Low-energy spectrum and dynamics of the weakly interacting Bose gas

Boßmann, Lea

doi:10.1063/5.0089983

We consider a gas of N bosons with interactions in the mean-field scaling regime. We review the proof of an asymptotic expansion of its low-energy spectrum, eigenstates, and dynamics, which provides corrections to Bogoliubov theory to all orders in 1/N. This is based on joint works with Petrat, Pickl, Seiringer, and Soffer. In addition, we derive a full asymptotic expansion of the ground state one-body reduced density matrix.

I. INTRODUCTION AND MAIN RESULTS

A. Introduction

Since the first experimental realization of Bose–Einstein condensation (BEC) in 1995, the experimental, theoretical, and mathematical investigation of systems of interacting bosons at low temperatures has become a very active field of research. In a typical experiment, the bosons are initially caught in an external trap. This situation is mathematically described by the N-body Hamiltonian

H_{N}^{t r a p} = \sum_{j = 1}^{N} (- Δ_{j} + V^{t r a p} (x_{j})) + \sum_{1 \leq i < j \leq N} v_{N} (x_{i} - x_{j})

(1.1)

for some confining potential V^trap and for some two-body interaction v_N, acting on the Hilbert space of square integrable, permutation symmetric functions on $R^{d N}$ ⁠,

H_{s y m}^{N} ≔ ⨂_{s y m}^{N} H, H ≔ L^{2} (R^{d}) .

The Bose gas is then cooled down to a low-energy eigenstate of $H_{N}^{t r a p}$ or to a superposition of such states. For simplicity, let us assume that the gas is prepared in the ground state $Ψ_{N}^{t r a p}$ of $H_{N}^{t r a p}$ ⁠, i.e.,

E_{N}^{t r a p} = \inf σ (H_{N}^{t r a p}), H_{N}^{t r a p} Ψ_{N}^{t r a p} = E_{N}^{t r a p} Ψ_{N}^{t r a p} .

(1.2)

Subsequently, the trap is switched off and the Bose gas propagates freely. Mathematically, this is described by the N-body Schrödinger equation with initial datum $Ψ_{N}^{t r a p}$ ⁠,

i \partial_{t} Ψ_{N} (t) = H_{N} Ψ_{N} (t), Ψ_{N} (0) = Ψ_{N}^{t r a p},

(1.3)

with the N-body Hamiltonian

H_{N} = \sum_{j = 1}^{N} (- Δ_{j}) + \sum_{1 \leq i < j \leq N} v_{N} (x_{i} - x_{j}) .

(1.4)

Given that the number of particles in such a gas is usually large, an exact (analytical or numerical) analysis of the system in the presence of interactions is, in general, impossible; an exception is the explicitly solvable Lieb–Liniger model, which describes a one-dimensional gas with delta interactions. Over the last two decades, there have been many works in the mathematical physics community devoted to a rigorous derivation of suitable approximations of the statical and dynamical properties of the gas for large N. These questions have been studied for different classes of interactions v_N, in particular, for the so-called mean-field (or Hartree) regime,

v_{N} = λ_{N} v, λ_{N} ≔ \frac{1}{N - 1},

(1.5)

describing the situation of weak and long-range interactions.

In this note, we consider interactions of the form (1.5). We present an asymptotic expansion of the low-energy spectrum and eigenstates of $H_{N}^{t r a p}$ and of the dynamics (1.3), which makes the model fully computationally accessible to any order in 1/N. This review is based on Ref. 1 (in collaboration with Petrat and Seiringer) and Ref. 2 (in collaboration with Petrat, Pickl, and Soffer).

B. Model and main results

We consider a system of N interacting bosons in $R^{d}$ ⁠, d ≥ 1, which are described by the N-body Hamiltonian (1.1) with interactions (1.5). We impose the following assumptions on the interaction v_N and the external potential V^trap:

Assumption 1.

Define v_N as in (1.5).

Let $v : R^{d} \to R$ be bounded with v(−x) = v(x) and v ≢ 0.
Assume that v is of positive type, i.e., that it has a non-negative Fourier transform.

Assumption 2.

Let $V^{t r a p} : R^{d} \to R$ be measurable, locally bounded, and non-negative, and let V^trap(x) tend to infinity as |x| → ∞.

Our first main result concerns the ground state $Ψ_{N}^{t r a p}$ of $H_{N}^{t r a p}$ ⁠: We construct a norm approximation of $Ψ_{N}^{t r a p}$ and of its energy $E_{N}^{t r a p}$ to any order in 1/N.

Theorem 1.

Let

a \in N_{0}

, let Assumptions 1 and 2 be satisfied, and choose N sufficiently large. Then, there exists a constant C(a) such that

{‖Ψ_{N}^{t r a p} - \sum_{ℓ = 0}^{a} λ_{N}^{\frac{ℓ}{2}} ψ_{N, ℓ}^{t r a p}‖}_{H^{N}} \leq C (a) λ_{N}^{\frac{a + 1}{2}}

(1.6)

and

|E_{N}^{t r a p} - N e_{H}^{t r a p} - \sum_{ℓ = 0}^{a} λ_{N}^{ℓ} E_{ℓ}^{t r a p}| \leq C (a) λ_{N}^{a + 1} .

(1.7)

The coefficients

ψ_{N, ℓ}^{t r a p} \in H_{s y m}^{N}

of expansion (1.6) and the coefficients

e_{H}^{t r a p}

,

E_{ℓ}^{t r a p} \in R

of expansion (1.7) are given in (2.46), (2.4), and (2.47), respectively.

Our result extends to the low-energy excitation spectrum of $H_{N}^{t r a p}$ and to a certain class of unbounded interaction potentials v, including the repulsive three-dimensional Coulomb potential (see Sec. II E). To leading order (a = 0), the statements (1.6) and (1.7) have been proven (for bounded interactions) by Seiringer on the torus³ and by Grech and Seiringer in the inhomogeneous setting.⁴ For our class of unbounded interactions, the leading order approximation was obtained by Lewin, Nam, Serfaty, and Solovej.⁵ The higher orders in (1.6) and (1.7) were, to the best of our knowledge, first rigorously derived in Ref. 1. Another approach was proposed by Pizzo in Refs. 6–8, who considers a Bose gas on a torus and constructs an expansion for the ground state, based on a multi-scale analysis in the number of excitations, around a product state using Feshbach maps. As a consequence of the norm approximation (1.6), one can derive an expansion of the ground state one-body reduced density matrix,

γ_{N}^{t r a p, (1)} ≔ {T r}_{H^{N - 1}} | Ψ_{N}^{t r a p} 〉 〈 Ψ_{N}^{t r a p} |,

(1.8)

in trace norm (see Sec. II D for a proof of this statement).

Corollary 1.1.

Let

a \in N_{0}

⁠, and let Assumptions 1 and 2 be satisfied. Denote by

γ_{N}^{t r a p, (1)}

the one-body reduced density matrix of

Ψ_{N}^{t r a p}

. Then, there exists a constant C(a) > 0 such that

T r |γ_{N}^{t r a p, (1)} - \sum_{ℓ = 0}^{a} λ_{N}^{ℓ} γ_{1, ℓ}^{t r a p}| \leq C (a) λ_{N}^{a + 1}

(1.9)

for sufficiently large N, where the coefficients

γ_{1, ℓ}^{t r a p} \in L (H)

are defined in (2.50).

Theorem 1 and Corollary 1.1 determine the ground state $Ψ_{N}^{t r a p}$ of $H_{N}^{t r a p}$ to arbitrary precision. Now, we remove the confining potential V^trap and take $Ψ_{N}^{t r a p}$ as initial datum for the time evolution (1.3). Since an eigenstate of $H_{N}^{t r a p}$ is not necessarily an eigenstate of H_N, this leads to some non-trivial dynamics, for which we provide an approximation in norm to any order in 1/N in our second main result.

Theorem 2.

Let

a \in N_{0}

,

t \in R

, let Assumption 1a hold, and denote by Ψ_N(t) the solution of (1.3). Then, there exists a constant C(a) > 0 such that

{‖Ψ_{N} (t) - \sum_{ℓ = 0}^{a} λ_{N}^{\frac{ℓ}{2}} ψ_{N, ℓ} (t)‖}_{H^{N}} \leq e^{C (a) t} λ_{N}^{\frac{a + 1}{2}}

(1.10)

for sufficiently large N, where the coefficients ψ_N,ℓ(t) are defined in (3.19).

Note that for the dynamical result, we do not require the interaction potential to be of positive type. Finally, we derive from expansion (1.10) a trace norm approximation of the time-evolved one-body reduced density matrix

γ_{N}^{(1)} (t) ≔ {T r}_{H^{N - 1}} | Ψ_{N} (t) 〉 〈 Ψ_{N} (t) |

(1.11)

to arbitrary precision.

Corollary 1.2.

Let

a \in N_{0}

,

t \in R

⁠, and let Assumption 1a be satisfied. Then, there exists a constant C(a) such that

T r |γ_{N}^{(1)} (t) - \sum_{ℓ = 0}^{a} λ_{N}^{ℓ} γ_{1, ℓ} (t)| \leq e^{C (a) t} λ_{N}^{a + 1}

(1.12)

for sufficiently large N, where the coefficients

γ_{1, ℓ} (t) \in L (H)

are defined in (3.26).

Below, we will provide and explain the explicit formulas for the coefficients in Theorems 1 and 2 and in Corollaries 1.1 and 1.2. Note that $e_{H}^{t r a p}$ ⁠, $E_{ℓ}^{t r a p}$ ⁠, $γ_{1, ℓ}^{t r a p}$ ⁠, and γ_1,ℓ(t) are completely independent of N. The N-body wave functions $ψ_{N, ℓ}^{t r a p}$ and ψ_N,ℓ(t) naturally depend on N; however, this N-dependence is trivial in a sense to be made precise below. In particular, the computational effort to obtain physical quantities, such as expectation values of bounded operators with respect to the (time-evolved) N-body state, does not scale with N.

Finally, let us remark that all constants C(a) grow rapidly in a. Hence, all statements are to be read as asymptotic expansions: given any order a of the approximation, one can choose N sufficiently large such that the estimates are meaningful, but we cannot simultaneously send a to infinity.

To prove the above results, we first remove the particles in the condensate from the description and focus only on the excitations from the condensate. Mathematically, this is done by conjugating the N-body Hamiltonians with a unitary map that maps from the N-body Hilbert space into a truncated Fock space, whose elements describe the excitations. The resulting operator is then expanded in the parameter $λ_{N}^{1 / 2}$ ⁠, which (formally) leads to a series of the form

H = H_{0} + \sum_{j \geq 1} λ_{N}^{\frac{j}{2}} H_{j},

where the leading order term $H_{0}$ is the well-known Bogoliubov Hamiltonian. Formally, our results can be obtained by perturbation theory around $H_{0}$ to any order; however, the rigorous proofs are much more involved, mainly since all operators are unbounded and non-commutative.

This Review is organized as follows: In Sec. II, we explain the results from Ref. 1 concerning the low-energy spectrum and eigenstates and give a proof of Corollary 1.1. Section III contains the results for the dynamics obtained in Ref. 2.

We use the following notations:

The notation A ≲ B indicates that there exists a constant C > 0 such that A ≤ CB.
For k ≥ 1 and $x_{j} \in R^{d}$ ⁠, we abbreviate x^(k) ≔ (x₁, ..., x_k) and dx^(k) ≔ dx₁ ⋯ dx_k.
We use the notation $a^{♯_{1}} ≔ a^{†}$ and $a^{♯_{- 1}} ≔ a$ ⁠.
Multi-indices are denoted as j = (j₁, ..., j_n) with |j| ≔ j₁ + ⋯ + j_n.

