Beyond Bogoliubov Dynamics

Lea Boßmann∗, Soren Petrat†, Peter Pickl‡, and Avy Soffer§

March 30, 2022

Abstract

We consider a system of N interacting bosons in the mean-field scaling regime and con-struct corrections to the Bogoliubov dynamics that approximate the trueN -body dynamicsin norm to arbitrary precision. The N -independent corrections are given in terms of the so-lutions of the Bogoliubov and Hartree equations and satisfy a generalized form of Wick’stheorem. We determine the n-point correlation functions of the excitations around thecondensate, as well as the reduced densities of the N -body system, to arbitrary accuracy,given only the knowledge of the two-point correlation functions of a quasi-free state andthe solution of the Hartree equation. In this way, the complex problem of computing alln-point correlation functions for an interacting N -body system is essentially reduced tothe problem of solving the Hartree equation and the PDEs for the Bogoliubov two-pointcorrelation functions.

MSC class: 35Q40, 35Q55, 81Q05, 82C10

1 Introduction

We consider a system of N weakly interacting bosons in Rd, d ≥ 1, which are initially preparedin the state

ΨNtrap ∈

N⊗sym

L2(Rd) =:N⊗

sym

H =: HNsym ,

which is either the ground state or an appropriate low-energy excited eigenstate of the Hamil-tonian

HNtrap :=

N∑j=1

(−∆j + Vtrap(xj)) + λN∑

1≤i<j≤Nv(xi − xj) . (1.1)

Here, Vtrap : Rd → R denotes a suitable trapping potential, v : Rd → R is some bounded pairinteraction, and

λN :=1

N − 1. (1.2)

∗Fachbereich Mathematik, Eberhard Karls Universitat Tubingen, Auf der Morgenstelle 10, 72076 Tubingen,Germany; and Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria.lea.bossmann@ist.ac.at

†Department of Mathematics and Logistics, Jacobs University Bremen, Campus Ring 1, 28759 Bremen,Germany; and University of Bremen, Department 3 – Mathematics, Bibliothekstr. 5, 28359 Bremen, Germany.s.petrat@jacobs-university.de

‡Mathematisches Institut, Ludwig-Maximilians-Universitat Munchen, Theresienstr. 39, 80333 Munchen,Germany. pickl@math.lmu.de

§Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA.soffer@math.rugers.edu

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This model describes bosons in the mean-field or Hartree regime, which is characterized byweak, long-range interactions. At time t = 0, we remove the trapping potential and study thedynamics generated by the Hamiltonian

HN :=

N∑j=1

(−∆j) + λN∑

1≤i<j≤Nv(xi − xj) , (1.3)

which is determined by the N -body Schrodinger equation,

i∂tΨN (t) = HNΨN (t) , ΨN (0) = ΨN

trap . (1.4)

The dynamics is non-trivial since the eigenstates of HNtrap are not eigenstates of HN anymore.

As a consequence of the high-dimensional configuration space RdN and since all particles be-come correlated under the time evolution, it is practically impossible to solve (1.4) analyticallyor numerically for any reasonably large particle number N . It is therefore of physical rele-vance as well as of mathematical interest to derive and analyze suitable approximations tothe solution ΨN (t) of (1.4). In this paper, we propose a perturbative scheme which yields anapproximation of ΨN (t) to any order in N−1. Our construction is such that all corrections tocorrelation functions and expectation values of bounded operators are given in terms of thetwo-point correlation functions of a quasi-free state. This crucially reduces the complexity ofthe N -body problem and makes it possible to numerically compute these physically relevantquantities to arbitrary precision.

It is well known that the ground state as well as the low-energy excited eigenstates of HNtrap

exhibit Bose–Einstein condensation in the state ϕtrap ∈ H, which is given by the minimizerof the corresponding Hartree energy functional [10, 55]. To construct a norm approximationof the many-body state, one decomposes ΨN

trap into the condensate part and excitations fromthe condensate as in [53],

ΨNtrap =

N∑k=0

ϕ⊗(N−k)trap ⊗s χ(k)

trap , χ≤Ntrap :=(χ

(k)trap

)Nk=0∈ F≤N⊥ϕtrap

, (1.5)

where ⊗s denotes the symmetric tensor product. The set of k-particle excitations χ(k)trap forms

a vector χ≤Ntrap in the truncated Fock space over the orthogonal complement of ϕtrap. In [74,

39, 53], it was shown that χ≤Ntrap is approximated in norm by an appropriate eigenstate χtrap0 of

the Bogoliubov Hamiltonian corresponding to HNtrap. By (1.5), this leads to an approximation

of ΨNtrap with respect to the HN -norm.

This analysis was extended in [13], where it was shown that the ground state and low-energy

excited states of the Hamiltonian HNtrap admit an asymptotic expansion in the parameter λ

1/2N ,

i.e., it holds that ∥∥∥χ≤Ntrap −a∑`=0

λ`2Nχ

trap`

∥∥∥F≤N⊥ϕtrap

≤ Cλa+12

N (1.6)

for N -independent coefficients χtrap` ∈ F⊥ϕtrap . Since the leading order χtrap

0 is an eigenstateof the Bogoliubov Hamiltonian associated with HN

trap, it is related via a Bogoliubov trans-

formation to a state with fixed particle number. In particular, if ΨNtrap is the ground state

of HNtrap, χtrap

0 is quasi-free, i.e., it arises as Bogoliubov transformation of the vacuum. The

2

higher orders in the expansion (1.6) can be written as

χtrap` =

∑0≤n≤3`n+` even

∑j∈−1,1n

∫dx(n) a

(j)`,n

(x(n)

)a]j1x1 ···a

]jnxn χ

trap0 (1.7)

for some known coefficients a(j)`,n, a multi-index j = (j1, ..., jn), and where we abbreviated

a]−1 := a (annihilation operator), a]1 := a† (creation operator) and x(n) := (x1, ..., xn).

The property of Bose–Einstein condensation is preserved by the time evolution (1.4), whichwas first shown in the 1970s and 1980s by Hepp, Ginibre, Velo, and Spohn [44, 34, 35, 77].Interest in the question was revived in the early 2000s through the work of Bardos, Golse,and Mauser [6], and a series of papers by Erdos, Schlein, and Yau [30, 25, 26, 27, 24, 28, 29].Since then, many more results have been obtained, e.g., in [1, 2, 31, 72, 32, 49, 48, 64, 65, 8,21, 75, 3, 45, 47, 15, 46], see also [9, 38] for overviews of the topic. The optimal convergencerate was proven in [24, 19, 20, 57], where it was shown that for any t ∈ R, there exists someC(t) > 0 such that

Tr∣∣∣γ(1)

ΨN(t)− |ϕ(t)〉〈ϕ(t)|

∣∣∣ ≤ C(t)N−1 , (1.8)

where γ(1)

ΨN(t) denotes the one-particle reduced density matrix of ΨN (t) and where ϕ(t) is the

solution of the Hartree equation (2.1) with initial datum ϕ(0) = ϕtrap. We refer to [54, 33, 51]for an overview of the results and for further references.

To characterize the N -body dynamics on the level of the wave function, one decomposes,analogously to (1.5),

ΨN (t) =N∑k=0

ϕ(t)⊗(N−k) ⊗s χ(k)(t) , χ≤N (t) :=(χ(k)(t)

)Nk=0∈ F≤N⊥ϕ(t) . (1.9)

In [52], it was shown that χ≤N (t) is approximated in norm by the solution χ0(t) of theBogoliubov (or Bogoliubov–de Gennes) equation,

i∂tχ0(t) = H(0)ϕ(t)χ0(t) , χ0(0) = χtrap

0 , (1.10)

where H(0)ϕ(t) denotes the Bogoliubov Hamiltonian corresponding to HN . The dynamics (1.10)

describes the formation of pair correlations, which gives the first order correction to the mean-field dynamics. Fluctuations around the mean-field were first analyzed by Ginibre and Velo[34, 35] and by Grillakis, Machedon, and Margetis [42, 43, 40, 41] in a slightly different setting,and further results in this direction were obtained, e.g., in [52, 59, 11, 57, 22, 60, 61, 50, 14, 63].Moreover, dynamical central limit theorems were derived in [7, 16, 69].

In our main result (Theorem 1), we prove that the bound (1.6) is preserved by the timeevolution (1.4): for any a ∈ N0, we show that∥∥∥χ≤N (t)−

a∑`=0

λ`2Nχ`(t)

∥∥∥F≤N⊥ϕ(t)

≤ Ca(t)λa+12

N (1.11)

for explicitly computable, N -independent coefficients χ`(t) ∈ F⊥ϕ(t). The higher order cor-rections (` ≥ 1) to the leading order term χ0(t) are given by

χ`(t) =∑

0≤n≤3`n+` even

∑j∈−1,1n

∫dx(n)C

(j)`,n(t;x(n)) a

]j1x1 ··· a

]jnxn χ0(t) (1.12)

3

for some explicitly known, N -independent coefficients C(j)`,n (see Corollary 3.4), which are con-

structed iteratively from the initial coefficients a(j)`,n from (1.7). To derive (1.12), it is crucial

to note that the unitary time evolution generated by the Bogoliubov Hamiltonian acts as aBogoliubov transformation on creation and annihilation operators.

The main advantage of (1.12) lies in the fact that it essentially reduces the computationof the higher order corrections χ`(t) to the problem of solving first the well-studied Hartreeequation, and second the Bogoliubov equation, which is equivalent to solving a 2 × 2 matrixdifferential equation (see Section 4). Hence, (1.12) solves the dynamics of the mean-field Bosegas as complete as seems possible, to any order in 1/N , in terms of functions that can beretrieved with a reasonable computational effort which is completely independent of N .

Although we decided to focus on the situation where the initial state ΨN (0) is an eigenstateof the Hamiltonian HN

trap, our results remain valid in a more general setting, also including,e.g., mixtures of such states. In particular, it is not necessary to assume that the interactionv is of positive type, which is required for the static result (1.6).

As one application, our approximation scheme yields computationally accessible higherorder corrections to the trace norm convergence (1.8) of the reduced density matrices. InTheorem 2, we prove that

Tr

∣∣∣∣∣γ(1)

ΨN(t)−

a∑`=0

λ`Nγ(1)` (t)

∣∣∣∣∣ ≤ Ca(t)λa+1N (1.13)

for N -independent one-body operators γ(1)` (t) (see (3.40) for a definition). While the leading

order in (1.13),

γ(1)0 (t) = |ϕ(t)〉〈ϕ(t)| , (1.14)

recovers Bose–Einstein condensation with optimal rate as in (1.8), none of the higher order

terms in the expansion of γ(1)

ΨN(t) has—to the best of our knowledge—been known so far,

neither in the mathematics nor in the physics community. For example, the first correction to

γ(1)0 is given by

γ(1)1 (t) = |ϕ(t)〉〈β0,1(t)|+ |β0,1(t)〉〈ϕ(t)|+ γχ0(t) − TrHγχ0(t)|ϕ(t)〉〈ϕ(t)| , (1.15)

where β0,1(t, x) is the solution of

i∂tβ0,1(t) =(hϕ(t) +K

(1)ϕ(t)

)β0,1(t) +K

(2)ϕ(t)β0,1(t)

+(K

(3)ϕ(t)

)∗αχ0(t) + Tr1

(K

(3)ϕ(t)γχ0(t)

)+ Tr2

(K

(3)ϕ(t)γχ0(t)

). (1.16)

Here, the integral operators K(i)ϕ(t) are defined in terms of ϕ(t) and v in (2.23), and we used

the notation Tr1A :=∫

dzA(z, · ; z) and Tr2A :=∫

dzA( · , z; z), for an operator A : H→ H2.Moreover, γχ0(t) and αχ0(t) are the Bogoliubov two-point correlation functions,

γχ0(t)(x, y) :=⟨χ0(t), a†yaxχ0(t)

⟩, αχ0(t)(x, y) := 〈χ0(t), axayχ0(t)〉 , (1.17)

which are determined by the initial data through a system of two coupled PDEs (3.26) onR2d. In particular, it is possible to solve the PDEs numerically to arbitrary accuracy.

4

Finally, (1.12), leads to an asymptotic expansion of the n-point correlation functions ofthe excitations, ⟨

a]j1x1 ··· a

]jnxn

⟩(t)

N:=⟨χ≤N (t), a

]j1x1 ··· a

]jnxn χ

≤N (t)⟩F≤N⊥ϕ(t)

. (1.18)

Let us for simplicity consider only the situation where ΨN (0) = ΨNtrap is the ground state of

HNtrap. In this case, we prove in Corollary 3.7 that

⟨a]j1x1 · · · a

]j2nx2n

⟩(t)

N=

a∑`=0

λ`N

2∑m=0

⟨χm(t), a

]j1x1 · · · a

]j2nx2n χ2`−m(t)

⟩+O(λa+1

N ) , (1.19a)

⟨a]j1x1 · · · a

]j2n+1x2n+1

⟩(t)

N=

a−1∑`=0

λ`+ 1

2N

2`+1∑m=1

⟨χm(t), a

]j1x1 · · · a

]j2n+1x2n+1 χ2`+1−m(t)

⟩+O(λ

a+ 12

N ) . (1.19b)

The expansion (1.19) essentially reduces the N -body problem to the problem of computingthe two-point correlation functions of the quasi-free state χ0(t), since (1.12) leads to a gener-alization of Wick’s rule for the “mixed” n-point correlation functions of the corrections χ`(t):If `+ n+ k is odd, it holds that⟨

χ`(t), a]j1x1 ···a

]jnxn χk(t)

⟩F⊥ϕ(t)

= 0 , (1.20a)

and if `+ n+ k is even,⟨χ`(t), a

]j1x1 · · · a

]jnxn χk(t)

⟩F⊥ϕ(t)

(1.20b)

=

n+3(`+k)∑b=neven

∑m∈−1,1b

∑σ∈Pb

b/2∏i=1

∫dy(b)D

(j;m)`,k,n;b(t;x

(n); y(b))⟨χ0(t), a

]mσ(2i−1)yσ(2i−1)

a]mσ(2i)yσ(2i) χ0(t)

⟩F⊥ϕ(t)

,

where D(j;m)`,k,n;b is determined by the coefficients C from (1.12) and Pb is a set of pairings (see

Corollary 3.5 for the precise statement). In particular, by (1.20b), the right-hand side of (1.19)is explicitly given in terms of the Bogoliubov two-point correlation functions (1.17), which areobtained by solving a system of two coupled PDEs.

A possible application of this result is a precise computation of the condensate depletion.By (1.19a),⟨

χ≤N (t),Nχ≤N (t)⟩F≤N

= 〈χ0(t),Nχ0(t)〉+ λN

(〈χ1(t),Nχ1(t)〉+ 〈χ0(t),Nχ2(t)〉+ 〈χ2(t),Nχ0(t)〉

)+O(λ2

N ). (1.21)

Making use of (1.20b), the next order correction beyond Bogoliubov theory can (analyticallyor numerically) be evaluated. For dilute Bose gases, the condensate depletion has been studiedin various settings, and the predictions of Bogoliubov theory were experimentally confirmed[78, 18, 56].

In conclusion, our approximation scheme provides a computationally efficient perturba-tive algorithm for the analysis of the mean-field dynamics, which yields N -independent cor-rections. In the NLS and Gross–Pitaevskii scaling regimes, where the interaction scales as

5

N−1+3βv(N βx) for 0 < β ≤ 1 (in three dimensions), the correlations are larger due to thesingular interaction. We conjecture that our method can be extended to regimes with β > 0,at least for small values of β. However, the N -independence of the corrections might be hardto achieve. We also expect that our results can be extended to a class of pair interactionsincluding the repulsive Coulomb potential in three dimensions, but this is certainly harder toprove.

We conclude with an overview of closely related results. To the best of our knowledge,the first derivation of higher order corrections is due to Ginibre and Velo [37, 36]. Theyconsider the classical field limit ~ → 0 of the dynamics generated by a Hamiltonian H onFock space, which is such that H

∣∣HN

= HN upon identification ~ ≡ 1/N . Working in the

Heisenberg picture, they expand H in a power series of 1√Na(t) − ϕ(t), effectively separating

the (classical) motion of the condensate from the excitations, and study the unitary timeevolution W (t, s) of the operators b(t) = C(

√Nϕ(0))∗

(a(t) −

√Nϕ(t)

)C(√Nϕ(0)). Here, C

is the Weyl operator, which implements the c-number substitution for coherent initial data(similar to (1.5)). The authors construct a Dyson expansion of the unitary group W (t, s)in terms of the time evolution generated by the Bogoliubov Hamiltonian and prove that theexpansion is Borel summable for bounded interaction potentials [37] and strongly asymptoticfor a class of unbounded potentials [36].

The main differences to our work are that Ginibre and Velo consider a Hamiltonian onFock space (instead of HN ), accordingly use coherent states as initial data, and expand thetime evolution operator W (t, s) in a perturbation series (instead of ΨN ). In contrast, weprovide explicit formulas for physically relevant initial data, compute correlation functionsand reduced densities, and make use of the connection between Bogoliubov transformationsand Bogoliubov maps to reduce the computational complexity.

