Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement

We consider the three-dimensional time-dependent Gross-Pitaevskii equation arising in the description of rotating Bose-Einstein condensates and study the corresponding scaling limit of strongly anisotropic confinement potentials. The resulting effective equations in one or two spatial dimensions, respectively, are rigorously obtained as special cases of an averaged three dimensional limit model. In the particular case where the rotation axis is not parallel to the strongly confining direction the resulting limiting model(s) include a negative, and thus, purely repulsive quadratic potential, which is not present in the original equation and which can be seen as an effective centrifugal force counteracting the confinement.


Introduction and main result
We are interested in the dimension reduction problem arising in the description of rotating Bose-Einstein condensates with strongly anisotropic confinement potential. In physics experiments such potentials are used to obtain effective onedimensional (called cigar-shaped) or two-dimensional (called pancake-shaped) condensates which, among other features, exhibit different stability and instability properties than the usual three dimensional case (for a general introduction to the physics of Bose-Einstein condensates, see, e.g., [16,17]). The present work aims to give a rigorous justification to the use of these approximate lower-dimensional models. In comparison with earlier studies in the mathematics literature, see [2,3,4,5,6], the main novelty in our work is the presence of an additional angular momentum rotation term, whose strong interaction with the confinement will, in general, result in a nontrivial effect within the limiting model obtained.
The starting point of our investigation is the three-dimensional Gross-Pitaevskii equation, describing the Bose-Einstein condensate in a mean-field approximation, cf. [14,15,17]. Rescaled into dimensionless form (see, e.g., [6]) and in a rotating reference frame (which is customary used throughout the literature), this nonlinear Schrödinger equation (NLS) reads with an initial data ψ(t = 0, x) = ε −1/2 ψ 0 (x, z/ε) and where Ω ≡ (Ω 1 , Ω 2 , Ω z ) is a given vector of R 3 , describing the rotation axis. Here, the exponent of the nonlinearity is assumed to be 1 ≤ σ < 2. In particular, this means that the nonlinearity is H 1 (R 3 )-subcritical, cf. [9]. Note that this includes the cubic case σ = 1, which is the physically most relevant situation. The space variable x ∈ R 3 splits into x = (x, z) ∈ R 2 × R, with x ≡ (x 1 , x 2 ). Finally, we assume that ε ∈ (0, 1] is a small parameter describing the anisotropy within the confining potential. In the following we shall be interested in the limit ε → 0 for solutions to (1.1). Note that we thereby assume that the initial wave function is already confined at the scale epsilon in the z-direction (an assumption which is consistent with the asymptotic limiting regime considered) and such that its total mass ψ(t = 0, ·) 2 L 2 = 1, uniformly in ε. Let us rescale the variables. We set x ′ , z ′ ) = ε 1/2 ψ t, x ′ , εz ′ and assume that β ε = λε σ , where λ ∈ R is fixed. We are thus in a weak interaction regime similar to [5,6]. Under this rescaling the NLS becomes (dropping the primes in the variables for simplicity) with ψ ε (t = 0, x, z) = ψ 0 (x, z). Here, and in the following, we denote where L z is the angular momentum operator associated to a rotation around the negative z-axis.
In order to get rid of the singular rotation term proportional to ε −1 in (1.2), we shall invoke the following (unitary) change of unknown. Setting ψ ε (t, x, z) = e iεz(Ω 1 x 2 −Ω 2 x 1 ) u ε (t, x, z), (1.3) we obtain subject to initial data Note that in (1.4), the only singular term left is ε −2 H z . We consequently expect that by filtering the associated rapid oscillations the function e itHz /ε 2 u ε (t) will converge to some finite limit φ(t), as ε → 0.
A suitable functional framework for the analysis of our problem is the scale of Sobolev spaces adapted to H z and H x . For any real number s ≥ 0, we denote According to [12,5], this Hilbert space can be equipped with the following equivalent norms: It is also usefull to recall that, for all 0 ≤ ℓ ≤ s, we have For s > 3/2, Σ s is an algebra. Moreover, the self-adjoint operators H z and H x generate the groups of isometries θ → e iθHz and θ → e iθHx on any Σ s , s ≥ 0.
