The excitation spectrum of Bose gases interacting through singular potentials
Chiara Boccato, Christian Brennecke, Serena Cenatiempo, Benjamin Schlein
aa r X i v : . [ m a t h - ph ] M a y The Excitation Spectrum of Bose GasesInteracting Through Singular Potentials
Chiara Boccato , Christian Brennecke , Serena Cenatiempo , Benjamin Schlein Institute of Mathematics, University of ZurichWinterthurerstrasse 190, 8057 Zurich, Switzerland Gran Sasso Science Institute, Viale Francesco Crispi 767100 L’Aquila, Italy May 18, 2017
Abstract
We consider systems of N bosons in a box with volume one, interacting througha repulsive two-body potential of the form κN β − V ( N β x ). For all 0 < β <
1, andfor sufficiently small coupling constant κ >
0, we establish the validity of Bogoliubovtheory, identifying the ground state energy and the low-lying excitation spectrumup to errors that vanish in the limit of large N . We consider systems of N bosons in the three dimensional box Λ = [ − /
2; 1 / × withperiodic boundary conditions. The Hamilton operator is given by H βN = N X j =1 − ∆ x j + κN N X i
1, where the range of the interaction potential shrinks tozero, as N → ∞ . This is the content of our main theorem. Theorem 1.1.
Let V ∈ L ( R ) be non-negative, spherically symmetric, compactly sup-ported and assume that the coupling constant κ > is small enough. Fix < β < andlet m β ∈ N be the largest integer with m β ≤ / (1 − β ) + min(1 / , β/ (1 − β )) . Then, inthe limit N → ∞ , the ground state energy E βN of the Hamilton operator H βN defined in E βN = 4 π ( N − a βN − X p ∈ Λ ∗ + " p + κ b V (0) − q | p | + 2 p κ b V (0) − κ b V (0)2 p + O ( N − α ) (1.4) for all < α < β such that α ≤ (1 − β ) / . Here we set Λ ∗ + = 2 π Z \{ } and we defined πa βN = κ b V (0) − N X p ∈ Λ ∗ + κ b V ( p/N β ) p + m β X k =2 ( − k (2 N ) k X p ∈ Λ ∗ + κ b V ( p/N β ) p × X q ,q ,...,q k − ∈ Λ ∗ + κ b V (( p − q ) /N β ) q k − Y i =1 κ b V (( q i − q i +1 ) /N β ) q i +1 ! κ b V ( q k − /N β )(1.5) Moreover, the spectrum of H βN − E βN below an energy ζ consists of eigenvalues given, inthe limit N → ∞ , by X p ∈ Λ ∗ + n p q | p | + 2 p κ b V (0) + O ( N − α (1 + ζ )) (1.6) for all < α < β such that α ≤ (1 − β ) / . Here n p ∈ N for all p ∈ Λ ∗ + and n p = 0 forfinitely many p ∈ Λ ∗ + only.Remarks:
1) The sum over p ∈ Λ ∗ + appearing on the r.h.s. of (1.4) converges (a careful analysisshows that the expression in the parenthesis decays as | p | − for large | p | ). It givestherefore a contribution of order one to the ground state energy E βN .2) The r.h.s. of (1.5) is N times a Born series expansion for the scattering length of thepotential κN β − V ( N β . ). A simple computation shows that the k -th term in thesum on the r.h.s. of (1.5) (including the term on the first line, which is associatedwith k = 1) is of the order N k ( β − , for all k ∈ N . Hence, it gives a contribution tothe ground state energy (1.4) of the order N kβ − ( k − which is negligible (vanishes,as N → ∞ ) if β < ( k − /k or, equivalently, if k > / (1 − β ). The truncation ofthe Born series at k = m β ≃ / (1 − β ) + min(1 / , β/ (1 − β )) is chosen to make surethat the error we do in neglecting terms with k > m β is of the order O ( N − α ), forall α ≤ min( β, (1 − β ) /
2) (and therefore it is negligible, compared to other errorsarising in the analysis). Notice that in the Gross-Pitaevskii regime β = 1, which3s not covered by Theorem 1.1, the situation is different. In this case, all terms inthe Born series are of order one; their sum reconstructs the scattering length a ofthe unscaled potential V . In this sense, (1.4) is consistent with the results of Lieband Yngvason in [16] and of Lieb, Seiringer and Yngvason in [14] which imply that E β =1 N = 4 πa N + o ( N ).3) As explained in [3, 13, 8], the validity of Bogoliubov’s approximation is restricted toregimes where the ratio µ = a/R between the scattering length a of the interactionand its range R is such that 1 ≫ µ ≫ p ρa ≫ µ (1.7)For the trapped gas described by (1.1), we have a ≃ N − , R ≃ N − β and ρ = N .Hence, (1.7) is only satisfied if β < /
2. In other words, for 1 / ≤ β <
1, Theorem1.1 establishes the validity of the predictions of Bogoliubov’s theory in regimeswhere Bogoliubov approximation fails (in fact, Bogoliubov theory is expected tohold in any dilute limit, with ρa ≪ ψ ∈ L s (Λ N ) can be represented uniquely as ψ = N X n =0 ψ ( n ) ⊗ s ϕ ⊗ ( N − n )0 for a sequence ψ n ∈ L ⊥ (Λ) ⊗ s n , where L ⊥ (Λ) is the orthogonal complement of ϕ ( x ) = 1in L (Λ) and L ⊥ (Λ) ⊗ s n is the symmetric tensor product of n copies of L ⊥ (Λ). Thisremark allows us to define a unitary map U : L s (Λ N ) → F ≤ N + = N M n =0 L ⊥ (Λ) ⊗ s n (1.8)4hrough U ψ = { ψ (0) , . . . , ψ ( N ) } ( F ≤ N + is the Fock space over L ⊥ (Λ), truncated to excludesectors with more than N particles). The map U factors out the condensate (particlesdescribed by the wave function ϕ ) and let us focus on its orthogonal excitations. Us-ing U , we can define a first excitation Hamiltonian L βN = U H βN U ∗ : F ≤ N + → F ≤ N + .Conjugating with U , we effectively extract, from the interaction term in the originalHamiltonian (1.1), contributions to L βN that are constant (c-numbers) and quadratic increation and annihilation operators. This is very much in the spirit of the Bogoliubov ap-proximation, where creation and annihilation operators involving the condensate mode ϕ are replaced by commuting numbers.In the mean-field regime (i.e. for β = 0), conjugation with U is sufficient to extractall contributions of the many-body interaction whose expectation in low energy statessurvives, as N → ∞ . In other words, in the mean-field regime, the excitation Hamil-tonian L β =0 N can be approximated by the sum of a constant and of a term quadratic increation and annihilation operators; the expected value of all other terms vanishes, as N → ∞ , when we consider low-energy states.For β >
0, the situation is different; some of the quartic terms in L βN that werenegligible for β = 0 are now important, in the limit N → ∞ . To better understand thispoint, let us observe that h Ω , L βN Ω i = h U ∗ Ω , H βN U ∗ Ω i = h ϕ ⊗ N , H βN ϕ ⊗ N i = ( N − b V (0)2According to (1.4), the difference between h Ω , L βN Ω i and the real ground state energyof (1.1) is of the order N β and diverges, as N → ∞ . To make up for this error, wehave to take into account correlations among particles. In [1], this goal was achievedby conjugating the excitation Hamiltonian with a unitary Bogoliubov transformation ofthe form e T = exp X p ∈ π Z ,p =0 (cid:0) η p a ∗ p a ∗− p − ¯ η p a p a − p (cid:1) (1.9)for appropriate coefficients η p = η − p (the context of [1] was slightly different; it focused onthe time-evolution for approximately coherent initial data on the Fock space). In (1.9),the operators a ∗ p and a p create and, respectively, annihilate a particle with momentum p ∈ π Z (see Section 2 below for precise definitions). While Bogoliubov transformationsof the form (1.9) work well on the Fock space, they do not map the space F ≤ N + into itself(because they do not preserve the constraint N ≤ N ).To obviate this problem, we follow the strategy used in [4, 2]. We introduce modifiedcreation and annihilation operators b ∗ p , b p for all p ∈ π Z . The creation operator b ∗ p creates a particle with momentum p but, at the same time, it removes a particle withmomentum zero from the condensate. Similarly, b p annihilates a particle with momentum p but, simultaneously, it creates a particles with momentum 0 in the condensate. Hence, b ∗ p and b p create and annihilate excitations but they do not change the total numberof particles in the system. As a consequence, when transformed with U , they map the5ilbert space F ≤ N + into itself. Using these operators, we can therefore define generalizedBogoliubov transformations of the form T = exp X p ∈ π Z (cid:0) η p b ∗ p b ∗− p − ¯ η p b p b − p (cid:1) (1.10)with appropriate coefficients η p = η − p . In contrast with (1.9), generalized Bogoliubovtransformations leave the space F ≤ N + invariant. This allows us to define a modifiedexcitation Hamiltonian having the form G βN = T ∗ U H N U ∗ T : F ≤ N + → F ≤ N + .With the right definition of the coefficients η p , we can show that the modified ex-citation Hamiltonian G βN can be approximated by the sum of a constant and of a termquadratic in creation and annihilation operators. To be more precise, we first prove, aswe recently did in [2] for the case β = 1, that, for sufficiently small κ > G βN satisfiesthe lower bound G βN − E βN ≥ N + − C (1.11)where N + denotes the number of particles operator on F ≤ N + . As we will show inProp. 4.1, Eq. (1.11) easily implies that states ψ N ∈ L s (Λ N ) with bounded excitationenergy can be written as ψ N = U T ξ N , for an excitation vector ξ N ∈ F ≤ N + satisfying h ξ N , N + ξ N i ≤ C (1.12)with a constant C > N . In other words, (1.11) shows that low-energystates exhibit complete Bose-Einstein condensation in a very strong sense: the numberof excitations, that is the number of particles that are not in the condensate, remainsbounded, uniformly in N . Notice that Bose-Einstein condensation in the ground stateof the Gross-Pitaevskii Hamiltonian (i.e. (1.1) for β = 1) has been known since the work[12] of Lieb and Seiringer; the novelty of (1.12) is the fact that it gives a bound, in factan optimal bound, on the number of excitations.Combining (1.11) with the commutator estimate ± (cid:2) G βN , i N + (cid:3) ≤ C ( H βN + 1) (1.13)where H βN denotes the Hamiltonian (1.1), rewritten as an operator on the Fock space,we show then that the excitation vector ξ N ∈ F ≤ N + associated with a low-energy statealso satisfy the stronger bound h ξ N , ( N + + 1)( H βN + 1) ξ N i ≤ C (1.14)for a constant C > N . Eq. (1.14) does not only provide control on theexpectation of the number of excitations, but also on their energy. It is worth noticingthat, like for (1.12), the improved estimate (1.14) does not require the assumption β < β = 1.6quipped with the bound (1.14) we go back to the excitation Hamiltonian G βN andwe show that, in low-energy states, the expectation of all terms that are not constant orquadratic in creation and annihilation operators vanish, in the limit of large N . Moreprecisely, we prove that G βN = C βN + Q βN + E βN (1.15)where C βN is a constant (to leading order, the ground state energy of H βN ), Q βN isquadratic in creation and annihilation operators, and E βN is such that ± E βN ≤ CN − α ( N + + 1)( H βN + 1) (1.16)for all 0 < α < β such that α ≤ (1 − β ) /
2. Combining (1.16) with the bound (1.12),it follows that, on low-energy states, G βN is dominated by its quadratic part. As aconsequence, to conclude the proof of Theorem 1.1, we only have to conjugate G βN witha second generalized Bogoliubov transformation, diagonalizing the quadratic operator Q βN .It is in the proof of (1.16) that the assumption β < β = 1,in the Gross-Pitaevskii regime, the error term E βN is not small; in this case, (1.16) onlyholds with α = 0. In other words, the excitation Hamiltonian G β =1 N contains cubic andquartic contributions that remain of order one in the limit of large N . This observationis not surprising. Already in [7] and more recently in [17, 18], it has been shown thatquasi-free states can only approximate the ground state of a dilute Bose gas up to anerror of order one. For this reason, when β = 1 we cannot expect to extract all relevantterms from the Hamiltonian (1.1) by applying Bogoliubov transformations. To proveTheorem 1.1 in the Gross-Pitaevskii regime, the Hamilton operator H β =1 N must insteadbe conjugated with more complicated maps. A first partial result in this direction is theupper bound for the ground state energy obtained by Yau and Yin in [24].The paper is organized as follows. In Section 2 we introduce the formalism of secondquantization, defining generalized Bogoliubov transformations and studying their prop-erties. The main results of this section are Lemma 2.5 and Lemma 2.6 (taken from [4])where we show how to expand the action of generalized Bogoliubov transformations inabsolutely convergent infinite series. In Section 3, we define the excitation Hamiltonian G βN and we state its most important properties (namely the bounds (1.11), (1.13) and(1.16)) in Theorem 3.2, whose long and technical proof is deferred to Section 7. InSection 4, we show how (1.11) and (1.13) can be used to show the bounds (1.12) and,more importantly, (1.14) for the excitation vectors of low-energy states. In Section 5, weshow how to diagonalize the quadratic part of G βN using a second generalized Bogoliubovtransformation. Using the results of Section 4 and Section 5, we prove our main result,Theorem 1.1, in Section 6. Acknowledgement.
B.S. gratefully acknowledge support from the NCCR SwissMAPand from the Swiss National Foundation of Science through the SNF Grant “Effectiveequations from quantum dynamics” and the SNF Grant “Dynamical and energetic prop-erties of Bose-Einstein condensates”. 7
Fock space
Let F = M n ≥ L s (Λ n ) = M n ≥ L (Λ) ⊗ s n be the bosonic Fock space over L (Λ), where L s (Λ n ) is the subspace of L (Λ n ) consistingof wave functions that are symmetric w.r.t. permutations. We use the notation Ω = { , , . . . } ∈ F for the vacuum vector in F .For g ∈ L (Λ), we define the creation operator a ∗ ( g ) and the annihilation operator a ( g ) by( a ∗ ( g )Ψ) ( n ) ( x , . . . , x n ) = 1 √ n n X j =1 g ( x j )Ψ ( n − ( x , . . . , x j − , x j +1 , . . . , x n )( a ( g )Ψ) ( n ) ( x , . . . , x n ) = √ n + 1 Z Λ ¯ g ( x )Ψ ( n +1) ( x, x , . . . , x n ) dx Notice that a ∗ ( g ) is the adjoint of a ( g ) and that creation and annihilation operatorssatisfy canonical commutation relations[ a ( g ) , a ∗ ( h )] = h g, h i , [ a ( g ) , a ( h )] = [ a ∗ ( g ) , a ∗ ( h )] = 0 (2.1)for all g, h ∈ L (Λ) (here h g, h i is the usual inner product on L (Λ)).Since we consider a translation invariant system, it will be convenient to work inmomentum space. From now on, let Λ ∗ = 2 π Z . For p ∈ Λ ∗ , we define the normalizedwave function ϕ p ( x ) = e − ip · x in L (Λ). We set a ∗ p = a ∗ ( ϕ p ) , and a p = a ( ϕ p ) (2.2)In other words, a ∗ p and a p create, respectively annihilate, a particle with momentum p .In some occasions, it will also be important to switch to position space (where it iseasier to use the positivity of the potential V ( x )). For this reason, we introduce operatorvalued distributions ˇ a x , ˇ a ∗ x defined so that a ( f ) = Z ¯ f ( x ) ˇ a x dx, a ∗ ( f ) = Z f ( x ) ˇ a ∗ x dx (2.3)On F , we also introduce the number of particles operator, defined by ( N Ψ) ( n ) = n Ψ ( n ) . Notice that N = X p ∈ Λ ∗ a ∗ p a p = Z ˇ a ∗ x ˇ a x dx . It is important to observe that creation and annihilation operators are bounded by thesquare root of the number of particles operator, i.e. k a ( f )Ψ k ≤ k f kkN / Ψ k , k a ∗ ( f )Ψ k ≤ k f kk ( N + 1) / Ψ k (2.4)8or all f ∈ L (Λ).We will often deal with quadratic (and translation invariant) expressions in creationand annihilation operators. For f ∈ ℓ (Λ ∗ ), we define A ♯ ,♯ ( f ) = X p ∈ Λ ∗ f p a ♯ α p a ♯ α p (2.5)where ♯ , ♯ ∈ {· , ∗} , and where we use the notation a ♯ = a , if ♯ = · , and a ♯ = a ∗ if ♯ = ∗ .Also, α j ∈ {± } is chosen so that α = 1, if ♯ = ∗ , α = − ♯ = · , α = 1 if ♯ = · and α = − ♯ = ∗ . Notice that, in position space A ♯ ,♯ ( f ) = Z dxdy ˇ f ( x − y ) ˇ a ♯ x ˇ a ♯ y with the inverse Fourier transformˇ f ( x ) = X p ∈ Λ ∗ f p e ip · x . In the next simple lemma, taken from [4], we show how to bound quadratic operators ofthe form (2.5).
Lemma 2.1.
Let f ∈ ℓ (Λ ∗ ) and, if ♯ = · and ♯ = ∗ assume additionally that j ∈ ℓ (Λ ∗ ) . Then we have, for any Ψ ∈ F , k A ♯ ,♯ ( f )Ψ k ≤ √ k ( N + 1)Ψ k · (cid:26) k f k + k f k if ♯ = · , ♯ = ∗k f k otherwise As already explained in the introduction, we will work on certain subspaces of F .Recall that ϕ ∈ L (Λ) is the constant wave function with ϕ ( x ) = 1 for all x ∈ Λ. Wedenote by L ⊥ (Λ) the orthogonal complement of the one dimensional space spanned by ϕ in L (Λ). We define F + = M n ≥ L ⊥ (Λ) ⊗ s n . as the Fock space constructed over L ⊥ (Λ), i.e. the Fock space generated by creation andannihilation operators a ∗ p , a p , with p ∈ Λ ∗ + := 2 π Z \{ } . On F + , we denote the numberof particles operator by N + = X p ∈ Λ ∗ + a ∗ p a p We will also need a truncated version of F + . For N ∈ N , we define F ≤ N + = N M n =0 L ⊥ (Λ) ⊗ s n . On F ≤ N + , we construct modified creation and annihilation operators. For f ∈ L ⊥ (Λ),we set b ( f ) = r N − N + N a ( f ) , and b ∗ ( f ) = a ∗ ( f ) r N − N + N
9e have b ( f ) , b ∗ ( f ) : F ≤ N + → F ≤ N + . As we will discuss in the next section, the impor-tance of these fields arises from the application of the map U , defined in (1.8), since, forinstance, U a ∗ ( f ) a ( ϕ ) U ∗ = a ∗ ( f ) p N − N + = √ N b ∗ ( f ) (2.6)Based on (2.6), we can interpret b ∗ ( f ) as an operator exciting a particle from the con-densate into its orthogonal complement. Compared with the standard fields a ∗ , a , themodified operators b ∗ , b have an important advantage; they create (or annihilate) exci-tations but, at the same time, they annihilate (or create) a particle in the condensate,preserving thus the total number of particles.It is convenient to introduce modified creation and annihilation operators in momen-tum space, setting b p = r N − N + N a p , and b ∗ p = a ∗ p r N − N + N for all p ∈ Λ ∗ + and operator valued distributions in position spaceˇ b x = r N − N + N ˇ a x , and ˇ b ∗ x = ˇ a ∗ x r N − N + N for all x ∈ Λ.Modified creation and annihilation operators satisfy the commutation relations[ b p , b ∗ q ] = (cid:18) − N + N (cid:19) δ p,q − N a ∗ q a p [ b p , b q ] = [ b ∗ p , b ∗ q ] = 0 (2.7)and, in position space, [ˇ b x , ˇ b ∗ y ] = (cid:18) − N + N (cid:19) δ ( x − y ) − N ˇ a ∗ y ˇ a x [ˇ b x , ˇ b y ] = [ˇ b ∗ x , ˇ b ∗ y ] = 0 (2.8)Furthermore [ˇ b x , ˇ a ∗ y ˇ a z ] = δ ( x − y )ˇ b z , [ˇ b ∗ x , ˇ a ∗ y ˇ a z ] = − δ ( x − z )ˇ b ∗ y (2.9)It follows that [ˇ b x , N + ] = ˇ b x , [ˇ b ∗ x , N + ] = − ˇ b ∗ x and, in momentum space, [ b p , N + ] = b p ,[ b ∗ p , N + ] = − b ∗ p . With (2.4), we obtain k b ( f ) ξ k ≤ k f k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N / (cid:18) N + 1 − N + N (cid:19) / ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k b ∗ ( f ) ξ k ≤ k f k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( N + + 1) / (cid:18) N − N + N (cid:19) / ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (2.10)10or all f ∈ L ⊥ (Λ) and ξ ∈ F ≤ N + . Since N + ≤ N on F ≤ N + , b ( f ) , b ∗ ( f ) are boundedoperators with k b ( f ) k , k b ∗ ( f ) k ≤ ( N + 1) / k f k .We will consider quadratic expressions in the b -fields. As we did in (2.5), we restrictour attention to translation invariant operators. For f ∈ ℓ (Λ ∗ + ), we let B ♯ ,♯ ( f ) = X p ∈ Λ ∗ f p b ♯ α p b ♯ α p with α = 1 if ♯ = ∗ , α = − ♯ = · , α = 1 if ♯ = · and α = − ♯ = ∗ . Byconstruction, B ♯ ,♯ ( f ) : F ≤ N + → F ≤ N + . In position space, we find B ♯ ,♯ ( f ) = Z ˇ f ( x − y ) b ♯ x b ♯ y dxdy From Lemma 2.1, we obtain the following bounds.
Lemma 2.2.