A similar result in the N -body setting was derived in [12] by expanding the N -bodytime evolution in a comparable Dyson series. Instead of the Bogoliubov time evolution, theexpansion is in terms of an auxiliary time evolution Uϕ(t, s) on HN , whose generator hasa quadratic structure comparable to the Bogoliubov Hamiltonian (sometimes called particlenumber preserving Bogoliubov Hamiltonian). However, the auxiliary time evolution Uϕ(t, s) isa rather inaccessible object, which implicitly still depends on N . In particular, it is not clear towhat extent the computation of physical quantities of interest is less complex with respect tothe time evolution Uϕ(t, s) than with respect to the full N -body problem. The approximationscheme we propose in this paper can be understood as an improvement of this result, wherewe modified the construction precisely such as to make it accessible to computations.

A related approach to obtain higher order corrections in the mean-field regime was in-troduced by Paul and Pulvirenti in [62]. In that work, the authors approach the problemfrom a kinetic theory perspective and consider the dynamics of the reduced density matricesof the N -body state. Their approach is formally similar to ours, since Bogoliubov theoryin the sense of linearization of the Hartree equation is used for the expansion (but withoutprojecting onto the excitation Fock space), an even-odd structure similar to (1.20) is observed,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 our approximations (1.12) arecompletely N -independent.

A multi-scale expansion of the ground state of the Bose gas in the mean-field limit wasintroduced by Pizzo [66, 67, 68] via a constructive approach using Feshbach maps. In [13],perturbation theory for the ground state and low-energy excited states corresponding to our

6

dynamical results is made rigorous (see the discussion above and in Section 3). We refer to[13] for further literature in the static case.

In addition to [62, 12], our work is particularly inspired by the works of Grillakis, Machedonand Margetis [42, 43, 40, 41], Nam and Napiorkowski [59, 60], and some of the classicalreferences [5, 71, 23], which emphasize the concepts of Bogoliubov transformations and quasi-free states. For example, in [59, 60] the authors prove that describing the excitations aroundthe Hartree solution ϕ(t) as solutions of the N -independent Bogoliubov equation yields acomputationally efficient approximation to the many-body dynamics: since the Bogoliubovtime evolution preserves quasi-freeness, all time evolved n-point correlation functions are—thanks to Wick’s rule—determined by the initial two-point correlation functions, up to orderN−1/2. The result presented here can be understood as a continuation of these ideas. Ourcorrections χ`(t) to the Bogoliubov description are still completely determined by the initialtwo-point correlation functions of the leading order contribution χ0(0), and in this sense, thecorrections χ`(t) can be understood as generalized quasi-free states which satisfy a generalizedform of Wick’s rule.

Notation

• Expressions that are independent of N, t,ΨN (0), ϕ(0), but may depend on all fixedquantities of the model such as v, are referred to as constants. While we explicitlyindicate the dependence of constants on the order a of the approximation by writing Caor C(a), note that these constants may vary from line to line. We use the notation

A . B (1.22)

to indicate that there exists a constant C > 0 such that A ≤ CB.

• Unless indicated otherwise, ϕ(t) denotes the solution of (2.1) with initial datum ϕ(0).

• If a vector in F is written as a direct sum, it is always understood with respect to thedecomposition F = F≤N ⊕F>N . For φ1,φ2,φ ∈ F , we use the notation

〈φ1,φ2〉F≤N :=N∑k=0

⟨φ

(k)1 , φ

(k)2

⟩Hk

, ‖φ‖2F≤N :=N∑k=0

‖φ(k)‖2Hk . (1.23)

• We abbreviatex(k) := (x1, ..., xk) , dx(k) := dx1 ··· dxk (1.24)

for k ≥ 1 and xj ∈ Rd, and use the notation

a]1 := a† , a]−1 := a . (1.25)

We denote by j = (j1, ..., jn) a multi-index and define |j| := j1 + · · ·+ jn.

• The set of pairings of 1, ..., 2a used for Wick’s rule is defined as

P2a := σ ∈ S2a : σ(2j − 1) < minσ(2j), σ(2j + 1) ∀ 1 ≤ j ≤ 2a, (1.26)

where S2a denotes the symmetric group on the set 1, ..., 2a.

• L(V,W ) denotes the space of bounded operators from V to W , and the correspondingnorm is sometimes abbreviated as ‖A‖op := ‖A‖L(V,W ). We denote the Hilbert–Schmidtnorm by ‖A‖HS.

7

2 Preliminaries

2.1 Framework

We begin with stating our assumptions on the Hamiltonian HN from (1.3) and on the initialcondensate wave function.

Assumption 0. Assume that the interaction potential v : Rd → R is bounded and even, andlet ϕ(0) ∈ H1(Rd) with ‖ϕ(0)‖H = 1.

As a consequence of Assumption 0, the N -body Hamiltonian HN generates a unique familyof unitary time evolution operators, which leaves D(HN ) = H2(Rd) invariant.

The evolution of the condensate is described by the solution ϕ(t) of the Hartree equation,

i∂tϕ(t) =(−∆ + v ∗ |ϕ(t)|2 − µϕ(t)

)ϕ(t) =: hϕ(t)ϕ(t), ϕ(0) = ϕtrap , (2.1)

where the real-valued phase factor µϕ(t) is given by

µϕ(t) = 12

∫Rd

(v ∗ |ϕ(t)|2

)(x)|ϕ(t, x)|2 dx . (2.2)

The solution of (2.1) in H1(Rd) is unique and exists globally [17, Corollaries 4.3.3 and 6.1.2].Moreover, it holds that ‖ϕ(t)‖H = ‖ϕ(0)‖H = 1.

We define the (truncated) Fock spaces

F≤N⊥ϕ(t) :=N⊕k=0

k⊗sym

ϕ(t)⊥ , F⊥ϕ(t) :=∞⊕k=0

k⊗sym

ϕ(t)⊥ , F(H) :=∞⊗k=0

k⊗sym

H , (2.3)

for ϕ⊥ :=φ ∈ H : 〈φ, ϕ〉H = 0

. Whenever the context is unambiguous, we abbreviate

F := F(H) . (2.4)

The Fock vacuum is denoted by |Ω〉 = (1, 0, 0, . . . ). Note that the subspaces F≤N⊥ϕ(t) andF⊥ϕ(t) depend on time via the condensate wave function, while the full Fock space F is time-

independent. We refer to F⊥ϕ(t) as excitation Fock space, and to F≤N⊥ϕ(t) as truncated excitation

Fock space. The one-particle sector ϕ(t)⊥ of the subspace F⊥ϕ(t) can be understood as ahyperplane in H, which, due to the time evolution of the condensate wave function ϕ(t), “ro-tates” through H; the higher sectors are (symmetrized) direct sums of these hyperplanes.

The creation and annihilation operators on F are defined as

(a†(f)φ)(k)(x1, ..., xk) =1√k

k∑j=1

f(xj)φ(k−1)(x1, ..., xj−1, xj+1, ..., xk) , k ≥ 1 , (2.5)

(a(f)φ)(k)(x1, ..., xk) =√k + 1

∫dxf(x)φ(k+1)(x1, ..., xk, x) , k ≥ 0 (2.6)

for f ∈ H and φ ∈ F . They can be expressed in terms of the operator-valued distributionsa†x, ax, satisfying the canonical commutation relations

[ax, a†y] = δ(x− y) , [ax, ay] = [a†x, a

†y] = 0 , (2.7)

8

as

a†(f) =

∫dxf(x) a†x , a(f) =

∫dxf(x) ax . (2.8)

The second quantization in F of a k-body operator A(k) is defined as

dΓ(A(k)) =

∫dx(k)

∫dy(k)A(k)(x(k); y(k)) a†x1 . . . a

†xkay1 . . . ayk , (2.9)

and the number operator on F is denoted by

N =

∫a†xax dx , (Nφ)(k) = kφ(k) . (2.10)

The number of excitations around the time-evolved condensate ϕ(t)⊗N is counted by theexcitation number operator,

N⊥ϕ(t) := N − a†(ϕ(t))a(ϕ(t)) = dΓ(qϕ(t)) , (2.11)

where qϕ(t) denotes the projection onto the orthogonal complement of ϕ(t), i.e.,

pϕ(t) := |ϕ(t)〉〈ϕ(t)| , qϕ(t) := 1H − pϕ(t) . (2.12)

Thus, when restricted to vectors φ(t) ∈ F⊥ϕ(t), N counts the number of excitations,

N|F⊥ϕ(t) = N⊥ϕ(t)|F⊥ϕ(t) . (2.13)

Note that N is not time-dependent, while N⊥ϕ(t) depends on time. The relation between the

N -body state ΨN (t) and the corresponding excitation vector χ≤N (t) is given by the unitarymap

UN,ϕ(t) : HNsym → F≤N⊥ϕ(t) , ΨN (t) 7→ UN,ϕ(t)Ψ

N (t) := χ≤N (t) . (2.14)

Equivalently, UN,ϕ(t) can be interpreted as a partial isometry from HNsym to F , where U∗N,ϕ(t)

is extended by zero outside the truncated excitation Fock space F≤N⊥ϕ(t). The product state

ϕ(t)⊗N is mapped to the vacuum, i.e.,

UN,ϕ(t)ϕ(t)⊗N = (1, 0, 0, . . . ) . (2.15)

Written in terms of creation and annihilation operators, the map UN,ϕ(t) acts as

UN,ϕ(t)ΨN (t) =

N⊕j=0

(qϕ(t)

)⊗j (a(ϕ(t))N−j√(N − j)!

ΨN (t)

)(2.16)

([53, Proposition 4.2]). This leads for f, g ∈ ϕ(t)⊥ to

UN,ϕ(t)a†(ϕ(t))a(ϕ(t))U∗N,ϕ(t) = N −N⊥ϕ(t) , (2.17a)

UN,ϕ(t)a†(f)a(ϕ(t))U∗N,ϕ(t) = a†(f)

√N −N⊥ϕ(t) , (2.17b)

UN,ϕ(t)a†(ϕ(t))a(g)U∗N,ϕ(t) =

√N −N⊥ϕ(t) a(g) , (2.17c)

UN,ϕ(t)a†(f)a(g)U∗N,ϕ(t) = a†(f)a(g) (2.17d)

as identities on F≤N⊥ϕ(t). Note that since N⊥ϕ(t) = N on F⊥ϕ(t), we can equivalently replace

N⊥ϕ(t) by N in (2.17).

9

2.2 Excitation Hamiltonian

The map UN,ϕ(t) decouples the evolution (2.1) of the condensate wave function from the

evolution of the excitations χ≤N (t) = UN,ϕ(t)ΨN (t), which is determined by

i∂tχ≤N (t) = H≤Nϕ(t)χ

≤N (t) , (2.18)

with excitation Hamiltonian

H≤Nϕ(t) = i(∂tUN,ϕ(t))U∗N,ϕ(t) + UN,ϕ(t)H

NU∗N,ϕ(t) : F≤N⊥ϕ(t) → F≤N . (2.19)

For later convenience, we write H≤Nϕ(t) as restriction to F≤N⊥ϕ(t) of a Hamiltonian Hϕ(t) which isdefined on the full Fock space F ,

H≤Nϕ(t) = Hϕ(t)

∣∣F≤N⊥ϕ(t)

. (2.20)

The transformation rules (2.17) can be used to obtain an explicit formula for Hϕ(t) from(2.19) (see, e.g., [53, Section 4] and [52, Section 4.2 and Appendix B]). Written in a way thatis more convenient for our analysis, it is given as

Hϕ(t) = K(0)ϕ(t) +

N −NN − 1

K(1)ϕ(t)

+

K(2)ϕ(t)

√[(N −N )(N −N − 1)]+

N − 1+

√[(N −N )(N −N − 1)]+

N − 1K(2)ϕ(t)

+

K(3)ϕ(t)

√[N −N ]+

N − 1+

√[N −N ]+

N − 1K(3)ϕ(t)

+1

N − 1K(4)ϕ(t) , (2.21)

where [·]+ denotes the positive part and where we used the shorthand notation

K(0)ϕ(t) =

∫dx a†xh

ϕ(t)x ax , (2.22a)

K(1)ϕ(t) =

∫dx1 dx2K

(1)ϕ(t)(x1;x2)a†x1ax2 , (2.22b)

K(2)ϕ(t) = 1

2

∫dx1 dx2K

(2)ϕ(t)(x1, x2)a†x1a

†x2 , (2.22c)

K(3)ϕ(t) =

∫dx(3)K

(3)ϕ(t)(x1, x2;x3)a†x1a

†x2ax3 (2.22d)

K(4)ϕ(t) = 1

2

∫dx(4)K

(4)ϕ(t)(x1, x2;x3, x4)a†x1a

†x2ax3ax4 , (2.22e)

K(j)ϕ(t) :=

(K(j)ϕ(t)

)∗, j = 2, 3 (2.22f)

with

K(1)ϕ(t) : ϕ(t)⊥ → ϕ(t)⊥, K

(1)ϕ(t) := qϕ(t)K

(1)ϕ(t)q

ϕ(t) , (2.23a)

K(2)ϕ(t) ∈ ϕ(t)⊥ ⊗ ϕ(t)⊥, K

(2)ϕ(t)(x1, x2) :=

(qϕ(t)1 q

ϕ(t)2 K

(2)ϕ(t)

)(x1, x2) , (2.23b)

K(3)ϕ(t) : ϕ(t)⊥ → ϕ(t)⊥ ⊗ ϕ(t)⊥,

10

ψ 7→ (K(3)ϕ(t)ψ)(x1, x2) := q

ϕ(t)1 q

ϕ(t)2 Wϕ(t)(x1, x2)ϕ(x1)(qϕ(t)ψ)(x2) , (2.23c)

K(4)ϕ(t) : ϕ(t)⊥ ⊗ ϕ(t)⊥ → ϕ(t)⊥ ⊗ ϕ(t)⊥,

ψ 7→ (K(4)ϕ(t)ψ)(x1, x2) := q

ϕ(t)1 q

ϕ(t)2 Wϕ(t)(x1, x2)(q

ϕ(t)1 q

ϕ(t)2 ψ)(x1, x2) . (2.23d)

Here, K(1)ϕ(t) is the operator with kernel

K(1)ϕ(t)(x1;x2) := ϕ(t, x2)v(x1 − x2)ϕ(t, x1) , (2.24)

K(2)ϕ(t) denotes the vector

K(2)ϕ(t)(x1, x2) := v(x1 − x2)ϕ(t, x1)ϕ(t, x2) , (2.25)

and Wϕ(t) is the multiplication operator on H⊗ H defined by

Wϕ(t)(x, y) := v(x− y)−(v ∗ |ϕ(t)|2

)(x)−

(v ∗ |ϕ(t)|2

)(y) + 2µϕ(t). (2.26)

The notation is understood such that the projections qϕ(t)1 , q

ϕ(t)2 act on the respective functions

on their right. For example, the function K(3)ϕ(t)ψ is obtained from ψ ∈ H by taking the tensor

product of qϕ(t)ψ and ϕ(t), acting on it with the multiplication operator Wϕ(t), and finally

projecting the resulting function on H⊗H onto the subspace ϕ(t)⊥⊗ϕ(t)⊥. In Appendix A,we provide a derivation of (2.21).

Let us remark that there is more than one way to extend H≤Nϕ(t) to the full Fock spaceF . In particular, we could have defined Hϕ(t) in terms of N⊥ϕ(t) instead of N since the two

operators coincide on F≤N⊥ϕ(t). We chose this way of defining Hϕ(t) for later convenience.

Expanding the N -dependent expressions in (2.21) in a Taylor series, we write the excitation

Hamiltonian Hϕ(t) as a power series in λ1/2N with N -independent (operator-valued) coefficients

(see Section 5.3 for a proof):

Lemma 2.1. Let a ∈ N0. In the sense of operators on F , it holds that

Hϕ(t) =

a∑n=0

λn2NH

(n)ϕ(t) + λ

a+12

N R(a) , (2.27)

where

H(0)ϕ(t) := K(0)

ϕ(t) + K(1)ϕ(t) + K(2)

ϕ(t) + K(2)ϕ(t) , (2.28a)

H(1)ϕ(t) := K(3)

ϕ(t) + K(3)ϕ(t) , (2.28b)

H(2)ϕ(t) := −(N − 1)K(1)

ϕ(t) −K(2)ϕ(t)(N −

12)− (N − 1

2)K(2)ϕ(t) + K(4)

ϕ(t) , (2.28c)

H(2n−1)ϕ(t) := cn−1

(K(3)ϕ(t)(N − 1)n−1 + (N − 1)n−1K(3)

ϕ(t)

), (2.28d)

H(2n)ϕ(t) :=

n∑ν=0

dn,ν

(K(2)ϕ(t)(N − 1)ν + (N − 1)νK(2)

ϕ(t)

)(2.28e)

11

for n ≥ 2 and with

c(`)0 := 1 , c(`)

n :=(`− 1

2)(`+ 12)(`+ 3

2)···(`+ n− 32)

n!, cn := c(0)

n (n ≥ 1) , (2.29)

dn,ν :=

ν∑`=0

c(0)` c

(0)ν−`c

(`)n−ν (n ≥ ν ≥ 0) . (2.30)

The remainder R(a) satisfies

‖R(a)φ‖F ≤ C(a)‖(N + 1)a+42 φ‖F (2.31)

for some constant C(a) > 0 and any φ ∈ F .

The leading order H(0)ϕ(t) is the Bogoliubov Hamiltonian. The higher orders H(n)

ϕ(t) containpowers of N , which originate from the Taylor expansions of the square roots in Hϕ(t).

