We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson’s upper bound, which provides the correct leading order asymptotics for hard spheres. Afterward, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross–Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross–Pitaevskii limit. Here, we only sketch the main ideas; details will appear elsewhere.

In 1957, in a short but very influential paper,1 Dyson proved the first rigorous estimate for the ground state energy of an interacting Bose gas at low density. He considered a gas of hard spheres, but his approach can be easily extended to more general repulsive, short-range interactions. To leading order, he obtained the correct upper bound for the ground state energy and a lower bound, off by a factor around 14.

Forty years later, in 1997, Lieb–Yngvason2 managed to show a lower bound for the ground state energy, matching Dyson’s upper bound. They considered a system of N bosons, moving in the box ΛL = [0;L]3 (with periodic boundary conditions) and interacting through a repulsive (i.e., non-negative), two body radial potential $V:R3→R$, with integrable decay at infinity. They proved that the ground state energy per unit particle at fixed density ρ = N/L3, in the limit N, L, is such that

$e(ρ)=4πaρ1+o(1)$
(1.1)

in the dilute limit $ρa3→0$. Here, $a$ denotes the scattering length of the potential V and is defined through the solution of the zero-energy scattering equation

$−Δ+12Vf=0$

with the boundary condition f(x) → 1, as |x| → , so that $f(x)≃1−a/|x|$ asymptotically for large |x|. For the hard-sphere interaction, $a$ coincides with the radius of the spheres.

The next order corrections to (1.1) have been predicted in the physics literature by Lee–Huang–Yang,3 who derived the expression

$e(ρ)=4πaρ1+12815π(ρa3)1/2+o(ρa3)1/2$

again for the dilute limit $ρa3≪1$. It is interesting to observe how (1.1) and also (1.2) only depend on the scattering length $a$, and not on further details of the interaction. A rigorous lower bound matching the Lee–Huang–Yang formula has been recently derived by Fournais–Solovej, first for integrable potentials in Ref. 4 and then also for hard-sphere interactions in Ref. 5. A rigorous upper bound for the ground state energy per particle matching (1.2), on the other hand, has been first derived by Yau–Yin in Ref. 6 for smooth interaction potentials (a quasi-free trial state, which only recovered the correct asymptotics in the limit of weak potential, was previously proposed in Ref. 7). More recently, a simpler trial state has been proposed in Ref. 8 for interaction potentials $V∈L3(R3)$. Still missing is an upper bound matching (1.2) for non-integrable interactions, such as hard-sphere potentials (ironically, the upper bound is still missing exactly for the hard-sphere interaction that was first considered by Dyson, who probably regarded it as the simplest case).

In experiments, Bose gases are often very dilute. This leads us to consider also scaling regimes in which the density is coupled with the number of particles N and tends to 0, as N. An important example is the Gross–Pitaevskii regime, where the density is 1/N2. Rescaling lengths, the Gross–Pitaevskii regime corresponds to systems of N particles moving in the fixed box Λ = [0; 1]3 (with periodic boundary conditions) and interacting through a repulsive (i.e., non-negative) rescaled potential N2V(N·), whose scattering length is given by $a/N$, with $a$ denoting the scattering length of V. Translating (1.1) to the Gross–Pitaevskii limit, it implies that the ground state energy in this limit is given by

$ENGP=4πaN+o(N)$
(1.2)

as N.

In the recent years, making use of a rigorous version of Bogoliubov theory,9 it was possible, at least for integrable potentials, to go beyond the leading order estimate (1.2) and to resolve the ground state energy and the low-energy excitation spectrum up to errors vanishing in the limit of large N. For particles interacting through a repulsive, radial, and compactly supported interaction $V∈L3(R3)$, the ground state energy in the Gross–Pitaevskii regime was shown in Ref. 10 to satisfy

$ENGP=4πa(N−1)+eΛa2−12∑p∈Λ+*p2+8πa−|p|4+16πap2−(8πa)22p2+O(N−1/4),$
(1.3)

where

$eΛ=2−limM→∞∑p∈Z3\{0}:|p1|,|p2|,|p3|≤Mcos(|p|)p2,$
(1.4)

and in particular, the limit can be proven to exist. The second line of (1.3) corresponds, in the thermodynamic limit, to the second term in (1.2) [in this regime, the sum in (1.3) can be replaced by an integral, leading to (1.2); see Eqs. (A.26)-(A.29) in Ref. 11]. The correction $eΛa2$, on the other hand, is a finite volume effect, arising because $a$ is defined through the scattering equation (1.2) on the whole space $R3$, rather than on the box Λ, and it has no analog in (1.2).

Additionally, after subtraction of the ground state energy, the excitation spectrum below a threshold ζ > 0 was proven in Ref. 10 to consist of eigenvalues having the form

$∑p∈2πZ3\{0}np|p|4+16πap2+O(N−1/4ζ3),$
(1.5)

with $np∈N$ for all momenta $p∈2πZ3\{0}$. In other words, excited eigenvalues are determined, in good approximation, by the sum of the energies of quantized excitations, labeled by their momentum and characterized by the dispersion law $ε(p)=|p|4+16πap2$. Again, we observe that (1.3) and (1.5) only depend on the interaction potential through its scattering length $a$.