II. LOW-ENERGY SPECTRUM AND EIGENSTATES

In this section, we consider the Hamiltonian $H_{N}^{t r a p}$ from (1.1) and explain the asymptotic expansion of its ground state $Ψ_{N}^{t r a p}$ ⁠, the ground state energy $E_{N}^{t r a p}$ ⁠, and the corresponding reduced density matrix $γ_{N}^{t r a p, (1)}$ ⁠. To keep the notation simple, we drop the superscript^trap.

A. Framework

1. Condensate

It is well known (see, e.g., Refs. 3–5 and 9) that the N-body ground state Ψ_N exhibits (complete asymptotic) BEC in the minimizer $φ \in H$ of the Hartree energy functional $E_{H}$ ⁠,

E_{H} [ϕ] ≔ \int_{R^{d}} (| \nabla ϕ (x) |^{2} + V (x) | ϕ (x) |^{2}) d x + \frac{1}{2} \int_{R^{2 d}} v (x - y) | ϕ (x) |^{2} | ϕ (y) |^{2} d x d y .

(2.1)

For potentials v and V satisfying Assumptions 1 and 2, the minimizer φ of $E_{H}$ is unique, strictly positive, and solves the stationary Hartree equation

h φ ≔ (- Δ + V + v * φ^{2} - μ_{H}) φ = 0

(2.2)

with the Lagrange parameter $μ_{H} ≔ ⟨φ, (- Δ + V + v * φ^{2}) φ⟩ \in R .$ We denote by p^φ and q^φ the projector onto φ and its orthogonal complement, i.e.,

p^{φ} ≔ | φ 〉 〈 φ |, q^{φ} ≔ 1 - p^{φ} .

(2.3)

The minimum of $E_{H}$ is given as

e_{H} ≔ E_{H} [φ] = ⟨φ, (- Δ + V + \frac{1}{2} v * φ^{2}) φ⟩ .

(2.4)

Heuristically, (complete asymptotic) BEC in the state φ means that $N - o (N)$ particles occupy the condensate state φ. Mathematically, this is reflected by the fact that the N-body wave function is determined by the one-body state φ in the sense of reduced densities, i.e.,

\lim_{N \to \infty} T r |γ_{N}^{(1)} - | φ 〉 〈 φ || = 0 .

(2.5)

The condensate determines the leading order of the ground state energy, namely,

E_{N} = N e_{H} + O (1) .

(2.6)

2. Excitations

The errors in (2.5) and (2.6) are caused by $O (1)$ particles that are excited from the condensate due to the inter-particle interactions. To describe these excitations, we decompose Ψ_N as

Ψ_{N} = \sum_{k = 0}^{N} φ^{⨂ (N - k)} ⨂_{s} χ^{(k)}, χ^{(k)} \in ⨂_{s y m}^{k} H_{⊥ φ},

(2.7)

with ⨂_s being the symmetric tensor product and where $H_{⊥ φ} ≔ \{ϕ \in H : {⟨ϕ, φ⟩}_{H} = 0\}$ denotes the orthogonal complement of φ in $H$ ⁠.⁵ The excitations

χ_{\leq N} ≔ {(χ^{(k)})}_{k = 0}^{N}

(2.8)

form a vector in the truncated (excitation) Fock space over $H_{⊥ φ}$ ⁠,

F_{⊥ φ}^{\leq N} = ⨁_{k = 0}^{N} ⨂_{s y m}^{k} H_{⊥ φ} \subset F_{⊥ φ} = ⨁_{k \geq 0}^{\infty} ⨁_{s y m} H_{⊥ φ} \subset F = ⨁_{k \geq 0} ⨂_{s y m}^{k} H,

(2.9)

which is a subspace of the Fock space $F$ over $H$ ⁠. The creation/annihilation operators a^†/a on $F$ are defined in the usual way, and we denote the second quantization in $F$ of an operator T on $H$ by dΓ(T). The number operator on $F_{⊥ φ}$ is given by

N_{⊥ φ} ≔ d Γ (q^{φ}) .

(2.10)

The relation between Ψ_N and the corresponding excitation vector χ_≤N is given by the unitary map

U_{N, φ} : H^{N} \to F_{⊥ φ}^{\leq N}, Ψ \mapsto U_{N, φ} Ψ = χ_{\leq N},

(2.11)

whose action is explicitly known [see Ref. 5 (Proposition 4.2)]. Conjugating H_N with $U_{N, φ}$ yields the operator

H_{\leq N} ≔ U_{N, φ} (H_{N} - N e_{H}) U_{N, φ}^{*}

(2.12)

on $F_{⊥ φ}^{\leq N}$ ⁠, whose ground state is given by χ_≤N. Hence, the ground state energy of $H_{\leq N}$ ⁠,

E_{\leq N} ≔ \inf σ (H_{\leq N}) = {⟨χ_{\leq N}, H_{\leq N} χ_{\leq N}⟩}_{F_{⊥ φ}^{\leq N}} = E_{N} - N e_{H},

(2.13)

is precisely the $O (1)$ -term in (2.6).

3. Excitation Hamiltonian

Making use of the explicit form of $U_{N, φ}$ [Ref. 5 (Proposition 4.2)], we can express $H_{\leq N}$ as

\begin{matrix} H_{\leq N} & = & K_{0} + (\frac{N - N_{⊥ φ}}{N - 1}) K_{1} + (K_{2} \frac{\sqrt{(N - N_{⊥ φ}) (N - N_{⊥ φ} - 1)}}{N - 1} + \frac{\sqrt{(N - N_{⊥ φ}) (N - N_{⊥ φ} - 1)}}{N - 1} K_{2}^{*}) \\ + (K_{3} \frac{\sqrt{N - N_{⊥ φ}}}{N - 1} + \frac{\sqrt{N - N_{⊥ φ}}}{N - 1} K_{3}^{*}) + \frac{1}{N - 1} K_{4} \end{matrix}

(2.14)

as an operator on $F_{⊥ φ}^{\leq N}$ ⁠, where we used the shorthand notation

K_{0} ≔ d Γ (h), K_{1} ≔ d Γ (K_{1}), K_{4} ≔ d Γ (K_{4}),

(2.15a)

K_{2} ≔ \frac{1}{2} \int d x_{1} d x_{2} K_{2} (x_{1}, x_{2}) a_{x_{1}}^{†} a_{x_{2}}^{†},

(2.15b)

K_{3} ≔ \int d x^{(3)} K_{3} (x_{1}, x_{2}; x_{3}) a_{x_{1}}^{†} a_{x_{2}}^{†} a_{x_{3}}

(2.15c)

for h as in (2.2) and where

K_{1} : H_{⊥ φ} \to H_{⊥ φ}, K_{1} ≔ q^{φ} K q^{φ},

(2.16a)

K_{2} \in H_{⊥ φ} \otimes H_{⊥ φ}, K_{2} (x_{1}, x_{2}) ≔ (q_{1}^{φ} q_{2}^{φ} K) (x_{1}, x_{2}),

(2.16b)

K_{3} : H_{⊥ φ} \to H_{⊥ φ} \otimes H_{⊥ φ}, (K_{3} ψ) (x_{1}, x_{2}) ≔ q_{1}^{φ} q_{2}^{φ} W (x_{1}, x_{2}) φ (x_{1}) (q_{2}^{φ} ψ) (x_{2}),

(2.16c)

K_{4} : H_{⊥ φ} \otimes H_{⊥ φ} \to H_{⊥ φ} \otimes H_{⊥ φ}, (K_{4} ψ) (x_{1}, x_{2}) ≔ q_{1}^{φ} q_{2}^{φ} W (x_{1}, x_{2}) (q_{1}^{φ} q_{2}^{φ} ψ) (x_{1}, x_{2}) .

(2.16d)

Here, K(x₁, x₂) is defined as

K (x_{1}; x_{2}) ≔ v (x_{1} - x_{2}) φ (x_{1}) φ (x_{2}),

(2.17)

K is the operator with kernel K(x₁, x₂), and W is the multiplication operator defined by

W (x_{1}, x_{2}) ≔ v (x_{1} - x_{2}) - (v * φ^{2}) (x_{1}) - (v * φ^{2}) (x_{2}) + ⟨φ, v * φ^{2} φ⟩ .

(2.18)

By construction, $H_{\leq N}$ is explicitly N-dependent. To extract its contributions to each order in λ_N, we first extend $H_{\leq N}$ trivially to an operator on $F_{⊥ φ}$ ⁠,

H ≔ H_{\leq N} \oplus c,

(2.19)

where the direct sum is with respect to the decomposition $F = F^{\leq N} \oplus F^{< N}$ ⁠. The constant c in (2.19) will later be chosen conveniently (see Sec. II C). Similarly, we extend χ_≤N to a vector $χ \in F_{⊥ φ}$ as

χ ≔ χ_{\leq N} \oplus 0

(2.20)

and denote the corresponding projectors on $F_{⊥ φ}$ by

P ≔ | χ 〉 〈 χ |, Q ≔ 1 - P .

(2.21)

A (formal) expansion of $H$ in powers of $λ_{N}^{1 / 2}$ yields

H = H_{0} + \sum_{j \geq 1} λ_{N}^{\frac{j}{2}} H_{j},

(2.22)

where

H_{0} ≔ K_{0} + K_{1} + K_{2} + K_{2}^{*},

(2.23a)

H_{1} ≔ K_{3} + K_{3}^{*},

(2.23b)

H_{2} ≔ - (N_{⊥ φ} - 1) K_{1} - (K_{2} (N_{⊥ φ} - \frac{1}{2}) + h . c .) + K_{4},

(2.23c)

H_{2 j - 1} ≔ c_{j - 1} (K_{3} {(N_{⊥ φ} - 1)}^{j - 1} + h . c .),

(2.23d)

H_{2 j} ≔ \sum_{ν = 0}^{j} d_{j, ν} (K_{2} {(N_{⊥ φ} - 1)}^{ν} + h . c .)

(2.23e)

for j ≥ 2, with $K_{j}$ as in (2.15). The coefficients c_j and d_j,ν are given as

c_{0}^{(ℓ)} ≔ 1, c_{j}^{(ℓ)} ≔ \frac{(ℓ - \frac{1}{2}) (ℓ + \frac{1}{2}) (ℓ + \frac{3}{2}) \cdot \cdot \cdot (ℓ + j - \frac{3}{2})}{j!}, c_{j} ≔ c_{j}^{(0)} (j \geq 1),

(2.24a)

d_{j, ν} ≔ \sum_{ℓ = 0}^{ν} c_{ℓ}^{(0)} c_{ν - ℓ}^{(0)} c_{j - ν}^{(ℓ)} (j \geq ν \geq 0) .

(2.24b)

4. Bogoliubov approximation

The leading order term $H_{0}$ in (2.22) is the well-known Bogoliubov Hamiltonian. We denote the unique ground state of $H_{0}$ and the ground state energy by

E_{0} ≔ \inf σ (H_{0}), H_{0} χ_{0} = E_{0} χ_{0},

(2.25)

and the corresponding projectors are defined as

P_{0} ≔ | χ_{0} 〉 〈 χ_{0} |, Q_{0} ≔ 1 - P_{0} .

