Derivation of the 2d Gross-Pitaevskii equation for strongly confined 3d bosons

We study the dynamics of a system of $N$ interacting bosons in a disc-shaped trap, which is realised by an external potential that confines the bosons in one spatial dimension to a region of order $\varepsilon$. The interaction is non-negative and scaled in such a way that its scattering length is of order $(N/\varepsilon)^{-1}$, while its range is proportional to $(N/\varepsilon)^{-\beta}$ with scaling parameter $\beta\in(0,1]$. The choice $\beta=1$ corresponds to the physically relevant Gross-Pitaevskii regime. We consider the simultaneous limit $(N,\varepsilon)\to(\infty,0)$ and assume that the system initially exhibits Bose-Einstein condensation. We prove that condensation is preserved by the $N$-body dynamics, where the time-evolved condensate wave function is the solution of a two-dimensional non-linear equation. The strength of the non-linearity depends on the scaling parameter $\beta$. For $\beta\in(0,1)$, we obtain a cubic defocusing non-linear Schr\"odinger equation, while the choice $\beta=1$ yields a Gross-Pitaevskii equation featuring the scattering length of the interaction. In both cases, the coupling parameter depends on the confining potential.


Introduction
Since two decades, it has been experimentally possible to realise quasi-two dimensional Bose gases in disc-shaped traps [14,31,33]. The study of such systems is physically of particular interest since they permit the detection of inherently two-dimensional effects and serve as models for different statistical physics phenomena [17,18,35]. In this article, our aim is to contribute to the mathematically rigorous understanding of such systems. We consider a Bose-Einstein condensate of N identical, non-relativistic, interacting bosons in a disc-shaped trap, which effectively confines the particles in one spatial direction to an interval of length ε. We study the dynamics of this system in the simultaneous limit (N, ε) → (∞, 0), where the Bose gas becomes quasi two-dimensional. To describe the N bosons, we use the coordinates z = (x, y) ∈ R 2+1 , where x denotes the two longitudinal dimensions and y is the transverse dimension. The confinement in the y-direction is modelled by the scaled potential 1 ε 2 V ⊥ y ε for 0 < ε 1 and some V ⊥ : R → R. In units such that = 1 and m = 1 2 , the Hamiltonian is given by where ∆ denotes the Laplace operator on R 3 and V : R × R 3 → R is an additional external potential, which may depend on time. The interaction w µ,β between the particles is purely repulsive and scaled in dependence of the parameters N and ε. In this paper, we consider two fundamentally different scaling regimes, corresponding to different choices of the scaling parameter β ∈ R: β ∈ (0, 1) yields the non-linear Schrödinger (NLS) regime, while β = 1 is known as the Gross-Pitaevskii regime. Making use of the parameter the Gross-Pitaevskii regime is realised by scaling an interaction w : R 3 → R, which is compactly supported, spherically symmetric and non-negative, as For the NLS regime, we will consider a more generic form of the interaction (see Definition 2.2). For the length of this introduction, let us focus on the special case with β ∈ (0, 1). Clearly, (2) equals (3) with the choice β = 1. Both scaling regimes describe very dilute gases, and we comment on their physical relevance below.
The transverse part χ ε is given by the normalised ground state of − d 2 dy 2 + 1 ε 2 V ⊥ ( y ε ), which is defined by Here, E 0 denotes the minimal eigenvalue of the unscaled operator − d 2 dy 2 + V ⊥ , corresponding to the normalised ground state χ. The relation of χ ε and χ is In this paper, we derive an effective description of the many-body dynamics ψ N,ε (t). We show that if the system initially forms a Bose-Einstein condensate with factorised condensate wave function, then the dynamics generated by H µ,β (t) preserve this property. Under the assumption that lim (N,ε)→(∞,0) where the limit (N, ε) → (∞, 0) is taken along a suitable sequence, we show that lim (N,ε)→(∞,0) with time-evolved condensate wave function ϕ ε (t) = Φ(t)χ ε . While the transverse part of the condensate wave function remains in the ground state, merely undergoing phase oscillations, the longitudinal part is subject to a non-trivial time evolution. We show that this evolution is determined by the two-dimensional non-linear equation The coupling parameter b β in (7) depends on the scaling regime and is given by R |χ(y)| 4 dy for β ∈ (0, 1), 8πa R |χ(y)| 4 dy for β = 1, where a denotes the scattering length of w (see Section 3.2 for a definition). The evolution equation (7) provides an effective description of the dynamics. Since the N bosons interact, it contains an effective one-body potential, which is given by the probability density N |Φ(t)| 2 times the two-body scattering process times a factor R |χ ε (y)| 4 dy from the confinement. At low energies, the scattering is to leading order described by the s-wave scattering length a µ,β of the interaction w µ,β , which scales as a µ,β ∼ µ for the whole parameter range β ∈ (0, 1] (see [11,Lemma A.1]) and characterises the length scale of the inter-particle correlations.
In the scaling regime β = 1, the first order Born approximation breaks down since a µ,1 ∼ µ, which implies that the correlations are visible on the length scale µ of the interaction even in the limit (N, ε) → (∞, 0). Consequently, the coupling parameter b 1 contains the full scattering length, which makes (7) a Gross-Pitaevskii equation.
Physically, the scaling β = 1 is relevant because it corresponds to an (N, ε)-independent interaction via a suitable coordinate transformation. The Gross-Pitaevskii regime is characterised by the requirement that the kinetic energy per particle (in the longitudinal directions) is of the same order of magnitude as the total energy per particle (without counting the energy from the confinement or the external potential). For N bosons which interact via a potential with scattering length A in a trap with longitudinal extension L and transverse size εL, the former scales as E kin ∼ L −2 . The latter can be computed as E total ∼ A 3d ∼ AN/(L 3 ε), where 3d denotes the particle density. Both quantities being of the same order implies the scaling condition A/L ∼ ε/N .