Let f ∈ ℓ (Λ ∗ + ) . If ♯ = · and ♯ = ∗ , we assume additionally that f ∈ ℓ (Λ ∗ + ) . Then k B ♯ ,♯ ( f ) ξ k (cid:13)(cid:13)(cid:13) ( N + + 1) (cid:16) N +2 −N + N (cid:17) ξ (cid:13)(cid:13)(cid:13) ≤ √ · (cid:26) k f k + k f k if ♯ = · , ♯ = ∗k f k otherwisefor all ξ ∈ F ≤ N . Since N + ≤ N on F ≤ N + , the operator B ♯ ,♯ ( f ) is bounded, with k B ♯ ,♯ ( f ) k ≤ √ N (cid:26) k f k + k f k if ♯ = · , ♯ = ∗k f k otherwise We will need to consider products of several creation and annihilation operators. Inparticular, two types of monomials in creation and annihilation operators will play an im-portant role in our analysis. For f , . . . , f n ∈ ℓ (Λ ∗ + ), ♯ = ( ♯ , . . . , ♯ n ) , ♭ = ( ♭ , . . . , ♭ n − ) ∈{· , ∗} n , we setΠ (2) ♯,♭ ( f , . . . , f n )= X p ,...,p n ∈ Λ ∗ b ♭ α p a ♯ β p a ♭ α p a ♯ β p a ♭ α p . . . a ♯ n − β n − p n − a ♭ n − α n − p n b ♯ n β n p n n Y ℓ =1 f ℓ ( p ℓ ) (2.11)where, for every ℓ = 0 , , . . . , n , we set α ℓ = 1 if ♭ ℓ = ∗ , α ℓ = − ♭ ℓ = · , β ℓ = 1if ♯ ℓ = · and β ℓ = − ♯ ℓ = ∗ . In (2.11), we impose the condition that for every j = 1 , . . . , n −
1, we have either ♯ j = · and ♭ j = ∗ or ♯ j = ∗ and ♭ j = · (so that theproduct a ♯ ℓ β ℓ p ℓ a ♭ ℓ α ℓ p ℓ +1 always preserves the number of particles, for all ℓ = 1 , . . . , n − (2) ♯,♭ ( f , . . . , f n ) maps F ≤ N + into itself.If, for some ℓ = 1 , . . . , n , ♭ ℓ − = · and ♯ ℓ = ∗ (i.e. if the product a ♭ ℓ − α ℓ − p ℓ a ♯ ℓ β ℓ p ℓ for11 = 2 , . . . , n , or the product b ♭ α p a ♯ β p for ℓ = 1, is not normally ordered) we requireadditionally that f ℓ ∈ ℓ (Λ ∗ + ). In position space, the same operator can be written asΠ (2) ♯,♭ ( f , . . . , f n ) = Z ˇ b ♭ x ˇ a ♯ y ˇ a ♭ x ˇ a ♯ y ˇ a ♭ x . . . ˇ a ♯ n − y n − ˇ a ♭ n − x n ˇ b ♯ n y n n Y ℓ =1 ˇ f ℓ ( x ℓ − y ℓ ) dx ℓ dy ℓ (2.12)An operator of the form (2.11), (2.12) with all the properties listed above, will be calleda Π (2) -operator of order n .For g, f , . . . , f n ∈ ℓ (Λ ∗ + ), ♯ = ( ♯ , . . . , ♯ n ) ∈ {· , ∗} n , ♭ = ( ♭ , . . . , ♭ n ) ∈ {· , ∗} n +1 , wealso define the operatorΠ (1) ♯,♭ ( f , . . . , f n ; g )= X p ,...,p n ∈ Λ ∗ b ♭ α ,p a ♯ β p a ♭ α p a ♯ β p a ♭ α p . . . a ♯ n − β n − p n − a ♭ n − α n − p n a ♯ n β n p n a ♭n ( g ) n Y ℓ =1 f ℓ ( p ℓ )(2.13)where α ℓ and β ℓ are defined as above. Also here, we impose the condition that, forall ℓ = 1 , . . . , n , either ♯ ℓ = · and ♭ ℓ = ∗ or ♯ ℓ = ∗ and ♭ ℓ = · . This implies thatΠ (1) ♯,♭ ( f , . . . , f n ; g ) maps F ≤ N + back into F ≤ N + . Additionally, we assume that f ℓ ∈ ℓ (Λ ∗ + ),if ♭ ℓ − = · and ♯ ℓ = ∗ for some ℓ = 1 , . . . , n (i.e. if the pair a ♭ ℓ − α ℓ − p ℓ a ♯ ℓ β ℓ p ℓ is not normallyordered). In position space, the same operator can be written asΠ (1) ♯,♭ ( f , . . . , f n ; g ) = Z ˇ b ♭ x ˇ a ♯ y ˇ a ♭ x ˇ a ♯ y ˇ a ♭ x . . . ˇ a ♯ n − y n − ˇ a ♭ n − x n ˇ a ♯ n y n ˇ a ♭n ( g ) n Y ℓ =1 ˇ f ℓ ( x ℓ − y ℓ ) dx ℓ dy ℓ (2.14)An operator of the form (2.13), (2.14) will be called a Π (1) -operator of order n . Operatorsof the form b ( f ), b ∗ ( f ), for a f ∈ ℓ (Λ ∗ + ), will be called Π (1) -operators of order zero.In the next lemma we show how to bound Π (2) - and Π (1) -operators. The simpleproof, based on Lemma 2.1, can be found in [4]. Lemma 2.3.
Let n ∈ N , g, f , . . . , f n ∈ ℓ (Λ ∗ + ) . Assume that Π (2) ♯,♭ ( f , . . . , f n ) and Π (1) ♯,♭ ( f , . . . , f n ; g ) are defined as in (2.11), (2.13). Then (cid:13)(cid:13)(cid:13) Π (2) ♯,♭ ( f , . . . , f n ) ξ (cid:13)(cid:13)(cid:13) ≤ n n Y ℓ =1 K ♭ ℓ − ,♯ ℓ ℓ (cid:13)(cid:13)(cid:13)(cid:13) ( N + + 1) n (cid:18) − N + − N (cid:19) ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Π (1) ♯,♭ ( f , . . . , f n ; g ) ξ (cid:13)(cid:13)(cid:13) ≤ n k g k n Y ℓ =1 K ♭ ℓ − ,♯ ℓ ℓ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( N + + 1) n +1 / (cid:18) − N + − N (cid:19) / ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (2.15) where K ♭ ℓ − ,♯ ℓ ℓ = (cid:26) k f ℓ k + k f ℓ k if ♭ ℓ − = · and ♯ ℓ = ∗k f ℓ k otherwise ince N + ≤ N on F ≤ N + , it follows that (cid:13)(cid:13)(cid:13) Π (2) ♯,♭ ( f , . . . , f n ) (cid:13)(cid:13)(cid:13) ≤ (12 N ) n n Y ℓ =1 K ♭ ℓ − ,♯ ℓ ℓ (cid:13)(cid:13)(cid:13) Π (1) ♯,♭ ( f , . . . , f n ; g ) (cid:13)(cid:13)(cid:13) ≤ (12 N ) n √ N k g k n Y ℓ =1 K ♭ ℓ − ,♯ ℓ ℓ Next, we introduce generalized Bogoliubov transformations and we discuss theirproperties. For η ∈ ℓ (Λ ∗ + ) with η − p = η p for all p ∈ Λ ∗ + , we define B ( η ) = 12 X p ∈ Λ ∗ + (cid:0) η p b ∗ p b ∗− p − ¯ η p b p b − p (cid:1) . (2.16)and we consider e B ( η ) = exp X p ∈ Λ ∗ + (cid:0) η p b ∗ p b ∗− p − ¯ η p b p b − p (cid:1) (2.17)Notice that B ( η ) , e B ( η ) : F ≤ N + → F ≤ N + . We refer to unitary operators of the form (2.17)as generalized Bogoliubov transformations. The name arises from the observation that,on states with N + ≪ N , we have b p ≃ a p , b ∗ p ≃ a ∗ p and therefore B ( η ) ≃ e B ( η ) = 12 X p ∈ Λ ∗ + (cid:0) η p a ∗ p a ∗− p − ¯ η p a p a − p (cid:1) (2.18)Since e B ( η ) is quadratic in creation and annihilation operators, exp( e B ( η )) is a stan-dard Bogoliubov transformation, whose action on creation and annihilation operators isexplicitly given by e − e B ( η ) a p e e B ( η ) = cosh( η p ) a p + sinh( η p ) a ∗− p . (2.19)As discussed in the introduction, (2.18) does not map F ≤ N into itself. For this reason,in the following it will be convenient for us to work with generalized Bogoliubov trans-formations of the form (2.17). The price we have to pay is the fact that there is noexplicit expression like (2.19) for the action of (2.17). We need other tools to control theaction of generalized Bogoliubov transformations.A first important observation is the following lemma, whose proof can be found in[4] (a similar result was previously established in [23]). Lemma 2.4.
Let η ∈ ℓ (Λ ∗ ) and B ( η ) be defined as in (2.16). Then, for every n , n ∈ Z , there exists a constant C > (depending on k η k ) such that, on F ≤ N + , e − B ( η ) ( N + + 1) n ( N + 1 − N + ) n e B ( η ) ≤ C ( N + + 1) n ( N + 1 − N + ) n . p ∈ Λ ∗ + , e − B ( η ) b p e B ( η ) = b p + Z ds dds e − sB ( η ) b p e sB ( η ) = b p − Z ds e − sB ( η ) [ B ( η ) , b p ] e sB ( η ) = b p − [ B ( η ) , b p ] + Z ds Z s ds e − s B ( η ) [ B ( η ) , [ B ( η ) , b p ] e s B ( η ) Iterating m times, we find e − B ( η ) b p e B ( η ) = m − X n =1 ( − n ad ( n ) B ( η ) ( b p ) n !+ Z ds Z s ds · · · Z s m − ds m e − s m B ( η ) ad ( m ) B ( η ) ( b p ) e s m B ( η ) (2.20)where we recursively definedad (0) B ( η ) ( A ) = A and ad ( n ) B ( η ) ( A ) = [ B ( η ) , ad ( n − B ( η ) ( A )]We will later show that, under appropriate assumptions on η , the norm of the error termon the r.h.s. of (2.20) vanishes, as m → ∞ . Hence, the action of e B ( η ) on b p , b ∗ p can bedescribed in terms of the nested commutators ad ( n ) B ( η ) ( b p ) and ad ( n ) B ( η ) ( b ∗ p ) for n ∈ N . Inthe next lemma, we give a detailed analysis of these operators; the proof can be foundin [2, Lemma 2.5]. Lemma 2.5.
Let η ∈ ℓ (Λ ∗ + ) be such that η p = η − p for all p ∈ ℓ (Λ ∗ ) . To simplifythe notation, assume also η to be real-valued (as it will be in applications). Let B ( η ) bedefined as in (2.16), n ∈ N and p ∈ Λ ∗ . Then the nested commutator ad ( n ) B ( η ) ( b p ) can bewritten as the sum of exactly n n ! terms, with the following properties.i) Possibly up to a sign, each term has the form Λ Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η sp ϕ αp ) (2.21) for some i, k, s ∈ N , j , . . . , j k ∈ N \{ } , ♯ ∈ {· , ∗} k , ♭ ∈ {· , ∗} k +1 and α ∈ {± } chosen so that α = 1 if ♭ k = · and α = − if ♭ k = ∗ (recall here that ϕ p ( x ) = e − ip · x ).In (2.21), each operator Λ w : F ≤ N → F ≤ N , w = 1 , . . . , i , is either a factor ( N − N + ) /N , a factor ( N − ( N + − /N or an operator of the form N − h Π (2) ♯ ′ ,♭ ′ ( η z , η z , . . . , η z h ) (2.22) for some h, z , . . . , z h ∈ N \{ } , ♯, ♭ ∈ {· , ∗} h . i) If a term of the form (2.21) contains m ∈ N factors ( N − N + ) /N or ( N − ( N + − /N and j ∈ N factors of the form (2.22) with Π (2) -operators of order h , . . . , h j ∈ N \{ } , then we have m + ( h + 1) + · · · + ( h j + 1) + ( k + 1) = n + 1 (2.23) iii) If a term of the form (2.21) contains (considering all Λ -operators and the Π (1) -operator) the arguments η i , . . . , η i m and the factor η sp for some m, s ∈ N , and i , . . . , i m ∈ N \{ } , then i + · · · + i m + s = n. iv) There is exactly one term having of the form (2.21) with k = 0 and such that all Λ -operators are factors of ( N − N + ) /N or of ( N + 1 − N + ) /N . It is given by (cid:18) N − N + N (cid:19) n/ (cid:18) N + 1 − N + N (cid:19) n/ η np b p (2.24) if n is even, and by − (cid:18) N − N + N (cid:19) ( n +1) / (cid:18) N + 1 − N + N (cid:19) ( n − / η np b ∗− p (2.25) if n is odd.v) If the Π (1) -operator in (2.21) is of order k ∈ N \{ } , it has either the form X p ,...,p k b ♭ α p k − Y i =1 a ♯ i β i p i a ♭ i α i p i +1 a ∗− p k η rp a p k Y i =1 η j i p i or the form X p ,...,p k b ♭ α p k − Y i =1 a ♯ i β i p i a ♭ i α i p i +1 a p k η r +1 p a ∗ p k Y i =1 η j i p i for some r ∈ N , j , . . . , j k ∈ N \{ } . If it is of order k = 0 , then it is either givenby η rp b p or by η r +1 p b ∗− p , for some r ∈ N .vi) For every non-normally ordered term of the form X q ∈ Λ ∗ η iq a q a ∗ q , X q ∈ Λ ∗ η iq b q a ∗ q X q ∈ Λ ∗ η iq a q b ∗ q , or X q ∈ Λ ∗ η iq b q b ∗ q appearing either in the Λ -operators or in the Π (1) -operator in (2.21), we have i ≥ . ℓ -norm k η k is small enough. The proof of the next Lemma is a simple adaptationof the proof of [4, Lemma 3.3] Lemma 2.6.
Let η ∈ ℓ (Λ ∗ ) be symmetric, with k η k sufficiently small. Then we have e − B ( η ) b p e B ( η ) = ∞ X n =0 ( − n n ! ad ( n ) B ( η ) ( b p ) e − B ( η ) b ∗ p e B ( η ) = ∞ X n =0 ( − n n ! ad ( n ) B ( η ) ( b ∗ p ) (2.26) and the series on the r.h.s. are absolutely convergent. We define the unitary operator U : L s (Λ N ) → F ≤ N + as in (1.8). In terms of creation andannihilation operators, the map U is given by U ψ = N M n =0 (1 − | ϕ ih ϕ | ) ⊗ n a ( ϕ ) N − n p ( N − n )! ψ for all ψ ∈ L s (Λ N ) (here we identify ψ ∈ L s (Λ N ) with the vector { , . . . , , ψ, , . . . } ∈F ). The map U ∗ : F ≤ N + → L s (Λ N ) is given, on the other hand, by U ∗ { ψ (0) , . . . , ψ ( N ) } = N X n =0 a ∗ ( ϕ ) N − n p ( N − n )! ψ ( n ) It is useful to compute the action of U on the product of a creation and an annihilationoperators. We find (see [11]): U a ∗ a U ∗ = N − N + U a ∗ p a U ∗ = a ∗ p p N − N + U a ∗ a p U ∗ = p N − N + a p U a ∗ p a q U ∗ = a ∗ p a q (3.1)for all p, q ∈ Λ ∗ + = 2 π Z \{ } . Writing (1.1) in momentum space, we find H βN = X p ∈ Λ ∗ p a ∗ p a p + κN X p,q,r ∈ Λ ∗ b V ( r/N β ) a ∗ p a ∗ q a q − r a p + r where b V ( q ) = Z R V ( x ) e − iq · x dx V , for all q ∈ R . With (3.1), we can conjugate H βN with the map U , defining L βN = U H βN U ∗ : F ≤ N + → F ≤ N + . We find L βN = L (0) β,N + L (2) β,N + L (3) β,N + L (4) β,N (3.2)with L (0) N,β = N − N b V (0)( N − N + ) + b V (0)2 N N + ( N − N + ) L (2) N,β = X p ∈ Λ ∗ + p a ∗ p a p + X p ∈ Λ ∗ + b V ( p/N β ) (cid:20) b ∗ p b p − N a ∗ p a p (cid:21) + 12 X p ∈ Λ ∗ + b V ( p/N β ) (cid:2) b ∗ p b ∗− p + b p b − p (cid:3) L (3) N,β = 1 √ N X p,q ∈ Λ ∗ + : p + q =0 b V ( p/N β ) (cid:2) b ∗ p + q a ∗− p a q + a ∗ q a − p b p + q (cid:3) L (4) N,β = 12 N X p,q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − p, − q b V ( r/N β ) a ∗ p + r a ∗ q a p a q + r (3.3)As explained in the introduction, for β = 0, conjugation with U does not yet pull allimportant contributions for low-energy states into the constant and the quadratic partsof the excitation Hamiltonian L βN . In other words, in contrast to the mean-field case β = 0, for β > L (3) N,β and L (4) N,β to be small on low-energy states,in the limit N → ∞ . For this reason, we need to conjugate L βN with an appropriategeneralized Bogoliubov transformation of the form (2.17).To choose the function η ∈ ℓ (Λ ∗ + ) entering (2.16) and (2.17), we fix a length 0 <ℓ < /
2, independently of N , and we consider the solution of the Neumann problem (cid:16) − ∆ + κ N β − V ( N β x ) (cid:17) f N,ℓ ( x ) = λ N,ℓ f N,ℓ ( x ) (3.4)on the ball | x | ≤ ℓ , with radial derivative ∂ | x | f N,ℓ ( x ) = 0 and with the normalization f N,ℓ ( x ) = 1 for | x | = ℓ (we omit the β -dependence in the notation for f N,ℓ and for λ N,ℓ ).The condition ℓ < / ℓ is contained in Λ. We extendthen f N,ℓ to Λ, by setting f N,ℓ ( x ) = 1 for all | x | > ℓ . Then, for all x ∈ Λ, we have (cid:16) − ∆ + κ N β − V ( N β x ) (cid:17) f N,ℓ ( x ) = λ N,ℓ f N,ℓ ( x ) χ ℓ ( x ) (3.5)where χ ℓ is the characteristic function of the ball of radius ℓ . It is also useful to define w N,ℓ = 1 − f N,ℓ (so that w N,ℓ ( x ) = 0 if | x | > ℓ ). Since w N,ℓ is compactly supported onΛ, it can be interpreted as a periodic function. Its Fourier coefficients are given by b w N,ℓ ( p ) = Z Λ w N,ℓ ( x ) e − ip · x dx p ∈ Λ ∗ . From (3.5), we find that − p b w N,ℓ ( p ) + κ N b V ( p/N β ) − κ N X q ∈ Λ ∗ b V (( p − q ) /N β ) b w N,ℓ ( q )= λ N,ℓ b χ ℓ ( p ) − λ N,ℓ X q ∈ Λ ∗ b χ ℓ ( p − q ) b w N,ℓ ( q ) (3.6)for all p ∈ Λ ∗ . In the next lemma we collect some important properties of λ N,ℓ and ofthe functions w N,ℓ , f
N,ℓ ; the proof can be found in [6, Lemma A.1] and in [4] (notice thatthis lemma is the reason we require V ∈ L ( R ); for the rest of the analysis, V ∈ L ( R )would suffice). Lemma 3.1.
Let V ∈ L ( R ) be non-negative, compactly supported and sphericallysymmetric. Fix < ℓ < / and let f N,ℓ denote the ground state solution of the Neumannproblem (3.4) .i) We have λ N,ℓ = 3 κ b V (0)8 πN ℓ (cid:16) O ( N β − ) (cid:17) ii) We have ≤ f N,ℓ , w
N,ℓ ≤ .iii) There exists a constant C > such that w N,ℓ ( x ) ≤ CκN ( | x | + N − β ) and |∇ w N,ℓ ( x ) | ≤ CκN ( x + N − β ) . (3.7) for all | x | ≤ ℓ . As a result Z Λ w N,ℓ ( x ) dx ≤ Cκℓ N iv) There exists a constant C > such that | b w N,ℓ ( p ) | ≤ CκN p for all p ∈ Λ ∗ + . Using the function w N,ℓ = 1 − f N,ℓ defined above, we define η : Λ ∗ + → R through η p = − N b w N,ℓ ( p ) (3.8)From Lemma 3.1, it follows that | η p | ≤ Cκp (3.9)18ence η ∈ ℓ (Λ ∗ + ), uniformly in N . With Lemma 3.1 (part iii)), we also obtain X p ∈ Λ ∗ + p | η p | = k∇ ˇ η k ≤ CN β κ (3.10)Sometimes, it is useful to define η also at the point p = 0. We set e η ( p ) = − N b w ℓ ( p ) forall p ∈ Λ ∗ . Then e η p = η p for all p = 0. By Lemma 3.1, part iii), we find | e η | ≤ N Z Λ w N,ℓ ( x ) dx ≤ Cκℓ (3.11)From (3.6), we obtain the following relation for the coefficients e η : p e η p + κ b V ( p/N β ) + κ N X q ∈ Λ ∗ b V (( p − q ) /N β ) e η q = N λ
N,ℓ b χ ℓ ( p ) + λ N,ℓ X q ∈ Λ ∗ b χ ℓ ( p − q ) e η q (3.12)With η ∈ ℓ (Λ ∗ + ), we construct, as in (2.17), the generalized Bogoliubov transformation e B ( η ) : F ≤ N + → F ≤ N + . Furthermore, we define the excitation Hamiltonian G βN : F ≤ N + →F ≤ N + by setting (recall the definition (3.2) of the operator L βN ) G βN = e − B ( η ) L βN e B ( η ) = e − B ( η ) U H βN U ∗ e B ( η ) (3.13)In the next theorem, we collect important properties of the self-adjoint operator G βN . Wewill use the notation K = X p ∈ Λ ∗ + p a ∗ p a p , and V N = κ N X p,q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − p, − q b V ( r/N β ) a ∗ p + r a ∗ q a p a q + r for the kinetic and potential energy operators on the excitation Fock space F ≤ N + . Wealso define H βN = K + V N . Theorem 3.2.
Let V ∈ L ( R ) be non-negative, compactly supported and sphericallysymmetric and assume that the coupling constant κ ≥ is small enough.a) Let E βN denote the ground state energy of the Hamilton operator (1.1). There existsa constant C > such that G βN − E βN ≥ H βN − C (3.14) and ± h i G βN , N + i ≤ C ( H βN + 1) (3.15)19 ) For p ∈ Λ ∗ + , we set σ p = sinh( η p ) and γ p = cosh( η p ) . Let C βN = ( N − κ b V (0)+ X p ∈ Λ ∗ + h p σ p + κ b V ( p/N β )( σ p + σ p γ p ) + κ N X q ∈ Λ ∗ + b V (( p − q ) /N β ) η p η q i (3.16) Moreover, for every p ∈ Λ ∗ + , we define F p = p ( σ p + γ p ) + κ b V ( p/N β )( σ p + γ p ) G p = 2 p σ p γ p + κ b V ( p/N β )( σ p + γ p ) + κN X q ∈ Λ ∗ b V (( p − q ) /N β ) e η q (3.17) We use the coefficients F p , G p to construct the operator Q βN = X p ∈ Λ ∗ + h F p b ∗ p b p + 12 G p ( b ∗ p b ∗− p + b p b − p ) i quadratic in the b, b ∗ -fields. We define the self-adjoint operator E βN through theidentity G βN = C βN + Q βN + E βN Then there exists a constant C such that, on F ≤ N + , ± E βN ≤ CN ( β − / ( N + + 1)( K + 1) (3.18)In the last term in the definition of G p , recall that e η q = − N b w N,ℓ ( q ) coincides with η q for all q = 0 (we find it more convenient to include the contribution with q = 0 in thedefinition of G p ). The proof of Theorem 3.2 represents the main technical part of ourpaper. It is deferred to Section 7 below. In the next three sections, on the other hand,we show how to use the statement of Theorem 3.2 to complete the proof of Theorem 1.1. In this section, we establish important bounds for excitation vectors of the form ξ N = e − B ( η ) U ψ N ∈ F ≤ N + associated with low energy states ψ N ∈ L s (Λ N ). We begin with asimple application of the bound (3.14) in Theorem 3.2. Proposition 4.1.