To derive the limit model as ε → 0, we need to introduce the following nonlinear function: and study the behavior of F t/ε 2 , u , as ε → 0. For s > 3/2, Σ s is an algebra and it is readily seen that F ∈ C(R × Σ s , Σ s ). Moreover, since the spectrum of the quantum harmonic oscillator H z is {n + 1/2, n ∈ N}, the operator e iθ(Hz−1/2) is 2π periodic with respect to θ, so F is also 2π periodic with respect to θ. Denoting the average of this function by with the initial data φ(t = 0) = ψ 0 . The Gross-Pitaevskii type energy associated to this equation is where here and in the following, Note that (1.9) is still a model in three spatial dimensions. Except for the nonlinear averaging operator F av (φ), however, the variable z only enters as a parameter and hence the linear part of the dynamics with respect to z is, in fact, trivial. This allows to derive from (1.9) an effective two-dimensional limiting model, provided the initial data is polarized on a single mode of H z , see Corollary 1.2 below.
One should also note that in (1.9) there is a non-trivial effect due the presence of the rotation. Indeed, in the case where the rotation axis is not parallel to the z-axis, i.e., if Ω 1 , Ω 2 = 0, a repulsive quadratic potential is present in the limiting model (see also the discussion at the beginning of Section 2.1 below). The reason for this effect becomes apparent from the scaling of equation (1.2), which includes a rotation term of order O(ε −1 ). The latter becomes large in the limit of strong confinement ε → 0, resulting in an effective centrifugal force counteracting the original trap. In the physics literature, it seems that it is almost always assumed that the rotation axis is equal to the z-axis, and hence, this effect is almost never considered. We finally remark that, at least formally, a second order averaging procedure (similar to [4,10]) can be used to derive (1.9) from (1.2) directly. In order to make this procedure rigorous, though, uniform (in ε) energy estimates are needed which seem to be rather difficult to obtain on the level of (1.2) (given its singular scaling). Thus, instead of working with (1.2) directly, we use the change of variables (1.3) which yields the same effect as the second order averaging and also allows us to use the better behaved model (1.4).
This result follows from Theorem 1.1 (ii) and the fact that (1.9) preserves the initial polarization, i.e., admits solutions of the form φ(t, x, z) = ϕ(t, x)χ n (z). To see this, recall that the eigenfunctions {χ m } L 2 form an orthonormal basis of L 2 (R). Using this we can write However, λ m − λ n ∈ N and thus, this integral is identically zero unless m = n, for which it is equal to 2π. Thus, the whole sum collapses to one term only and we consequently obtain that in the case of polarized solutions, (1.9) reduces to (1.10). The latter is an effective two-dimensional model describing the degrees of freedom in the unconstrained direction.
The paper is organized as follows: In Section 2 we shall, as a first step, establish well-posedness of the Cauchy problem corresponding to both the three dimensional NLS (1.2) and the averaged limiting model (1.9). Once this is done, rigorous error estimates between the exact and the approximate solution will be established in Section 3. In there, we shall also indicate how to obtain an improved error estimate, provided ψ ε satisfies sufficiently strong regularity assumptions. Finally, in Section 4 we shall show how to adapt our results to the situation with strong confinement in two spatial dimensions, and derive the associated limiting model.

Analysis of the Cauchy problems
In this section we shall prove local and global well-posedness results for equation (1.2), i.e., the original NLS in d = 3 dimensions, and for the formal limiting model (1.9). The analysis of the former is relatively standard and follows along the lines of [1]. We shall therefore only sketch the main ideas and rather focus on the Cauchy problem corresponding to (1.9).
2.1. The Cauchy problem corresponding to the averaged NLS model. In this subsection we shall analyze the Cauchy problem with some general initial data φ(t = 0) = φ 0 ∈ Σ 1 . Recall that φ depends on the space variables x ∈ R 2 and z ∈ R, but in this problem dispersive effects only occur in the x direction (due to the lack of a second order derivative in z). The basic existence proof therefore requires several changes from the standard approach.
To this end, let us first derive Strichartz estimates adapted to the situation at hand. We recall that, in dimension two, a pair (q, r) is said to be admissible if 2 ≤ r < ∞ and We denote by U (t) = e itH the strongly continuous group of unitary operators generated by the Hamiltonian This operator can be seen as a special case of the Weyl-quantization of a (realvalued) second order polynomial H(x, ξ). It is thus essentially self-adjoint on [13]. In the case without rotation Ω z = 0, H is of the form of an anisotropic harmonic oscillator with potential Clearly, this potential becomes repulsive if Ω 1 , Ω 2 > 1. Physically speaking this results in a loss of confinement, and thus, the destruction of the condensate. On the other hand, by means of Young's inequality, one easily sees that a sufficient condition for confinement, i.e., V (x) → +∞ as |x| → ∞, is In this case H has only pure point spectrum with no finite accumulation points. Similarly, in the case with rotation Ω z = 0 the operator H remains confining provided Ω z is sufficiently small (with respect to Ω 1 , Ω 2 ). This can be seen by rewriting (2.2) in the form of a magnetic Schrödinger operator where V is as before and A(x) = Ω z x ⊥ with x ⊥ = (x 2 , −x 1 ). In this form, the effect of the rotation term L z has been split into Coriolis and centrifugal forces. The latter is seen to act as a repulsive quadratic potential, counteracting the confinement. Depending on the size of Ω 1 , Ω 2 , Ω z we thus might have de-confinement due to the combined effects of the rotation and the strong confinement. This also has an influence on the question of global existence of solutions to the NLS, see the Remark 2.3 below.