The operator Hϕ(t) preserves the truncation of F≤N (see Lemma 5.3), whereas this property

is lost upon expansion of the square roots. For example, the second term of H(0)ϕ(t) acting on the

N -particle sector creates two new particles, resulting in a non-zero (N + 2)-component. Thesingle “fluxes” out of the truncated Fock space cancel with the remainder, such that the wholeexpression is truncation preserving. Moreover, the effect is very small since the occupation ofthe N -particle sector is negligible for large N (see Lemma 5.5b).

2.3 Assumptions

In this section, we present and discuss our assumptions on the initial N -body wave function.

Assumption 1. Let Assumption 0 hold and let a ∈ N0. Let ΨN (0) ∈ HNsym, define χ≤N (0) =

UN,ϕ(0)ΨN (0), and assume that there exists a constant C(a) > 0 such that∥∥∥∥∥χ≤N (0)−

a∑`=0

λ`2Nχ`(0)

∥∥∥∥∥F≤N

≤ C(a)λa+12

N , (2.32)

where the functions χ`(0) are defined as follows:

• Let ν ∈ N0, let UV0 be a Bogoliubov transformation on F⊥ϕ(0), and let fjνj=1 ⊂ ϕ(0)⊥be some orthonormal system (or the empty set for ν = 0). Define

χ0(0) := UV0a†(f1

)··· a†

(fν)|Ω〉 . (2.33)

• For 1 ≤ ` ≤ a, define iteratively

χ`(0) :=∑n=1

∑0≤m≤n+2m+n even

m∑µ=0

∫dx(µ) dy(m−µ)a(`)

n,m,µ

(x(µ); y(m−µ)

)×a†x1 ···a

†xµay1 ···aym−µ χ`−n(0) , (2.34)

where a(`)n,m,µ

(x(µ); y(m−µ)

)are the (Schwartz) kernels of some N -independent operators.

12

We stated (2.34) in this iterative form for later convenience (in particular, this simplifiesthe proof of Corollary 3.4). By solving the iteration and bringing the result into normal order,one can show that (2.34) is equivalent to the statement that

χ`(0) =∑

0≤m≤3`m+` even

m∑µ=0

∫dx(µ) dy(m−µ)a(`)

m,µ

(x(µ); y(m−µ)

)a†x1 ···a

†xµay1 ···aym−µ χ0(0) (2.35)

for some bounded operators a(`)m,µ : Hm−µ → Hµ.

Since the operators a(`)n,m,µ are N -independent, the only N -dependence in the initial state

is due to the powers of λ1/2N in the series expansion. While χ0(0) is normalized, this is not

the case for χ`(0) for ` ≥ 1, hence ΨN (0) need not be normalized. To recover a normalizedinitial N -body wave function ΨN (0), it suffices to divide χ≤N (0) and all corrections χ`(0) bya normalization factor ‖χ≤N (0)‖F≤N = ‖χ≤N (t)‖F≤N . This situation is often encountered inperturbation theory (see, e.g., [73]).

Our analysis generalizes to the case where χ0(0) is given as a linear combination of Bo-goliubov transformed states with different particle numbers ν. We refrain from including suchinitial states in Assumption 1 to simplify the formulas (especially Corollary 3.5).

In [13], it is shown that Assumption 1 is satisfied if ΨN (0) is the ground state or a suitablelow-energy eigenstate of HN

trap as in (1.1). We formulate the precise statement and discussdegeneracies and an example in Appendix B.

By Assumption 1, finite moments of N with respect to χ`(0) and χ≤N (0) are boundeduniformly in N (see Section 5.2 for a proof):

Lemma 2.2. Let Assumption 1 hold for some a ∈ N0 and denote χ≤N (0) = UN,ϕ(0)ΨN (0).

(a) Then for all b ≥ 0 and 0 ≤ ` ≤ a there exists a constant C(`, b) > 0 such that⟨χ`(0), (N + 1)bχ`(0)

⟩F≤ C(`, b) . (2.36)

(b) For all 0 ≤ b ≤ a+ 1 there exists a constant C(b) > 0 such that⟨χ≤N (0), (N + 1)bχ≤N (0)

⟩F≤N

≤ C(b) . (2.37)

3 Results: Dynamical Perturbation Theory

3.1 Norm Approximation to Any Order

By Assumption 1, the initial excitation vector χ≤N (0)⊕ 0 admits an expansion in powers of

λ1/2N . With the formal ansatz

χ≤N (t)⊕ 0 =

∞∑`=0

λ`2Nχ`(t) , (3.1)

the Schrodinger equation (2.18) leads to the set of equations

i∂tχ`(t) = H(0)ϕ(t)χ`(t) +

∑n=1

H(n)ϕ(t)χ`−n(t) . (3.2)

13

The decomposition of χ≤N (t)⊕ 0 into the wave functions χ`(t) is according to orders in λ1/2N

and does not relate to the number of excitations contained in χ`(t), as χ`(t) is not necessarilya state with a fixed particle number. Due to the inhomogeneity in (3.2), the norm of χ`(t)changes with time for ` ≥ 1. However, the norm of the perturbation series is conserved in anyorder, i.e.,

∑`m=0〈χ`−m(t),χm(t)〉 does not change in time.

For ` = 0, (3.2) recovers the Bogoliubov equation (1.10). The time evolution generated by

H(0)ϕ(t) is well-posed and acts as a Bogoliubov transformation UV(t,s) on F . The corresponding

Bogoliubov map V(t, s) on H⊕ H is determined by the differential equation

i∂tV(t, s) = A(t)V(t, s) , V(s, s) = 1 (3.3)

with

V(t, s) =

(Ut,s V t,s

Vt,s U t,s

), A(t) =

hϕ(t) +K(1)ϕ(t) −K(2)

ϕ(t)

K(2)ϕ(t) −

(hϕ(t) +K

(1)ϕ(t)

) , (3.4)

see Lemma 4.8. In Section 4, we give an overview of Bogoliubov transformations and collectthe corresponding rigorous results. Written in integral form, (3.2) motivates the followingdefinition:

Definition 3.1. Let V(t, s) be the solution of (3.3), denote by UV(t,s) the corresponding Bo-

goliubov transformation on F , and define H(n)ϕ(t) as in (2.28). For any ` ∈ N0, we define

iteratively

χ`(t) := UV(t,0)χ`(0)− i∑n=1

t∫0

UV(t,s) H(n)ϕ(s)χ`−n(s) ds . (3.5)

By unitarity of UV(t,s), this is equivalent to the formula

χ`(t) = UV(t,0)χ`(0)− i∑n=1

t∫0

H(n)t,s UV(t,s)χ`−n(s) ds, (3.6)

whereH(n)t,s := UV(t,s)H

(n)ϕ(s) U

∗V(t,s) . (3.7)

Iterating (3.6) yields the following result:

Proposition 3.2. Let ` ∈ N0, let Assumption 0 hold and let χn(0) ∈ F⊥ϕ(0) ∩ D(N 3(`−n)/2)for 0 ≤ n ≤ `. Then χ`(t) ∈ F⊥ϕ(t) and

χ`(t) = UV(t,0)χ`(0)

+

`−1∑n=0

`−n∑m=1

∑j∈Nm|j|=`−n

(−i)mt∫

0

ds1

s1∫0

ds2 ···sm−1∫0

dsm H(j1)t,s1··· H(jm)

t,sm UV(t,0)χn(0) , (3.8)

where H(n)t,s from (3.7) can be computed as

H(n)t,s =

∑2≤p≤n+2p+n even

∑j∈−1,1p

∫dx(p)A(j)

n,p(t, s;x(p)) a

]j1x1 ··· a

]jpxp . (3.9)

14

The coefficients A(j)n,p are given in terms of the kernels of K

(1)ϕ(t) to K

(4)ϕ(t) and the matrix entries

Ut,s and Vt,s of V(t, s). They are explicitly stated in (5.29).

We postpone the proof of Proposition 3.2 to Section 5.3. The addition of sufficiently

many corrections λ`/2N χ`(t) approximates the excitation vector χ≤N (t) in norm to arbitrary

precision. This is our main result:

Theorem 1. Let a ∈ N0, let Assumption 1 hold for a = a and denote χ≤N (t) = UN,ϕ(t)ΨN (t).

For χ`(t) as in Definition 3.1, there exists a constant C(a) > 0 such that∥∥∥χ≤N (t)−a∑`=0

λ`2Nχ`(t)

∥∥∥F≤N

≤ eC(a)tλa+12

N (3.10)

for all t ∈ R and sufficiently large N . Consequently, the solution ΨN (t) of (1.4) is approx-imated with increasing accuracy by the sequence

ΨN` (t)

`⊂ HN of N -body wave functions

ΨN` (t) :=

N∑k=0

ϕ(t)⊗(N−k) ⊗s χ(k)` (t) , (3.11)

i.e., ∥∥∥ΨN (t)−a∑`=0

λ`2NΨN

` (t)∥∥∥HN≤ eC(a)tλ

a+12

N (3.12)

for sufficiently large N .

We prove Theorem 1 in Section 5.4 via a Gronwall argument.

Remark 3.3.

(a) As explained in the introduction, Theorem 1 is comparable to the results obtained in[37, 36, 12], all of which correspond to expansions of the unitary group governing thedynamics of the excitations. In contrast, we focus on the dynamics of initial data sat-isfying Assumption 1. This simplifies the approximation since fewer terms are requiredat a given order a because the state is expanded simultaneously with the Hamiltonian.

(b) More precisely, the error in (3.12) is of the form

(C1(a+ 1))C2(a+1)2eC3(a+1)2tλa+12

N

for some constants C1, C2, C3 > 0. Hence, (3.12) is not uniform in a, and, in particular,does not imply convergence of the perturbation series for fixed N as a→∞. Since ourexpansion is designed as a large N expansion, we do not expect this to hold true. Weremark that Borel summability was shown for a comparable but combinatorially simplerexpansion in [37].

(c) Theorem 1 holds under more general conditions than Assumption 1. More precisely,it suffices to assume that the initial excitation vector χ≤N (0) admits an asymptoticexpansion of the form (2.32), where the coefficients χ`(0) must satisfy (2.36) for b = 4a+4but need not be given by (2.33) and (2.34).

(d) We expect that our result can be extended to interactions of the form λNNdβv(N βx)

for sufficiently small values of β, comparable to the range β ∈ [0, 1/(4d)) covered in [12].

15

(e) We made no effort to use dispersive estimates and always bound the solution ϕ(t) of(2.1) by its H-norm, which is possible since ‖v‖∞ . 1. As a consequence, our estimateswork in all dimensions d ≥ 1 and apply in the focusing as well as in the defocusing case.

(f) In (3.10), one could equivalently identify χ≤N with χ≤N⊕0 and use the norm on the fullFock space since the contribution of the sectors with more than N particles to ‖χ`(t)‖Fis negligible (Lemma 5.5b).

3.2 Simplified Form of the Perturbation Series

By Assumption 1, Proposition 3.2 yields a formula for the corrections χ`(t) in terms of onlythe initial condition χ0(0).

Corollary 3.4. Let Assumption 1 hold for some a ∈ N0 and let ` ≤ a. Then it holds for χ`(t)as in Definition 3.1 that

χ`(t) =∑

0≤n≤3`n+` even

∑j∈−1,1n

∫dx(n)C

(j)`,n(t;x(n)) a

]j1x1 ··· a

]jnxn χ0(t) , (3.13)

where χ0(t) is the solution of the Bogoliubov equation (1.10) and where the N -independent

coefficients C(j)`,n are defined in (5.53).

We prove Corollary 3.4 in Section 5.5 by iteratively constructing the coefficients C(j)`,n. They

are given in terms of the kernels K(1)ϕ(t) to K

(4)ϕ(t) and the matrix entries Ut,s and Vt,s of the

solution V(t, s) of (3.3). For example, the first order correction to χ0(t) is given as

χ1(t) =

∑j∈−1,1

∫dxC

(j)1,1(t;x) a

]jx +

∑j∈−1,13

∫dx(3)C

(j)1,3(t;x(3)) a

]j1x1 a

]j2x2 a

]j3x3

χ0(t)

(3.14)with

C(j)1,1(t;x) =

∫dy(a

(1)1,1,0(y)ω

(−1,j)t,0 (y;x) + a

(1)1,1,1(y)ω

(1,j)t,0 (y;x)

), (3.15a)

C(j)1,3(t;x(3)) =

3∑n=0

∫dy(3)a

(1)1,3,n(y(3))

n∏ν=1

ω(1,jν)t,0 (yν ;xν)

3∏µ=ν+1

ω(−1,jµ)t,0 (yµ;xµ)

−i

t∫0

ds

∫dy(3)

(K

(3)ϕ(s)(y

(3))ω(1,j1)t,s (y1;x1)ω

(1,j2)t,s (y2;x2)ω

(−1,j3)t,s (y3;x3)

+(K

(3)ϕ(s)

)∗(y(3))ω

(1,j1)t,s (y1;x1)ω

(−1,j2)t,s (y2;x2)ω

(−1,j3)t,s (y3;x3)

)(3.15b)

for a(1)n,m,µ from Assumption 1. Here, we define

ω(−1,−1)t,s (x; y) := U∗t,s(x; y) , ω

(−1,1)t,s (x; y) := V ∗t,s(x; y) , (3.16a)

ω(1,−1)t,s (x; y) := Vt,s(y;x) , ω

(1,1)t,s (x; y) := Ut,s(y;x) . (3.16b)

16

Making use of (2.33), one can equivalently express (3.13) as

χ`(t) = UV(t,0)V0

∑0≤n≤3`n+` even

∑j∈−1,1n

∫dx(n+ν)C

(j)`,n(t;x(n+ν)) a

]j1x1 ··· a

]jnxn a

†xn+1

··· a†xn+ν |Ω〉,

(3.17)

where the coefficients C(j)`,n additionally depend on the functions f1, ..., fν from Assumption 1.

Hence, χ`(t) is a Bogoliubov transformed sum of states with finitely many particles.

3.3 Generalized Wick Rule

In this section, we study the mixed n-point correlation functions⟨a]1x1 · · · a

]nxn

⟩(t)

`,k:=⟨χ`(t), a

]1x1 · · · a

]nxnχk(t)

⟩F. (3.18)

For example, in the simplest case where χ0(0) is quasi-free, Corollary 3.4 and Wick’s ruleyield⟨

a]jx

⟩(t)

0,1

=∑

m∈−1,1

∫dy C

(m)1,1 (t; y)

⟨a]jx a

]my

⟩(t)

0,0+

∑m∈−1,13

∫dy(3)C

(m)1,3 (t; y(3))

×(⟨a]jx a

]m1y1

⟩(t)

0,0

⟨a]m2y2 a

]m3y3

⟩(t)

0,0+⟨a]jx a

]m2y2

⟩(t)

0,0

⟨a]m1y1 a

]m3y3

⟩(t)

0,0+⟨a]jx a

]m3y3

⟩(t)

0,0

⟨a]m1y1 a

]m2y2

⟩(t)

0,0

)

=∑b=2,4

m∈−1,1b

∫dy(b)C

(m2,...,mb)1,b−1 (t; y2, ..., yb)δ(y1 − x)δj,m1

∑σ∈Pb

b/2∏i=1

⟨a]mσ(2i−1)yσ(2i−1)

a]mσ(2i)yσ(2i)

⟩(t)

0,0(3.19)

and ⟨a]j1x1 a

]j2x2

⟩(t)

0,1= 0 . (3.20)

To simplify the notation in (3.19), we integrate/sum also over the fixed variable/index x/j,which is taken into account by the delta distribution/Kronecker delta. This leads to thefollowing generalization of Wick’s theorem, which is proven in Section 5.6.

Corollary 3.5 (Generalized Wick Rule). Let Assumption 1 hold for some a ∈ N0 and letn ∈ N, k, ` ≤ a and j ∈ −1, 1n.

(a) If k + `+ n odd, then ⟨a]j1x1 · · · a

]jnxn

⟩(t)

`,k= 0 . (3.21)

(b) Let ν = 0 in Assumption 1, i.e., χ0(0) is quasi-free. Then for k + `+ n even⟨a]j1x1 · · · a

]jnxn

⟩(t)

`,k

=

n+3(`+k)∑b=neven

∑m∈−1,1b

∑σ∈Pb

b/2∏i=1

∫dy(b)D

(j;m)`,k,n;b(t;x

(n); y(b))

⟨a]mσ(2i−1)yσ(2i−1)

a]mσ(2i)yσ(2i)

⟩(t)

0,0

(3.22)

17

for Pb the set of pairings defined in (1.26) and with

D(j;m)`,k,n;b(t;x

(n); y(b)) :=

min3k,b−n∑q=max0,b−n−3`

q+k even

C(−mb−n−q ,...,−m1)`,b−n−q (t; yb−n−q, ..., y1)

×C(mb−q+1,...,mb)k,q (t; yb−q+1, ..., yb)

×n∏µ=1

δ(yb−n−q+µ − xµ)δmb−n−q+µ, jµ . (3.23)

We stated part (b) only for the case ν = 0. If ν > 0, one obtains a similar but more com-

plicated formula for 〈a]1x1 · · · a]nxn〉

(t)`,k in terms of the two-point correlation functions of UV0 |Ω〉,

where UV0 denotes the Bogoliubov transformation from Assumption 1. Moreover, one mayconsider initial data that are finite superpositions of states with different ν, but in this casethe mixed n-point correlation functions corresponding to k + ` + n odd do not necessarilyvanish.