The results that we discussed in the last two paragraphs, concerning particles trapped in the unit torus Λ in the Gross–Pitaevskii limit, can also be extended to Bose gases on $R3$, confined by external trapping potentials Vext. In this case, Lieb–Seiringer–Yngvason proved in Ref. 12 that the ground state energy per particle is given, to leading order, by

$limN→∞ENGP(Vext)N=minφ∈L2(R3):‖φ‖=1EGP(φ),$

with the Gross–Pitaevskii energy functional

$EGP(φ)=∫|∇φ(x)|2+Vext(x)|φ(x)|2+4πa|φ(x)|4dx.$

More precise bounds, analogous to (1.3) and (1.5), have been recently established, for integrable potentials, in Refs. 13 and 14.

Estimates (1.3) and (1.5) have been proven in Ref. 10 for $V∈L3(R3)$. As indicated in Ref. 14, it would be relatively easy to extend them to $V∈L1(R3)$. On the other hand, extension to hard-spheres (or to more general non-integrable potentials with a hard-core) requires substantial new ideas. The goal of these notes is to present an upper bound for the ground state energy in the Gross–Pitaevskii regime, for particles interacting through a hard-sphere potential, consistent with (1.3); a complete proof of this estimate will appear in Ref. 15; here, we will only sketch some of the main ideas.

Before stating and discussing our new estimate in Sec. IV, we are going to review some tools that play a crucial role in its proof. In Sec. II, we will briefly recall Dyson’s upper bound for the leading order contribution to the ground state energy of a dilute gas of hard spheres. In Sec. III, on the other hand, we will review some of the main steps of the rigorous version of Bogoliubov theory for integrable potentials that led to (1.3) and (1.5).

In this section, we review Dyson’s upper bound for the ground state energy of hard-spheres in the dilute limit. Since the argument plays an important role in the derivation of our new estimate for the Gross–Pitaevskii regime, we are going to sketch a slightly different proof, compared with Dyson’s original work (in fact, our trial state is closer to the one proposed by Jastrow in Ref. 16).

We consider a gas of N hard spheres in the three dimensional torus ΛL = [0;L]3. We are looking for an upper bound to the ground state energy

$EN,L=inf⟨Ψ,∑j=1N−ΔxjΨ⟩‖Ψ‖2,$
(2.1)

with the infimum taken over all $Ψ∈L2(ΛLN)$, symmetric with respect to permutations of the N particles and satisfying the hard-sphere condition Ψ(x1, …, xN) = 0 if there exist i, j ∈ {1, …, N}, ij, with $|xi−xj|. Here (and in the following), we indicate with |xixj| the distance between xi and xj on the torus. We are interested in the thermodynamic limit, where N, L with fixed ρ = N/L3, and we focus on the low density regime, where $ρa3≪1$.

To get an upper bound for (2.1), we have to evaluate the energy of an appropriate trial state. To this end, we modify the non-interacting ground state $ΨN,La=0(x1,…,xN)≡1$ by adding correlations among particles. Since correlations are produced mainly by two-body scattering events, it seems natural to consider trial states having the form

$ΨN,L(x1,…,xN)=∏i
(2.2)

where $a≪ℓ≪L$ is a parameter that will be fixed later and f is meant to describe two-body correlations up to distance . Product trial functions like (2.2) have been first considered by Jastrow in Ref. 16 (Dyson, on the other hand, worked in Ref. 1 with a non-symmetric trial state, describing only nearest neighbor correlations). More precisely, in (2.2), we choose f as the ground state solution of the Neumann problem

$−Δfℓ=λℓfℓ,$
(2.3)

on the ball |x| ≤ , with the hard-sphere condition f(x) = 0 for $|x| and with the normalization f(x) = 1 for |x| = . We can then extend f to a function over the torus ΛL, setting f(x) = 1 for |x| > . With standard tools in analysis [see, for example, Ref. 18 (Lemma A.1)], we can study the eigenvalue problem (2.3) and prove that

$λℓ=3aℓ31+O(a/ℓ).$
(2.4)

Moreover, denoting by χ the characteristic function of the ball |x| ≤ , we find

$|∇fℓ(x)|≤Caχℓ(x)|x|2,0≤1−fℓ2(x)≤Caχℓ(x)|x|.$
(2.5)

To compute the energy of the trial state (2.2), we observe that

$−ΔxjΨN(x1,…,xN)∏i
(2.6)

where the sum in the last term runs over i, j, m ∈ {1, …, N} all different. From (2.3), we obtain, with the notation $x=(x1,…,xN)∈ΛLN$,

$Ψ,∑j=1N−ΔxjΨ=2λℓ∑i
(2.7)

Thus, using the permutation symmetry, we find

$⟨Ψ,∑j=1N−ΔxjΨ⟩‖Ψ‖2=N(N−1)λℓ∫χℓ(x1−x2)∏i

Defining $uℓ(x)=1−fℓ2(x)$ and estimating

$1−∑j=2Nuℓ(x1−xj)≤∏j=2Nfℓ2(x1−xj)≤1$

in the numerator and in the denominator of A, we can decouple the variable x1; with (2.4) and (2.5), we obtain

$A≤N2λℓ∫χℓ(x)dxL3−N∫uℓ(x)dx≤4πaρN⋅1+Caℓ+Cρaℓ2$

under the assumption that $a/ℓ,ρaℓ2≪1$. Similarly, using again (2.5), we can bound [this estimate could be improved using that, by symmetry, the integral of ∇f(x1x2) · ∇f(x2x3) over x1, x2, x3 vanishes]

$B≤CN3∫|∇fN,ℓ(x)|dx2L6−CNL3∫uℓ(x)dx≤CN3a2ℓ2L6(1−Cρaℓ2)≤CNρaρaℓ2.$

Thus, we conclude that

$EN,L≤4πaρN⋅1+Caℓ+Cρaℓ2.$

Choosing = ρ−1/3, we obtain the upper bound

$EN,L≤4πaρN⋅1+Caρ1/3,$

which captures the correct leading order of the ground state energy EN,L in the dilute limit.