Let V ∈ L ( R ) be non-negative, compactly supported and sphericallysymmetric and assume that the coupling constant κ ≥ is small enough. Let E βN be theground state energy of the Hamilton operator (1.1). Let ψ N ∈ L s (Λ N ) be a normalizedwave function, with h ψ N , H βN ψ N i ≤ E βN + ζ or some ζ > . Let ξ N = e − B ( η ) U ψ N be the excitation vector associated with ψ N (sothat ψ N = U ∗ e B ( η ) ξ N ). Then there exists a constant C > such that h ξ N , N + ξ N i ≤ C (1 + ζ ) h ξ N , H βN ξ N i ≤ C (1 + ζ ) (4.1) Proof.
Since, on F ≤ N + , N + ≤ (2 π ) − K ≤ (2 π ) − H βN , it is enough to show the secondbound in (4.1). From (3.14), we find h ξ N , H βN ξ N i ≤ C + 2 h ξ N , ( G βN − E βN ) ξ N i = C + 2 h h ξ N , e − B ( η ) U H N U ∗ e B ( η ) ξ N i − E βN i = C + 2 h h ψ N , H N ψ N i − E βN i ≤ C (1 + ζ )To control the expectation of the error term in (3.18), we need stronger estimates onexcitation vectors associated with low-energy states. We prove the required bounds inthe next proposition, combining (3.14) with the commutator estimate (3.15). We remarkthat the proposition also holds with the same proof in the case β = 1. Proposition 4.2.
Let V ∈ L ( R ) be non-negative, compactly supported and sphericallysymmetric and assume that the coupling constant κ ≥ is small enough. Let E βN be theground state energy of the Hamilton operator (1.1). Let ψ N ∈ L s (Λ N ) with k ψ N k = 1 belong to the spectral subspace of the Hamiltonian (1.1), with energies below E βN + ζ , forsome ζ > . In other words, assume that ψ N = ( −∞ ; E βN + ζ ] ( H βN ) ψ N Let ξ N = e − B ( η ) U ψ N be the excitation vector associated with ψ N . Then there exists aconstant C > such that h ξ N , ( N + + 1)( K + 1) ξ N i ≤ h ξ N , ( N + + 1)( H βN + 1) ξ N i ≤ C (1 + ζ ) Proof.
The first inequality follows from V N ≥ K , V N both commute with N + . We focus on the second inequality. From (3.14), we find h ξ N , ( N + + 1)( H βN + 1) ξ N i = h ξ N ( N + + 1) / ( H βN + 1)( N + + 1) / ξ N i≤ h ξ N , ( N + + 1) / ( e G βN + C )( N + + 1) / ξ N i where we introduced the notation e G βN = G βN − E βN . Next, we commute the operator( e G βN + C ) to the right, through the factor ( N + + 1) / . We obtain h ξ N , ( N + + 1)( H βN + 1) ξ N i ≤ h ξ N , ( N + + 1)( e G βN + C ) ξ N i + 2 D ξ N , ( N + + 1) / h G βN , ( N + + 1) / i ξ N E (4.2)21ith Cauchy-Schwarz, the first term on the r.h.s. of (4.2) can be estimated by (cid:12)(cid:12)(cid:12) h ξ N , ( N + + 1)( e G βN + C ) ξ N i (cid:12)(cid:12)(cid:12) ≤ h ξ N , ( N + + 1)( e G βN + C ) − ( N + + 1) ξ N i / h ξ N , ( e G βN + C ) ξ N i / Since, by (3.14), ( e G βN + C ) ≥ c ( N + + 1) for some c > C > ξ N = e − B ( η ) U ψ N is in the spectral subspace of e G βN , associated with the interval[0; ζ ], we conclude that (cid:12)(cid:12)(cid:12) h ξ N , ( N + + 1)( e G βN + C ) ξ N i (cid:12)(cid:12)(cid:12) ≤ h ξ N , ( N + + 1) ξ N i / ( ζ + C ) / ≤ C (1 + ζ ) (4.3)where we used Prop. 4.1.As for the commutator term on the r.h.s. of (4.2), we use the representation1 √ z = 1 π Z ∞ √ t t + z dt . We find i h G βN , ( N + + 1) / i = 1 π Z ∞ dt √ t t + N + + 1 i [ G βN , N + ] 1 t + N + + 1= 1 π Z ∞ dt √ t t + N + + 1 ( H βN + 1) / A ( H βN + 1) / t + N + + 1where we defined the operator A = ( H βN + 1) − / i [ G βN , N + ]( H βN + 1) − / . It follows from(3.15) that A is a bounded operator, with norm kAk ≤ C , uniformly in N . Hence, wehave (since [ H βN , N + ] = 0) (cid:12)(cid:12)(cid:12) h ξ N , ( N + + 1) / [ G βN , ( N + + 1) / ] ξ N i (cid:12)(cid:12)(cid:12) ≤ π Z ∞ dt √ t (cid:12)(cid:12)(cid:12)D ξ N , ( N + + 1) / ( H βN + 1) / t + N + + 1 A ( H βN + 1) / t + N + + 1 ξ N E(cid:12)(cid:12)(cid:12) ≤ π Z ∞ dt √ t t + 1) (cid:13)(cid:13)(cid:13) ( N + + 1) / ( H βN + 1) / ξ N (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ( H βN + 1) / ξ N (cid:13)(cid:13)(cid:13) Therefore, for every δ >
C > (cid:12)(cid:12)(cid:12) h ξ N , ( N + + 1) / [ G βN , ( N + + 1) / ] ξ N i (cid:12)(cid:12)(cid:12) ≤ δ h ξ N , ( N + + 1)( H βN + 1) ξ N i + C h ξ N , ( H βN + 1) ξ N i≤ δ h ξ N , ( N + + 1)( H βN + 1) ξ N i + C (1 + ζ )by Prop. 4.1. Choosing δ = 1 /
2, we conclude from (4.2) and (4.3) that h ξ N , ( N + + 1)( H βN + 1) ξ N i ≤ C (1 + ζ )22 Diagonalization of quadratic Hamiltonian
From Theorem 3.2, we recall that the excitation Hamiltonian G βN = e − B ( η ) U H N U ∗ e B ( η ) can be decomposed as G βN = C βN + Q βN + E βN (5.1)with the constant C βN defined in (3.16), the quadratic part Q βN = X p ∈ Λ ∗ + h F p b ∗ p b p + 12 G p ( b ∗ p b ∗− p + b p b − p ) i (5.2)with the coefficients F p , G p defined in (3.17) and with the error term E βN satisfying ± E βN ≤ CN ( β − / ( N + + 1)( K + 1) , (5.3)The goal of this section is to diagonalize the quadratic operator (5.2). To this end,we will conjugate the excitation Hamiltonian G βN with one more generalized Bogoliubovtransformation.In order to define the Bogoliubov transformation that is going to diagonalize Q βN weneed, first of all, to establish some properties of the coefficients F p , G p defined in (3.17). Lemma 5.1.
Let V ∈ L ( R ) be non-negative, compactly supported and sphericallysymmetric. If the coupling constant κ ≥ is small enough, we find a constant C > such that p / ≤ F p ≤ C (1 + p ) , | G p | ≤ Cκp (5.4) and | G p | F p ≤ C | p | ≤
12 (5.5) for all p ∈ Λ ∗ + .Proof. Since σ p + γ p ≥
1, and since there is a constant
C > | b V ( p/N β ) | ≤ C and | σ p | , γ p ≤ C for all p ∈ Λ ∗ + (using the boundedness (3.9) of the coefficients η p ), weeasily find that F p ≥ p − Cκ ≥ p /
2, if κ > | p | > (2 π ) onΛ ∗ + ). To bound G p , we write G p = 2 p η p + κ b V ( p/N β ) + κN X q ∈ Λ ∗ b V (( p − q ) /N β ) e η q + e G N,p (5.6)where e G N,p is such that | e G N,p | ≤
Cκp − for all p ∈ Λ ∗ + . Here we used the fact that | σ p γ p − η p | = | sinh( η p ) cosh( η p ) − η p | = (cid:12)(cid:12)(cid:12)(cid:12)
12 sinh(2 η p ) − η p (cid:12)(cid:12)(cid:12)(cid:12) ≤ X n ≥ n +1 | η p | n +1 (2 n + 1)! ≤ Cκ | p | | ( σ p + γ p ) − | ≤ Cκp To estimate the other terms in (5.6), we use the relation (3.12). We obtain that G p = 2 N λ
N,ℓ b χ ℓ ( p ) + 2 λ N,ℓ X q ∈ Λ ∗ b χ ℓ ( p − q ) e η q + e G N,p (5.7)From Lemma 3.1, part i), we have
N λ
N,ℓ ≤ Cκ . A simple computation shows that b χ ℓ ( p ) = Z | x |≤ ℓ e − ip · x dx = 4 π | p | (cid:18) sin( ℓ | p | ) | p | − ℓ cos( ℓ | p | ) (cid:19) (5.8)which, in particular, implies that | b χ ℓ ( p ) | ≤ C | p | − . Similarly, we find λ N X q ∈ Λ ∗ b χ ℓ ( p − q ) e η q = N λ
N,ℓ Z Λ χ ℓ ( x ) w N,ℓ ( x ) e − ip · x dx = N λ
N,ℓ Z | x |≤ ℓ w N,ℓ ( x ) e − ip · x dx Switching to spherical coordinates and integrating by parts, we find (abusing slightlythe notation by writing w N,ℓ ( r ) to indicate w N,ℓ ( x ) for | x | = r ), Z | x |≤ ℓ w N,ℓ ( x ) e − ip · x dx = 2 π Z ℓ dr r w N,ℓ ( r ) Z π dθ sin θ e − i | p | r cos θ = 4 π | p | Z ℓ dr rw N,ℓ ( r ) sin( | p | r )= − π | p | lim r → rw N,ℓ ( r ) + 4 π | p | Z ℓ dr ddr ( rw N,ℓ ( r )) cos( | p | r )With (3.7) and using again the bound N λ
N,ℓ ≤ Cκ , we conclude that there is a constant C > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ N X q ∈ Λ ∗ b χ ℓ ( p − q ) e η q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cκp (5.9)for all p ∈ Λ ∗ + . From (5.7), we obtain that there is C > | G p | ≤ Cκ/p .Together with the estimate | F p | ≥ p /
2, we find the desired bound, choosing κ > | G p | /F p ≤ / p ∈ Λ ∗ + , we can define asequence τ p by setting tanh(2 τ p ) = − G p F p for all p ∈ Λ ∗ + . Equivalently, τ p = 14 log 1 − G p /F p G p /F p (5.10)24his easily implies that | τ p | ≤ C | G p | F p ≤ Cκ | p | (5.11)for all p ∈ Λ ∗ + . Let us stress the fact that the fast decay of τ for large momenta (whichwill be crucial below) is a consequence of the fact that the coefficients η p satisfy therelation (3.12).We use the coefficients τ p (which are, by definition, real) to define a new generalizedBogoliubov transformation. As in (2.17), we construct the antisymmetric operator B ( τ ) = 12 X p ∈ Λ ∗ + τ p ( b ∗ p b ∗− p − b p b − p )and the generalized Bogoliubov transformation e B ( τ ) = exp X p ∈ Λ ∗ + τ p ( b ∗ p b ∗− p − b p b − p ) (5.12)With (5.12), we define a new excitation Hamiltonian M βN : F ≤ N + → F ≤ N + by setting M βN = e − B ( τ ) e − B ( η ) U H N U ∗ e B ( η ) e B ( τ ) = e − B ( τ ) G βN e B ( τ ) = C βN + e − B ( τ ) Q βN e B ( τ ) + e − B ( τ ) E βN e B ( τ ) (5.13)In the next lemma we show that, with (5.10), the action of the generalized Bogoliubovtransformation (5.12) approximately diagonalizes the quadratic operator Q βN . Lemma 5.2.
Let V ∈ L ( R ) be non-negative, compactly supported and sphericallysymmetric and assume that the coupling constant κ ≥ is small enough, so that thebounds of Lemma 5.1 hold true. Let Q βN be defined as in (5.2) and τ p as in (5.10). Then e − B ( τ ) Q βN e B ( τ ) = 12 X p ∈ Λ ∗ + h − F p + q F p − G p i + X p ∈ Λ ∗ + q F p − G p a ∗ p a p + δ N,β where the self-adjoint operator δ N,β is such that ± δ N,β ≤ CN − ( N + + 1)( K + 1) (5.14) Proof.
For p ∈ Λ ∗ + , we define a remainder operator d p through e − B ( τ ) b p e B ( τ ) = cosh( τ p ) b p + sinh( τ p ) b ∗− p + d p (5.15)With (5.15) and using the short-hand notation e γ p = cosh τ p , e σ p = sinh τ p , we can write e − B ( τ ) Q βN e B ( τ ) = X p ∈ Λ ∗ + (cid:0) F p e σ p + G p e γ p e σ p (cid:1) + X p ∈ Λ ∗ + h F p ( e γ p + e σ p ) + 2 G p e σ p e γ p i b ∗ p b p + 12 X p ∈ Λ ∗ + h F p e γ p e σ p + G p ( e γ p + e σ p ) i ( b p b − p + b ∗ p b ∗− p ) + e δ N,β (5.16)25here e δ N,β = X p ∈ Λ ∗ + F p d ∗ p e − B ( τ ) b p e B ( τ ) + X p ∈ Λ ∗ + F p ( e γ p b ∗ p + e σ p b p ) d p + 12 X p ∈ Λ ∗ + G p h d ∗ p e − B ( τ ) b ∗− p e B ( τ ) + h.c. i + 12 X p ∈ Λ ∗ + G p h ( e γ p b ∗ p + e σ p b − p ) d ∗− p + h.c. i (5.17)With the definition (5.10), (5.16) simplifies, after a lengthy but straightforward compu-tation, to e − B ( τ ) Q βN e B ( τ ) = 12 X p ∈ Λ ∗ + h − F p + q F p − G p i + X p ∈ Λ ∗ + q F p − G p b ∗ p b p + e δ N,β
From the bound F p ≤ C (1 + p ) in Lemma 5.1 we obtain (cid:12)(cid:12)(cid:12) X p ∈ Λ ∗ + q F p − G p h h ξ, b ∗ p b p ξ i − h ξ, a ∗ p a p ξ i i(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) N X p ∈ Λ ∗ + q F p − G p h ξ, a ∗ p N + a p ξ i (cid:12)(cid:12)(cid:12) ≤ N X p ∈ Λ ∗ + ( p + 1) k a p ( N + + 1) / ξ k = 1 N h ξ, ( N + + 1)( K + 1) ξ i for all ξ ∈ F ≤ N + . Hence, the claim follows if we can show that the operator e δ N,β definedin (5.17) satisfies (5.14). To reach this goal we notice that, by Lemma 2.6, e − B ( τ ) b p e B ( τ ) = X n ∈ N ( − n n ! ad ( n ) B ( τ ) ( b p )and therefore d p = X n ∈ N n )! h ad (2 n ) B ( τ ) ( b p ) − τ np b p i − X n ∈ N n + 1)! h ad (2 n +1) B ( τ ) ( b p ) − τ n +1 p b ∗− p i Let us now consider the expectation of the first term on the r.h.s. of (5.17). We find (cid:12)(cid:12)(cid:12) X p ∈ Λ ∗ + F p h d p ξ, e − B ( τ ) b p e B ( τ ) ξ i (cid:12)(cid:12)(cid:12) ≤ X n,m ∈ N n ! m ! X p ∈ Λ ∗ + F p k ( N + + 1) − / (cid:2) ad ( n ) B ( τ ) ( b p ) − τ np b ♯ n α n p (cid:3) ξ k× k ( N + + 1) / ad ( m ) B ( τ ) ( b p ) ξ k (5.18)where α n = 1 and ♯ n = · if n is even while α n = − ♯ n = ∗ if n is odd.26rom Lemma 2.5 it follows that, for any m ∈ N , ad ( m ) B ( τ ) ( b p ) is given by the sum of2 m m ! terms of the form Λ . . . Λ i N − k Π (1) ♯,♭ ( τ j , . . . , τ j k ; τ ℓ p ) (5.19)where i , k , ℓ ∈ N , j , . . . , j k ∈ N \{ } , and where each Λ j is either a factor ( N −N + ) /N , ( N + 1 − N + ) /N or a Π (2) -operator having the form N − p Π (2) ♯,♭ ( τ q , . . . , τ q p ) (5.20)for some p, q , . . . , q p ∈ N \{ } . Distinguishing the cases ℓ ≥ ℓ = 0, this impliesthat k ( N + + 1) / ad ( m ) B ( τ ) ( b p ) ξ k ≤ C m κ m m ! h | p | − k ( N + + 1) ξ k + k b p ( N + + 1) / ξ k i (5.21)Similarly, the operator ad ( n ) B ( τ ) ( b p ) can be expanded in the sum of 2 n n ! contributionsof the form (5.19). Part iv) of Lemma 2.5 implies that exactly one of these contributionswill have the form (cid:18) N − N + N (cid:19) n/ (cid:18) N + 1 − N + N (cid:19) n/ τ np b p (5.22)if n is even or the form − (cid:18) N − N + N (cid:19) ( n +1) / (cid:18) N + 1 − N + N (cid:19) ( n − / τ np b ∗− p (5.23)if n is odd. All other terms will have either k = 0 or at least one of the Λ-operatorhaving the form (5.20). Notice that the main part of the contribution (5.22), (5.23)is exactly τ np b p if n is even and − τ np b ∗− p if n is odd and it is canceled exactly by thesubtraction of τ np b ♯ n α n p . We obtain k ( N + + 1) − / (cid:2) ad ( n ) B ( τ ) ( b p ) − τ np b ♯ n α n p (cid:3) ξ k≤ C n κ n n ! N − h | p | − k ( N + + 1) ξ k + k b p ( N + + 1) / ξ k i (5.24)Inserting the last inequality and (5.21) in (5.18), and using the estimate F p ≤ C ( p + 1)from Lemma 5.1, we conclude that the expectation of the first term on the r.h.s. of(5.17) is bounded by (cid:12)(cid:12)(cid:12) X p ∈ Λ ∗ + F p h d p ξ, e − B ( τ ) b p e B ( τ ) ξ i (cid:12)(cid:12)(cid:12) ≤ CN − h ξ, ( N + + 1)( K + 1) ξ i The expectation of the second term on the r.h.s. of (5.17) can be bounded similarly.27o bound the expectation of the third term on the r.h.s. of (5.17) we expand (cid:12)(cid:12)(cid:12) X p ∈ Λ ∗ + G p h d p ξ, e − B ( τ ) b ∗− p e B ( τ ) ξ i (cid:12)(cid:12)(cid:12) = X p ∈ Λ ∗ + | G p |k ( N + + 1) − / d p ξ kk ( N + + 1) / e − B ( τ ) b ∗− p e B ( τ ) ξ k≤ Cκ k ( N + + 1) ξ k X n ≥ n ! X p ∈ Λ ∗ + | p | − (cid:13)(cid:13)(cid:13) ( N + + 1) − / (cid:2) ad ( n ) B ( τ ) ( b p ) − τ np b ♯ n α n p (cid:3) ξ (cid:13)(cid:13)(cid:13) where we used (twice) Lemma 2.4 and the bound | G p | ≤ Cκ | p | − from Lemma 5.1.Inserting (5.24), we find (cid:12)(cid:12)(cid:12) X p ∈ Λ ∗ + G p h d p ξ, e − B ( τ ) b ∗− p e B ( τ ) ξ i (cid:12)(cid:12)(cid:12) ≤ CN − k ( N + + 1) ξ k if κ > M βN , as defined in (5.13). Lemma 5.3.