The choice A ∼ 1 entails L ∼ N/ε, which corresponds to an (N, ε)-independent interaction potential. Hence, to capture N bosons in a strongly asymmetric trap while remaining in the Gross-Pitaevskii regime, one must increase the longitudinal length scale of the trap as N/ε and the transverse scale as N . For our analysis, we choose to work instead in a setting where L ∼ 1, thus we consider interactions with scattering length A ∼ ε/N . Both choices are related by the coordinate transform z → (ε/N )z, which comes with the time rescaling t → (ε/N ) 2 t in the N -body Schrödinger equation (4).
For the scaling regime β ∈ (0, 1), there is no such coordinate transform relating w µ,β to a physically relevant (N, ε)-independent interaction. We consider this case mainly because the derivation of the Gross-Pitaevskii equation for β = 1 relies on the corresponding result for β ∈ (0, 1). The central idea of the proof is to approximate the interaction w µ by an appropriate potential with softer scaling behaviour covered by the result for β ∈ (0, 1), and to control the remainders from this substitution. We follow the approach developed by Pickl in [30], which was adapted to the problem with strong confinement in [4] and [5], where an effectively onedimensional NLS resp. Gross-Pitaevskii equation was derived for three-dimensional bosons in a cigar-shaped trap. The model considered in [4,5] is analogous to our model (1) but with a two-dimensional confinement, i.e., where (x, y) ∈ R 1+2 . Since many estimates are sensitive to the dimension and need to be reconsidered, the adaptation to our problem with one-dimensional confinement is non-trivial. A detailed account of the new difficulties is given in Remarks 4 and 5.
To the best of our knowledge, the only existing derivation of a two-dimensional evolution equation from the three-dimensional N -body dynamics is by Chen and Holmer in [8]. Their analysis is restricted to the range β ∈ (0, 2 5 ), which in particular does not include the physically relevant Gross-Pitaevskii case. In this paper, we extend their result to the full regime β ∈ (0, 1] and include a larger class of confining traps as well as a possibly time-dependent external potential. We impose different conditions on the parameters N and ε, which are stronger than in [8] for small β but much less restrictive for larger β (see Remark 3). Related results for a cigar-shaped confinement were obtained in [4,5,9,22].
Regarding the situation without strong confinement, the first mathematically rigorous justification of a three-dimensional NLS equation from the quantum many-body dynamics of three-dimensional bosons with repulsive interactions was by Erdős, Schlein and Yau in [11], who extended their analysis to the Gross-Pitaevskii regime in [12]. With a different approach, Pickl derived effective evolution equations for both regimes [30], providing also estimates of the rate of convergence. Benedikter, De Oliveira and Schlein proposed a third and again different strategy in [3], which was then adapted by Brennecke and Schlein in [6] to yield the optimal rate of convergence. For two-dimensional bosons, effective NLS dynamics of repulsively interacting bosons were first derived by Kirkpatrick, Schlein and Staffilani in [23]. This result was extended to more singular scalings of the interaction, including the Gross-Pitaevskii regime, by Leopold, Jeblick and Pickl in [20], and two-dimensional attractive interactions were covered in [10,21,24].
The dimensional reduction of non-linear one-body equations was studied in [2] by Ben Abdallah, Méhats, Schmeiser and Weishäupl, who consider an n + d-dimensional NLS equation with a d-dimensional quadratic confining potential. In the limit where the diameter of this confinement converges to zero, they obtain an effective n-dimensional NLS equation. A similar problem for a cubic NLS equation in a quantum waveguide, resulting in a limiting one-dimensional equation, was covered by Méhats and Raymond in [28].
The remainder of the paper is structured as follows: in Section 2, we state our assumptions and present the main result. The strategy of proof for the NLS scaling is explained in Section 3.1, while the Gross-Pitaevskii scaling is covered in Section 3.2. Section 3.3 contains the proof of our main result, which depends on five propositions. Section 4 collects some auxiliary estimates, which are used in Sections 5 and 6 to prove the propositions for β ∈ (0, 1) and β = 1, respectively.
Notation. We use the notations A B, A B and A ∼ B to indicate that there exists a constant C > 0 independent of ε, N, t, ψ N,ε 0 , Φ 0 such that A ≤ CB, A ≥ CB or A = CB, respectively. This constant may, however, depend on the quantities fixed by the model, such as V ⊥ , χ and V . Besides, we will exclusively use the symbol · to denote the weighted many-body operators from Definition 3.1 and use the abbreviations Finally, we write x + and x − to denote (x + σ) and (x − σ) for any fixed σ > 0, which is to be understood in the following sense: Let the sequence (N n , ε n ) n∈N → (∞, 0). Then for sufficiently large n , f (N, ε) ε x − :⇔ for any σ > 0, f (N n , ε n ) ε x−σ n for sufficiently large n , f (N, ε) µ x − :⇔ for any σ > 0, f (N n , ε n ) µ x−σ n for sufficiently large n .
Note that these statements concern fixed σ in the limit (N, ε) → (∞, 0) and do in general not hold uniformly as σ → 0.