If χ0(0) is quasi-free, 〈a]1x1 · · · a]nxn〉

(t)`,k is given explicitly in terms of the two-point correlation

functions of χ0(t),

γχ0(t)(x, y) =⟨χ0(t), a†yaxχ0(t)

⟩F, αχ0(t)(x, y) = 〈χ0(t), axayχ0(t)〉F , (3.24)

which can be obtained from the two-point correlation functions of χ0(0). Evaluating theaction of the Bogoliubov transformation UV(t,0) on creation and annihilation operators (seeSection 4), one computes

γχ0(t)(x, y) =⟨χ0(0), U∗V(t,0) a

†y UV(t,0) U∗V(t,0) ax UV(t,0)χ0(0)

⟩=

(V t,0γ

Tχ0(0)V

∗t,0 + Ut,0γχ0(0)U

∗t,0 − V t,0α

∗χ0(0)U

∗t,0 − Ut,0αχ0(0)V

∗t,0

)(x, y)

+(V t,0V

∗t,0

)(x, y) , (3.25a)

αχ0(t)(x, y) =(Ut,0αχ0(0)U

∗t,0 + V t,0α

∗χ0(0)V

∗t,0 − Ut,0γχ0(0)V

∗t,0 − V t,0γ

Tχ0(0)U

∗t,0

)(x, y)

+(Ut,0V

∗t,0

)(x, y) , (3.25b)

where Ut,0 and Vt,0 denote the matrix entries of the solution V(t, 0) of (3.3). Alternatively,one can obtain γχ0(t) and αχ0(t) by solving the system of differential equations

i∂tγχ0(t) =(hϕ(t) +K

(1)ϕ(t)

)γχ0(t) − γχ0(t)

(hϕ(t) +K

(1)ϕ(t)

)+K

(2)ϕ(t)α

∗χ0(t) − αχ0(t)

(K

(2)ϕ(t)

)∗, (3.26a)

i∂tαχ0(t) =(hϕ(t) +K

(1)ϕ(t)

)αχ0(t) + αχ0(t)

(hϕ(t) +K

(1)ϕ(t)

)T+K

(2)ϕ(t) +K

(2)ϕ(t)γ

Tχ0(t) + γχ0(t)K

(2)ϕ(t) , (3.26b)

which is not restricted to quasi-free initial states χ0(0) (see [59, Proposition 4(i)] and [40, Eq.(17b-c)]).

18

3.4 Perturbation Series for Correlation Functions

Theorem 1 and Corollary 3.5a imply an approximation of the n-point correlation functions ofthe excitations, which are defined as follows:

Definition 3.6. Let χ ∈ F and n ∈ N. The n-point correlation functions of χ are defined as⟨χ, a]1x1 · · · a

]nxnχ

⟩F, (3.27)

where a]j ∈ a†, a for j ∈ 1, ..., n. For χ≤N (t) = UN,ϕ(t)ΨN (t), we use the short-hand

notation ⟨a]1x1 · · · a

]nxn

⟩(t)

N:=⟨χ≤N (t), a]1x1 · · · a

]nxnχ

≤N (t)⟩F≤N

. (3.28)

Formally, the expansion of χ≤N (t)⊕ 0 from (3.1) yields

⟨a]1x1 · · · a

]nxn

⟩(t)

N=

⟨ ∞∑`=0

λ`2Nχ`(t), a

]1x1 · · · a

]nxn

∞∑m=0

λm2N χm(t)

⟩F

=a∑`=0

λ`2N

∑m=0

⟨χm(t), a]1x1 · · · a

]nxnχ`−m(t)

⟩F

+O(λa+12

N

). (3.29)

By the generalized Wick rule (Corollary 3.5), all contributions 〈a]x1 ··· a]xn〉

(t)m,`−m with ` + n

odd vanish. This is made rigorous in the following corollary (see Section 5.7 for a proof).

Corollary 3.7. Let Assumption 1 hold for some a ∈ N0 and let n, p ∈ N0 with n+ p ≤ a+ 1,t ∈ R and B ∈ L(Hp,Hn). Let a ≤ a−1

2 if n+p even and a ≤ a2 if n+p odd. Then there exists

some constant C(a, n, p) > 0 such that for n+ p even,∣∣∣∣∣∫

dx(n) dy(p)B(x(n); y(p))

(⟨a†x1 ··· a

†xn ay1 ··· ayp

⟩(t)

N

−a∑`=0

λ`N

2∑m=0

⟨a†x1 ··· a

†xn ay1 ··· ayp

⟩(t)

m,2`−m

)∣∣∣∣∣ ≤ ‖B‖L(Hp,Hn) eC(a,n,p)t λa+1N , (3.30a)

and for n+ p odd,∣∣∣∣∣∫

dx(n) dy(p)B(x(n); y(p))

(⟨a†x1 ··· a

†xn ay1 ··· ayp

⟩(t)

N

−a−1∑`=0

λ`+ 1

2N

2`+1∑m=0

⟨a†x1 ··· a

†xn ay1 ··· ayp

⟩(t)

m,2`+1−m

)∣∣∣∣∣ ≤ ‖B‖L(Hp,Hn) eC(a,n,p)t λa+ 1

2N ,(3.30b)

where for a = 0 the sum∑a−1

`=0 is interpreted as zero.

If χ0(0) is quasi-free (ν = 0 in Assumption 1), the approximation can be simplified furtherby Corollary 3.5b. For example, one obtains for n+ p even∫

dx(n+p)B(x(n+p))⟨a†x1 ··· a

†xn axn+1 ··· axn+p

⟩(t)

N

19

=a∑`=0

λ`N

n+p+6`∑b=n+pb even

∑µ∈−1,1b

∑σ∈Pb

b/2∏i=1

∫dx(n+p+b)D

(j,µ)n,p;`,b(t;x

(n+p+b))

⟨a]µσ(2i−1)xσ(2i−1)

a]µσ(2i)xµσ(2i)

⟩(t)

0,0

+O(λa+1N )eC(n,p,a)t (3.31)

for j = 1n × −1p and where the coefficients D(j,µ)n,p;`,b are given in terms of the kernel of B

and (3.23).

Remark 3.8.

(a) We stated Corollary 3.7 only for normal-ordered correlation functions since any correla-tion function can be normal-ordered using (2.7).

(b) Corollary 3.7 implies for n+ p even the L2-bound∥∥∥∥∥⟨a†x1 ··· a†xn ay1 ··· ayp⟩(t)

N

−a∑`=0

λ`N

2∑m=0

⟨a†x1 ··· a

†xn ay1 ··· ayp

⟩(t)

m,2`−m

∥∥∥∥∥Hn+p

≤ eC(a,n,p)tλa+1N (3.32)

and analogously for n + p odd. This follows directly from choosing B as the Hilbert-Schmidt operator with kernel

B(x(n); y(p)) =⟨a†x1 ··· a

†xn ay1 ··· ayp

⟩(t)

N−

a∑`=0

λ`N

2∑m=0

⟨a†x1 ··· a

†xn ay1 ··· ayp

⟩(t)

m,2`−m

and from the fact that ‖B‖op ≤ ‖B‖HS = ‖B(· ; ·)‖Hn+p .

(c) The 2n-point correlation function 〈a†y1 ··· a†ynax1 ··· axn〉

(t)N can be understood as the inte-

gral kernel γ(n)

χ≤N (t)(x(n); y(n)) of the reduced n-particle density matrix of χ≤N (t). Since

the trace class operators are the dual of the compact operators, Corollary 3.7 impliesthat

Tr

∣∣∣∣∣γ(n)

χ≤N (t)−

a∑`=0

λ`N

2∑m=0

γ(n)m,2`−m(t)

∣∣∣∣∣ ≤ eC(a,n)tλa+1N , (3.33)

where γ(n)m,2`−m(t) is the operator on Hn with kernel 〈a†y1 ··· a

†ynax1 ··· axn〉

(t)m,2`−m.

Making use of Proposition 3.2, the mixed n-point correlation functions 〈a†x1 ··· a†xn ay1 ··· ayp〉

(t)`,k

can be computed explicitly and independently of N , given their initial values and the solu-tions V(t, s) and ϕ(t) of the two-body problem (3.3) and the one-body problem (2.1), re-spectively. Since the actual correlation functions of the N -body problem are determined by

〈a†x1 ··· a†xn ay1 ··· ayp〉

(t)`,k to any order in N−1/2, this implies a drastic reduction of complexity.

Alternatively, the functions 〈a†x1 ··· a†xn ay1 ··· ayp〉

(t)`,k can be obtained from solving a system

of PDEs. A straightforward computation yields

i∂t∑m=0

⟨a]1x1 ··· a

]nxn

⟩(t)

m,`−m=

∑m=0

⟨[a]1x1 ··· a

]nxn ,H

(0)ϕ(t)

]⟩(t)

m,`−m(3.34a)

20

+

`−1∑k=0

k∑m=0

⟨[a]1x1 ··· a

]nxn ,H

(`−k)ϕ(t)

]⟩(t)

m,k−m. (3.34b)

The commutator on the right-hand side of (3.34a) contains again n creation and annihilationoperators and is evaluated in the same m, ` − m inner product as the left-hand side. Thecommutators in (3.34b) contain at most n+ `− k creation and annihilation operators and aredetermined by a PDE analogous to (3.34). Since they are evaluated only in inner productswith χ0(t), . . . ,χ`−1(t) and not χ`(t), iterating this procedure yields a closed system of coupledPDEs. For the simplest example with ` = 1 and n = 1, see Section 3.5.

As a consequence of Corollary 3.7, the expectation value of any bounded k-body operatorA(k) with respect to ΨN (t) can be computed to arbitrary precision. To see this, recall that⟨

ΨN (t), A(k)ΨN (t)⟩HN

=(Nk

)−1⟨χ≤N (t),UN,ϕ(t) dΓ(A(k))U∗N,ϕ(t)χ

≤N (t)⟩F≤N

, (3.35)

and the latter is as a sum of expressions covered by Corollary 3.7. Equivalently, this yields an

expansion of the reduced k-body density matrices γ(k)ΨN

(t) of ΨN (t), which is discussed in thenext section for the simplest case k = 1.

3.5 One-Particle Reduced Density Matrix

Corollary 3.7 implies an asymptotic expansion of the k-particle reduced density matrices

γ(k)

ΨN(t) in terms of ϕ(t) and the mixed n-point correlation functions 〈a] ···a]〉(t)`,k. We restrict

to the case k = 1. First, one observes that

γ(1)

ΨN(t) = pϕ(t)γ

(1)

ΨN(t)pϕ(t) + qϕ(t)γ

(1)

ΨN(t)qϕ(t) +

(pϕ(t)γ

(1)

ΨN(t)qϕ(t) + h.c.

)=

1

Npϕ(t)

⟨ΨN (t), a†(ϕ(t))a(ϕ(t))ΨN (t)

⟩HN

+1

Nγχ≤N (t)

+

(1

N

∑`≥1

|ϕ(t)〉〈ϕ`(t)|⟨

ΨN (t), a†(ϕ`(t))a(ϕ(t))ΨN (t)⟩HN

+ h.c.

)(3.36)

for any orthonormal basis ϕ`(t)`≥0 of H with ϕ0(t) = ϕ(t), and where γχ≤N (t) denotes the

one-body reduced density matrix of χ≤N (t) with kernel γχ≤N (t)(x; y) = 〈χ≤N (t), a†yaxχ≤N (t)〉.

Hence, the substitution rules (2.17) yield

γ(1)

ΨN(t) = pϕ(t) +

1√N

(|ϕ(t)〉〈βχ≤N (t)|+ |βχ≤N (t)〉〈ϕ(t)|

)+

1

N

(γχ≤N (t) − pϕ(t)

⟨χ≤N (t),Nχ≤N (t)

⟩F≤N

), (3.37)

where

βχ≤N (t)(x) :=

⟨χ≤N (t),

√1− N

Naxχ

≤N (t)

⟩F≤N

. (3.38)

Expanding the N -dependent expressions in (3.37) in powers of λ1/2N and applying Corollary 3.7

leads to the following result, whose proof is postponed to Section 5.8.

Theorem 2. Let Assumption 1 hold for some a ∈ N and let a ≤ a−12 and t ∈ R. Then

Tr∣∣∣γ(1)

ΨN(t)−

a∑`=0

λ`Nγ(1)` (t)

∣∣∣ ≤ eC(a)tλa+1N (3.39)

21

for some constant C(a) > 0, where

γ(1)0 (t;x; y) := ϕ(t, x)ϕ(t, y) , (3.40a)

γ(1)` (t;x; y) :=

∑m=1

[`−m∑k=0

2m−1∑n=0

c`−m,k

(ϕ(t, x)

⟨a†y(N − 1)k

⟩(t)

n,2m−n−1

+⟨

(N − 1)kax

⟩(t)

n,2m−n−1ϕ(t, y)

)+

2m−2∑n=0

c`−m

(⟨a†yax

⟩(t)

n,2m−n−2− ϕ(t, x)ϕ(t, y) 〈N〉(t)n,2m−n−2

)](3.40b)

for ` ≥ 1, where c` = (−1)`c(3/2)` and c`,k := c`−kc

(0)k with c

(n)` as in (2.29).

Theorem 2 implies for any bounded operator A : H→ H that∣∣∣∣∣TrHAγ(1)

ΨN(t)−

a∑`=0

λ`NTrHAγ(1)` (t)

∣∣∣∣∣ ≤ ‖A‖op eC(a)tλa+1N . (3.41)

For a = 0, we recover the well-known statement that γ(1)

ΨN(t) ≈ pϕ(t) up to an error of order

N−1 [24, 19, 20, 57]. In [72, 64], an error estimate of order N−1/2 was proven by estimatingβχ≤N (t) uniformly in N without making use of the Bogoliubov approximation. In this case, theBogoliubov approximation improves the convergence rate but does not give itself a correctionto the one-body reduced density matrix.

The next order correction to the leading order γ(1)0 (t) = pϕ(t) is

γ(1)1 (t;x; y) = ϕ(t, x)

(〈a†y〉

(t)0,1 + 〈a†y〉

(t)1,0

)+(〈ax〉(t)1,0 + 〈ax〉(t)0,1

)ϕ(t, y) + γχ0(t)(x; y)

−(TrHγχ0(t)

)ϕ(t, x)ϕ(t, y) . (3.42)

The two-point correlation function γχ0(t) is given by the solution of (3.26), or, if χ0(0) is quasi-

free, directly by (3.25a). The mixed correlation functions 〈a]〉(t)1,0 and 〈a]〉(t)0,1 can be obtainedfrom this by Corollary 3.5. Alternatively,

β0,1(t, x) := 〈ax〉(t)0,1 + 〈ax〉(t)1,0 (3.43)

is determined by a partial differential equation. As a consequence of (3.2),

i∂tβ0,1(t, x) =⟨[ax,H

(0)ϕ(t)

]⟩(t)

1,0+⟨[ax,H

(0)ϕ(t)

]⟩(t)

0,1+⟨[ax,H

(1)ϕ(t)

]⟩(t)

0,0

= hϕ(t)x β0,1(t, x) +

∫dyK

(1)ϕ(t)(x; y)β0,1(t, y) +

∫dy K

(2)ϕ(t)(x; y)β0,1(y)

+

∫dy(2)

(K

(3)ϕ(t)(x, y1; y2) +K

(3)ϕ(t)(y1, x; y2)

)γχ0(t)(y2; y1)

+

∫dy(2)

(K

(3)ϕ(t)

)∗(x; y1, y2)αχ0(t)(y1, y2) , (3.44)

which is (1.16) from the introduction.

22

4 Bogoliubov Transformations and Quasi-free States

In this section, we briefly recall the concepts of Bogoliubov transformations, Bogoliubov maps,and quasi-free states, and prove some of their properties. Our main references are [76, 52, 59].Let us consider

F = f ⊕ Jg = f ⊕ g =

(fg

)∈ H⊕ H , (4.1)

where J : H → H, (Jf)(x) = f(x), denotes the complex conjugation map. The generalizedcreation and annihilation operators A(F ) and A†(F ) are defined as

A(F ) = a(f) + a†(g) , A†(F ) = A(JF ) = a†(f) + a(g) (4.2)

for J =(

0 JJ 0

). If an operator V on H ⊕ H is such that the map F 7→ A(VF ) has the same

properties as the map F 7→ A(F ), i.e., if

A†(VF ) = A(VJF ) , [A(VF1), A†(VF2)] = [A(F1), A†(F2)] , (4.3)

the operator V is called a (bosonic) Bogoliubov map. This requirement is equivalent to thefollowing definition:

Definition 4.1. A bounded operator

V : H⊕ H→ H⊕ H (4.4)

is called a Bogoliubov map if it satisfies

V∗SV = S = VSV∗ , JVJ = V , (4.5)

where S :=(

1 00 −1

). Equivalently, V is of the block form

V :=

(U V

V U

), U, V : H→ H , (4.6)

where U and V satisfy the relations

U∗U = 1 + V ∗V , UU∗ = 1 + V V∗, V ∗U = U∗V , UV ∗ = V U

∗. (4.7)

We denote the set of Bogoliubov maps on H⊕ H as

V(H) := V ∈ L (H⊕ H) | V is a Bogoliubov map . (4.8)

Bogoliubov maps can be implemented on Fock space in the following sense:

Lemma 4.2. Let V ∈ V(H). Then there exists a unitary transformation UV : F(H)→ F(H)such that

UVA(F )U∗V = A(VF ) (4.9)

for all F ∈ H⊕ H if and only if

‖V ‖2HS(H) := Tr(V ∗V ) <∞ (4.10)

(Shale–Stinespring condition). In this case, V is called (unitarily) implementable.