Let V ∈ L ( R ) be non-negative, compactly supported and sphericallysymmetric and assume that the coupling constant κ ≥ is small enough, so that thebounds of Lemma 5.1 hold true (with F p , G p defined as in (3.17)). Suppose that C βN isdefined as in (3.16). Then, for N → ∞ , C βN + 12 X p ∈ Λ ∗ + h − F p + q F p − G p i = 4 π ( N − a βN + 12 X p ∈ Λ ∗ + " − p − κ b V (0) + q | p | + 2 | p | κ b V (0) + κ b V (0)2 p + O ( N − α )(5.25) with a βN as defined in (1.5) and for all < α < β such that α ≤ (1 − β ) / . Furthermore,on F ≤ N + , we have X p ∈ Λ ∗ + q F p − G p a ∗ p a p = X p ∈ Λ ∗ + q p + 2 p κ b V (0) a ∗ p a p + ϑ N,β (5.26) where ± ϑ N,β ≤ CN − α ( N + + 1) for all α ≤ min( β, (1 − β )) . roof. From (3.16) and from the definition of the coefficients F p , G p in (3.17) we obtain C βN − X p ∈ Λ ∗ + F p = ( N − κ b V (0) − X p ∈ Λ ∗ + h p + κ b V ( p/N β ) i + κ N X p,q ∈ Λ ∗ + b V ( p/N β ) η p η q On the other hand, setting A p = − h κ b V ( p/N β )( γ p + σ p ) + 2 p γ p σ p i κN X q ∈ Λ ∗ b V (( p − q ) /N β ) e η q − h κN X q ∈ Λ ∗ b V (( p − q ) /N β ) e η q i (5.27)we find that F p − G p = | p | + 2 p κ b V ( p/N β ) + A p (5.28)Notice that with (3.9) and (3.11), we have (cid:12)(cid:12)(cid:12) κN X q ∈ Λ ∗ b V (( p − q ) /N β ) e η q (cid:12)(cid:12)(cid:12) ≤ C κ N X q ∈ Λ ∗ | b V (( p − q ) /N β ) | q + 1 ≤ Cκ N β − (5.29)which implies that | A p | ≤ CN β − (5.30)for every fixed p ∈ Λ ∗ . Choosing κ > | p | + 2 p κ b V ( p/N β ) and | p | +2 p κ b V ( p/N β ) + A p are positive and bounded away from 0, uniformly in p ∈ Λ ∗ + , weobserve that q | p | + 2 p κ b V ( p/N β ) + A p = q | p | + 2 p κ b V ( p/N β )+ A p q | p | + 2 p κ b V ( p/N β ) + A p + q | p | + 2 p κ b V ( p/N β )The denominator in the last term is such that2 p ≤ q | p | + 2 p κ b V ( p/N β ) + A p + q | p | + 2 p κ b V ( p/N β ) ≤ p " C A p | p | + κ b V ( p/N β ) p ! This implies that A p p " − C A p | p | + κ b V ( p/N β ) p ! ≤ A p q | p | + 2 p κ b V ( p/N β ) + A p + q | p | + 2 p κ b V ( p/N β ) ≤ A p p p ∈ Λ ∗ + . Since, from (5.30), X p ∈ Λ ∗ + A p | p | ≤ CN β − , and X p ∈ Λ ∗ + A p | b V ( p/N β ) || p | ≤ CN β − we conclude that C βN + 12 X p ∈ Λ ∗ + h − F p + q F p − G p i = ( N − κ b V (0) + 12 X p ∈ Λ ∗ + (cid:20) − p − κ b V ( p/N β ) + q | p | + 2 p κ b V ( p/N β ) (cid:21) + X p ∈ Λ ∗ + A p p + κ N X q ∈ Λ ∗ + b V (( p − q ) /N β ) η p η q + O ( N β − ) (5.31)We still have to computeB := X p ∈ Λ ∗ + A p p + κ N X q ∈ Λ ∗ + b V (( p − q ) /N β ) η p η q (5.32)To this end, we decompose A p = A ,p + A ,p with A ,p = − h κ b V ( p/N β ) + 2 p η p + κ N X q ∈ Λ ∗ b V (( p − q ) /N β ) e η q ih κN X q ∈ Λ ∗ + b V (( p − q ) /N β ) e η q i In other words, we define A ,p by replacing, in (5.27), ( γ p + σ p ) by 1 and γ p σ p by η p ;recalling the bound (5.29), we conclude that the rest term A ,p is such that X p ∈ Λ ∗ + A ,p p ≤ CN β − (5.33)From (5.32), we obtainB = − X p ∈ Λ ∗ + p h κ N X q ∈ Λ ∗ b V (( p − q ) /N β ) e η q i × h κ b V ( p/N β ) + p η p + κ N X q ∈ Λ ∗ b V (( p − q ) /N β ) e η q i + O ( N β − )Notice here that, in contrast with (5.32), the sum on the r.h.s. includes the point q = 030which gives a contribution of order N β − ). Using the relation (3.12), we findB = − X p ∈ Λ ∗ + p h κ N X q ∈ Λ ∗ b V (( p − q ) /N β ) e η q i × h κ b V ( p/N β ) + N λ
N,ℓ b χ ℓ ( p ) + λ N,ℓ X q ∈ Λ ∗ b χ ℓ ( p − q ) e η q i + O ( N β − )With (5.8) and the bounds (5.9) and (5.29), we can simplify the last identity toB = − X p ∈ Λ ∗ + κ b V ( p/N β )2 p κ N X q ∈ Λ ∗ b V (( p − q ) /N β ) e η q + O ( N β − )= − X p ∈ Λ ∗ + κ b V ( p/N β )2 p κ N X q ∈ Λ ∗ + b V (( p − q ) /N β ) η q + O ( N β − ) (5.34)since the contribution from the term with q = 0 is of the order N β − (and since e η q = η q for q = 0). The r.h.s. is of the order N β − (the sum over q is of the order N β − , but itdoes not decay in p ; summing over p produces an additional factor N β ). For β < / N → ∞ . For β ≥ /
2, on the other hand, wehave to expand it further. To this end, we use again the relation (3.12) to write q η q = − κ b V ( q/N β ) − κ N X q ∈ Λ ∗ b V (( q − q ) /N β ) e η q + N λ
N,ℓ b χ ℓ ( q ) + λ N,ℓ X q ∈ Λ ∗ b χ ℓ ( q − q ) e η q (5.35)Inserting this identity in the r.h.s. of (5.34) we notice that the contribution of the lasttwo terms on the r.h.s. is negligible, in the limit of large N (after summing over p, q, q ,it is of the order N β − ≪ q = 0 in the secondterm on the r.h.s. of (5.35) vanishes, as N → ∞ (it is of order N − β ) ). We arrive atB = X p ∈ Λ ∗ + κ b V ( p/N β )2 p κ N X q ∈ Λ ∗ b V (( p − q ) /N β ) κ b V ( q/N β )2 q + X p ∈ Λ ∗ + κ b V ( p/N β )2 p κ N X q ∈ Λ ∗ b V (( p − q ) /N β ) κ q N X q ∈ Λ ∗ + b V (( q − q ) /N β ) η q + O ( N β − ) (5.36)While the first term on the r.h.s. is of the order N β − , the second term is now of theorder N β − . If β < /
3, it is negligible. If instead β ≥ /
3, we iterate again the same31rocedure, expressing η q using (5.35). After k iterations, we obtainB = k X j =1 ( − j +1 κ j +2 j +2 N j X p,q ,...,q j ∈ Λ ∗ + b V ( p/N β ) p b V (( p − q ) /N β ) q b V (( q − q ) /N β ) q . . .. . . b V (( q j − − q j ) /N β ) q j b V ( q j /N β )+ O ( N ( k +1) β − k ) + O ( N β − ) . (5.37)Choosing k = m β the largest integer with m β ≤ / (1 − β ) + min(1 / , β/ (1 − β )), weobtain that ( k + 1) β − k < − min((1 − β ) / , β ). Inserting (5.37) in (5.31), we obtain C βN + 12 X p ∈ Λ ∗ + h − F p + q F p − G p i = ( N − κ b V (0) + m β X j =1 ( − j +1 κ j +2 j +2 N j × X p,q ,...,q j ∈ Λ ∗ + V ( p/N β ) p b V (( p − q ) /N β ) q . . . b V (( q j − − q j ) /N β ) q j b V ( q j /N β )+ 12 X p ∈ Λ ∗ + h − p − κ b V ( p/N β ) + q | p | + 2 p κ b V ( p/N β ) i + O ( N − α )for all α < min( β, (1 − β ) / P p ∈ Λ ∗ + κ b V ( p/N β ) / (4 p ) andcomparing with the definition (1.5) of a βN , we get C βN + 12 X p ∈ Λ ∗ + h − F p + q F p − G p i = 4 π ( N − a βN + 12 X p ∈ Λ ∗ + h − p − κ b V ( p/N β ) + q | p | + 2 p κ b V ( p/N β ) + κ b V ( p/N β )2 p i + O ( N − α )(5.38)for every α < min( β, (1 − β ) / q | p | + 2 p κ b V ( p/N β )= p ( κ b V ( p/N β ) p − κ b V ( p/N β )2 | p | + 3 κ b V ( p/N β ) | p | Z ds s Z ds s Z ds h κs s s b V ( p/N β ) p i / (5.39)32t is easy to check that (cid:12)(cid:12)(cid:12) − p − κ b V ( p/N β ) + q | p | + 2 p κ b V ( p/N β ) + κ b V ( p/N β )2 p (cid:12)(cid:12)(cid:12) ≤ C | p | uniformly in N and, comparing (5.39) with a similar expansion with b V ( p/N β ) replacedby b V (0), that (cid:12)(cid:12)(cid:12)h − p − κ b V ( p/N β ) + q | p | + 2 p κ b V ( p/N β ) + κ b V ( p/N β )2 p i − h − p − κ b V (0) + q | p | + 2 p κ b V (0) + κ b V (0)2 p i(cid:12)(cid:12)(cid:12) ≤ CN − β | p | − Here, we used the fact that κ > p ∈ Λ ∗ + . Separating the sumin two regions | p | ≤ N β and | p | ≥ N β , we conclude that (cid:12)(cid:12)(cid:12) X p ∈ Λ ∗ + " − p − κ b V ( p/N β ) + q | p | + 2 p κ b V ( p/N β ) + κ b V ( p/N β )2 p − X p ∈ Λ ∗ + " − p − κ b V (0) + q | p | + 2 p κ b V (0) + κ b V (0)2 p ≤ CN − α for every α < β . Inserting in (5.38), we obtain (5.25).Let us now prove (5.26). From (5.28), we find X p ∈ Λ ∗ + q F p − G p a ∗ p a p = X p ∈ Λ ∗ + q | p | + 2 p κ b V ( p/N β ) + A p a ∗ p a p = X p ∈ Λ ∗ + q | p | + 2 p κ b V ( p/N β ) a ∗ p a p + X p ∈ Λ ∗ + A p q | p | + 2 p κ b V ( p/N β ) + A p + q | p | + 2 p κ b V ( p/N β ) a ∗ p a p =: B + B (5.40)With (5.30), we find |h ξ, B ξ i| ≤ X p ∈ Λ ∗ + | A p | q | p | + 2 p κ b V ( p/N β ) + A p + q | p | + 2 p κ b V ( p/N β ) k a p ξ k ≤ CN β − h ξ, N + ξ i (5.41)33s for B , we writeB = X p ∈ Λ ∗ + s | p | + 2 p κ b V (0) + 2 N − β p κ Z ds p · ∇ b V ( s p/N β ) a ∗ p a p = X p ∈ Λ ∗ + q | p | + 2 p κ b V (0) a ∗ p a p + X p ∈ Λ ∗ N − β p κ R ds p · ∇ b V ( s p/N β ) q | p | + 2 p κ b V (0) + q | p | + 2 p κ b V ( p/N β ) a ∗ p a p = X p ∈ Λ ∗ + q | p | + 2 p κ b V (0) a ∗ p a p + e B The expectation of the second term can be bounded by |h ξ, e B ξ i| ≤ CN − β X p ∈ Λ ∗ + | p |k a p ξ k ≤ CN − β X p ∈ Λ ∗ + (1 + p ) k a p ξ k ≤ CN − β h ξ, ( K + 1) ξ i Combining the last bound with (5.40) and (5.41) we obtain X p ∈ Λ ∗ + q F p − G p a ∗ p a p = X p ∈ Λ ∗ + q | p | + 2 κp b V (0) a ∗ p a p + ϑ N,β where ϑ N,β is such that ± ϑ N,β ≤ CN − α ( H βN + 1)( N + + 1)for all α ≤ min( β, (1 − β )).To show that the error term E N appearing in the decomposition (5.1) of G βN remainsnegligible after conjugation with the generalized Bogoliubov transformation e B ( τ ) , weuse the following lemma. Lemma 5.4.
Let V ∈ L ( R ) be non-negative, spherically symmetric, compactly sup-ported and suppose that the coupling constant κ > is small enough. Suppose that, for p ∈ Λ ∗ + , τ p is defined as in (5.10). Then there exists C > such that e − B ( τ ) ( N + + 1)( H βN + 1) e B ( τ ) ≤ C ( N + + 1)( H βN + 1) (5.42) Proof.
We apply Gronwall’s inequality. For ξ ∈ F ≤ N + and s ∈ R , we consider ∂ s h ξ, e − sB ( τ ) ( H βN + 1)( N + + 1) e sB ( τ ) ξ i = −h ξ, e − sB ( τ ) [ B ( τ ) , ( H βN + 1)( N + + 1)] e sB ( τ ) ξ i We have[ B ( τ ) , ( H βN + 1)( N + + 1)] = [ B ( τ ) , K ]( N + + 1)+ [ B ( τ ) , V N ]( N + + 1) + ( H βN + 1)[ B ( τ ) , N + ] (5.43)34onsider first the last term on the r.h.s. of (5.43). With[ B ( τ ) , N + ] = X p ∈ Λ ∗ + τ p ( b p b − p + b ∗ p b ∗− p )we find h ξ, e − sB ( τ ) ( H βN + 1)[ B ( τ ) , N + ] e sB ( τ ) ξ i = X p,q ∈ Λ ∗ + τ p q h ξ, e − sB ( τ ) a ∗ q a q ( b p b − p + b ∗ p b ∗− p ) e sB ( τ ) ξ i + X p ∈ Λ ∗ + τ p h ξ, e − sB ( τ ) V N ( b p b − p + b ∗ p b ∗− p ) e sB ( τ ) ξ i = I + II (5.44)where | I | ≤ X p,q ∈ Λ ∗ + τ p q k a q ( N + + 1) / e sB ( τ ) ξ k× (cid:2) k ( b p b − p + b ∗ p b ∗− p )( N + + 1) − / a q e sB ( τ ) ξ k + δ p,q k ξ k (cid:3) ≤ C X p,q ∈ Λ ∗ + τ p q k a q ( N + + 1) / e sB ( τ ) ξ k + X p τ p p k a p ( N + + 1) / ξ kk ξ k≤ C kK / ( N + + 1) / e sB ( τ ) ξ k + k ( N + + 1) e sB ( τ ) ξ kk ξ k≤ C h e sB ( τ ) ξ, ( N + + 1)( K + 1) e sB ( τ ) ξ i and, expressing the potential energy operator in position space, | II | ≤ X p ∈ Λ ∗ + τ p Z dxdy N − β V ( N β ( x − y )) (cid:12)(cid:12)(cid:12) h ˇ a x ˇ a y e sB ( τ ) ξ, ˇ a x ˇ a y ( b p b − p + b ∗ p b ∗− p ) e sB ( τ ) ξ i (cid:12)(cid:12)(cid:12) ≤ X p ∈ Λ ∗ + τ p Z dxdy N − β V ( N β ( x − y )) k ˇ a x ˇ a y ( N + + 1) / e sB ( τ ) ξ k× h k ( b p b − p + b ∗ p b ∗− p )( N + + 1) − / ˇ a x ˇ a y e sB ( τ ) ξ k + k ˇ a y e sB ( τ ) ξ k + k ˇ a x e sB ( τ ) ξ k i ≤ C Z dxdy N − β V ( N β ( x − y )) k ˇ a x ˇ a y ( N + + 1) / e sB ( τ ) ξ k + C k ( N + + 1) / e sB ( τ ) ξ k ≤ C h ξ, e − sB ( τ ) ( V N + 1)( N + + 1) e sB ( τ ) ξ i Here we used (5.11) to conclude that P p ∈ Λ ∗ + | τ p | < ∞ . From (5.44) we obtain that (cid:12)(cid:12)(cid:12) h ξ, e − sB ( τ ) ( H βN + 1)[ B ( τ ) , N + ] e sB ( τ ) ξ i (cid:12)(cid:12)(cid:12) ≤ C h ξ, e − sB ( τ ) ( H βN + 1)( N + + 1) e sB ( τ ) ξ i (5.45)35et us now consider the first term on the r.h.s. of (5.43). Since[ B ( τ ) , K ] = X p ∈ Λ ∗ + p τ p (cid:0) b p b − p + b ∗ p b ∗− p (cid:1) we obtain by Cauchy-Schwarz that (cid:12)(cid:12)(cid:12) h ξ,e − sB ( τ ) [ B ( τ ) , K ]( N + + 1) e sB ( τ ) ξ i (cid:12)(cid:12)(cid:12) ≤ X p ∈ Λ ∗ + p | τ p |k b p b − p e sB ( τ ) ξ kk ( N + + 1) e sB ( τ ) ξ k≤ C k ( N + + 1) e sB ( τ ) ξ k h X p ∈ Λ ∗ + k b p ( N + + 1) / e sB ( τ ) ξ k i / ≤ C h ξ, e − sB ( τ ) ( N + + 1) e sB ( τ ) ξ i ≤ C h ξ, e − sB ( τ ) ( H βN + 1)( N + + 1) e sB ( τ ) ξ i (5.46)Here, we used the estimate (5.11) (to make sure that P p ∈ Λ ∗ + p τ p < ∞ ) and again thefact that, on F ≤ N + , N + ≤ C H βN .Finally, let us consider the second term on the r.h.s. of (5.43). It is convenient toexpress the potential energy operator V N in position space. We find h ξ,e − sB ( τ ) [ B ( τ ) , V N ]( N + + 1) e sB ( τ ) ξ i = κ N Z Λ × Λ dxdy N β V ( N β ( x − y ))ˇ τ ( x − y ) h e sB ( τ ) ξ, ( b ∗ x b ∗ y + b x b y )( N + + 1) e sB ( τ ) ξ i + κN Z Λ × Λ dxdyN β V ( N β ( x − y )) h e sB ( τ ) ξ, (cid:2) b ∗ x b ∗ y a ∗ (ˇ τ y )ˇ a x + h.c. (cid:3) ( N + + 1) e sB ( τ ) ξ i = III + IVwhere we set ˇ τ ( x ) = P p ∈ Λ ∗ + τ p e ip · x . Since k ˇ τ k ∞ ≤ k τ k ≤ C < ∞ uniformly in N , it issimple to estimate | I | ≤ C h ξ, e − sB ( τ ) ( V N + 1)( N + + 1) e sB ( τ ) ξ i Similarly, since k ˇ τ y k = k ˇ τ k = k τ k ≤ C < ∞ independently of y ∈ Λ and of N , wefind | II | ≤ C h ξ, e − sB ( τ ) ( V N + 1)( N + + 1) e sB ( τ ) ξ i We conclude therefore that (cid:12)(cid:12)(cid:12) h ξ, e − sB ( τ ) [ B ( τ ) , V N ]( N + + 1) e sB ( τ ) ξ i (cid:12)(cid:12)(cid:12) ≤ C h ξ, e − sB ( τ ) ( V N + 1)( N + + 1) e sB ( τ ) ξ i Combining this bound with (5.45) and (5.46), we obtain from (5.43) that (cid:12)(cid:12)(cid:12) ∂ s h ξ, e − sB ( τ ) ( H βN + 1)( N + + 1) e sB ( τ ) ξ i (cid:12)(cid:12)(cid:12) ≤ C h ξ, e − sB ( τ ) ( H βN + 1)( N + + 1) e sB ( τ ) ξ i By Gronwall’s inequality, we arrive at (5.42).36n the next corollary, we summarize the properties of the excitation Hamiltonian M βN defined in (5.13) that follow from Lemma 5.2, Lemma 5.3 and Lemma 5.4 above.This corollary will be the starting point for the proof of Theorem 1.1 in the next section. Corollary 5.5.
Fix < β < . Let V ∈ L ( R ) be non-negative, spherically symmetricand compactly supported with sufficiently small coupling constant κ > . Then theexcitation Hamiltonian M βN = e − B ( τ ) e − B ( η ) U H N U ∗ e B ( η ) e B ( τ ) : F ≤ N + → F ≤ N + is suchthat M βN =4 π ( N − a βN + 12 X p ∈ Λ ∗ + " − p − κ b V (0) + q | p | + 2 | p | κ b V (0) + κ b V (0)2 p + X p ∈ Λ ∗ + q p + 2 p κ b V (0) a ∗ p a p + ρ N,β (5.47) where, for all < α < β such that α ≤ (1 − β ) / there exists C > with ± ρ N,β ≤ CN − α ( N + + 1)( H βN + 1) Furthermore, let E βN be the ground state energy of the Hamiltonian H βN and let ψ N ∈ L s ( R N ) with k ψ N k = 1 belong to the spectral subspace of H βN with energies below E βN + ζ ,for some ζ > . In other words, assume that ψ N = ( −∞ ; E βN + ζ ] ( H βN ) ψ N Let ξ N = e − B ( τ ) e − B ( η ) U ψ N ∈ F ≤ N + . Then there exists a constant C > such that h ξ N , ( N + + 1)( H βN + 1) ξ N i ≤ C (1 + ζ ) Let e E βN = 4 π ( N − a βN + 12 X p ∈ Λ ∗ + " − p − κ b V (0) + q | p | + 2 | p | κ b V (0) + κ b V (0)2 p with a βN defined as in (1.5). To prove Theorem 1.1, we will compare the eigenvalues of M βN − e E βN (which of course coincide with the eigenvalues of H βN − e E βN ) with those of thediagonal quadratic operator D = X p ∈ Λ ∗ + ε p a ∗ p a p , (6.1)acting on F ≤ N + . Here we defined ε p = ( | p | + 2 p κ b V (0)) / for all p ∈ Λ ∗ + . For m ∈ N ,let λ m denote the m -th eigenvalue of M βN − e E βN and e λ m the m -th eigenvalue of D (eigenvalues are counted with multiplicity). We will show that | λ m − e λ m | ≤ CN − α (1 + ζ ) (6.2)37or all 0 < α < β such that α ≤ (1 − β ) / m ∈ N such that e λ m < ζ .Since e λ = 0, (6.2) implies first of all that E βN = e E βN + O ( N − α ), for all 0 < α < β such that α ≤ (1 − β ) /
2. Furthermore, since the eigenvalues e λ m of (6.1) have the form k X j =1 n j ε p j for k ∈ N , n , . . . , n k ∈ N and p , . . . , p k ∈ Λ ∗ + , (6.2) also implies the relation (1.6) forthe low-lying excitation energies of H βN .To show (6.2), we first prove an upper bound for λ m , valid for all m ∈ N with e λ m < ζ .To this end, we use the min-max principle, which implies that λ m = inf Y ⊂F ≤ N + :dim Y = m sup ξ ∈ Y : k ξ k =1 h ξ, ( M βN − e E βN ) ξ i ≤ sup ξ ∈ Y m D : k ξ k =1 h ξ, ( M βN − e E βN ) ξ i (6.3)where Y m D denotes the space spanned by normalized eigenvectors ξ , . . . , ξ m of D , as-sociated with the eigenvalues e λ ≤ · · · ≤ e λ m < ζ . Without loss of generality, since D commutes with N + we may assume that ξ , . . . , ξ m are also eigenvectors of N + ; we de-note the corresponding eigenvalue by r , . . . , r m ∈ N , i.e. N + ξ j = r j ξ j . Since D ≥ c N + ,we find r j ≤ Cζ . From Lemma 7.3 and since K ≤ D we obtain h ξ, ( N + + 1)( H βN + 1) ξ i ≤ C h ξ, ( N + + 1) ( K + 1) ξ i ≤ C h ξ, ( N + + 1) ( D + 1) ξ i ≤ C (1 + ζ )for all ξ ∈ Y m D . With (5.47), we conclude that h ξ, ( M βN − e E βN ) ξ i ≤ h ξ, D ξ i + CN − α (1 + ζ )for all ξ ∈ Y m D and all 0 < α < β such that α ≤ (1 − β ) /
2. From (6.3), we obtain λ m ≤ sup ξ ∈ Y m D : k ξ k =1 h ξ, D ξ i + CN − α (1 + ζ ) ≤ e λ m + CN − α (1 + ζ )again for all 0 < α < β such that α ≤ (1 − β ) / λ m . From the upper bound above and since weassumed that e λ m < ζ , we find that λ m ≤ ζ if N is large enough. Denoting by P ζ thespectral projection of M βN − e E βN associated with the interval ( −∞ ; ζ ], we find λ m = inf Y ⊂F ≤ N + :dim Y = m sup ξ ∈ Y : k ξ k =1 h ξ, ( M βN − e E βN ) ξ i≥ inf Y ⊂ P ζ ( F ≤ N + ):dim Y = m sup ξ ∈ Y : k ξ k =1 h ξ, D ξ i − CN − α (1 + ζ ) ≥ inf Y ⊂F ≤ N + :dim Y = m sup ξ ∈ Y : k ξ k =1 h ξ, D ξ i − CN − α (1 + ζ )= e λ m − CN − α (1 + ζ )38or all 0 < α < β such that α ≤ (1 − β ) /
2. This concludes the proof of (6.2) and theproof of Theorem 1.1.
Remark:
Theorem 1.1 states the convergence of low-lying eigenvalues of the Hamiltonoperator (1.1) towards the eigenvalues of the quadratic Hamiltonian Q ∞ = E βN + X p ∈ Λ ∗ + q | p | + 2 p κ b V (0) a ∗ p a p (6.4)In fact, using ideas from [9], one can also show convergence of the corresponding eigen-vectors. More precisely, for a fixed j ∈ N , let P ( j ) H βN denote the orthogonal projectiononto the subspace of L s ( R N ) spanned by the eigenvectors associated with the j small-est eigenvalues of H βN . Similarly, let P ( j ) Q denote the orthogonal projection onto thesubspace of F ≤ N + spanned by the eigenvectors associated with the j smallest eigenvalues E βN = µ ≤ µ ≤ · · · ≤ µ j of the quadratic Hamiltonian Q ∞ . Then, assuming that µ j +1 > µ j , we find (cid:13)(cid:13)(cid:13) e − B ( τ ) e − B ( η ) U P ( j ) H βN U ∗ e B ( η ) e B ( τ ) − P ( j ) Q ∞ (cid:13)(cid:13)(cid:13) ≤ Cµ j +1 − µ j N − α (6.5)for all 0 < α < β such that α ≤ (1 − β ) /
2. In particular, if ψ βN denotes the groundstate of the Hamiltonian H βN defined in (1.1), then there exists an appropriate phase θ ∈ [0; 2 π ) such that (cid:13)(cid:13) ψ βN − e iθ U ∗ e B ( η ) e B ( τ ) Ω (cid:13)(cid:13) ≤ Cµ − µ N − α for all 0 < α < β such that α ≤ (1 − β ) /
2. The proof of (6.5) follows very closely thearguments used in Section 7 of [9].
The goal of this section is to show Theorem 3.2. We decompose G βN = G (0) N,β + G (2) N,β + G (3) N,β + G (4) N,β (7.1)with G ( j ) N,β = e − B ( η ) L ( j ) N,β e B ( η ) and with L ( j ) N,β as defined in (3.3), for j = 0 , , ,
4. We study the four contributions onthe r.h.s. of (7.1) in the following subsections.First, in the next three lemmas, we collect some preliminary results that will be usedlater to analyze the operators G ( j ) N,β , j = 0 , , ,
4. In the first lemma, we show how tobound typical operators arising from expansions of nested commutators, as described inLemma 2.5 above. 39 emma 7.1.