(i) For any admissible pair (q, r), there exists C such that, for all φ ∈ L 2 (R 3 ), Proof. This lemma relies on the usual Strichartz estimates for the group U (t) acting on functions depending only on x and on Minkowski's inequality. The existence of Strichartz estimates for U (t) thereby follows from the results in [13] which can be directly applied to the Hamiltonian given in (2.2). The fact that H in general will have eigenvalues, implies that dispersive effects will only be present for small |t| < δ, preventing the existence of global-in-time Strichartz estimates, cf. [7] for a more detailed discussion on this issue. Let us introduce a Hilbert basis (e p ) p∈N of L 2 z (R). Decomposing the function φ ∈ L 2 (R 3 ) as φ(x, z) = p φ p (x)e p (z), one obtains: . Here, we have used twice the Minkowski's inequality [11] in the third line (notice that q/2 ≥ 1 and r/2 ≥ 1) and the usual Strichartz estimate for each U (t)φ p in the fourth line. This proves (2.4).
Let us prove (2. we estimate similarly (we have again q/2 > 1 and r/2 ≥ 1) Here, we have used, in the fourth line, the Strichartz inequality for each g j and, in the fifty line, the reverse Minkowski's inequality [11] (note that we have necessarily γ ′ /2 < 1 and ρ ′ /2 < 1). The proof of Lemma 2.1 is complete.
Moreover, the following conservation laws hold Note that for φ(t) ∈ Σ 2 we have the strong form of the conservation law, i.e., be two weak solutions of (2.1), where, without loss of generality, we can assume that and then, by (2.5), for any admissible pairs (q, r) and (γ, ρ), Let us choose r > 2 and ρ > 2 such that 1 Then, denoting v = e iθHz u and v = e iθHz u, one gets by Hölder, Here, we have used the unitarity of e iθHz in L 2 z and Σ 1 , and the embeddings ) (see the Appendix of [6]). Since u and u belong to L ∞ ((0, T ), Σ 1 ), this yields Since γ ′ < 2 < q, this inequality is enough to conclude that u = u, see Lemma 4.2.2 in [9].
Step 2: local existence. Let us adapt to (2.1) the proof of well-posedness of NLS with a quadratic potential of [7], using Kato's strategy (see for instance [9]) and the vectorial Strichartz estimates given in Lemma 2.1. The main technical difficulty here is the fact that we are working with an NLS in three spatial dimensions, but we can only utilize the dispersive properties of the two-dimensional Schrödinger group U (t) = e itH . In order to remedy this, an important ingredient will be the anisotropic Sobolev imbeddings proved in [6].
With this in mind, local in-time existence for solutions in φ(t) ∈ Σ 1 can be proved by means of a fixed point theorem applied to Duhamel's representation of (1.9), i.e., where, as before, U (t) = e itH and H is given by (2.2). We want to show that for φ 0 ∈ Σ 1 and sufficiently small T > 0, Φ is a contraction mapping in the complete metric space z . The real numbers r, q, ρ, γ are taken as in Step 1 above and T , M are to be chosen later. To prove that X T,M is stable by Φ, one first checks that the commutator where V is as in (2.3), and hence ∇ x V is in fact linear in x. We consequently obtain that the combination of φ, xφ and ∇φ form a closed coupled system of equations. Therefore, by applying the operators x and ∇ to (2.1) and by using Lemma 2.1, we obtain for all φ ∈ X T,M , where a b stands for a ≤ Cb for some constant C > 0. Denoting v = e iθHz φ, one gets where we have used the fact that e iθHz is unitary in L 2 z and Σ 1 , together with a Hölder estimate and the embedding H 1 (R 3 ) ֒→ L 4 x (R 2 , L ∞ z (R)). Hence, Similarly, we obtain To estimate zF av (u) and ∂ z F av (u), we use (1.6) several times: which yields, again Finally, we have proved that We now set M = 2C φ 0 Σ 1 and choose T small enough so that It follows that Φ(φ) ∈ X T,M . The contraction property can then be proved by following the same lines as the proof of uniqueness in Step 1: it can be proved that, for T small enough, we have for all φ, φ ∈ X T,M . Hence, by Banach's fixed point theorem, Φ has a unique fixed point, which is a mild solution of (2.1).