23

A proof of the lemma is given, for instance, in [76, Theorem 9.5]. In the following, werefer to the unitary implementation UV : F(H) → F(H) of a Bogoliubov map V ∈ V(H)as Bogoliubov transformation. Some relevant properties of Bogoliubov transformations aresummarized in the following lemma, which is easily verified by direct computations using(4.7).

Lemma 4.3. The Bogoliubov maps V(H) form a subgroup of the group of isomorphisms onH⊕H. In particular, the adjoint and inverse of V ∈ V(H) with block form (4.6) are given as

V∗ =

(U∗ V ∗

V∗

U∗

), V−1 = SV∗S =

(U∗ −V ∗−V ∗ U

∗

). (4.11)

If V is a Hilbert–Schmidt operator, the set of all Bogoliubov transformations,

UV : F(H)→ F(H) | V ∈ V(H) , (4.12)

forms a subgroup of the group of unitary maps on F(H). Moreover, the map V 7→ UV is agroup homomorphism, which, in particular, implies that

UV−1 = (UV)−1 = U∗V . (4.13)

We write the operators U and V as integral operators with (Schwartz) kernels U(x; y) andV (x; y), i.e., for any f ∈ H,

(Uf)(x) :=

∫U(x; y)f(y) dy , (V f)(x) :=

∫V (x; y)f(y) dy , (4.14)

and the operators V and U are to be understood as integral operators with kernels V (x; y)and U(x; y). The transformation rule (4.9) corresponds to

UV ax U∗V =

∫dy U(y;x) ay +

∫dy V (y;x) a†y , (4.15a)

UV a†x U∗V =

∫dy V (y;x) ay +

∫dy U(y;x) a†y. (4.15b)

The inverse relation to (4.9) is given by

U∗VA(F )UV = A(V−1F ) , (4.16)

which, by (4.11), leads to the relations

U∗V ax UV =

∫dy U(x; y)ay −

∫dy V (x; y)a†y , (4.17a)

U∗V a†x UV = −∫

dyV (x; y)ay +

∫dy U(x; y)a†y . (4.17b)

The Bogoliubov transformed number operator can be bounded as follows:

Lemma 4.4. Let b ∈ N, let V ∈ V(H) be unitarily implementable, and denote by UV thecorresponding Bogoliubov transformation on F . Then it holds in the sense of operators on Fthat

UV(N + 1)b U∗V ≤ CbV bb(N + 1)b (4.18)

withCV := 2‖V ‖2HS + ‖U‖2op + 1 (4.19)

for V =

(U V

V U

).

24

Proof. Let ` ∈ R. As a consequence of (4.15), it follows that

(N + 1)` UV(N + 1)U∗V

=

∫dy dz(U V

∗)(y; z)ayaz(N − 1)` +

∫dy dz(UV ∗)(y; z)a†ya

†z(N + 3)`

+

(∫dy dz(V V

∗+ UU∗)(y; z)a†yaz + ‖V ‖2HS(H) + 1

)(N + 1)` . (4.20)

By Lemma 5.1 and (4.7), this yields for ` ∈ R+0 , b ∈ N and φ ∈ F

‖(N + 1)` UV(N + 1)b U∗Vφ‖F= ‖(N + 1)` UV(N + 1)U∗V UV(N + 1)b−1 U∗Vφ‖F

≤ CV

∥∥∥(N + 2)(N + 3)` UV(N + 1)U∗V UV(N + 1)b−2 U∗Vφ∥∥∥F

≤ CbV‖(N + 2)(N + 4)···(N + 2b)(N + 2b+ 1)`φ‖F . (4.21)

The choice ` = 0 proves the lemma for b even. For b odd, note first that

‖(N + 1)12 U∗Vφ‖2F

=

(⟨φ,

∫dy dz(U V

∗)(y; z)ayazφ

⟩F

+ h.c.

)+

∫dz

⟨∫dy(V V ∗ + U U

∗)(y; z)ayφ, azφ

⟩F

+

(‖V ‖2HS + 1

)‖φ‖2F

≤ CV‖(N + 1)12φ‖2F , (4.22)

where we used that⟨φ,

∫dy dzf(y; z)ayazφ

⟩F≤ ‖f‖H2

∑k≥0

‖√k + 1φ(k)‖Hk‖

√k + 2φ(k+2)‖Hk+2

≤ ‖f‖H2‖(N + 1)12φ‖2F , (4.23)

and that ‖∫

dy(V V ∗)(y; z)ayφ‖F ≤ ‖V V ∗‖op‖azφ‖F , etc., and∫

dz‖azφ‖2F = ‖N12φ‖2F .

Thus, we find

‖(N + 1)2b+1

2 U∗Vφ‖F = ‖(N + 1)12 U∗V UV(N + 1)b U∗Vφ‖F

≤ Cb+ 1

2V (2b+ 1)b+

12 ‖(N + 1)b+

12φ‖F (4.24)

by (4.22) and (4.21) with ` = 12 .

A concept that is closely connected to Bogoliubov transformations is the notion of quasi-free states.

Definition 4.5. A state φ ∈ F(H) with ‖φ‖F(H) = 1 is called a quasi-free (pure) state if itcan be expressed as a Bogoliubov transformation of the vacuum, i.e., if there exists a V ∈ V(H)such that

φ = UV |Ω〉 . (4.25)

The set of all quasi-free states on F(H) is denoted by

Q(H) := φ ∈ F(H) : ∃V ∈ V(H) s.t. φ = UV |Ω〉 ⊂ F(H) . (4.26)

25

Our definition of quasi-free states is sometimes referred to as even quasi-free [23]. Wedefine the set of quasi-free excitation vectors at time t ∈ R as

Q⊥ϕ(t) := Q(H) ∩ F⊥ϕ(t) =φ ∈ F⊥ϕ(t) : ∃V ∈ V(H) s.t. φ = UV |Ω〉

. (4.27)

Clearly, any Bogoliubov transformation leaves the set of quasi-free states invariant, i.e.,

UVQ(H) = Q(H) (4.28)

for all V ∈ V(H).

Definition 4.6. Define the generalized one-particle density matrix Γφ : H ⊕ H → H ⊕ H ofφ ∈ F(H) as

Γφ :=

(γφ αφα∗φ 1 + γTφ

), (4.29)

where the one-body density matrix γφ : H→ H and the pairing density αφ : H→ H are definedas the operators with kernels

γφ(x, y) =⟨φ, a†y axφ

⟩F, αφ(x, y) = 〈φ, ax ay φ〉F . (4.30)

Quasi-free states are uniquely determined by their generalized one-particle density matrix,or alternatively by Wick’s rule.

Lemma 4.7. Let φ ∈ F(H) with ‖φ‖F = 1. Then the following are equivalent:

(a)φ ∈ Q(H) . (4.31)

(b)〈φ,Nφ〉F(H) <∞ (4.32)

and, for all a] ∈ a†, a, n ≥ 1 and f1, ..., f2n ∈ H,⟨φ, a](f1)···a](f2n−1)φ

⟩F(H)

= 0 , (4.33a)⟨φ, a](f1)···a](f2n)φ

⟩F(H)

=∑σ∈P2n

n∏j=1

⟨φ, a](fσ(2j−1))a

](fσ(2j))φ⟩F(H)

,(4.33b)

where P2n denotes the set of pairings as in (1.26). The property (4.33) is known asWick’s rule.

(c)γφαφ = αφγ

Tφ , αφα

∗φ = γφ(1 + γφ) . (4.34)

(d)ΓφSΓφ = −Γφ . (4.35)

The equivalence of (a ⇔ b) is proven in [58, Theorem 1.6], (a ⇔ d) in [58, Theorem 1.6]or [76, Theorem 10.4], and (c⇔ d) is a simple computation.

Let us now make the connection to the time-dependent setting. The crucial observation

is that the time evolution operator U(0)ϕ (t, s) of the Bogoliubov equation (1.10) is a time-

dependent Bogoliubov transformation.

26

Lemma 4.8. Let s, t ∈ R, φ(s) ∈ F(H), ϕ(s) ∈ H1(Rd) with ‖ϕ(s)‖H = 1, and denote byϕ(t) the unique solution of (2.1) with initial datum ϕ(s).

(a) The Bogoliubov equation (1.10) with initial datum φ(0) in the quadratic form domainQ(dΓ(1−∆)) has a unique solution φ ∈ C0([0,∞),F(H))∩L∞loc([0,∞), Q(dΓ(1−∆))).

We denote by U(0)ϕ (t, s) the corresponding unitary time evolution on F(H), i.e.,

φ(t) = U (0)ϕ (t, s)φ(s) . (4.36)

(b) If φ(s) is quasi-free, then φ(t) is quasi-free, i.e.,

U (0)ϕ (t, s)Q(H) ⊆ Q(H) . (4.37)

(c) Let φ(s) ∈ F⊥ϕ(s) such that 〈φ(s),Nφ(s)〉 <∞. Then φ(t) ∈ F⊥ϕ(t), which, in partic-ular, implies that

U (0)ϕ (t, s)Q⊥ϕ(s) ⊆ Q⊥ϕ(t) . (4.38)

(d) U(0)ϕ (t, s) is a Bogoliubov transformation on F(H), i.e., there exists a two-parameter

group of Bogoliubov maps V(t, s) ∈ V(H) such that

U (0)ϕ (t, s) = UV(t,s) . (4.39)

Moreover, V(t, s) is the weak1 solution of the differential equationi∂tV(t, s) = A(t)V(t, s)

V(s, s) = 1 ,(4.40)

where

A(t) =

hϕ(t) +K(1)ϕ(t) −K(2)

ϕ(t)

K(2)ϕ(t) −

(hϕ(t) +K

(1)ϕ(t)

) . (4.41)

Proof. Parts (a) and (c) are shown in [52, Theorem 7] (see also [34, 35, 59, 4]), and assertion(b) is proven in [59, Proposition 4, Step 5]. Part (d) was shown, in a slightly different setting,in [7, Theorem 2.2], and a similar result was obtained in [4, Lemma 3.10]. We give a simpleproof here. Without loss of generality, let us assume that s = 0. We prove Lemma 4.8d inthree steps.

Claim 1. Let φ0 ∈ Q(H). Then there exists a unitarily implementable Vt ∈ V(H) such that

φ(t) := U(0)ϕ (t, 0)φ0 = UVtφ0.

Proof. Since φ0 is quasi-free, there exist Bogoliubov transformations UV0 and UVt such thatφ0 = UV0 |Ω〉 and φ(t) = UVt |Ω〉 by Lemma 4.8b. This proves the claim with UVt := UVtV−1

0.

Claim 2. The Bogoliubov map Vt is the unique weak solution of (4.40) in V(H).

1With “weak” solution we mean here that V(t, s) is strongly continuous and i∂t⟨F,V(t, s)G

⟩=⟨

F,A(t)V(t, s)G⟩

for all G ∈ L2(Rd)⊕ L2(Rd) and F ∈ H2(Rd)⊕H2(Rd).

27

Proof. By [59, Proposition 4, Step 3], the pair (γφ(t), αφ(t)) of density matrices is the uniqueweak solution of the set of equations (3.26) with initial condition (γφ0

, αφ0). Equivalently, by

Definition 4.6, Γφ(t) is the unique weak solution of the equation

i∂tΓ(t) = A(t)∗Γ(t)− Γ(t)A(t) , Γ(0) = Γφ0, (4.42)

for A(t) as in (4.41). As⟨F1,Γφ(t)F2

⟩H⊕H =

⟨φ(t), A†(F2)A(F1)φ(t)

⟩F(H)

for F1, F2 ∈ H⊕H,

one easily verifies thatV∗t Γφ(t)Vt = Γφ0

, (4.43)

which implies that i∂t(V∗t Γφ(t)Vt

)= 0 and consequently

V∗t Γφ(t) (i∂tVt −A(t)Vt)− h.c. = 0 . (4.44)

This proves the claim because there is a one-to-one correspondence between Vt and Γφ(t). Notethat the weak well-posedness as in Footnote 1 follows from [70, Theorem X.69] by a Dysonexpansion in the interaction picture.

Claim 3. In the sense of operators on F(H), it holds that UVt = U(0)ϕ (t, 0).

Proof. Let φ0 ∈ Q(H). We prove by induction over n ∈ N that

i∂t UVtA(F1)···A(Fn)φ0 = H(0)ϕ(t) UVtA(F1)···A(Fn)φ0 (4.45)

for any n ∈ N and F1, ..., Fn ∈ H⊕ H. By Lemma 4.8a, this implies that

U (0)ϕ (t, 0)A(F1) · · ·A(Fn)φ0 = UVtA(F1) · · ·A(Fn)φ0 , (4.46)

and then the claim follows by density. By Claim 1, i∂t UVtφ0 = H(0)ϕ(t) UVtφ0, and one easily

verifies that [A(F ),H(0)

ϕ(t)

]= A(A(t)F ) , i∂t UVtA(F )U∗Vt =

[H(0)ϕ(t), A(VtF )

], (4.47)

which yields (4.45) for n = 1. Given (4.45) for n− 1,

i∂t UVtA(F1)···A(Fn)φ0 =[H(0)ϕ(t), A(VtF1)

]UVtA(F2)···A(Fn)φ0

+A(VtF1)H(0)ϕ(t) UVtA(F2)···A(Fn)φ0

= H(0)ϕ(t) UVtA(F1)···A(Fn)φ0 , (4.48)

which completes the induction.

Finally, we provide estimates on the time dependence of the solution V(t, s) of (4.40):

Lemma 4.9. Let V(t, 0) denote the (weak) solution of (4.40) with initial condition V(0, 0) = 1,i.e.,

V(t, 0) =

(Ut,0 V t,0

Vt,0 U t,0

), V(0, 0) =

(1H 00 1H

). (4.49)

Then there exists a constant C > 0 such that

‖Ut,0‖op . eCt , ‖Vt,0‖HS . eCt . (4.50)

28

Proof. Define u(t, s) : H → H as the unitary two-parameter group (in the weak sense as

discussed in Footnote 1) generated by the self-adjoint operator hϕ(t) +K(1)ϕ(t), i.e.,

i∂su(t, s) = −u(t, s)(hϕ(s) +K

(1)ϕ(s)

). (4.51)

Then it follows from (4.40) that

i∂t (u(0, t)Ut,0) = −u(0, t)K(2)ϕ(t)Vt,0 , (4.52)

for K(2)ϕ(t) the Hilbert–Schmidt operator with kernel K

(2)ϕ(t)(x1, x2), hence

‖Ut,0‖op = ‖u(0, t)Ut,0‖op =∥∥∥U0,0 + i

t∫0

u(0, s)K(2)ϕ(s)Vs,0 ds

∥∥∥op≤ 1 +

t∫0

‖K(2)ϕ(s)Vs,0‖op ds .

(4.53)Since

‖K(2)ϕ(s)Vs,0‖

2op ≤ ‖K(2)

ϕ(s)Vs,0‖2HS

= Tr(U s,0U

∗s,0

(K

(2)ϕ(s)

)∗K

(2)ϕ(s)

)− Tr

((K

(2)ϕ(s)

)∗K

(2)ϕ(s)

)≤ ‖Us,0‖2op‖K

(2)ϕ(s)(·, ·)‖

2H2 . ‖Us,0‖2op (4.54)

by (4.7) and Lemma 5.2, the first statement follows with Gronwall’s lemma. For the secondpart of the lemma, one defines u(t, s) : H → H as the unitary group generated by −

(hϕ(t) +

K(1)ϕ(t)

), and analogously to above obtains

u(0, t)Vt,0 = −i

t∫0

u(0, s)K(2)ϕ(s)Us,0 ds , (4.55)

hence the previous result implies that

‖Vt,0‖HS ≤t∫

0

‖K(2)ϕ(s)‖HS‖Us,0‖op ds . eCt . (4.56)

5 Proofs

In the following, we will abbreviate ‖·‖ ≡ ‖·‖F and 〈· , ·〉 ≡ 〈· , ·〉F .

5.1 Auxiliary Estimates

Recall that for any F : R+0 → R and j ∈ −1, 1, it holds that

a]jx F (N ) = F (N − j)a]jx , F (N )a

]jx = a

]jx F (N + j) . (5.1)

Second-quantized operators can be bounded in terms of N :

29

Lemma 5.1. Let n, p ≥ 0, f : Hp → Hn be a bounded operator with (Schwartz) kernelf(x(n); y(p)), and φ ∈ F . Then∥∥∥∥∫ dx(n) dy(p)f(x(n); y(p))a†x1 ··· a

†xn ay1 ··· aypφ

∥∥∥∥ ≤ ‖f‖Hp→Hn‖(N + n)n+p2 φ‖

≤ nn+p2 ‖f‖Hp→Hn‖(N + 1)

n+p2 φ‖ . (5.2)

Proof. Abbreviating z(n)j = (zj1 , ..., zjn), we compute for k ≥ n∥∥∥∥∥

[∫dx(n) dy(p)f(x(n); y(p))a†x1 ···a

†xnay1 ···aypφ

](k)∥∥∥∥∥Hk

≤√

(k − n+ p)!

k!