• β = 1 : By imposing the admissibility condition, we ensure that the diameter ε of the confining potential does not shrink too slowly compared to the range µ β of the interaction. Consequently, the energy gap above the transverse ground state, which scales as ε −2 , is always large enough to sufficiently suppress transverse excitations. Equivalently, the condition can be written as for sufficiently large N and small ε. Clearly, it is necessary to choose Θ > 1, and the condition is weaker for larger Θ. In the proof, we require the admissibility condition to control the orthogonal excitations in the transverse direction (see Remark 4), which results in the respective upper bound for Θ. The threshold Θ = 3 + admits N ∼ ε −2 , which has a physical implication: if the confinement is realised by a harmonic trap V ⊥ (y) = ω 2 y 2 , the frequency ω ε of the rescaled oscillator ε −2 V ⊥ (y/ε) scales as ω ε = ωε −2 . Hence, Θ = 3 + means that the frequency of the confining trap grows proportionally to N . The moderate confinement condition implies that, for sufficiently large N and small ε, Moderate confinement means that ε does not shrink too fast compared to µ β . For β ∈ (0, 1), it implies that the interaction is always supported well within trap. This is automatically true for β = 1 because µ/ε = N −1 , but we require a somewhat stronger condition to handle the Gross-Pitaevskii scaling (see Remark 5). This leads to the additional moderate confinement condition for β = 1 with parameter γ > 1, which is clearly a weaker restriction for smaller γ.
The upper bound Γ < Θ is necessary to ensure the mutual compatibility of admissibility and moderate confinement. From a technical point of view, the moderate confinement condition allows us to compensate for certain powers of ε −1 in terms of powers of N −1 , while the admissibility condition admits the control of powers of N by powers of ε.
To visualise the restrictions due to admissibility and moderate confinement, we plot in Figure 1 the largest possible subset of the parameter space N × [0, 1] which can be covered by our analysis. A sequence (N, ε) → (∞, 0) passes through this space from the top right to the bottom left corner. The two boundaries correspond to the two-stage limits where first N → ∞ at constant ε and subsequently ε → 0, and vice versa. The edge cases are not contained in our model.
The sequences (N, ε) → (∞, 0) within the dark grey region in Figure 1 are covered by our analysis and yield an NLS or Gross-Pitaevskii equation, respectively. Naturally, these restrictions are meaningful only for sufficiently large N and small ε, which implies that mainly the section of the plot around the bottom left corner is of importance. The white region in figures (a) to (c) is excluded from our analysis by the admissibility condition. In figure (d), there is an additional prohibited region due to moderate confinement. Note that Chen and Holmer impose constraints which are weaker for small β and stronger for larger β ∈ (0, 2 5 ), which are discussed in Remark 3 and plotted in Figure 2.
The light grey region in Figure 1, which is present for β ∈ (0, 1), is not contained in Theorem 1 as a consequence of the moderate confinement condition. We expect the dynamics in this region to be described by an effective equation with coupling parameter b β = 0 since it corresponds to the condition ε/µ β 1, implying that the the confinement shrinks much faster than the interaction. Consequently, the interaction is predominantly supported in a region that is essentially inaccessible to the bosons, which results in a free evolution equation. For β < 1 3 and a cigar-shaped confinement by Dirichlet boundary conditions, this was shown in [22].
As mentioned above, we will consider interactions in the NLS scaling regime β ∈ (0, 1) which are of a more generic form than (3).
Throughout the paper, we will use two notions of one-particle energies: To make the moderate confinement condition Γ = 1 + for β = 1 visible, we implemented it as Γ = 1.01. Theorem 1 applies in the dark grey area, while the white region is excluded from our analysis. In the light grey part, we expect the dynamics to be effectively described by a free evolution equation. Plotted with Matplotlib [19].
• The "renormalised" energy per particle: for ψ ∈ D(H µ,β (t) where E 0 denotes the lowest eigenvalue of − d 2 dy 2 + V ⊥ (y). By rescaling, the lowest eigenvalue of − d 2 • The effective energy per particle: for Φ ∈ H 1 (R 2 ) and b ∈ R, We can now state our assumptions: A1 Interaction potential.
A2 Confining potential. Let V ⊥ : R → R such that − d 2 dy 2 + V ⊥ is self-adjoint and has a non-degenerate ground state χ with energy E 0 < inf σ ess (−∆ y + V ⊥ ). Assume that the negative part of V ⊥ is bounded and that χ ∈ C 2 b (R), i.e., χ is bounded and twice continuously differentiable with bounded derivatives. We choose χ normalised and real.
Further, assume that for each fixed t ∈ R, A4 Initial data. Let (N, ε) → (∞, 0) be admissible and moderately confining with parameters (Θ, Γ) β as in (8). Assume that the family of initial data ψ N,ε Further, let In our main result, we prove the persistence of condensation in the state ϕ ε (t) = Φ(t)χ ε for initial data ψ N,ε 0 from A4. Naturally, we are interested in times for which the condensate wave function Φ(t) exists, and, moreover, we require H 4 (R 2 )-regularity of Φ(t) for the proof. Let us therefore introduce the maximal time of H 4 (R 2 )-existence, where Φ(t) is the solution of (7) with initial datum Φ 0 from A4. Remark 1. The regularity of the initial data is for many choices of V propagated by the evolution (7). For several classes of external potentials, global existence in H 4 (R 2 )-sense and explicit bounds on the growth of Φ(t) H 4 (R 2 ) are known: • The case without external field, V = 0, was covered in [34, Corollary 1.3]: for initial for all t ∈ R. If the initial data are further restricted to the set the bound is even uniform in t ∈ R. This is, for Φ 0 ∈ Σ k , there exists C > 0 such that for all t ∈ R [7, Section 1.2].
• For time-dependent external potentials V (t, (x, 0)) that are at most quadratic in x uniformly in time, global existence of H k (R 2 )-solutions with double exponential growth was shown in [7,Corollary 1.4] for initial data Φ 0 ∈ Σ k : for all t ∈ R. In case of a time-independent harmonic potential and initial data Φ 0 ∈ Σ k , this can be improved to an exponential rather than double exponential bound. Note, however, that unbounded potentials V (t, z) are excluded by assumption A3.
where the limits are taken along the sequence from A4. Here, Φ(t) is the solution of (7) with initial datum Φ(0) = Φ 0 from A4 and with coupling parameter is exponentially localised on a scale of order ε for any potential V ⊥ satisfying A2. Valid examples for V ⊥ are harmonic potentials or smooth, bounded potentials that admit at least one bound state below the essential spectrum.