Remark.

In (2.3), we cutoff correlations by choosing f as the solution of the Neumann problem on the ball |x| ≤ . Compared with the simpler truncation of the zero-energy scattering equation used in,1 our procedure has the advantage that the computation of the many-body kinetic energy (2.7) produces a nice two-body potential (in,1 the correct energy emerges as a boundary term, corresponding to a potential supported on |x| = ). This will play an important role in Section IV, where we will discuss how to modify (2.2)) to obtain better estimates.

In this section, we consider a gas of N bosons moving in the unit torus Λ = [0; 1]3 and interacting through a repulsive (i.e., non-negative) potential with scattering length of the order 1/N (Gross–Pitaevskii regime). The Hamilton operator takes the form

$HNGP=∑j=1N−Δxj+∑i
(3.1)

and acts on the Hilbert space $Ls2(ΛN)$, the subspace of L2N) consisting of functions that are symmetric with respect to permutations of the N particles. Here, we assume $V:R3→R$ to be a repulsive, radial, and compactly supported potential.

In fact, it is convenient to embed (3.1) in a family of Hamilton operators for trapped Bose gases, all acting on $Ls2(ΛN)$, given by

$HNβ=∑j=1N−Δxj+1N∑i
(3.2)

and depending on a parameter β ∈ [0; 1]. For β = 1, we recover the Gross–Pitaevskii Hamiltonian (3.1). For β = 0, on the other hand, (3.2) describes a mean-field scaling, where particles interact weakly through a potential varying on the same length scale L = 1 characterizing the box Λ. Mathematically, the mean-field scaling is more approachable, but from the point of view of physics, the Gross–Pitaevskii regime is, of course, more relevant.

Extending the bound (1.2) to all β ∈ [0; 1], we find that the ground state energy of (3.2) is given, to leading order, by

$limN→∞ENβN=4πlimN→∞aNβ,$

with $aNβ$ denoting N times the scattering length of the rescaled interaction potential N3β−1V(Nβ·). For β = 1, simple scaling of the zero-energy scattering Eq. (1.2) implies that $aNβ=1=a$; for β < 1, on the other hand, $aNβ$ converges toward its first Born approximation $V̂(0)/8π$ [the Born series for $aNβ$ is given in (3.14)]. We conclude that

$limN→∞ENβN=V̂(0)/2ifβ∈[0;1)4πaifβ=1.$
(3.3)

In the last decade, there has been progress in the mathematical understanding of the properties of trapped Bose gases described by (3.2), based on rigorous versions of Bogoliubov theory, that made it possible to go beyond the leading order estimate (3.3), at least for certain classes of integrable interactions. In the rest of this section, we are going to describe some of the main new ideas and results in this area. We focus here on integrable interaction potentials, excluding hard-spheres (we will come back to this point at the very end of this section and in Sec. IV).

The first important observation in Bogoliubov theory is that, close to the ground state energy, trapped Bose gases exhibit complete Bose–Einstein condensation. All particles, up to a fraction vanishing as N, are described by the same orbital φ0 defined by φ0(x) = 1 for all x ∈ Λ. In the mean-field limit and assuming the interaction potential to be bounded and positive definite, Bose–Einstein condensation follows from

$0≤∫dxdyV(x−y)∑j=1Nδ(x−xj)−N∑i=1Nδ(y−xi)−N=∑i,j=1NV(xi−xj)−N2V̂(0)=2∑i

which immediately gives

$HNβ=0≥N2V̂(0)+∑j=1N−Δxj−C.$

This operator inequality implies that for every approximate ground state ψN, i.e., for every $ψN∈Ls2(ΛN)$ satisfying

$ψN,HNβ=0−NV̂(0)/2ψN≤C,$

the expected number of orthogonal excitations of the condensate is bounded by

$⟨ψN,q1+⋯+qNψN⟩≤C,$
(3.4)

where we introduced the notation $q=1−|φ0〉〈φ0|$ for the projection on the orthogonal complement of the condensate wave function φ and where we used the gap in the Laplace operator to estimate $−Δ≥Cq$. Remark that (3.4) implies condensation in a strong sense, proving that the number of excitations remains bounded, as N.

Verifying the existence of Bose–Einstein condensation in the Gross–Pitaevskii regime (more generally, for β > 1/3, i.e., if the range of the potential is much smaller than the typical distance between particles) is much more challenging. The first proof was given by Lieb–Seiringer. In Refs. 19 and 20, they showed that every approximate ground state wave function of (3.1) is such that

$1N⟨ψN,q1+⋯+qNψN⟩→0$

as N. A similar result was later obtained in Ref. 21, with different tools. More recently, the stronger estimate (3.4), giving optimal bounds on the number of excitations, was proven to hold in the Gross–Pitaevskii regime and, in fact, for all β ∈ (0; 1], in Refs. 10 and 22. Different proofs and extensions to the case of particles trapped by external potentials have been obtained in Refs. 23–25.

After establishing Bose–Einstein condensation, the next step in Bogoliubov theory consists in factoring out the condensate and in focusing instead on its orthogonal excitations. To this end, we proceed as in Ref. 26, and we observe that every $ψN∈Ls2(ΛN)$ can be written as

$ψN=α0φ0⊗N+α1⊗sφ⊗(N−1)+⋯+αN−1⊗sφ0+αN$

with uniquely determined $αj∈L⊥2(Λ)⊗sj$ for j = 0, …, N (here, ⊗s denotes the symmetric tensor product). Defining UNψN = {α0, …, αN}, we map the Hilbert space $Ls2(ΛN)$ into the truncated Fock space

$F+≤N=⊕n=0NL⊥2(Λ)⊗sn$

constructed over the orthogonal complement $L⊥2(Λ)$ of the condensate wave function φ0. Through UN, we can define the excitation Hamiltonian $LNβ=UNHNβUN*$, acting on the excitation Hilbert space $F+≤N$.

