Bose-Einstein Condensation with Optimal Rate for Trapped Bosons in the Gross-Pitaevskii Regime
aa r X i v : . [ m a t h - ph ] F e b Bose-Einstein Condensation with Optimal Rate forTrapped Bosons in the Gross-Pitaevskii Regime
Christian Brennecke , Benjamin Schlein , Severin Schraven Department of Mathematics, Harvard University,One Oxford Street, Cambridge MA 02138, USA Institute of Mathematics, University of Zurich,Winterthurerstrasse 190, 8057 Zurich, Switzerland , February 23, 2021
Abstract
We consider a Bose gas consisting of N particles in R , trapped by an externalfield and interacting through a two-body potential with scattering length of order N ´ . We prove that low energy states exhibit complete Bose-Einstein condensa-tion with optimal rate, generalizing previous work in [1, 4], restricted to translationinvariant systems. This extends recent results in [15], removing the smallness as-sumption on the size of the scattering length. We consider a system of N P N bosons trapped by an external potential in the Gross-Pitaevskii regime; the particles interact through a repulsive two-body potential withscattering length of order N ´ . The Hamilton operator as the form H N “ N ÿ j “ “ ´ ∆ x j ` V ext p x j q ‰ ` ÿ ď i ă j ď N N V p N p x i ´ x j qq . (1.1)and it acts on a dense subspace of the Hilbert space L s p R N q , the subspace of L p R N q consisting of functions that are symmetric with respect to permutations of the N parti-cles. The confining potential V ext P L loc p R q diverge to infinity, as | x | Ñ 8 (more preciseconditions on V ext will be introduced later on). Furthermore, we assume V P L p R q to be pointwise non-negative, spherically symmetric and compactly supported (but ourresults could easily be extended to potentials decaying sufficiently fast at infinity).The scattering length of V is defined through the zero-energy scattering equation „ ´ ∆ ` V p x q f p x q “ f p x q Ñ
1, as | x | Ñ 8 . For | x | large enough (outside thesupport of V ) we find f p x q “ ´ a | x | where the constant a ą V . A simple computationshows that 8 π a “ ż R V p x q f p x q dx. (1.3)Moreover, by scaling, (1.2) implies that „ ´ ∆ ` N V p N x q f p N x q “ N V p N. q is given by a { N .From [14], it is known that the ground state energy E N of the Hamilton operator(1.1) satisfies lim N Ñ8 E N N “ inf ψ P H p R q : } ψ } “ E GP p ψ q , (1.4)where E GP denotes the Gross-Pitaevskii energy functional E GP p ψ q “ ż R ` | ∇ ψ p x q| ` V ext p x q| ψ p x q| ` π a | ψ p x q| ˘ dx. (1.5)For the rest of the paper, we will lighten the notation and write ş instead of ş R .Furthermore, we will write } ¨ } for the L -norm and indicate other L p -norms by asuitable subscript. The Gross–Pitaevskii functional E GP admits a unique normalized,strictly positive minimizer ϕ P L p R q . It satisfies the Euler-Lagrange equation ´ ∆ ϕ ` V ext ϕ ` π a | ϕ | ϕ “ ε GP ϕ (1.6)with the Lagrange multiplier ε GP : “ E GP p ϕ q ` π a } ϕ } . As first shown in [12], theground state of (1.1) exhibits complete Bose-Einstein condensation in the state ϕ . Moreprecisely, if γ p q N “ tr ,...,N | ψ N yx ψ N | denotes the one-particle reduced density associatedwith the ground state of (1.1), thenlim N Ñ8 x ϕ, γ p q N ϕ y “ . (1.7)This implies that, in the ground state of (1.1), the fraction of particles in the state ϕ approaches one, as N Ñ 8 . The convergence in (1.7) was later extended in [13, 16] toany sequence ψ N of approximate ground states, satisfyinglim N Ñ8 N x ψ N , H N ψ N y “ E GP p ϕ q . For translation invariant systems (particles trapped in the box Λ “ r
0; 1 s , withperiodic boundary conditions), (1.4) (stating, in this case, that E N { N Ñ π a ) and21.7) (establishing Bose-Einstein condensation in the zero-momentum mode ϕ p x q “ x P Λ) have been proved in [1, 4, 10] to hold with the optimal rate of convergence.This result was recently generalised in [15] (extending the approach of [6, 8]) to trappedsystems described by the Hamilton operator (1.1), under the assumption of sufficientlysmall scattering length a .Our goal in this paper is to obtain optimal bounds on the rate of convergence in (1.4)and (1.7), with no restriction on the size of the scattering length. To reach this goal, weadapt and extend the approach developed in [4] for the translation invariant case. Tothis end, we will impose, throughout the rest of this paper, the following conditions: p q V P L p R q , V p x q ě x P R , V spherically symmetric, supp p V q compact , p q V ext P C p R ; R q , V ext p x q Ñ 8 as | x | Ñ 8 , V ext spherically symmetric, D C ą @ x, y P R : V ext p x ` y q ď C p V ext p x q ` C qp V ext p y q ` C q , ∇ V ext , ∆ V ext have at most exponential growth as | x | Ñ 8 . (1.8)Note in particular that all polynomials in x with positive leading coefficient satisfycondition p q in (1.8). The assumptions that V ext p x ` y q ď C p V ext p x q ` C qp V ext p y q ` C q and that ∇ V ext , ∆ V ext grow at most exponentially allow for certain simplifications of ouranalysis and are needed for technical reasons only. Theorem 1.1.
Assume (1.8) and let E N denote the ground state energy of (1.1). Then,there exists a constant C ą such that E N ě N E GP p ϕ q ´ C (1.9) and H N ě N E GP p ϕ q ` C ´ N ÿ i “ ` ´ | ϕ yx ϕ | i ˘ ´ C. In particular, if ψ N P L s p R N q with } ψ N } “ is an sequence of approximate groundstates such that x ψ N , H N ψ N y ď N E GP p ϕ q ` ζ, for a ζ ą , then the reduced density γ p q N associated with ψ N satisfies ´ x ϕ, γ p q N ϕ y ď p C ` ζ q N .
Remark:
Our techniques could also be used to prove an upper bound for E N matching(1.9), implying that | E N ´ N E GP p ϕ q| ď C . We do not show it, because it would requiresome non-trivial additional work (this is a consequence of our choice, leading to sometechnical simplifications, to work on the Fock space F ď N rather than on F ď N K ϕ , where weimpose orthogonality to ϕ ; we will explain this point in the next section) and becauseit is already established in [15] (the upper bound there does not require restrictions onthe size of the potential) 3 emark: We plan to apply Theorem 1.1 in a separate paper to determine the low-energy spectrum of (1.1) and to establish the validity of the predictions of Bogoliubovtheory, extending recent results obtained in [2, 3] for the translation invariant setting.
We introduce the bosonic Fock space F “ à n ě L s p R n q “ à n ě L p R q b s n On F , we consider creation and annihilation operators, satisfying the canonical commu-tation relations r a p g q , a ˚ p h qs “ x g, h y , r a p g q , a p h qs “ r a ˚ p g q , a ˚ p h qs “ g, h P L p R q . We also introduce position and momentum-space operator-valueddistributions a x , a ˚ x and ˆ a p , ˆ a ˚ p , for x, p P R , so that a p f q “ ż ¯ f p x q a x dx “ ż ¯ˆ f p p q ˆ a p dp, a ˚ p f q “ ż f p x q a ˚ x dx “ ż ˆ f p p q ˆ a ˚ p dp. In terms of these operator-valued distributions, the number of particles operator N ,defined by p N Ψ q p n q “ n Ψ p n q for every Ψ P F , takes the form N “ ż a ˚ x a x dx “ ż ˆ a ˚ p ˆ a p dp. More generally, given an operator A : L p R q Ñ L p R q with kernel A p x ; y q , we defineits second quantization d Γ p A q acting in F through d Γ p A q “ ż dxdy A p x ; y q a ˚ x a y . In particular, N “ d Γ p q is the second quantization of the identity operator.It is simple to check that creation and annihilation operators are bounded, withrespect to N { . In fact, we find that } a p f q Ψ } ď } f } } N { Ψ } , } a ˚ p f q Ψ } ď } f } }p N ` q { Ψ } . (2.2)To describe excitations of the Bose-Einstein condensate, we also define the truncatedFock spaces F ď N “ N à n “ L p R q b s n , and F ď N K ϕ “ N à n “ L K ϕ p R q b s n defined over L p R q and, respectively, over L K ϕ p R q , the orthogonal complement of thecondensate wave function ϕ in L p R q ( ϕ is the unique minimizer of (1.5)). Since they4o not preserve the number of particles, creation and annihilation operators are not well-defined on F ď N and F ď N K ϕ (but notice that products of a creation and an annihilationoperators are well-defined). They are replaced by modified creation and annihilationoperators b ˚ p f q “ a ˚ p f q c N ´ N N , b p f q “ c N ´ N N a p f q . For every f P L p R q , b p f q and b ˚ p f q map F ď N to F ď N . If moreover f K ϕ , we alsofind b p f q , b ˚ p f q : F ď N K ϕ Ñ F ď N K ϕ . From (2.2) we have } b p f q Ψ } ď } f }} N { Ψ } , } b ˚ p f q Ψ } ď } f }}p N ` q { Ψ } . Also here, it is convenient to introduce operator-valued distributions b x , b ˚ x , for any x P R , and, in momentum space ˆ b p , ˆ b ˚ p , for any p P R . They satisfy the commutationrelations (focussing here on position space operators) r b x , b ˚ y s “ ˆ ´ N N ˙ δ p x ´ y q ´ N a ˚ y a x , r b x , b y s “ r b ˚ x , b ˚ y s “ r b x , a ˚ y a z s “ δ p x ´ y q b z , r b ˚ x , a ˚ y a z s “ ´ δ p x ´ z q b ˚ y . (2.4)In particular, it follows that r b x , N s “ b x and r b ˚ x , N s “ ´ b ˚ x .We factor out the Bose-Einstein condensate applying a unitary map U N : L s p R N q Ñ F ď N K ϕ , first introduced in [11]. To define U N , we observe that any ψ N P L s p R N q can beuniquely decomposed as ψ N “ α ϕ b N ` α b s ϕ bp N ´ q ` ¨ ¨ ¨ ` α N with α j P L K ϕ p R q b s j for every j “ , . . . , N . Thus, we can set U N ψ N “ t α , α , . . . , α N u P F ď N K ϕ . It is then possible to show that U N is unitary; see [11] for details.With U N , we can define the excitation Hamiltonian L N “ U N H N U ˚ N , acting on adense subspace of F ď N K ϕ . The action of U N on creation and annihilation operators isgiven by U N a ˚ p ϕ q a p ϕ q U ˚ N “ N ´ N ,U N a ˚ p f q a p ϕ q U ˚ N “ a ˚ p f q? N ´ N “ ? N b ˚ p f q ,U N a ˚ p ϕ q a p g q U ˚ N “ ? N ´ N a p g q “ ? N b p g q ,U N a ˚ p f q a p g q U ˚ N “ a ˚ p f q a p g q (2.5)for all f, g P L K ϕ p R q , where N denotes the number of particles operator in F ď N K ϕ .Writing H N in second quantized form and using (2.5), we proceed as in [11, Section 4]and find that L N “ L p q N ` L p q N ` L p q N ` L p q N ` L p q N , F ď N K ϕ , we have that L p q N “ @ ϕ, “ ´ ∆ ` V ext ` ` N V p N ¨q ˚ | ϕ | ˘‰ ϕ D p N ´ N q´ @ ϕ, ` N V p N ¨q ˚ | ϕ | ˘ ϕ D p N ` qp ´ N { N q , L p q N “ ? N b `` N V p N ¨q ˚ | ϕ | ´ π a | ϕ | ˘ ϕ ˘ ´ N ` ? N b `` N V p N ¨q ˚ | ϕ | ˘ ϕ ˘ ` h.c. , L p q N “ ż dx p ∇ x a ˚ x ∇ x a x ` V ext p x q a ˚ x a x q` ż dxdy N V p N p x ´ y qq| ϕ p y q| ´ b ˚ x b x ´ N a ˚ x a x ¯ ` ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q ´ b ˚ x b y ´ N a ˚ x a y ¯ ` ż dxdy N V p N p x ´ y qq ϕ p y q ϕ p x q ´ b ˚ x b ˚ y ` h.c. ¯ , L p q N “ ż N { V p N p x ´ y qq ϕ p y q ` b ˚ x a ˚ y a x ` h.c. ˘ dxdy, L p q N “ ż dxdy N V p N p x ´ y qq a ˚ x a ˚ y a y a x . (2.6)While L N maps F ď N K ϕ to itself, the operators L p j q N , j P t , , , , u are also welldefined as operators on F ď N . In the following, it will be technically convenient toidentify the L p j q N , through Eq. (2.6), as acting in F ď N and in this case we will denotethem by r L p j q N . Moreover, we define r L N “ r L p q N ` r L p q N ` r L p q N ` r L p q N ` r L p q N (2.7)as a self-adjoint operator acting on a dense subspace in F ď N . In particular, if Γ p q q isthe orthogonal projection from F ď N onto F ď N K ϕ , defined by p Γ p q q Ψ q p n q “ q b n Ψ p n q forevery n P t , , . . . , N u , with q “ ´ | ϕ yx ϕ | , then L N “ Γ p q q r L N Γ p q q in F ď N K ϕ .The vacuum expectation of L N is given, up to terms of order one, by N ż „ | ∇ ϕ p x q| ` V ext p x q| ϕ p x q| ` p V p q| ϕ p x q| dx Since ˆ V p q ą π a , the difference between x Ω , L N Ω y and the ground state energy of L N (or of H N ) is very large, of order N , in the limit N Ñ 8 . The point is that, throughthe map U N , we expand H N around the energy of the pure condensate wave function ϕ b N P L s p R N q . In the Gross-Pitaevskii regime, however, it is well-known that shortscale correlations among particles play a crucial role; they even affect the energy toleading order. To extract the missing correlation energy, we will conjugate L N with6wo unitary operator, given by the exponential of a quadratic and a cubic operator in(modified) creation and annihilation operators. We follow here the basic strategy of [4].Loosely speaking, through unitary conjugation, we renormalize the singular interaction,producing a soft, mean-field potential whose ultraviolet behavior is easy to control. Thisprocedure extracts the missing correlation energy from the higher order terms in L N and, at the same time, it generates the coercivity needed to show Theorem 1.1.In our first renormalization step, we will conjugate L N with a generalized Bogoliubovtransformation. To define the kernel of the quadratic phase, we consider the ground statesolution of the Neumann problem „ ´ ∆ ` V f ℓ “ λ ℓ f ℓ (2.8)on the ball | x | ď N ℓ , for some 0 ă ℓ ă
1. For simplicity, we omit here the N -dependencein the notation for f ℓ and for λ ℓ . By radial symmetry of the interaction V , f ℓ is radiallysymmetric and we normalize it such that f ℓ p x q “ | x | “ N ℓ . By scaling, f ℓ p N. q solves „ ´ ∆ ` N V p N x q f ℓ p N x q “ N λ ℓ f ℓ p N x q on the ball where | x | ď ℓ . Later in our analysis, we will choose the parameter ℓ ą N . We thenextend f ℓ p N. q to R , by setting f N,ℓ p x q “ f ℓ p N x q if | x | ď ℓ and f N,ℓ p x q “ x P R with | x | ą ℓ . Thus, f ℓ p N. q solves the equation ˆ ´ ∆ ` N V p N. q ˙ f N,ℓ “ N λ ℓ f N,ℓ χ ℓ (2.9)where χ ℓ denotes the characteristic function of the ball B ℓ p q of radius ℓ , centered atthe origin in R . Finally, we denote by w ℓ the function w ℓ “ ´ f ℓ . Notice that byscaling, w ℓ p N. q has compact support in B ℓ p q , for all N P N sufficiently large. Definingthe Fourier transform of w ℓ through p w ℓ p p q “ ż dx w ℓ p x q e ´ πipx , we see that w ℓ p N. q has Fourier transform ż dx w ℓ p N x q e ´ πipx “ N p w ℓ p p { N q and we record that (2.9) implies that ´ π p p w ℓ p p { N q ` N p p V p . { N q ˚ p f N,ℓ qp p q “ N λ ℓ p p χ ℓ ˚ p f N,ℓ qp p q . The next lemma collects important properties of f ℓ , w ℓ and the Neumann eigenvalue λ ℓ .7 emma 2.1. Let V P L p R q be non-negative, compactly supported and sphericallysymmetric. Fix ℓ ą and let f ℓ denote the solution of (2.8) .i) We have that λ ℓ “ a p ℓN q ´ ` O ´ a ℓN ¯¯ (2.10) ii) We have ď f ℓ , w ℓ ď and there exists a constant C ą such that ˇˇˇˇż R V p x q f ℓ p x q dx ´ π a ˇˇˇˇ ď C a ℓN (2.11) for all ℓ P p
0; 1 q , N P N .iii) There exists a constant C ą such that w ℓ p x q ď C | x | ` and | ∇ w ℓ p x q| ď C | x | ` . (2.12) for all x P R , ℓ P p
0; 1 q and N P N large enough. Moreover, ˇˇˇ p N ℓ q ż R w ℓ p x q dx ´ π a ˇˇˇ ď C a N ℓ for all ℓ P p
0; 1 q and N P N large enough.iv) There exists a constant C ą such that | p w ℓ p p q| ď C | p | (2.13) for all p P R , ℓ P p
0; 1 q and N P N large enough.Proof. This has already been proved in [3, Appendix B], based on [7, Lemma A.1] and[4, Lemma 4.1].Let us now define the correlation kernel for the generalized Bogoliubov transformationthat we will be using. We denote by G : R Ñ R the rescaled function G p x q “ ´ N w ℓ p N x q (2.14)which has compact support in B ℓ p q and which, by (2.13), satisfies for all p P R | p G p p q| ď C | p | . (2.15)We denote by χ H the characteristic function of t p P R : | p | ě ℓ ´ α u , for fixed α ą
0, andby q χ H its inverse Fourier transform. Finally, we define η H P L p R ˆ R q by η H p x, y q “ p G ˚ q χ H qp x ´ y q ϕ p x q ϕ p y q . (2.16)The next lemma summarizes basic properties of the kernel η H P L p R ˆ R q .8 emma 2.2. Assume (1.8) , let ℓ P p
0; 1 q and let α ą . Set η H,x p y q “ η H p x ; y q for x P R . Then, there exists C ą , uniform in N , ℓ P p
0; 1 q and x P R , such that } η H } ď Cℓ α { , } η H,x } ď Cℓ α { | ϕ p x q| , } ∇ η H } , } ∇ η H } ď C ? N . (2.17)
Furthermore, identifying η H p x ; y q with the kernel of a Hilbert-Schmidt operator on L p R q and η p n q H p x ; y q with the kernel of its n -th power, we have for n ě and x, y P R that | η p n q H p x ; y q| ď } η H,x } ¨ } η H,y } ¨ } η H } n ´ ď Cℓ α } η H } n ´ | ϕ p x q| ¨ | ϕ p y q| (2.18) and, for all N sufficiently large, that | η H p x ; y q| ď CN | ϕ p x q|| ϕ p y q| ď CN. (2.19)
Proof.
By Theorem A.1, } ϕ } ď C ă 8 so that together with Eq. (2.15), we find that } η H,x } “ ż dy |p G ˚ q χ H qp x ´ y q| ¨ | ϕ p x q| ¨ | ϕ p y q| ď } ϕ } ¨ | ϕ p x q| ¨ } G ˚ q χ H } ď C | ϕ p x q| ż | p |ě ℓ ´ α dp | p G p p q| ď C | ϕ p x q| ż | p |ě ℓ ´ α dp | p | “ Cℓ α | ϕ p x q| . This concludes the first two bounds in (2.17), once we integrate the right hand side ofthe last estimate. To bound the gradient, we proceed similarly and find for i “ , } ∇ i η H } ď ż dxdy | ∇ p G ˚ q χ H qp x ´ y q| ¨ | ϕ p x q| ¨ | ϕ p y q| ` ż dxdy |p G ˚ q χ H qp x ´ y q| ¨ | ∇ ϕ p x q| ¨ | ϕ p y q| ď C }p G ˚ q χ H q} H ď C } G } H ď C ż dx N p| N x | ` q “ CN.
Notice that we used the pointwise estimate (2.12) in the second to last step. The esti-mates in (2.18) follow directly from (2.17) and Cauchy-Schwarz. Finally, (2.19) followsfrom } ϕ } ď C, } G } ď N as well as } G ˚ } χ H c } ď } G } } } χ H c } ď Cℓ ´ α { ď CN for N large enough. Here, q χ H c denotes the inverse Fourier transform of the characteristicfunction on P cH .With the kernel η H , we consider the quadratic expression B “ ż dxdy η H p x ; y q “ b ˚ x b ˚ y ´ b x b y ‰ (2.20)and the unitary operator e B : F ď N Ñ F ď N (since η H is real, the operator B is antisym-metric). Because of its similarity with a Bogoliubov transformation (which would have9 x , a y , instead of the modified fields b x , b y in (2.20)), we call e B a generalized Bogoliubovtransformation. Notice that we do not project η H into the orthogonal complement ofthe condensate wave function ϕ . As a consequence, B and e B do not map F ď N K ϕ intoitself. This is not a problem for us, because we perform our analysis on the larger space F ď N and only at the end (in Section 3) we switch back to the right space. An importantproperties of the unitary operator e B is that it preserves the number of particles, up tocorrections of order one. The following lemma was proved in [5, Lemma 3.1]; it is basedon the observation that } η H } ď Cℓ α { ď C . Lemma 2.3.