∑j1 6=... 6=jn∈1,...,k

(∫dz(k)

∣∣∣∣∫ dy(p)f(z(n)j ; y(p))φ(k−n+p)

(z(k), y(p) \ z(n)

j

)∣∣∣∣2) 1

2

≤ (k + p)p+n2 ‖f‖Hp→Hn‖φ(k−n+p)‖Hk−n+p , (5.3)

and summing over k ≥ 0 completes the proof.

We now collect some useful properties of Hϕ(t), H(n)ϕ(t) and K(j)

ϕ(t).

Lemma 5.2. (a) For K(1)ϕ(t) to K

(4)ϕ(t) as defined in (2.23), it holds that

‖K(1)ϕ(t)‖H→H ≤ ‖v‖∞ , ‖K(2)

ϕ(t)‖H2 ≤ ‖v‖∞ ,

‖K(3)ϕ(t)‖H→H2 . ‖v‖∞ , ‖K(4)

ϕ(t)‖H2→H2 . ‖v‖∞ .(5.4)

(b) Let b ≥ 0. For K(j)ϕ(t) as in (2.22) and any φ ∈ F , it holds that

‖(N + 1)bK(1)ϕ(t)φ‖ . ‖(N + 1)b+1φ‖ (5.5a)

‖(N + 1)bK(j)ϕ(t)φ‖ ≤ C(b)‖(N + 1)b+

j2φ‖ for j = 2, 3, 4 , (5.5b)

‖(N + 1)bK(j)ϕ(t)φ‖ ≤ C(b)‖(N + 1)b+

j2φ‖ for j = 2, 3 (5.5c)

for some constant C(b) > 0.

(c) Let n ∈ N and b ≥ 0. For H(n)ϕ(t) as in (2.28) and any φ ∈ F ,

‖(N + 1)bH(n)ϕ(t)φ‖ . C(n, b)‖(N + 1)b+

n+22 φ‖ (5.6)

for some constant C(n, b) > 0.

Proof. Since ‖ϕ(t)‖H = 1 and ‖Wϕ(t)‖∞ . ‖v‖∞, part (a) follows directly from the definition(2.23), and parts (b) and (c) are implied by part (a) and Lemma 5.1.

Finally, note that Hϕ(t) leaves the subspace F≤N invariant because the first and last linein (2.21) preserve the particle number and the remaining terms are zero on the sectors with

more than N particles. Moreover, all terms in Hϕ(t) except for K(0)ϕ(t) preserve the subspace

F⊥ϕ(t), hence Hϕ(t) leads out of the subspace F⊥ϕ(t) but the time evolution generated by H≤Nϕ(t)

maps F≤N⊥ϕ(s) into F≤N⊥ϕ(t). We summarize these observations in the following lemma:

30

Lemma 5.3. For Hϕ(t) as in (2.21), K(j)ϕ(t) as in (2.22) and j 6= 0,

Hϕ(t)F≤N ⊆ F≤N , (5.7)

K(j)ϕ(t)F⊥ϕ(t) , K

(j)ϕ(t)F⊥ϕ(t) ⊆ F⊥ϕ(t) , (5.8)(

H≤Nϕ(t) −K(0)ϕ(t)

)F≤N⊥ϕ(t) ⊆ F≤N⊥ϕ(t) . (5.9)

5.2 Proof of Lemma 2.2

Combining (2.35), Lemma 5.1 and (2.33), we find for part (a) that

‖(N + 1)b2χ`(0)‖) . C(`, b)‖(N + 1)

3`+b2 UV0a†(f1)···a†(fν)|Ω〉‖ . C(`, b) (5.10)

by Lemma 4.4. Part (b) follows from part (a) and (2.32) by a simple triangle argument since∥∥∥(N + 1)b2χ≤N (0)

∥∥∥F≤N

≤

∥∥∥∥∥(N + 1)b2

(χ≤N (0)−

a∑`=0

λ`2Nχ`(0)

)∥∥∥∥∥F≤N

+a∑`=0

λ`2N

∥∥∥(N + 1)b2χ`(0)

∥∥∥F≤N

. C(b)(

(N + 1)b2N−

a+12 + 1

)(5.11)

as N ≤ N in the sense of operators on F≤N .

5.3 Proof of Proposition 3.2

First, we expand the N -dependent square roots in (2.21) in a Taylor series and estimate theremainders.

Lemma 5.4. Let a ∈ N0 and define c(n)` and d`,j as in (2.29) and (2.30). Then

|c`| ≤1

2`, |c(j)

` | ≤ 2j+`−1 , |d`,j | ≤ 2`(j + 1) . (5.12)

(a) Define the operator R(3)a on F via the identity√[N −N

]+

N − 1=

a∑`=0

c` λ`+ 1

2N (N − 1)` + λ

a+ 32

N R(3)a . (5.13)

Then [R(3)a ,N ] = 0 and it holds for any φ ∈ F that

‖R(3)a φ‖ ≤ 2a+1‖(N + 1)a+1φ‖ . (5.14)

(b) Define the operator R(2)a on F through√[

(N −N )(N −N − 1)]+

N − 1=

a∑`=0

λ`N∑j=0

d`,j(N − 1)j + λa+1N R(2)

a . (5.15)

Then [R(2)a ,N ] = 0 and it holds for any φ ∈ F that

‖R(2)a φ‖ ≤ (a+ 1)24a+1‖(N + 1)a+1φ‖ . (5.16)

31

To obtain (5.14) and (5.16), note that the sums in (5.13) and (5.15) are the Taylor expan-

sions of the corresponding square roots, which converge on F≤N⊥ (for R(3)a ) and F≤N−1

⊥ (for

R(2)a ), respectively. On F≥N and F>N−1

⊥ , respectively, one observes that N ≥ λ−1N . The full

proof is given in Appendix C.

Next, we make use of this result to prove the expansion of Hϕ(t) from Lemma 2.1.

Proof of Lemma 2.1. By (2.21), the Taylor expansion of Lemma 5.4 leads to (2.27) with

H(n)ϕ(t) as in (2.28). The remainders in (2.27) are given as

R(0) :=

K(3)ϕ(t)

√[N −N ]+N − 1

+ h.c.

+λ

12N

(K(4)ϕ(t) − (N − 1)K(1)

ϕ(t) +(K(2)ϕ(t)R

(2)0 + h.c.

)), (5.17a)

R(1) := K(4)ϕ(t) − (N − 1)K(1)

ϕ(t) +(K(2)ϕ(t)R

(2)0 + h.c.

)+ λ

12N

(K(3)ϕ(t)R

(3)0 + h.c.

), (5.17b)

R(2n) := K(3)ϕ(t)R

(3)n−1 + λ

12NK

(2)ϕ(t)R

(2)n + h.c., (5.17c)

R(2n+1) := K(2)ϕ(t)R

(2)n + λ

12NK

(3)ϕ(t)R

(3)n + h.c. (5.17d)

for n ≥ 1, with R(2)j and R

(3)j from Lemma 5.4. Hence, Lemmas 5.2b and 5.4 imply

‖R(2n)φ‖ ≤ C(n)

(‖(N + 1)n+ 3

2φ‖+ λ12N‖(N + 1)n+2φ‖

), (5.18)

‖R(2n+1)φ‖ ≤ C(n)

(‖(N + 1)n+2φ‖+ λ

12N‖(N + 1)n+ 5

2φ‖)

(5.19)

for any n ∈ N0, where we used that

√[N−N ]+N−1 . 1 in the sense of operators on F .

Proof of Proposition 3.2. First, we construct the explicit form (3.8) of χ`(t) (Step 1).Second, we conclude that χ`(t) ∈ F⊥ϕ(t) (Step 2), and finally we derive the expression (3.9)

for the operators H(n)t,s (Step 3).

Step 1. Iterating (3.6) and using that UV(t,s1)UV(s1,s2) = UV(t,s2), we find that

χ`(t) = UV(t,0)χ`(0) + (−i)∑j1=1

t∫0

ds1 H(j1)t,s1UV(t,s1)χ`−j1(s1)

= UV(t,0)χ`(0) + (−i)∑j1=1

t∫0

ds1 H(j1)t,s1UV(t,0)χ`−j1(0)

+(−i)2∑j1=1

`−j1∑j2=1

t∫0

ds1

s1∫0

ds2 H(j1)t,s1

H(j2)t,s2UV(t,s2)χ`−j1−j2(s2) , (5.20)

and successive iterations yield

χ`(t)− UV(t,0)χ`(0) =∑m=1

∑n=m

∑j∈Nm|j|=n

I(j)t χ`−n(0) =

`−1∑n=0

`−n∑m=1

∑j∈Nm|j|=`−n

I(j)t χn(0) , (5.21)

32

where we abbreviated

I(j)t := (−i)mt∫

0

ds1 ···sm−1∫0

dsmH(j1)t,s1··· H(jm)

t,sm UV(t,0) . (5.22)

Step 2. Combining Lemmas 4.4 and 5.2c, we find that

‖(N + 1)bH(j)t,sφ‖ = ‖(N + 1)b UV(t,s)H

(j)ϕ(s) U

∗V(t,s)φ‖

≤ C(b, j)C2b+1+ j

2

V(t,s) ‖(N + 1)b+1+ j2φ‖

. eC(b,j)t‖(N + 1)b+1+ j2φ‖ , (5.23)

where we used in the last step that

CV(t,s) = 2‖Vt,s‖2HS + ‖Ut,s‖2op + 1 . eCt (5.24)

by Lemma 4.9. Consequently, for any j ∈ Nm with |j| = `− n,

‖H(j1)t,s1··· H(jm)

t,sm UV(t,0)χn(0)‖ . eC(`)t‖(N + 1)`−n2

+mχn(0)‖ (5.25)

and we conclude that

‖χ`(t)‖ .`−1∑n=0

`−n∑m=1

∑j∈Nm|j|=`−n

tmeC(`)t‖(N + 1)`−n2

+mχn(0)‖ <∞ (5.26)

for χn(0) ∈ D(N32

(`−n)). Finally, χ`(t) ∈ F⊥ϕ(t) because χn(0) ∈ F⊥ϕ(0) and since UV(s1,s2)

maps F⊥ϕ(s1) to F⊥ϕ(s2) by Lemma 4.8.

Step 3. Recall that by (4.15), UV(t,s) transforms creation and annihilation operators as

UV(t,s)a]`x U∗V(t,s) =

∑j∈−1,1

∫dy ω

(`,j)t,s (x; y)a

]jy , ` ∈ −1, 1 , (5.27)

with ω(`,j)t,s as in (3.16). Thus, abbreviating zj := (yj ;xj), we find

UV(t,s)N U∗V(t,s) =∑

j∈−1,12

∫dx(2) dy ω

(1,j1)t,s (y;x1)ω

(−1,j2)t,s (y;x2)a

]j1x1 a

]j2x2 , (5.28a)

UV(t,s)K(1)ϕ(s) U

∗V(t,s) =

∑j∈−1,12

∫dx(2) dy(2)K

(1)ϕ(s)(y

(2))ω(1,j1)t,s (z1)ω

(−1,j2)t,s (z2)a

]j1x1 a

]j2x2 , (5.28b)

and analogously for K(2)ϕ(t) to K(4)

ϕ(t). Using that UV(t,s1) UV(s1,s2) = UV(t,s2) and abbreviating

ν(k) :=⌊k−2

2

⌋for brc = maxz ∈ Z : z ≤ r, we obtain (3.9) with the coefficients

A(j)2n−1,k(t, s;x

(k))

= (−1)n−ν(k)−1cn−1

(n−1ν(k)

)[ ∫dy(3)K

(3)ϕ(s)(y

(3))ω(1,j1)t,s (z1)ω

(1,j2)t,s (z2)ω

(−1,j3)t,s (z3)

33

×(k−1)/2∏µ=2

(∫dy2µω

(1,j2µ)t,s (z2µ)ω

(−1,j2µ+1)t,s (y2µ;x2µ+1)

)

+

(k−3)/2∏µ=1

(∫dy2µ−1ω

(1,j2µ−1)t,s (z2µ−1)ω

(−1,j2µ)t,s (y2µ−1;x2µ)

)×∫

dyk−2 dyk−1 dyk(K

(3)ϕ(s)

)∗(yk−2, yk−1, yk)

×ω(1,jk−2)t,s (zk−2)ω

(−1,jk−1)t,s (zk−1)ω

(−1,jk)t,s (zk)

](5.29a)

for n ≥ 1, and

A(j)2n,k(t, s;x

(k))

= 12

n∑µ=ν(k)

((−1)µ−ν(k)dn,µ

( µν(k)

))[ ∫dy(2)K

(2)ϕ(s)(y

(2))ω(1,j1)t,s (z1)ω

(1,j2)t,s (z2)

×(k−2)/2∏m=1

(∫dy2m+1ω

(1,j2m+1)t,s (z2m+1)ω

(−1,j2m+2)t,s (y2m+1;x2m+2)

)

+

(k−2)/2∏m=1

(∫dy2m−1ω

(1,j2m−1)t,s (z2m−1)ω

(−1,j2m)t,s (y2m−1;x2m)

)

×∫

dyk−1 dykK(2)ϕ(s)(yk−1, yk)ω

(−1,jk−1)t,s (zk−1)ω

(−1,k)t,s (zk)

](5.29b)

for n ≥ 2. The remaining coefficients can be expressed as

A(j)2,2(t, s;x(2))

=

∫dy(2)

[K

(1)ϕ(s)(y

(2))ω(1,j1)t,s (z1)ω

(−1,j2)t,s (z2) + 1

4K(2)ϕ(s)(y

(2))ω(1,j1)t,s (z1)ω

(1,j2)t,s (z2)

+14K

(2)ϕ(s)(y

(2))ω(−1,j1)t,s (z1)ω

(−1,j2)t,s (z2)

], (5.29c)

A(j)2,4(t, s;x(4))

= −∫

dy1 dy3 dy4

[ω

(1,j1)t,s (z1)ω

(−1,j2)t,s (y1;x2)K

(1)ϕ(s)(y3, y4)ω

(1,j3)t,s (z3)ω

(−1,j4)t,s (z4)

+12K

(2)ϕ(s)(y1, y3)ω

(1,j1)t,s (z1)ω

(1,j2)t,s (y3;x2)ω

(1,j3)t,s (y4;x3)ω

(−1,j4)t,s (z4)

+12ω

(1,j1)t,s (z1)ω

(−1,j2)t,s (y1;x2)K

(2)ϕ(s)(y3, y4)ω

(−1,j3)t,s (z3)ω

(−1,j4)t,s (z4)

]

+12

∫dy(4)K

(4)ϕ(s)(y

(4))ω(1,j1)t,s (z1)ω

(1,j2)t,s (z2)ω

(−1,j3)t,s (z3)ω

(−1,j4)t,s (z4) . (5.29d)

34

5.4 Proof of Theorem 1

We begin with proving that under Assumption 1, moments of N with respect to χ`(t) andχ≤N (t) remain bounded uniformly in N under the time evolution.

Lemma 5.5. Let Assumption 1 hold for some a ∈ N0 and let t ∈ R, b ∈ N0 and 0 ≤ ` ≤ a.

(a) There exists a constant C(`, b) > 0 such that⟨χ`(t), (N + 1)bχ`(t)

⟩F

. eC(`,b)t . (5.30)

(b) Let c ∈ N0. There exists a constant C(`, b, c) > 0 such that⟨χ`(t), (N + 1)bχ`(t)

⟩F≥N

. N−c eC(`,b,c)t . (5.31)

(c) There exists a constant C(b) > 0 such that for all 0 ≤ b ≤ a+ 1,⟨χ≤N (t), (N + 1)bχ≤N (t)

⟩F≤N

. eC(b)t . (5.32)

Part (a) for ` = 0 and part (c) are standard results, which are proven, e.g., in [72, Propo-sition 3.3], [57, Lemma 2.3], and [59, Proposition 4 (iii)].

Proof. For part (a), we infer from (5.21), (5.25) and Lemma 4.4 that

‖(N + 1)b2χ`(t)‖ ≤ ‖(N + 1)

b2 UV(t,0)χ`(0)‖+

`−1∑n=0

`−n∑m=1

∑j∈Nm|j|=`−n

‖(N + 1)b2 I(j)t χn(0)‖

. eC(`,b)t∑n=0

‖(N + 1)b+3(`−n)

2 χn(0)‖ (5.33)

for I(j)t as in (5.22). For part (b), note that N ≥ N on F≥N , hence

∥∥∥(N + 1)b2χ`(t)

∥∥∥F≥N

≤

∥∥∥∥∥(N + 1)b+c2

Nc2

χ`(t)

∥∥∥∥∥F≥N

. N−c2 eC(b,`,c)t (5.34)

for any c ∈ N0 by part (a).