To compute $LNβ$, it is convenient to rewrite (3.2) in momentum space using the formalism of second quantization with the creation and annihilation operators $ap*,ap$, defined for every $p∈Λ*=2πZ3$, satisfying canonical commutation relations

$ap,aq*=δp,q,ap,aq=ap*,aq*=0.$

We find

$HNβ=∑p∈Λ+*p2ap*ap+12N∑p,q,r∈Λ*V̂(r/Nβ)ap+r*aq*aq+rap.$

Denoting by $N+$ the number of particles operator on $F+≤N$ ($N+$ measures the number of excitations of the condensate), we have, from Ref. 26, the rules

$UNa0*a0UN*=N−N+,UNa0*apUN*=N−N+ap=:Nbp,UNap*a0UN*=ap*N−N+=:Nbp*,UNap*aqUN=ap*aq$
(3.5)

for all $p,q∈Λ+*=2πZ3\{0}$. Thus, we obtain

$LN,β=UNHNβUN*=LN,β(0)+LN,β(2)+LN,β(3)+LN,β(4),$

where we defined

$LN,β(0)=N−12NV̂(0)(N−N+)+V̂(0)2NN+(N−N+),LN,β(2)=∑p∈Λ+*p2ap*ap+∑p∈Λ+*V̂(p/Nβ)ap*N−N+−1Nap+12∑p∈Λ+*V̂(p/Nβ)(bp*b−p*+h.c.),LN,β(3)=1N∑p,q∈Λ+*,p+q≠0V̂(p/Nβ)bp+q*a−p*aq+aq*a−pbp+q,LN,β(4)=12N∑p,q∈Λ+*,r≠p,−qV̂(r/Nβ)ap+r*aq*apaq+r.$
(3.6)

In (3.5), we introduced operators $bp*,bp$ creating and, respectively, annihilating an excitation with momentum p (removing and, respectively, adding a particle to the condensate). On states with only few excitations, i.e., with $N+≪N$, we have $bp*≃ap*$ and bpap.

The cubic and the quartic terms in (3.6) look small in the limit N. In the mean-field regime, one can prove that, on states with few excitations, their contribution is indeed negligible. In other words, for β = 0, we can approximate

$LN,0≃LN,0(0)+LN,0(2)=(N−1)2V̂(0)+∑p∈Λ+*p2+V̂(p)ap*ap+12∑p∈Λ+*V̂(p)(bp*b−p*+h.c.).$

To diagonalize the resulting quadratic operator, we conjugate it with a (generalized) Bogoliubov transformation having the form

$Tτ=exp12∑p∈Λ+*τp(bp*b−p*−bpb−p).$
(3.7)

On states with few excitations, we expect (3.7) to act almost as a standard Bogoliubov transformation (with $b±p♯$ replaced by $a±p♯$), i.e.,

$Tτ*apTτ≃cosh(τp)ap+sinh(τp)a−p*.$

Choosing $τ∈ℓ2(Λ+*)$ so that $tanh(2τp)=−V̂(p)/(p2+V̂(p))$, we obtain, therefore,

$Tτ*LN,0Tτ≃(N−1)2V̂(0)−12∑p∈Λ+*p2+V̂(p)−|p|4+2p2V̂(p)+∑p∈Λ+*|p|4+2p2V̂(p)ap*ap.$

This shows that the wave function $UN*TτΩ∈Ls2(ΛN)$ is a good approximation for the ground state of $HNβ=0$, and it allows us to read off the ground state energy

$ENβ=0≃(N−1)2V̂(0)−12∑p∈Λ+*p2+V̂(p)−|p|4+2p2V̂(p).$
(3.8)

Moreover, it implies that the low-energy excitation spectrum of $HNβ=0−ENβ=0$ consists (up to corrections vanishing as N) of the eigenvalues

$∑p∈Λ+*np|p|4+2p2V̂(p),$
(3.9)

with $np∈N$, for all $p∈Λ+*$. Rigorous justifications of (3.8) and (3.9) for the mean-field regime have been obtained in Refs. 26–29 (recently, higher order expansions of the energy have been given in Refs. 30 and 31).

For β > 0, the analysis is more difficult. In fact, for β ≥ 1/2, cubic and quartic terms produce relevant contributions to the energy and can certainly not be neglected. The main difference with respect to the case β = 0 is that correlations among particles, which are negligible in the mean-field limit, become here more important. To model correlations, we fix 0 > 0 small enough (but independent of N), and we consider [similarly to (2.3)] the ground state solution of the Neumann problem

$−Δ+N3β2NV(Nβx)fN,ℓ0=λℓ0fN,ℓ0$
(3.10)

on the ball |x| ≤ 0, with the normalization $fN,ℓ0(x)=1$ if |x| = 0. We extend $fN,ℓ0$ to Λ, setting $fN,ℓ0(x)=1$ for |x| > 0, and we define $wN,ℓ0=1−fN,ℓ0$. For $p∈Λ+*$, we consider the coefficients $ηp=−NŵN,ℓ0(p)$, and we introduce the generalized Bogoliubov transformation

$Tη=exp12∑p∈Λ+*ηp(bp*b−p*−bpb−p).$
(3.11)

To understand (at least heuristically) the choice of the sequence η, we can write

$TηΩ≃C⁡exp12∫dxdyη̌(x−y)ax*ay*Ω$

for a constant $C∈R$, making sure that normalization is preserved. This leads to

$UN*TηΩ(x1,…,xN)≃Ce1N∑i
(3.12)

and [comparing with a wave function of the form (2.2)] explains the choice of η.