Step 3: blow-up alternative. From the uniqueness result and from the fact that the existence time in Step 1 only depends on φ 0 Σ 1 , one can define the maximal solution φ ∈ C([0, T max ), Σ 1 ) and obtain a first blow-up alternative in terms of the whole Σ 1 norm: Then, we compute from (2.1) Hence, a bound on ∇ x φ L 2 yields a bound on xφ(t) L 2 by the Gronwall lemma.
Since the L 2 norm of φ is conserved, it is clear that lim t→Tmax φ(t) Σ 1 = +∞ implies that lim t→Tmax ∇φ(t) L 2 = +∞. We have proved the blow-up alternative as it is stated in the Proposition.
Step 4: conservation laws. In order to prove the conservation laws stated above, it is enough to consider the case of local-in-time solutions φ(t) which are sufficiently smooth and decaying. By following a standard regularization procedure (as given in, e.g., [9]), this can then be extended to general φ(t) ∈ Σ 1 . Conservation of mass then follows from the fact that H is self-adjoint and hence d dt However, z , e iθHz ] = 0, see [5]. Finally, in order to prove the conservation of the energy it is useful to first note that E(φ) is formally equal to We then compute d dt Hφ, φ L 2 = 2 Im F av (φ), From here, it follows that E(φ) is conserved provided that This, however, can be seen by a direct computation, using the definition of F av and N av , i.e., Step 5: Σ 2 regularity. Assume that φ 0 ∈ Σ 2 . Since Σ 2 is an algebra and e iθHz is unitary on Σ 2 , it is easy to see that F av is locally Lipschitz continuous on Σ 2 provided σ ≥ 1. Hence, by a standard fixed-point technique, we can show the existence of a unique maximal solution φ ∈ C([0, T 1 ), Σ 2 ) ∩ C 1 ([0, T 1 ), L 2 ), with 0 < T 1 ≤ T max . Let us prove that T 1 = T max by contradiction. To this end, we assume that T 1 < T max and consequently deduce that Therefore, it suffices to find τ < T 1 such that φ(t) Σ 2 is bounded on the interval (τ, T 1 ) to have the desired contradiction. Let τ ≥ 0, to be fixed later, be such that 0 < T 1 − τ < δ (δ is defined in Lemma 2.1). By differentiating (2.1) with respect to time, we obtain that Denoting ψ = e −iθHz φ, we compute now and, using the same admissible pairs (q, r) and (γ, ρ) as in Step 1, we estimate similarly

Using the Strichartz estimates (2.4) and (2.5), this yields
where the time integral is computed on the interval I = (T 1 − τ, T 1 ). Since γ ′ < q, if T 1 − τ is small enough, this implies ∂ t φ L q t L r x L 2 z ≤ C ∂ t φ(τ ) L 2 and, applying again the Strichartz estimates, we get For the last inequality, we used (2.1) at t = τ and the following estimate deduced from a Sobolev inequality and from interpolation inequality with s = 3σ 2σ+1 and α = max(σ − 1, 0) < 1. Now, we remark that, by integrations by parts, a direct calculation gives so, using directly (2.1), we estimate and then, using the bounds of φ Σ 1 and ∂ t φ L 2 , we deduce that, for all t ∈ I Next we can proceed similarly as above to estimate |x| 2 φ(t) L 2 . We have Hence, using that and that, by (2.7) and (1.7), we get by Strichartz estimates From γ ′ < q, it is easy to conclude that, for T 1 − τ small enough, Finally, consider the equation satisfied by H z φ. We have and, denoting again ψ = e −iθHz φ, one computes where we used the following Gagliardo-Nirenberg inequality in dimension 1: L ∞ . Hence, as above, we get which enables to conclude again with Strichartz inequalities that, for (2.10) From (2.8), (2.9) and (2.10), we deduce that φ(t) Σ 2 is uniformly bounded on the interval I = (τ, T 1 ): the proof is complete.
There are certainly situations for which T max = +∞. In view of the discussion at the beginning of Section 2.1, this will be true, in particular, if Ω 2 1 + Ω 2 2 < 1 and the nonlinearity is defocusing λ > 0, since in this case, the results given in [1] apply. However, if the effective centrifugal force is too big, the resulting repulsive quadratic potential requires particular techniques, see [8] which would need to be combined with the effect of the rotation term.