(b) Due to assumptions A1-A3, the Hamiltonian H µ,β (t) is for any t ∈ R self-adjoint on its time-independent domain D(H µ,β ). Since we assume continuity of t → V (t) ∈ L(L 2 (R 3 )), [16] implies that the family {H µ,β (t)} t∈R generates a unique, strongly continuous, unitary time evolution that leaves D(H µ,β ) invariant. By imposing the further assumptions on V , we can control the growth of the one-particle energies and the interactions of the particles with the external potential. Note that it is physically important to include time-dependent external traps, since this admits non-trivial dynamics even if the system is initially prepared in an eigenstate.
(c) Assumption A4 states that the system is initially a Bose-Einstein condensate which factorises in a longitudinal and a transverse part. In [32, Theorems 1.1 and 1.3], Schnee and Yngvason prove that both parts of the assumption are fulfilled by the ground state of H µ,β (0) for β = 1 and V (0, z) = V (x) with V locally bounded and diverging as |x| → ∞.
(d) The situation of a strong confinement in two directions is studied in [4,5]. Our proof can be understood as an adaptation of these works, and we summarise the mathematical differences in Remarks 4 and 5.
(e) Our proof yields an estimate of the rate of the convergence (16). Since we did not focus on obtaining an optimal rate, we do not state it explicitly. However, it can be recovered from the bounds in Propositions 3.6 and 3.11 by optimising over the parameters.  (8)). To conclude this section, let us discuss these constraints: • By (8), the weakest possible constraints are given by (Θ, Γ) β = ( 3 β − , 1 β ) for β ∈ (0, 1) and (Θ, Γ) 1 = (3, 1 + ) for β = 1. Instead of choosing these least restrictive values, we present Theorem 1 and all estimates in explicit dependence of the parameters Θ and Γ, making it more transparent where the conditions enter the proof. Moreover, the rate of convergence improves for more restrictive choices of the parameters Γ and Θ.
• In [8], Chen and Holmer prove Theorem 1 for the regime β ∈ (0, 2 5 ) under different assumptions on the sequence (N, ε). The subset of the parameter range N×[0, 1] covered by their analysis is visualised in Figure 2. While no admissibility condition is required for their proof, they impose a moderate confinement condition which is equivalent to our condition for β ∈ (0, 3 11 ]. For larger β ∈ ( 3 11 , 2 5 ), they restrict the parameter range much stronger 1 , and their condition becomes so restrictive with increasing β that it limitates the range of scaling parameters to β ∈ (0, 2 5 ).
• No restriction comparable to the admissibility condition is needed for the ground state problem in [32]. Given the work [28] where the strong confinement limit of the threedimensional NLS equation is taken, this suggests that our result should hold without any such restriction. However, for the present proof, the condition is indispensable (see Remarks 4 and 5).
• As argued above, the moderate confinement condition for β ∈ (0, 1) is optimal, in the sense that we expect a free evolution equation if µ β /ε → ∞. For β = 1, we require that µ/ε γ → 0 for γ > 1. Note that the choice γ = 1 would mean no restriction at all because µ/ε = N −1 . Our proof works for γ that are arbitrarily close to 1. However, since the estimates are not uniform in γ, the case γ = 1 is excluded.
• Although no moderate confinement condition is required to derive the one-dimensional Gross-Pitaevskii equation in the cigar-shaped case [5], our analysis covers a considerably larger subset of the parameter space N × [0, 1] than is included in [5]. In that work, the admissibility condition is given as N ε 2 5 − → 0, which is much more restrictive than our condition.

Proof of the main result
The proof of Theorem 1, both for the NLS scaling β ∈ (0, 1) and the Gross-Pitaevskii case β = 1, follows the approach developed by Pickl in [30]. The main idea is to avoid a direct estimate of the differences in (16) and (17), but instead to define a functional in such a way that Physically, the functional α < ξ, w µ,β measures the part of the wave function ψ N,ε (t) that remains outside the condensed phase ϕ ε (t), and is therefore also referred to as a counting functional. The index ξ is a parameter which is required for technical reasons and will be defined below. The index w µ,β indicates that the evolutions of ψ N,ε (t) and ϕ ε (t) are generated by H µ,β (t) and h β (t), which depend, directly or indirectly, on the interaction w µ,β . To define the functional α < ξ, w µ,β , we recall the projectors onto the condensate wave function that were introduced in [29,22]: is the solution of the NLS equation (7) with initial datum Φ 0 from A4 and with χ ε as in (6). Let where we drop the t-and ε -dependence of p in the notation. For i ∈ {1, . . . , N }, define the projection operators on L 2 (R 3N ) Further, define the orthogonal projections on L 2 (R 3 ) and define p Φ j , q Φ j , p χ ε j and q χ ε j on L 2 (R 3N ) analogously to p j and q j . Finally, for 0 ≤ k ≤ N , define the many-body projections P k = q 1 · · · q k p k+1 · · · p N sym := J⊆{1,...,N } |J|=k j∈J q j l / ∈J p l and P k = 0 for k < 0 and k > N . Further, for any function f : In the sequel, we will make use of the following weight functions: and, for some ξ ∈ (0, 1 2 ), Further, define the weight functions m : The corresponding weighted many-body operators are denoted by m . Finally, define Note that m equals n with a smooth, ξ-dependent cut-off to soften the singularity of dn dk for small k.