With Tη, we define the renormalized excitation Hamiltonian

$GN,β=Tη*LN,βTη=Tη*UNHNβUN*Tη.$

Similarly to what we did in (3.6) for $LN,β$, we can also decompose the operator $GN,β$ in constant, quadratic, cubic, and quartic terms. Through conjugation with Tη, we extracted important contributions to the energy from the quartic terms in (3.6). As a consequence, the constant term in $GN,β$ is now closer to the true ground state energy, compared with the constant term in (3.6); the vacuum expectations of $LN,β$ and of $GN,β$ are given, to leading order, by $NV̂(0)/2$ and $4πNaNβ$, respectively; the difference is of order Nβ, which is exactly the energy carried by the correlations generated by Tη. [With $η̌$ (x) ≃ C χ (|x| ≤ 0)/(|x| + N− β), we expect that Tη creates order $‖η‖22$ ≃ 1 excitations, with an energy of order $‖η‖H12$Nβ.]

For β ∈ (0; 1), it is possible to show that cubic and quartic terms in $GN,β$ are negligible in the limit of large N. Thus, in this case, the renormalized excitation Hamiltonian $GN,β$ can be approximated by its quadratic component and can therefore be diagonalized by means of another (generalized) Bogoliubov transformation Tτ, similar to (3.7), correcting the energy at order one. Following this strategy, it was shown in Ref. 33 that, for β ∈ (0; 1), the wave function $UN*TηTτΩ∈Ls2(ΛN)$ is a good approximation for the ground state vector of $HNβ$, that the ground state energy is given by

$ENβ≃4π(N−1)aNβ−12∑p∈Λ+*p2+V̂(0)−|p|4+2p2V̂(0)−V̂2(0)2p2,$
(3.13)

and that the low-energy excited eigenvalues of $HNβ−ENβ$ have the form

$∑p∈Λ+*np|p|4+2p2V̂(0),$

with $np∈N$ for all $p∈Λ+*$, up to errors that vanish as N. Here, $aNβ$ denotes N times the scattering length of the potential N3β−1V(Nβ·) and can be recovered through the finite Born series inserted in (3.13).

$8πaNβ=V̂(0)−12N∑p∈Λ+*V̂2(p/Nβ)p2+∑k=2mβ(−1)k(2N)k∑p∈Λ+*V̂(p/Nβ)p2×∑q1,q2,…,qk−1∈Λ+*V̂((p−q1)/Nβ)q12∏i=1k−2V̂((qi−qi+1)/Nβ)qi+12V̂(qk−1/Nβ),$
(3.14)

where the order mβ is chosen so large that the error is much smaller than 1/N and therefore only produces negligible contributions when inserted in (3.13). Comparing with (3.8), we conclude that contributions to $aNβ$ associated with k ≥ 2, which are relevant for β ≥ 1/2, emerge from cubic and quartic terms in (3.6).

In the Gross–Pitaevskii regime (i.e. for β = 1), even after renormalization with the Bogoliubov transformation Tη, cubic and quartic terms in $GNGP=GN,β=1$ are still important, and they cannot be neglected. In this case, following Ref. 33, we need to perform a second renormalization of the excitation Hamiltonian, this time conjugating it with a unitary operator having the form

$S=exp1N∑r∈PH,v∈PLηrsinh(ηv)br+v*b−r*b−v*+cosh(ηv)br+v*b−r*bv−h.c.$

given by the exponential of a cubic expression in (modified) creation and annihilation operators (here, PH, PL are appropriately defined sets of high and low momenta). This leads to the (twice) renormalized excitation Hamiltonian $MNGP=S*Tη*UNHNGPUNTηS$. Through conjugation with S, we extracted additional contributions from the cubic and the quartic terms in $GNGP$. At this point, the remaining cubic and quartic terms in $MNGP$ are small, and we can focus on its quadratic part. Diagonalization through a last Bogoliubov transformation Tτ produces the good ansatz $UN*TηSTτΩ∈Ls2(ΛN)$ for the ground state vector and leads to the bounds (1.3) for the ground state energy and (1.5) for the excited eigenvalues.

The approach that we discussed in this section and that led to the estimates (1.3) and (1.5) requires $V∈L3(R3)$ (as well as the standard assumptions that V is non-negative, radial, and of short range). As recently shown in Ref. 14 (for the case of a Bose gas trapped by an external potential), it could be extended to $V∈L1(R3)$ with a slightly different choice of the coefficients ηp in (3.11) [the stronger condition $V∈L3(R3)$ is only important to control properties of the solution of the Neumann problem (3.10)]. On the other hand, this approach cannot be easily extended to particles interacting through a hard-sphere potential. The problem is that it is difficult to impose the hard sphere condition on states that are defined through the action of the unitary transformations Tη, Tτ, S. Keeping, for example, $η̌=−NwN,ℓ0$, as defined after (3.10), we would have $η̌(x)=−N$ for $|x|, and according to the heuristic identity (3.12), $(UN*TηΩ)(x1,…,xN)$ would be far from zero, even when particles are closer than allowed by the hard-sphere condition [in this case, of course, the last approximation of the exponential in (3.12) is not valid].