(2.26)

It is well known^3–5 that the ground state χ_≤N of $H_{\leq N}$ and the ground state energy E_≤N converge to χ₀ and E₀, respectively, i.e.,

\lim_{N \to \infty} E_{\leq N} = E_{0}, \lim_{N \to \infty} ‖ χ_{\leq N} - χ_{0} ‖_{F_{⊥ φ}^{\leq N}} = 0 .

(2.27)

Consequently, E₀ gives the next-to-leading order term in (1.7); analogously, the leading order contribution in (1.6) is given by $ψ_{N, 0} = U_{N, φ}^{*} χ_{0} |_{F_{⊥ φ}^{\leq N}}$ ⁠.

The Bogoliubov Hamiltonian $H_{0}$ is a very useful approximation of $H$ because it is much simpler than the full problem: it is quadratic in the number of creation/annihilation operators and can be diagonalized by Bogoliubov transformations.

Let us briefly recall the concept of Bogoliubov transformations. For $F = f \oplus J g \in H \oplus H$ ⁠, where $J : H \to H$ denotes complex conjugation, one defines the generalized creation and annihilation operators A(F) and A^†(F) as

A (F) = a (f) + a^{†} (g), A^{†} (F) = A (J F) = a^{†} (f) + a (g)

(2.28)

for $J = (\begin{matrix} 0 & J \\ J & 0 \end{matrix})$ ⁠. An operator $V$ on $H \oplus H$ such that $F \mapsto A (V F)$ has the same properties as F ↦ A(F), i.e., $A^{†} (V F) = A (V J F)$ and $[A (V F_{1}), A^{†} (V F_{2})] = [A (F_{1}), A^{†} (F_{2})],$ is called a (bosonic) Bogoliubov map and can be written in block form as

V ≔ (\begin{matrix} U & \bar{V} \\ V & \bar{U} \end{matrix}), U, V : H_{⊥ φ} \to H_{⊥ φ} .

(2.29)

If V is Hilbert–Schmidt, the Bogoliubov map $V$ can be unitarily implemented on $F$ ⁠, i.e., there exists a unitary transformation $U_{V} : F \to F$ (called a Bogoliubov transformation) such that $U_{V} A (F) U_{V}^{*} = A (V F)$ for all $F \in H \oplus H$ ⁠. This implies the transformation rule

U_{V} a (f) U_{V}^{*} = a (U f) + a^{†} (\bar{V f}), U_{V} a^{†} (f) U_{V}^{*} = a^{†} (U f) + a (\bar{V f}) .

(2.30)

A normalized state $ϕ \in F_{⊥ φ}$ that can be written as

ϕ = U_{V} | Ω 〉

(2.31)

for some Bogoliubov map $V$ is called a quasi-free state. Quasi-free states have a finite expectation value of the number operator and satisfy Wick’s rule, i.e.,

{⟨ϕ, a^{♯} (f_{1}) \cdot \cdot \cdot a^{♯} (f_{2 n - 1}) ϕ⟩}_{F_{⊥ φ}} = 0,

(2.32a)

{⟨ϕ, a^{♯} (f_{1}) \cdot \cdot \cdot a^{♯} (f_{2 n}) ϕ⟩}_{F_{⊥ φ}} = \sum_{σ \in P_{2 n}} \prod_{j = 1}^{n} {⟨ϕ, a^{♯} (f_{σ (2 j - 1)}) a^{♯} (f_{σ (2 j)}) ϕ⟩}_{F_{⊥ φ}}

(2.32b)

for a^♯ ∈ {a^†, a}, $n \in N$ ⁠, and $f_{1}, . . ., f_{2 n} \in H_{⊥ φ}$ ⁠. Here, P_2n denotes the set of pairings

P_{2 n} ≔ {σ \in S_{2 n} : σ (2 a - 1) < \min {σ (2 a), σ (2 a + 1)} \forall a \in {1, 2, . . ., 2 n}},

(2.33)

for $S_{2 n}$ the symmetric group on the set {1, 2, ..., 2n}. In particular, the ground state χ₀ of $H_{0}$ is a quasi-free state,

χ_{0} = U_{V_{0}}^{*} | Ω 〉,

(2.34)

where $U_{V_{0}}$ is the Bogoliubov transformation that diagonalizes $H_{0}$ ⁠.

B. Expansion of the ground state

To prove Theorem 1, we show that the projector $P$ from (2.21) admits a series expansion in powers of $λ_{N}^{1 / 2}$ in the following sense:

Proposition 2.1.

Let Assumptions 1 and 2 hold, let

A \in L (F_{⊥ φ})

be a bounded operator on

F_{⊥ φ}

⁠, and let

a \in N_{0}

. Then, there exists some constant C(a) such that

|T r A P - \sum_{ℓ = 0}^{a} λ_{N}^{\frac{ℓ}{2}} T r A P_{ℓ}| \leq C (a) λ_{N}^{\frac{a + 1}{2}} ‖ A ‖_{o p}

(2.35)

for sufficiently large N, where ‖·‖_op denotes the operator norm. The coefficients

P_{ℓ}

are defined as

P_{ℓ} ≔ \{\begin{cases} P_{0} & i f ℓ = 0, \\ - \sum_{ν = 1}^{ℓ} \sum_{\begin{array}{c} j \in N^{ν} \\ | j | = ℓ \end{array}} \sum_{\begin{array}{c} k \in N_{0}^{ν + 1} \\ | k | = ν \end{array}} O_{k_{1}} H_{j_{1}} O_{k_{2}} H_{j_{2}} \cdot \cdot \cdot O_{k_{ν}} H_{j_{ν}} O_{k_{ν + 1}} & i f ℓ \geq 1, \end{cases})

(2.36)

with

P_{0}

as in (2.26) and

H_{j}

as in (2.23) and where we abbreviated

O_{k} ≔ \{\begin{cases} - P_{0} & k = 0, \\ \frac{Q_{0}}{{(E_{0} - H_{0})}^{k}} & k > 0 . \end{cases})

(2.37)

The growth of the constant C(a) in the order a of the approximation can be estimated as

C (a) \leq C {(a + 1)}^{{(a + 6)}^{2}},

which we expect to be far from optimal. By means of Bogoliubov transformations, the operators $P_{ℓ}$ can be brought into a more explicit form. For example, the first order correction $P_{1}$ is given by

P_{1} = U_{V_{0}}^{*} (U_{V_{0}} O_{1} U_{V_{0}}^{*}) (U_{V_{0}} H_{1} U_{V_{0}}^{*}) | Ω 〉 〈 χ_{0} | + h . c .,

(2.38)

where $U_{V_{0}}$ is the Bogoliubov transformation diagonalizing $H_{0}$ such that $χ_{0} = U_{V_{0}}^{*} | Ω 〉$ ⁠. To simplify (2.38), one notes that $U_{V_{0}} H_{1} U_{V_{0}}^{*} | Ω 〉$ is a superposition of one- and three-particle states and that $U_{V_{0}} O_{1}^{(0)} U_{V_{0}}^{*}$ is particle-number preserving. Hence, $P_{1}$ can be expressed as

P_{1} = U_{V_{0}}^{*} (\int d x Θ_{1} (x) a_{x}^{†} | Ω 〉 + \int d x^{(3)} Θ_{3} (x^{(3)}) a_{x_{1}}^{†} a_{x_{2}}^{†} a_{x_{3}}^{†} | Ω 〉) 〈 χ_{0} | + h . c .,

(2.39)

where the functions Θ₁ and Θ₃ can be retrieved by diagonalizing $H_{0}$ and computing the Bogoliubov transformation of $H_{1}$ under $U_{V_{0}}$ ⁠.

From Proposition 2.1, we deduce three consequences.

1. Ground state wave function

As an immediate consequence of Proposition 2.1, we find that

T r |P - \sum_{ℓ = 0}^{a} λ_{N}^{\frac{ℓ}{2}} P_{ℓ}| \leq C (a) λ_{N}^{\frac{a + 1}{2}} .

(2.40)

Since $P = | χ 〉〈 χ |$ is a rank one projector, expansion (2.40) implies an expansion of the excitation wave function χ,

{‖χ - \sum_{ℓ = 0}^{a} λ_{N}^{\frac{ℓ}{2}} χ_{ℓ}‖}_{F} \leq C (a) λ_{N}^{\frac{a + 1}{2}}

(2.41)

[see Ref. 1 (Appendix B) for a proof of this statement in a general Hilbert space setting]. The coefficients of the expansion are given by

χ_{ℓ} ≔ \sum_{j = 0}^{ℓ} α_{j} {\tilde{χ}}_{ℓ - j} (ℓ \geq 1),

(2.42)

where

{\tilde{χ}}_{ℓ} ≔ \sum_{ν = 1}^{ℓ} \sum_{\begin{array}{c} j \in N^{ν} \\ | j | = ℓ \end{array}} P_{j_{1}} \cdot \cdot \cdot P_{j_{ν}} χ_{0} (ℓ \geq 1),

(2.43)

with $P_{ℓ}$ as in (2.36) and χ₀ as in (2.25), and for n ≥ 1,

α_{0} ≔ 1, α_{2 n - 1} ≔ 0, α_{2 n} ≔ - \frac{1}{2} \sum_{\begin{array}{c} j \in N_{0}^{4} \\ j_{1}, j_{2} < 2 n \\ | j | = 2 n \end{array}} α_{j_{1}} α_{j_{2}} ⟨{\tilde{χ}}_{j_{3}}, {\tilde{χ}}_{j_{4}}⟩ .

(2.44)

For example,

χ_{1} = \frac{Q_{0}}{E_{0} - H_{0}} H_{1} χ_{0} = U_{V_{0}}^{*} (\int d x Θ_{1} (x) a_{x}^{†} | Ω 〉 + \int d x^{(3)} Θ_{3} (x^{(3)}) a_{x_{1}}^{†} a_{x_{2}}^{†} a_{x_{3}}^{†} | Ω 〉)

(2.45)

for Θ₁ and Θ₃ as in (2.39). Finally, the coefficients ψ_N,ℓ in expansion (1.6) of the N-body ground state Ψ_N (Theorem 1) are constructed from this by inserting (2.42) into (2.7), i.e.,

ψ_{N, ℓ} ≔ \sum_{k = 0}^{N} φ^{\otimes (N - k)} \otimes_{s} {(χ_{ℓ})}^{(k)} .

(2.46)

The functions ψ_N,ℓ depend on N by construction. However, this N-dependence is trivial, since it comes only from the splitting into condensate φ and excitations χ. The coefficients χ_ℓ in expansion (1.6) of the excitations χ are completely independent of N.

2. Ground state energy

Another consequence of Proposition 2.1 is expansion (1.7) of the ground state energy $E_{N}$ (Theorem 1). The coefficients E_ℓ in (1.7) are given as

E_{ℓ} ≔ \sum_{ν = 1}^{2 ℓ} \sum_{\begin{array}{c} j \in N^{ν} \\ | j | = 2 ℓ \end{array}} \sum_{\begin{array}{c} m \in N_{0}^{ν - 1} \\ | m | = ν - 1 \end{array}} \frac{1}{κ (m)} T r P_{0} H_{j_{1}} O_{m_{1}} \cdot \cdot \cdot H_{j_{ν - 1}} O_{m_{ν - 1}} H_{j_{ν}}

(2.47)

for $P_{0}$ as in (2.26), $H_{j}$ as in (2.23), $O_{m}$ as in (2.37) and where

κ (m) ≔ 1 + |\{μ : m_{μ} = 0\}| \in {1, . . ., ν - 1}

(2.48)

is the number of operators $P_{0}$ within the trace. This confirms the predictions of (formal) Rayleigh–Schrödinger perturbation theory. For example, the first coefficient in (2.47) simplifies to

E_{1} = ⟨χ_{0}, H_{2} χ_{0}⟩ + ⟨χ_{0}, H_{1} \frac{Q_{0}}{E_{0} - H_{0}} H_{1} χ_{0}⟩ .