Let B be the antisymmetric operator defined in (2.20). For every n P Z there exists a constant C ą such that e ´ B p N ` q n e B ď C p N ` q n as an operator inequality on F ď N . Other important properties of the generalized Bogoliubov transformation e B will bediscussed at the beginning of Section 4 (in particular, we will show there that, on stateswith few excitations, e B acts like a standard Bogoliubov transformation, up to smallerrors).With r L N from (2.7), we can now define the quadratically renormalized excitationHamiltonian G N “ e ´ B r L N e B . (2.21)The next proposition summarizes important properties of G N . Before stating it, let usintroduce the notation K “ ż dx a ˚ x p´ ∆ x q a x , V N “ ż dxdy N V p N p x ´ y qq a ˚ x a ˚ y a y a x , V ext “ ż dx V ext p x q a ˚ x a x , H N “ K ` V ext ` V N Proposition 2.4.
Assume (1.8) and let η H be defined as in (2.16) , with an α ą . Let G eff N “ N E GP p ϕ q ´ ε GP N ` π a } ϕ } N { N ` H N ` p V p q ż dx | ϕ p x q| b ˚ x b x ` πa ż dxdy q χ H c p x ´ y q ϕ p x q ϕ p y qp b ˚ x b ˚ y ` h.c. q` ? N ż dxdy N V p N p x ´ y qq ϕ p x qp b ˚ x a ˚ y a x ` h.c. q (2.22) where ε GP has been introduced in (1.6) and χ H c denotes the characteristic function ofthe set t p P R : | p | ď ℓ ´ α u . Then we have G N “ G eff N ` E G N , where we can bound ˘ E G N ď Cℓ p α ´ q{ p N ` K ` V N q ` Cℓ ´ α { N ´ p N ` q ` Cℓ ´ α for a constant C , independent of N P N and ℓ P p
0; 1 q . emark: In (2.20), we could have projected the kernel η H orthogonally to ϕ . In thisway, we could have defined G N as an operator on F ď N K ϕ (conjugating L N , rather than r L N , with e B ). Also with this definition of G N , we could have proven bounds similar tothose in Prop. 2.4, at the expenses of a longer proof (this procedure would also give anupper bound for the ground state energy, as in the remark after Theorem 1.1).While the vacuum expectation of G N gives now, to leading order, the correct groundstate energy of the Hamiltonian (1.1), the estimates in Prop. 2.4 are still not enough toshow a bound for N in low-energy states. The main issue is the cubic term in (2.22).With Cauchy-Schwarz we can bound it with the positive quartic term V N (containedin H N ), but this produces a negative quadratic contribution that kills coercivity. Toovercome this problem, we have to renormalize the excitation Hamiltonian G N again(actually, only its main part G eff N ), but this time through the exponential of an expressioncubic in creation and annihilation operators.If χ H denotes the characteristic function of the set t p P R : | p | ě ℓ ´ α u and G isdefined as in (2.14), we define the kernel ν H p x ; y q “ p G ˚ q χ H qp x ´ y q ϕ p y q . (2.23)For our analysis below, it will also be useful to introduce r k p x ; y q “ ´ N w ℓ p N p x ´ y qq ϕ p y q so that in particular ν H p x ; y q “ r k p x ; y q ´ p G ˚ q χ H c qp x ´ y q ϕ p y q . The next lemma sum-marizes important properties of ν H and its relation to r k . Lemma 2.5.
Assume (1.8) and let ℓ P p
0; 1 q . Then ν H satisfies } ν H } ď Cℓ α { , } ν H,x } ď Cℓ α { , } ν H,y } ď Cℓ α { | ϕ p y q| (2.24) for all x, y P R and for all N sufficiently large. Moreover, denoting by p ν H p p ; q q “ p G p p q χ H p p q p ϕ p p ` q q the Fourier transform of ν H as a function in L p R ˆ R q , we havefor all p, q P R that } p ν H,p } ď C | p | χ H p p q , } p ν H,q } ď Cℓ α { . (2.25) Proof.
Using (2.15) we find that } ν H } ď } G ˚ q χ H } } ϕ } ď C } p Gχ H } ď Cℓ α { as well as } ν H,x } ď } ϕ } } G ˚ q χ H } ď Cℓ α { , } ν H,y } “ | ϕ p y q|} G ˚ q χ H } ď Cℓ α { | ϕ p y q| . Finally, (2.25) follows from p ν H p p ; q q “ p G p p q χ H p p q p ϕ p p ` q q and the estimate (2.15).11 part from χ H , we will also need a second cutoff, localising on small momenta (weare constructing the cubic operator (2.28), involving three particles; one particle shouldhave small momentum, the other two large momenta, similarly to [4, Eq. (5.1)] in thetranslation invariant case). Here, it is convenient to use the Gaussian function g L p p q “ e ´p ℓ β p q (2.26)for an exponent 0 ă β ă α ( g L localises on momenta | p | À ℓ ´ β ! ℓ ´ α ). Notice that theinverse Fourier transform of g L is given byˇ g L p x q “ p? πℓ ´ β q e ´p πℓ ´ β x q . In particular, it satisfies } ˇ g L } “ , } ˇ g L } “ Cℓ ´ β . (2.27)With ν H defined as in (2.23) and g L from (2.26), we introduce the operator A “ ? N ż dxdydz ν H p x ; y q ˇ g L p x ´ z q ` b ˚ x a ˚ y a z ´ h.c. ˘ . (2.28)Since A is anti-symmetric, e A : F ď N Ñ F ď N is a unitary map. An important observa-tion is that conjugation with e A only increases the number of particles by a constant oforder one, independent of N (this result is similar to Lemma 2.3, for the action of thegeneralized Bogoliubov transform e B ). Lemma 2.6.
Assume (1.8) , let ℓ P p
0; 1 q , t P r´
1; 1 s , α ą and let k P Z . Then, thereexists a constant C “ C p k q ą such that in the sense of forms on on F ď N , we have e ´ tA p N ` q k e tA ď C p N ` q k . (2.29) Proof.
The proof is based on a Gronwall argument. Given ξ P F ď N , we define f ξ by f ξ p s q “ x ξ, e ´ sA p N ` q k e sA ξ y . Taking its derivative yields B s f ξ p s q “ x ξ, e ´ sA rp N ` q k , A s e sA ξ y and it is straight-forward to verify that rp N ` q k , A s “ ? N ż dxdydz ν H p x ; y q ˇ g L p x ´ z q b ˚ x a ˚ y a z ` p N ` q k ´ p N ` q k ˘ ` h.c.By the mean value theorem, there exists a function Θ : N Ñ p
0; 1 q such that p N ` q k ´ p N ` q k “ k p N ` Θ p N q ` q k ´ . |B s f ξ p s q|ď k ? N ż dxdydz | ν H p x ; y q| ˇ g L p x ´ z q} a x a y p N ` q k ´ e sA ξ }} a z p N ` q k ´ e sA ξ }ď C ? N ˆ ż dxdz } ν H,x } | ˇ g L p x ´ z q|} a z p N ` q k ´ e sA ξ } ˙ { ˆ ˆ ż dxdydz | ˇ g L p x ´ z q|} a x a y p N ` q k ´ e sA ξ } ˙ { ď Cℓ α { ? N }p N ` q k e sA ξ }}p N ` q k ` e sA ξ } ď C }p N ` q k e sA ξ } “ Cf ξ p s q . Since C is indepedent of ξ P F ď N , the claim follows from Gronwall’s inequality.Now, recalling the definition of G eff N in (2.22), let us define the cubically renormalizedexcitation Hamiltonian J N through J N “ e ´ A G eff N e A . (2.30) Proposition 2.7.
Assume (1.8) , let ℓ P p
0; 1 q and fix β ą α ą β { as well as α ą .Moreover, assume that N P N is sufficiently large and that ℓ P p
0; 1 q is sufficientlysmall (but fixed, independently of N ). Then, there exist κ ą and a constant C ą ,independent of N and ℓ , such that J N ě N E GP p ϕ q ` d Γ p´ ∆ x ` V ext p x q ` π a | ϕ p x q| ´ ε GP q´ Cℓ κ N ´ Cℓ ´ α p N ` q { N ´ Cℓ ´ β . (2.31) It is enough to prove that there exist constants c, C ą
0, independent of N P N , suchthat for all sufficiently large N , we have in the sense of forms in F ď N K ϕ that L N ě N E GP p ϕ q ` c N ´ C. (3.1)To prove (3.1), we first localize the operator r L N defined in (2.7), based on an argumentfrom [11] (see, in particular, [11, Prop. 6.1]). To this end, let δ P p
0; 1 q (it will bedetermined below) and let 0 ď f, g ď f ` g “ f p x q “ | x | ď { f p x q “ | x | ě
1. With the notation f δN “ f p N { δN q , g δN “ g p N { δN q , we observe that r L N “ f δN r L N f δN ` g δN r L N g δN ` ` r f δN , r f δN , r L N ss ` r g δN , r g δN , r L N ss ˘ , F ď N , we have L N “ Γ p q q f δN r L N f δN Γ p q q ` Γ p q q g δN r L N g δN Γ p q q`
12 Γ p q q ` r f δN , r f δN , r L N ss ` r g δN , r g δN , r L N ss ˘ Γ p q q . (3.2)Recall here that q “ ´ | ϕ yx ϕ | and that Γ p q q is the orthogonal projection from F ď N onto F ď N K . The second line in (3.2) contains error terms and they can be controlled asfollows. Observing that r f δN , r f δN , r L N ss“ ? N b ˚ `` N V p N ¨q ˚ | ϕ | ´ π a | ϕ | ˘ ϕ ˘ ` f δN p N ` q ´ f δN p N q ˘ ´ b ˚ `` N V p N ¨q ˚ | ϕ | ˘ ϕ ˘ N ` ? N ` f δN p N ` q ´ f δN p N q ˘ ` ż dxdy N V p N p x ´ y qq ϕ p y q ϕ p x q b ˚ x b ˚ y ` f δN p N ` q ´ f δN p N q ˘ ` ż dxdy N { V p N p x ´ y qq ϕ p y q b ˚ x a ˚ y a x ` f δN p N ` q ´ f δN p N q ˘ ` h.c. , and similarly for r g δN , r g δN , r L N ss , a straight-forward application of Cauchy-Schwarz, } ϕ } ď C, } N V p N. q ˚ | ϕ | } ď C and the mean value theorem implies that r f δN , r f δN , r L N ss ` r g δN , r g δN , r L N ss ě ´ C ` } f } ` } g } ˘ p V N ` N q δ N . Hence, we find the lower bound L N ě Γ p q q f δN r L N f δN Γ p q q ` Γ p q q g δN r L N g δN Γ p q q ´ C Γ p q q V N δ N Γ p q q ´ C. (3.3)In the next step, we control the first contribution on the r.h.s. in (3.3) throughProp. 2.4 and Prop. 2.7. We define G N as in (2.21), J N as in (2.30) and we choosethe parameters α, β ą β ą α ą β { α ą
5. Moreover, we assume inthe following that N is sufficiently large and that ℓ P p
0; 1 q is sufficiently small. Theseassumptions ensure that the conditions in Prop. 2.4 and Prop. 2.7 are satisfied. Then,by Prop. 2.4 and Lemma 2.3, we find that f δN r L N f δN “ f δN e B G eff N e ´ B f δN ` f δN e B E G N e ´ B f δN ě N E GP p ϕ q f δN ` f δN e B ` G eff N ´ N E GP p ϕ q ˘ e ´ B f δN ´ Cℓ κ f δN e B ` N ` K ` V N ˘ e ´ B f δN ´ Cℓ ´ κ N ´ p N ` q f δN ´ Cℓ ´ κ ě N E GP p ϕ q f δN ` f δN e B ` G eff N ´ N E GP p ϕ q ˘ e ´ B f δN ´ Cℓ κ f δN e B ` N ` K ` V N ˘ e ´ B f δN ´ Cℓ ´ κ δ N f δN ´ Cℓ ´ κ for suitable κ , κ ą
0. Before we can continue and apply Prop. 2.7, we need to boundthe error proportional to p N ` K ` V N q in terms of G eff N , as defined in (2.22). To this14nd we observe that, since χ H c p p q ě ż dxdy q χ H c p x ´ y q ϕ p x q ϕ p y qp b ˚ x ` b x qp b ˚ y ` b y q ě , Denoting by A the operator with kernel A p x ; y q “ ˇ χ H c p x ´ y q ϕ p x q ϕ p y q , and observingthat, with (A.2), } A } ď sup x ş | A p x ; y q| dy ď C , uniformly in ℓ P p
0; 1 q , we conclude that ż dxdy q χ H c p x ´ y q ϕ p x q ϕ p y q ` b ˚ x b ˚ y ` h.c. ˘ ě ´ d Γ p A q ´ q χ H c p q ě ´ C N ´ Cℓ ´ α With this bound (and with Cauchy-Schwarz), (2.22) then easily implies that G eff N ´ N E GP p ϕ q ě H N ´ C N ´ Cℓ ´ α . Hence, we obtain that f δN r L N f δN ě N E GP p ϕ q f δN ` p ´ Cℓ κ q f δN e B ` G eff N ´ N E GP p ϕ q ˘ e ´ B f δN ´ C ` ℓ κ ` ℓ ´ κ δ ˘ N f δN ´ Cℓ ´ κ . (3.4)Now, we apply Prop. 2.7, Lemma 2.3 and Lemma 2.6 which yields similarly as above f δN e B ` G eff N ´ N E GP p ϕ q ˘ e ´ B f δN “ f δN e B e A ` J N ´ N E GP p ϕ q ˘ e ´ A e ´ B f δN ě f δN e B e A d Γ p´ ∆ x ` V ext p x q ` π a | ϕ p x q| ´ ε GP q e ´ A e ´ B f δN ´ C ` ℓ κ ` ℓ ´ κ δ ˘ N f δN ´ Cℓ ´ κ . From (1.6) and under the assumptions (1.8), the operator h GP “ ´ ∆ ` V ext p x q ` π a | ϕ p x q| ´ ε GP has the simple eigenvalue 0 at the bottom of its spectrum (witheigenvector ϕ ) and a positive gap λ ą f δN e B ` G eff N ´ N E GP p ϕ q ˘ e ´ B f δN ě λ f δN e B e A p N ´ a ˚ p ϕ q a p ϕ qq e ´ A e ´ B f δN ´ C ` ℓ κ ` ℓ ´ κ δ ˘ N f δN ´ Cℓ ´ κ (3.5)With the commutators ” a ˚ p ϕ q a p ϕ q , A ı “ ? N ż dxdydz ν H p x ; y q ˇ g L p x ´ z q ϕ p x q b ˚ p ϕ q a ˚ y a z ` ? N ż dxdydz ν H p x ; y q ˇ g L p x ´ z q ϕ p y q b ˚ x a ˚ p ϕ q a z ´ ? N ż dxdydz ν H p x ; y q ˇ g L p x ´ z q ϕ p z q b ˚ x a ˚ y a p ϕ q ` h.c. ” a ˚ p ϕ q a p ϕ q , B ı “ ż dxdy η H p x ; y q ϕ p y q b ˚ x b ˚ p ϕ q ` h.c.15nd with Lemma 2.5 and Lemma 2.6, we find ´ e B e A a ˚ p ϕ q a p ϕ q e ´ A e ´ B ě ´ a ˚ p ϕ q a p ϕ q ´ Cℓ α { p N ` q . Thus, inserting in (3.5) and then in (3.4), and using again Lemmas 2.3 and 2.6, we arriveat f δN r L N f δN ě N E GP p ϕ q f δN ` r c p ´ Cℓ κ q N f δN ´ λ p ´ Cℓ κ q a ˚ p ϕ q a p ϕ q f δN ´ C ` ℓ κ ` ℓ ´ κ δ ˘ N f δN ´ Cℓ ´ κ , where the constants r c, C ą ℓ P p
0; 1 q , δ P p
0; 1 q and N P N .We now set δ “ ℓ κ and choose ℓ sufficiently small, so that the last bound implies f δN r L N f δN ě N E GP p ϕ q f δN ` c N f δN ´ λ a ˚ p ϕ q a p ϕ q f δN ´ C. (3.6)Here, the positive constant c ą ℓ P p
0; 1 q , δ P p
0; 1 q and N P N . Forthe rest of the proof, we fix this choice of δ “ ℓ κ and this (sufficiently small) value of ℓ P p
0; 1 q so that (3.6) holds true. Then, substituting (3.6) into (3.3), we get L N ě N E GP p ϕ q f δN Γ p q q ` c N f δN Γ p q q ` g δN Γ p q q r L N Γ p q q g δN ´ C Γ p q q V N δ N Γ p q q ´ C in the sense of forms in F ď N K ϕ . Notice that we used that Γ p q q a ˚ p ϕ q a p ϕ q Γ p q q “ r Γ p q q , N s “
0. Since Γ p q q r L N Γ p q q “ L N in the sense of forms in F ď N K ϕ and sinceΓ p q q | F ď N K ϕ “ id | F ď N K ϕ , the previous bound translates to L N ě N E GP p ϕ q f δN ` c N f δN ` g δN L N g δN ´ C V N δ N ´ C (3.7)in the sense of forms in F ď N K ϕ Next, we notice that the contribution proportional to g δN L N g δN in (3.7) can becontrolled as explained in [4], using the results from [13, 16], i.e. (1.7). This argumentshows that there exists some constant c ą N P N g δN L N g δN ě N E GP p ϕ q g δN ` « inf ξ P F ď N K ϕ X F ě δN { K ϕ : } ξ }“ ´ N x ξ, L N ξ y ´ E GP p ϕ q ¯ff N g δN ě N E GP p ϕ q g δN ` c N g δN (3.8)where F ě δN { K ϕ “ t ξ P F K ϕ : χ p N ě δN { q ξ “ ξ u . Hence, plugging (3.8) into (3.7), weconclude that L N ě N E GP p ϕ q ` c N f δN ` c N g δN ´ C V N δ N ´ C ě N E GP p ϕ q ` r c N ´ C V N δ N ´ C
16n the sense of forms in F ď N K ϕ , where r c “ min p c , c q ą V N , we recall thedefinitions (2.6) and a simple computation involving Cauchy-Schwarz shows that L N ě H N ´ CN ě V N ´ CN for some constant C ą
0, independent of N P N . Therefore, we find L N ě N E GP p ϕ q ` r c N ´ C L N δ N ´ C, so that, by choosing N P N sufficiently large, we have proved that L N ě N E GP p ϕ q ` c N ´ C for some constant c ą N . G N In this section we analyse the operator G N “ e ´ B r L N e B , as defined in (2.21). To computethe action of the generalized Bogoliubov transform e B on r L N , we are going to compareit with the action of a standard Bogoliubov transformation. Interpreting (2.16) as theintegral kernel of a Hilbert-Schmidt operator on L p R q , we definesinh η H “ ÿ j “ η p k ` q H p k ` q ! , cosh η H “ ÿ j “ η p k q H p k q ! . In addition, we define the Hilbert-Schmidt operators p η H and r η H byp η H “ sinh η H ´ η H “ ÿ j “ η p k ` q H p k ` q ! , r η H “ cosh η H ´ id “ ÿ j “ η p k q H p k q ! . (4.1)Using (2.17) one obtains for ℓ P p
0; 1 q small enough } p η H } , } r η H } ď Cℓ α , | p η H p x, y q| , | r η H p x, y q| ď Cℓ α ϕ p x q ϕ p y q . (4.2)The following lemma, whose proof is an adaptation of the translation-invariant case[4, Lemma 3.4], shows that, on states with few excitations, e B acts approximately like astandard Bogoliubov transformation. Lemma 4.1.