Proof of Theorem 1. Let us abbreviate

χa(t) := χ≤N (t)⊕ 0−a∑`=0

λ`2Nχ`(t) . (5.35)

For simplicity of presentation, we first do a formal computation. Note that (H≤Nϕ(t)χ≤N )⊕ 0 =

Hϕ(t)(χ≤N ⊕ 0), hence (2.18) and (3.2) imply

∂t‖χa(t)‖2F≤N

= 2 Im

⟨χa(t),

(Hϕ(t)χ

≤N (t)⊕ 0−a∑`=0

λ`2N

∑n=0

H(n)ϕ(t)χ`−n(t)

)⟩F≤N

35

= 2 Im

⟨χa(t),

(Hϕ(t)

(χa(t) +

a∑`=0

λ`2Nχ`(t)

)−

a∑`=0

λ`2N

∑n=0

H(n)ϕ(t)χ`−n(t)

)⟩F≤N

= 2 Im⟨χa(t),Hϕ(t)χa(t)

⟩F≤N

+ 2a∑`=0

λ`2N Im

⟨χa(t),

(Hϕ(t)χ`(t)−

∑n=0

H(n)ϕ(t)χ`−n(t)

)⟩F≤N

. (5.36)

By self-adjointness of Hϕ(t) and since [Hϕ(t),1F≤N ] = 0, the first summand vanishes. Ina similar way, using only the (well-defined) integral form of (2.18) and Definition 3.1, werigorously obtain the integrated version of (5.36), viz.,

‖χa(t)‖2F≤N − ‖χa(0)‖2F≤N

= 2a∑`=0

λ`2N Im

∫ t

0ds

⟨χa(s),

((Hϕ(s) −H(0)

ϕ(s)

)χ`(s)−

∑n=1

H(n)ϕ(s)χ`−n(s)

)⟩F≤N

. (5.37)

By reordering the summation and using Cauchy-Schwarz we find

‖χa(t)‖2F≤N − ‖χa(0)‖2F≤N

= 2

a∑`=0

λ`2N Im

∫ t

0ds

⟨χa(s),

((Hϕ(s) −H(0)

ϕ(s)

)χ`(s)−

a−∑n=1

λn2NH

(n)ϕ(s)χ`(s)

)⟩F≤N

≤ 2

∫ t

0ds ‖χa(s)‖F≤N

a∑`=0

λ`2N

∥∥∥∥∥(Hϕ(s) −H(0)

ϕ(s) −a−∑n=1

λn2NH

(n)ϕ(s)

)χ`(s)

∥∥∥∥∥F≤N

= 2λa+12

N

∫ t

0ds ‖χa(s)‖F≤N

a∑`=0

‖Ra−`χ`(s)‖F≤N , (5.38)

with the definition of Ra from Lemma 2.1. Then the remainder estimate from Lemma 2.1,and Lemma 5.5a yield

‖χa(t)‖2F≤N − ‖χa(0)‖2F≤N ≤ 2λa+12

N C(a)

∫ t

0ds ‖χa(s)‖F≤N

a∑`=0

‖(N + 1)a−`+4

2 χ`(s)‖

. λa+12

N

∫ t

0ds eC(a)s‖χa(s)‖F≤N . (5.39)

Since ‖χa(0)‖2F≤N ≤ C(a)N−(a+1) by Assumption 1, Gronwall’s lemma implies that

‖χa(t)‖2F≤N . λa+1N eC(a)t . (5.40)

5.5 Proof of Corollary 3.4

The proof is an inductive construction of the coefficients in (3.13). We proceed by provingfour auxiliary claims; Claim 4 is the statement of the Corollary.

Claim 1. (Bogoliubov time evolution of the initial data).

UV(t,0)χ`(0) =∑n=1

n+2∑m=0

m+n even

∑j∈−1,1m

∫dx(m)A

(j)`,n,m

(t;x(m)

)a]j1x1 ··· a

]jmxm UV(t,0)χ`−n(0) ,

(5.41)

36

where

A(j)`,n,m

(t;x(m)

)=

m∑µ=0

∫dy(m)a(`)

n,m,µ(y(m))

µ∏p=1

ω(1,jp)t,0 (yp;xp)

m∏q=µ+1

ω(−1,jq)t,0 (yq;xq) . (5.42)

Proof. Analogous to Step 3 in the proof of Proposition 3.2.

Claim 2. (Induction base case).

χ1(t) =∑q=1,3

∑j∈−1,1q

∫dx(q)C

(j)1,q(t;x

(q))a]j1x1 ···a

]jqxq χ0(t) , (5.43)

whereC

(j)1,q

(t;x(q)

)= B

(j)1,q

(t;x(q)

)+ B

(j)1,q

(t;x(q)

)(5.44)

with

B(j)1,1(t;x) = 0 , B

(j)1,3(t;x(3)) = −i

t∫0

A(j)1,3

(t, s;x(3)

)ds , (5.45a)

B(j)1,q(t;x

(q)) = A(j)1,1,q(t;x

(q)) (5.45b)

for A(j)n,p as in (5.29) and A

(j)`,n,m as in (5.42).

Proof. By (3.6), we can decompose

χ1(t) = UV(t,0)χ1(0) + χint1 (t) , χint

1 (t) := −i

t∫0

H(1)t,s dsχ0(t) . (5.46)

Insertion of (5.41) and (3.9) for n = 1 yields

χint1 (t) =

∑m=1,3

∑j∈−1,1m

∫dx(m)B

(j)1,m

(t;x(m)

)a]j1x1 ··· a

]jmxm χ0(t) , (5.47a)

UV(t,0)χ1(0) =∑m=1,3

∑j∈−1,1m

∫dx(m)B

(j)1,m

(t;x(m)

)a]j1x1 ··· a

]jmxm χ0(t) (5.47b)

with B(j)1,m and B

(j)1,m as in (5.45).

Claim 3. (Induction for the Bogoliubov time evolved initial data)

UV(t,0)χ`(0) =∑

0≤q≤3`q+` even

∑j∈−1,1q

∫dx(q)B

(j)`,q

(t;x(q)

)a]j1x1 ··· a

]jqxq χ0(t) , (5.48)

where the coefficients B(j)`,q are determined by the iteration rule

B0,0(t) := 1 , (5.49a)

B(j)`,q

(t;x(q)

):= 0 if q > 3` or q + ` odd , (5.49b)

37

and otherwise

B(j)`,q

(t;x(q)

)=

min`, 3`+2−q2∑

n=1

minq,n+2∑m=max0,q−3(`−n)

m+n even

A(j1,...,jm)`,n,m

(t;x(m)

)B

(jm+1,...,jq)`−n,q−m

(t;xm+1, ..., xq

)(5.49c)

with A(j)`,n,m as in (5.42).

Proof. We prove the hypothesis (5.48) by induction over ` ∈ N. The base case ` = 1 isestablished by (5.47b). Assume (5.48) holds for 1, 2, , ..., `− 1. Then it follows from Claim 1that

UV(t,0)χ`(0) =∑n=1

n+2∑m=0

m+n even

∑0≤q≤3(`−n)q+`−n even

∑j∈−1,1m+q

∫dx(m+q)A

(j1,...,jm)`,n,m

(t;x(m)

)

×B(jm+1,...,jm+q)`−n,q

(t;xm+1, ..., xm+q

)a]j1x1 ··· a

]jm+qxm+q χ0(t) . (5.50)

We now rearrange the sums in (5.50). Abbreviating the integrand as I(j)n;m;q, we find that

∑n=1

∑0≤m≤n+2m+n even

∑0≤q≤3(`−n)q+`−n even

∑j∈−1,1m+q

∫dx(m+q)I(j)

n;m;q

=∑n=1

∑0≤m≤n+2m+n even

∑m≤q≤m+3(`−n)

q+` even

∑j∈−1,1q

∫dx(q)I

(j)n;m;q−m

=∑n=1

∑0≤q≤3`+2−2n

q+` even

minq,n+2∑m=max0,q−3(`−n)

m+n even

∑j∈−1,1q

∫dx(q)I

(j)n;m;q−m

=∑

0≤q≤3`q+` even

∑j∈−1,1q

min 3`+2−q2

,`∑n=1

minq,n+2∑m=max0,q−3(`−n)

m+n even

∫dx(q)I

(j)n;m;q−m . (5.51)

This closes the induction.

Claim 4. (Induction for the full corrections).

χ`(t) =∑

0≤q≤3`q+` even

∑j∈−1,1q

∫dx(q)C

(j)`,q (t;x(q)) a

]j1x1 ···a

]jqxq χ0(t) , (5.52)

whereC

(j)`,q

(t;x(q)

):= B

(j)`,q

(t;x(q)

)(5.53a)

if q = 0, 1, and otherwise

C(j)`,q

(t;x(q)

)= B

(j)`,q

(t;x(q)

)− i

min`, 3`+2−q2∑

n=1

minq,n+2∑p=max2,q−3(`−n)

p+n even

∑m∈−1,1q−p

t∫0

ds

∫dy(q−p)

38

×A(j1,...,jp)n,p (t, s;x(p))C

(m)`−n,q−p(s; y

(q−p))

q−p∏µ=1

ω(mµ,jp+µ)t,s (yµ, xp+µ) (5.53b)

with A(j)n,p as in (5.42) and B

(j)`,q as in (5.49).

Proof. We prove the hypothesis (5.52) by induction over ` ∈ N. The base case ` = 1 isestablished in Claim 2. By (3.6), it follows that

χ`(t) = UV(t,0)χ`(0) + χint` (t) , χint

` (t) := −i∑n=1

t∫0

ds H(n)t,s UV(t,s)χ`−n(s) , (5.54)

and the first term is given by Claim 3. Assume (5.52) holds for 1, 2, ..., `− 1. Then by (3.9),

χint` (t) = −i

∑n=1

t∫0

ds∑

2≤p≤n+2p+n even

∑0≤q≤3(`−n)q+`−n even

∑j∈−1,1p+q

∫dx(p+q)A

(j1,...,jp)n,p (t, s;x(p))a

]j1x1 ···a

]jpxp

×C(jp+1,...,jp+q)`−n,q (s;xp+1, ..., xp+q)UV(t,s)a

]jp+1xp+1 U∗V(t,s) ··· UV(t,s)a

]jp+qxp+q U∗V(t,s)χ0(t)

=∑n=1

∑2≤p≤n+2p+n even

∑0≤q≤3(`−n)q+`−n even

∑j∈−1,1p+q

∫dx(p+q)

(− i

∑m∈−1,1q

t∫0

ds

∫dy(q)

×A(j1,...,jp)n,p (t, s;x(p))C

(m)`−n,q(s; y

(q))

( q∏µ=1

ω(mµ,jp+µ)t,s (yµ;xp+µ)

))

×a]j1x1 ···a]jp+qxp+q χ0(t) , (5.55)

where we used that

UV(t,s)a]jp+1xp+1 U∗V(t,s) ··· UV(t,s)a

]jp+qxp+q U∗V(t,s)

=∑

m∈−1,1q

∫dy(q)

( q∏µ=1

ω(jp+µ,mµ)t,s (xp+µ; yµ)

)a]m1y1 ···a

]mqyq (5.56)

and subsequently renamed the variables xp+1, ..., xp+q ↔ y1, ..., yq as well as the indicesjp+1, ..., jp+q ↔ m1, ...,mq. Reordering the sums as in Claim 3 and adding (5.48) completesthe induction.

5.6 Proof of Corollary 3.5

As a consequence of Corollary 3.4 and since (a]j )† = a]−j , we obtain⟨a]j1x1 · · · a

]jnxn

⟩(t)

`,k=

∑0≤p≤3`p+` even

∑0≤q≤3kq+k even

∑m∈−1,1pµ∈−1,1q

∫dy(p) dz(q)C

(m)`,p (t; y(p))C

(µ)k,q (t; z(q))

×⟨a]−mpyp ··· a]−m1

yp a]j1x1 ···a

]jnxn a

]µ1z1 ··· a

]µqzq

⟩(t)

0,0

=∑

0≤p≤3`p+` even

∑0≤q≤3kq+k even

∑m∈−1,1n+p+q

Ip,q

39

=∑

n≤b≤n+3(`+k)b+k+l+n even

∑m∈−1,1b

min3k,b−n∑q=max0,b−n−3`

q+k even

Ib−n−q,q , (5.57)

where we abbreviated

Ip,q :=

∫dy(n+p+q)C

(−mp,...,−m1)`,p (t; yp, ..., y1)C

(mn+p+1,...,mn+p+q)k,q (t; yn+p+1, ..., yn+p+q)

×n∏µ=1

δ(yp+µ − xµ)δmp+µ,jµ

⟨a]m1y1 ··· a

]mn+p+qyn+p+q

⟩(t)

0,0. (5.58)

Consequently,⟨a]j1x1 · · · a

]jnxn

⟩(t)

`,k=

∑n≤b≤n+3(`+k)b+k+l+n even

∑m∈−1,1b

∫dy(b)D

(j,m)`,k,n;b(t;x

(n); y(b))⟨a]m1y1 ··· a

]mbyb

⟩(t)

0,0

(5.59)

for D(j,m)`,k,n;b as defined as in (3.23). By Assumption 1, the inner product in (5.59) is the b+ 2ν-

point correlation function of a quasi-free state. By Lemma 4.7b, it vanishes for b odd (or,equivalently, k + ` + n odd) and decomposes into the sum over all possible pairings for beven.

5.7 Proof of Corollary 3.7

With the abbreviation (5.35), we find⟨n∏i=1

a†xi

p∏j=1

ayj

⟩(t)

N

=

⟨χa(t),

n∏i=1

a†xi

p∏j=1

ayjχ≤N (t)

⟩F≤N

+a∑`=0

λ`2N

⟨p−1∏j=0

a†yp−j

n−1∏i=0

axn−iχ`(t), χa−`(t)

⟩

+

a∑`=0

λ`2N

∑m=0

⟨χm(t),

n∏i=1

a†xi

p∏j=1

ayjχ`−m(t)

⟩. (5.60)

Therefore, using Cauchy-Schwarz and Lemma 5.1, it holds for any bounded B : Hp → Hn that∣∣∣∣∣∫

dx(n) dy(p)B(x(n); y(p))

(⟨n∏i=1

a†xi

p∏j=1

ayj

⟩(t)

−a∑`=0

λ`2N

∑m=0

⟨n∏i=1

a†xi

p∏j=1

ayj

⟩(t)

m,`−m

)∣∣∣∣∣≤ ‖χa(t)‖F≤N

∥∥∥∥∫ dx(n) dy(p)B(x(n); y(p))n∏i=1

a†xi

p∏j=1

ayjχ≤N (t)

∥∥∥∥F≤N

+a∑`=0

λ`2N

∥∥∥∥∫ dx(n) dy(p)B∗(y(p);x(n))

p∏j=1

a†yj

n∏i=1

axiχ`(t)

∥∥∥∥‖χa−`(t)‖≤ C(n, p)‖B‖op

(‖χa(t)‖F≤N

∥∥∥(N + 1)n+p2 χ≤N (t)

∥∥∥F≤N

+a∑`=0

λ`2N‖χa−`(t)‖

∥∥∥(N + 1)n+p2 χ`(t)

∥∥∥)

40

. eC(n,p,a)t‖B‖op

(λa+12

N +a∑`=0

λ`2Nλ

a−`+12

N

), (5.61)

where we used for the last estimate Lemmas 5.5a and 5.5c and Theorem 1. Finally, thestatement follows by expanding up to 2a+ 1 for n+p even and up to 2a for n+p odd becauseall contributions to the last line of (5.60) with n+ p+ ` odd vanish by Corollary 3.5a.

5.8 Proof of Theorem 2

We start from the expression (3.37) for the one-particle reduced density matrix, i.e.,

γ(1)

ΨN(t) = pϕ(t) +

1√N

(|ϕ(t)〉〈βχ≤N (t)|+ |βχ≤N (t)〉〈ϕ(t)|

)+

1

N

(γχ≤N (t) − pϕ(t)

⟨χ≤N (t),Nχ≤N (t)

⟩F≤N

), (5.62)

where

βχ≤N (t)(x) :=

⟨χ≤N (t),

√1− N

Naxχ

≤N (t)

⟩F≤N

. (5.63)

The idea of the proof is to use expansions of√

1− NN and 1N in powers of λN , estimate the

remainder terms, and use Corollary 3.7 to estimate the difference of the microscopic andeffective n-point correlation functions. Following the calculations in Appendix C, we obtain

1

N= λN

1

1 + λN=

a−1∑`=0

c`λ1+`N + λa+1

N R1,a , (5.64a)√[N −N ]+

N=

a∑`=0

λ`+ 1

2N

∑k=0

c`,k(N − 1)k + λa+ 3

2N R2,a , (5.64b)

with c` := (−1)`c(3/2)` , c`,k := c`−kc

(0)k , |R1,a| ≤ C(a) and ‖R2,aφ‖ ≤ C(a)‖(N + 1)a+1φ‖ for

any φ ∈ F and some constant C(a) > 0. For the |ϕ(t)〉〈βχ≤N (t)| term in (5.62) we then find,for any A ∈ L(H),

1√N

∫dx dyA(x; y)ϕ(t, x)βχ≤N (t)(y)

=a∑`=0

∑k=0

λ`+ 1

2N c`,k

∫dx(A∗ϕ)(t, x)

⟨(N − 1)kax

⟩(t)

N+R(1)

A

=

a∑`=0

∑k=0

a−`−1∑m=0

2m+1∑n=0

λ`+m+1N c`,k

∫dx(A∗ϕ)(t, x)

⟨(N − 1)kax

⟩(t)

n,2m+1−n+R(1)

A +R(2)A

=

a−1∑`=0

λ`+1N

∑m=0

`−m∑k=0

2m+1∑n=0

c`−m,k

∫dx(A∗ϕ)(t, x)

⟨(N − 1)kax

⟩(t)

n,2m+1−n+R(1)

A +R(2)A (5.65)

with

R(1)A := λ

a+ 32

N

∫dx(A∗ϕ)(t, x) 〈R2,aax〉(t)N

41

R(2)A :=

a∑`=0

λ`+ 1

2N

∑k=0

c`,k

∫dx(A∗ϕ)(t, x)

[⟨(N − 1)kax

⟩(t)

N

−a−`−1∑m=0

2m+1∑n=0

λm+ 1

2N

⟨(N − 1)kax

⟩(t)

n,2m+1−n

]. (5.66)

Analogously, we compute for the second line in (5.62),

1

N

∫dx dyA(x; y)

(⟨a†xay

⟩(t)

N+ pϕ(t)(y;x) 〈N〉(t)N

)=

a−1∑`=0

λ`+1N

∑m=0

2m∑n=0

c`−m

∫dx dyA(x; y)

(⟨a†xay

⟩(t)

n,2m−n+ pϕ(t)(y;x) 〈N〉(t)n,2m−n

)+R(3)

A +R(4)A , (5.67)

where

R(3)A := λa+1

N R1,a

∫dx dyA(x; y)

(⟨a†xay

⟩(t)

N+ pϕ(t)(y;x) 〈N〉(t)N

), (5.68)

R(4)A :=

a−1∑`=0

c`λ`+1N

∫dx dyA(x; y)

[⟨a†xay

⟩(t)

N−a−`−1∑m=0

λmN

2m∑n=0

⟨a†xay

⟩(t)

n,2m−n

]

+ 〈ϕ(t), Aϕ(t)〉Ha−1∑`=0

c`λ`+1N

[〈N〉(t)N −

a−`−1∑m=0

λmN

2m∑n=0

〈N〉(t)n,2m−n

].