In Sec. IV, we are going to present an alternative approach that we recently developed to obtain an upper bound for the ground state energy matching (1.3) for Bose gases interacting through hard-sphere potential.

In this section, we consider a gas of N hard spheres, with radius $aN=a/N$, moving on the three-dimensional unit torus Λ = [0; 1]3. We are interested in the ground state energy of the system, which is defined as

$ENHS=infΨN,∑j=1N−ΔxjΨN‖ΨN‖2,$

with the infimum taken over $ΨN∈Ls2(ΛN)$, ΨN ≠ 0, satisfying the hard-sphere condition

$ΨN(x1,…,xN)=0, if there exist i,j∈{1,2,…,N} with i≠j and |xi−xj|≤a/N.$
(4.1)

Theorem 4.1.
There existC, ɛ > 0 such that
$ENHS≤4πa(N−1)+eΛa2−12∑p∈Λ+*p2+8πa−|p|4+16πap2−(8πa)22p2+CN−ε,$
(4.2)
witheΛas defined in(1.4).

As usual for upper bounds, to prove Theorem 4.1, we need to find a trial state $ΨN∈Ls2(ΛN)$ satisfying the hard-sphere condition (4.1), whose energy matches the rhs of (4.2).

Summarizing the content of Secs. II and III, we have discussed two possible approaches to construct an appropriate trial state. Following the Dyson–Jastrow approach presented in Sec. II, we could consider a trial function having the form

$ΨN(x1,…,xN)=∏i
(4.3)

with fN, describing two-body correlations, up to the distance 1/N ≪ 1. A nice feature of (4.3) is the fact that it automatically satisfies the hard-sphere condition (4.1) [assuming, of course, that fN,(x) = 0 for $|x|]. On the other hand, to show a bound of the form (4.2), resolving the energy up to errors that vanish in the limit N, one would need to choose comparable with the size of the box of order one. For such values of , it seems extremely difficult to control product (4.3) and to compute its energy with sufficient accuracy. Following the Bogoliubov approach, we could instead consider a trial state of the form

$ΨN=UN*TηSTτΩ$
(4.4)

like those that have been considered in Refs. 10 and 33 to prove (1.3) for integrable potentials. Because of the nice algebraic properties of Bogoliubov transformations, these states are much more accessible to computation than (4.3). In particular, they allow us to create and to control correlations at all length scales (this is the reason why this approach has been successfully applied in Ref. 8 and, in a somehow different form, also in Refs. 6, 7, and 35 to compute the upper bound for integrable potentials in the thermodynamic limit). As indicated at the end of Sec. III, however, it seems very difficult to impose the hard-sphere condition on (4.4).

Our idea to show Theorem 4.1 is to combine the Dyson–Jastrow and the Bogoliubov approaches using a Jastrow factor to capture correlations at short distances and a Bogoliubov transformation to describe them at large distances. In other words, we consider a trial state having the form

$ΨN(x1,…,xN)=ΦN(x1,…,xN)⋅∏i
(4.5)

where fN, is chosen [similarly to (2.3), but now the hard core has the radius $a/N$] as the ground state of the Neumann problem

$−ΔfN,ℓ=λN,ℓfN,ℓ,$
(4.6)

on the ball |x| ≤ , with fN,(x) = 0 for all $|x|≤a/N$. Choosing N−1 but small enough, we take care of the hard-sphere conditions, and at the same time, we keep computations involving the Jastrow factor simple. The wave function ΦN, on the other hand, will be defined through appropriate (generalized) Bogoliubov transformations; to resolve the ground state energy to the desired precision, it must produce the correct correlations on all length scales larger than . Because of the presence of the Jastrow factor, however, it does not need to satisfy hard-sphere conditions. We find it convenient to choose ΦN with ‖ΦN‖ = 1; as a consequence, ΨN will not be normalized in $Ls2(ΛN)$ (but we will show later that ‖ΨN‖ → 1, as N).

Let us now compute the energy of the trial state (4.5). We find

$−ΔxjΨN(x1,…,xN)∏i

where the sum in the last term runs over i, j, m ∈ {1, …, N} all different [note that, for ΦN ≡ 1, the first term vanishes and the other two appeared already in (2.6)]. Integrating by parts and using (4.6), we conclude that

$ΨN,∑j=1N−ΔxjΨN=∑j=1N∫|∇xjΦN(x)|2∏i
(4.7)

where x = (x1, …, xN) ∈ ΛN. The three-body term on the last line turns out to be negligible, with the appropriate choice of and of the wave function ΦN. In fact, using permutation invariance and the operator inequality

$±W(x1−x2)W(x1−x3)≤C‖W‖r2(1−Δx1)(1−Δx2)(1−Δx3)$

valid for all r > 1, we can bound its contribution by

$1‖ΨN‖2∑i,j,m∫∇fN,ℓ(xj−xi)fN,ℓ(xj−xi)⋅∇fN,ℓ(xj−xm)fN,ℓ(xj−xm)|ΦN(x)|2∏i
(4.8)

For a rough estimate, we can approximate $fN,ℓ≃1−aχℓ(x)/(N|x|)$ (which is what we would find cutting off the solution of the zero-energy scattering equation at |x| = ). This gives

$‖∇fN,ℓ‖r2≤Cℓ6−4r/N2$
(4.9)

for 1 ≤ r < 3/2.