Let n P Z , and let f P L p R q . Let d η H p f q as well as d η H ,x be defined as e ´ B b p f q e B “ b p cosh η H p f qq ` b ˚ p sinh η H p f qq ` d η H p f q , (4.3) respectively e ´ B b x e B “ b p cosh η H ,x q ` b ˚ p sinh η H ,x q ` d η H ,x . (4.4)17 hen, there exists a constant C ą such that }p N ` q n { d η H p f q ξ } ď Cℓ α { N } f }}p N ` q p n ` q{ ξ } , }p N ` q n { d η H p f q ˚ ξ } ď Cℓ α { N } f }}p N ` q p n ` q{ ξ } (4.5) and such that, for all x P R , we have that }p N ` q n { d η H ,x ξ } ď CN ” ℓ α { } a x p N ` q p n ` q{ ξ } ` } η H,x }}p N ` q p n ` q{ ξ } ı . (4.6) Furthermore, if we set d η H ,x “ d η H,x ` p N { N q b ˚ p η H,x q , it holds true that }p N ` q n { a y d η H ,x ξ }ď CN ” } η H,x }} η H,y }}p N ` q p n ` q{ ξ } ` ℓ α { | η H p x ; y q|}p N ` q p n ` q{ ξ }` } η H,y }} a x p N ` q p n ` q{ ξ } ` ℓ α { } η H,x }} a y p N ` q p n ` q{ ξ }` ℓ α { } a x a y p N ` q p n ` q{ ξ } ı (4.7) and, finally, we have that }p N ` q n { d η H ,x d η H ,y ξ }ď CN ” } η H,x }} η H,y }}p N ` q p n ` q{ ξ } ` ℓ α { | η H p x ; y q|}p N ` q p n ` q{ ξ }` ℓ α { } η H,y }|} a x p N ` q p n ` q{ q ξ } ` ℓ α { } η H,x }} a y p N ` q p n ` q{ ξ }` ℓ α } a x a y p N ` q p n ` q{ ξ } ı . (4.8)From the decomposition (2.7), we can write G N “ G p q N ` G p q N ` G p q N ` G p q N ` G p q N , where, for j P t , , , , u , we set G p j q N “ e ´ B r L p j q N e B . In the following subsections, we will analyze the main contributions G p j q N separately and,in Section 4.6, we combine these results to conclude Proposition 2.4. G p q N From (2.6), we recall that r L p q N “ @ ϕ, “ ´ ∆ ` V ext ` ` N V p N ¨q ˚ | ϕ | ˘‰ ϕ D p N ´ N q´ @ ϕ, ` N V p N ¨q ˚ | ϕ | ˘ ϕ D p N ` qp ´ N { N q . (4.9)18 roposition 4.2. There exists a constant C ą such that G p q N “ @ ϕ, “ ´ ∆ ` V ext ` ` N V p N ¨q ˚ | ϕ | ˘‰ ϕ D p N ´ N q´ @ ϕ, ` N V p N ¨q ˚ | ϕ | ˘ ϕ D p N ` qp ´ N { N q ` E p q N,ℓ , where the self-adjoint operator E p q N,ℓ satisfies ˘ E p q N,ℓ ď Cℓ α { p N ` q for all α ą and ℓ P p
0; 1 q .Proof. We start with the observation that e ´ B N e B ´ N “ ż ds ˆ ż dxdy η H p x ; y q e ´ sB b ˚ x b ˚ y e sB ˙ ` h.c.This implies together with Lemma 2.3 and Cauchy-Schwarz that ˘ ` e ´ B N e B ´ N ˘ ď Cℓ α { p N ` q . Similarly, it is straight-forward to prove that ˘ ` e ´ B N e B ´ N ˘ ď Cℓ α { p N ` q . If we use these two observations together with the bounds |x ϕ, p´ ∆ ` V ext q ϕ y| ď E GP p ϕ q ď C, x ϕ, p N V p N ¨q ˚ | ϕ | q ϕ D | ď } V } } ϕ } ď C, the proposition follows directly from the definition of r L p q N in Eq. (4.9). G p q N From (2.6), we recall that r L p q N “ ? N b `` N V p N. q ˚ | ϕ | ´ π a | ϕ | ˘ ϕ ˘ ´ N ` ? N b `` N V p N ¨q ˚ | ϕ | ˘ ϕ ˘ ` h.c. (4.10)For the statement of the next proposition, let us define h N P L p R q X L p R q by h N “ ` N p V w ℓ qp N. q ˚ | ϕ | ˘ ϕ. (4.11) Proposition 4.3.
There exists a constant C ą such that G p q N “ “ ? N b p cosh η H p h N qq ` ? N b ˚ p sinh η H p h N qq ` h.c. ‰ ` E p q N,ℓ , where the self-adjoint operator E p q N,ℓ satisfies ˘ E p q N,ℓ ď Cℓ α { p N ` q ` Cℓ ´ α { N ´ p N ` q ` Cℓ ´ . for all α ą and ℓ P p
0; 1 q . roof. First of all, we notice that a simple application of Cauchy-Schwarz and the factthat }p N V p N ¨q ˚ | ϕ | q ϕ } ď C imply that ˘ ˆ N ` ? N b `` N V p N. q ˚ | ϕ | ˘ ϕ ˘ ` h.c. ˙ ď Cℓ α { p N ` q ` Cℓ ´ α { N ´ p N ` q . By Lemma 2.3, we therefore obtain that ˘ e ´ B ˆ N ` ? N b `` N V p N. q ˚ | ϕ | ˘ ϕ ˘ ` h.c. ˙ e B ď Cℓ α { p N ` q ` Cℓ ´ α { N ´ p N ` q . This controls the conjugation of the second contribution to r L p q N in (4.10). To deal withthe first term on the right hand side of (4.10), we first observe that ` N V p N. q ˚ | ϕ | ´ π a | ϕ | ˘ ϕ “ h N ` ` N p V f ℓ qp N. q ˚ | ϕ | ´ π a | ϕ | ˘ ϕ and we find with Lemma 2.1 ii q that } N p V f ℓ qp N. q ˚ | ϕ | ´ π a | ϕ | } ď sup x P R ż dy p V f ℓ qp y q ˇˇ ϕ p x ´ y { N q ´ ϕ p x q ˇˇ ` CℓN ď C } ∇ ϕ } } ϕ }} V } N ´ ` Cℓ ´ N ´ ď Cℓ ´ N ´ . Note that } ∇ ϕ } ď C by Appendix A. By Cauchy-Schwarz, this implies that ˘ e ´ B ` ? N b `` N p V f ℓ qp N. q ˚ | ϕ | ´ π a | ϕ | ˘ ϕ ˘ ` h.c. ˘ e B ď Cℓ ´ and it only remains to control e ´ B ? N b p h N q e B ` h.c. “ ? N b p cosh η H p h N qq ` ? N b ˚ p sinh η H p h N qq ` ? N d η H p h N q ` h.c.However, by Eq. (4.5) from Lemma 4.1, we know that for all ξ P F ď N we have that |x ξ, ? N d η H p h N q ξ y| ď Cℓ α { N ´ { } h N }}p N ` q { ξ }}p N ` q ξ } ď Cℓ α { x ξ, p N ` q ξ y , and hence, collecting the previous estimates, we conclude the proposition. G p q N From (2.6), we recall that r L p q N “ K ` V ext ` r L p ,V q N , where K “ ż dx ∇ x a ˚ x ∇ x a x , V ext “ ż dx V ext p x q a ˚ x a x r L p ,V q N “ ż dx ` N V p N. q ˚ | ϕ | ˘ p x q ´ b ˚ x b x ´ N a ˚ x a x ¯ ` ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q ´ b ˚ x b y ´ N a ˚ x a y ¯ ` ż dxdy N V p N p x ´ y qq ϕ p y q ϕ p x q ´ b ˚ x b ˚ y ` h.c. ¯ .
20n the following, we will analyze the contributions e ´ B K e B , e ´ B V ext e B and e ´ B r L p ,V q N e B separately. The analysis of the kinetic energy is the most involved so let us first treatthe contributions e ´ B V ext e B and e ´ B r L p ,V q N e B . Proposition 4.4.
There exists a constant C ą such that e ´ B V ext e B “ V ext ` E p ext q N,ℓ , where the self-adjoint operator E p ext q N,ℓ satisfies ˘ E p ext q N,ℓ ď Cℓ α { p N ` q . for all α ą and ℓ P p
0; 1 q .Proof. We start with the observation that e ´ B V ext e B ´ V ext “ ż ds ˆ ż dx V ext p x q e ´ sB b ˚ x b ˚ p η H,x q e sB ˙ ` h.c.In the appendix we show that the assumptions (2) in (1.8) imply that the external poten-tial V ext p x q has at most exponential growth as | x | Ñ 8 while, by (A.2), the minimizer ϕ p x q has exponential decay as | x | Ñ 8 with arbitrary rate. This implies in particu-lar that ş dx V ext p x q| ϕ p x q| ď C. As a consequence, Cauchy-Schwarz, Lemma 2.3 andLemma 2.2 imply that ˇˇˇˇ ż dx V ext p x qx ξ, e ´ sB b ˚ x b ˚ p η H,x q e sB ξ y ˇˇˇˇ ď ˆ ż dx V ext p x q} η H,x } }p N ` q { e sB ξ } ˙ { ˆ ż dx } a x e sB ξ } ˙ { ď Cℓ α { ˆ ż dx V ext p x q| ϕ p x q| ˙ { x ξ, p N ` q ξ y ď Cℓ α { x ξ, p N ` q ξ y uniformly in s P r
0; 1 s , which proves the claim. Proposition 4.5.
There exists a constant C ą such that e ´ B r L p ,V q N e B “ ż dx ` N V p N. q ˚ | ϕ | ˘ p x q b ˚ x b x ` ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q b ˚ x b y ` ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q ` b ˚ x b ˚ y ` b x b y ˘ ` ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q η H p x ; y q ˆ N ´ N N ˙ ˆ N ´ N ´ N ˙ ` E p ,V q N,ℓ , where the self-adjoint operator E p ,V q N,ℓ satisfies ˘ E p ,V q N,ℓ ď Cℓ α { p N ` V N ` q . for all α ą , ℓ P p
0; 1 q and N P N sufficiently large. roof. We split e ´ B L p ,V q N e B “ ř j “ F j , setting F : “ ż dx ` N V p N. q ˚ | ϕ | ˘ p x q e ´ B b ˚ x b x e B ,F : “ ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q e ´ B b ˚ x b y e B ,F : “ ´ N ż dx ` N V p N. q ˚ | ϕ | ˘ p x q e ´ B a ˚ x a x e B ,F : “ ´ N ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q e ´ B a ˚ y a x e B ,F : “ ż dxdy N V p N p x ´ y qq ϕ p y q ϕ p x q e ´ B ` b ˚ x b ˚ y ` b x b y ˘ e B . (4.12)To estimate these terms, we use the decomposition (4.4) and the bounds in Lemma 4.1.With Lemma 2.3 and Cauchy-Schwarz, we obtain ˘ ˆ F ´ ż dx ` N V p N. q ˚ | ϕ | ˘ p x q b ˚ x b x ˙ ď Cℓ α { p N ` q , ˘ ˆ F ´ ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q b ˚ x b y ˙ ď Cℓ α { p N ` q (4.13)and, similarly, that ˘ F , ˘ F ď N ´ } V } } ϕ } p N ` q ď Cℓ α { p N ` q (4.14)for all N sufficiently large. Hence, let us focus on the analysis of F . We split it into F “ F ` F ` F , where F : “ ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y qˆ ` b p cosh η H ,x q ` b ˚ p sinh η H ,x q ˘` b p cosh η H ,y q ` b ˚ p sinh η H ,y q ˘ ` h.c. ,F : “ ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q “ d η H ,x b p cosh η H ,y q ` d η H ,x b ˚ p sinh η H ,y q` b p cosh η H ,x q d η H ,y ` b ˚ p sinh η H ,x q d η H ,y ‰ ` h.c. ,F : “ ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q d η H ,x d η H ,y ` h.c. (4.15)and consider the different contributions separately. We use sinh η H “ η H ` p η H andcosh η H “ ` r η H and the estimates in Lemma 4.1 to rewrite F as F “ ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q ` b x b y ` b ˚ x b ˚ y ˘ ` ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q η H p x ; y q ˆ N ´ N N ˙ ` E V (4.16)22ith ˘ E V ď Cℓ α { p N ` q . (4.17)Let us switch to F , defined in (4.15). We write F “ F ` F ` F ` F with F : “ ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q d η H ,x b p cosh η H ,y q ` h.c. ,F : “ ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q d η H ,x b ˚ p sinh η H ,y q ` h.c. ,F : “ ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q b ˚ p sinh η H ,x q d η H ,y ` h.c. ,F : “ ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q b p cosh η H ,x q d η H ,y ` h.c.Now, by applying (4.6), we find that |x ξ, d η H ,x b p cosh η H ,y q ξ y| ď Cℓ α { }p N ` q { ξ } ´ N ´ { } a x a y ξ } ` | ϕ p x q|} a y ξ } ¯ ` Cℓ α { } r η H ,y }}p N ` q { ξ } ´ } a x ξ } ` | ϕ p x q|}p N ` q { ξ } ¯ and, similarly, that |x ξ, d η H ,x b ˚ p sinh η H ,y q ξ y|ď Cℓ α { }p N ` q ξ } ´ N ´ | sinh η H p x ; y q| ¨ }p N ` q ξ } ¯ ` Cℓ α { } sinh η H ,y }}p N ` q ξ } ´ } a x ξ } ` | ϕ p x q|}p N ` q { ξ } ¯ . Together with the fact thatsup x,y P R N ´ | sinh η H p x ; y q| ď N ´ ` | η H p x ; y q| ` C ˘ ď C by (2.19), this yields together with the estimates in Lemma 4.1 ˘ F ď Cℓ α { p N ` V N ` q , ˘ F ď Cℓ α { p N ` q . (4.18)A similar (but simpler) argument involving (4.6) also shows that ˘ F ď Cℓ α { p N ` q . (4.19)Next, let us switch to F , defined above. Here, we first compute b x N b ˚ p η H,y q “ b ˚ p η H,y q b x p N ` q ` η H p x ; y qp ´ N { N qp N ` q ´ a ˚ p η H,y q a x p N ` q{ N. If we recall the notation d η H ,y “ d η H ,y ` p N { N q b ˚ p η H,y q , we therefore obtain F “ ´ ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q η H p x ; y q ˆ N ´ N N ˙ ˆ N ` N ˙ ` E p V q E p V q “ ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q “ b x d η H ,y ` b p r η H ,x q d η H ,y ´ N ´ b ˚ p η H,y q b x p N ` q ` N ´ a ˚ p η H,y q a x p N ` q ‰ ` h.c.Now, using (4.6) and (4.7) and proceeding as in the previous steps, we find that ˘ E p V q ď Cℓ α { ` N ` V N ` ˘ . (4.20)Collecting the previous bounds, this controls the contribution F , defined in (4.15).Finally, to control the last contribution F , defined in (4.15), we use (4.8) and thefact that N ´ | η H p x ; y q| ď C so that ˘ F ď Cℓ α { ` N ` V N ` ˘ . (4.21)In summary, if we combine (4.12), (4.13), (4.14), (4.16), (4.17), (4.18), (4.19), (4.20) and(4.21), we obtain the claim.Finally, we study the kinetic energy. We start with two auxiliary lemmas. Lemma 4.6.
Assume (1.8) , let ℓ P p
0; 1 q be sufficiently small and let α ą . Then,there exists C ą , independent of N and ℓ , such that for all n ě and j P t , u} ∇ j η p n q H } , } ∆ j η p n q H } ď Cℓ α { } η H } n ´ . (4.22) As a consequence, we have that } ∇ j p η H } , } ∆ j p η H } , } ∇ j r η H } , } ∆ j r η H } ď Cℓ α { . (4.23) Proof.
Notice first of all that (4.23) follows indeed directly from (4.22) and the definition(4.1). To prove (4.22), let us first consider } ∆ η p n q H } . Since η H is symmetric, this takesalso care of the case j “
2. Moreover, note that it suffices to consider the case n “ } ∆ η p q H } ď ˜ż dxdz ˇˇˇˇż dy p ∆ G ˚ q χ H qp x ´ y q ϕ p x q ϕ p y q η H p y ; z q ˇˇˇˇ ¸ { ` ˜ ż dxdz ˇˇˇˇż dy p ∇ G ˚ q χ H qp x ´ y q ¨ ∇ ϕ p x q ϕ p y q η H p y ; z q ˇˇˇˇ ¸ { ` ˜ż dxdz ˇˇˇˇż dy p G ˚ q χ H qp x ´ y q ∆ ϕ p x q ϕ p y q η H p y ; z q ˇˇˇˇ ¸ { “ : T ` T ` T . T . After switching to Fourier space, using the bound (A.5) and using(2.15) that | y ∆ G p p q| “ | p | | p G p p q| ď C , we get T “ ż dr dr ds ds p G p r q χ H p r q p G p r q χ H p r q y | ϕ | p s ` s q y | ϕ | p r ´ s qˆ y | ϕ | p r ´ s q y | ϕ | p r ` r q y ∆ G p s q χ H p s q y ∆ G p s q χ H p s qď C ż | r | , | r | , | s | , | s |ě ℓ ´ α dr dr ds ds | r | | r | p ` | s ` s |q p ` | r ´ s |q ˆ p ` | r ´ s |q p ` | r ` r |q ď C ż | r | , | r |ě ℓ ´ α dr dr | r | | r | p ` | r ` r |q ď C ż | r |ě ℓ ´ α dr | r | “ Cℓ α . For the second term T , we use that ››{ | ∇ ϕ | ›› ď }| ∇ ϕ | } ď C by Lemma A.2 and that | y ∇ G p p q| ď ℓ α C for | p | ě ℓ ´ α , which yields by similar computation T ď Cℓ α . Similarlyone deals with T and the bounds on } ∇ j η p n q H } can be proved analogously. Lemma 4.7.
Assume (1.8) , let ℓ P p
0; 1 q be sufficiently small and let α ą . Then,there exists a constant C ą such that ˘ ż dx a ˚ p ∇ x η H,x q a p ∇ x η H,x q ď Cℓ α { p N ` q . (4.24) Moreover, recalling (4.3) , (4.4) and d η,x “ d η,x ` p N { N q b ˚ p η x q , we have that ˆż dx }? N p N ` q ´ { ∇ x d sη H ,x ξ } ˙ { ď C }p N ` K ` q { ξ } , ˆż dx }? N p N ` q ´ { ∇ x d sη H ,x ξ } ˙ { ď Cℓ α { }p N ` K ` q { ξ } (4.25) and that ˆż dx }? N p N ` q ´ { d sη H p ∇ x η H,x q ξ } ˙ { ď Cℓ α { }p N ` q { ξ } , ˆż dx }? N p N ` q ´ { d ˚ sη H p ∇ x η H,x q ξ } ˙ { ď Cℓ α { }p N ` q { ξ } (4.26) for all ξ P F ď N and all s P r
0; 1 s .Proof. We first note that, using arguments very similar to those in the proof of theprevious Lemma 4.6, it is simple to show that ż dydz ˇˇˇˇż dx ∇ x η H p y ; x q ¨ ∇ x η H p z ; x q ˇˇˇˇ ď Cℓ α . ˘ ˆ ż dx a ˚ p ∇ x η H,x q a p ∇ x η H,x q ˙ “ ˘ ˆ ż dydz „ ż dx ∇ x η H p y ; x q ∇ x η H p z ; x q a ˚ y a z ˙ ď Cℓ α p N ` q . This proves the first bound (4.24). To prove the bounds (4.25) and (4.26), we proceedsimilar as in the proof of Lemma 4.1 (which can be found in [4, Lemma 3.4]) and useLemma 2.2, Lemma 4.6 as well as the bound (4.24); we skip the details.We are now ready to analyze the kinetic energy.
Proposition 4.8.