(2.49)

3. Ground state reduced density

Finally, Proposition 2.1 implies an asymptotic expansion of the one-body reduced density $γ_{N}^{(1)}$ of Ψ_N (Corollary 1.1). The coefficients in (1.9) are given by the trace class operators with kernels

γ_{1, 0} (x; y) ≔ φ (x) φ (y),

(2.50a)

\begin{align} γ_{1, ℓ} (x; y) & ≔ & \sum_{n = 0}^{ℓ - 1} \sum_{k = 0}^{ℓ - n - 1} {\tilde{c}}_{ℓ - n - 1, k} (φ (x) T r P_{2 n + 1} a_{y}^{†} {(N_{⊥ φ} - 1)}^{k} + φ (y) T r P_{2 n + 1} {(N_{⊥ φ} - 1)}^{k} a_{x}) \\ + \sum_{n = 0}^{ℓ - 1} {\tilde{c}}_{ℓ - n - 1} (T r P_{2 n} a_{y}^{†} a_{x} - φ (x) φ (y) T r P_{2 n} N_{⊥ φ} ()), \end{align}

(2.50b)

with

{\tilde{c}}_{ℓ} ≔ {(- 1)}^{ℓ} c_{ℓ}^{(3 / 2)}, {\tilde{c}}_{ℓ, k} ≔ {\tilde{c}}_{ℓ - k} c_{k}^{(0)}

(2.51)

for $c_{j}^{(n)}$ as in (2.24a). For example, the leading order is γ_1,0 = p^φ, which recovers the well-known fact that the ground state exhibits BEC with optimal rate. The first correction to this is given by

\begin{aligned} γ_{1, 1} (x; y) = & φ (x) T r P_{1} a_{y}^{†} + φ (y) T r P_{1} a_{x} \\ + T r P_{0} a_{y}^{†} a_{x} - φ (x) φ (y) T r P_{0} N_{⊥ φ} . \end{aligned}

(2.52)

For the ground state of a homogeneous Bose gas on the torus, γ_1,1 was recently derived in Ref. 10 using different methods. In that case, the first line in (2.52) vanishes by translation invariance. We prove Corollary 1.1 in Sec. II D.

C. Strategy of proof

The first step is to express $P$ and $P_{0}$ as contour integrals around the resolvents of $H$ and $H_{0}$ ⁠, respectively, i.e.,

P = {\frac{1}{2 π i} \oint}_{γ} \frac{1}{z - H} d z, P_{0} = {\frac{1}{2 π i} \oint}_{γ} \frac{1}{z - H_{0}} d z .

(2.53)

The contour γ is chosen such that its length is $O (1)$ and that it encloses both the ground state energy E_≤N of $H_{\leq N}$ and the Bogoliubov ground state energy E₀ but leaves the remaining spectra of $H$ and $H_{0}$ outside. Since E_≤N converges to E₀ as N → ∞ by (2.27), such a contour exists if the constant c in $H = H_{\leq N} \oplus c$ from (2.19) is chosen a finite distance away from the spectrum of $H_{0}$ ⁠. This implies that $H$ has precisely one (infinitely degenerate) additional eigenvalue c compared to $H_{\leq N}$ ⁠. For simplicity, we place c at some finite distance below E₀ (see Fig. 1).

FIG. 1.

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Low-energy spectra of $H_{0}$ (drawn in black) and $H$ (drawn in gray). The additional eigenvalue c of $H$ is placed a finite distance below E₀. For sufficiently large N, the contour γ around E₀ encloses the ground state energy E_≤N of $H_{\leq N}$ ⁠.

The next step is to expand $H$ as¹¹

H = \sum_{j = 0}^{a} λ_{N}^{\frac{j}{2}} H_{j} + λ_{N}^{\frac{a + 1}{2}} R_{a},

(2.54)

with $H_{j}$ as in (2.23). The remainders $R_{a}$ ⁠, which are essentially the remainders of the Taylor series expansion of the square roots in (2.14), can be bounded above by powers of the number operator. Making use of expansion (2.54), we expand the resolvent of $H$ around the resolvent of $H_{0}$ and integrate along the contour γ, which finally yields

P = \sum_{ℓ = 0}^{a} λ_{N}^{\frac{ℓ}{2}} P_{ℓ} + λ_{N}^{\frac{a + 1}{2}} (B_{P} (a) + B_{Q} (a))

(2.55)

for $P_{ℓ}$ as in (2.36) and where

B_{P} (a) = \sum_{ν = 0}^{a} \sum_{m = 1}^{a - ν} \sum_{\begin{array}{c} j \in N^{m} \\ | j | = a - ν \end{array}} \frac{1}{2 π i} \oint_{γ} \frac{P}{z - H^{<}} R_{ν} \frac{1}{z - H_{0}} H_{j_{1}} \frac{1}{z - H_{0}} \cdot \cdot \cdot H_{j_{m}} \frac{1}{z - H_{0}} d z

(2.56)

and

B_{Q} (a) = \sum_{ν = 0}^{a} \sum_{m = 1}^{a - ν} \sum_{\begin{array}{c} j \in N^{m} \\ | j | = a - ν \end{array}} \sum_{ℓ = 0}^{m} \sum_{\begin{array}{c} k \in {0, 1}^{m + 1} \\ | k | = ℓ \end{array}} \frac{1}{2 π i} \oint_{γ} \frac{Q}{z - H^{<}} R_{ν} \frac{I_{k_{1}}}{z - H_{0}} H_{j_{1}} \cdot \cdot \cdot H_{j_{m}} \frac{I_{k_{m + 1}}}{z - H_{0}} d z

(2.57)

for $I_{k} = P_{0}$ if k = 0 and $I_{k} = Q_{0}$ if k = 1. To control the error terms, we estimate the operators $H_{j}$ and $R_{ν}$ in terms of powers of $(N_{⊥ φ} + 1)$ ⁠, prove a uniform bound on moments of the number operator with respect to χ, i.e.,

⟨χ, {(N_{⊥ φ} + 1)}^{b} χ⟩ \leq C (b),

(2.58)

and control alternating products of number operators and resolvents of $H_{0}$ by means of the estimate

‖{(N_{⊥ φ} + 1)}^{b + 1} \frac{1}{z - H_{0}} ϕ‖ \leq C (b) ‖ {(N_{⊥ φ} + 1)}^{b} ϕ ‖ .

(2.59)

To derive expansion (2.61) of the ground state energy, we observe that

T r H P = \frac{1}{2 π i} T r \oint_{γ} \frac{H}{z - H} d z = E_{0} + \frac{1}{2 π i} T r \oint_{γ} \frac{z - E_{0}}{z - H} d z

(2.60)

and derive from this the expansion

T r H P = E_{0} + \sum_{ℓ = 1}^{a} λ_{N}^{\frac{ℓ}{2}} \sum_{ν = 1}^{ℓ} \sum_{\begin{array}{c} j \in N^{ν} \\ | j | = ℓ \end{array}} T r \frac{1}{2 π i} \oint_{γ} \frac{1}{z - H_{0}} H_{j_{1}} \frac{1}{z - H_{0}} \cdot \cdot \cdot H_{j_{ν}} \frac{z - E_{0}}{z - H_{0}} d z + O (λ_{N}^{\frac{a + 1}{2}}) .

(2.61)

All half-integer powers of λ_N in (2.61) vanish by parity, which can be seen by conjugating with the unitary map $U_{P}$ acting as $U_{P} a^{†} (f) U_{P} = - a^{†} (f)$ [recall from (2.23) that $H_{j}$ contains an even/odd number of creation/annihilation operators for j even/odd]. After some lengthy computations, this yields (2.47).

D. Proof of corollary 1.1

To prove Corollary 1.1, one first observes that $γ_{N}^{(1)}$ can be decomposed as

γ_{N}^{(1)} = p^{φ} + \frac{1}{\sqrt{N}} (| φ 〉 〈 β_{χ} | + | β_{χ} 〉 〈 φ |) + \frac{1}{N} (γ_{χ} - p^{φ} T r P N_{⊥ φ}),

(2.62)

where γ_χ denotes the one-body reduced density matrix of χ with kernel $γ_{χ} (x; y) = ⟨ χ, a_{y}^{†} a_{x} χ ⟩$ and where $β_{χ} : R^{d} \to C$ is defined as

β_{χ} (x) ≔ T r P \sqrt{1 - \frac{N_{⊥ φ}}{N}} a_{x}

(2.63)

[see Ref. 2 (Sec. 3.5)]. Next, one expands the N-dependent expressions in (2.62) in powers of $λ_{N}^{1 / 2}$ and estimates the remainders using (a generalized version of) Proposition 2.1. We will show this for a = 1; the higher orders follow similarly using estimates from Ref. 1.

For $A \in L (H)$ ⁠, (2.62) yields

\begin{align} |T r A γ_{N}^{(1)} - T r A γ_{1, 0} - λ_{N} T r A γ_{1, 1}| \leq & |T r P a^{†} (A φ) \frac{\sqrt{N - N_{⊥ φ}}}{N} - λ_{N} T r P_{1} a^{†} (A φ)| \end{align}

(2.64a)

+ |T r P \frac{\sqrt{N - N_{⊥ φ}}}{N} a (A φ) - λ_{N} T r P_{1} a (A φ)|

(2.64b)

+ |\frac{1}{N} T r d Γ (A) P - λ_{N} T r d Γ (A) P_{0}|

(2.64c)

+ |⟨φ, A φ⟩| |\frac{1}{N} T r P N_{⊥ φ} - λ_{N} T r P_{0} N_{⊥ φ}| .

