y )v N x x )φx )shy 3, x )}. 74f) The rest can be reduced to linear or eliminated, for example 70d) can be eliminated. For the cubic terms in the first set. From the commutator relation a y Q y y 3 = δy y )a y 3 + δy y 3 )a y modulo a terms which kill 0 ) we reduce them to linear and here is the list, { N / dx dx shy, x ) sh sh ) x, x ) φx )v N x x ) + 75a) shy, x ) sh sh ) x, x ) φx )v N x x ) + 75b) chy, x ) ch sh ) x, x ) φx } )v N x x ). 75c) From the cubic terms in the second list we have the commutators D y y a y 3 = δy y 3 )a y and D y y a y 3 = δy y 3 )a y modulo a terms) hence from 70a),70b) as well as from 70e) we obtain N / dx dx { chy, x )sh sh)x, x )φx )v N x x ) + 76a) chy, x )sh sh)x, x )φx )v N x x ) + 76b) shy, x ) sh ch ) } x, x )φx )v N x x ). 76c) In this section we prove 6. Estimates for the error terms

34 34 M. GRILLAKIS AND M. MACHEDON Proposition 6.. The following estimates for the error terms holds: N / e B [A, V]e B 0 F N 3β log 3 + t) t 3/ N e B Ve B 0 F CN 3β log 4 + t) Proof. All estimates are straightforward, based on Corollary 4.) and Lemma 3.4), and we only include a few typical ones from each category. Estimate for 7a): N v N y y )shk)y 3, y )shk)y, y 4 ) L dy dy dy 3 dy 4 ) N v N L shk) L CN 3β log + t) Estimate for 7d): N dx dx p y, x )p x, y )v N x x )shy 3, x )shx, y 4 ) L dy dy dy 3 dy 4 ) N v N L shk) L p L CN 3β log 4 + t) Estimate for 7f), keeping only the δ contribution in chk): N shk)y, y )v N y y ) L dy dy ) v N L shk) L CN 3β log + t) Estimate for 74a), : N / v N y y )φy )shk)y 3, y ) L dy dy dy 3 ) φ L v N L shk) L CN 3β log + t) t 3/ Estimates for 76b): N / dx dx p y, x )shk) shk))x, x )φx )v N x x ) L dy ) N / p L shk) shk)x, x) L dx) φ L v N L N / p L shk) L φ L v N L CN 3β log 3 + t) t 3/ 7. Appendix The purpose of this appendix is to make certain connections with our previous work [0], [] and to provide some useful information. We would like to explain first the Lie algebra isomorphism which is crucial in our work. This was explained in [0] but we include it here

35 PAIR EXCITATIONS,I 35 for completeness and for the convenience of the reader. Let us define first ax f x) 0 A x := and fx) := and J = f x) 0 a x and using these form A f ) := dx { A T x fx) } = dx {f x)a x + f x)a x}. 77) It is straightforward to check the commutation, [ ] Af), Ag) = dx {f x)g x) f x)g x)} = dxdy { f T x)δx y)jgy) }. 78) For L dxdy) kernel functions dt, x, y) and kt, x, y), lt, x, y) such that k and l are symmetric in the x, y) variables form the symplectic matrix d k Sd, k, l) := l d T. 79) We will write Sx, y) when convenient, suppressing the t dependence. Next we define the map I : spc) Quad from the space of complex, L symplectic matrices to quadratic expressions in a, a ) as follows : IS) := dxdy { } A T x Sx, y)ja y. 80) Theorem 7.. Let fx) a vector function and Af) the corresponding expression 77). Then the following commutations relations hold [ ] IS), Af ) = A S f 8a) e IS) Af)e IS) = A e S f ) 8b) where for example) S f = dx{sx, y)fy)} etc. Formula 8b) holds for any complex symplectic S sp c R) = C T ) spr)c T where C is the change-of-basis matrix C = I ii I ii and spr) is the Lie algebra of real symplectic matrices. This condition insures that e S is unitary.

36 36 M. GRILLAKIS AND M. MACHEDON Proof. The commutation relation in 8a) can be easily checked, but we point that it follows from 78). For any rank one quadratics we have using 78) [ ] gf Af)Ag), Ah) = A T + fg ) T Jh). Thus for any R we have [ dxdy { } ] R A T x Rx, y)a y, Af) = A ) + R T Jf. Now specialize to R = /)SJ with S sp and use S T = JSJ to complete the proof. For the second formula, introduce a complex parameter t and take ψ a Fock space vector with finitely many non-zero components. It is trivial to check, using 8a), that all t derivatives of e tis) Af)e tis) ψ and A e ts f ) ψ agree when t = 0, and both the left hand side and right hand side are analytic if t is sufficiently small. Thus they agree for all t complex, sufficiently small. To take t large, we have to restrict ourselves to real t and use the group properties of the unitary family e ts. A more formal but convincing) proof follows from e IS) Ce IS) = C + [IS), C] +![ IS), [IS), C] ] +... Theorem 7.. The map I : spc) Quad defined in 80) is a Lie algebra isomorphism. Moreover for S, R sp c R) we have the formulas t I e S) ) e S = t e IS)) e IS) 8a) I e S R e S) = e IS) IR)e IS). 8b) Proof. We point out that 8b) implies 8b), at least when R is rank one, R := fx)g T y). Notice that 8b) can be written e IS) dx { } f T x)a x e IS) = } dxdy {f T x)e ST A y

37 [Nx x, N y y ] = δx y )N y x δx y )N x y PAIR EXCITATIONS,I 37 from which we have e IS) dxdy { A T x fx)g T y)ja y } e IS) = e IS) Af)e IS) e IS) = A e S f ) dydz { } g T y)je JSJ A z = dxdy { } A T x e S Re S JA y dy { g T y)ja y } e IS) since S T = JSJ and Je JSJ = e S J. A direct proof of the Lie algebra isomorphism follows from the following elementary computations. Define first Q xy := a x a y and Q xy := a xa y and N xy := ax a y + a ya x ). It is straightforward to verify, [ Qx x, Q y y ] = δx y )N x y + δx y )N x y + δx y )N x y + δx y )N x y [Qx x, N y y ] = δx y )Q x y + δx y )Q x y [Nx x, Q y y ] = δx y )Q x y + δx y )Q x y and using these we can directly verify that [ { } { } ] dx dx kx, x )Q x x, dy dy ly, y )Q y y = dxdy { k l ) } x, y)n xy which corresponds to the relation [ ] 0 k 0 0 k l 0, = 0 0 l 0 0 l k The other cases are checked in a similar manner. To prove 8a), expand both the left and right-hand side as I t e S) e S) = I Ṡ + ) [S, Ṡ] +... = İS) + [ IS), İS) ] +... = t e IS)) e IS). The proof of 8b) is along the same lines. The proofs can be made rigorous by an analyticity argument on the dense subset of vectors with finitely many non-zero components..