There exists a constant C ą such that e ´ B K e B “ K ` ż dx “ b p ∇ x η H,x q ∇ x b x ` h.c. ‰ ` } ∇ η H } p ´ N { N qp ´ N { N ´ { N q ` E p K q N,ℓ where the self-adjoint operator E p K q N,ℓ satisfies ˘ E p K q N,ℓ ď Cℓ p α ´ q{ p N ` K ` V N ` q ` Cℓ ´ α { N ´ p N ` q for all α ą , ℓ P p
0; 1 q and all N large enough.Proof. Using a first order Taylor expansion and the identity (4.3), we have that e ´ B K e B ´ K “ ˆ ż ds ż dx “ b p cosh sη H p ∇ x η H,x qq ` b ˚ p sinh sη H p ∇ x η H,x q ‰ ˆ “ ∇ x b p cosh sη H ,x q ` ∇ x b ˚ p sinh sη H ,x q ‰ ` h.c. ˙ ` ˆ ż ds ż dx ”` b p cosh sη H p ∇ x η H,x qq ` b ˚ p sinh sη H p ∇ x η H,x qq ˘ ∇ x d sη H ,x ` d sη H p ∇ x η H,x q ` ∇ x b p cosh sη H ,x q ` ∇ x b ˚ p sinh sη H ,x q ˘ı ` h.c. ˙ ` ˆ ż ds ż dx d sη H p ∇ x η H,x q ∇ x d sη H ,x ` h.c. ˙ “ : G ` G ` G . (4.27)Let us start to analyze G . Integrating by parts and using (4.2), Lemma 4.6 as wellas the bound (4.24) we conclude thatG “ ż dx “ b p ∇ x η H,x q ∇ x b x ` h.c. ‰ ` } ∇ η H } p ´ N { N q ` E , (4.28)26here the error E satisfies ˘ E ď Cℓ α { p N ` q . Next we extract the relevant terms from G , defined in (4.27). We split G intoG “ ż ds ż dx “ b ˚ p sinh sη H p ∇ x η H,x qq ∇ x d sη H ,x ` h.c. ‰ ` ż ds ż dx “ d sη H p ∇ x η H,x q ∇ x b p cosh sη H ,x q ` h.c. ‰ ` ż ds ż dx “ b p cosh sη H p ∇ x η H,x qq ∇ x d sη H ,x ` h.c. ‰ ` ż ds ż dx “ d sη H p ∇ x η H,x q ∇ x b ˚ p sinh sη H ,x q ` h.c. ‰ “ : G ` G ` G ` G . From Lemma 4.6, Lemma 4.7 and } sinh η H p ∇ x η H,x q} ď C } ∇ x η p q H } , we easily find that ˘ G ď Cℓ α { p N ` K ` q , ˘ G ď Cℓ α { p N ` K ` q , so let us continue with the analysis of G . We split it into G “ ż ds ż dx “ b ` r sη H p´ ∆ x η H,x q ˘ d sη H ,x ` h.c. ‰ ` ż ds ż dx “ b p ∇ x η H,x q ∇ x d sη H ,x ` h.c. ‰ ´ ż ds ż dx r b p ∇ x η H,x qp N { N q b ˚ p ∇ x sη H,x q ` h.c. s “ : G ` G ` G and, similarly as above, it is simple to see that ˘ G ď Cℓ α { p N ` q . To control the contribution G , we use that ´ ∆ x η H p y ; x q “ ´ ∆ G p y ´ x q ϕ p x q ϕ p y q ` ∇ G p y ´ x q ∇ ϕ p x q ϕ p y q´ G p y ´ x q ∆ ϕ p x q ϕ p y q ` ∆ x “ p G ˚ q χ H c qp y ´ x q ϕ p x q ϕ p y q ‰ so that integrating by parts implies G “ ż ds ż dxdy p´ ∆ G qp y ´ x q ϕ p x q ϕ p y q b y d sη H ,x ` ż ds ż dxdy G p y ´ x q ∆ ϕ p x q ϕ p y q b y d sη H ,x ` ż ds ż dxdy G p y ´ x q ∇ ϕ p x q ϕ p y q b y ∇ x d sη H ,x ` ż ds ż dxdy ∆ x ` p G ˚ q χ H c qp y ´ x q ϕ p x q ϕ p y q ˘ b y d sη H ,x ` h.c.27sing the scattering equation (2.9), we have that ż ds ż dxdy p´ ∆ G qp y ´ x q ϕ p x q ϕ p y q b y d sη H ,x “ ´ ż ds ż dxdy “ N V p N p y ´ x qq ´ N λ ℓ f ℓ p N p y ´ x qq ‰ χ ℓ p x ´ y q ϕ p x q ϕ p y q b y d sη H ,x With Lemma 2.1 and Lemma 4.1, we conclude, proceeding in the usual way, ˇˇˇˇ ż ds ż dxdy p´ ∆ G qp y ´ x q ϕ p x q ϕ p y qx ξ, b y d sη H ,x ξ y ˇˇˇˇ ď Cℓ α { ż dxdy “ N V p N p x ´ y qq ` Cℓ ´ χ ℓ p x ´ y q ‰ | ϕ p x q|| ϕ p y q|}p N ` q { ξ }ˆ N ´ “ ℓ α { }p N ` q { ξ } ` | η H p x ; y q|}p N ` q { ξ } ` ℓ α { } a x ξ }` ℓ α { } a y p N ` q ξ } ` } a x a y p N ` q { ξ } ı ď Cℓ p α ´ q{ x ξ, p N ` q ξ y ` Cℓ α { x ξ, V N ξ y . Using once more Lemma 4.1 and Lemma 4.7, we also find that ˇˇˇˇ ż ds ż dxdy G p y ´ x q ∆ ϕ p x q ϕ p y qx ξ, b y d sη H ,x ξ y ˇˇˇˇ ` ˇˇˇˇ ż ds ż dxdy G p y ´ x q ∇ ϕ p x q ϕ p y qx ξ, b y ∇ x d sη H ,x ξ y ˇˇˇˇ ď C ż ds }p N ` q { ξ } „ ˆż dx } ∇ x d sη H ,x ξ } ˙ { ` ˆż dx } d sη H ,x ξ } ˙ { ď Cℓ α { x ξ, p N ` K ` q ξ y and, since | ∆ x rp G ˚ q χ H c qp x ´ y q ϕ p x q ϕ p y qs| ď Cℓ ´ α r| ϕ p x q| ` | ∇ x ϕ p x q| ` | ∆ x ϕ p x q|s| ϕ p y q| , we have furthermore by Cauchy-Schwarz and (4.7) that ˇˇˇˇ ż ds ż dxdy ∆ x ` p G ˚ q χ H c qp y ´ x q ϕ p x q ϕ p y q ˘ x ξ, b y d sη H ,x ξ y ˇˇˇˇ ď Cℓ ´ α { N x ξ, p N ` q ξ y . In summary, this proves that ˘ G ď Cℓ p α ´ q{ p N ` q ` Cℓ α { p K ` V N q ` Cℓ ´ α { N ´ p N ` q . and since G “ ´} ∇ η H } N ` N N ´ N N ´ ż dx a ˚ p ∇ x η H,x q a p ∇ x η H,x q N ` N N ´ N N ,
28e easily deduce, with Lemma 4.7, that ˘ ˆ G ` } ∇ η H } N ` N N ´ N N ˙ ď Cℓ p α ´ q{ p N ` q ` Cℓ α { p K ` V N q ` Cℓ ´ α { N ´ p N ` q . With very similar arguments, one can show that ˘ G ď Cℓ α { p N ` q so that, in summary, we haveG “ ´} ∇ η H } N ` N N ´ N N ` E (4.29)for an error E that satisfies ˘ E ď Cℓ p α ´ q{ p N ` q ` Cℓ α { p K ` V N q ` Cℓ ´ α { N ´ p N ` q . Going back to (4.27), we finally use once more Lemmas 4.1 and 4.7 to deduce ˘ G ď Cℓ α { p N ` K ` q . (4.30)Hence, collecting (4.28), (4.29) and (4.30) proves the proposition. G p q N From (2.6), we recall that r L p q N “ ż dxdy N { V p N p x ´ y qq ϕ p y q ` b ˚ x a ˚ y a x ` h.c. ˘ Let us also recall the definition of h N “ ` N p V w ℓ qp N. q ˚ | ϕ | ˘ ϕ from (4.11). Proposition 4.9.
There exists a constant C ą such that G p q N “ ż dxdy N { V p N p x ´ y qq ϕ p y q ` b ˚ x a ˚ y a x ` h.c. ˘ ´ ” ? N b ` cosh η H p h N q ˘ ` ? N b ˚ ` sinh η H p h N q ˘ ` h.c. ı ` E p q N,ℓ , where the self-adjoint operator E p q N,ℓ satisfies ˘ E p q N,ℓ ď Cℓ α { p N ` V N ` q ` CN ´ { p N ` q { ` Cℓ ´ α . for all α ą , ℓ P p
0; 1 q and N large enough. roof. We use the identity e ´ B a ˚ y a x e B “ a ˚ y a x ` ż ds e ´ sB “ b p η H,y q b x ` b ˚ y b ˚ p η H,x q ‰ e sB to split G p q N into G p q N “ J ` J ` J ` h.c., whereJ : “ ż dxdy N { V p N p x ´ y qq ϕ p y q e ´ B b ˚ x e B a ˚ y a x , J : “ ż dxdy N { V p N p x ´ y qq ϕ p y q e ´ B b ˚ x e B ż ds e ´ sB b p η H,y q b x e sB , J : “ ż dxdy N { V p N p x ´ y qq ϕ p y q e ´ B b ˚ x e B ż ds e ´ sB b ˚ y b ˚ p η H,x q e sB . (4.31)We start with the analysis of J . Using (4.3), we have thatJ “ ż dxdy N { V p N p x ´ y qq ϕ p y q b ˚ x a ˚ y a x ` ż dxdy N { V p N p x ´ y qq ϕ p y q “ b ˚ p r η H ,x q ` b p p η H ,x q ‰ a ˚ y a x ` ż dxdy N { V p N p x ´ y qq ϕ p y q d ˚ η H ,x a ˚ y a x ` ż dxdy N { V p N p x ´ y qq ϕ p y q b p η H,x q a ˚ y a x “ : ż dxdy N { V p N p x ´ y qq ϕ p y q b ˚ x a ˚ y a x ` J ` J ` J . (4.32)First, it is simple to see that ˘ J ď Cℓ α p N ` q and, by (4.2) and the bound (4.7) from Lemma 4.1, we also find that |x ξ, J ξ y| ď ż dxdy N { V p N p x ´ y qq| ϕ p y q|} a y d η H ,x ξ }} a x ξ }` ż dxdy N { V p N p x ´ y qq| ϕ p y q|} a y p N { { N q a ˚ p η H,x q ξ }} N { a x ξ }ď Cℓ α { }p N ` V N ` q { ξ } ` CN ´ { }p N ` q { ξ } . Note that we used | η H p x ; y q| ď CN for all N large enough, by (2.19).Going back to (4.32) and recalling the definition (4.11), we finally see thatJ “ ż dxdy N { V p N p x ´ y qq ϕ p y q “ η H p y ; x q b x ` a ˚ y a x b p η H,x q ‰ “ ´? N b p h N q ´ ż dxdy N { V p N p x ´ y qqp G ˚ q χ H c qp x ´ y q ϕ p y q ϕ p x q b x ` ż dxdy N { V p N p x ´ y qq ϕ p y q a ˚ y a x b p η H,x q , q χ H c denotes the inverse Fourier transform of the characteristic function of the set t p P R : | p | ď ℓ ´ α u . Using that }p G ˚ q χ H c q} ď Cℓ ´ α , by (2.15), we deduce that ˘ ` J ` ? N b p h N q ˘ ď Cℓ ´ α ` Cℓ α { p N ` q and hence, if we collect the previous estimates, we have proved that ˘ ˆ J ´ ż dxdy N { V p N p x ´ y qq ϕ p y q b ˚ x a ˚ y a x ` ? N b p h N q ˙ ď Cℓ α { p N ` V N ` q ` Cℓ ´ α N ´ { p N ` q { ` Cℓ ´ α . (4.33)Next, we bound J , defined in (4.31). We apply as usual the identity (4.3), Lemma4.1 and Cauchy-Schwarz to estimate |x ξ, J ξ y| ď ż dxdy N { V p N p x ´ y qq| ϕ p y q|} η H,y }ˆ ż ds } b p cosh η H ,x q ξ ` b ˚ p sinh η H ,x q ξ }}p N ` q { b x e sB ξ }` ż dxdy N { V p N p x ´ y qq| ϕ p y q|} η H,y }ˆ ż ds } d η H ,x ξ }}p N ` q { b x e sB ξ }ď Cℓ α { }p N ` q { ξ } ` Cℓ α ż dxdy N { V p N p x ´ y qqˆ ż ds “ } a x ξ } ` | ϕ p x q|}p N ` q { ξ } ‰ }p N ` q { b x e sB ξ }ď Cℓ α { }p N ` q { ξ } . (4.34)Finally, let us analyze the contribution J , defined in (4.31). We split this contribu-tion into J “ J ` J ` J , whereJ “ ż dxdy N { V p N p x ´ y qq ϕ p y q e ´ B b ˚ x e B ˆ ż ds ´ e ´ sB b ˚ y e sB ´ b ˚ y ¯ e ´ sB b ˚ p η H,x q e sB , J “ ż dxdy N { V p N p x ´ y qq ϕ p y q ´ e ´ B b ˚ x e B ´ b p η H,x q ¯ b ˚ y ˆ ż ds e ´ sB b ˚ p η H,x q e sB , J “ ż dxdy N { V p N p x ´ y qq ϕ p y q b p η H,x q b ˚ y ż ds e ´ sB b ˚ p η H,x q e sB .
31o control the error terms J and J , we proceed as before to bound |x ξ, J ξ y| ď ż dxdy N { V p N p x ´ y qq| ϕ p y q|} b x e B ξ }ˆ ż ds ››“ b ˚ p r sη H ,y q ` b p sinh sη H ,y q ‰ e ´ sB b ˚ p η H,x q e sB ξ ›› ` Cℓ α { ż dxdy N { V p N p x ´ y qq| ϕ p y q|| ϕ p x q|}p N ` q { ξ ›› ˆ ż ds ›› d sη H ,y “ b p cosh η H ,x q ` b ˚ p sinh η H ,x q ` d η H ,x ‰ ξ ›› ď Cℓ α { x ξ, p N ` V N ` q ξ y as well as |x ξ, J ξ y| ď ż dxdy N { V p N p x ´ y qq| ϕ p y q|} b x e B ξ }ď Cℓ α { ż dxdy N { V p N p x ´ y qq| ϕ p y q|| ϕ p x q|}p N ` q { ξ ›› ˆ ›› b y “ b p cosh η H ,x q ` b ˚ p p η H ,x q ` d η H ,x ´ N N b ˚ p η H,x q ‰ ξ ›› ď Cℓ α { x ξ, p N ` V N ` q ξ y . Here we used in the in the last step (4.7) to bound the term with d and further used b y b ˚ p η H,x q “ a ˚ p η H,x q a y p ´ N { N q ` η H p x, y q p ´ N { N q and } η H } ď CN . To controlthe last term J , on the other hand, we first rewrite it asJ “ ´? N ż dxdy N p V w ℓ qp N p x ´ y qq| ϕ p y q| ϕ p x qˆ ż ds “ b ˚ p cosh sη H p η H,x qq ` b p sinh sη H p η H,x qq ‰ ´ ż dxdy N { V p N p x ´ y qqp G ˚ q χ H c qp x ´ y q| ϕ p y q| ϕ p x qˆ ż ds “ b ˚ p cosh sη H p η H,x qq ` b p sinh sη H p η H,x qq ‰ ` ż dxdy N { V p N p x ´ y qq ϕ p y q η H p x ; y q ż ds d ˚ sη H p η H,x q` ż dxdy N { V p N p x ´ y qq ϕ p y q a ˚ y a p η H,x qp ´ N { N q ż ds e ´ sB b ˚ p η H,x q e sB , where q χ H c denotes the inverse Fourier transform of the characteristic function of the set32 p P R : | p | ą ℓ ´ α u . With very similar arguments as before, we find that ˘ ˆ J ` ? N ż dxdy N p V w ℓ qp N p x ´ y qq| ϕ p y q| ϕ p x qˆ ż ds “ b ˚ p cosh sη H p η H,x qq ` b p sinh sη H p η H,x qq ‰˙ ď Cℓ α { p N ` q ` Cℓ ´ α . Finally, if we recall the definition (4.11), we observe that ż dxdy N p V w ℓ qp N p x ´ y qq| ϕ p y q| ϕ p x q ż ds “ b ˚ p cosh sη H p η H,x qq ` b p sinh sη H p η H,x qq ‰ “ ż dxdy N p V w ℓ qp N p x ´ y qq| ϕ p y q| ϕ p x q “ b ˚ p sinh η H ,x q ` b p r η H ,x q ‰ “ ” b ` cosh η H p h N q ˘ ` b ˚ ` sinh η H p h N q ˘ı ´ b p h N q . Hence, the previous bounds together with (4.33) and (4.34) prove the proposition. G p q N From (2.6), we recall that r L p q N “ ż dxdy N V p N p x ´ y qq a ˚ x a ˚ y a y a x . For the analysis of G p q N “ e ´ B r L p q N e B , we will use the following Lemma which is astraightforward consequence of Lemmas 2.2, 4.1 and the decomposition (4.3); we omitits proof. Lemma 4.10.
Assume (1.8) and let ℓ P p
0; 1 q be sufficiently small. Then, there existsa constant C ą such that }p N ` q n { e ´ sB b x b y e sB ξ }ď Cℓ α | ϕ p x q|| ϕ p y q|}p N ` q p n ` q{ ξ } ` Cℓ α { | ϕ p y q|} a x p N ` q p n ` q{ ξ }` Cℓ α { | ϕ p x q|} a y p N ` q p n ` q{ ξ } ` | η H p x ; y q|}p N ` q n { ξ } ` } a x a y p N ` q n { ξ } for all ξ P F ď N and all s P r
0; 1 s . Proposition 4.11.
There exists a constant C ą such that G p q N “ V N ´ ż dxdy N p V w ℓ qp N p x ´ y qq ϕ p x q ϕ p y q ` b x b y ` b ˚ x b ˚ y ˘ ` N ż dxdy N V p N p x ´ y qq w ℓ p N p x ´ y qq| ϕ p x q| | ϕ p y q| ˆ p ´ N { N qp ´ N { N ´ { N q ` E p q N,ℓ , here the self-adjoint operator E p q N,ℓ satisfies ˘ E p q N,ℓ ď Cℓ α { p N ` V N ` q ` Cℓ ´ α . for all α ą and ℓ P p
0; 1 q .Proof. Using the identity r a ˚ x a ˚ y a y a x , b u b v s “ ´ ´ δ p x ´ u q a ˚ y a v ` δ p x ´ v q a ˚ y a u ` δ p x ´ u q δ p y ´ v q` δ p x ´ v q δ p u ´ y q ` δ p y ´ u q a ˚ x a v ` δ p y ´ v q a ˚ x a u ¯ b x b y , we have that G p q N,ℓ “ e ´ B p η H q L p q N e B “ V N ` ż ds ż dxdy N V p N p x ´ y qq e ´ sB “ a ˚ x a ˚ y a y a x , B ‰ e sB “ V N ` ż ds ż dxdy N V p N p x ´ y qq η H p x ; y q ` e ´ sB b x b y e sB ` h.c. ˘ ` ż ds ż dxdy N V p N p x ´ y qq ` e ´ sB a ˚ y a p η H,x q b x b y e sB ` h.c. ˘ . For the conjugation of the quartic term, we use furthermore that e ´ sB a ˚ y a u e sB “ a ˚ y a u ` ż s dτ e ´ τB “ a ˚ y a u , B ‰ e τB “ a ˚ y a u ` ż s dτ e ´ τB ´ b ˚ y b ˚ p η H,u q ` b p η H,y q b u ¯ e τB so that G p q N “ V N ` ż ds ż dxdy N V p N p x ´ y qq η H p x ; y q ` e ´ sB b x b y e sB ` h.c. ˘ ` ż ds ż dxdy N V p N p x ´ y qq ` a ˚ y a p η H,x q e ´ sB b x b y e sB ` h.c. ˘ ` ż ds ż s dτ ż dxdy N V p N p x ´ y qqˆ ` e ´ τB b p η H,y q b p η H,x q e τB e ´ sB b x b y e sB ` h.c. ˘ ` ż ds ż s dτ ż dxdy N V p N p x ´ y qqˆ ´ e ´ τB b ˚ y b ˚ p η p q H,x q e τB e ´ sB b x b y e sB ` h.c. ¯ “ : V N ` W ` W ` W ` W . ˘ W ď Cℓ α { p N ` V N ` q , ˘ W ď Cℓ α p N ` V N ` q , ˘ W ď Cℓ α p N ` V N ` q . We omit the details and focus on the only relevant term W which can be written asW “ ż ds ż dxdy N V p N p x ´ y qq η H p x ; y q ` b p cosh sη H ,x q ` b ˚ p sinh sη H ,x q ` d sη H ,x ˘ ˆ ` b p cosh sη H ,y q ` b ˚ p sinh sη H ,y q ` d sη H ,y ˘ ` h.c. “ ż ds ż dxdy N V p N p x ´ y qq η H p x ; y q ` b p cosh sη H ,x q b p cosh sη H ,y q ` h.c. ˘ ` ż ds ż dxdy N V p N p x ´ y qq η H p x ; y q ` b p cosh sη H ,x q b ˚ p sinh sη H ,y q ` h.c. ˘ ` ż ds ż dxdy N V p N p x ´ y qq η H p x ; y q ` b p cosh sη H ,x q d sη H ,y ` h.c. ˘ ` E p q “ : W ` W ` W ` E p q , where E p q “ ż ds ż dxdy N V p N p x ´ y qq η H p x ; y q ` b ˚ p sinh sη H ,x q ` d sη H ,x ˘ ˆ ` b p cosh sη H ,y q ` b ˚ p sinh sη H ,y q ` d sη H ,y ˘ ` h.c.Using (4.6), (4.7) and N ´ | η H p x ; y q| ď C | ϕ p x q|| ϕ p y q| by (2.19), we get ˇˇ x ξ, E p q ξ y ˇˇ ď Cℓ α { ż ds ż dxdy N V p N p x ´ y qq| η H p x ; y q|}p N ` q ξ }ˆ ” | ϕ p x q|} p b p cosh sη H ,y q ` b ˚ p sinh sη H ,y q ` d sη H ,y q ξ }` N ´ }p N ` q { a x p b p cosh sη H ,y q ` b ˚ p sinh sη H ,y q ` d sη H ,y q ξ } ı ď Cℓ α { x ξ, p N ` V N ` q ξ y . Next, let us analyze the contributions W , W and W . We writeW “ ż dxdy N V p N p x ´ y qq η H p x ; y q ` b x b y ` b ˚ x b ˚ y ˘ ` E p q for an error E p q that satisfies |x ξ, E p q ξ y| ď C ż ds ż dxdy N V p N p x ´ y qq| η H p x ; y q|}p N ` q ξ }ˆ }p N ` q ´ { p b x b p p sη H ,y q ` b p p sη H ,x q b y ` b p p sη H ,x q b p p sη H ,y qq ξ }ď Cℓ α { x ξ, p N ` q ξ y . “ ż dxdy N V p N p x ´ y qq η H p x ; y q p ´ N { N q ` E p q , where E p q “ ż ds ż dxdy N V p N p x ´ y qq η H p x ; y qp ´ N { N qˆ r a ˚ p sinh sη H ,y q a p cosh sη H ,x q ` p sη H p x ; y q ` x r sη H ,x , sinh sη H ,y y ` h.c. s and thus ˘ E p q ď Cℓ α { p N ` q .Finally, we have thatW “ ´ ż dxdy N V p N p x ´ y qq η H p x ; y q ˆ ´ N N ˙ N ` N ` E p q (4.35)where, by the bound (4.7), it is simple to see that |x ξ, E p q ξ y| ď C ż ds ż dxdy N V p N p x ´ y qq| η H p x ; y q| ¨ }p N ` q { ξ }ˆ ” }p N ` q ´ { a ˚ p sη H,y q a x ξ } ` ℓ α { | ϕ p x q|} d sη H ,y ξ }` }p N ` q ´ { a x d sη H ,y ξ } ı ď Cℓ α { x ξ, p N ` V N ` q ξ y . In summary, the analysis from above proves that G p q N “ V N ` ż dxdy N V p N p x ´ y qq η H p x ; y q ` b x b y ` b ˚ x b ˚ y ˘ ` ż dxdy N V p N p x ´ y qq η H p x ; y q p ´ N { N qp ´ N { N ´ { N q ` r E p q N,ℓ , for an error r E p q N,ℓ satisfies ˘ r E p q N,ℓ ď Cℓ α { p N ` V N ` q . Replacing finally η H p x ; y q by G p x ´ y q ϕ p x q ϕ p y q “ ´ N w ℓ p N p x ´ y qq ϕ p x q ϕ p y q in the firsttwo contributions on the right hand side of the last equation for G p q N , we conclude theproposition, using that } G ˚ q χ H c } ď Cℓ ´ α and N ´ | η H p x ; y q| ď C , by (2.19). Collecting the results from the previous subsections, we are now ready to prove Propo-sition 2.4. Since the proof is similar to the proof of [5, Theorem 4.4] and [4, Prop. 4.2],we explain the main steps only. 36 roof of Proposition 2.4.