Let ξ ∈ F ≤ N , p, q ∈ Λ ∗ + , i , i , k , k , ℓ , ℓ ∈ N , j , . . . , j k , m , . . . , m k ∈ N \{ } and let α ℓ i = ( − ℓ i , for i = 1 , . For every s = 1 , . . . , max { i , i } , let Λ s , Λ ′ s beeither a factor ( N − N + ) /N , a factor ( N + 1 − N + ) /N or a Π (2) -operator of the form N − h Π (2) ♯,♭ ( η z , . . . , η z h ) (7.2) for some h ∈ N \{ } and z , . . . , z h ∈ N \{ } . Suppose that the operators Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p ϕ α ℓ p )Λ ′ . . . Λ ′ i N − k Π (1) ♯ ′ ,♭ ′ ( η m , . . . , η m k ; η ℓ q ϕ α ℓ q ) appear in the expansion of ad ( n ) B ( η ) ( b p ) and of ad ( k ) B ( η ) ( b q ) , as described in Lemma 2.5, forsome n, k ∈ N .i) For any β ∈ Z , letB = ( N + + 1) ( β − / Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p ϕ α ℓ p ) ξ (7.3) and e B = ( N + + 1) ( β − / N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p ϕ α ℓ p ) ∗ Λ ∗ i . . . Λ ∗ ξ Then, we have k B k , k e B k ≤ C n κ n p − ℓ k ( N + + 1) β/ ξ k (7.4) If ℓ is even, we also find k B k ≤ C n κ n p − ℓ k a p ( N + + 1) ( β − / ξ k (7.5) ii) For β ∈ Z , letD = ( N + + 1) ( β − / Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p ϕ α ℓ p ) × Λ ′ . . . Λ ′ i N − k Π (1) ♯ ′ ,♭ ′ ( η m , . . . , η m k ; η ℓ q ϕ α ℓ q ) ξ (7.6) Then, we have k D k ≤ C n + k κ n + k p − ℓ q − ℓ k ( N + + 1) ( β +1) / ξ k (7.7) If ℓ is even, we find k D k ≤ C n + k κ n + k p − ℓ q − ℓ k a q ( N + + 1) β/ ξ k (7.8) If ℓ is even, we have k D k ≤ C n + k kN − κ n + k p − ℓ +1) q − ℓ k ( N + + 1) ( β +1) / ξ k + C n + k κ n + k p − ℓ + ℓ ) µ ℓ δ p, − q k ( N + + 1) ( β − / ξ k + C n + k κ n + k p − ℓ q − ℓ k a p ( N + + 1) β/ ξ k (7.9)40 here µ ℓ = 1 if ℓ is odd and µ ℓ = 0 if ℓ is even. If ℓ is even and either k > or k > or there is at least one Λ - or Λ ′ -operator having the form (7.2), weobtain the improved bound k D k ≤ C n + k kN − κ n + k p − ℓ +1) q − ℓ k ( N + + 1) ( β +1) / ξ k + C n + k N − κ n + k p − ℓ + ℓ ) µ ℓ δ p, − q k ( N + + 1) ( β +1) / ξ k + C n + k κ n + k p − ℓ q − ℓ k a p ( N + + 1) β/ ξ k (7.10) Finally, if ℓ = ℓ = 0 , we can writeD = D ( p, q ) + D a p a q ξ (7.11) where k D ( p, q ) k ≤ C n + k kN − κ n + k p − k a q ( N + + 1) β/ ξ k and D is a bounded operator on F ≤ N + with k D ♮ ζ k ≤ C n + k κ n + k k ( N + + 1) ( β − / ζ k (7.12) for ♮ ∈ {· , ∗} and for all ζ ∈ F ≤ N + . If k > or k > or at least one of the Λ - or Λ ’-operators has the form (7.2), we also have the improved bound k D ♮ ζ k ≤ C n + k N − κ n + k k ( N + + 1) ( β +1) / ζ k (7.13) for ♮ ∈ {· , ∗} and all ζ ∈ F ≤ N + .iii) All the bounds in part ii) remain true if, in the definition of D, we replace theoperator Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p ϕ α ℓ p ) by the operator η n b ♮ n α n p , where ♮ n = · and α n = 1 if n is even while ♮ n = ∗ and α n = − if n is odd (in this case, ℓ = n ). The proof of Lemma 7.1, part i) and ii) can be found in [2, Lemma 4.1]. The proofof part iii) is very similar to the proof of part ii). Notice that part iii) states essentiallythat all bounds in part ii) remain true if in the definition of D, we replace all operatorsΛ , . . . , Λ i by the identity). We will use part iii) of Lemma 7.1 in the proof of Prop.7.5 and Prop. 7.6 below. In some occasions, it will also be important to bound vectorsof the form (7.6), expressed as functions in position space. To this end, we will use thefollowing lemma, whose proof follows closely the proof of Lemma 5.2 in [4]. Lemma 7.2.
Let ξ ∈ F ≤ N , β ∈ N , i , i , k , k , ℓ , ℓ ∈ N , j , . . . , j k , m , . . . , m k ∈ N \{ } , For every s = 1 , . . . , max { i , i } , let Λ s , Λ ′ s be either a factor ( N − N + ) /N , ( N + 1 − N + ) /N or a Π (2) -operator of the form N − h Π (2) ♯,♭ ( η z , . . . , η z h ) (7.14)41 or some h, z , . . . , z h ∈ N \{ } . Suppose that the operators Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; ˇ η ℓ x )Λ ′ . . . Λ ′ i N − k Π (1) ♯ ′ ,♭ ′ ( η m , . . . , η m k ; ˇ η ℓ y ) appear in the expansion of ad ( n ) B ( η ) (ˇ b x ) , ad ( k ) B ( η ) (ˇ b y ) , respectively, for some n, k ∈ N . Herewe use the notation ˇ η ℓ x for the function z → ˇ η ℓ ( x − z ) , where ˇ η ℓ denotes the Fouriertransform of the function η ℓ defined on Λ ∗ + . LetS = ( N + + 1) β/ Λ ′ . . . Λ ′ i N − k Π (1) ♯ ′ ,♭ ′ ( η m , . . . , η m k ; ˇ η ℓ y ) × Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; ˇ η ℓ x ) ξ Then we have the following bounds. If ℓ , ℓ ≥ , k S k ≤ C n + k κ n + k k ( N + + 1) ( β +2) / ξ k (7.15) If ℓ = 0 and ℓ ≥ , k S k ≤ C n + k κ n + k k ˇ a x ( N + + 1) ( β +1) / ξ k If ℓ ≥ and ℓ = 0 , k S k ≤ C n + k κ n + k nN − k ( N + + 1) ( β +2) / ξ k + C n + k κ n + k − ℓ µ ℓ | ˇ η ℓ ( x − y ) |k ( N + + 1) β/ ξ k + C n + k κ n + k k ˇ a y ( N + + 1) ( β +1) / ξ k (7.16) where µ ℓ = 1 if ℓ is odd, while µ ℓ = 0 if ℓ is even. If ℓ ≥ and ℓ = 0 and weadditionally assume that k > or k > or at least one of the Λ - or Λ ′ -operators is a Π (2) -operator of the form (7.14), we obtain the improved estimate k S k ≤ C n + k κ n + k nN − k ( N + + 1) ( β +2) / ξ k + C n + k κ n + k − ℓ µ ℓ N − | ˇ η ℓ ( x − y ) |k ( N + + 1) ( β +2) / ξ k + C n + k κ n + k k ˇ a y ( N + + 1) ( β +1) / ξ k (7.17) Finally, if ℓ = ℓ = 0 , k S k ≤ C n + k κ n + k nN − k ˇ a x ( N + + 1) ( β +1) / ξ k + C n + k κ n + k k ˇ a x ˇ a y ( N + + 1) β/ ξ k Finally, in the next lemma, we show that the expectation of the potential energyoperator is small, of the order N β − , on states with bounded expectation of ( N + +1)( K + 1). This lemma will be important to show that, asymptotically the quadraticpart of the generator G βN is dominant. 42 emma 7.3. Suppose V ∈ L ( R ) . Then there exists C > such that h ξ, V N ξ i = κ N X p,q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − p, − q b V ( r/N β ) a ∗ p + r a ∗ q a q + r a p ≤ CκN β − k ( K + 1) / ( N + + 1) / ξ k for every ξ ∈ F ≤ N + . Here K = P p ∈ Λ ∗ + p a ∗ p a p is the kinetic energy operator.Proof. We observe that h ξ, V N ξ i ≤ κ N X p,q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − p, − q | b V ( r/N β ) | | p + r || q + r | k a p + r a q ξ k | q + r || p + r | k a q + r a p ξ k≤ κN X p,q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − p, − q | b V ( r/N β ) | ( q + r ) ( p + r ) k a p + r a q ξ k ≤ κ sup q ∈ Λ ∗ + N X r ∈ Λ ∗ : r = − q | b V ( r/N β ) | ( q + r ) kN / K / ξ k The claim follows from the estimate1 N X r ∈ Λ ∗ : r = − q | b V ( r/N β ) | ( q + r ) ≤ k b V k ∞ N X r ∈ Λ ∗ : | r + q |≤ N β r + q ) + 1 N h X r ∈ Λ ∗ | b V ( r/N β ) | i / h X r ∈ Λ ∗ : | r + q | >N β | r + q | i / ≤ CN β − (7.18)uniformly in q ∈ Λ ∗ + . G (0) N From (3.3), we have G (0) N,β = e − B ( η ) L (0) N,β e B ( η ) = ( N − κ b V (0) + E (0) N,β where E (0) N,β = κ b V (0)2 N e − B ( η ) N + e B ( η ) − κ b V (0)2 N e − B ( η ) N e B ( η ) We collect the properties of E (0) N,β in the next proposition.
Proposition 7.4.
Under the assumptions of Theorem 3.2, there exists
C > such that,on F ≤ N + , ±E (0) N,β ≤ CκN ( N + + 1) ≤ Cκ ( N + + 1) ± [ E (0) N,β , i N + ] ≤ C ( N + + 1) (7.19)43 roof. The first bound in (7.19) follows directly from Lemma 2.4. To prove the secondestimate in (7.19), we write e − B ( η ) N + e B ( η ) = N + + X p ∈ Λ ∗ + Z ds e − sB ( η ) [ a ∗ p a p , B ( η )] e sB ( η ) = N + + X p ∈ Λ ∗ + η p Z ds e − sB ( η ) ( b p b − p + b ∗ p b ∗− p ) e sB ( η ) With Lemma 2.6, we obtain e − B ( η ) N + e B ( η ) = N + + X n,m ≥ ( − n + m n ! m !( n + m + 1) X p ∈ Λ ∗ + η p (cid:16) ad ( n ) B ( η ) ( b p )ad ( m ) B ( η ) ( b − p ) + h.c. (cid:17) (7.20)It follows from Lemma 2.5 that the operator X p ∈ Λ ∗ + η p ad ( n ) B ( η ) ( b p )ad ( m ) B ( η ) ( b − p )can be written as the sum of 2 n n ! terms of the formE = X p ∈ Λ ∗ + η p Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p ϕ α ℓ p ) × Λ ′ . . . Λ ′ i N − k Π (1) ♯,♭ ( η m , . . . , η m k ; η ℓ p ϕ − α ℓ p ) (7.21)where i , i , k , k , ℓ , ℓ ∈ N , j , . . . , j k , m , . . . , m k ∈ N \{ } , α ℓ = ( − ℓ and whereeach Λ r , Λ ′ r is either a factor ( N − N + ) /N , a factor ( N + 1 − N + ) /N or a Π (2) -operatorof the form N − h Π (2) ♯,♭ ( η z , . . . , η z h )with h, z , . . . , z h ∈ N \{ } . Lemma 7.1, part ii), allows us to bound matrix-elements of(7.21) by |h ξ , E ξ i| ≤ X p ∈ Λ ∗ + | η p |k ( N + + 1) / ξ k× k ( N + + 1) − / Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ ϕ α ℓ p ) × Λ ′ . . . Λ ′ i N − k Π (1) ♯,♭ ( η m , . . . , η m k ; η ℓ ϕ − α ℓ p ) ξ k≤ C n + m κ n + m +1 k ( N + + 1) / ξ k X p ∈ Λ ∗ + n k p | − k ( N + + 1) / ξ k + | p | − k a p ξ k o ≤ C n + m κ n + m +1 k ( N + + 1) / ξ kk ( N + + 1) / ξ k (7.22)44ince [ N + , E] has again the form E, up to a multiplicative constant bounded by ( n + m ),the bound (7.22), with (7.20), also implies that (cid:12)(cid:12)(cid:12) h ξ , h e − B ( η ) N + e B ( η ) , N + i ξ i (cid:12)(cid:12)(cid:12) ≤ Cκ k ( N + + 1) / ξ kk ( N + + 1) / ξ k (7.23)for all ξ , ξ ∈ F ≤ N + . With Lemma 2.4, we obtain (cid:12)(cid:12)(cid:12)(cid:10) ξ, h e − B ( η ) N e B ( η ) , N + i ξ (cid:11)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) h ξ, e − B ( η ) N + e B ( η ) h e − B ( η ) N + e B ( η ) , N + i ξ i + h ξ, h e − B ( η ) N + e B ( η ) , N + i e − B ( η ) N + e B ( η ) ξ i (cid:12)(cid:12)(cid:12) ≤ Cκ k ( N + + 1) / e − B ( η ) N + e B ( η ) ξ kk ( N + + 1) / ξ k≤ Cκ k ( N + + 1) / ξ kk ( N + + 1) / ξ k≤ CκN k ( N + + 1) / ξ k Together with (7.23), this concludes the proof of the second estimate in (7.19). G (2) N From (3.3), we have G (2) N,β = e − B ( η ) L (2) N,β e B ( η ) = e − B ( η ) K e B ( η ) + e − B ( η ) L ( V ) N,β e B ( η ) where K = P p ∈ Λ ∗ + p a ∗ p a p and L ( V ) N,β = X p ∈ Λ ∗ + κ b V ( p/N β ) (cid:20) b ∗ p b p − N a ∗ p a p (cid:21) + κ X p ∈ Λ ∗ + b V ( p/N β ) (cid:2) b ∗ p b ∗− p + b p b − p (cid:3) (7.24)We study first the contribution arising from the kinetic energy operator K . We definethe operator e E ( K ) N,β through e − B ( η ) K e B ( η ) = K + X p ∈ Λ ∗ + p η p + X p ∈ Λ ∗ + p η p (cid:2) b ∗ p b ∗− p + b p b − p (cid:3) + e E ( K ) N,β (7.25)To prove part b) of Theorem 3.2, we need to keep track of more order one terms arisingfrom the conjugation of K . We define the operator E ( K ) N,β through e − B ( η ) K e B ( η ) = K + X p ∈ Λ ∗ + h p σ p + p σ p γ p (cid:0) b p b − p + b ∗ p b ∗− p (cid:1) + 2 p σ p b ∗ p b p i + E ( K ) N,β (7.26)In the next proposition, we study the properties of the error terms e E ( K ) N,β , E ( K ) N,β .45 roposition 7.5.
Under the assumptions of Theorem 3.2, for every δ > there exists C > such that, on F ≤ N + , ± e E ( K ) N,β ≤ δ H βN + Cκ ( N + + 1) ± h e E ( K ) N,β , i N + i ≤ C ( H βN + 1) (7.27) Furthermore, there exists
C > such that ± E ( K ) N,β ≤ CN β − ( N + + 1)( K + 1) (7.28) Proof.