The expression ⟪ψ N,ε (t), mψ N,ε (t)⟫ is a suitably weighted sum of the expectation values of P k ψ N,ε (t), i.e., of the parts of ψ N,ε (t) with k particles outside ϕ ε (t). As m(0) ≈ 0 and m is increasing, P k ψ N,ε (t) with larger k contribute more to α < ξ, w µ,β (t) than P k ψ N,ε (t) with smaller k. It is well known that the convergence ⟪ψ N,ε (t), mψ N,ε (t)⟫ → 0 is equivalent to the convergence (16) of the one-particle reduced density matrix of ψ N,ε (t) to |ϕ ε (t) ϕ ε (t)|. Hence, the convergence α < ξ, w µ,β (t) → 0 is equivalent to (16) and (17). The relation between the respective rates of convergence is stated in the following lemma, whose proof is given in [4, Lemma 3.6]: The strategy of our proof is to derive a bound for | d dt α < ξ, w µ,β (t)|, which leads to an estimate of α < ξ, w µ,β (t) by means of Grönwall's inequality. The first step is therefore to characterise the expressions arising from this derivative.
The term γ a,< summarises all contributions from interactions between the particles and the external field V , while γ b,< collects all contributions from the mutual interactions between the bosons. The latter can be subdivided into four parts: b,< and γ (4) b,< contain the quasi two-dimensional interaction w µ,β (x 1 − x 2 ) resulting from integrating out the transverse degrees of freedom in w µ,β , which is given as b,< and γ (4) b,< can be understood as two-dimensional analogue of the corresponding expressions in the three-dimensional problem without confinement [30,Lemma A.4], and the estimates are inspired by [30]. Note that γ (1) b,< contains the difference between the quasi two-dimensional interaction potential w µ,β and the effective one-body potential b β |Φ(t)| 2 , which means that it vanishes in the limit (N, ε) → (∞, 0) only if (7) with coupling parameter b β is the correct effective equation. The last line (32) of γ (4) b,< contains merely the effective interaction potential b β |Φ(t)| 2 instead of the pair interaction w µ,β , hence, it is easily controlled.
• γ (2) b,< and γ (3) b,< are remainders from the replacement w µ,β → w µ,β , hence they have no three-dimensional equivalent. They are comparable to the expression γ in [4] from the analogous replacement of the originally three-dimensional interaction by its quasi one-dimensional counterpart.
The second step is to control γ a,< to γ (4) b,< in terms of α < ξ, w µ,β (t) and by expressions that vanish in the limit (N, ε) → (∞, 0). To write the estimates in a more compact form, let us define the function where Φ(t) denotes the solution of (7) with initial datum Φ 0 from A4. Note that e β (t) is bounded uniformly in N and ε because the only (N, The function e β is particularly useful since for any t ∈ [0, T ex V ) by the fundamental theorem of calculus. Note that for a time-independent external field V , e 2 β (t) 1 as a consequence of Remark 1, hence E . Proposition 3.6. Let β ∈ (0, 1) and assume A1 -A4 with parameters β and η in A1 and Then, for sufficiently small µ, the terms γ a,< to γ The estimates of γ a,< , γ (1) b,< and γ (2) b,< work analogously to the corresponding bounds in [4] and are briefly summarised in Sections 5.2.1 and 5.2.2. While γ a,< is easily bounded since it contains only one-body contributions, the key for the estimate of γ is that for sufficiently large N and small ε, due to sufficient regularity of ϕ ε and since the support of w µ,β shrinks as µ β . For this argument, it is crucial that the sequence (N, ε) is moderately confining.
The main idea to control γ (2) b,< is an integration by parts, exploiting that the antiderivative of w µ,β is less singular than w µ,β and that ∇ψ N,ε (t) can be controlled in terms of the To this end, we define the function h ε as the solution of the equation ∆h ε = w µ,β on a three-dimensional ball with radius ε and Dirichlet boundary conditions and integrate by parts on that ball. To prevent contributions from the boundary, we insert a smoothed step function whose derivative can be controlled (Definition 5.1). To make up for the factors ε −1 from the derivative, one observes that all expressions in γ , which follows since the spectral gap between ground state and excitation spectrum grows proportionally to ε −2 , the projections q χ ε provide the missing factors ε. The second main ingredient is the admissibility condition, which allows us to cancel small powers of N by powers of ε gained from q χ ε .
(b) For γ (3) b,< , this strategy of a three-dimensional integration by parts does not work: whereas q χ ε cancels the factor ε −1 from the derivative, we do not gain sufficient powers of ε to compensate for all positive powers of N . Note that this problem did not occur in [4], where the ratio of N and ε was different. 2 To cope with γ (3) b,< , note that both (28) and (29) contain the expression p χ ε 1 w (12) µ,β p χ ε 1 , which, analogously to w µ,β , defines a function w µ,β (x 1 − x 2 , y 2 ) where one of the yvariables is integrated out (Definition 5.4). We integrate by parts only in the x-variable, which has the advantages that ∇ x does not generate factors ε −1 and that the x-antiderivative of w µ,β (·, y) diverges only logarithmically in µ −1 (Lemma 5.6b). Due to admissibility and moderate confinement condition, this can be cancelled by any positive power of ε or N −1 . In distinction to γ (2) b,< , we do not integrate by parts on a ball with Dirichlet boundary conditions but instead add and subtract suitable counter-terms as in [30] and integrate over R 2 . Note that one obtains the same result when choosing the other path, but in this way the estimates are easily transferable to γ (4) b,< (see below). 2 In the 3d → 1d case [4], the range of the interaction scales as µ β 1d = (ε 2 /N ) β , besides χ ε 1d (y) = ε −1 χ 1d (y/ε), and the admissibility condition reads ε 2 /µ β 1d → 0. These slightly different formulas lead to the esti- (Lemma 5.2). Following the same path as in γ (2) b,< , e.g., for (28) (corresponding to (21) in [4]), we obtain in the 1d problem the estimate ∼ N β 2 ε 1−β = (ε 2 /µ β 1d ) 1 2 , which can be controlled by the respective admissibility condition. As opposed to this, we compute in our case that (28) 2 , which diverges due to moderate confinement.