(2.64d)

In the first line, we expand $\sqrt{N - N_{⊥ φ}} / N = λ_{N}^{1 / 2} + λ_{N}^{3 / 2} R$ ⁠, where $R$ is a function of $N_{⊥ φ}$ such that $‖ R ϕ ‖ ≲ ‖ (N_{⊥ φ} + 1) ϕ ‖$ for any $ϕ \in F$ [see Ref. 2, Sec. 5H, Eq. (5-64b)]. By parity,

T r P_{0} a^{†} (A φ) R = T r P_{0} a^{†} (A φ) = 0,

(2.65)

and hence,

\begin{matrix} (2.64 a) & \leq & λ_{N}^{\frac{1}{2}} |T r P a^{†} (A φ) - T r (P_{0} + λ_{N}^{\frac{1}{2}} P_{1}) a^{†} (A φ)| \\ + λ_{N}^{\frac{3}{2}} |T r P a^{†} (A φ) R - T r P_{0} a^{†} (A φ) R| . \end{matrix}

(2.66)

Since

‖ a^{†} (A φ) R ϕ ‖ \leq ‖ A ‖_{o p} ‖ {(N_{⊥ φ} + 1)}^{\frac{3}{2}} ϕ ‖, ‖ a^{†} (A φ) ϕ ‖ \leq ‖ A ‖_{o p} ‖ {(N_{⊥ φ} + 1)}^{\frac{1}{2}} ϕ ‖,

(2.67)

one shows as in the proof of [Ref. 1 (Theorem 1)] that $(2.66) ≲ λ_{N}^{2} ‖ A ‖_{o p}$ ⁠. The estimate of (2.64b) works analogously. For the third line in (2.64), one notes that $| 1 / N - λ_{N} | ≲ λ_{N}^{2}$ and that $T r P_{1} d Γ (A) = 0$ by parity, and hence,

(2.64 c) ≲ λ_{N}^{2} |T r A γ_{χ}| + λ_{N} |T r (P - P_{0} - λ_{N}^{\frac{1}{2}} P_{1}) d Γ (A)| ≲ λ_{N}^{2} ‖ A ‖_{o p}

(2.68)

as above, where we used that $‖ d Γ (A) ϕ ‖ \leq ‖ A ‖_{o p} ‖ (N_{⊥ φ} + 1) ϕ ‖$ for any $ϕ \in F$ ⁠. Analogously, we derive the bound $(2.64 d) ≲ λ_{N}^{2} ‖ A ‖_{o p}$ ⁠, making use of the fact that finite moments of $N_{⊥ φ}$ with respect to χ₀ and χ are bounded uniformly in N Ref. 1 [Lemmas 4.7(d) and 5.6(a)]. This concludes the proof of Corollary 1.1 by duality of compact and trace class operators. □

E. Extensions

The results proven in Ref. 1 are more general than what we have presented so far. In this section, we briefly comment on some extensions of Theorem 1.

1. Unbounded interaction potentials

One extension concerns unbounded interaction potentials, including the three-dimensional repulsive Coulomb potential. In fact, we can replace Assumption 1 by the following assumption:

Assumption 1’.

Let

v : R^{d} \to R

be measurable with v(−x) = v(x) and v ≢ 0, and assume that there exists a constant C > 0 such that in the sense of operators on

Q (- Δ) = H^{1} (R^{d})

,

| v |^{2} \leq C (1 - Δ) .

(2.69)

In addition, assume that v is of positive type.

In this situation, we require one additional assumption, ensuring that the N-body state exhibits complete BEC with not too many particles outside the condensate.

Assumption 3.

Assume that there exist constants C₁ ≥ 0, 0 < C₂ ≤ 1, and a function

ε : N \to R_{0}^{+}

with

\lim_{N \to \infty} N^{- \frac{1}{3}} ε (N) \leq C_{1}

such that

H_{N} - N e_{H} \geq C_{2} \sum_{j = 1}^{N} h_{j} - ε (N)

(2.70)

in the sense of operators on

D (H_{N})

.

Under these more general assumptions, several new issues arise, at the core of which is the problem that dΓ(v) cannot be bounded by powers of $N_{⊥ φ} + 1$ alone. This affects the Proof of Proposition 2.1 at multiple points; most notably, it becomes considerably more difficult to obtain the uniform bound on moments of the number operator (2.58).

2. Excited states

The analysis in Ref. 1 extends to the low-energy eigenstates of H_N, i.e., it includes all eigenstates with an energy of order one above the ground state energy. In this situation, the expansion must be done more carefully, since the excited eigenvalues $E_{0}^{(n)} > E_{0}$ of $H_{0}$ can be degenerate, and the degeneracy of eigenvalues of $H_{\leq N}$ may change in the limit N → ∞. For instance, an eigenvalue $E_{0}^{(n)}$ of $H_{0}$ could be twice degenerate, with two distinct eigenvalues $E_{\leq N}^{(n_{1})} \neq E_{\leq N}^{(n_{2})}$ of $H_{\leq N}$ such that

\lim_{N \to \infty} E_{\leq N}^{(n_{1})} = E_{0}^{(n)} = \lim_{N \to \infty} E_{\leq N}^{(n_{2})} .

In this case, we expand the projector

P^{(n)} = {\frac{1}{2 π i} \oint}_{γ^{(n)}} \frac{1}{z - H} d z

(2.71)

around

P_{0}^{(n)} = {\frac{1}{2 π i} \oint}_{γ^{(n)}} \frac{1}{z - H_{0}} d z,

(2.72)

where γ⁽ⁿ⁾ is a $O (1)$ contour around $E_{0}^{(n)}$ with a finite distance to the remaining spectrum of $H_{0}$ ⁠. Since γ⁽ⁿ⁾ encloses both poles $E_{\leq N}^{(n_{1})}$ and $E_{\leq N}^{(n_{2})}$ of ${(z - H)}^{- 1}$ ⁠, the contour integral (2.71) gives precisely the sum of the two spectral projectors of $H$ corresponding to $E_{\leq N}^{(n_{1})}$ and $E_{\leq N}^{(n_{2})}$ ⁠.

In Ref. 1, we show that there is a constant C(a, n), which, in particular, depends on $| E_{0}^{(n)} |$ ⁠, such that

|T r A P^{(n)} - \sum_{ℓ = 0}^{a} λ_{N}^{\frac{ℓ}{2}} T r A P_{ℓ}^{(n)}| \leq C (a, n) λ_{N}^{\frac{a + 1}{2}} ‖ A ‖_{o p}

(2.73)

for sufficiently large N. The coefficients $P_{ℓ}^{(n)}$ are defined analogously to $P_{ℓ}$ from (2.36) but with $P_{0}$ replaced by $P_{0}^{(n)}$ ⁠. Note that the statement is non-trivial only for states with an energy of order one above the ground state energy because the constant C(a, n) depends on |E₀|.

To state the generalization of expansion (1.7) to the low-energy spectrum of H_N, we need some more notation. We denote by

E_{N} \equiv E_{N}^{(0)} < E_{N}^{(1)} < \dots < E_{N}^{(ν)} < \dots

the eigenvalues of H_N and by $δ_{N}^{(ν)}$ the degeneracy of $E_{N}^{(ν)}$ (we follow the convention of counting eigenvalues without multiplicity). Given an eigenvalue $E_{0}^{(n)}$ of $H_{0}$ ⁠, we collect the indices ν of the eigenvalues $E_{N}^{(ν)}$ that converge to $N e_{H} + E_{0}^{(n)}$ for some given n in the index set

ι^{(n)} ≔ \{ν \in N_{0} : \lim_{N \to \infty} (E_{N}^{(ν)} - N e_{H}) = E_{0}^{(n)}\} .

(2.74)

The generalization of (1.7) to excited eigenvalues $E_{N}^{(n)}$ is then given by

|\sum_{ν \in ι^{(n)}} δ_{N}^{(ν)} E_{N}^{(ν)} - δ_{0}^{(n)} N e_{H} - \sum_{ℓ = 0}^{a} λ_{N}^{ℓ} E_{ℓ}^{(n)}| \leq C (a, n) λ_{N}^{a + 1},

(2.75)

where $δ_{0}^{(n)}$ denotes the degeneracy of $E_{0}^{(n)}$ and where $E_{ℓ}^{(n)}$ is defined as in (2.47) but with $P_{0}$ replaced by $P_{0}^{(n)}$ ⁠. The constant C(a, n) depends on $| E_{0}^{(n)} |$ ⁠.

3. Expectation values of unbounded operators

Finally Ref. 1 yields an asymptotic expansion of expectation values of self-adjoint m-body operators A^(m), which are relatively bounded with respect to $\sum_{j = 1}^{m} (- Δ_{j} + V (x_{j}))$ ⁠, i.e.,

‖ A^{(m)} ψ ‖_{H^{m}} \leq C {‖\sum_{j = 1}^{m} (- Δ_{j} + V (x_{j}) + 1) ψ‖}_{H^{m}} for ψ \in D (\sum_{j = 1}^{m} (- Δ_{j} + V (x_{j}))) .

(2.76)

For $A_{N}^{(m)}$ ⁠, the symmetrized version of A^(m),

A_{N}^{(m)} ≔ {(\binom{N}{m})}^{- 1} \sum_{1 \leq j_{1} < \dots < j_{m} \leq N} A_{j_{1}, . . ., j_{m}}^{(m)},

(2.77)

we prove that there exists a constant C(m, a) such that

|⟨Ψ_{N}, A_{N}^{(m)} Ψ_{N}⟩ - \sum_{ℓ = 0}^{a} λ_{N}^{\frac{ℓ}{2}} T r ((U_{N, φ} A_{N}^{(m)} U_{N, φ}^{*} \oplus 0) P_{ℓ}^{(n)})| \leq C (m, a) λ_{N}^{\frac{a + 2}{2}}

(2.78)

for sufficiently large N. The statement extends to excited states as explained in Sec. II E 2.

The rate in (2.78) is by a factor $λ_{N}^{1 / 2}$ better than the error estimate in Proposition 2.1. To see this, one considers the operator

A_{r e d}^{(m)} = U_{N, φ} (A_{N}^{(m)} - ⟨φ^{\otimes N}, A_{N}^{(m)} φ^{\otimes N}⟩) U_{N, φ}^{*} \oplus 0,

where we have subtracted the condensate expectation value of $A_{N}^{(m)}$ (which is of order one). Because of this subtraction, one can show that $A_{r e d}^{(m)}$ satisfies the estimate

‖ A_{r e d}^{(m)} ϕ ‖_{F_{⊥ φ}} ≲ λ_{N}^{\frac{1}{2}}, Φ \in {χ, χ_{0}},

(2.79)

and Proposition 2.1 for $A_{r e d}^{(m)}$ concludes the proof.

III. DYNAMICS

In the remaining part of this Review, we study the dynamics generated by the Hamiltonian H_N from (1.4) and explain expansions (1.10) and (1.12) of the time-evolved N-body wave function Ψ_N and of the reduced one-body density $γ_{N}^{(1)}$ ⁠. We use the superscript^trap wherever it applies.

A. Framework

We study the solutions Ψ_N(t) of the time-dependent N-body Schrödinger equation (1.3) generated by the Hamiltonian H_N from (1.4), which describes a system of N interacting bosons without an external trapping potential. As the initial state, we take

Ψ_{N} (0) = Ψ_{N}^{t r a p},

where $Ψ_{N}^{t r a p}$ is the ground state of $H_{N}^{t r a p}$ ⁠.

1. Condensate

As explained above, $Ψ_{N}^{t r a p}$ exhibits BEC in the Hartree minimizer φ^trap, and it is well known that this property is preserved by the time evolution. More precisely,

T r |γ_{N}^{(1)} (t) - | φ (t) 〉 〈 φ (t) || \leq \frac{C (t)}{N}

(3.1)

(see, e.g., Refs. 12 and 13), where φ(t) is the solution of the Hartree equation,

i \partial_{t} φ (t) = (- Δ + v * | φ (t) |^{2} - μ^{φ (t)}) φ (t) = : h^{φ (t)} φ (t), φ (0) = φ^{t r a p},

(3.2)

with the phase factor $μ^{φ (t)} = \frac{1}{2} \int_{R^{d}} (v * | φ (t) |^{2}) (x) | φ (t, x) |^{2} d x$ ⁠. The solution of (3.2) in $H^{1} (R^{d})$ is unique and exists globally. We define the projectors p^φ(t) and q^φ(t) analogously to (2.3).