Let us collect the results of Propositions 4.2, 4.3, 4.4, 4.5, 4.8,4.9 and 4.11, noting that there is a cancellation between the linear main contributionsfrom G p q N in Prop. 4.3 with those of G p q N in Prop. 4.9. We find that G N “ @ ϕ, “ ´ ∆ ` V ext ` ` N V p N ¨q ˚ | ϕ | ˘‰ ϕ D p N ´ N q´ @ ϕ, ` N V p N ¨q ˚ | ϕ | ˘ ϕ D p N ` qp ´ N { N q` ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q η H p x ; y qp ´ N { N qp ´ N { N ´ { N q` ż dxdy p´ ∆ x η H p x ; y qqq η H p x ; y qp ´ N { N qp ´ N { N ´ { N q` N ż dxdy N V p N p x ´ y qq w ℓ p N p x ´ y qq| ϕ p x q| | ϕ p y q| ˆ p ´ N { N qp ´ N { N ´ { N q` ż dx ` N V p N. q ˚ | ϕ | ˘ p x q b ˚ x b x ` ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q b ˚ x b y ` ż dxdy ` N p V f ℓ qp N p x ´ y qq ϕ p x q ϕ p y q ´ ∆ y η H p x ; y q ˘` b ˚ x b ˚ y ` b x b y ˘ ` ż dxdy N { V p N p x ´ y qq ϕ p y q ` b ˚ x a ˚ y a x ` h.c. ˘ ` K ` V ext ` V N ` r E p q G N , (4.36)where the error r E p q G N satisfies the estimate ˘ r E p q G N ď Cℓ p α ´ q{ p N ` K ` V N ` q ` Cℓ ´ α { N ´ p N ` q ` Cℓ ´ α . To prove Proposition 2.4, we need to simplify G N ´ r E p q G N further.We start with the terms on the first six lines of (4.36). Recalling that ´ ∆ x η H p x ; y q “ ´ ∆ G p x ´ y q ϕ p x q ϕ p y q ´ ∇ G p x ´ y q ∇ ϕ p x q ϕ p y q´ G p x ´ y q ∆ ϕ p x q ϕ p y q ` ∆ x “ p G ˚ q χ H c qp x ´ y q ϕ p x q ϕ p y q ‰ an application of the scattering equation (2.9) together with the bounds (2.10), (2.12)from Lemma 2.1 as well as the pointwise bounds } G ˚ q χ H c } ď Cℓ ´ α and ˇˇ ∇ x “ p G ˚ q χ H c qp x ´ y q ϕ p x q ϕ p y q ˇˇ ď Cℓ ´ α | ϕ p y q| “ | ϕ p x q| ` | ∇ ϕ p x q| ` | ∆ ϕ p x q| ‰ , ˇˇ ∆ x “ p G ˚ q χ H c qp x ´ y q ϕ p x q ϕ p y q ˇˇ ď Cℓ ´ α | ϕ p y q| “ | ϕ p x q| ` | ∇ ϕ p x q| ` | ∆ ϕ p x q| ‰ @ ϕ, “ ´ ∆ ` V ext ` ` N V p N ¨q ˚ | ϕ | ˘‰ ϕ D p N ´ N q´ @ ϕ, ` N V p N ¨q ˚ | ϕ | ˘ ϕ D p N ` qp ´ N { N q` ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q η H p x ; y qp ´ N { N qp ´ N { N ´ { N q` ż dxdy p´ ∆ x η H p x ; y qqq η H p x ; y qp ´ N { N qp ´ N { N ´ { N q` N ż dxdy N V p N p x ´ y qq w ℓ p N p x ´ y qq| ϕ p x q| | ϕ p y q| ˆ p ´ N { N qp ´ N { N ´ { N q“ @ ϕ, “ ´ ∆ ` V ext ` ` N p V f ℓ qp N ¨q ˚ | ϕ | ˘‰ ϕ D p N ´ N q´ @ ϕ, ` N p V f ℓ qp N ¨q ˚ | ϕ | ˘ ϕ D N p ´ N { N q ` r E p q G N , for an error that satisfies ˘ r E p q G N ď C p ℓ p α ´ q{ ` ℓ ´ α q . Using the Gross-Pitaevskii equation(A.1), the bound (2.11) from Lemma 2.1, a simple application of the mean value theoremshows furthermore that @ ϕ, “ ´ ∆ ` V ext ` ` N p V f ℓ qp N ¨q ˚ | ϕ | ˘‰ ϕ D p N ´ N q´ @ ϕ, ` N p V f ℓ qp N ¨q ˚ | ϕ | ˘ ϕ D N p ´ N { N q“ N E GP p ϕ q ´ ε GP N ` π a } ϕ } N { N ` r E p q G N , up to an error that satisfies ˘ r E p q G N ď Cℓ ´ . This shows that G N “ N E GP p ϕ q ´ ε GP N ` π a } ϕ } N { N ` ż dx ` N V p N. q ˚ | ϕ | ˘ p x q b ˚ x b x ` ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q b ˚ x b y ` ż dxdy ` N p V f ℓ qp N p x ´ y qq ϕ p x q ϕ p y q ´ ∆ y η H p x ; y q ˘` b ˚ x b ˚ y ` b x b y ˘ ` ż dxdy N { V p N p x ´ y qq ϕ p y q ` b ˚ x a ˚ y a x ` h.c. ˘ ` K ` V ext ` V N ` r E p q G N ` r E p q G N ` r E p q G N . Now, let us simplify the quadratic contributions on the right hand side of the lastequation. First of all, another application of the mean value theorem shows that ˘ ˆ ż dx ` N V p N. q ˚ | ϕ | ˘ p x q b ˚ x b x ´ p V p q ż | ϕ p x q| b ˚ x b x ` ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q b ˚ x b y ´ p V p q ż | ϕ p x q| b ˚ x b x ˙ ď CN ´ p N ` K ` q . | ˆ V p p { N q ´ ˆ V p q| ď C | p |{ N and that ż dxdy N V p N p x ´ y qq ϕ p x q ϕ p y q b ˚ y b x “ ż dx ˆ V p q ϕ p x q b ˚ x b x ` ż dp ´ ˆ V p p { N q ´ ˆ V p q ¯ p a ˚ p p ϕ p q ˆ ´ N N ˙ p a p p ϕ p q , where p a p p ϕ p q “ ş dy e πipx ϕ p x q a x . The previous bound follows from ż | p | } p a p p ϕ p q ξ } ď ż | ∇ ϕ p x q| } a x ξ } ` ż dxϕ p x q} ∇ x a x ξ } ď C }p N ` K q { ξ } (4.37)because } ϕ } , } ∇ ϕ } ď C by (A.4).This controls the diagonal terms. For the non-diagonal term, we use once more thescattering equation (2.9), the simple identity p χ ℓ p p q “ π ´ sin p ℓ | p |q{| p | ´ π ´ ℓ cos p ℓ | p |q{| p | so that | p χ ℓ p p q| ď Cℓ | p | ´ and the pointwise bounds similarly as above to deduce that ˘ ˆ ż dxdy ` N p V f ℓ qp N p x ´ y qq ϕ p x q ϕ p y q ´ ∆ y η H p x ; y q ˘` b ˚ x b ˚ y ` b x b y ˘ ´ ż dxdy ` N p V f ℓ qp N. q ˚ q χ H c ˘ p x ´ y q ϕ p x q ϕ p y q ` b ˚ x b ˚ y ` b x b y ˘˙ ď C p ℓ α { ` ℓ p α ´ q{ qp N ` K ` q . Finally, switching to Fourier space, we see from Eq. (2.11) that for all p P R | p V f ℓ qp p { N q ´ πa | ď ż dx p V f ℓ qp x q ¨ | e πixp { N ´ | ` ˇˇˇˇż dx p V f ℓ qp x q ´ πa ˇˇˇˇ ď CN ´ | p | ` CN ´ ℓ ´ (4.38)and therefore ˘ ˆ ż dxdy ` p N p V f l qp N. q ˚ q χ H c ˘ p x ´ y q ϕ p x q ϕ p y q ` b x b y ` b ˚ x b ˚ y ˘ ´ πa ż dxdy q χ H c p x ´ y q ϕ p x q ϕ p y q ` b x b y ` b ˚ x b ˚ y ˘˙ ď C ›› ` p N p V f l qp N. q ˚ q χ H c ˘ ´ πa q χ H c ›› p N ` qď CN ´ ż | p |ď ℓ ´ α dp ` | p | ` ℓ ´ ˘ p N ` q ď C ` ℓ ´ α ` ℓ ´ α ´ ˘ . G N “ N E GP p ϕ q ´ ε GP N ` π a } ϕ } N { N ` ż dx a ˚ x “ ´ ∆ x ` V ext p x q ‰ a x ` p V p q ż | ϕ p x q| b ˚ x b x ` πa ż dxdy q χ H c p x ´ y q ϕ p x q ϕ p y q ` b x b y ` b ˚ x b ˚ y ˘ ` ż dxdy N { V p N p x ´ y qq ϕ p y q ` b ˚ x a ˚ y a x ` h.c. ˘ ` ż dxdy N V p N p x ´ y qq a ˚ x a ˚ y a x a y ` r E p q G N ` r E p q G N ` r E p q G N ` r E p q G N , where the error E G N : “ r E p q G N ` r E p q G N ` r E p q G N ` r E p q G N satisfies ˘ E G N ď C ` ℓ p α ´ q{ q ` ℓ p α ´ q{ ˘ p N ` K ` V N ` q ` Cℓ ´ α { N ´ p N ` q ` C p ℓ p α ´ q{ ` ℓ ´ α ` ℓ ´ α ´ ` ℓ ´ ˘ . Choosing α ą
3, this concludes the proof of Proposition 2.4. J N The goal of this section is to show Prop. 2.7 for the excitation Hamiltonian J N “ e ´ A G eff N e A , where G eff N has been introduced in (2.22) and can be decomposed as G eff N “ D N ` Q N ` C N ` H N (5.1)with H N “ K ` V ext ` V N and where D N “ N E GP p ϕ q ´ ε GP N ` π a } ϕ } N { N, Q N “ V p q ż dx | ϕ p x q| b ˚ x b x ` π a ż dxdy ˇ χ H c p x ´ y q ϕ p x q ϕ p y qp b ˚ x b ˚ y ` h.c. q , C N “ ? N ż dxdy N V p N p x ´ y qq ϕ p x qr b ˚ x a ˚ y a x ` h.c. s . (5.2)In the next subsections, we will study the action of the unitary operator e A on theseterms, where we recall from (2.28) that A “ ? N ż dxdydz ν H p x ; y q ˇ g L p x ´ z q ` b ˚ x a ˚ y a z ´ h.c. ˘ with ν H and g L as defined in (2.23) and (2.26), respectively, with parameters α ą β ą A .40 .1 Preliminary estimates First of all, with the next lemma we control the growth of the expectation of the externalpotential.
Lemma 5.1.
Assume (1.8) and let α ą β ą . Then there exists C ą such that forall ξ P F ď N ` , t P r
0; 1 s , ℓ P p
0; 1 q and N P N large enough, ˇˇˇ x ξ, e ´ tA V ext e tA ξ y ´ x ξ, V ext ξ y ˇˇˇ ď Cℓ α { @ ξ, “ V ext ` N ` ‰ ξ D . (5.3) Proof.
We compute r V ext , A s “ ? N ż dxdydz r V ext p x q ` V ext p y q ´ V ext p z qs ν H p x ; y q ˇ g L p x ´ z q b ˚ x a ˚ y a z ` h.c. Using (2.27) and the fact that, by (A.3), } V ext ϕ } ď C we get ˇˇˇˇ ? N ż dxdydz V ext p y q ν H p x ; y q ˇ g L p x ´ z qx ξ, b ˚ x a ˚ y a z ξ y ˇˇˇˇ ď C ? N ż dxdydz |p G ˚ q χ H qp x ´ y q| ˇ g L p x ´ z q} a x a y ξ }} a z ξ } ď Cℓ α }p N ` q ξ } . (5.4)Furthermore, recalling the assumption V ext p x ` y q ď C p V ext p x q ` C qp V ext p y q ` C q from(1.8), and using (2.24) as well as (2.27), we find that ˇˇˇˇ ? N ż dxdydz V ext p x q ν H p x ; y q ˇ g L p x ´ z qx ξ, b ˚ x a ˚ y a z ξ y ˇˇˇˇ ď ? N ˆż dxdydz V ext p x q ˇ g L p x ´ z q| ν H p x ; y q| } a z ξ } ˙ ˆ ˆż dxdydz V ext p x q ˇ g L p x ´ z q} a x a y ξ } ˙ ď Cℓ α ? N ˆż dxdz V ext p x ` z q ˇ g L p x q} a z ξ } ˙ } V ext p N ` q ξ }ď Cℓ α ? N p} V ext ξ } ` }p N ` q ξ }q } V ext p N ` q ξ } ď Cℓ α x ξ, p V ext ` N ` q ξ y . Notice that we used in the last step the bound } V ext ˇ g L } ď C , which follows from thefact that V ext grows at most exponentially (see Appendix A) and the explicit formulaˇ g L p x q “ p? πℓ ´ β q e ´p πℓ ´ β x q . Similarly, we get ˇˇˇˇ ? N ż dxdydz V ext p z q ν H p x ; y q ˇ g L p x ´ z qx ξ, b ˚ x a ˚ y a z ξ y ˇˇˇˇ ď Cℓ α x ξ, p V ext ` N ` q ξ y . Thus, we have ˘r V ext , A s ď Cℓ α p V ext ` N ` q . With (2.29), Eq. (5.3) follows now by Gronwall’s lemma, applied to the function f p t q “x ξ, e ´ tA V ext e tA ξ y . 41ext, we need to control the growth of the kinetic and potential energy. To estimatecontributions arising from the kinetic energy, we will often need to switch to momentumspace. We will use the formal notation ˆ a p “ a p e πip ¨ x q and ˆ a ˚ p “ a ˚ p e πip ¨ x q to indicatecreation and annihilation operators in momentum space. For f P L p R q (interpreted asa function of momentum), we set ˆ a p f q “ ş ¯ f p p q ˆ a p and similarly for ˆ a ˚ p f q . It is useful tokeep in mind that ż dy e πip ¨ y ϕ p y q a y “ ˆ a p ˆ ϕ p q (5.5)where ˆ ϕ p p q q : “ ˆ ϕ p p ´ q q .We will often encounter operators as in (5.5), with ϕ the minimizer (or the square ofthe minimizer) of the Gross-Pitaevskii energy functional. Switching to position space,we can bound ż dp } ˆ a p ˆ ϕ p q ξ } “ ż dx | ϕ p x q| } a x ξ } ď } ϕ } } N { ξ } ď C } N { ξ } (5.6)and, as already discussed in (4.37), ż dp p } ˆ a p ˆ ϕ p q ξ } ď C }p K ` N q { ξ } . (5.7) Lemma 5.2.