(c) Finally, to estimate γ (4) b,< (Section 5.2.5), we define w µ,β as above and integrate by parts in x, using an auxiliary potential v ρ analogously to v ρ (Definition 5.4). To cope with the logarithmic divergences from the two-dimensional Green's function, we integrate by parts twice, following an idea from [30]. This is the reason why we defined h β ,ρ and h β ,ρ on R 2 and not on a ball, which would require the use of a smoothed step function. While the results are the same when integrating by parts only once, it turns out that the additional factors ρ −1 from a second derivative hitting the step function cannot be controlled sufficiently well.
For (31), the bound ∇ x 1 ψ N,ε (t) 2 1 from a priori energy estimates is insufficient, comparable to the situation in [30] and [4]. Instead, we require an improved bound on the kinetic energy of the part of ψ N,ε (t) with at least one particle orthogonal to Φ(t), . The rigorous proof of this bound (Lemma 5.7) is an adaptation of the corresponding Lemma 4.21 in [4] and requires the new strategies described above, as well as both moderate confinement and admissibility condition.

The Gross-Pitaevskii case β = 1
For an interaction w µ in the Gross-Pitaevskii scaling regime, the previous strategy, i.e., deriving an estimate of the form | d dt α < ξ,wµ (t)| α < ξ,wµ (t) + O(1), cannot work. To understand this, let us analyse the term γ (1) b,< , which contains the difference between the quasi two-dimensional interaction w µ,β and the effective potential b 1 |Φ(t)| 2 . As pointed out in Remark 4a, the basic idea here is to expand |ϕ ε (z 1 − z 2 )| 2 around z 2 , which can be made rigorous for sufficiently regular ϕ ε and yields Whereas this equals (at least asymptotically) the coupling parameter b β for β ∈ (0, 1), the situation is now different since b 1 = 8πa |χ(y)| 4 dy. In order to see that (34) and b 1 are not asymptotically equal, but actually differ by an error of O(1), let us briefly recall the definition of the scattering length and its scaling properties.
The zero energy scattering equation for the interaction w µ = µ −2 w(·/µ) is By [27, Theorems C.1 and C.2], the unique solution j µ ∈ C 1 (R 3 ) of (35) is spherically symmetric, non-negative and non-decreasing in |z|, and satisfies The parameter a µ ∈ R in (36) defines the scattering length of w µ . Equivalently, From the scaling behaviour of (35), it is obvious that j µ (z) = j µ=1 (z/µ) and that where a denotes the scattering length of the unscaled interaction w. Returning to the original question, this implies that and consequently where we have used that w µ L 1 (R 3 ) = µ w L 1 (R 3 ) and that j µ (z) is continuous and nondecreasing, hence j µ (z) ≤ j µ (µ) for z ∈ supp w µ and 1 − j µ (µ) ≈ a. In conclusion, the contribution from γ b,< does not vanish if b 1 is the coupling parameter in [4]. Naturally, one could amend this by taking |χ(y)| 4 dy w L 1 (R 3 ) instead of b 1 as parameter in the non-linear equation. However, for this choice, the contributions from γ (2) b,< to γ (4) b,< would not vanish in the limit (N, ε) → (∞, 0), as can easily be seen by setting β = 1 in Proposition 3.6.
The physical reason why the Gross-Pitaevskii scaling is fundamentally different -and why it requires a different strategy of proof -is the fact that the length scale a µ of the inter-particle correlations is of the same order as the range µ of the interaction. In contrast, for β ∈ (0, 1), the relation a µ,β µ β implies that j µ,β ≈ 1 on the support of w µ,β , hence the first order Born approximation 8πa µ,β ≈ w µ,β L 1 (R 3 ) applies in this case.
Before explaining the strategy of proof for the Gross-Pitaevskii scaling, let us introduce the auxiliary function f β ∈ C 1 (R 3 ). This function will be defined in such a way that it asymptotically coincides with j µ on supp w µ but, in contrast to j µ , satisfies f β (z) = 1 for sufficiently large |z|, which has the benefit of 1 − f β and ∇f β being compactly supported. To construct f β , we define the potential U µ, β such that the scattering length of w µ − U µ, β equals zero, and define f β as the solution of the corresponding zero energy scattering equation: where β is the minimal value in (µ β , ∞] such that the scattering length of w µ − U µ, β equals zero. Further, let f β ∈ C 1 (R 3 ) be the solution of and define In the sequel, we will abbreviate In [5,Lemma 4.9], it is shown by explicit construction that a suitable β exists and that it is of order µ β . Note that Definition 3.7 implies in particular that which is an equivalent way of expressing that the scattering length of w µ − U µ, β equals zero. Heuristically, one may think of the condensed N -body state as a product state that is overlaid with a microscopic structure described by f β , i.e., For β ∈ (0, 1), it holds that f β ≈ 1, i.e., the condensate is approximately described by the product (ϕ ε ) ⊗N -which is precisely the state onto which the operator P 0 = p 1 ···p N projects. For the Gross-Pitaevskii scaling, however, f β is not approximately constant, and the product state is no appropriate description of the condensed N -body wave function. The idea in [30] is to account for this in the counting functional by replacing the projection P 0 onto the product state by the projection onto the correlated state ψ cor . In this spirit, one substitutes the expression ⟪ψ, mψ⟫ in α < ξ, w µ,β (t) by ⟪ψ, where we expanded f β = 1 − g β and kept only the terms which are at most linear in g β . This leads to the following definition: The new functional α ξ,wµ (t) equals α < ξ, wµ (t) up to a correction term. Since the convergence of α < ξ, wµ (t) is equivalent to (16) and (17), an estimate of α ξ,wµ (t) is only meaningful if this correction converges to zero as (N, ε) → (∞, 0). This is the reason why we defined it using the operator r (Definition 3.2) instead of m: as r contains additional projections p 1 and p 2 , we can use the estimate g 2). In the following proposition, it is shown that this suffices for the correction term to vanish in the limit.