We compute e − B ( η ) K e B ( η ) = K + Z ds dds e − sB ( η ) K e sB ( η ) = K + Z ds e − sB ( η ) [ K , B ( η )] e sB ( η ) = K + Z ds X p ∈ Λ ∗ + p η p e − sB ( η ) (cid:0) b p b − p + b ∗ p b ∗− p (cid:1) e sB ( η ) With Lemma 2.6 we find e − B ( η ) K e B ( η ) = K + Z ds X n,m ≥ ( − n + m n ! m ! X p ∈ Λ ∗ + p η p h ad ( n ) sB ( η ) ( b p )ad ( m ) sB ( η ) ( b − p ) + h.c. i = K + Z ds X n,m ≥ ( − n + m n ! m ! X p ∈ Λ ∗ + p η p × nh s n η np b ♯ n α n + ad ( n ) sB ( η ) ( b p ) − s n η np b ♯ n α n i × h s m η mp b ♯ m α m + ad ( m ) sB ( η ) ( b p ) − s m η mp b ♯ m α m i + h.c. o where we defined α n = +1 and ♯ n = · if n is even while α n = − ♯ n = ∗ if n is odd.Integrating over s , and using η p Z (cid:0) cosh ( sη p ) + sinh ( sη p ) (cid:1) ds = cosh( η p ) sinh( η p )2 η p Z sinh( sη p ) cosh( sη p ) ds = sinh ( η p )46e easily find, with the notation γ p = cosh η p and σ p = sinh η p , e − B ( η ) K e B ( η ) = K + X p ∈ Λ ∗ + p σ p + X p ∈ Λ ∗ + p γ p σ p (cid:0) b p b − p + b ∗ p b ∗− p (cid:1) + 2 X p ∈ Λ ∗ + p σ p b ∗ p b p + X n,m ≥ ( − n + m n ! m !( n + m + 1) X p ∈ Λ ∗ + p η n +1 p b ♯ n α n p h ad ( m ) B ( η ) ( b − p ) − η mp b ♯ m − α m p i + h.c.+ X n,m ≥ ( − n + m n ! m !( n + m + 1) X p ∈ Λ ∗ + p h ad ( n ) B ( η ) ( b p ) − η np b ♯ n α n p i η m +1 p b ♯ m − α m p + h.c.+ X n,m ≥ ( − n + m n ! m !( n + m + 1) X p ∈ Λ ∗ + p η p h ad ( n ) B ( η ) ( b p ) − η np b ♯ n α n p i × h ad ( m ) B ( η ) ( b − p ) − η mp b ♯ m − α m p i + h.c.=: K + X p ∈ Λ ∗ + h p σ p + p γ p σ p (cid:0) b p b − p + b ∗ p b ∗− p (cid:1) + 2 p σ p b ∗ p b p i + E ( K )1 + E ( K )2 + E ( K )3 . Comparing with (7.25) and (7.26), we conclude that E ( K ) N,β = E ( K )1 + E ( K )2 + E ( K )3 and e E ( K ) N,β = X p ∈ Λ ∗ + p (cid:2) σ p − η p (cid:3) + 2 p σ p b ∗ p b p + p (cid:2) σ p γ p − η p (cid:3)(cid:2) b ∗ p b ∗− p + b p b − p (cid:3) + E ( K )1 + E ( K )2 + E ( K )3 =: E ( K )0 + E ( K )1 + E ( K )2 + E ( K )3 Since, by (3.9), | σ p − η p | ≤ Cκ | p | − , p σ p ≤ Cκ and | σ p γ p − η p | ≤ Cκ | p | − , it is easyto check that |h ξ, E ( K )0 ξ i| ≤ Cκ k ( N + + 1) / ξ k |h ξ, [ E ( K )0 , N + ] ξ i| ≤ Cκ k ( N + + 1) / ξ k Hence, Proposition 7.5 follows if we can show that the three error terms E ( K )1 , E ( K )2 , E ( K )3 satisfy the three bounds in (7.27), (7.28).We consider first the term E ( K )1 . According to Lemma 2.5, the operator X p ∈ Λ ∗ + p η n +1 p b ♯ n α n p (cid:2) ad ( m ) B ( η ) ( b − p ) − η mp b ♯ m − α m p (cid:3) (7.29)47s given by the sum of one term of the formF = X p ∈ Λ ∗ + p η m + n +1 p b ♯ n α n p × (cid:18) N − N + N (cid:19) m +(1 − αm ) / (cid:18) N + 1 − N + N (cid:19) m − (1 − αm ) / − b ♯ m − α m p (7.30)and of 2 m m ! − = X p ∈ Λ ∗ + p η n +1 p b ♯ n α n p Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p ϕ − α ℓ p ) (7.31)where i , k , ℓ ∈ N , j , . . . , j k ∈ N \{ } , α ℓ = ( − ℓ and where each Λ r is either afactor ( N − N + ) /N , ( N + 1 − N + ) /N or a Π (2) -operator of the form N − h Π (2) ♯,♭ ( η z , . . . , η z h ) (7.32)with h, z , . . . , z h ∈ N \{ } . Furthermore, since we are considering the term (7.30) sepa-rately, each term of the form (7.31) must have either k > p > f ( N + ) = − (cid:18) N − N + N (cid:19) m +(1 − αm ) / (cid:18) N + 1 − N + N (cid:19) m − (1 − αm ) / and we notice that − Cm/N ≤ f ( N + ) ≤ Cm N + /N (7.33)Since f ( N + ) = 0 when m = 0, distinguishing the two cases n + m ≥ n = 0 , m = 1we conclude that |h ξ, F ξ i| ≤ C n + m +1 κ n + m +1 X p ∈ Λ ∗ + (cid:26) ( m + 1) N | p | k ( N + + 1) ξ k + mN p k b p ( N + + 1) / ξ k (cid:27) + mN k ( N + + 1) / ξ k X p ∈ Λ ∗ + p η p ≤ C n + m +1 κ n + m +1 ( m + 1) n N − k ( N + + 1) ξ k + N β − k ( N + + 1) / ξ k o (7.34)for all n, m ∈ N (the second line bounds the term with the commutator [ b p , b ∗ p ] arisingwhen n = 0 and m = 1). Since N + ≤ N on F ≤ N + , (7.34) also implies that |h ξ, F ξ i| ≤ C n + m +1 κ n + m +1 ( m + 1) k ( N + + 1) / ξ k (7.35)48q. (7.34) will be used in the proof of (7.28), while (7.35) will be used to show (7.27).Let us now consider the expectation of (7.31). First, assume that ℓ + n ≥
1. Then,Lemma 7.1, part iii), implies that |h ξ, F ξ i| ≤ X p ∈ Λ ∗ + p | η p |k ( N + + 1) / ξ k× k ( N + + 1) − / b ♯ n α n p Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p ϕ − α ℓ p ) ξ k≤ C n + m κ n + m +1 k ( N + + 1) / ξ k× X p ∈ Λ ∗ + (cid:26) (1 + m/N ) | p | k ( N + + 1) / ξ k + 1 | p | k a p ξ k (cid:27) + C n + m κ n + m − k ( N + + 1) / ξ k N X p ∈ Λ ∗ + p η p ≤ C n + m κ n + m +1 ( m + 1) k ( N + + 1) / ξ k (7.36)by (3.10), which will be used in the proof of (7.27). Also here we will need a slightlydifferent estimate to show (7.28). Using again Lemma 7.1, part iii), under the assumption ℓ + n ≥
1, we find |h ξ, F ξ i| ≤ X p ∈ Λ ∗ + p | η p |k ( N + + 1) ξ k× k ( N + + 1) − b ♯ n α n p Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p ϕ − α ℓ p ) ξ k≤ C n + m κ n + m +1 N k ( N + + 1) ξ k× X p ∈ Λ ∗ + (cid:26) (1 + m ) | p | k ( N + + 1) ξ k + 1 | p | k a p ( N + + 1) / ξ k (cid:27) + C n + m κ n + m − N k ( N + + 1) ξ kk ξ k X p ∈ Λ ∗ + p η p ≤ C n + m κ n + m +1 ( m + 1) N β − k ( N + + 1) ξ k (7.37)In the case n = ℓ = 0, Lemma 7.1, part iii), allows us to write h ξ, F ξ i = X p ∈ Λ ∗ + p η p h ξ, D ( p ) i + X p ∈ Λ ∗ + p η p h ξ, D a p a − p ξ i where k ( N + + 1) − D ( p ) k ≤ C m κ m mN − | p | − k a p ( N + + 1) − / ξ k k D ∗ ξ k ≤ C m κ m N − k ( N + + 1) ξ k . Hence, in this case, |h ξ, F ξ i| ≤ C m κ m +1 mN k ( N + + 1) ξ k X p ∈ Λ ∗ + | p | − k a p ( N + + 1) − / ξ k + (cid:12)(cid:12)(cid:12) X p ∈ Λ ∗ + p η p h ξ, D a p a − p ξ i (cid:12)(cid:12)(cid:12) ≤ C m κ m +1 mN − k ( N + + 1) ξ k + (cid:12)(cid:12)(cid:12) X p ∈ Λ ∗ + p η p h D ∗ ξ, a p a − p ξ i (cid:12)(cid:12)(cid:12) To control the last term, we use (3.12) to replace p η p = − κ b V ( p/N β ) − κ N X q ∈ Λ ∗ b V (( p − q ) /N β ) e η q + N λ
N,ℓ b χ ℓ ( p ) + λ N,ℓ X q ∈ Λ ∗ b χ ℓ ( p − q ) e η q (7.38)To bound the contribution proportional to κ b V ( p/N β ), we switch to position space. Wefind (cid:12)(cid:12)(cid:12) κ X p ∈ Λ ∗ + b V ( p/N β ) h D ∗ ξ, a p a − p ξ i (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) κ Z Λ × Λ dxdyN β V ( N β ( x − y )) h D ∗ ξ, ˇ a x ˇ a y ξ i (cid:12)(cid:12)(cid:12)(cid:12) ≤ C m κ m +1 N Z Λ × Λ dxdyN β V ( N β ( x − y )) k ( N + + 1) ξ kk ˇ a x ˇ a y ξ k≤ C m κ m +1 / N − / kV / N ξ kk ( N + + 1) ξ k The contribution of the other terms on the r.h.s. of (7.38) can be bounded similarly. Weconclude that, for n = ℓ = 0, |h ξ, F ξ i| ≤ C m κ m +1 ( m + 1) N k ( N + + 1) ξ k + C m κ m +1 / √ N k ( N + + 1) ξ kkV / N ξ k Since N + ≤ N on F ≤ N + , the last estimate also implies that |h ξ, F ξ i| ≤ C m κ m +1 ( m + 1) k ( N + + 1) / ξ k + C m κ m +1 / k ( N + + 1) / ξ kkV / N ξ k Combining the last two bounds with (7.36) and (7.37) we obtain that, for every n, m ∈ N , |h ξ, F ξ i| ≤ C n + m κ n + m +1 ( m +1) k ( N + +1) / ξ k + C n + m κ n + m +1 / k ( N + +1) / ξ kkV / N ξ k (7.39)and, with Lemma 7.3, |h ξ, F ξ i| ≤ C n + m κ n + m +1 ( m + 1) N β − k ( N + + 1) ξ k + C n + m κ n + m kV / N ξ k ≤ C n + m κ n + m ( m + 1) N β − k ( N + + 1) / ( K + 1) / ξ k . (7.40)50rom (7.35) and (7.39) we conclude that, if κ > |h ξ, E ( K )1 ξ i| ≤ Cκ k ( N + + 1) / ξ k + Cκ / k ( N + + 1) / ξ kkV / N ξ k Hence, for every δ >
C > |h ξ, E ( K )1 ξ i| ≤ δ kV / N ξ k + Cκ k ( N + + 1) / ξ k From (7.40) and (7.35), we can also estimate, if κ > |h ξ, E ( K )1 ξ i| ≤ CN β − k ( N + + 1) / ( K + 1) / ξ k This proves that the error term E ( K )1 satisfies the first bound in (7.27) and (7.28). Infact, it also satisfies the second bound in (7.27), because the commutator of every term ofthe form (7.29) with N + has again the same form, up to multiplication with a constant,bounded by C ( m + 1) (because the difference between the number of creation and thenumber of annihilation operators in (7.30), (7.31) is at most proportional to m ).The error term E ( K )2 can be controlled exactly as we did with E ( K )1 . Also the errorterm E ( K )3 can be controlled similarly. The difference is that, now, the operator X p ∈ Λ ∗ + p η p (cid:2) ad ( n ) B ( η ) ( b p ) − η np b ♯ n α n p (cid:3)(cid:2) ad ( m ) B ( η ) ( b − p ) − η mp b ♯ m − α m p (cid:3) can be written as the sum of (2 m m ! − n n ! −
1) terms of the formF = X p ∈ Λ ∗ + p η p Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p ϕ α ℓ p ) × Λ ′ . . . Λ ′ i N − k Π (1) ♯ ′ ,♭ ′ ( η m , . . . , η m k ; η ℓ p ϕ − α ℓ p ) (7.41)of (2 m m ! −
1) terms of the formF = X p ∈ Λ ∗ + p η p (cid:18) N − N + N (cid:19) n +(1 − αn ) / (cid:18) N + 1 − N + N (cid:19) n − (1 − αn ) / − b ♯ n α n p × Λ ′ . . . Λ ′ i N − k Π (1) ♯ ′ ,♭ ′ ( η m , . . . , η m k ; η ℓ p ϕ − α ℓ p ) (7.42)of (2 n n ! −
1) terms of the formF = X p ∈ Λ ∗ + p η p Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p ϕ α ℓ p ) × (cid:18) N − N + N (cid:19) m +(1 − αm ) / (cid:18) N + 1 − N + N (cid:19) m − (1 − αm ) / − b ♯ m − α m p (7.43)51nd of one term of the formF = X p ∈ Λ ∗ + p η p (cid:18) N − N + N (cid:19) n +(1 − αn ) / (cid:18) N + 1 − N + N (cid:19) n − (1 − αn ) / − b ♯ n α n p × (cid:18) N − N + N (cid:19) m +(1 − αm ) / (cid:18) N + 1 − N + N (cid:19) m − (1 − αm ) / − b ♯ m − α m p (7.44)where i , i , k , k , ℓ , ℓ ∈ N , j , . . . , j k , m , . . . , m k ∈ N \{ } , α r = ( − r and whereeach Λ r - and Λ ′ r -operator is either a factor ( N − N + ) /N , a factor ( N + 1 − N + ) /N or aΠ (2) -operator of the form (7.32). Furthermore, in (7.41), we must have k > k > ′ -operator of the form(2.11). Similarly, in (7.42) we must have k > ′ -operator of the form(2.11) and in (7.43) we must have k > defined in (7.30) and the terms F defined in (7.31).We omit the details.Next, we focus on the quadratic terms in (7.24). We define the operator e E ( V ) N through e − B ( η ) L ( V ) N,β e B ( η ) = X p ∈ Λ ∗ + " κ b V ( p/N β ) η p + κ b V ( p/N β )2 ( b p b − p + b ∗ p b ∗− p ) + e E ( V ) N,β (7.45)To prove part b) of Theorem 3.2, we will need to keep track of more contributions to L ( V ) N , so that the error has a vanishing expectation, in the limit of large N , on low-energystates. We define therefore the operator E ( V ) N,β through e − B ( η ) L ( V ) N,β e B ( η ) = X p ∈ Λ ∗ + h κ b V ( p/N β ) σ p + κ b V ( p/N β ) σ p γ p i + X p ∈ Λ ∗ + κ b V ( p/N β )( γ p + σ p ) b ∗ p b p + 12 X p ∈ Λ ∗ + κ b V ( p/N β )( γ p + σ p ) ( b p b − p + b ∗ p b ∗− p ) + E ( V ) N,β (7.46)In the next proposition, we establish bounds for the error terms e E ( V ) N,β and E ( V ) N,β . Proposition 7.6.
Under the assumptions of Theorem 3.2, for every δ > there exists C > such that, on F ≤ N + , ± e E ( V ) N,β ≤ δ V N + Cκ ( N + + 1) ± h e E ( V ) N,β , i N + i ≤ C ( H βN + 1) Furthermore, ±E ( V ) N,β ≤ CN β − k ( N + + 1) / ( K + 1) / ξ k roof. From the definition of L ( V ) N,β in (7.24), we find G (2) N,β = κ X p ∈ Λ ∗ + b V ( p/N β ) e − B ( η ) b ∗ p b p e B ( η ) − κN X p ∈ Λ ∗ + b V ( p/N β ) e B ( η ) a ∗ p a p e − B ( η ) + κ X p ∈ Λ ∗ + b V ( p/N β ) e − B ( η ) (cid:2) b p b − p + b ∗ p b ∗− p (cid:3) e B ( η ) =: G (2 , N,β + G (2 , N,β + G (2 , N,β (7.47)From Lemma 2.6, the term G (2 , N,β can be written (using again the notation γ p = cosh η p , σ p = sinh η p ) as G (2 , N,β = X m,n ≥ ( − m + n m ! n ! κ X p ∈ Λ ∗ + b V ( p/N β ) h ad ( m ) B ( η ) ( b ∗ p ) − η mp b ¯ ♯ m α m p + η mp b ¯ ♯ m α m p i × h ad ( n ) B ( η ) ( b p ) − η np b ♯ n α n p + η np b ♯ n α n p i = X m,n ≥ ( − m + n m ! n ! κ X p ∈ Λ ∗ + b V ( p/N β ) η m + np b ¯ ♯ m α m p b ♯ n α n p + E ( V )1 = κ X p ∈ Λ ∗ + b V ( p/N β ) (cid:2) γ p b ∗ p + σ p b − p (cid:3)(cid:2) γ p b p + σ p b ∗− p ] + E ( V )1 (7.48)with α n = 1 and ♯ n = · if n is even while α n = − ♯ n = ∗ if n is odd (and ¯ ♯ n = ∗ if ♯ n = · and ¯ ♯ n = · if ♯ n = ∗ ) and with the error term E ( V )1 = X m,n ≥ ( − m + n m ! n ! κ X p ∈ Λ ∗ + b V ( p/N β ) η mp b ¯ ♯ m α m p (cid:2) ad ( n ) B ( η ) ( b p ) − η np b ♯ n α n p (cid:3) + X m,n ≥ ( − m + n m ! n ! κ X p ∈ Λ ∗ + b V ( p/N β ) (cid:2) ad ( m ) B ( η ) ( b ∗ p ) − η mp b ¯ ♯ m α m p (cid:3) η np b ♯ n α n p + X m,n ≥ ( − m + n m ! n ! κ X p ∈ Λ ∗ + b V ( p/N β ) (cid:2) ad ( m ) B ( η ) ( b ∗ p ) − η mp b ¯ ♯ m α m p (cid:3)(cid:2) ad ( n ) B ( η ) ( b p ) − η np b ♯ n α n p (cid:3) (7.49)According to Lemma 2.5, the operator κ X p ∈ Λ ∗ + b V ( p/N β ) η mp b ¯ ♯ m α m p (cid:2) ad ( n ) B ( η ) ( b p ) − η np b ♯ n α n p (cid:3) = κ X p ∈ Λ ∗ + b V ( p/N β ) η m + np b ¯ ♯ m α m p × (cid:18) N − N + N (cid:19) n +(1 − αn ) / (cid:18) N + 1 − N + N (cid:19) n − (1 − αn ) / − b ♯ n α n p (7.50)and of 2 n n ! − = κ X p ∈ Λ ∗ + b V ( p/N β ) η mp b ¯ ♯ m α m p Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p ϕ α ℓ p ) (7.51)with i , k , ℓ ∈ N , j , . . . , j k ∈ N \{ } , α r = ( − r and where each Λ r is either a factor( N − N + ) /N , a factor ( N + 1 − N + ) /N or a Π (2) -operator of the form N − h Π (2) ♯,♭ ( η z , . . . , η z p ) (7.52)for h, z , . . . , z p ∈ N \{ } . Furthermore, each operator of the form (7.51) must have either k > = 0 if n = 0, the expectation of (7.50) can be bounded by |h ξ, G ξ i| ≤ C n + m κ n + m +1 N k ( N + + 1) ξ k× X p ∈ Λ ∗ + | b V ( p/N β ) | p (cid:20) k a p ( N + + 1) / ξ k + 1 p k ( N + + 1) ξ k (cid:21) ≤ C n + m κ n + m +1 N k ( N + + 1) ξ k As for the term G defined in (7.51), its expectation can be bounded with Lemma 7.1part iii) by |h ξ, G ξ i| ≤ C n + m κ n + m +1 N X p ∈ Λ ∗ + | b V ( p/N β ) | (cid:26) k a p ( N + + 1) / ξ k + 1 p k ( N + + 1) ξ k (cid:27) ≤ C n + m κ n + m +1 N k ( N + + 1) ξ k for all ξ ∈ F ≤ N + . The expectation of the operators appearing on the second and thirdline in (7.49) can be controlled similarly, using again Lemma 7.1. Therefore, if κ > m, n ∈ N , and from (7.49) we conclude that |h ξ, E ( V )1 ξ i| ≤ CκN k ( N + + 1) ξ k ≤ Cκ k ( N + + 1) / ξ k . (7.53)54ince commutators of N + with operators of the form (7.50), (7.51) have again the sameform (up to a multiplicative constant bounded by C ( n + 1)), we also find |h ξ, (cid:2) N + , E ( V )1 (cid:3) ξ i| ≤ Cκ k ( N + + 1) / ξ k . (7.54)Let us now consider the second contribution to G (2) N,β on the r.h.s. of (7.47). Weobserve that −G (2 , N,β = κN X p ∈ Λ ∗ + b V ( p/N β ) e − B ( η ) a ∗ p a p e B ( η ) = κN X p ∈ Λ ∗ + b V ( p/N β ) (cid:20) a ∗ p a p + Z ds e − sB ( η ) [ a ∗ p a p , B ( η )] e sB ( η ) (cid:21) = κN X p ∈ Λ ∗ + b V ( p/N β ) a ∗ p a p + Z ds κN X p ∈ Λ ∗ + b V ( p/N β ) e − sB ( η ) ( b p b − p + b ∗ p b ∗− p ) e sB ( η ) = κN X p ∈ Λ ∗ + b V ( p/N β ) a ∗ p a p + X n,m ≥ ( − m + n m ! n !( m + n + 1) κN X p ∈ Λ ∗ + b V ( p/N β ) h ad ( n ) B ( η ) ( b p )ad ( m ) B ( η ) ( b − p ) + h.c. i (7.55)The first term on the r.h.s. of (7.55) is clearly bounded by Cκ N + /N . Let us focus nowon the sum over m, n . By Lemma 2.5, the operator κN X p ∈ Λ ∗ + b V ( p/N β )ad ( n ) B ( η ) ( b p )ad ( m ) B ( η ) ( b − p )can be written as the sum of 2 n + m n ! m ! terms of the formL = κN X p ∈ Λ ∗ + b V ( p/N β )Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . η j k ; η ℓ p ϕ α ℓ p ) × Λ ′ . . . Λ ′ i N − k Π (1) ♯ ′ ,♭ ′ ( η m , . . . , η m k ; η ℓ p ϕ − α ℓ p ) (7.56)with i , i , k , k , ℓ , ℓ ∈ N , j , . . . , j k , m , . . . , m k ∈ N \{ } , α r = ( − r and whereeach Λ r and Λ ′ r -operator is either a factor ( N − N + ) /N , a factor ( N + 1 − N + ) /N or aΠ (2) -operator of the form (7.52).If ℓ + ℓ ≥
1, Lemma 7.1, part ii), implies that |h ξ, L ξ i| ≤ C n + m κ n + m +1 N X p ∈ Λ ∗ + | b V ( p/N β ) | p k ( N + + 1) / ξ k ≤ C n + m κ n + m +1 N β − k ( N + + 1) / ξ k (7.57)55f instead ℓ = ℓ = 0, we use Lemma 7.1, part ii), to write h ξ, L ξ i = κN X p ∈ Λ ∗ + b V ( p/N β ) h ξ, D ( p ) i + κN X p ∈ Λ ∗ + b V ( p/N β ) h ξ, D a p a − p ξ i (7.58)with k ( N + + 1) − / D ( p ) k ≤ C m + n κ m + n mN p k a p ξ k and k D k ≤ C m + n κ m + n . (7.59)Switching to position space to estimate the second term on the r.h.s. of (7.58) we find,for ℓ = ℓ = 0, |h ξ, L ξ i| ≤ C m + n κ m + n +1 mN k ( N + + 1) / ξ k + (cid:12)(cid:12)(cid:12)(cid:12) κN Z Λ × Λ N β V ( N β ( x − y )) h ξ, D ˇ a x ˇ a y ξ i (cid:12)(cid:12)(cid:12)(cid:12) ≤ C m + n κ m + n +1 mN k ( N + + 1) / ξ k + C m + n κ m + n +1 N Z Λ × Λ N β V ( N β ( x − y )) k ξ kk ˇ a x ˇ a y ξ k≤ C m + n κ m + n +1 mN k ( N + + 1) / ξ k + C m + n κ m + n +1 / √ N kV / N ξ kk ξ k (7.60)Combining (7.57) with (7.60) we conclude that |h ξ, L ξ i| ≤ C m + n κ m + n +1 ( m + 1) N β − k ( N + + 1) / ξ k + C m + n κ m + n +1 / √ N k ξ kkV / N ξ k Hence, for κ > m, n ∈ N ), (7.55) implies |h ξ, G (2 , N,β ξ i| ≤ CκN β − k ( N + + 1) / ξ k + Cκ / √ N k ξ kkV / N ξ k (7.61)This shows, on the one hand, that for every δ > C > |h ξ, G (2 , N,β ξ i| ≤ δ kV / N ξ k + Cκ k ( N + + 1) / ξ k (7.62)and, since as usual the commutator of N + with operators of the form (7.56) has againthe same form, |h ξ, (cid:2) G (2 , N,β , N + (cid:3) ξ i| ≤ δ kV / N ξ k + Cκ k ( N + + 1) / ξ k (7.63)On the other hand, taking into account Lemma 7.3, (7.61) also proves that |h ξ, G (2 , N,β ξ i| ≤ CN β − k ( N + + 1) / ( K + 1) / ξ k G (2) N,β on the r.h.s. of (7.47). Wehave, with Lemma 2.6, G (2 , N,β = κ X p ∈ Λ ∗ + b V ( p/N β ) e − B ( η ) (cid:2) b p b − p + b ∗ p b ∗− p (cid:3) e B ( η ) = X m,n ≥ ( − m + n m ! n ! κ X p ∈ Λ ∗ + b V ( p/N β ) h ad ( m ) B ( η ) ( b p )ad ( n ) B ( η ) ( b − p ) + h.c. i = κ X p ∈ Λ ∗ + b V ( p/N β ) × (cid:8)(cid:2) γ p b p + σ p b ∗− p (cid:3) (cid:2) γ p b − p + σ p b ∗ p (cid:3) + (cid:2) γ p b ∗ p + σ p b − p (cid:3) (cid:2) γ p b ∗− p + σ p b p (cid:3)(cid:9) + E ( V )3 (7.64)with the error term E ( V )3 = X m,n ≥ ( − m + n m ! n ! κ X p ∈ Λ ∗ + b V ( p/N β ) η mp b ♯ m α m p (cid:2) ad ( n ) B ( η ) ( b − p ) − η np b ♯ n − α n p (cid:3) + X m,n ≥ ( − m + n m ! n ! κ X p ∈ Λ ∗ + b V ( p/N β ) (cid:2) ad ( m ) B ( η ) ( b p ) − η mp b ♯ m α m p (cid:3) η np b ♯ n − α n p + X m,n ≥ ( − m + n m ! n ! κ X p ∈ Λ ∗ + b V ( p/N β ) × (cid:2) ad ( m ) B ( η ) ( b p ) − η mp b ♯ m α m p (cid:3) (cid:2) ad ( n ) B ( η ) ( b − p ) − η np b ♯ n − α n p (cid:3) + h.c. (7.65)We consider the first sum on the r.h.s. of (7.65). According to Lemma 2.6, the operator κ X p ∈ Λ ∗ + b V ( p/N β ) η mp b ♯ m α m p (cid:2) ad ( n ) B ( η ) ( b − p ) − η np b ♯ n − α n p (cid:3) can be written as the sum of the one termM = κ X p ∈ Λ ∗ + b V ( p/N β ) η m + np b ♯ m α m p (cid:18) N − N + N (cid:19) n +(1 − αn ) / (cid:18) N + 1 − N + N (cid:19) n − (1 − αn ) / − b ♯ n − α n p (7.66)and of 2 n n ! − = κ X p ∈ Λ ∗ + b V ( p/N β ) η mp b ♯ m α m p Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p ϕ − α ℓ p ) (7.67)57here i , k , ℓ ∈ N , j , . . . , j k ∈ N \{ } and where each Λ r -operator is either a factor( N − N + ) /N , a factor ( N + 1 − N + ) /N or a Π (2) -operator of the form (7.52). In everyterm of the form (7.67) we have k > |h ξ, M ξ i| ≤ C m + n κ m + n +1 ( n + 1) k ( N + + 1) / ξ k (7.68)and also |h ξ, M ξ i| ≤ C m + n κ m + n +1 ( n + 1) N β − k ( N + + 1) ξ k (7.69)Next, we bound the expectation of the term M , defined in (7.67). If m + ℓ ≥
1, wecan use Lemma 7.1, part iii), to estimate |h ξ, M ξ i| ≤ κ X p ∈ Λ ∗ + | b V ( p/N β ) || η p | m k ( N + + 1) / ξ k× k ( N + + 1) − / b ♯ m α m p Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p ϕ − α ℓ p ) ξ k≤ C n + m κ n + m +1 k ( N + + 1) / ξ k X p ∈ Λ ∗ + | b V ( p/N β ) |× (cid:26) (1 + m/N ) | p | k ( N + + 1) / ξ k + 1 | p | k a p ξ k + 1 N | p | k ( N + + 1) / ξ k (cid:27) ≤ C n + m κ n + m +1 k ( N + + 1) / ξ k Alternatively, again for m + ℓ ≥
1, we can also use Lemma 7.1, part iii), to show thebound |h ξ, M ξ i| ≤ κ X p ∈ Λ ∗ + | b V ( p/N β ) || η p | m k ( N + + 1) ξ k× k ( N + + 1) − b ♯ m α m p Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p ϕ − α ℓ p ) ξ k≤ C m + n κ m + n +1 ( m + 1) N k ( N + + 1) ξ k× X p ∈ Λ ∗ + | b V ( p/N β ) | (cid:20) | p | k ( N + + 1) ξ k + 1 p k a p ( N + + 1) / ξ k + 1 p k ξ k (cid:21) ≤ C m + n κ m + n +1 N β − k ( N + + 1) ξ k If now m = ℓ = 0, we write h ξ, M ξ i = κ X p ∈ Λ ∗ + b V ( p/N β ) h ξ, D ( p ) i + κ X p ∈ Λ ∗ + b V ( p/N β ) h ξ, D a p a − p ξ i (7.70)58ith D ( p ) and the operator D satisfying k ( N + + 1) − / D ( p ) k ≤ C n κ n +1 nN p k a p ξ k and k D ∗ ξ k ≤ C n κ n +1 N k ( N + + 1) ξ k . (7.71)Switching to position space to estimate the second term on the r.h.s. of (7.70), weconclude that |h ξ, M ξ i| ≤ C n κ n +1 nN k ( N + + 1) / ξ k + C n κ n +1 N Z Λ × Λ dxdyN β V ( N β ( x − y )) k ˇ a x ˇ a y ξ kk ( N + + 1) ξ k≤ C n κ n +1 N k ( N + + 1) / ξ k + C n κ n +1 / √ N kV / N ξ kk ( N + + 1) ξ k (7.72)With (7.68) and (7.72), we can control the first sum on the r.h.s. of (7.65). The secondand third sum can be controlled similarly. We conclude that, if κ > n, m ∈ N ), |h ξ, E ( V )3 ξ i| ≤ Cκ k ( N + + 1) / ξ k + Cκ / k ( N + + 1) / ξ kkV / N ξ k Hence, for every δ >
C > |h ξ, E ( V )3 ξ i| ≤ δ kV / N ξ k + Cκ k ( N + + 1) / ξ k (7.73)and (since the commutator with N + of every term of the form (7.66), (7.67) is again anoperator with the same form, up to a constant bounded by C ( n + 1)) |h ξ, (cid:2) N + , E ( V )3 (cid:3) ξ i| ≤ δ kV / N ξ k + Cκ k ( N + + 1) / ξ k (7.74)Combining (7.69) with (7.72), we arrive moreover with Lemma 7.3 at the bound |h ξ, E ( V )3 ξ i| ≤ CN β − k ( N + + 1) / ( K + 1) / ξ k . (7.75)From (7.47), (7.48), (7.64) and from the definition (7.46), we obtain E ( V ) N,β = E ( V )1 + G (2 , N,β + E ( V )3 Hence, from the bounds (7.53), (7.62) and (7.75), we conclude that |h ξ, E ( V ) N,β ξ i| ≤ CN β − k ( K + 1) / ( N + + 1) / ξ k Furthermore, with the definition (7.45), we find that e E ( V ) N,β = E ( V )0 + E ( V )1 + G (2 , N,β + E ( V )3 E ( V )0 = X p ∈ Λ ∗ + κ b V ( p/N β )[ σ p + σ p γ p − η p ] + X p ∈ Λ ∗ + κ b V ( p/N β )( γ p + σ p ) b ∗ p b p + κ X p ∈ Λ ∗ + b V ( p/N β )[( γ p + σ p ) − b p b − p + b ∗ p b ∗− p )Since | σ p + σ p γ p − η p | ≤ Cκ | p | − , | γ p + σ p | ≤ C and | ( γ p + σ p ) − | ≤ Cκ | p | − , weeasily find that |h ξ, E ( V )0 ξ i| ≤ Cκ k ( N + + 1) / ξ k |h ξ, (cid:2) N + , E ( V )0 (cid:3) ξ i| ≤ Cκ k ( N + + 1) / ξ k for all ξ ∈ F ≤ N + . Together with the estimates (7.53), (7.54), (7.62), (7.63), (7.73), (7.74),we conclude that for every δ > C > |h ξ, e E ( V ) N,β ξ i| ≤ δ kV / N ξ k + Cκ k ( N + + 1) / ξ k |h ξ, (cid:2) N + , e E ( V ) N,β (cid:3) ξ i| ≤ δ kV / N ξ k + Cκ k ( N + + 1) / ξ k G (3) N Recall from (3.3) that G (3) N,β = e − B ( η ) L (3) N,β e B ( η ) = 1 √ N X p,q ∈ Λ ∗ + : p + q =0 b V ( p/N β ) e − B ( η ) (cid:2) b ∗ p + q a ∗− p a q + a ∗ q a − p b p + q (cid:3) e B ( η ) (7.76)In the next proposition, we show how to control the operator G (3) N,β . Proposition 7.7.