Although we still need to define ΦN, we already know that it will have to describe two-body correlations on scales |x| ≥ . To get an idea of the size of the expectation on the rhs of (4.8), we can therefore replace ΦN [after integrating out the other (N − 3) variables] by the three-body wave function

$1−aN(|x1−x2|+ℓ)1−aN(|x2−x3|+ℓ)1−aN(|x1−x3|+ℓ).$
(4.10)

$⟨ΦN,(−Δx1)ΦN⟩≲1Nℓ,⟨ΦN,(−Δx1)(−Δx2)ΦN⟩≲1N2ℓ3,⟨ΦN,(−Δx1)(−Δx2)(−Δx3)ΦN⟩≲1N3ℓ4.$
(4.11)

Since ΦN describes two-body correlations, it is not surprising that adding a second Laplacian costs more in the estimates than introducing the first and the third. Of course, so far (4.11) are only heuristic bounds; to make our arguments rigorous, after proper choice of ΦN, we should verify that (4.11) really holds true (the rigorous bounds are proven in Ref. 15).

As for the norm of ΨN, appearing in the denominator in (4.8), we set $uN,ℓ=1−fN,ℓ2$ and we bound

$∏i
(4.12)

This implies that

$‖ΨN‖2≥1−N(N−1)2∫uN,ℓ(x1−x2)|ΦN(x)|2dx.$
(4.13)

With the heuristic approximation $fN,ℓ≃1−aχℓ(x)/(N|x|)$, we find ‖uN,1Cℓ2/N. From the operator inequality,

$±W(x1−x2)≤C‖W‖1(1−Δx1)3/4+δ/2(1−Δx2)3/4+δ/2$

valid for any δ > 0, and using again (4.10) to heuristically estimate

$⟨ΦN,(−Δx1)3/4+δ/2(−Δx2)3/4+δ/2ΦN⟩≲1N2ℓ2+δ,$

we conclude that

$N(N−1)2∫uN,ℓ(x1−x2)|ΦN(x)|2dx≤CN2‖uN,ℓ‖1⟨ΦN,(1−Δx1)3/4+δ(1−Δx2)3/4+δΦN⟩≤CNℓ21+1N2ℓ2+2δ,$
(4.14)

which, in particular, implies (choosing δ > 0 small enough) that ‖ΨN‖ ≥ 1/2.

From (4.8), we expect, therefore, that for every δ > 0, there exists C > 0 with

$1‖ΨN‖2∑i,j,m∫∇fN,ℓ(xj−xi)fN,ℓ(xj−xi)⋅∇fN,ℓ(xj−xm)fN,ℓ(xj−xm)|ΦN(x)|2∏i

Fixing δ > 0 small enough, we conclude that this term is negligible in the limit N for every choice of satisfying N−1N−1/2.

Hence, let us focus on the first two terms on the rhs of (4.7). We define

$Ekin(ΦN)=∑j=1N∫|∇xjΦN(x)|2∏i
(4.15)

Complementing (4.12) with the upper bound

$∏i

the kinetic term defined in (4.15) can be estimated by

$Ekin(ΦN)≤N∫|∇x1ΦN(x)|2dx−N∫|∇x1ΦN(x)|2∑i
(4.16)

with a small error $Ekin$, which can be estimated arguing similarly to what we did to bound the rhs of (4.8) and vanishes, as N, if N−1N−2/3. Instead of (4.9), we use here the fact [which can be justified by $fN,ℓ≃1−aχℓ(x)/(N|x|)$] that for any 1 ≤ r < 3,

$‖uN,ℓ‖r≤Cℓ3/r−1/N.$

Moreover, to control ΦN, we need estimates similar to (4.11) but with derivatives hitting more particles (up to five, in fact); see Ref. 15 (Sec. 4) for more details.

As for the potential term in (4.15), we obtain

$Epot(ΦN)≤N(N−1)λN,ℓ∫χℓ(x1−x2)fN,ℓ2(x1−x2)|ΦN(x)|2dx−N(N−1)(N−2)(N−3)2λN,ℓ∫χℓ(x1−x2)fN,ℓ2(x1−x2)|ΦN(x)|2uN,ℓ(x3−x4)dx+Epot$
(4.17)

for another error $Epot$, vanishing as N (for N−1N−3/4 but small enough); again, details can be found in Ref. 15 (Sec. 4).

Let us introduce the effective N-particle Hamilton operator

$HNeff=∑j=1N−Δxj+2∑i
(4.18)

Then, combining (4.16) and (4.17) with (4.13), we conclude from (4.7) that

$ΨN,∑j=1N−ΔxjΨN‖ΨN‖2≤1+N(N−1)2∫uN,ℓ(x1−x2)|ΦN(x)|2dx×⟨ΦN,HNeffΦN⟩−N(N−1)2⟨ΦN,HN−2eff⊗uN,ℓ(xN−1−xN)ΦN⟩+E$

for an error $E$, vanishing as N, if N−1N−3/4. Here, we brought the second term on the rhs of (4.13) to the numerator using (4.14) to show that its square is negligible, even after multiplication with quantities of order N. We obtain

$ΨN,∑j=1N−ΔxjΨN‖ΨN‖2≤〈ΦN,HNeffΦN〉−N(N−1)2ΦN,HN−2eff−〈ΦN,HNeffΦN〉⊗uN,ℓ(xN−1−xN)ΦN+E.$
(4.19)

Equation (4.19) dictates the choice of the wave function ΦN. To get the best possible upper bound, we should take ΦN so that the expectation of $HNeff$ is as small as possible, making sure that the regularity bounds in (4.11) [and the additional bounds needed to control the error terms in (4.16) and (4.17)] are satisfied.