Proposition 3.9. Assume A1 -A4. Then . By adding the correction term to α < ξ, wµ (t), we effectively replace w µ by U µ, β f β in the time derivative of α < ξ, wµ (t). To explain what is meant by this statement, let us analyse the contributions to the time derivative of α ξ,wµ (t), which are collected in the following proposition: γ e (t) : Here, we have used the abbreviations The proof of this proposition is given in Section 6.5. Note that the contributions to the derivative d dt α ξ,wµ (t) fall into two categories: • The terms (42)-(43) in γ < equal γ a,< from Proposition 3.5, and (44) is exactly γ b,< with interaction potential U µ, β f β . Hence, estimating γ < is equivalent to estimating the functional α < ξ,U µ, β f β (t), which arises from α < ξ, wµ (t) by replacing the interaction w µ by U µ, β f β . Since U µ, β f β ∈ W β,η for any η ∈ (0, 1 − β) (Lemma 6.4), this is an interaction in the NLS scaling regime, which was covered in the previous section.
• γ a to γ f can be understood as remainders from this substitution. γ a collects the contributions coming from the fact that the N -body wave function interacts with a threedimensional external trap V , while only V evaluated on the plane y = 0 enters in the effective equation (7). Since this is an effect of the strong confinement, it has no equivalent in the three-dimensional problem [30], but the same contribution occurs in the situation of a cigar-shaped confinement [5]. The terms γ b to γ f are analogous to the corresponding expressions in [30] and [5].
The physical idea behind the replacement is that low-energy scattering at any potential is to leading order described by the scattering length. Note that f β ≈ 1 on supp U µ, β , hence U µ, β ≈ U µ, β f β and consequently the scattering length of w µ,β − U µ, β f β is approximately zero by construction (40). This implies that a sufficiently distant test particle with very low energy cannot resolve the difference between the two potentials. .

t). On the other hand, heuristic arguments
For the Gross-Pitaevskii scaling of the interaction, ∇ x 1 q Φ 1 ψ N,ε (t) 2 is not asymptotically zero because the microscopic structure described by f β lives on the same length scale as the interaction and thus contributes a kinetic energy of O(1). However, as this kinetic energy is concentrated around the scattering centres, one can show a similar bound for the kinetic energy on a subset A 1 of R 3N , where appropriate holes around these centres are cut out (Definition 6.5). This is done in Section 6.3, where we show in Lemma 6.7 that The proof of this lemma is similar to the corresponding proof in [5,Lemma 4.12], which, in turn, adjusts ideas from [30] to the problem with dimensional reduction. However, since one key tool for the estimate is the Gagliardo-Nirenberg-Sobolev inequality in the x-coordinates, the estimates depend in a non-trivial way on the dimension of x. As one consequence, our estimate requires the moderate confinement condition with parameter γ > 1, where no such restriction was needed in [5].
Finally, we adapt the estimate of (31). In distinction to the corresponding proof in [5, Section 4.5.1], we need to integrate by parts in two steps to be able to control the logarithmic divergences that are due to the two-dimensional Green's function. Inspired by an idea in [30], we introduce two auxiliary potentials v µ β 2 and ν 1 3 See [5, pp. 1019-1020]. Essentially, when evaluated on the trial function ψcor from (41), the energy difference is to leading order given by N ⟪ψcor(t), (w The expressions depending on ν 1 can be controlled immediately, while we integrate the remainders by parts in x, making use of different properties of h β ,µ β 2 and h µ β 2 ,1 (Lemma 5.6b). Subsequently, we insert identi- On the one hand, this yields , which can be controlled by the new energy lemma (Lemma 6.7).
On the other hand, we obtain terms containing 1 A 1 , which we estimate by exploiting the smallness of A 1 . The full argument is given in Section 6.6.1.
(b) The remainders γ a to γ f are estimated in Sections 6.6.2, and work, for the most part, analogously to the corresponding proofs in [5, Sections 4.5.2 -4.5.7]. The only exception is γ c , where the strategy from [5] produces too many factors ε −1 . Instead, we estimate the x-and y-contributions to the scalar product (∇g β ) · ∇ r = (∇ x g β ) · ∇ x r + (∂ y g β )∂ y r separately. To control the y-part, we integrate by parts in y and use the moderate confinement condition with γ > 1. Again, this is different from the situation in [5], where the corresponding term γ c could be estimated without any restriction on the sequence (N, ε).

Preliminaries
We will from now on always assume that assumptions A1 -A4 are satisfied. , Proof.
(b) Define Q 0 := p j , Q 1 := q j , Q 0 := p i p j , Q 1 ∈ {p i q j , q i p j } and Q 2 := q i q j . Let S j be an operator acting non-trivially only on coordinate j and T ij only on coordinates i and j. Then for µ, ν ∈ {0, 1, 2} Proof. [4], Lemma 4.2.
(a) The operators P k and f are continuously differentiable as functions of time, i.e., where h (j) β (t) denotes the one-particle operator corresponding to h β (t) from (7) acting on the j th coordinate.