2. Excitations

Analogously to (2.7), we decompose the time-evolved N-body state Ψ_N(t) into the condensate φ(t) and excitations χ_≤N(t) from the condensate. The excitation vector χ_≤N(t) is an element of the (truncated) excitation Fock space $F_{⊥ φ (t)}^{\leq N} \subset F_{⊥ φ (t)} \subset F$ defined analogously to (2.9). When restricted to the time-dependent excitation Fock space $F_{⊥ φ (t)}$ ⁠, the number operator $N$ on the (time-independent) Fock space $F$ counts the number of excitations around the time-evolved condensate φ(t)^⊗N. As before, the relation between Ψ_N(t) and χ_≤N(t) is given by the (now time-dependent) unitary map $U_{N, φ (t)}$ defined analogously to (2.11), namely,

χ_{\leq N} (t) = U_{N, φ (t)} Ψ_{N} (t) .

(3.3)

The evolution of the excitations is determined by the Schrödinger equation

i \partial_{t} χ_{\leq N} (t) = H_{\leq N}^{φ (t)} χ_{\leq N} (t), χ_{\leq N} (0) = U_{N, φ^{t r a p}} Ψ_{N}^{t r a p}

(3.4)

on $F_{⊥ φ (t)}^{\leq N}$ ⁠, generated by the excitation Hamiltonian

H_{\leq N}^{φ (t)} = i (\partial_{t} U_{N, φ (t)}) U_{N, φ (t)}^{*} + U_{N, φ (t)} H_{N} U_{N, φ (t)}^{*} .

(3.5)

For convenience, we write $H_{\leq N}^{φ (t)}$ as restriction to $F_{⊥ φ (t)}^{\leq N}$ of a Hamiltonian $H^{φ (t)}$ on $F$ ⁠, which can be expressed, analogously to (2.14), in terms of N, $N$ ⁠, and operators $K_{j}^{φ (t)}$ ⁠, which are defined analogously¹⁴ to (2.15). Expanding the N-dependent expressions in a Taylor series yields (formally) the power series

H^{φ (t)} = H_{0}^{φ (t)} + \sum_{n \geq 1} λ_{N}^{\frac{n}{2}} H_{n}^{φ (t)},

(3.6)

with coefficients $H_{j}^{φ (t)}$ analogously to (2.23). Note that the operator $H^{φ (t)}$ preserves the truncation of $F^{\leq N}$ ⁠, whereas this property is lost when truncating the expansion after finitely many terms.

3. Bogoliubov approximation

The leading order $H_{0}^{φ (t)}$ in (3.6) is the time-dependent Bogoliubov Hamiltonian, which generates the Bogoliubov time evolution,

i \partial_{t} χ_{0} (t) = H_{0}^{φ (t)} χ_{0} (t), χ_{0} (0) = χ_{0}^{t r a p} .

(3.7)

It is well known that the solution of (3.7) approximates the solution χ_≤N(t) of (3.4) to leading order, i.e.,

\lim_{N \to \infty} ‖ χ_{\leq N} (t) - χ_{0} (t) ‖_{F_{⊥ φ (t)}^{\leq N}} = 0

(3.8)

(see, e.g., Refs. 15 and 16). This is a very useful approximation because the time evolution generated by $H_{0}^{φ (t)}$ acts as a Bogoliubov transformation $U_{V (t, s)}$ on $F$ ⁠. This means a huge simplification compared with the full N-body dynamics because it essentially reduces the N-body problem to the problem of solving a 2 × 2 matrix differential equation: the corresponding Bogoliubov map $V (t, s)$ on $H \oplus H$ is determined by the differential equation

i \partial_{t} V (t, s) = A (t) V (t, s), V (s, s) = 1,

(3.9)

with

V (t, s) = (\begin{matrix} U_{t, s} & {\bar{V}}_{t, s} \\ V_{t, s} & {\bar{U}}_{t, s} \end{matrix}), A (t) = (\begin{matrix} h^{φ (t)} + K_{1}^{φ (t)} & - K_{2}^{φ (t)} \\ \bar{K_{2}^{φ (t)}} & - (h^{φ (t)} + \bar{K_{1}^{φ (t)}}) \end{matrix}) .

(3.10)

Since it is a Bogoliubov transformation, the Bogoliubov time evolution preserves quasi-freeness. Hence, χ₀(t) is uniquely determined by its two-point functions,

γ_{χ_{0} (t)} (x, y) = {⟨χ_{0} (t), a_{y}^{†} a_{x} χ_{0} (t)⟩}_{F}, α_{χ_{0} (t)} (x, y) = {⟨χ_{0} (t), a_{x} a_{y} χ_{0} (t)⟩}_{F},

(3.11)

which can be computed directly from the two-point functions of χ₀(0) as

γ_{χ_{0} (t)} (x, y) = ({\bar{V}}_{t, 0} γ_{χ_{0} (0)}^{T} {\bar{V}}_{t, 0}^{*} + U_{t, 0} γ_{χ_{0} (0)} U_{t, 0}^{*} - {\bar{V}}_{t, 0} α_{χ_{0} (0)}^{*} U_{t, 0}^{*} - U_{t, 0} α_{χ_{0} (0)} {\bar{V}}_{t, 0}^{*}) (x, y) + ({\bar{V}}_{t, 0} {\bar{V}}_{t, 0}^{*}) (x, y),

(3.12a)

α_{χ_{0} (t)} (x, y) = (U_{t, 0} α_{χ_{0} (0)} {\bar{U}}_{t, 0}^{*} + {\bar{V}}_{t, 0} α_{χ_{0} (0)}^{*} V_{t, 0}^{*} - U_{t, 0} γ_{χ_{0} (0)} V_{t, 0}^{*} - {\bar{V}}_{t, 0} γ_{χ_{0} (0)}^{T} {\bar{U}}_{t, 0}^{*}) (x, y) + (U_{t, 0} V_{t, 0}^{*}) (x, y) .

(3.12b)

Alternatively, one obtains $γ_{χ_{0} (t)}$ and $α_{χ_{0} (t)}$ by solving the system of differential equations

i \partial_{t} γ_{χ_{0} (t)} = (h^{φ (t)} + K_{1}^{φ (t)}) γ_{χ_{0} (t)} - γ_{χ_{0} (t)} (h^{φ (t)} + K_{1}^{φ (t)}) + K_{2}^{φ (t)} α_{χ_{0} (t)}^{*} - α_{χ_{0} (t)} {(K_{2}^{φ (t)})}^{*},

(3.13a)

i \partial_{t} α_{χ_{0} (t)} = (h^{φ (t)} + K_{1}^{φ (t)}) α_{χ_{0} (t)} + α_{χ_{0} (t)} {(h^{φ (t)} + K_{1}^{φ (t)})}^{T} + K_{2}^{φ (t)} + K_{2}^{φ (t)} γ_{χ_{0} (t)}^{T} + γ_{χ_{0} (t)} K_{2}^{φ (t)}

(3.13b)

(see Refs. 16 and 17).

B. Expansion of the dynamics

1. Expansion of the time-evolved wave function

With the formal ansatz

χ_{\leq N} (t) \oplus 0 = \sum_{ℓ = 0}^{\infty} λ_{N}^{\frac{ℓ}{2}} χ_{ℓ} (t),

(3.14)

the Schrödinger equation (3.4) leads to the set of equations

i \partial_{t} χ_{ℓ} (t) = H_{0}^{φ (t)} χ_{ℓ} (t) + \sum_{n = 1}^{ℓ} H_{n}^{φ (t)} χ_{ℓ - n} (t) .

(3.15)

Motivated by (3.15), we define iteratively

χ_{ℓ} (t) : = U_{V (t, 0)} χ_{ℓ} (0) - i \sum_{n = 1}^{ℓ} \int_{0}^{t} U_{V (t, s)} H_{n}^{φ (s)} χ_{ℓ - n} (s) d s,

(3.16)

where $U_{V (t, s)}$ denotes the Bogoliubov time evolution, i.e., the Bogoliubov transformation corresponding to the solution $V (t, s)$ of (3.9). To prove Theorem 2, we show that these functions χ_ℓ are the coefficients in an asymptotic expansion of χ_≤N.

Proposition 3.1.

Let Assumption 1a be satisfied, let

a \in N_{0}

⁠, and denote by χ_≤N(t) the solution of (3.4). Then,

χ_{ℓ} (t) \in F_{⊥ φ (t)}

and there exists a constant C(a) such that

{‖χ_{\leq N} (t) - \sum_{ℓ = 0}^{a} λ_{N}^{\frac{ℓ}{2}} χ_{ℓ} (t)‖}_{F^{\leq N}} \leq e^{C (a) t} λ_{N}^{\frac{a + 1}{2}}

(3.17)

for all

t \in R

and sufficiently large N.

The growth of the constant C(a) in a can be estimated as

C (a) \leq C a^{2} \ln a .

(3.18)

We do not expect this to be optimal, especially since Borel summability was shown for a comparable expansion in Ref. 18. As a consequence of Proposition 3.1, the coefficients Ψ_N,ℓ(t) of expansion (1.10) of Ψ_N(t) are given by

Ψ_{N, ℓ} (t) ≔ \sum_{k = 0}^{N} {φ (t)}^{\otimes (N - k)} \otimes_{s} {(χ_{ℓ} (t))}^{(k)} .

(3.19)

The higher orders χ_ℓ(t) are completely determined by the solution χ₀(t) of the Bogoliubov equation as

χ_{ℓ} (t) = \sum_{\begin{array}{c} 0, \leq, n \leq 3 ℓ \\ n + ℓ even \end{array}} \sum_{j \in {- 1, 1}^{n}} \int d x^{(n)} C_{ℓ, n}^{(j)} (t; x^{(n)}) a_{x_{1}}^{♯_{j_{1}}} \cdot \cdot \cdot a_{x_{n}}^{♯_{j_{n}}} χ_{0} (t),

(3.20)

where we used the notation

a_{x}^{♯_{- 1}} ≔ a_{x}, a_{x}^{♯_{1}} ≔ a_{x}^{†} .

(3.21)

The N-independent functions $C_{ℓ, n}^{(j)}$ are given in terms the matrix entries U_t,s and V_t,s of the solution $V (t, s)$ of (3.9) and the initial data. For example,

C_{1, 1}^{(1)} (t) = (U_{t, 0} {(U_{0}^{t r a p})}^{*} - {\bar{V}}_{t, 0} {(\bar{V_{0}^{t r a p}})}^{*}) Θ_{1}^{t r a p},

(3.22a)

C_{1, 1}^{(- 1)} (t) = (V_{t, 0} {(U_{0}^{t r a p})}^{*} - {\bar{U}}_{t, 0} {(\bar{V_{0}^{t r a p}})}^{*}) Θ_{1}^{t r a p}

(3.22b)

for $Θ_{1}^{t r a p}$ as in (2.39). Here, $U_{0}^{trap}$ and $V_{0}^{trap}$ denote the matrix entries of the Bogoliubov map corresponding to the Bogoliubov transformation $U_{V_{0}}^{t r a p}$ that diagonalizes $H_{0}^{t r a p}$ ⁠. The coefficients $C_{ℓ, n}^{(j)}$ with larger indices are constructed from this in a systematic iterative procedure. Since the general formula is very long and not particularly insightful, we refrain from stating it here and refer to Ref. 2 [Eq. (5.51)].

The higher orders χ_ℓ(t) satisfy a generalized Wick rule for the “mixed” correlation functions,

{⟨a_{x_{1}}^{♯_{1}} \dots a_{x_{n}}^{♯_{n}}⟩}_{ℓ, k}^{(t)} ≔ ⟨χ_{ℓ} (t), a_{x_{1}}^{♯_{1}} \dots a_{x_{n}}^{♯_{n}} χ_{k} (t)⟩ .