Since uN, is small, unless particles are very close, we can think of $HNeff$ as a perturbation of the many-body Hamiltonian

$H̃Neff=∑j=1N−Δxj+2λN,ℓ∑i
(4.20)

Recalling, from (2.4) (but now with $a$ replaced by $a/N$), that $λN,ℓ≃3a/Nℓ3$, we can consider (4.20), in good approximation, as an Hamiltonian of the form (3.2), with β ∈ (0; 1) chosen so that = Nβ. As discussed in Sec. III, we know how to approximate the ground state energy and the ground state wave function of Hamilton operators of this form; we need to introduce Bogoliubov transformations Tη, Tτ as defined in (3.11) and (3.7), and we have to consider states of the form $UN*TηTτΩ∈Ls2(ΛN)$. This leads to expression (3.13) for the ground state energy.

Unfortunately, considering the Hamilton operator (4.20) is not enough, and the difference to (4.18) is not small and needs to be taken into account. However, we can apply the rigorous version of Bogoliubov theory that has been developed in Ref. 36 to determine the spectrum of (4.20) and also to study the ground state energy of (4.18) and to construct an approximation for its ground state vector.

Since $HNeff$ corresponds to an intermediate regime in (3.2), with 0 < β < 1, the renormalization of the excitation Hamiltonian only involves generalized Bogoliubov transformations, and no cubic renormalization is required. The presence of the second term on the rhs of (4.18), however, affects the choice of the sequence η, needed in (3.11) to define Tη. Let 0 > 0 be sufficiently small but fixed, of order one. It turns out that for every momentum $p∈Λ+*$, one can take ηp as the Fourier coefficient of the function $η̌=−N(1−gN,ℓ0)$, with $gN,ℓ0=fN,ℓ0/fN,ℓ$, given by the ratio of the two solutions of (4.6), defined on balls of radii 0 and (recall that N−1N−3/4, while 0 is small but fixed, of order one). It is then easy to verify that $gN,ℓ0$ satisfies the partial differential equation

$−Δ−2∇fN,ℓfN,ℓ⋅∇gN,ℓ0+λN,ℓχℓfN,ℓ2gN,ℓ0=λN,ℓ0χℓ0gN,ℓ0.$
(4.21)

To understand this choice of η, recall from (3.12) that, at least on the heuristic level, $UN*TηΩ$ is an approximation for the product

$∏i

This is exactly what is needed, in (4.5), to replace fN, by $fN,ℓ0$. This procedure introduces, in our trial state, the missing two-body correlations, up to the scale 0.

Apart from this heuristic explanation, the choice of the sequence η is determined by the computation of the renormalized excitation Hamiltonian $GNeff=Tη*UNHNeffUN*Tη$, acting on the truncated Fock space $F+≤N$. Compared with the analysis in Ref. 36 [where the initial Hamilton operator has essentially the form (4.20)], the second term on the rhs of (4.18) and the presence of the factor $fN,ℓ2$ in the third term on the rhs of (4.18) produce new large contributions to $GNeff$. The condition that these terms cancel [when combined with the large contributions arising from the conjugation of (4.20)] so that $GNeff$ can be well approximated by a quadratic operator fixes the correct form of η [the condition appears essentially as Eq. (4.21)].

Finally, we need to diagonalize the quadratic part of $GNeff$. As explained in Sec. III, this can be achieved through conjugation with a second generalized Bogoliubov transformation Tτ. This leads us to the trial state $ΦN=UN*TηTτΩ∈Ls2(ΛN)$ for the ground state wave function of the effective Hamilton operator (4.18). While Tη takes care of correlations on length scales between and 0, the final conjugation with Tτ introduces the last missing two-body correlations, on scales larger than 0.

The details of this part of the analysis can be found in Ref. 15 (Secs. 5 and 6). At the end, with this definition of ΦN, we obtain, on the one hand, the estimate

$⟨ΦN,HNeffΦN⟩=4πa(N−1)+eΛa2−12∑p∈Λ+*p2+8πa−|p|4+16πap2−(8πa)22p2+CN−ε.$
(4.22)

On the other hand, we can show that ΦN satisfies the regularity bounds (4.11) [and also the more involved bounds needed to control error terms arising from (4.16) and (4.17)]; see Ref. 15 (Sec. 7).

Inserting (4.22) on the rhs of (4.19), the Proof of Theorem 4.1 is almost complete. What is still missing is a bound showing that the second contribution on the rhs of (4.19) is negligible in the limit N. This requires some additional work because the second term on the rhs (4.18) affects the coercivity of the excitation Hamiltonian $UNHN−2effUN*$. We skip here further details, which can be found in Ref. 15 (Sec. 8).

This paper is dedicated to the memory of Freeman Dyson, whose work has been inspiring the mathematical physics community for the last 70 years.

A.O., G.P., and B.S. gratefully acknowledge support from the European Research Council through the ERC Advanced Grant CLaQS. Additionally, B.S. acknowledges partial support from the NCCR SwissMAP and from the Swiss National Science Foundation through the Grant “Dynamical and energetic properties of Bose–Einstein condensates.” G.B., S.C., and A.O. acknowledge the GNFM Gruppo Nazionale per la Fisica Matematica—INDAM.

The authors have no conflicts to disclose.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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With $η̌(x)≃Cχ(|x|≤ℓ0)/(|x|+N−β)$, we expect that Tη creates order $‖η‖22≃1$ excitations, with an energy of order $‖η‖H12≃Nβ$.
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Comparing with (3.8), we conclude that contributions to $aNβ$ associated with k ≥ 2, which are relevant for β ≥ 1/2, emerge from cubic and quartic terms in (3.6).
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