Lemma 5.1 and the bound on moments of N + 1 from Lemma 5.5c imply∣∣R(1)A

∣∣ ≤ λa+ 3

2N C(a)‖A‖op‖(N + 1)a+1χ≤N (t)‖F≤N ‖N

12χ≤N (t)‖F≤N

≤ λa+ 3

2N ‖A‖op eC(a)t , (5.69)∣∣R(3)

A

∣∣ ≤ λa+1N |R1,a|

(‖χ≤N (t)‖F≤N

∥∥∥∫ dx dyA(x; y)a†xayχ≤N (t)

∥∥∥F≤N

+ 〈ϕ(t), Aϕ(t)〉H⟨χ≤N (t),Nχ≤N (t)

⟩ )≤ λa+1

N ‖A‖op eC(a)t . (5.70)

In order to bound R(2)A and R(4)

A , we bring N k into normal order by iteratively applying (5.1),i.e.,

N k =

k−1∑`1=0

(k−1`1

) ∫dx1a

†x1N

`1ax1 =

k∑`=1

Ck,`

∫dx(`) a†x1 ··· a

†x`ax1 ··· ax` (5.71)

for some constants Ck,`. Consequently, for 〈·〉(t) ∈〈·〉(t)`,k , 〈·〉

(t)N

,∫

dx(A∗ϕ)(t, x)⟨

(N − 1)kax

⟩(t)

=

k∑`=1

Ck,`

∫dx(`) dy(`+1)B(t, x(`); y(`+1))

⟨a†x1 ··· a

†x`ay1 ··· ay`+1

⟩(t), (5.72)

42

where B(t, x(`); y(`+1)) := δ(y1 − x1)···δ(y` − x`)(A∗ϕ)(t, x`+1) is the Schwartz integral ker-nel of the operator B : H`+1 → H`, ψ 7→ (Bψ)(x(`)) =

∫dx`+1(A∗ϕ)(t, x`+1)ψ(x`+1) with

‖B‖L(H`+1,H`) ≤ ‖A‖op. Hence, Corollary 3.7 leads to the bound∣∣R(j)A

∣∣ ≤ ‖A‖op eC(a)tλa+1N , j = 2, 4 . (5.73)

Finally, Theorem 2 follows by duality of compact and trace class operators.

Acknowledgments

We are grateful for the hospitality of Central China Normal University (CCNU),where parts of this work were done, and thank Phan Thanh Nam, Simone

Rademacher, Robert Seiringer and Stefan Teufel for helpful discussions. L.B. gratefully ac-knowledges the support by the German Research Foundation (DFG) within the ResearchTraining Group 1838 “Spectral Theory and Dynamics of Quantum Systems”, and the fundingfrom the European Union’s Horizon 2020 research and innovation programme under the MarieSk lodowska-Curie Grant Agreement No. 754411.

A Derivation of the Excitation Hamiltonian

Let us write the Hamiltonian (1.3) as

HN =

N∑j=1

hϕ(t)j + λN

∑1≤i<j≤N

Wϕ(t)(xi, xj) , (A.1)

with Wϕ(t) as in (2.26). Since χ≤N (t) := UN,ϕ(t)ΨN (t), we find with (2.16) that

i∂tχ≤N (t) = i∂t

N⊕j=0

(qϕ(t)

)⊗j (a(ϕ(t))N−j√(N − j)!

ΨN (t)

)= dΓ

(hϕ(t)

)χ≤N (t) + UN,ϕ(t)

HN −N∑j=1

hϕ(t)j

U∗N,ϕ(t)χ≤N (t) . (A.2)

In some orthonormal basis ϕn(t)n≥0 with ϕ0(t) = ϕ(t), we define the matrix elements

Wijk` :=

∫dx dy ϕi(t, x)ϕj(t, y)Wϕ(t)(x, y)ϕk(t, x)ϕ`(t, y) , (A.3)

omitting the time dependence for ease of notation. Then, denoting a]i := a](ϕi(t)),

HN −N∑j=1

hϕ(t)j =

1

2λN

∑i,j,k,`≥0

Wijk`a†ia†jaka`

= λN∑j,k>0

W0jk0 a†0a†jaka0 +

λN2

∑i,j>0

Wij00 a†ia†ja0a0 + h.c.

+λN

∑i,j,k>0

Wijk0 a†ia†jaka0 + h.c.

+λN2

∑i,j,k,`>0

Wijk` a†ia†jaka` (A.4)

43

since W0000 = W000` = Wi0k0 = 0 etc. for all i, k, ` > 0 by definition of Wϕ(t). Finally, (2.21)

follows directly from the transformation rules (2.17) because W0jk0 =⟨ϕj(t),K

(1)ϕ(t)ϕk(t)

⟩,

Wij00 =⟨ϕi(t)⊗ ϕj(t),K(2)

ϕ(t)

⟩, and Wijk0 =

⟨ϕi(t)⊗ ϕj(t),K(3)

ϕ(t)(t)ϕk(t)⟩

for all i, j, k > 0.

B Assumption 1 for Trapped Initial Data

Let us discuss in more detail how the results of [13] provide a natural class of initial datasatisfying Assumption 1. If ΨN (0) is the ground state or a low-energy eigenstate of HN

trap asin (1.1), the initial condensate wave function ϕ(0) is given by the normalized minimizer ϕtrap

of the Hartree energy functional

Etrap[ϕ] =

∫Rd

(|∇ϕ(x)|2 + Vtrap(x)|ϕ(x)|2

)dx+ 1

2

∫R2d

v(x− y)|ϕ(x)|2|ϕ(y)|2 dx dy , (B.1)

whose minimum is denoted by etrap := Etrap[ϕtrap]. More precisely, Assumption 1 is satisfied

for each eigenstate ΨNtrap of HN

trap associated with an eigenvalue Etrap ∈ Eζtrap for some ζ ≥ 0.

Here, Eζtrap is the set of eigenvalues of HNtrap such that |E(n)

trap−Netrap| ≤ ζ for any E(n)trap ∈ Eζtrap

and such thatlimN→∞

(Netrap − E(n1)trap ) 6= lim

N→∞(Netrap − E(n2)

trap ) (B.2)

for any E(n1)trap , E

(n2)trap ∈ Eζtrap with n1 6= n2, where the eigenvalues are counted with multiplicity.

Hence, Eζtrap contains all non-degenerate eigenvalues of HNtrap with an energy of order one above

the ground state energy, which additionally satisfy the condition that no two elements of Eζtrap

converge to the same eigenvalue of the Bogoliubov Hamiltonian corresponding to HNtrap

2. The

state χ0(0) as in (2.33) is a normalized eigenstate of this Bogoliubov Hamiltonian. If ΨN (0)is the ground state of HN

trap, it follows that ν = 0 in (2.33), i.e., χ0(0) is quasi-free.

Proposition B.1. Let v : Rd → R be measurable, even, and of positive type (i.e., v hasa non-negative Fourier transform). Let Vtrap : Rd → R be measurable, locally bounded and

non-negative, and let Vtrap tend to infinity as |x| → ∞. Let Etrap ∈ Eζtrap for some ζ ≥ 0 and

denote by ΨNtrap ∈ HNsym the associated normalized eigenstate of HN

trap. Let ΨN (0) = ΨNtrap and

ϕ(0) = ϕtrap. Then Assumption 1 is satisfied for any a ∈ N0.

Proposition B.1 is proven in [13, Theorem 3]. The coefficients a(`)n,m,µ can be retrieved from

[13]. For example, for ΨN (0) the ground state of HNtrap, the first order correction χ1(0) to the

Bogoliubov ground state χ0(0) is given as

χ1(0) =1− |χ0(0)〉〈χ0(0)|E

(0)trap −H(0)

trap

H(1)trapχ0(0) , (B.3)

2The result in [13] is essentially a perturbative expansion of the spectral projectors of H≤Ntrap around thespectral projectors of the Bogoliubov Hamiltonian. Due to the strategy of proof (the spectral projectors areexpressed as integrals of the resolvent of H≤Ntrap along a contour enclosing the appropriate Bogoliubov eigenvalue),

one cannot separate projectors of H≤N corresponding to eigenvalues with equality in (B.2). Moreover, in thecase of degenerate eigenvalues, there is no one-to-one correspondence between the spectral projector and a wavefunction. We expect that suitable functions χ`(t) can be defined in the context of degenerate perturbationtheory.

44

where the operators H(0)trap and H(1)

trap (corresponding to HNtrap) are defined analogously to the

operators H(0)ϕ(0) and H(1)

ϕ(0) (corresponding to HN ), and where E(0)trap denotes the ground state

energy of H(0)trap. For Utrap the Bogoliubov transform diagonalizing H(0)

trap, it holds that

χ0(0) = U∗trap|Ω〉 (B.4)

andχ1(0) = U∗trap

(⊕m≥0

O(m))Utrap H

(1)trap U∗trap|Ω〉 , (B.5)

where

Utrap1− |χ0(0)〉〈χ0(0)|E

(0)trap −H(0)

trap

U∗trap =: O =⊕m≥0

O(m) (B.6)

because O is particle number preserving. The m-body operators O(m) are given by

O(m) = −∑j∈Nm

j1≤···≤jm

1∑mk=1 ejk

a†(ξj1)···a†(ξjm)|Ω〉〈Ω|a(ξjm)···a(ξj1) , (B.7)

where ξjj≥1 denotes a complete set of normalized eigenfunctions of the one-body operator

resulting from the diagonalization of H(0)trap. The corresponding eigenvalues are denoted by ej .

One computes

Utrap H(1)trap U∗trap|Ω〉 =

∫dxf

(1)trap(x)a†x|Ω〉+

∫dx(3)f

(3)trap(x(3))a†x1a

†x2a†x3 |Ω〉 (B.8)

with

f(1)trap(x) :=

∫dy(A

(−1,1,1)trap;1,3 (y, y, x) + A

(−1,1,1)trap;1,3 (y, x, y) + A

(1,−1,1)trap;1,3 (x, y, y)

), (B.9a)

f(3)trap(x(3)) := A

(1,1,1)trap;1,3(x(3)) (B.9b)

for A(j)trap;1,3 defined analogously to (5.29). Consequently,

χ1(0) = U∗trap

∫dx(O(1)f

(1)trap

)(x)a†x Utrapχ0(0)

+U∗trap

∫dx(3)

(O(3)f

(3)trap

)(x(3))a†x1a

†x2a†x3 Utrapχ0(0) (B.10)

for f(3)trap(x(3)) = (3!)−1/2

∑σ∈S3

f(3)trap(xσ(1), xσ(2), xσ(3)) the symmetrized version of f

(3)trap. Fi-

nally, the coefficients a(1)1,m,µ follow from this by (4.17), e.g.,

a(1)1,1,0(x) = −

(V∗trapO(1)f

(1)trap

)(x) , a

(1)1,1,1(x) =

(U∗trapO(1)f

(1)trap

)(x) (B.11)

for Utrap and Vtrap the matrix entries of Utrap as in (4.6).

45

C Proof of Lemma 5.4.

We estimate the Taylor series remainders for the functions f(0)0 and f

(0)µ f

(0)0 , where

f (n)µ : [−1, 1− µ]→ [0,∞) , x 7→ f (n)

µ (x) := (1− x− µ)12−n , (C.1)

for µ ∈ [0, 14) and n ≥ 0. For c

(n)` as in (2.29), we find that

f (n)µ (x) =:

a∑`=0

c(n)`

(1− µ)n+`− 12

x` +R(n)a,µ(x) , (C.2)

(f (0)µ f

(0)0

)(x) =:

a∑`=0

∑m=0

c(0)m c

(0)`−m

(1

1− µ

)m− 12

x` + Ra,µ(x) . (C.3)

Estimates for f(0)0 . The remainder R

(n)a,0(x) is given as

R(n)a,0(x) = (a+ 1) c

(n)a+1

x∫0

1

(1− t)n+ 12

(x− t1− t

)adt = c

(n)a+1

(1

1− ξ

)n+a+ 12

xa+1 (C.4)

for some ξ ∈ (0, x). For x ∈ [−1, 12 ], the second equality yields |R(0)

a,0(x)| ≤ 2a|x|a+1 since

1− ξ > 12 and by (5.12). For x ∈ (1

2 , 1], |R(0)a,0(x)| ≤ 1 ≤ 2a+1|x|a+1 by the first equality.

Estimates for f(0)λNf

(0)0 . By (C.3) for µ = λN and x = λN (k − 1) with 0 ≤ k ≤ N − 1,

(f

(0)λNf

(0)0

)(λN (k − 1)) =

a∑`=0

∑n=0

c(0)n c

(0)`−nf

(n)0 (λN )λ`N (k − 1)` + Ra,λN (λN (k − 1)) . (C.5)

By (C.2) and (C.4),

f(n)0 (λN ) =

a−∑ν=0

c(n)ν λνN + c

(n)a−`+1(1− ξ)`−a−n−

12λa−`+1

N (C.6)

for some ξ ∈ (0, λN ). Inserting this formula into (C.5) yields(f

(0)λNf

(0)0

)(λN (k − 1))

=a∑`=0

∑n=0

a−∑ν=0

c(0)n c

(0)`−nc

(n)ν (k − 1)`λν+`

N

+λa+1N

a∑`=0

∑n=0

c(0)n c

(0)`−nc

(n)a−`+1

(1

1− ξ

)a−`+n+ 12

(k − 1)` + Ra,λN (λN (k − 1))

=a∑`=0

λ`N∑m=0

d`,m(k − 1)m + λa+1N r(k, a) + Ra,λN (λN (k − 1)) , (C.7)

where we re-ordered the triple sum using d`,m from (2.30), and abbreviated the double sum

as r(k, a). Since ξ ∈ (0, λN ) and n ≤ ` ≤ a, it follows that (1− ξ)−(a−`+n+ 12

) ≤ (N−1N−2)a+ 1

2 and

46

consequently |r(k, a)| ≤ (a+ 1)22a(N−1N−2

)a+ 12

(k + 1)a. The remainder Ra,λN is given as

Ra,λN (x) =

a+1∑n=0

c(0)n c

(0)a+1−n

(1

1− ξ − λN

)n− 12(

1

1− ξ

)a−n+ 12

xa+1

= (a+ 1)a+1∑n=0

c(0)n c

(0)a+1−n

x∫0

(1

1− t− λN

)n− 12(

1

1− t

)a−n+ 12

(x− t)a dt (C.8)

for some ξ ∈ (0, x). For x ∈ [−1, 12 ], the first equality yields |Ra,λN (x)| ≤ a 4a|x|a+1, where we

used that that 2 > 1 − ξ > 12 and 2 > 1 − ξ − λN > 1

2N−3N−1 . For x ∈ (1

2 , 1 − λN ], the second

equality leads to the estimate |Ra,λN (x)| ≤ (a+ 1)22a+2|x|a+1.

Proof of Lemma 5.4. Since

√[N−N ]+N−1 = λ

12Nf

(0)0 (λN (N −1)) on F≤N , λ−1

N = N −1 ≤ N −1

on F≥N , and

√[(N−N )(N−N−1)]+

N−1 =(f

(0)λNf

(0)0

)(λN (N − 1)) on F≤N−1, we find

‖R(3)a φ‖F≤N ≤ 2a+1‖(N + 1)a+1φ‖F≤N , (C.9a)

‖R(3)a φ‖F≥N ≤ (a+ 1)‖(N + 1)a+1φ‖F>N , (C.9b)

‖R(2)a φ‖F≤N−1 ≤ 4a(a+ 1)2‖(N + 1)a+1φ‖F≤N−1 , (C.9c)

‖R(2)a φ‖F≥N ≤ 2a(a+ 1)3‖(N + 1)a+1φ‖F≥N . (C.9d)

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