(3.23)

Proposition 3.2

(generalized Wick rule).

If k + ℓ + n odd,
${⟨a_{x_{1}}^{♯_{j_{1}}} \dots a_{x_{n}}^{♯_{j_{n}}}⟩}_{ℓ, k}^{(t)} = 0 .$
(3.24)
If k + ℓ + n even,
${⟨a_{x_{1}}^{♯_{j_{1}}} \dots a_{x_{n}}^{♯_{j_{n}}}⟩}_{ℓ, k}^{(t)} = \sum_{\begin{array}{c} b = n \\ even \end{array}}^{n + 3 (ℓ + k)} \sum_{m \in {- 1, 1}^{b}} \sum_{σ \in P_{b}} \prod_{i = 1}^{b / 2} \int d y^{(b)} D_{ℓ, k, n; b}^{(j; m)} (t; x^{(n)}; y^{(b)}) {⟨a_{y_{σ (2 i - 1)}}^{♯_{m_{σ (2 i - 1)}}} a_{y_{σ (2 i)}}^{♯_{m_{σ (2 i)}}}⟩}_{0, 0}^{(t)}$
(3.25)
for P_b being the set of pairings defined in (2.33). The functions $D_{ℓ, k, n; b}^{(j; m)}$ are determined by the coefficients $C$ from (3.20) [see Ref. 2 (Corollary 3.5) for the precise formula].

2. Expansion of the one-body reduced density matrix

As an application of (3.17), we derive expansion (1.12) of the one-body reduced density matrix. The coefficients $γ_{N, ℓ}^{(1)}$ in (1.12) are given by the trace class operators with kernels

γ_{1, 0} (t; x; y) ≔ φ (t, x) \bar{φ (t, y)},

(3.26a)

\begin{align} γ_{1, ℓ} (t; x; y) & ≔ & \sum_{m = 1}^{ℓ} [\sum_{k = 0}^{ℓ - m} \sum_{n = 0}^{2 m - 1} {\tilde{c}}_{ℓ - m, k} (φ (t, x) {⟨a_{y}^{†} {(N - 1)}^{k}⟩}_{n, 2 m - n - 1}^{(t)})) \\ + ({⟨{(N - 1)}^{k} a_{x}⟩}_{n, 2 m - n - 1}^{(t)} \bar{φ (t, y)}) \\ + (\sum_{n = 0}^{2 m - 2} {\tilde{c}}_{ℓ - m} ({⟨a_{y}^{†} a_{x}⟩}_{n, 2 m - n - 2}^{(t)} - φ (t, x) \bar{φ (t, y)} {⟨N⟩}_{n, 2 m - n - 2}^{(t)})], \end{align}

(3.26b)

with ${\tilde{c}}_{ℓ}$ and ${\tilde{c}}_{ℓ, k}$ as in (2.51) and where we used the notation (3.23). For example, the leading order of the expansion is $γ_{0}^{(1)} (t) = p^{φ (t)}$ ⁠, which recovers (3.1). The next-to-leading order is given by

γ_{1}^{(1)} (t) = | φ (t) 〉 〈 β_{0, 1} (t) | + | β_{0, 1} (t) 〉 〈 φ (t) | + γ_{χ_{0} (t)} - T r γ_{χ_{0} (t)} p^{φ (t)},

(3.27)

where the function $β_{0, 1} : R^{d} \to C$ is the solution of

\begin{matrix} i \partial_{t} β_{0, 1} (t) & = & (h^{φ (t)} + K_{1}^{φ (t)}) β_{0, 1} (t) + K_{2}^{φ (t)} \bar{β_{0, 1} (t)} \\ + {(K_{3}^{φ (t)})}^{*} α_{χ_{0} (t)} + {T r}_{1} (K_{3}^{φ (t)} γ_{χ_{0} (t)}) + {T r}_{2} (K_{3}^{φ (t)} γ_{χ_{0} (t)}) . \end{matrix}

(3.28)

Here, $γ_{χ_{0} (t)}$ and $α_{χ_{0} (t)}$ are the Bogoliubov two-point functions as in (3.11), and we used the notation Tr₁A ≔ ∫dzA(z, ·; z) and Tr₂A ≔ ∫ dzA(·, z; z) for an operator $A : H \to H^{2}$ ⁠.

C. Strategy of proof

To prove Proposition 3.1, we first show that the functions χ_ℓ(t) defined in (3.16) are elements of $F_{⊥ φ (t)}$ by proving that

{⟨χ_{ℓ} (t), {(N + 1)}^{b} χ_{ℓ} (t)⟩}_{F} ≲ e^{C (ℓ, b) t}

(3.29)

for any $b \in N_{0}$ ⁠. To this end, we re-write χ_ℓ(t) as

χ_{ℓ} (t) = U_{V (t, 0)} χ_{ℓ} (0) + \sum_{n = 0}^{ℓ - 1} \sum_{m = 1}^{ℓ - n} \sum_{\begin{array}{c} j \in N^{m} \\ | j | = ℓ - n \end{array}} {(- i)}^{m} \int_{0}^{t} d s_{1} \int_{0}^{s_{1}} d s_{2} \cdot \cdot \cdot \int_{0}^{s_{m - 1}} d s_{m} {\tilde{H}}_{t, s_{1}}^{(j_{1})} \cdot \cdot \cdot {\tilde{H}}_{t, s_{m}}^{(j_{m})} U_{V (t, 0)} χ_{n} (0),

(3.30)

with

{\tilde{H}}_{t, s}^{(n)} ≔ U_{V (t, s)} H_{n}^{φ (s)} U_{V (t, s)}^{*},

(3.31)

bound the operators ${\tilde{H}}_{t, s}^{(n)}$ by powers of $(N + 1)$ ⁠, and make use of the fact that any finite moment of $N$ with respect to χ_n(0) is bounded since $χ_{n} (0) = χ_{n}^{t r a p}$ from (2.42). To prove (3.17), we expand $H^{φ (t)}$ in a Taylor series with remainder analogously to (2.54), prove an estimate the remainder in terms of $N$ ⁠, and make use of (3.29) to close a Gronwall argument for the function ${\tilde{χ}}_{a} (t) = χ_{\leq N} (t) \oplus 0 - \sum_{ℓ = 0}^{a} λ_{N}^{ℓ / 2} χ_{ℓ} (t)$ ⁠.

To prove Corollary 1.2, one decomposes $γ_{N}^{(1)} (t)$ analogously to (2.62) and expands it in powers of $λ_{N}^{1 / 2}$ ⁠, which yields expressions containing correlation functions of χ_≤N,

{⟨a_{x_{1}}^{♯_{1}} \dots a_{x_{n}}^{♯_{n}}⟩}_{N}^{(t)} ≔ {⟨χ_{\leq N} (t), a_{x_{1}}^{♯_{1}} \dots a_{x_{n}}^{♯_{n}} χ_{\leq N} (t)⟩}_{F^{\leq N}} .

(3.32)

Finally, we show that, in a suitable sense,

{⟨a_{x_{1}}^{♯_{1}} \dots a_{x_{n}}^{♯_{n}}⟩}_{N}^{(t)} = \sum_{ℓ = 0}^{a} λ_{N}^{\frac{ℓ}{2}} \sum_{m = 0}^{ℓ} {⟨χ_{m} (t), a_{x_{1}}^{♯_{1}} \dots a_{x_{n}}^{♯_{n}} χ_{ℓ - m} (t)⟩}_{F} + O (λ_{N}^{\frac{a + 1}{2}}),

(3.33)

where all half-integer powers of λ_N vanish by the generalized Wick rule (Proposition 3.2).

D. Extensions

The results proven in Ref. 2 are more general than what was stated so far, namely, they admit a larger class of initial data. It is not necessary to start the time evolution in the ground state $Ψ_{N}^{t r a p}$ of the trapped system (or in any low-energy eigenstate of $H_{N}^{t r a p}$ ⁠), but it suffices if the initial state satisfies the following assumption:

Assumption 4.

Let

\tilde{a} \in N_{0}

. Let

Ψ_{N} (0) \in D (H_{N})

, define

χ_{\leq N} (0) = U_{N, φ (0)} Ψ_{N} (0)

, and assume that there exists a constant

C (\tilde{a}) > 0

such that

{‖χ_{\leq N} (0) - \sum_{ℓ = 0}^{\tilde{a}} λ_{N}^{\frac{ℓ}{2}} χ_{ℓ} (0)‖}_{F^{\leq N}} \leq C (\tilde{a}) λ_{N}^{\frac{\tilde{a} + 1}{2}},

(3.34)

where the functions χ_ℓ(0) are defined as follows:

Let $\tilde{ν} \in N_{0}$ , let $U_{V_{0}}$ be a Bogoliubov transformation on $F_{⊥ φ (0)}$ , and let ${f_{j}}_{j = 1}^{\tilde{ν}} \subset {φ (0)}^{⊥}$ be some orthonormal system. Define
$χ_{0} (0) : = U_{V_{0}} a^{†} (f_{1}) \cdot \cdot \cdot a^{†} (f_{\tilde{ν}}) | Ω 〉 .$
(3.35)
For $1 \leq ℓ \leq \tilde{a}$ , let
$χ_{ℓ} (0) = \sum_{\begin{array}{c} 0 \leq m \leq 3 ℓ \\ m + ℓ even \end{array}} \sum_{μ = 0}^{m} \int d x^{(μ)} d y^{(m - μ)} {\tilde{a}}_{m, μ}^{(ℓ)} (x^{(μ)}; y^{(m - μ)}) a_{x_{1}}^{†} \cdot \cdot \cdot a_{x_{μ}}^{†} a_{y_{1}} \cdot \cdot \cdot a_{y_{m - μ}} χ_{0} (0),$
(3.36)
where $a_{n, m, μ}^{(ℓ)} (x^{(μ)}; y^{(m - μ)})$ are the kernels of some N-independent bounded operators.

Moreover, our analysis generalizes to the case where χ₀(0) is given as a linear combination of Bogoliubov transformed states with different particle numbers $\tilde{ν}$ ⁠. It is clear that this is satisfied by any superposition of low-energy eigenstates of $H_{N}^{t r a p}$ ⁠.

E. Related results

We conclude with a brief overview of closely related results in the literature. The first derivation of higher order corrections is due to Ginibre and Velo,^18,19 who consider the classical field limit ℏ → 0 of the dynamics generated by a Hamiltonian on Fock space with coherent states as initial data. They construct a Dyson expansion of the unitary group W(t, s) in terms of the time evolution generated by the Bogoliubov Hamiltonian; moreover, they prove that the expansion is Borel summable for bounded interaction potentials.¹⁸ The main difference to our work (apart from the Fock space setting) is that the authors expand the time evolution operator W(t, s) in a perturbation series (and not the wave function). In contrast, we derive an expansion of the time-evolved wave function for a specific, physically relevant choice of initial data. This simplifies the approximation since fewer terms are required at a given order of the approximation because the state is expanded simultaneously with the Hamiltonian.

Another approach to higher order corrections in the mean-field regime in the N-body setting was proposed by Paul and Pulvirenti.²⁰ In that work, the authors approach the problem from a kinetic theory perspective and consider the dynamics of the reduced density matrices of the N-body state. Their approach is formally similar to ours, since Bogoliubov theory in the sense of linearization of the Hartree equation is used for the expansion and an a-dependent but N-independent number of operations is required for the construction. In comparison, the main advantage of our approach is that the coefficients χ_ℓ in our approximation are completely independent of N.