Under the assumptions of Theorem 3.2, for every δ > there exists C > such that, on F ≤ N + , ±G (3) N,β ≤ δ V N + Cκ ( N + + 1) ± h G (3) N,β , i N + i ≤ C ( H βN + 1) Furthermore, we have ±G (3) N,β ≤ CN ( β − / ( K + 1)( N + + 1)60 roof. With Lemma 2.6, we write e − B ( η ) a ∗− p a q e B ( η ) = a ∗− p a q + Z ds e − sB ( η ) [ a ∗− p a q , B ( η )] e sB ( η ) = a ∗− p a q + Z e − sB ( η ) ( η q b ∗− p b ∗− q + η p b q b p ) e sB ( η ) = a ∗− p a q + X n,k ≥ ( − n + k n ! k !( n + k + 1) h η q ad ( n ) B ( η ) ( b ∗− p )ad ( k ) B ( η ) ( b ∗− q ) + η p ad ( n ) B ( η ) ( b q )ad ( k ) B ( η ) ( b p ) i Inserting this identity in (7.76), we find G (3) N,β = G (3 , N,β + G (3 , N,β + G (3 , N,β (7.77)with G (3 , N,β = X r ≥ ( − r r ! κ √ N X p,q ∈ Λ ∗ + : p + q =0 b V ( p/N β )ad ( r ) B ( η ) ( b ∗ p + q ) a ∗− p a q + h.c. G (3 , N,β = X n,k,r ≥ ( − n + k + r n ! k ! r !( n + k + 1) × κ √ N X p,q ∈ Λ ∗ + ,p + q =0 b V ( p/N β ) η q ad ( r ) B ( η ) ( b ∗ p + q )ad ( n ) B ( η ) ( b ∗− p )ad ( k ) B ( η ) ( b ∗− q ) + h.c. G (3 , N,β = X n,k,r ≥ ( − n + k + r n ! k ! r !( n + k + 1) × κ √ N X p,q ∈ Λ ∗ + ,p + q =0 b V ( p/N β ) η p ad ( r ) B ( η ) ( b ∗ p + q )ad ( n ) B ( η ) ( b p )ad ( k ) B ( η ) ( b q ) + h.c.(7.78)Let us consider first the term G (3 , N,β . With Lemma 2.5, the operator κ √ N X p,q ∈ Λ ∗ + ,p + q =0 b V ( p/N β ) η p ad ( r ) B ( η ) ( b ∗ p + q )ad ( n ) B ( η ) ( b p )ad ( k ) B ( η ) ( b q ) (7.79)can be expanded in the sum of 2 n + k + r n ! k ! r ! terms having the formP = κ √ N X p,q ∈ Λ ∗ + ,p + q =0 b V ( p/N β ) η p Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p + q ϕ α ℓ ( p + q ) ) ∗ Λ ∗ i . . . Λ ∗ i × Λ ′ . . . Λ ′ i N − k Π (1) ♯ ′ ,♭ ′ ( η m , . . . , η m k ; η ℓ p ϕ α ℓ p ) × Λ ′′ . . . Λ ′′ i N − k Π (1) ♯ ′′ ,♭ ′′ ( η s , . . . , η s k ; η ℓ ϕ α ℓ q ) (7.80)61or i , i , i , k , k , k , ℓ , ℓ , ℓ ∈ N , j , . . . , j k , m , . . . , m k , s , . . . , s k ∈ N \{ } , α ℓ i =( − ℓ i and where each Λ i , Λ ′ i , Λ ′′ i is either a factor ( N − N + ) /N , a factor ( N + 1 − N + ) /N or a Π (2) -operator of the form N − h Π (2) ♯,♭ ( η z , . . . , η z s ) (7.81)for some h, z . . . , z h ∈ N \{ } . We bound the expectation of (7.80) by |h ξ, P ξ i| ≤ κ √ N X p,q ∈ Λ ∗ + : p = − q | b V ( p/N β ) | η p × k Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p + q ϕ α ℓ ( p + q ) ) ξ k× k Λ ′ . . . Λ ′ i N − k Π (1) ♯ ′ ,♭ ′ ( η m , . . . , η m k ; η ℓ p ϕ α ℓ p ) × Λ ′′ . . . Λ ′′ i N − k Π (1) ♯ ′′ ,♭ ′′ ( η s , . . . , η s k ; η ℓ ϕ α ℓ q ) ξ k From Lemma 7.1, part i) and ii), we conclude that |h ξ, P ξ i|≤ C n + k + r κ n + k + r +2 × √ N X p,q ∈ Λ ∗ + : p = − q p n p + q ) k ( N + + 1) / ξ k + k a p + q ξ k o × n (1 + r/N ) p q k ( N + + 1) ξ k + (1 + r/N ) p k a q ( N + + 1) / ξ k + 1 q k a p ( N + + 1) / ξ k + k a p a q ξ k o ≤ C n + k + r (1 + r ) κ n + k + r +2 √ N k ( N + + 1) ξ kk ( N + + 1) / ξ k Hence, for κ > (cid:12)(cid:12) h ξ, G (3 , N,β ξ i (cid:12)(cid:12) ≤ Cκ √ N k ( N + + 1) ξ kk ( N + + 1) / ξ k (7.82)Next, we consider the term G (3 , N,β in (7.78) (we take its hermitian conjugate). Sincewe will use the potential energy operator to control this term, it is convenient to switchto position space. We write κ √ N X p,q ∈ Λ ∗ + ,p + q =0 b V ( p/N β ) η q ad ( r ) B ( η ) ( b − q )ad ( n ) B ( η ) ( b − p )ad ( k ) B ( η ) ( b p + q )= κ √ N Z Λ × Λ dxdy N β V ( N β ( x − y ))ad ( r ) B ( η ) ( b (ˇ η ℓ x )ad ( n ) B ( η ) (ˇ b y )ad ( k ) B ( η ) (ˇ b x ) (7.83)where we used the notation ˇ η s to indicate the Fourier transform of the sequence Λ ∗ ∋ p → η sp , and ˇ η sx denotes the function (or the distribution, if s = 0) z → ˇ η sx ( z ) = ˇ η s ( z − x ).62ith Lemma 2.5, the r.h.s. of (7.83) can be written as the sum of 2 n + k + r n ! k ! r ! terms,all having the formQ = κ √ N Z Λ × Λ dxdy N β V ( N β ( x − y )) Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; ˇ η ℓ x ) × Λ ′ . . . Λ ′ i N − k Π (1) ♯ ′ ,♭ ′ ( η m , . . . , η m k ; ˇ η ℓ y ) × Λ ′′ . . . Λ ′′ i N − k Π (1) ♯ ′′ ,♭ ′′ ( η s , . . . , η s k ; ˇ η ℓ x ) (7.84)where i , i , i , k , k , k , ℓ , ℓ , ℓ ∈ N , j , . . . , j k , m , . . . , m k , s , . . . , s k ∈ N \{ } andwhere each operator Λ i , Λ ′ i , Λ ′′ i is either a factor ( N − N + ) /N , a factor ( N + 1 − N + ) /N or a Π (2) -operator of the form (7.81). To estimate the expectation of (7.84), we firstassume that ( ℓ , ℓ ) = (0 , |h ξ, Q ξ i| ≤ κ √ N Z Λ × Λ dxdy N β V ( N β ( x − y )) × k N − k Π (1) ♯,♭ ( η j , . . . , η j k ; ˇ η ℓ +1 x ) ∗ Λ ∗ i . . . Λ ∗ ξ k× (cid:13)(cid:13)(cid:13) Λ ′ . . . Λ ′ i N − k Π (1) ♯ ′ ,♭ ′ ( η m , . . . , η m k ; ˇ η ℓ y ) × Λ ′′ . . . Λ ′′ i N − k Π (1) ♯ ′′ ,♭ ′′ ( η s , . . . , η s k ; ˇ η ℓ x ) ξ (cid:13)(cid:13)(cid:13) (7.85)With Lemma 7.2 we estimate k N − k Π (1) ♯,♭ ( η j , . . . , η j k ; ˇ η ℓ +1 x ) ∗ Λ ∗ i . . . Λ ∗ ξ k ≤ C r κ r +1 k ( N + + 1) / ξ k (7.86)and, using the condition ( ℓ , ℓ ) = (0 , (cid:13)(cid:13)(cid:13) Λ ′ . . . Λ ′ i N − k Π (1) ♯ ′ ,♭ ′ ( η m , . . . , η m k ; ˇ η ℓ y )Λ ′′ . . . Λ ′′ i N − k Π (1) ♯ ′′ ,♭ ′′ ( η s , . . . , η s k ; ˇ η ℓ x ) ξ (cid:13)(cid:13)(cid:13) ≤ C n + k κ n + k n (1 + k/N ) k ( N + + 1) ξ k + (1 + k/N ) k ˇ a x ( N + + 1) / ξ k + k ˇ a y ( N + + 1) / ξ k + k ˇ a x ˇ a y ξ k o . Inserting these bounds in (7.85), we arrive at |h ξ, Q ξ i| ≤ C n + k + r κ n + k + r +2 (1 + k ) k ( N + + 1) / ξ k× √ N Z Λ × Λ dxdy N β V ( N β ( x − y )) × n k ( N + + 1) ξ k + k ˇ a x ( N + + 1) / k + k ˇ a y ( N + + 1) / ξ k + k ˇ a x ˇ a y ξ k o ≤ C n + k + r κ n + k + r +2 (1 + k ) √ N k ( N + + 1) ξ kk ( N + + 1) / ξ k + C n + k + r κ n + k + r +1 (1 + k ) kV / N ξ kk ( N + + 1) / ξ k . (7.87)63or ( ℓ , ℓ ) = (0 ,
1) we can proceed similarly. The only additional remark is that, in thiscase, the the commutator [ˇ a y , a ∗ (ˇ η x )] = ˇ η ( x − y )between the annihilation operator associated with the second Π (1) -factor (the one con-taining ˇ η ℓ y ) and the creation operator a ∗ (ˇ η x ) associated with the third Π (1) -operator,gives a vanishing contribution to the expectation h ξ, Q ξ i , for all ξ ∈ F ≤ N + (because ofthe assumption that ξ is orthogonal to ϕ ).With (7.87) we conclude that, if κ > (cid:12)(cid:12) h ξ, G (3 , N,β ξ i (cid:12)(cid:12) ≤ Cκ √ N k ( N + + 1) / ξ kk ( N + + 1) ξ k + Cκ kV / N ξ kk ( N + + 1) / ξ k (7.88)Finally, we consider the term G (3 , N,β in (7.78). From Lemma 2.5, each operator κ √ N X p,q ∈ Λ ∗ + : p + q =0 b V ( p/N β )ad ( r ) B ( η ) ( b ∗ p + q ) a ∗− p a q (7.89)can be written as the sum of 2 r r ! terms having the formR = κ √ N X p,q ∈ Λ ∗ + : p + q =0 b V ( p/N β ) N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ p + q ϕ α ℓ ( p + q ) ) ∗ Λ ∗ i . . . Λ ∗ a ∗− p a q (7.90)for i , k , ℓ ∈ N , j , . . . , j k ∈ N \{ } , α ℓ = ( − ℓ , and where each Λ j operator is eithera factor ( N − N + ) /N , a factor ( N + 1 − N + ) /N of a Π (2) -operator of the form (7.81). If ℓ ≥
2, we use Lemma 7.1, part iii), to bound |h ξ, R ξ i| ≤ Cκ √ N X p,q ∈ Λ ∗ + : p = − q | η p + q | ℓ k a q ξ k× k a − p Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; ϕ α ℓ ( p + q ) ) ξ k≤ C r κ r +1 √ N X p,q ∈ Λ ∗ + : p = − q p + q ) k a q ξ k (cid:26) k a − p ( N + + 1) / ξ k + rN p k ( N + + 1) ξ k (cid:27) ≤ C r κ r +1 (1 + r ) √ N k ( N + + 1) / ξ kk ( N + + 1) ξ k (7.91)If ℓ = 1, we commute the operator a − ( p + q ) (or the b − ( p + q ) operator) appearing inthe Π (1) -operator in (7.90) to the right, and the operator a ∗− p to the left (it is important64o note that [ a − ( p + q ) , a ∗− p ] = 0 since q = 0). Lemma 7.1, part iii), implies that |h ξ, R ξ i| ≤ C r κ r +1 √ N X p,q ∈ Λ ∗ + : p = − q | b V ( p/N β ) | p + q ) × n rN p k ( N + + 1) ξ kk a q ξ k + 1 N ( p + q ) k a − p ( N + + 1) / ξ kk a q ξ k + k a − p ξ kk a − ( p + q ) a q ξ k o ≤ C r κ r +1 √ N k ( N + + 1) ξ kk ( N + + 1) / ξ k (7.92)Finally, if ℓ = 0, we commute a ∗− p to the left. With Lemma 7.1, we find h ξ, R ξ i = κ √ N X p,q ∈ Λ ∗ + : p + q =0 b V ( p/N β ) h D ( p, q ) , a q ξ i + κ √ N X p,q ∈ Λ ∗ + : p + q =0 b V ( p/N β ) h D a − p a p + q ξ, a q ξ i (7.93)where k D ( p, q ) k ≤ C r κ r rN p k a p + q ( N + + 1) / ξ k and k D k ≤ C r κ r . Switching to position space to control the second term on the r.h.s.of (7.93), we conclude therefore that |h ξ, R ξ i| ≤ C r κ r +1 rN / X p,q ∈ Λ ∗ + | b V ( p/N β ) | p k a q + p ( N + + 1) / ξ kk a q ξ k + C r κ r +1 √ N Z Λ × Λ dxdy N β V ( N β ( x − y )) k ˇ a x ˇ a y ξ kk ˇ a y ξ k≤ C r κ r +1 r √ N k ( N + + 1) ξ kk ( N + + 1) / ξ k + C r κ r +1 / kV / N ξ kk ( N + + 1) / ξ k Together with (7.91) and (7.92), the last estimate implies that, if κ > |h ξ, G (3 , N,β ξ i| ≤ Cκ √ N k ( N + + 1) / ξ kk ( N + + 1) ξ k + Cκ / kV / N ξ kk ( N + + 1) / ξ k (7.94)Combining the last bound with (7.82) and (7.88) (and using the fact that N + ≤ N on F ≤ N + ), we easily obtain that for every δ > C > ±G (3) N,β ≤ δ V N + Cκ ( N + + 1)As usual, we can show the same bound for the commutator of G (3) N,β with N + (simplybecause the commutator of N + with all terms of the form (7.80), (7.84) and (7.90) hasagain the same form, up to a constant bounded by C ( n + k + r )), i.e. ± (cid:2) G (3) N,β , i N + (cid:3) ≤ δ V N + Cκ ( N + + 1)65inally, combining (7.82), (7.88) and (7.94) with Lemma 7.3, we also obtain ±G (3) N,β ≤ CN ( β − / ( N + + 1)( K + 1) . G (4) N From (3.3) we have G (4) N,β = e − B ( η ) L (4) N,β e B ( η ) = 12 N X p,q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − p, − q b V ( r/N β ) e − B ( η ) a ∗ p + r a ∗ q a p a q + r e B ( η ) We define the operator E (4) N through G (4) N,β = V N + 12 N X p,q ∈ Λ ∗ + b V (( p − q ) /N β ) η p η q + 12 N X p,q ∈ Λ ∗ + b V (( p − q ) /N β ) η q ( b ∗ p b ∗− p + b p b − p ) + E (4) N,β (7.95)In the next proposition, we estimate the error term E (4) N,β . Proposition 7.8.
Under the assumptions of Theorem 3.2, for every δ > there exists C > such that, on F ≤ N + , ±E (4) N,β ≤ δ V N + Cκ ( N + + 1) ± h E (4) N,β , i N + i ≤ C ( H βN + 1) Furthermore, we find ±E (4) N,β ≤ CN ( β − / ( H βN + 1)( N + + 1) Proof.