Let α ą , ă β ă α . Then we can write r K , A s “ T ` S ` S ` δ K (5.8) where T “ ´ ? N ż dxdydz ∇ x ν H p x ; y q ∇ x ˇ g L p x ´ z qr b ˚ x a ˚ y a z ` h.c. s ,S “ ´ ? N ż dxdydz N p V f ℓ qp N p x ´ y qq ϕ p y q ˇ g L p x ´ z qr b ˚ x a ˚ y a z ` h.c. s ,S “ ? N ż dxdydz rp N p V f ℓ qp N ¨qq ˚ q χ H c sp x ´ y q ϕ p y q ˇ g L p x ´ z qr b ˚ x a ˚ y a z ` h.c. s and where |x ξ, δ K ξ y| ď Cℓ p α ´ q{ x ξ, N ξ y ` Cℓ α { } K { ξ }}p N ` q { ξ } . Moreover, we find ˘ T ď Cℓ α { K , ˘ S ď C p V N ` N ` q , ˘ S ď C K ` Cℓ ´ α p N ` q , (5.9) and thus ˘ r K , A s ď C p K ` V N q ` Cℓ ´ α p N ` q . (5.10)42 roof. With the commutation relations (2.1), (2.4) and integration by parts, we obtain r K , A s “ ? N ż dxdydz ν H p x ; y q ˇ g L p x ´ z qr´ ∆ x b ˚ x a ˚ y a z ´ b ˚ x ∆ y a ˚ y a z ` b ˚ x a ˚ y ∆ z a z ` h.c. s“ ´ ? N ż dxdydz r ∆ x ν H p x ; y q ` ∆ y ν H p x ; y qs ˇ g L p x ´ z qr b ˚ x a ˚ y a z ` h.c. s´ ? N ż dxdydz ∇ x ν H p x ; y q ∇ x ˇ g L p x ´ z qr b ˚ x a ˚ y a z ` h.c. s“ : T ` T . We used here the identity ´ ∆ x ˇ g L p x ´ z q ` ∆ z ˇ g L p x ´ z q “
0. We rewrite T as T “ ´ ? N ż dxdydz p ∆ G ˚ q χ H qp x ´ y q ϕ p y q ˇ g L p x ´ z qr b ˚ x a ˚ y a z ` h.c. s` ? N ż dxdydz p G ˚ q χ H qp x ´ y q ∆ ϕ p y q ˇ g L p x ´ z qr b ˚ x a ˚ y a z ` h.c. s` ? N ż dxdydz p G ˚ q χ H qp x ´ y q ∇ ϕ p y q ˇ g L p x ´ z qr b ˚ x ∇ y a ˚ y a z ` h.c. s“ : T ` T ` T . Using Young’s inequality and (2.27) we obtain |x ξ, T ξ y| ď C ? N } G ˚ q χ H } ż dxdz ˇ g L p x ´ z q} a x p N ` q ξ }} a z ξ }ď Cℓ α }p N ` q ξ } and |x ξ, T ξ y| ď C ? N } G ˚ q χ H } ż dxdz ˇ g L p x ´ z q} K a x ξ }} a z ξ }ď Cℓ α }p N ` q ξ }} K ξ } . We are left with T . For this term we use the scattering equation (2.9) and get T “ ´ ? N ż dxdydz N p V f ℓ qp N p x ´ y qq ϕ p y q ˇ g L p x ´ z qr b ˚ x a ˚ y a z ` h.c. s` ? N ż dxdydz rp N p V f ℓ qp N ¨qq ˚ q χ H c sp x ´ y q ϕ p y q ˇ g L p x ´ z qr b ˚ x a ˚ y a z ` h.c. s` ? N ż dxdydz “ N λ ℓ p f ℓ p N ¨q χ ℓ q ˚ q χ H ‰ p x ´ y q ϕ p y q ˇ g L p x ´ z qr b ˚ x a ˚ y a z ` h.c. s“ : S ` S ` S . An explicit calculation shows that | ˆ χ ℓ p p q| “ ℓ | ˆ χ p ℓp q| ď Cℓ | p | ´ . With (2.15), we find }p f ℓ p N. q χ ℓ q ˚ ˇ χ H } ď ℓ ` α { (for N large enough). From (2.10), (2.27), we obtain |x ξ, S ξ y| ď C ? N N λ ℓ }p f ℓ p N ¨q χ ℓ q ˚ q χ H } ż dxdz ˇ g L p x ´ z q} a x p N ` q ξ }} a z ξ }ď Cℓ p α ´ q }p N ` q ξ } . T “ ´ ? N ż dxdydz p G ˚ q χ H qp x ´ y q ∇ ϕ p y q ˇ g L p x ´ z qr b ˚ x a ˚ y ∇ z a z ` h.c. s´ ? N ż dxdydz p G ˚ q χ H qp x ´ y q ϕ p y q ˇ g L p x ´ z qr b ˚ x ∇ y a ˚ y ∇ z a z ` h.c. s . With } G ˚ ˇ χ H } “ } ˆ Gχ H } ď ℓ α { and (2.27) we get |x ξ, T ξ y| “ Cℓ α ? N ż dxdz | ˇ g L p x ´ z q|} N a x ξ }} ∇ z a z ξ }` Cℓ α ? N ż dxdz | ˇ g L p x ´ z q|} K a x ξ }} ∇ z a z ξ }ď Cℓ α ? N } ˇ g L } } K p N ` q ξ }} K ξ } ď Cℓ α { } K { ξ } . By (2.27), we have |x ξ, S ξ y| ď ? N ż dxdydz N V p N p x ´ y qq ϕ p y q ˇ g L p x ´ z q} a x a y ξ }} a z ξ }ď C ˆż dxdydz N V p N p x ´ y qq ˇ g L p x ´ z q} a x a y ξ } ˙ ˆ ˆż dxdydz N V p N p x ´ y qq ˇ g L p x ´ z q} a z ξ } ˙ ď C } V N ξ }}p N ` q ξ } . (5.11)For S we change to momentum space to get, using (5.7) |x ξ, S ξ y| “ ˇˇˇˇ ? N ż dpdq { p V f ℓ qp p { N q χ H c p p q g L p q qx ˆ a p ˆ ϕ p q ˆ b q ´ p ξ, ˆ a q ξ y ˇˇˇˇ ď C ? N „ż dpdq p } ˆ a p ˆ ϕ p q ˆ a q ´ p ξ } { «ż | p |ď ℓ ´ α dpdq | p | ´ } ˆ a q ξ } ff { ď Cℓ ´ α { }p K ` N q { ξ }}p N ` q { ξ } . This concludes the proof of (5.9) and thus of (5.10).To bound the commutator of A with the interaction energy operator V N , it is usefulto introduce, for any θ ą
0, the notation K θ “ ż | p |ď θ p ˆ a ˚ p ˆ a p (5.12)for the kinetic energy of particles having momentum below θ .44 emma 5.3. Let α ě β ą and ε P p , α ´ β q . Then there exists C ą such that for ℓ P p
0; 1 q sufficiently small we have r V N , A s “ ? N ż dxdydz ν H p x ; y q ˇ g L p x ´ z q N V p N p x ´ y qqr b ˚ x a ˚ y a z ` h.c. s ` δ V , (5.13) where ˘ δ V ď Cℓ p α ´ β q “ K ℓ ´ β ´ ε ` V N ` N ` ‰ . Estimating the term on the r.h.s. of (5.13), we conclude that ˘ r V N , A s ď C “ K ℓ ´ β ´ ε ` V N ` N ` ‰ . (5.14) Proof.
With the commutation relations (2.1), (2.4), we find r V N , A s “ ´ ? N ż dxdydzdu ν H p x ; y q ˇ g L p x ´ z q N V p N p u ´ z qqr b ˚ u a ˚ x a ˚ y a z a u ` h.c. s` ? N ż dxdydzdu ν H p x ; y q ˇ g L p x ´ z q N V p N p u ´ y qqr b ˚ u a ˚ y a ˚ x a u a z ` h.c. s` ? N ż dxdydzdu ν H p x ; y q ˇ g L p x ´ z q N V p N p x ´ u qqr b ˚ x a ˚ u a ˚ y a u a z ` h.c. s` ? N ż dxdydz ν H p x ; y q ˇ g L p x ´ z q N V p N p x ´ y qqr b ˚ x a ˚ y a z ` h.c. s“ : ÿ j “ V j . Switching partially to momentum space and using (5.6), (5.7), we have that |x ξ, V ξ y| ď C ? N ż dpdqdudz N V p N p u ´ z qq| p G p p q| χ H p p q g L p q q} a u ˆ a p ` q ˆ a p x ϕ p q ξ }} a z a u ξ }ď Cℓ α { ? N ˆ ż dpdq p } ˆ a p ` q ˆ a p x ϕ p q ξ } ˙ { ˆ ż dq | g L p q q| ˙ { } V { N ξ }ď Cℓ p α ´ β q{ x ξ, p K ` V N ` N q ξ y . We have } ν H,x } ď } G ˚ q χ H } ¨ } ϕ } ď Cℓ α . Thus, we get using (2.27) |x ξ, V ξ y| ď Cℓ α ? N } ˇ g L } ż dzdu N V p N p u ´ z qq} a u ξ } ¨ } a z a u ξ }ď Cℓ α N } V N ξ } ¨ }p N ` q ξ } . Writing ν H p x, y q “ p G ˚ ˇ χ H qp x ´ y q ϕ p y q and expanding p G ˚ ˇ χ H q and also ˇ g L in45ourier space we can estimate |x ξ, V ξ y| ď ? N „ż dσdpdudy N V p N p u ´ y qq g L p p q p } a u a y ˆ a σ ` p ξ } { ˆ „ż dσdpdudy | ˆ G p σ q χ H p σ q| N V p N p u ´ y qq g L p p q p } a u ˆ a p ξ } { ď Cℓ p α ´ β q{ } V { N ξ } }p K ℓ ´ β ´ ε ` N ` q { ξ } , where, to bound the parenthesis in the second line, we divided the p -integral in the twodomains | p | ă ℓ ´ β ´ ε (where we can use the operator K ℓ ´ β ´ ε ) and | p | ě ℓ ´ β ´ ε , wherewe use the estimate sup | p |ą ℓ ´ β ´ ε p e ´ p ℓ β ă C . The term V can be bounded similarly.This proves (5.13). To show (5.14), we use | ν H p x ; y q| ď CN ϕ p y q , (2.27) and (2.29) toestimate |x ξ, V ξ y| ď C ż dxdydz N V p N p x ´ y qq ˇ g L p x ´ z q ϕ p y q} a x a y ξ } ¨ } a z ξ }ď C } ˇ g L } ¨ } V N ξ } ˆż dxdydz N V p N p x ´ y qq ˇ g L p x ´ z q} a z ξ } ˙ ď C } V N ξ } ¨ }p N ` q ξ } . Combining the last three lemmas, we obtain a bound for the growth of the Hamiltonoperator H N “ K ` V ext ` V N . Lemma 5.4.
Assume (1.8) . Let α ą β ą and α ě . There exists C ą such thatfor all t P r
0; 1 s , all ℓ P p
0; 1 q and all N P N large enough e ´ tA H N e tA ď C H N ` Cℓ ´ α p N ` q . (5.15) Proof.
We define f ξ p s q : “ x ξ, e ´ sA r K ` V N s e sA ξ y . Then B s f ξ p s q “ x ξ, e ´ sA r K ` V N , A s e sA ξ y . Inserting the bounds (5.10), (5.14) (with K ℓ ´ β ´ ε ď K ) and applying (2.29), we obtain B s f ξ p s q ď Cf ξ p s q ` Cℓ ´ α x ξ, e ´ sA p N ` q e sA ξ yď Cf ξ p s q ` Cℓ ´ α x ξ, e ´ sA p N ` q e sA ξ y ď Cf ξ p s q ` Cℓ ´ α x ξ, p N ` q ξ y for all s P r
0; 1 s . With Gronwall’s Lemma and using (5.3) to estimate the growth of theexternal potential, we arrive at (5.15).The estimate (5.15) is still not optimal (because of the large factor ℓ ´ α in front of p N ` ` q ). To improve this bound, we first have to study the growth of the operator K θ ,defined in (5.12) measuring the kinetic energy of particles with momenta smaller than θ ă ℓ ´ α ´ ℓ ´ β . 46 emma 5.5. Assume (1.8) . Let ă β ă α ď β , α ě , ε P p α ´ β q . Thenthere exists a constant C such that for all s P r
0; 1 s all ℓ ą small enough and all ℓ ´ β ´ ε ď θ ă ℓ ´ α ´ ℓ ´ β ´ ε , we have e ´ sA K θ e sA ď C K θ ` Cℓ p α ´ β ´ ε q p H N ` N ` q . (5.16) Proof.
In momentum space, we have A “ ? N ż dudvdw p ν H p v ; w q g L p u qr ˆ b ˚ u ` v ˆ a ˚ w ˆ a u ´ h.c. s“ ? N ż dudvdw ˆ G p v q χ H p v q ˆ ϕ p v ` w q g L p u qr ˆ b ˚ u ` v ˆ a ˚ w ˆ a u ´ h.c. s . (5.17)Thus, we find r K θ , A s “ ´ ? N ż | u |ď θ dudvdw p ν H p v ; w q g L p u q| u | r ˆ b ˚ u ` v ˆ a ˚ w ˆ a u ` h.c. s` ? N ż | u ` v |ď θ dudvdw p ν H p v ; w q g L p u q| u ` v | r ˆ b ˚ u ` v ˆ a ˚ w ˆ a u ` h.c. s` ? N ż | w |ď θ dudvdw p ν H p v ; w q g L p u q| w | r ˆ b ˚ u ` v ˆ a ˚ w ˆ a u ` h.c. s“ : A ` A ` A . Let ε P p α ´ β q . For | u | ď ℓ ´ β ´ ε and | v | ě ℓ ´ α we have (since α ą β ` ε ) | u ` v | ě ℓ ´ α { ℓ small enough. Therefore, by (2.25), (A.5), we find |x ξ, A ξ y| ď C ? N ż | u |ď θ, | v |ě ℓ ´ α dudvdw | v | | p ϕ p v ` w q| g L p u q| u | } ˆ a w ˆ a u ` v ξ }} ˆ a u ξ }ď C ? N ℓ α „ż dudvdw | ˆ ϕ p v ` w q|p u ` v q } ˆ a w ˆ a u ` v ξ } { ˆ «ż | u |ď θ, | v |ě ℓ ´ α dudvdw | v | | ˆ ϕ p v ` w q| g L p u q u } ˆ a u ξ } ff { ď Cℓ α } K { ξ }}p N ` q { ξ } ` Cℓ p α ´ β ´ ε q{ } K { ξ }} K { θ ξ } , (5.18)where, in the last step, we separated the u -integral in the second parenthesis between | u | ą ℓ ´ β ´ ε (where we control | u | with g L p u q ) and | u | ă ℓ ´ β ´ ε (where we use K θ ).47ow we turn to A . We bound |x ξ, A ξ y| ď C ? N θℓ α ż | w |ď θ, | v |ě ℓ ´ α dudvdw | ˆ ϕ p v ` w q| g L p u q| u | | w |} ˆ a w ˆ a u ` v ξ }| u |} ˆ a u ξ }ď C ? N θℓ α «ż | w |ď θ dudvdw g L p u q| u | | w | } ˆ a w ˆ a u ` v ξ } ff { ˆ «ż | w |ď θ, | v |ě ℓ ´ α dudvdw | ˆ ϕ p v ` w q| g L p u q| u | } ˆ a u ξ } ff { ď Cℓ α } K { θ ξ }} N { ξ } ` Cθ { ℓ α ` β ` ε { } K { θ ξ } ď Cℓ α } K { θ ξ }} N { ξ } ` Cℓ p β ` ε ´ α q{ } K { θ ξ } , (5.19)where we divided the u -integral in the third line an the integral over | u | ą ℓ ´ β ´ ε (herewe can control factors of u and extract arbitrary polynomial decay in ℓ from g L ) and anintegral over | u | ď ℓ ´ β ´ ε (here we used the fact that | v ` w | ě | v | ´ | w | ě ℓ ´ β and thebound (A.5) for the decay of ˆ ϕ ).We are left with A . Observing that | u | ą | v | ´ | u ` v | ě ℓ ´ β ´ ε if | u ` v | ď θ ď ℓ ´ α ´ ℓ ´ β ´ ε and | v | ě ℓ ´ α , we can extract arbitrary polynomial decay in ℓ from g L .Thus, we easily get |x ξ, A ξ y| ď Cℓ α }p N ` q { ξ } . (5.20)From (5.18), (5.19), (5.20) and the assumption β ă α ď β , we conclude that |x ξ, r K θ , A s ξ y| ď C } K { θ ξ } ` Cℓ α ´ β ´ ε }p K ` N ` q { ξ } . Defining f ξ p s q “ x ξ, e ´ sA K θ e sA ξ y , we conclude that |B s f ξ p s q| ď Cf ξ p s q ` Cℓ α ´ β ´ ε }p K ` N ` q { e sA ξ } . Estimating K ď H N and applying (5.15) and Lemma 2.6, Gronwall leads us to (5.16).Lemma 5.5 can be used to improve the estimate (5.15) on the growth of the Hamil-tonian. We begin with the potential energy operator V N . Lemma 5.6.
Assume (1.8) . Let ă β ă α ď β , α ě . Then there exists C ą suchthat for ℓ ą small enough and all s P r
0; 1 s , we have e ´ sA V N e sA ď C p H N ` N ` q . (5.21) Proof.
Eq. (5.21) follows from (5.14), using (2.29), (5.16) and Gronwall’s lemma.We conclude this subsection with an improvement on the growth of the kinetic energyoperator K . 48 emma 5.7. Assume (1.8) . Let ă β ă α ď β , α ě , ă ε ă α ´ β . Then thereexists C ą such that for ℓ P p
0; 1 q small enough, we have for all s P r
0; 1 s e ´ sA K e sA ď Cℓ ´ p α ` β ` ε q p H N ` N ` q . (5.22) Proof.
From (5.8) and (5.11), we find that r K , A s “ S ` δ K , with ˘ δ K ď Cℓ α { K ` C p V N ` N ` q and S “ ? N ż dxdydz rp N p V f ℓ qp N ¨qq ˚ q χ H c sp x ´ y q ϕ p y q ˇ g L p x ´ z qr b ˚ x a ˚ y a z ` h.c. s . Switching to Fourier space, we have |x ξ, S ξ y| ď C ? N ż | p |ď ℓ ´ α dpdq g L p q q} ˆ a p ˆ ϕ p q ˆ a q ´ p ξ }} ˆ a q ξ }ď Cℓ α } N { ξ } ` C ? N ż | p |ď ℓ ´ α , | q |ă ℓ ´ β ´ ε dpdq } ˆ a p ˆ ϕ p q ˆ a q ´ p ξ }} ˆ a q ξ } , where we used g L to extract decay in ℓ for the region | q | ą ℓ ´ β ´ ε . To control the lastintegral, we split the region where | q ´ p | ă ℓ ´ α ` ℓ ´ β ´ ε in two parts, the first with | q ´ p | ă ℓ ´ α ´ ℓ ´ β ´ ε “ : θ and the second with θ ď | q ´ p | ă ℓ ´ α ` ℓ ´ β ´ ε . We obtain |x ξ, S ξ y| ď Cℓ α } N { ξ } ` C ? N «ż | q ´ p |ď θ dpdq p q ´ p q } ˆ a p ˆ ϕ p q ˆ a q ´ p ξ } ff { «ż | q ´ p |ď θ dpdq p q ´ p q ´ } ˆ a q ξ } ff { ` C ? N „ż dpdq p q ´ p q } ˆ a p ˆ ϕ p q ˆ a q ´ p ξ } { ˆ «ż θ ď| q ´ p |ď ℓ ´ α ` ℓ ´ β ´ ε dpdq p q ´ p q ´ } ˆ a q ξ } ff { ď Cℓ α } N { ξ } ` Cℓ ´ α { } K { θ ξ }} N { ξ } ` Cℓ ´p β ` ε q{ } K { ξ }} N { ξ } . We conclude that ˇˇˇ dds x ξ, e sA K e ´ sA ξ y ˇˇˇ ď Cℓ α { x ξe ´ sA K e sA ξ y ` Cℓ ´ α { x ξ, e ´ sA p K θ ` V N ` N ` q e sA ξ y` Cℓ ´p β ` ε q{ } K { e sA ξ }}p N ` q { e sA ξ } . With (2.29), (5.16), (5.21), and with (5.15) (estimating K ď H N ), we obtain ˇˇˇ dds x ξ, e sA K e ´ sA ξ y ˇˇˇ ď Cℓ ´p α ` β ` ε q{ x ξ, p H N ` N ` q ξ y . Integrating over s yields (5.22). 49 .2 Analysis of e ´ A D N e A In this section we study the contribution arising from the operator D N , defined in (5.2). Lemma 5.8.
There is C ą s.t. for all F P L p R q , α, β ą , ℓ P p
0; 1 q , we have ˇˇˇˇż du F p u qx ξ , ` e ´ A a ˚ u a u e A ´ a ˚ u a u ˘ ξ y ˇˇˇˇ ď Cℓ α } F } }p N ` q ξ }}p N ` q ξ } , ˇˇˇˇż du F p u qx ξ , ` e ´ A b ˚ u b u e A ´ b ˚ u b u ˘ ξ y ˇˇˇˇ ď Cℓ α } F } }p N ` q ξ }}p N ` q ξ } . (5.23) Proof.
With (2.4), we find ż du F p u q ` e ´ A a ˚ u a u e A ´ a ˚ u a u ˘ “ ? N ż ds ż dxdydz p F p x q ` F p y q ´ F p z qq ν H p x ; y q ˇ g L p x ´ z q “ e ´ sA b ˚ x a ˚ y a z e A ` h.c. ‰ . Using (2.27) and (2.29) we obtain ˇˇˇˇ ? N ż ds ż dxdydz p F p x q ` F p y q ´ F p z qq ν H p x ; y q ˇ g L p x ´ z qx ξ , e ´ sA b ˚ x a ˚ y a z e sA ξ y ˇˇˇˇ ď } F } ? N ż ds ż dxdydz | ν H p x ; y q|| ˇ g L p x ´ z q|} a x a y e sA ξ }} a z e sA ξ }ď C } F } ℓ α }p N ` q ξ }}p N ` q ξ } . Interchanging ξ and ξ yields the same estimate for the hermitian conjugate and impliesthe first estimate in (5.23). Proceeding similarly, we obtain also the second bound.Using Lemma 5.8, we can easily control the action of e A on the operator D N . Theproof of the next lemma follows very closely the proof of [4, Prop. 8.7]. Lemma 5.9.
Let α, β ą . Then there exists C ą such that for all ℓ P p
0; 1 q holds e ´ A D N e A “ D N ` δ D N , (5.24) where ˘ δ D N ď Cℓ α p N ` q . e ´ A Q N e A In this section we study the contribution to J N arising from the operator Q N , definedin (5.2). 50 emma 5.10. Let ă β ă α ď β , α ě and ε P p α ´ β q . Then there exists C ą such that for ℓ P p
0; 1 q small enough and N large enough, we have e ´ A Q N e A “ Q N ` δ Q N , (5.25) where ˘ δ Q N ď C “ ℓ p α ´ β ´ ε q{ ` ℓ p α ´ β ´ ε q{ ‰ p H N ` N ` q . Proof.