Finally, a similar result in the N-body setting was obtained in a joint work with Pavlović, Pickl, and Soffer²¹ In this paper, we expand the N-body time evolution in a Dyson series comparable to (3.16) but with one crucial difference: instead of using the Bogoliubov time evolution, the expansion is in terms of an auxiliary time evolution ${\tilde{U}}_{φ} (t, s)$ on $H^{N}$ ⁠, whose generator has a quadratic structure comparable to the Bogoliubov Hamiltonian (sometimes called the particle number preserving Bogoliubov Hamiltonian).

Unfortunately, this auxiliary time evolution ${\tilde{U}}_{φ} (t, s)$ is a rather inaccessible object, which implicitly still depends on N. In particular, it is not clear to what extent computations are less complex with respect to the time evolution ${\tilde{U}}_{φ} (t, s)$ than with respect to the full N-body problem. This problem was the original motivation for the work,² where we modified the construction precisely such as to make the approximations completely N-independent and accessible to computations. Eventually, this also led to Ref. 1, which was partially intended as a rigorous motivation of the assumptions on the initial data in Ref. 2.

IV. OPEN PROBLEMS

There are several open questions related to the results presented here. First, it would be interesting to generalize the dynamical analysis (Theorem 2) to the class of unbounded interaction potentials considered in Sec. II E 1 for the static problem, which, in particular, includes the Coulomb potential.

In addition, one can attempt to push the analysis to singular interactions of the type

v_{N, β} (x) = N^{- 1 + d β} v (N^{β} x), β \in [0, 1],

for some bounded and compactly supported interaction potential v, where β is a scaling parameter interpolating from the Hartree (β = 0) to the Gross–Pitaevskii regime (β = 1). We expect the analysis to become harder with increasing β, mainly because of the emergence of an N-dependent short-scale correlation structure. Whereas new ideas are needed to cope with the extremely singular Gross–Pitaevskii regime, we expect our analysis to extend to a certain range of positive β.

Another interesting open problem is proving Borel summability of the asymptotic series in Theorems 1 and 2, at least for bounded interaction potentials. This property was established in Ref. 18 for the corresponding dynamical problem on Fock space described in Sec. III E; hence, we conjecture that it should hold true also in the N-body setting, at least for bounded interaction potentials. As our current estimates of the growth of the error C(a) in the parameter a are insufficient, new ideas are needed to improve this.

Finally, we expect our asymptotic expansions to be useful in answering various open problems related to the mean-field Bose gas. For instance, one should be able to derive effective interactions between the quasi-particles as discussed in Ref. 22 and to prove corrections to the central limit theorem obtained in Ref. 23.

ACKNOWLEDGMENTS

The author thanks Nataša Pavlović, Sören Petrat, Peter Pickl, Robert Seiringer, and Avy Soffer for the collaboration on Refs. 1, 2 and 21. Funding from the European Union’s Horizon 2020 Research and Innovation Programme under Marie Skℓodowska-Curie Grant Agreement No. 754411 is gratefully acknowledged.

AUTHOR DECLARATIONS

Conflict of Interest

The author has no conflicts to disclose.

DATA AVAILABILITY

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

REFERENCES

1.

L.

Boßmann

,

S.

Petrat

, and

R.

Seiringer

, “

Asymptotic expansion of low-energy excitations for weakly interacting bosons

,”

Forum Math. Sigma

9

,

e28

(

2021

).

https://doi.org/10.1017/fms.2021.22

Google Scholar

Crossref

2.

L.

Boßmann

,

S.

Petrat

,

P.

Pickl

, and

A.

Soffer

, “

Beyond Bogoliubov dynamics

,”

Pure Appl. Anal.

3

(

4

),

677

–

726

(

2021

).

https://doi.org/10.2140/paa.2021.3.677

Google Scholar

Crossref

3.

R.

Seiringer

, “

The excitation spectrum for weakly interacting bosons

,”

Commun. Math. Phys.

306

(

2

),

565

–

578

(

2011

).

https://doi.org/10.1007/s00220-011-1261-6

Google Scholar

Crossref

4.

P.

Grech

and

R.

Seiringer

, “

The excitation spectrum for weakly interacting bosons in a trap

,”

Commun. Math. Phys.

322

(

2

),

559

–

591

(

2013

).

https://doi.org/10.1007/s00220-013-1736-8

Google Scholar

Crossref

5.

M.

Lewin

,

P. T.

Nam

,

S.

Serfaty

, and

J. P.

Solovej

, “

Bogoliubov spectrum of interacting Bose gases

,”

Commun. Pure Appl. Math.

68

(

3

),

413

–

471

(

2015

).

https://doi.org/10.1002/cpa.21519

Google Scholar

Crossref

6.

A.

Pizzo

, “

Bose particles in a box I. A convergent expansion of the ground state of a three-modes Bogoliubov Hamiltonian

,” arXiv:1511.07022 (

2015

).

Google Scholar

7.

A.

Pizzo

, “

Bose particles in a box II. A convergent expansion of the ground state of the Bogoliubov Hamiltonian in the mean field limiting regime

,” arXiv:1511.07025 (

2015

).

Google Scholar

8.

A.

Pizzo

, “

Bose particles in a box III. A convergent expansion of the ground state of the Hamiltonian in the mean field limiting regime

,” arXiv:1511.07026 (

2015

).

Google Scholar

9.

M.

Lewin

,

P. T.

Nam

, and

N.

Rougerie

, “

Derivation of Hartree’s theory for generic mean-field Bose systems

,”

Adv. Math.

254

,

570

–

621

(

2014

).

https://doi.org/10.1016/j.aim.2013.12.010

Google Scholar

Crossref

10.

P. T.

Nam

and

M.

Napiórkowski

, “

Two-term expansion of the ground state one-body density matrix of a mean-field Bose gas

,”

Calculus Var. Partial Differ. Equations

60

(

3

),

99

(

2021

).

https://doi.org/10.1007/s00526-021-01954-2

Google Scholar

Crossref

11.

For technical reasons, we split $H = H^{<} + H^{>}$ ⁠, where $H^{>} ≔ 0 \oplus (c - K_{0} - (\frac{N - N_{⊥ φ}}{N - 1}) K_{1} - \frac{1}{N - 1} K_{4})$ ⁠, and expand only $H^{<}$ ⁠. To keep the notation simple, we will ignore this subtlety for the sketch of the proof. All details can be found in Ref. 1.

12.

L.

Chen

,

J. O.

Lee

, and

B.

Schlein

, “

Rate of convergence towards Hartree dynamics

,”

J. Stat. Phys.

144

(

4

),

872

–

903

(

2011

).

https://doi.org/10.1007/s10955-011-0283-y

Google Scholar

Crossref

13.

D.

Mitrouskas

,

S.

Petrat

, and

P.

Pickl

, “

Bogoliubov corrections and trace norm convergence for the Hartree dynamics

,”

Rev. Math. Phys.

31

(

8

),

1950024

(

2019

).

https://doi.org/10.1142/s0129055x19500247

Google Scholar

Crossref

14.

To obtain the time-dependent operators $K_{j}^{φ (t)}$ from (2.15), one replaces φ by φ(t), h by h^φ(t), μ by μ^φ(t), and K₁ by $K_{1}^{φ (t)} = q^{φ (t)} {\tilde{K}}^{φ (t)} q^{φ (t)}$ with ${\tilde{K}}^{φ (t)} (x_{1}; x_{2}) = \bar{φ (t, x_{2})} v (x_{1} - x_{2}) φ (t, x_{1})$ ⁠.

15.

M.

Lewin

,

P. T.

Nam

, and

B.

Schlein

, “

Fluctuations around Hartree states in the mean field regime

,”

Am. J. Math.

137

(

6

),

1613

–

1650

(

2015

).

https://doi.org/10.1353/ajm.2015.0040

Google Scholar

Crossref

16.

P. T.

Nam

and

M.

Napiórkowski

, “

Bogoliubov correction to the mean-field dynamics of interacting bosons

,”

Adv. Theor. Math. Phys.

21

(

3

),

683

–

738

(

2017

).

https://doi.org/10.4310/atmp.2017.v21.n3.a4

Google Scholar

Crossref

17.

M.

Grillakis

and

M.

Machedon

, “

Pair excitations and the mean field approximation of interacting bosons, I

,”

Commun. Math. Phys.

324

(

2

),

601

–

636

(

2013

).

https://doi.org/10.1007/s00220-013-1818-7

Google Scholar

Crossref

18.

J.

Ginibre

and

G.

Velo

, “

The classical field limit of nonrelativistic bosons. I. Borel summability for bounded potentials

,”

Ann. Phys.

128

(

2

),

243

–

285

(

1980

).

https://doi.org/10.1016/0003-4916(80)90320-6

Google Scholar

Crossref

19.

J.

Ginibre

and

G.

Velo

, “

The classical field limit of non-relativistic bosons. II. Asymptotic expansions for general potentials

,”

Ann. Inst. Henri Poincare Phys. Theor.

33

(

4

),

363

–

394

(

1980

), available at http://www.numdam.org/article/AIHPA_1980__33_4_363_0.pdf.

Google Scholar

20.

T.

Paul

and

M.

Pulvirenti

, “

Asymptotic expansion of the mean-field approximation

,”

Discrete Contin. Dyn. Syst. A

39

(

4

),

1891

–

1921

(

2019

).

https://doi.org/10.3934/dcds.2019080

Google Scholar

Crossref

21.

L.

Boßmann

,

N.

Pavlović

,

P.

Pickl

, and

A.

Soffer

, “

Higher order corrections to the mean-field description of the dynamics of interacting bosons

,”

J. Stat. Phys.

178

(

6

),

1362

–

1396

(

2020

).

https://doi.org/10.1007/s10955-020-02500-8

Google Scholar

Crossref

22.

T.

Sugiyama

, “

Microscopic derivation of the roton-roton interaction

,”

Prog. Theor. Phys.

63

(

4

),

1170

–

1180

(

1980

).

https://doi.org/10.1143/ptp.63.1170

Google Scholar

Crossref

23.

G.

Ben Arous

,

K.

Kirkpatrick

, and

B.

Schlein

, “

A central limit theorem in many-body quantum dynamics

,”

Commun. Math. Phys.

321

,

371

–

417

(

2013

).

https://doi.org/10.1007/s00220-013-1722-1

Google Scholar

Crossref

2022

Author(s)

Low-energy spectrum and dynamics of the weakly interacting Bose gas

I. INTRODUCTION AND MAIN RESULTS

A. Introduction

B. Model and main results

II. LOW-ENERGY SPECTRUM AND EIGENSTATES

A. Framework

1. Condensate

2. Excitations

3. Excitation Hamiltonian

4. Bogoliubov approximation

B. Expansion of the ground state

1. Ground state wave function

2. Ground state energy

3. Ground state reduced density

C. Strategy of proof

D. Proof of corollary 1.1

E. Extensions

1. Unbounded interaction potentials

2. Excited states

3. Expectation values of unbounded operators

III. DYNAMICS

A. Framework

1. Condensate

2. Excitations

3. Bogoliubov approximation

B. Expansion of the dynamics

1. Expansion of the time-evolved wave function

2. Expansion of the one-body reduced density matrix

C. Strategy of proof

D. Extensions

E. Related results

IV. OPEN PROBLEMS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

DATA AVAILABILITY

REFERENCES

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