We have G (4) N,β = κ N X p,q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − p,q b V ( r/N ) e − B ( η ) a ∗ p a ∗ q a q − r a p + r e B ( η ) = V N + κ N X p,q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − p,q b V ( r/N ) Z ds e − sB ( η ) (cid:2) a ∗ p a ∗ q a q − r a p + r , B ( η ) (cid:3) e sB ( η ) = V N + κ N X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q b V ( r/N ) η q + r Z ds (cid:16) e − sB ( η ) b ∗ q b ∗− q e sB ( η ) + h.c. (cid:17) + κN X p,q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = p, − q b V ( r/N ) η q + r Z ds (cid:16) e − sB ( η ) b ∗ p + r b ∗ q a ∗− q − r a p e sB ( η ) + h.c. (cid:17) (7.96)66xpanding again e − sB ( η ) a ∗− q − r a p e sB ( η ) = a ∗− q − r a p + Z s dτ e − τB ( η ) (cid:2) a ∗− q − r a p , B ( η ) (cid:3) e − τB ( η ) = a ∗− q − r a p + Z s dτ e − τB ( η ) (cid:0) η p b ∗− p b ∗− q − r + η q + r b p b q + r (cid:1) e − τB ( η ) and using Lemma 2.6, we obtain G (4) N,β − V N = W + W + W + W (7.97)where we definedW = ∞ X n,k =0 ( − n + k n ! k !( n + k + 1) × κ N X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q b V ( r/N β ) η q + r (cid:16) ad ( n ) B ( η ) ( b q )ad ( k ) B ( η ) ( b − q ) + h.c. (cid:17) W = ∞ X n,k =0 ( − n + k n ! k !( n + k + 1) × κN X p,q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = p, − q b V ( r/N β ) η q + r (cid:16) ad ( n ) B ( η ) ( b ∗ p + r )ad ( k ) B ( η ) ( b ∗ q ) a ∗− q − r a p + h.c. (cid:17) (7.98)andW = ∞ X n,k,i,j =0 ( − n + k + i + j n ! k ! i ! j !( i + j + 1)( n + k + i + j + 2) κN X p,q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − p − q b V ( r/N β ) η q + r η p × (cid:16) ad ( n ) B ( η ) ( b ∗ p + r )ad ( k ) B ( η ) ( b ∗ q )ad ( i ) B ( η ) ( b ∗− p )ad ( j ) B ( η ) ( b ∗− q − r ) + h.c. (cid:17) W = ∞ X n,k,i,j =0 ( − n + k + i + j n ! k ! i ! j !( i + j + 1)( n + k + i + j + 2) × κN X p,q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − p − q b V ( r/N β ) η q + r × (cid:16) ad ( n ) B ( η ) ( b ∗ p + r )ad ( k ) B ( η ) ( b ∗ q )ad ( i ) B ( η ) ( b p )ad ( j ) B ( η ) ( b q + r ) + h.c. (cid:17) (7.99)In W , we isolate the contributions associated to ( n, k ) = (0 , , (0 , = κ N X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q b V ( r/N β ) η r + q ( b q b − q + h.c.) − κ N X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q b V ( r/N β ) η q + r ( b q [ B ( η ) , b − q ] + h.c.) + f W = κ N X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q b V ( r/N β ) η r + q ( b q b − q + h.c.) − κ N X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q b V ( r/N β ) η q + r η q + T + f W where we defined f W = ∗ X n,k ( − n + k n ! k !( n + k + 1) × κ N X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q b V ( r/N β ) η q + r (cid:16) ad ( n ) B ( η ) ( b q )ad ( k ) B ( η ) ( b − q ) + h.c. (cid:17) (7.100)with the sum P ∗ n,k running over all pairs ( n, k ) = (0 , , (0 , − κ N X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q b V ( r/N β ) η q + r ( b q [ B ( η ) , b − q ] + h.c.)+ κ N X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q b V ( r/N β ) η q + r η q =: T + T + T (7.101)with T = κN X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q b V ( r/N β ) η r + q η q (2 N + + 1 + N + /N + N /N )T = 2 κN X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q b V ( r/N β ) η r + q η q a ∗ q a q (cid:18) − N + + 1 N (cid:19) T = κN X q,m ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q b V ( r/N β ) η r + q η m a ∗ m a ∗− m a q a − q In the computation of T, we used the fact that[ B ( η ) , b − q ] = − η q (1 − N + /N ) b ∗ q + 1 N X m ∈ Λ ∗ + η m b ∗ m a ∗− m a − q Comparing with (7.95), we arrive at E (4) N,β = T + f W + W + W + W (7.102)68et us start by analyzing the operator T, defined in (7.101). Using (3.10), we estimate1 N X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q | b V ( r/N β ) || η r + q || η q | ≤ CN β − N X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q | b V ( r/N β ) | q ( q + r ) / Next, we observe that1 N X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q | b V ( r/N β ) | q ( q + r ) ≤ CN X r ∈ Λ ∗ | b V ( r/N β ) | | r | + 1 ≤ C k b V k ∞ N X | r |≤ N β | r | + 1 + C k b V ( ./N β ) k N β ≤ CN β − Hence, we conclude that1 N X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q | b V ( r/N β ) || η r + q || η q | ≤ CN β − (7.103)With this bound, we easily arrive at |h ξ, T ξ i| , |h ξ, T ξ i| ≤ Cκ N β − k ( N + + 1) / ξ k (7.104)To bound T , we switch to position space. We obtainT = κN X q,m ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q b V ( r/N β ) η r + q η m a ∗ m a ∗− m a q a − q = κN Z Λ × Λ dxdy N β V ( N β ( x − y ))ˇ η ( x − y )Dˇ a x ˇ a y where D = P m ∈ Λ ∗ + η m a ∗ m a ∗− m . Since k D ∗ ξ k ≤ Cκ k ( N + + 1) ξ k , we find |h ξ, T ξ i| ≤ Cκ N − k ( N + + 1) ξ k Z Λ × Λ dxdy N β V ( N β ( x − y )) | ˇ η ( x − y ) |k ˇ a x ˇ a y ξ k≤ Cκ N − β k ( N + + 1) ξ k Z Λ × Λ dxdy N β V ( N β ( x − y )) k ˇ a x ˇ a y ξ k≤ Cκ / N − k ( N + + 1) / ξ kkV / N ξ k Together with (7.104), we conclude that |h ξ, T ξ i| ≤ Cκ N β − k ( N + + 1) / ξ k + Cκ / N − k ( N + + 1) / ξ kkV / N ξ k . (7.105)Let us now consider the operator f W , defined in (7.100). According to Lemma 2.5,the operator κN X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q b V ( r/N β ) η q + r ad ( n ) ( b q )ad ( k ) ( b − q )69an be written as the sum of 2 n + k n ! k ! term s having the formX = κN X q ∈ Λ ∗ + ,r ∈ Λ ∗ : r = − q b V ( r/N β ) η q + r Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; η ℓ q ϕ α ℓ q ) × Λ ′ . . . Λ ′ i N − k Π (1) ♯ ′ ,♭ ′ ( η m , . . . , η m k ; η ℓ q ϕ − α ℓ q )where i , i , k , k , ℓ , ℓ ∈ N , j , . . . , j k , m , . . . , m k ∈ N \{ } , α ℓ i = ( − ℓ i and whereeach operator Λ r , Λ ′ r is either a factor ( N − N + ) /N , a factor ( N + 1 − N + ) /N or aΠ (2) -operator of the form N − h Π (2) ♯,♭ ( η z , . . . , η z p ) . (7.106)for h, z , . . . , z h ∈ N \{ } . To bound the expectation of the operator X, we distinguishtwo cases. If ℓ + ℓ ≥
1, we use Lemma 7.1, part ii), to estimate |h ξ, X ξ i| ≤ C n + k κ n + k +2 N k ( N + + 1) / ξ k× X q,r ∈ Λ ∗ + : r = − q | b V ( r/N β ) | ( q + r ) (cid:26) q (1 + k/N ) k ( N + + 1) / ξ k + 1 q k a q ξ k (cid:27) + C n + k κ n + k k ( N + + 1) / ξ k N X q,r ∈ Λ ∗ + : r = − q | b V ( r/N β ) || η q + r || η q | Here we used the fact that we excluded the pairs ( n, k ) = (0 , , (0 ,
1) to make sure that,if ℓ = 0 and ℓ = 1, then either k > k > ′ has to be a Π (2) -operator. From (7.103) and since, as we already showed in (7.18),sup q ∈ Λ ∗ + N X r ∈ Λ ∗ : r = − q | b V ( r/N β ) | q + r ) ≤ CN β − we conclude that, for ℓ + ℓ ≥ |h ξ, X ξ i| ≤ C n + k κ n + k +2 N β − k ( N + + 1) / ξ k (7.107)For ℓ = ℓ = 0, we use Lemma 7.1, part ii), to writeX = κN X q ∈ Λ ∗ + ,r ∈ Λ ∗ b V ( r/N β ) η q + r [D ( q ) + D a q a − q ] =: X + X where |h ξ, D ( q ) ξ i| ≤ C n + k κ n + k kN q k ( N + + 1) / ξ k and (since we excluded the term with ( n, k ) = (0 , k D ∗ ξ k ≤ C n + k N − κ n + k k ( N + + 1) ξ k
70e immediately obtain, using again (7.103), that |h ξ, X ξ i| ≤ C n + k κ n + k +2 N X q ∈ Λ ∗ + ,r ∈ Λ ∗ b V ( r/N β ) 1( q + r ) q k ( N + + 1) / ξ k ≤ C n + k κ n + k +2 N β − k ( N + + 1) / ξ k Switching to position space, we also find |h ξ, X ξ i| = (cid:12)(cid:12)(cid:12) κN Z Λ × Λ dxdy N β V ( N β ( x − y ))ˇ η ( x − y ) h D ∗ ξ, ˇ a x ˇ a y ξ i (cid:12)(cid:12)(cid:12) ≤ κN Z Λ × Λ dxdyN β V ( N β ( x − y )) | ˇ η ( x − y ) |k ˇ a x ˇ a y ξ kk D ∗ ξ k≤ C n + k κ n + k +2 N − β k ( N + + 1) ξ k Z Λ × Λ dxdyN β V ( N β ( x − y )) k ˇ a x ˇ a y ξ k≤ C n + k κ n + k +3 / N β − k ( N + + 1) / ξ kkV / N ξ k Combining the last two bounds with (7.107), and then summing over all n, k , we find |h ξ, f W ξ i| ≤ Cκ N β − k ( N + + 1) / ξ k + Cκ / N β − k ( N + + 1) / ξ kkV / N ξ k . (7.108)Next, we consider the expectation of the operator W , defined in (7.98). Since wewill need the potential energy operator to bound this term, it is convenient to switch toposition space. On F + , we findW = ∞ X n,k =0 ( − n + k n ! k !( n + k + 1) × κN Z Λ × Λ dxdyN β V ( N β ( x − y )) (cid:16) ad ( n ) B ( η ) (ˇ b ∗ x )ad ( k ) B ( η ) (ˇ b ∗ y ) a ∗ (ˇ η x )ˇ a y + h.c. (cid:17) (7.109)with the notation ˇ η x ( z ) = ˇ η ( x − z ). With Cauchy-Schwarz, we find (cid:12)(cid:12)(cid:12) κN Z Λ × Λ dxdy N β V ( N β ( x − y )) h ξ, ad ( n ) B ( η ) (ˇ b ∗ x )ad ( k ) B ( η ) (ˇ b ∗ y ) a ∗ (ˇ η x )ˇ a y ξ i (cid:12)(cid:12)(cid:12) ≤ κN Z Λ × Λ dxdy N β V ( N β ( x − y )) × k ( N + + 1) / ad ( k ) B ( η ) (ˇ b y )ad ( n ) B ( η ) (ˇ b x ) ξ kk ( N + + 1) − / a ∗ (ˇ η x )ˇ a y ξ k (7.110)We bound k ( N + + 1) − / a ∗ (ˇ η x )ˇ a y ξ k ≤ Cκ k ˇ a y ξ k (7.111)71ith Lemma 2.5, we estimate k ( N + +1) / ad ( k ) B ( η ) (ˇ b y )ad ( n ) B ( η ) (ˇ b x ) ξ k by the sum of 2 n + k n ! k !terms of the formZ = (cid:13)(cid:13)(cid:13) ( N + + 1) / Λ . . . Λ i N − k Π (1) ♯,♭ ( η j , . . . , η j k ; ˇ η ℓ y ) × Λ ′ . . . Λ ′ i N − k Π (1) ♯,♭ ( η m , . . . , η m k ; ˇ η ℓ x ) ξ (cid:13)(cid:13)(cid:13) (7.112)with i , i , k , k , ℓ , ℓ ≥ j , . . . , j k , m , . . . , m k ≥ i and Λ ′ i opera-tor is either a factor ( N − N + ) /N , ( N + 1 − N + ) /N or a Π (2) -operator of the form (7.106)(here ˇ η ℓ indicates the function with Fourier coefficients given by η ℓ p , for all p ∈ Λ ∗ + ).With Lemma 7.2, we findZ ≤ ( n + 1) C k + n κ k + n n k ( N + + 1) / ξ k + k ˇ a y ( N + + 1) ξ k + k ˇ a x ( N + + 1) ξ k + N β k ( N + + 1) / ξ k + √ N k ˇ a x ˇ a y ξ k o (7.113)Inserting (7.111) and (7.113) into (7.110) we obtain, for any ξ ∈ F ≤ N + , (cid:12)(cid:12)(cid:12)(cid:12) κN Z Λ × Λ dxdyN β V ( N β ( x − y )) h ξ, ad ( n ) B ( η ) (ˇ b ∗ x )ad ( k ) B ( η ) (ˇ b ∗ y ) a ∗ (ˇ η x )ˇ a y ξ i (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( n + 1)! k ! C n + k κ n + k +2 N Z dxdy N β V ( N β ( x − y )) k ˇ a y ξ k× n N k ( N + + 1) / ξ k + N k ˇ a y ξ k + N k ˇ a x ξ k + √ N k ˇ a x ˇ a y ξ k o ≤ ( n + 1)! k ! C n + k κ n + k +2 k ( N + + 1) / ξ k + ( n + 1)! k ! C n + k κ n + k +3 / k ( N + + 1) / ξ kkV / N ξ k Therefore, if κ > δ >
0, a constant
C > |h ξ, W ξ i| ≤ δ kV / N ξ k + Cκ k ( N + + 1) / ξ k (7.114)On the other hand, inserting (7.111) and (7.113) into (7.110), we also arrive at (cid:12)(cid:12)(cid:12)(cid:12) κN Z Λ × Λ dxdyN β V ( N β ( x − y )) h ξ, ad ( n ) B ( η ) (ˇ b ∗ x )ad ( k ) B ( η ) (ˇ b ∗ y ) a ∗ (ˇ η x )ˇ a y ξ i (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( n + 1)! k ! C n + k κ n + k +2 N Z Λ × Λ dxdy N β V ( N β ( x − y )) k ˇ a y ξ k× n ( √ N + N β ) k ( N + + 1) ξ k + √ N k ˇ a x ( N + + 1) / ξ k + √ N k ˇ a y ( N + + 1) / ξ k + √ N k ˇ a x ˇ a y ξ k o ≤ ( n + 1)! k ! C n + k κ n + k +2 N − min(1 − β, / k ( N + + 1) ξ k + ( n + 1)! k ! C n + k κ n + k +3 / k ( N + + 1) / ξ kkV / N ξ k κ > |h ξ, W ξ i| ≤ CN ( β − / k ( N + + 1) / ( H βN + 1) / ξ k . (7.115)Next, let us consider the term W , defined in (7.99). As above, we switch to positionspace. We findW = ∞ X n,k,i,j =0 ( − n + k + i + j n ! k ! i ! j !( i + j + 1)( n + k + i + j + 2) × κN Z dxdy N β V ( N β ( x − y )) × (cid:16) ad ( n ) (ˇ b ∗ x )ad ( k ) B ( η ) (ˇ b ∗ y )ad ( i ) B ( η ) ( b ∗ (ˇ η x ))ad ( j ) B ( η ) ( b ∗ (ˇ η y )) + h.c. (cid:17) (7.116)With Cauchy-Schwarz, we have (cid:12)(cid:12)(cid:12)(cid:12) κN Z dxdyN β V ( N β ( x − y )) h ξ, ad ( n ) B ( η ) (ˇ b ∗ x )ad ( k ) B ( η ) (ˇ b ∗ y )ad ( i ) B ( η ) (ˇ b ∗ (ˇ η x ))ad ( j ) B ( η ) (ˇ b (ˇ η y )) ξ i (cid:12)(cid:12)(cid:12)(cid:12) ≤ κN Z dxdy N β V ( N β ( x − y )) k ( N + + 1) / ad ( k ) B ( η ) (ˇ b y )ad ( n ) B ( η ) (ˇ b x ) ξ k× k ( N + + 1) − / ad ( i ) B ( η ) ( b (ˇ η x ))ad ( j ) ( b (ˇ η y )) ξ k Expanding ad ( i ) B ( η t ) ( b (ˇ η x ))ad ( j ) ( b (ˇ η y )) as in Lemma 2.5 and using Lemma 7.2, we obtain k ( N + + 1) − / ad ( i ) B ( η ) ( b (ˇ η x ))ad ( j ) ( b (ˇ η y )) ξ k ≤ i ! j ! C i + j κ i + j +2 k ( N + + 1) / ξ k (7.117)As for the norm k ( N + + 1) / ad ( k ) B ( η ) (ˇ b y )ad ( n ) B ( η ) (ˇ b x ) ξ k , we can estimate by the sum of2 n + k n ! k ! contributions of the form (7.112). With the bound (7.113), we can argue as inthe analysis of the term W . Similarly to (7.115) and (7.114), we conclude that, if κ > δ >
0, there exists
C > |h ξ, W ξ i| ≤ δ kV / N ξ k + Cκ k ( N + + 1) / ξ k |h ξ, W ξ i| ≤ CN ( β − / k ( N + + 1) / ( H βN + 1) / ξ k (7.118)The term W in (7.99) can be bounded similarly. First, we switch to position space:W = ∞ X n,k,i,j =0 ( − n + k + i + j n ! k ! i ! j !( i + j + 1)( n + k + i + j + 2) × κN Z dxdy N β V ( N β ( x − y )) (cid:16) ad ( n ) (ˇ b x )ad ( k ) (ˇ b y )ad ( i ) ( b (ˇ η x ))ad ( j ) (ˇ b y ) + h.c. (cid:17) (7.119)73he expectation of the operators on the r.h.s. of (7.119) can be bounded similarly as wedid for the operators on the r.h.s. of (7.116). The only difference is the fact that nowwe have to replace the estimate (7.117) with k ( N + + 1) − / ad ( i ) ( b (ˇ η x ))ad ( j ) (ˇ b y ) ξ k ≤ i ! j ! C i + j κ i + j +2 h k ( N + + 1) / ξ k + k ˇ a y ξ k i Hence, we obtain that, for every δ > |h ξ, W ξ i| ≤ δ kV / N ξ k + Cκ k ( N + + 1) / ξ k |h ξ, W ξ i| ≤ CN ( β − / k ( N + + 1) / ( H βN + 1) / ξ k . (7.120)Combining the bounds (7.105), (7.108), (7.114), (7.115), (7.118) and (7.120) weconclude, by (7.102), that, for all δ > C > ± E (4)
N,β ≤ δ V N + Cκ ( N + + 1) (7.121)and that, furthermore, ±E (4) N,β ≤ CN ( β − / ( N + + 1)( H βN + 1)As usual, the bound for the commutator of E (4) N,β with N + can be proven exactly as weproved (7.121). Combining the results of Prop. 7.4, Prop. 7.5, Prop. 7.6, Prop. 7.7 and Prop. 7.8, weconclude that G βN = ( N − κ b V (0) + X p ∈ Λ ∗ + p η p + κ b V ( p/N β ) η p + 12 N X q ∈ Λ ∗ + b V (( p − q ) /N β ) η p η q + H βN + X p ∈ Λ ∗ + p η p + κ b V ( p/N β )2 + 12 N X q ∈ Λ ∗ + b V (( p − q ) /N β ) η q (cid:2) b ∗ p b ∗− p + b p b − p (cid:3) + e E N,β where the error e E N,β = E (0) N,β + e E ( K ) N,β + e E ( V ) N,β + G (3) N,β + E (4) N,β is such that, for every δ >
C > ± e E N,β ≤ δ H βN + Cκ ( N + + 1) ± [ e E N,β , i N + ] ≤ C ( H βN + 1)Using the relation (3.12), we can rewrite G βN = ( N − κ b V (0) + κ X p ∈ Λ ∗ + b V ( p/N β ) η p + H βN + E ′ N,β (7.122)74ith E ′ N,β = X p ∈ Λ ∗ + η p h N λ
N,ℓ b χ ℓ ( p ) + λ N,ℓ X q ∈ Λ ∗ b χ ℓ ( p − q ) e η q − κ N b V ( p/N β ) e η i + X p ∈ Λ ∗ + h N λ
N,ℓ b χ ℓ ( p ) + λ N,ℓ X q ∈ Λ ∗ b χ ℓ ( p − q ) e η q − κ N b V ( p/N β ) e η i(cid:2) b ∗ p b ∗− p + b p b − p (cid:3) + e E N,β
Since, by Lemma 3.1,
N λ
N,ℓ ≤ Cκ uniformly in N , since | e η q | ≤ Cκ/ ( | q | + 1) from (3.9)and (3.11) and since | b χ ℓ ( p ) | ≤ C | p | − (see (5.8)), we conclude easily that for every δ > C > ±E ′ N,β ≤ δ H βN + Cκ ( N + + 1) ± (cid:2) E ′ N,β , i N + (cid:3) ≤ C ( H βN + 1) (7.123)Eq. (7.122) implies, in particular, that the ground state energy of the Hamiltonian (1.1)is such that E βN ≤ h Ω , G βN Ω i ≤ ( N − κ b V (0) + κ X p ∈ Λ ∗ + b V ( p/N β ) η p + C for a constant C > N . Inserting in (7.122) and using the first boundin (7.123) (taking for example δ = 1 /
4) we conclude that, for κ small enough, G βN − E βN ≥ H βN − C Furthermore, (7.122) and the second bound in (7.123) immediately give ± (cid:2) G βN , i N + (cid:3) ≤ C ( H βN + 1)which concludes the proof of part a) of Theorem 3.2. To show part b), we notice thatProp. 7.4, Prop. 7.5, Prop. 7.6, Prop. 7.7 and Prop. 7.8 also imply that G βN = ( N − κ b V (0)+ X p ∈ Λ ∗ + h p σ p + κ b V ( p/N β )( σ p + σ p γ p ) + κ N X q ∈ Λ ∗ + b V (( p − q ) /N β ) η p η q i + H βN + X p ∈ Λ ∗ + (cid:2) p σ p + κ b V ( p/N β )( γ p + σ p ) (cid:3) b ∗ p b p + X p ∈ Λ ∗ + h p σ p γ p + κ b V ( p/N β )( γ p + σ p ) + κ N X q ∈ Λ ∗ + b V (( p − q ) /N β ) η q i × ( b ∗ p b ∗− p + b p b − p )+ b E βN b E βN = E (0) N,β + E ( K ) N,β + E ( V ) N,β + G (3) N,β + E (4) N,β is such that ± b E βN ≤ CN ( β − / ( N + + 1)( K + 1) (7.124)Comparing with (3.16) and (3.17), we obtain that G βN = C βN + Q βN + E βN with E βN = b E βN + V N + κ e η N X p ∈ Λ ∗ + b V ( p/N β ) e η ( b ∗ p b ∗− p + b p b − p ) (7.125)Switching to position space, we have κ N X p ∈ Λ ∗ + b V ( p/N β ) e η h ξ, b p b − p ξ i = κ e η N Z Λ × Λ dxdy N β V ( N β ( x − y )) h ξ, ˇ b x ˇ b y ξ i Since | e η | ≤ C from (3.11), we find (cid:12)(cid:12)(cid:12) κ N X p ∈ Λ ∗ + b V ( p/N β ) e η h ξ, b p b − p ξ i (cid:12)(cid:12)(cid:12) ≤ CN Z Λ × Λ dxdy N β V ( N β ( x − y )) k ˇ a x ˇ a y ξ kk ξ k≤ CN − / kV / N ξ kk ξ k≤ CN − / k ( N + + 1) / ( K + 1) / ξ k where we used Lemma 7.3. Combining the last estimate with (7.124) and again withLemma 7.3, Eq. (7.125) implies that ±E βN ≤ CN ( β − / ( N + + 1)( K + 1)This completes the proof of part b) of Theorem 3.2. References [1] N. Benedikter, G. de Oliveira and B. Schlein. Quantitative derivation of the Gross-Pitaevskii equation.
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