Lemma 4.6. Let Γ, Λ ∈ L 2 (R 3N ) ∈ H M such that j / ∈ M and k, l ∈ M with j = k = l = j. Let O j,k be an operator acting non-trivially only on coordinates j and k, denote by r k and s k operators acting only on k th coordinate, and let F : Proof. [4], Lemma 4.8 and [5], Lemma 4.4.
Then for sufficiently small ε, Proof. Part (a) follows from the Sobolev embedding theorem [1, Theorem 4.12, Part I A] and by definition of e β . Part (b) is an immediate consequence of (6), and part (c) is implied by (a) and (b).

Proof of Proposition 3.5
The proof works analogously to the proof of Proposition 3.7 in [4] and we provide only the main steps for convenience of the reader. From now on, we will drop the time dependence of Φ, ϕ ε and ψ N,ε in the notation and abbreviate ψ N,ε ≡ ψ. The time derivative of α < ξ, w µ,β (t) is bounded by For the second term in (56), note that which follows from Lemmas 4.3 and 4.4. Expanding q = q χ ε + p χ ε q Φ in (58) to (60) and subsequently estimating N m a −1 ≤ l and N m b −2 ≤ l for l ∈ L from (19) concludes the proof.

Proof of Proposition 3.6
In this section, we will again drop the time dependence of ψ N,ε (t), ϕ ε (t) and Φ(t) and abbreviate ψ N,ε ≡ ψ. Besides, we will always take l ∈ L from (19), hence Lemma 4.2 implies the bounds l op N ξ , l d q 1 ψ 1 for d ∈ Z.

Estimate of γ
The key idea for the estimate γ (2) b,< (t) is to integrate by parts on a ball with radius ε, using a smooth cut-off function to prevent contributions from the boundary.
We now use this lemma to estimate γ where the boundary terms upon integration by parts vanish because H ε (|z|) = 0 for |z| = ε, and where we have used Lemmas 4.6, 4.2, 4.8, 4.9a and 5.2. Similarly, one computes The bound for γ (2) b,< follows from this because N ξ µ and since the admissibility condition implies for ξ ≤ 3−δ 2 · β δ−β that

Preliminary estimates for the integration by parts
To control γ b,< (t) and γ (4) b,< (t), we define the quasi two-dimensional interaction potentials w µ,β (x 1 − x 2 , y 1 ) and w µ,β (x 1 − x 2 ), which result from integrating out one or both transverse variables of the three-dimensional pair interaction w µ,β (z 1 − z 2 ), and integrate by parts in x. In this section, we provide the required lemmas and definitions in a somewhat generalised form, which allows us to directly apply the results in Sections 5.2.4, 5.2.5, 5.3 and 6.6.1.
Further, define the set Note that supp ω σ ⊆ {x ∈ R 2 : |x| ≤ σ} and, since χ ε is normalised, the estimates for the norms of ω σ coincide with the respective estimates for ω σ . Next, we define the quasi twodimensional interaction potentials w µ,β and w µ,β as well as the auxiliary potentials needed for the integration by parts, and show that they are contained in the sets V σ and V σ , respectively, for suitable choices of σ.
Definition 5.4. Let w µ,β ∈ W β,η for some η > 0 and define It can easily be verified that w µ,β and v ρ can equivalently be written as Besides, note that Lemma 5.5. For w µ,β , w µ,β , v ρ and v ρ from Definition 5.4, it holds that Proof. It suffices to derive the respective estimates for w µ,β (·, y) and v ρ (·, y) uniformly in y ∈ R. For instance, Lemma 4.7 and (55) yield and the remaining parts are verified analogously.

Estimate of γ
To derive a bound for γ (3) b,< , observe first that both terms (28) and (29) contain the interaction w µ,β . We add and subtract v ρ from Definition 5.4 for suitable choices of ρ, i.e., by Lemma 5.6, which is applicable by Lemma 5.5.

Microscopic structure
This section collects properties of the scattering solution f β and its complement g β .
6.2 Characterisation of the auxiliary potential U µ, β In this section, we show that both U µ, β f β and U µ, β from Definition 3.7 are contained in the set W β,η from Definition 2.2, which admits the transfer of results obtained in Section 5 to these interaction potentials.
Proof. As before, it only remains to show that U µ, β f β satisfies part (d) of Definition 2.2. To see this, observe that R |χ(y)| 4 dy. By Lemma 6.1b, this implies and 6.3 Estimate of the kinetic energy for β = 1 The main goal of this section is to provide a bound for the kinetic energy of the part of ψ N,ε (t) with at least one particle orthogonal to Φ(t). Since the predominant part of the kinetic energy is caused by the microscopic structure and thus concentrated in neighbourhoods of the scattering centres, we will consider the part of the kinetic energy originating from the complement of these neighbourhoods and prove that it is subleading. The first step is to define the appropriate neighbourhoods C j as well as sufficiently large balls A j ⊃ C j around them.
Definition 6.5. Let max γ+1 2γ , 5 6 < d < β, j, k ∈ {1, ..., N }, and define Then the subsets A j , B j , C j and A and their complements are denoted by A j , B j , C j and A x j , e.g., A j := R 3N \ A j .
The sets A j and A x j contain all N -particle configurations where at least one other particle is sufficiently close to particle j or where the projections in the x-direction are close, respectively. The sets B j consist of all N -particle configurations where particles can interact with particle j but are mutually too distant to interact among each other.
Note that the characteristic functions 1 A x 1 and 1 A x 1 do not depend on any y-coordinate, and 1 B 1 and 1 B 1 are independent of z 1 . Hence, the multiplication operators corresponding to these functions commute with all operators that act non-trivially only on the y-coordinates or on z 1 , respectively. Some useful properties of these cut-off functions are collected in the following lemma.