The first contribution to Q N in (5.2) can be handled with (5.23); it produces anerror term bounded by Cℓ α p N ` q . As for the second contribution to Q N , we compute,with the commutation relations (2.3), (2.4),4 π a ż dudv q χ H c p u ´ v q ϕ p u q ϕ p v qr b ˚ u b ˚ v , A s“ ´ πa ? N ż dxdydzdu q χ H c p u ´ z q ϕ p u q ϕ p z q ν H p x ; y q ˇ g L p x ´ z q b ˚ x b ˚ y b ˚ u ` πa ? N ż dxdydzdu q χ H c p u ´ y q ϕ p u q ϕ p y q ν H p x ; y q ˇ g L p x ´ z q b ˚ z b ˚ u b x ` πa ? N ż dxdydzdu q χ H c p u ´ x q ϕ p u q ϕ p x q ν H p x ; y q ˇ g L p x ´ z q b ˚ z a ˚ u a y ˆ ´ N ` N ˙ ` πa ? N ż dxdydz q χ H c p x ´ y q ϕ p x q ϕ p y q ν H p x ; y q ˇ g L p x ´ z q b ˚ z ˆ ´ N ` N ˙ ´ πa N ? N ż dxdydzdudv q χ H c p u ´ v q ϕ p u q ϕ p v q ν H p x ; y q ˇ g L p x ´ z q b ˚ z a ˚ u a ˚ v a x a y ´ πa N ? N ż dxdydzdu q χ H c p u ´ y q ϕ p u q ϕ p y q ν H p x ; y q ˇ g L p x ´ z q b ˚ z a ˚ u a x “ : ÿ j “ τ j . Switching to momentum space we get, with the notation introduced in (5.5), |x ξ, τ ξ y| ď C ? N ż dpdsdt χ H c p p q| ˆ G p s q| χ H p s q g L p t q| ˆ ϕ p p ` t q|ˆ }p N ` q ´ ˆ a p ˆ ϕ p q ˆ a p ˆ ϕ ´ s q ˆ a s ` t ξ }}p N ` q ξ }ď Cℓ p α ´ β q }p K ` N q ξ }}p N ` q ξ } , (5.26)where we used the bound (5.7), } g L } ď Cℓ ´ β { and ş | s |ě ℓ ´ α | s | ´ ď Cℓ α . Similarly, |x ξ, τ ξ y| , |x ξ, τ ξ y| ď Cℓ p α ´ β q }p K ` N q ξ }}p N ` q ξ } . (5.27)Moreover, we can easily bound |x ξ, τ ξ y| , |x ξ, τ ξ y| ď Cℓ α }p N ` q { ξ } (5.28)51or N P N large enough (here, we can use the small factors N ´ { and N ´ { to gainarbitrary decay in ℓ ). As for τ , switching to momentum space we estimate |x ξ, τ ξ y| ď CN { ż dpdsdt χ H c p p q| ˆ G p s q| χ H p s q g L p t q } ˆ a t ˆ a p ˆ ϕ p q ˆ a p ˆ ϕ ´ p q ξ }} ˆ a s ` t ˆ a p ˆ ϕ s q ξ }ď CN «ż | s |ě ℓ ´ α dpdsdt p | s | } ˆ a t ˆ a p ˆ ϕ p q ξ } ff { «ż | p |ď ℓ ´ α dpdsdt s p } ˆ a s ` t ˆ a p ˆ ϕ s q ξ } ff { ď Cℓ α }p K ` N q ξ } , (5.29)where we used | ˆ G p s q| ď C { s and the bound (5.7). Combining (5.26), (5.27), (5.28) and(5.29) we conclude that |x ξ, r Q N , A s ξ y| ď Cℓ α }p K ` N q { ξ } ` Cℓ p α ´ β q{ }p K ` N q { ξ }}p N ` q { ξ } . With (2.29),(5.22), we obtain ˇˇˇ x ξ, e ´ A Q N e A ξ y ´ x ξ, Q N ξ y ˇˇˇ ď ż |x ξ, e ´ sA r Q N , A s e sA ξ y| ds ď C “ ℓ p α ´ β ´ ε q{ ` ℓ p α ´ β ´ ε q{ ‰ x ξ, p H N ` N ` q ξ y . e ´ A K e A To control the action of e A on the kinetic energy operator K , we need better estimateson the commutator r K , A s and also on a term arising from the second commutator rr K , A s , A s . Lemma 5.11.
Let ă β ă α ď β , α ą , and ε P p α ´ β q . Then there exists aconstant C ą such that for all ℓ ą sufficiently small and for N sufficiently large wehave r K , A s “ ´ ? N ż dxdydz N p V f ℓ qp N p x ´ y qq ϕ p y q ˇ g L p x ´ z qr b ˚ x a ˚ y a z ` h.c. s` ? N ż dxdydz πa q χ H c p x ´ y q ˇ g L p x ´ z q ϕ p y qr b ˚ x a ˚ y a z ` h.c. s ` r δ K (5.30) where |x ξ, r δ K ξ y| ď Cℓ p α ´ q }p N ` q ξ } ` Cℓ α }p K ` N q { ξ }}p K ℓ ´ β ´ ε ` N ` q { ξ } . Moreover, ˇˇˇ ? N ż dxdydz πa q χ H c p x ´ y q ˇ g L p x ´ z q ϕ p y qx ξ, r b ˚ x a ˚ y a z , A s ξ y ˇˇˇ ď Cℓ p α ´ β q }p K ` N q ξ }}p N ` q ξ } ` Cℓ α }p K ` N q ξ } ` Cℓ p α ´ β q } K ℓ ´ β ´ ε ξ }}p N ` q ξ } . (5.31)52 roof. We start with the decomposition r K , A s “ T ` S ` S ` δ K from (5.8). Theterm S corresponds exactly to the first term on the r.h.s. of (5.30). To show (5.30),we prove that T is small and that S corresponds to the second term on the r.h.s. of(5.30), up to small corrections. Switching to momentum space, we get T “ ´ ? N ż dpdq p ¨ q ˆ G p p q χ H p p q g L p q qr ˆ b ˚ p ` q ˆ a p ˆ ϕ ´ p q ˆ a q ` h.c. s . We estimate |x ξ, T ξ y| ď C ? N ż dpdq | p || q || ˆ G p p q| χ H p p q g L p q q} ˆ a p ` q ˆ a p ˆ ϕ ´ p q ξ }} ˆ a q ξ }ď C ? N „ż dpdq | p | } ˆ a p ` q ˆ a p ˆ ϕ ´ p q ξ } „ż dpdq | ˆ G p p q| χ H p p q g L p q q | q | } ˆ a q ξ } ď Cℓ α }p K ` N q ξ } „ } K ℓ ´ β ´ ε ξ } ` ℓ α } N ξ } , where in the second integral we separated | q | ă ℓ ´ β ´ ε (where we can use K ℓ ´ β ´ ε ) and | q | ą ℓ ´ β ´ ε (where g L is effective). Now we consider S , as defined in (5.9). Switchingto Fourier space and subtracting the second term on the r.h.s. of (5.30), we can bound ˇˇˇ x ξ, S ξ y ´ π a ? N ż dxdydz ˇ χ H c p x ´ y q ˇ g L p x ´ z q ϕ p y q x ξ, “ b ˚ x a ˚ y a z ` h.c. ‰ ξ y ˇˇˇ ď ? N ż dpdq ż dpdq | y V f ℓ p q { N q ´ π a | χ H c p q q g L p p q} ˆ a p ´ q ˆ a p ˆ ϕ q q ξ }} ˆ a p ξ }ď CN { ż dpdq p ℓ ´ ` | q |q χ H c p q q g L p p q} ˆ a p ´ q ˆ a p ˆ ϕ q q ξ }} ˆ a p ξ } ď Cℓ α }p N ` q { ξ } where we used (4.38) and we chose N large enough to obtain the desired decay in ℓ .This concludes the proof of (5.30).To show (5.31), we write, in momentum space, A as in (5.17) and8 π a ? N ż dxdydz ˇ χ H c p x ´ y q ˇ g L p x ´ z q ϕ p y q b ˚ x a ˚ y a z “ π a ? N ż dqdp χ H c p q q g L p p q ˆ b ˚ q ` p ˆ a ˚ p ˆ ϕ ´ q q ˆ a p . Using the relations (2.3), (2.4) (translated to momentum space), we compute8 π a ? N ż dxdydz ˇ χ H c p x ´ y q ˇ g L p x ´ z q ϕ p y q “ b ˚ x a ˚ y a z , A ‰ “ ÿ j “ Y j with the operators t Y , . . . , Y u as given below. We bound each term separately.We start with Y “ ´ π a N ż dqdpds χ H c p q q g L p p q ˆ G p s q χ H p s q g L p p ` q q ˆ b ˚ s ` p ` q ˆ b ˚ p ˆ ϕ ´ s q ˆ a ˚ p ˆ ϕ ´ q q ˆ a p . | ˆ G p s q| ď C | s | ´ , we find |x ξ, Y ξ y| ď CN „ż dqdpds s } ˆ a s ` p ` q ˆ a p ˆ ϕ ´ s q ˆ a p ˆ ϕ ´ q q ξ } ˆ «ż | s |ě ℓ ´ α dqdpds | s | ´ g L p q q } ˆ a p ξ } ff ď Cℓ p α ´ β q{ }p K ` N q { ξ }} N { ξ } . We continue with Y “ π a N ż dqdpdsdt χ H c p q q g L p p q ˆ G p s q χ H p s q ˆ ϕ p s ´ p ´ q q g L p t qˆ p ´ N { N q ˆ a ˚´ t ˆ a ˚ p ˆ ϕ ´ q q ˆ a t ´ s ˆ a p which can be bounded by |x ξ, Y ξ y| ď Cℓ α }p N ` q { ξ } ` CN «ż | s |ą ℓ ´ α , | t |ă ℓ ´ β ´ ε dqdpdsdt | s | ´ | ˆ ϕ p s ´ p ´ q q| t } ˆ a ´ t ˆ a p ˆ ϕ ´ q q ξ } ff { ˆ „ż dqdpdsdt g L p t q t | ˆ ϕ p s ´ p ´ q q| } ˆ a t ´ s ˆ a p ξ } { ď Cℓ α }p N ` q { ξ } ` Cℓ p α ´ β q{ } K { ℓ ´ β ´ ε ξ }}p N ` q { ξ } where we used the decay of g L to handle the region | t | ą ℓ ´ β ´ ε . Next, we consider Y “ π a N ż dpdsdt χ H c p p ` s ` t q ˇ g L p p q ˆ G p s q χ H p s q g L p t qˆ p ´ N { N q ˆ a ˚´ t ˆ a ˚ p ˆ ϕ p ` s ` t q ˆ a p ˆ ϕ s q ˆ a p whose contribution is estimated similarly as the one of Y by |x ξ, Y ξ y| ď Cℓ α }p N ` q { ξ } ` CN «ż | s |ą ℓ ´ α , | t |ă ℓ ´ β ´ ε dpdsdt | s | ´ t } ˆ a ´ t ˆ a p ˆ ϕ p ` s ` t q ξ } ff { ˆ „ż dpdsdt g L p t q t } ˆ a p ˆ ϕ s q ˆ a p ξ } { ď Cℓ α }p N ` q { ξ } ` Cℓ p α ´ β q{ } K { ℓ ´ β ´ ε ξ }}p N ` q { ξ } . Y “ π a N ż dqdpdsdt χ H c p q q g L p p q ˆ G p s q χ H p s q g L p t q ˆ ϕ p q ` s ` t q ˆ ϕ p q ´ p qˆ p ´ N { N q ˆ a ˚´ t ˆ a p ,Y “ π a N ż dpdsdt χ H c p p ` s ` t q g L p p q ˆ G p s q χ H p s q g L p t q x ϕ p p ` t qp ´ N { N q ˆ a ˚´ t ˆ a p ,Y “ ´ π a N ż dqdpdsdt χ H c p q q g L p p q ˆ G p s q χ H p s q g L p t q ˆ ϕ p q ´ s ´ t q ˆ a ˚´ t ˆ a ˚ q ` p ˆ a p ˆ ϕ s q ˆ a p ,Y “ ´ π a N ż dqdpdsdt χ H c p q q g L p p q ˆ G p s q χ H p s q g L p t q x ϕ p q ` s q ˆ a ˚´ t ˆ a ˚ q ` p ˆ a ´ s ´ t ˆ a p can be bounded by ˘ Y , ˘ Y , ˘ Y , ˘ Y ď Cℓ α p N ` q for N large enough (the additional factor 1 { N can be used to gain arbitrary decay in ℓ ).As for Y “ ´ π a N ż dqdpdsdt χ H c p q q g L p p q ˆ G p s q χ H p s q g L p t q ˆ a ˚´ t ˆ a ˚ p ˆ ϕ ´ q q ˆ a ˚ q ` p ˆ a ´ s ´ t ˆ a p ˆ ϕ s q ˆ a p , we estimate, again with (5.6), (5.7) and | ˆ G p s q| ď C | s | ´ , |x ξ, Y ξ y| ď CN «ż | s |ą ℓ ´ α dqdpdsdt | s | ´ q } ˆ a ´ t ˆ a p ˆ ϕ ´ q q ˆ a q ` p ξ } ff { ˆ «ż | q |ď ℓ ´ α dqdpdsdt | q | ´ s } ˆ a ´ s ´ t ˆ a p ˆ ϕ s q ˆ a p ξ } ff { ď Cℓ α }p K ` N q { ξ } . To bound Y “ π a N ż dqdpdsdt χ H c p q q g L p p q ˆ G p s q χ H p s q g L p t q ˆ ϕ p p ` s q ˆ b ˚ p ˆ ϕ ´ q q ˆ b ˚ q ` p ˆ a ˚ s ` t ˆ a t , we first apply Cauchy-Schwarz: |x ξ, Y ξ y| ď CN ż dqdpdsdt χ H c p q q g L p p q| ˆ G p s q| χ H p s q g L p t q| ˆ ϕ p p ` s q|ˆ } ˆ a p ˆ ϕ q ` p q ˆ a s ` t ξ }} ˆ a ˚ p ˆ ϕ ´ q q ˆ a t ξ } . With (2.1), we find } ˆ a ˚ p ˆ ϕ ´ q q ˆ a t ξ } ď } ˆ a p ˆ ϕ ´ q q ˆ a t ξ } ` } ˆ a t ξ } . For the contribution from } ˆ a t ξ } , we can extract arbitrary decay in ℓ from the factor N ´ . For | t | ą ℓ ´ β ´ ε , we can55ain decay in ℓ from the Gaussian g L . We find |x ξ, Y ξ y| ď Cℓ α }p N ` q { ξ } ` CN „ż dqdpdsdt g L p t q t | ˆ ϕ p p ` s q|} ˆ a p ˆ ϕ q ` p q ˆ a s ` t ξ } { ˆ «ż | s |ą ℓ ´ α , | t |ă ℓ ´ β ´ ε dqdpdsdt | s | | ˆ ϕ p p ` s q| t } ˆ a p ˆ ϕ ´ q q ˆ a t ξ } ff { ď Cℓ α }p N ` q { ξ } ` Cℓ p α ´ β q{ } K { ℓ ´ β ´ ε ξ }}p N ` q { ξ } . Next, consider Y “ π a N ż dqdsdt χ H c p q q g L p s ` t q ˆ G p s q χ H p s q g L p t q ˆ b ˚ q ` s ` t ˆ a ˚ p ˆ ϕ ´ s q ˆ b ˚ p ˆ ϕ ´ q q ˆ a t . We proceed as for Y , with } ˆ a ˚ p ˆ ϕ ´ q q ˆ a t ξ } ď } ˆ a p ˆ ϕ ´ q q ˆ a t ξ } ` } ˆ a t ξ } . Again, to bound thecontribution arising from } ˆ a t ξ } we can extract decay in ℓ from N ´ . We find |x ξ, Y ξ y| ď Cℓ α }p N ` q { ξ } ` CN «ż | q |ď ℓ ´ α dqdsdt | q | ´ s } ˆ a q ` s ` t ˆ a p ˆ ϕ ´ s q ξ } ff { ˆ «ż | q |ď ℓ ´ α , | s |ą ℓ ´ α dqdsdt | s | ´ q } ˆ a p ˆ ϕ ´ q q ˆ a t ξ } ff { ď Cℓ α }p N ` q { ξ } ` Cℓ α }p K ` N q { ξ } . As for Y “ ´ π a N ż dqdpdsdt χ H c p q q g L p p q ˆ G p s q χ H p s q g L p t q ˆ ϕ p q ` t q ˆ b ˚ q ` p ˆ a ˚ s ` t ˆ a ˚ p ˆ ϕ ´ s q ˆ a p , we have |x ξ, Y ξ y| ď CN „ż dqdpdsdt | ˆ ϕ p q ` t q| s } ˆ a p ˆ ϕ ´ s q ˆ a s ` t ˆ a q ` p ξ } { ˆ «ż | s |ą ℓ ´ α dqdpdsdt | s | ´ | ˆ ϕ p q ` t q| g L p t q } ˆ a p ξ } ff { ď Cℓ p α ´ β q{ }p K ` N q { ξ }}p N ` q { ξ } . On the other hand, Y “ π a N ż dqdpdsdt χ H c p q q g L p p q ˆ G p s q χ H p s q g L p t q x ϕ p q ` s q ˆ b ˚ q ` p ˆ a ˚´ t ˆ b ´ s ´ t ˆ a p | t | ą ℓ ´ β ´ ε with χ L ă ℓ ) by |x ξ, Y ξ y| ď Cℓ α }p N ` q { ξ } ` CN «ż | s |ą ℓ ´ α , | t |ă ℓ ´ β ´ ε dqdpdsdt | s | ´ | x ϕ p q ` s q| t } ˆ a ´ t ˆ a q ` p ξ } ff { ˆ „ż dqdpdsdt | x ϕ p q ` s q| g L p t q t } ˆ a ´ s ´ t ˆ a p ξ } { ď Cℓ α }p N ` q { ξ } ` Cℓ p α ´ β q{ } K { ℓ ´ β ´ ε ξ }}p N ` q { ξ } . Also the term Y “ π a N ż dqdpdsdt χ H c p q q g L p p q ˆ G p s q χ H p s q g L p t q ˆ ϕ p q ´ s ´ t q ˆ b ˚ q ` p ˆ a ˚´ t ˆ b p ˆ ϕ s q ˆ a p can be bounded similarly: |x ξ, Y ξ y| ď Cℓ α }p N ` q { ξ } ` CN «ż | s |ą ℓ ´ α , | t |ă ℓ ´ β ´ ε dqdpdsdt | s | ´ | ˆ ϕ p q ´ s ´ t q| t } ˆ a ´ t ˆ a q ` p ξ } ff { ˆ „ż dqdpdsdt | ˆ ϕ p q ´ s ´ t q| g L p t q t } ˆ a p ˆ ϕ s q ˆ a p ξ } { ď Cℓ α }p N ` q { ξ } ` Cℓ p α ´ β q{ } K { ℓ ´ β ´ ε ξ }}p N ` q { ξ } . Finally, we bound Y “ ´ π a N ż dqdsdt χ H c p q q g L p t q ˆ G p s q χ H p s q ˆ b ˚ q ´ t ˆ a ˚ p ˆ ϕ ´ q q ˆ b ´ s ´ t ˆ a p ˆ ϕ s q as follows: |x ξ, Y ξ y| ď CN «ż | s |ą ℓ ´ α dqdsdt | s | ´ q } ˆ a q ´ t ˆ a p ˆ ϕ ´ q q ξ } ff { ˆ «ż | q |ď ℓ ´ α dqdsdt | q | ´ s } ˆ a p ˆ ϕ s q ˆ a ´ s ´ t ξ } ff { ď Cℓ α }p K ` N q { ξ } . Combining the estimates for t Y , . . . , Y u , we obtain (5.31). e ´ A C N e A To control the action of e A on the cubic term C N in (5.2), we need precise estimates forthe commutator r C N , A s . 57 emma 5.12. Let ă β ă α and ε P p α ´ β q . Then there exists a constant C ą such that for all ℓ P p
0; 1 q sufficiently small and for N sufficiently large we have r C N , A s “ N ż dxdydz N V p N p x ´ y qq ν H p x ; y q ϕ p y q ˇ g L p x ´ z qr a ˚ z a y ` h.c. s ˆ ´ N N ˙ ` N ż dxdydz N V p N p x ´ y qq ν H p x ; y q ϕ p x q ˇ g L p x ´ z qr a ˚ z a x ` h.c. s ˆ ´ N N ˙ ` δ C N , (5.32) where |x ξ, δ C N ξ y| ď Cℓ p α ´ β q }p K ` N q ξ }} N ξ } ` Cℓ p α ´ β q }p K ℓ ´ β ´ ε ` V N ` N q ξ } . Proof.
A long but straightforward computation using (2.3), (2.4) shows that r C N , A s “ N ż dxdydz N V p N p x ´ y qq ν H p x ; y q ϕ p y q ˇ g L p x ´ z qr a ˚ z a y ` h.c. s ˆ ´ N N ˙ ` N ż dxdydz N V p N p x ´ y qq ν H p x ; y q ϕ p x q ˇ g L p x ´ z qr a ˚ z a x ` h.c. s ˆ ´ N N ˙ ` ÿ j “ C j , with the error terms t C , . . . , C u are listed and bounded below.We begin with C “ ´ N ż dxdydzdv N V p N p z ´ v qq ϕ p z q ν H p x ; y q ˇ g L p x ´ z q b ˚ x b ˚ y a ˚ v a z which can be bounded, switching to momentum space, by |x ξ, C ξ y| ď N ż dpdsdt | ˆ V p p { N q|| ˆ G p s q| χ H p s q g L p t q} ˆ a s ` t ˆ a p ˆ ϕ ´ s q ˆ a ´ p ξ }} ˆ a p ˆ ϕ t ´ p q ξ }ď CN „ż dpdsdt s } ˆ a s ` t ˆ a p ˆ ϕ ´ s q ˆ a ´ p ξ } { ˆ «ż | s |ą ℓ ´ α dpdsdt | s | ´ g L p t q } ˆ a p ˆ ϕ t ´ p q ξ } ff { ď Cℓ p α ´ β q{ }p K ` N q { ξ }} N { ξ } . Also for the term C “ N ż dxdydzdv N V p N p y ´ v qq ϕ p y q ν H p x ; y q ˇ g L p x ´ z q ˆ ´ N N ˙ a ˚ z a ˚ v a x a y
58e switch to momentum space. We find |x ξ, C ξ y| ď N ż dpdsdt | ˆ V p p { N q|| ˆ G p s q| χ H p s q g L p t q} ˆ a ´ t ˆ a ´ p ξ }} ˆ a p x ϕ s ´ p q ˆ a ´ s ´ t ξ }ď Cℓ α }p N ` q { ξ } ` CN «ż | s |ą ℓ ´ α , | t |ă ℓ ´ β ´ ε dpdsdt | s | ´ t } ˆ a ´ t ˆ a ´ p ξ } ff { ˆ „ż dpdsdt g L p t q t } ˆ a p x ϕ s ´ p q ˆ a ´ s ´ t ξ } { ď Cℓ α } N { ξ } ` Cℓ p α ´ β q{ } K { ℓ ´ β ´ ε ξ }} N { ξ } , where the first term on the r.h.s. arises from | t | ą ℓ ´ β ´ ε , where we can use g L p t q toextract (arbitrary polynomial) decay in ℓ . The term C “ N ż dxdydzdv N V p N p x ´ v qq ϕ p x q ν H p x ; y q ˇ g L p x ´ z q ˆ ´ N N ˙ a ˚ z a ˚ v a y a x can be handled similarly to C . We find |x ξ, C ξ y| ď Cℓ α } N { ξ } ` Cℓ p α ´ β q{ } K { ℓ ´ β ´ ε ξ }} N { ξ } . On the other hand, to estimate C “ ´ N ż dxdydzdudv N V p N p u ´ v qq ϕ p u q ν H p x ; y q ˇ g L p x ´ z q a ˚ u a ˚ z a ˚ v a x a y a u , we switch only partially to Fourier space (keeping V in position space). We obtain |x ξ, C ξ y| ď ż dudvdsdt N V p N p u ´ v qq| ˆ G p s q| χ H p s q g L p t q} ˆ a ´ t a u a v ξ }} ˆ a p ˆ ϕ s q ˆ a ´ s ´ t a u ξ }ď CN { «ż | s |ą ℓ ´ α dudvdsdt N V p N p u ´ v qq | s | ´ } ˆ a ´ t a u a v ξ } ff { ˆ „ż dudvdsdt N V p N p u ´ v qq} ˆ a p ˆ ϕ s q ˆ a ´ s ´ t a u ξ } { ď Cℓ α { } V { N ξ }} N { ξ } . The terms C “ ´ N ż dxdydzdu N V p N p u ´ x qq ϕ p u q ν H p x ; y q ˇ g L p x ´ z q a ˚ u a ˚ z a y a u C “ ´ N ż dxdydzdu N V p N p u ´ y qq ϕ p u q ν H p x ; y q ˇ g L p x ´ z q a ˚ u a ˚ z a x a u N ´ to gain arbitrary decay in ℓ . We easily find ˘ C , ˘ C ď Cℓ α p N ` q . As for C “ N ż dxdydzdv N V p N p y ´ v qq ϕ p y q ν H p x ; y q ˇ g L p x ´ z q b ˚ y b ˚ v a ˚ x a z , we stay in position space and estimate |x ξ, C ξ y| ď ż dxdydzdv N V p N p y ´ v qq| ϕ p y q|| ν H p x ; y q| ˇ g L p x ´ z q} a y a v a x ξ }} a z ξ }ď C „ż dxdydzdv N V p N p y ´ v qq ˇ g L p x ´ z q} a y a v a x ξ } { ˆ „ż dxdydzdv N V p N p y ´ v qq| ν H p x ; y q| ˇ g L p x ´ z q} a z ξ } { ď Cℓ α { } V { N ξ }} N { ξ } , using that (2.27) and (2.24). We can proceed very similarly to bound C “ N ż dxdydzdv N V p N p x ´ v qq ϕ p x q ν H p x ; y q ˇ g L p x ´ z q b ˚ x b ˚ v a ˚ y a z . We find |x ξ, C ξ y| ď Cℓ α { } V { N ξ }} N { ξ } Also C “ ´ N ż dxdydzdv N V p N p z ´ v qq ϕ p z q ν H p x ; y q ˇ g L p x ´ z q b ˚ z a ˚ v b x b y can be handled analogously, estimating |x ξ, C ξ y| ď C ? N „ż dxdydzdv N V p N p z ´ v qq| ν H p x ; y q| ˇ g L p x ´ z q} a z a v ξ } { ˆ „ż dxdydzdv N V p N p z ´ v qq ˇ g L p x ´ z q} a x a y ξ } { ď Cℓ α { } V { N ξ }} N { ξ } . To bound C “ ´ N ż dxdydzdu N V p N p u ´ z qq ϕ p u q ν H p x ; y q ˇ g L p x ´ z q b ˚ u b ˚ x a ˚ y a u , we switch partially to momentum space: |x ξ, C ξ y| ď ż dudzdsdt N V p N p u ´ z qq| ˆ G p s q| χ H p s q g L p t q} a u ˆ a p ˆ ϕ ´ s q ˆ a s ` t ξ }} a u ξ }ď CN „ż dudsdt s } a u ˆ a p ˆ ϕ ´ s q ˆ a s ` t ξ } «ż | s |ą ℓ ´ α dudsdt | s | ´ g L p t q } a u ξ } ff ď Cℓ p α ´ β q{ }p K ` N q { ξ }} N { ξ } . C “ N ż dxdydzdu N V p N p u ´ y qq ϕ p u q ν H p x ; y q ˇ g L p x ´ z q b ˚ u a ˚ z b x a u we switch partially to momentum space. We find |x ξ, C ξ y| ď ż dudydsdt N V p N p u ´ y qq| ˆ G p s q| χ H p s q g L p t q} a u ˆ a ´ t ξ }} a u ˆ a ´ s ´ t ξ }ď Cℓ α } N { ξ } ` CN «ż | t |ă ℓ ´ β ´ ε dudsdt | s | ´ t } a u ˆ a t ξ } ff ˆ «ż | s |ą ℓ ´ α dudsdt g L p t q t } a u ˆ a ´ s ´ t ξ } ff ď Cℓ α } N { ξ } ` Cℓ p α ´ β q{ } K { ℓ ´ β ´ ε ξ }} N { ξ } , where the first term on the r.h.s. arises from the region | t | ą ℓ ´ β ´ ε , where we can use g L to extract decay in ℓ . As for the term C “ N ż dxdydzdu N V p N p u ´ x qq ϕ p u q ν H p x ; y q ˇ g L p x ´ z q b ˚ u a ˚ z b y a u , it can be bounded similarly as C . We find |x ξ, C ξ y| ď Cℓ α } N { ξ } ` Cℓ p α ´ β q{ } K { ℓ ´ β ´ ε ξ }} N { ξ } . Recalling (5.1) and applying (5.3), (5.24) and (5.25), we obtain J N “ D N ` Q N ` C N ` H N ` ż ds e ´ sA r C N ` K ` V N , A s e sA ` δ , (5.33)where ˘ δ ď C ” ℓ p α ´ β ´ ε q{ ` ℓ p α ´ β ´ ε q{ ı p H N ` N ` q . Combining (5.13) with the first term on the r.h.s. of (5.30) (and recalling the defini-tion of C N in (5.2)), we obtain r K ` V N , A s“ ´ C N ` ? N ż dxdydz N V p N p x ´ y qq ϕ p y q p ´ g L qp x ´ z qr b ˚ x a ˚ y a z ` h.c. s´ ? N ż dxdydz p G ˚ q χ H c qp x ´ y q ϕ p y q ˇ g L p x ´ z q N V p N p x ´ y qqr b ˚ x a ˚ y a z ` h.c. s` πa ? N ż dxdydz q χ H c p x ´ y q ˇ g L p x ´ z q ϕ p y qr b ˚ x a ˚ y a z ` h.c. s ` δ , (5.34)61here |x ξ, δ ξ y| ď C ” ℓ p α ´ q{ ` ℓ p α ´ β q{ ı }p K ℓ ´ β ´ ε ` N ` V N ` q { ξ } ` Cℓ α { } K { ξ }}p K ℓ ´ β ´ ε ` N ` q { ξ } . Next, we observe that the second term on the r.h.s. of (5.34) can be bounded, usingthat 0 ď ´ g L p p q ď max p , ℓ β p q , by ˇˇˇ ? N ż dxdydz N V p N p x ´ y qq ϕ p y q p ´ g L qp x ´ z qx ξ, b ˚ x a ˚ y a z ξ y ˇˇˇ ď } V N ξ } „ż dx } a p p ´ g L q x ξ } “ } V N ξ } „ż dp | ´ g L p p q| } ˆ a p ξ } { ď Cℓ β } V N ξ }} K { ξ } . As for the third term on the r.h.s. of (5.34), we can first extract } G ˚ ˇ χ H c } ď } G }} χ H c } ď Cℓ ´ α and then use the factor N ´ { to gain arbitrary decay in ℓ . We obtain ˇˇˇ ? N ż dxdydz p G ˚ q χ H c qp x ´ y q ϕ p y q ˇ g L p x ´ z q N V p N p x ´ y qqx ξ, r b ˚ x a ˚ y a z ` h.c. s ξ y ˇˇˇ ď Cℓ α } V { N ξ }}p N ` q { ξ } . Finally, the fourth term on the r.h.s. of (5.34) can be bounded by ˇˇˇ π a ? N ż dxdydz ˇ χ H c p x ´ y q ˇ g L p x ´ z q ϕ p y qx ξ, r b ˚ x a ˚ y a z ` h.c. s ξ y ˇˇˇ ď C ? N „ż dxdydz ˇ g L p x ´ z q} a x a y ξ } { „ż dxdydz | ˇ χ H c p x ´ y q| ˇ g L p x ´ z q} a z ξ } { ď Cℓ ´ α { ? N } N ξ }} N { ξ } . Thus r K ` V N , A s “ ´ C N ` δ with |x ξ, δ ξ y| ď C ” ℓ p α ´ q{ ` ℓ p α ´ β q{ ı }p K ℓ ´ β ´ ε ` N ` V N ` q { ξ } ` C ” ℓ α { ` ℓ β ı } K { ξ }}p K ℓ ´ β ´ ε ` V N ` N ` q { ξ } ` Cℓ ´ α } N ξ } { N. Inserting into (5.33) and using (2.29), (5.16), (5.21), (5.22), we obtain J N “ D N ` Q N ` C N ` H N ´ ż ds e ´ sA C N e sA ` ż dse ´ sA r C N , A s e ´ sA ` δ “ D N ` Q N ` H N ` ż ds s e ´ sA r C N , A s e sA ` δ (5.35)62ith ˘ δ ď C ” ℓ p α ´ q{ ` ℓ p α ´ β ´ ε q{ ` ℓ p α ´ β ´ ε q{ ` ℓ p β ´ α ´ ε q{ ı p H N ` N ` q` Cℓ ´ α N { N. We computed the commutator r C N , A s in (5.32). To deal with the two main contributionson the r.h.s. of (5.32), we switch to momentum space. For the first term, we find1 N ż dxdydz N V p N p x ´ y qq ν H p x ; y q ϕ p y q ˇ g L p x ´ z qr a ˚ z a y ` h.c. s ˆ ´ N N ˙ “ ż dp « N ż | q |ą ℓ ´ α dq ˆ V pp p ´ q q{ N q ˆ G p q q ff g L p p q „ ˆ a ˚ p ˆ a p x ϕ p q ˆ ´ N N ˙ ` h.c. . (5.36)With (2.11) , we can estimate ˇˇˇ N ż | q |ą ℓ ´ α dq ˆ V pp p ´ q q{ N q ˆ G p q q ´ p π a ´ ˆ V p qq ˇˇˇ ď C p| p | ` ℓ ´ α q{ N. Since moreover (using 0 ď ´ g L p p q ď max p , ℓ β p q ) ż dp | ´ g L p p q||x ξ, ˆ a ˚ p ˆ a p x ϕ p qp ´ N { N q ξ y| ď ż dp | p | ℓ β } ˆ a p ξ }} ˆ a p x ϕ p q ξ } ď ℓ β } K { ξ }} N { ξ } , we conclude from (5.36), switching back to position space, that1 N ż dxdydz N V p N p x ´ y qq ν H p x ; y q ϕ p y q ˇ g L p x ´ z qr a ˚ z a y ` h.c. s ˆ ´ N N ˙ “ p π a ´ ˆ V p qq ż dp “ ˆ a p ˆ a ˚ p x ϕ p q ` h.c. s ` δ “ p π a ´ ˆ V p qq ż dx ϕ p x q a ˚ x a x ` δ , where |x ξ, δ ξ y| ď Cℓ β } K { ξ }} N { ξ } ` C } N ξ } { N. Similarly, we can also handle the second term on the r.h.s. of (5.32). We conclude that r C N , A s “ p π a ´ ˆ V p qq ż dx ϕ p x q a ˚ x a x ` δ with |x ξ, δ ξ y| ď C p ℓ β ` ℓ p α ´ β q{ q}p K ` N q { ξ }} N { ξ }` Cℓ p α ´ β q{ }p K ℓ ´ β ´ ε ` V N ` N q { ξ } ` C } N ξ } { N. Inserting in (5.35), we find, with (5.23), (2.29), (5.16), (5.21), (5.22), J N “ D N ` Q N ` H N ` p π a ´ ˆ V p qq ż dx ϕ p x q a ˚ x a x ` δ ˘ δ ď C ” ℓ p α ´ q{ ` ℓ p α ´ β ´ ε q{ ` ℓ p α ´ β ´ ε q{ ` ℓ p β ´ α ´ ε q{ ı p H N ` N ` q` Cℓ ´ α N { N. Under the assumption α ą
4, 7 β { ă α ă β , we can find ε ą κ “ min pp α ´ q{ , p α ´ β ´ ε q{ , p α ´ β ´ ε q{ , p β ´ α ´ ε q{ q ą
0. Inserting D N , Q N as in (5.2), we arrive therefore at J N ě N E GP p ϕ q ´ ε GP N ` π a ż dxdy q χ H c p x ´ y q ϕ p x q ϕ p y qr b x b y ` b ˚ x b ˚ y s` π a ż dx ϕ p x q a ˚ x a x ` p ´ Cℓ κ q H N ´ Cℓ κ p N ` q ´ Cℓ ´ α p N ` q { N . (5.37)Next, we observe that0 ď π a ż dxdy ˇ χ H c p x ´ y q ϕ p x q ϕ p y qr b x ` b ˚ x sr b ˚ y ` b y s . This implies that4 π a ż dxdy ˇ χ H c p x ´ y q ϕ p x q ϕ p y qr b x b y ` b ˚ x b ˚ y sě ´ π a ż dxdy ˇ χ H c p x ´ y q ϕ p x q ϕ p y q b ˚ x b y ´ π a ż dxdy ˇ χ H c p x ´ y q ϕ p x q ϕ p y qr b x , b ˚ y sě ´ π a ż dxdy ˇ χ H c p x ´ y q ϕ p x q ϕ p y q a ˚ x a y ´ Cℓ ´ α ´ C N { N, where in the last step we used the commutation relations (2.3) (and we replaced b ˚ x , b y by a ˚ x , a y ). Since, switching to momentum space, ż dxdy ˇ χ H p x ´ y q ϕ p x q ϕ p y qx ξ, a ˚ x a y ξ y“ ż dpχ H p p q} ˆ a p ˆ ϕ p q ξ } ď ℓ α ż dp p } ˆ a p ˆ ϕ p q ξ } ď ℓ α }p K ` N q { ξ } , with (5.7), we conclude that4 π a ż dxdy ˇ χ H c p x ´ y q ϕ p x q ϕ p y qr b x b y ` b ˚ x b ˚ y sě ´ π a ż dx ϕ p x q a ˚ x a x ´ Cℓ ´ α ´ C N { N ´ Cℓ α p K ` N q . Inserting in (5.37), we arrive at J N ě N E GP p ϕ q ` p ´ Cℓ κ q d Γ p´ ∆ ` V ext ` π a | ϕ p x q| ´ ε GP q´ Cℓ κ N ´ Cℓ ´ α ´ Cℓ ´ α N { N, which implies (2.31), if ℓ ą Properties of the Gross-Pitaevskii Functional
In this appendix we collect several well-known results about the Gross-Pitaevskii func-tional E GP , defined in equation (1.5). Let us recall that E GP : D GP Ñ R is given by E GP p ϕ q “ ż R ` | ∇ ϕ p x q| ` V ext p x q| ϕ p x q| ` πa | ϕ p x q| ˘ dx with domain D GP “ ϕ P H p R q X L p R q : V ext | ϕ | P L p R q ( . Recall, moreover, assumption p q in Eq. (1.8) on the external potential V ext . Thefollowing was proved in [14, Theorems 2.1, 2.5 & Lemma A.6]. Theorem A.1.
There exists a minimizer ϕ P D GP with } ϕ } “ such that inf ψ P D GP : } ψ } “ E GP p ψ q “ E GP p ϕ q . The minimizer ϕ is unique up to a complex phase, which can be chosen so that ϕ isstrictly positive. Furthermore, the minimizer ϕ solves the Gross-Pitaevskii equation ´ ∆ ϕ ` V ext ϕ ` πa | ϕ | ϕ “ ε GP ϕ, (A.1) with µ given by ε GP “ E GP p ϕ q ` πa } ϕ } . Moreover, ϕ P L p R q X C p R q and for every ν ą there exists C ν (which onlydepends on ν and a ) such that for all x P R it holds true that | ϕ p x q| ď C ν e ´ ν | x | . (A.2)We denote by ϕ in the following the unique, strictly positive minimizer of E GP ,subject to the contraint } ϕ } “
1. In addition to Theorem A.1, we need to collect a fewadditional facts about the regularity of ϕ . Before we do so, notice that the assumption V ext p x ` y q ď C p V ext p x q ` C qp V ext p y q ` C q implies that V ext has at most exponential growth, as | x | Ñ 8 . Indeed, by (1.8), we find R ą V ext p x q ą | x | ě R . Let ˜ C be the maximum of V ext in the ball ofradius 2 R around the origin. For | x | ě R , we pick n P N such that nR ď | x | ă p n ` q R and obtain | V ext p x q| ď C n p V ext p x { n q ` C q n ď p C p ˜ C ` C qq n ď p C p ˜ C ` C qq | x |{ R . Hence, V ext grows at most exponentially. In particular, by (A.2), this implies that } V ext ϕ } ď C. (A.3)65 emma A.2. Let V ext satisfy the assumptions in (1.8) . Then ϕ P H p R q X C p R q and for every ν ą there exists C ν ą such that for every x P R we have | ∇ ϕ p x q| , | ∆ ϕ p x q| ď C ν e ´ ν | x | . (A.4) Moreover, if p ϕ denotes the Fourier transform of ϕ , we have for all p P R that | p ϕ p p q| ď C p ` | p |q . (A.5) Proof.
By the previous Theorem A.1, the Gross-Pitaevskii equation (A.1), Eq. (A.2) andthe fact that V ext , ∇ V ext grow at most exponentially (by the assumptions (1.8) and theprevious remark), we obtain the exponential decay of ∆ ϕ . Moreover, since ϕ P L p R q and local H¨older continuity of V ext , we get the local H¨older continuity of ∆ ϕ . Ellipticregularity then implies that ϕ P C p R , R q .Next, by [9, Theorem 3.9], if u P C p B p y qq solves ∆ u “ f , then there exists aconstant C ą
0, independent of y P R , such that } ∇ u } L p B p y qq ď C ` } u } L p B p y qq ` } ∆ f } L p B p y qq ˘ . Here, B p y q and B p y q denote the open balls of radius one and two, respectively, centeredat y P R . Applying this last bound to ϕ and using the exponential decay of ϕ, ∆ ϕ impliesthat also ∇ ϕ has exponential decay.Finally, let us prove the decay estimate (A.5). By Theorem (A.1), Eq. (A.1) andelliptic regularity theory, we conclude ϕ P C p R q and that∆ ϕ “ ´ ∆ p ϕV ext q ` ε GP ∆ ϕ ` πa p ϕ ∆ ϕ ` ϕ | ∇ ϕ | q . Since, on the one hand, ϕ, ∇ ϕ and ∆ ϕ all have exponential decay with arbitrary ratewhile V ext , ∇ V ext , ∆ V ext grow at most exponentially by assumption (1.8), we concludethat ∆ ϕ P L p R q . This implies the estimate (A.5) by switching to Fourier space. References [1] C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein. Complete Bose-Einstein con-densation in the Gross-Pitaevskii regime.
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