The Cauchy problem for the NLS equation in 3D.
Before we can proceed to the proof of convergence for solutions ψ ε of (1.2) as ε → 0, we need, as a final preparatory step, a suitable existence result for the three-dimensional Cauchy problem (1.2) Proposition 2.4. Let ψ 0 ∈ Σ 1 . Then, for any fixed ε > 0, there exists T ε 1 ∈ (0, +∞] such that (1.2) admits a unique maximal solution The proof follows along the same lines as the one for Proposition 2.2 above, i.e., through a fixed point argument for Duhamel's formula in a suitable metric space. Indeed, it is even easier, since (1.2) is a standard three-dimensional NLS equation with quadratic potential and rotation term. For such equations, the local and global existence theory in Σ 1 , based on Strichartz estimates and the use of energy-methods, has been studied in detail in [1] (see also [7]). Additional smoothness for ψ 0 ∈ Σ s with s > 1, then follows by the same arguments as given in Step 5 in the proof of Proposition 2.2.
One might be concerned that, as ε → 0, the existence time T ε 1 → 0, but our convergence proof below will show that this is indeed not the case. We finally, note that (1.2) admits the usual conservation laws for the mass and the total energy. The latter, however, is in general indefinite, due to the appearance of the angular momentum operator. Since, we shall not use any of these conservation laws in the following, we omit a more detailed discussion of these issues and refer the reader to [1].

Convergence proof and error estimates
This section is devoted to the proof of our main Theorem 1.1. Item (i) of this theorem is a consequence of Proposition 2.2.We prove items (ii) in Subsection 3.2 after we have obtained some uniform estimates.
From a Gagliardo-Nirenberg inequality, we get Lemma 3.1. There exists a constant C M such that, for 0 < ε < ε 1 and for all Proof. We first recall from (1.6), that We will now derive suitable a priori estimates for these three parts of the Σ 2 -norm. To this end, we first multiply (1.4) by u ε , integrate over R 3 , and take the real part of the resulting expression. This yields d dt u ε 2 L 2 = 0, i.e., the conservation of mass. For the other two parts of the Σ 2 -norm, we first compute the commutation relations (where in the third expression we have used that H x is rotationally symmetric), as well as [H z , z] = −∂ z , and [H z , z 2 ] = −(1 + 2z∂ z ). Keeping these relations in mind, we can thus apply H z to (1.4), commute, and, after multiplying by H z u, integrate over R 3 . Taking the real part of the resulting expression, we obtain d dt After several integrations by parts and using Cauchy-Schwarz, this yields the following estimate d dt where C 1 , C 2 , C 3 are some constants depending only on Ω 1 , Ω 2 , and Ω z , but not on ε. Similarly, a computation shows , and using again Cauchy-Schwarz yields the analogous estimate for H x u ε L 2 . Combining the three estimates obtained above, allows us to write d dt where K 1,2 = K 1,2 (ε, Ω 1 , Ω 2 , Ω z ) > 0 are both bounded as ε → 0. Now, we use the fact that, by Sobolev's imbedding, Gronwall's lemma consequently implies that u ε Σ 2 stays bounded for all t ∈ [0, T ε ].
We note that an important consequence of this lemma is that if T ε < +∞ then T ε < T ε 1 and u ε (T ε ) L ∞ = 2M.

.1) and (3.3) imply that
and (3.6) and (3.7) give Let us estimate A 3 . To this aim, we introduce the following function, defined on R × Σ 2 , Since F (·, u) is 2π-periodic and F av is its average, F(θ, u) is periodic with respect to θ. Hence, it is readily seen that this function satisfies the following properties: Recall that U (t) = e itH , where H is the Hamiltonian defined by (2.2). Hence
We note that under under sufficiently high regularity assumptions on ψ ε (t), a slightly stronger approximation result can be proved. In this case, one can show that, indeed, A 4 L 2 ≤ Cε 2 , and not only O(ε) as shown above. To see this, we expand where J is as above. Here, we have also used that z|χ m | 2 is odd and thus only indices m = n appear in the double sum above and R ε 1 is consequently seen to be highly oscillatory. After an integration by parts in time (which requires the improved regularity of v ε ), one obtains that R ε 1 = O(ε 2 ). Using this one can show that, for 0 < T < T max and 0 < ε ≤ ε T , the following improved error estimate holds where φ ε is the solution of (1.9) with initial data (3.10) Note, however, that if we apply this ε-correction to the Cauchy data, the solution φ ε does not remain polarized any more, in which case the reduced model is still posed in three spatial dimensions.

The case of strong two-dimensional confinement
In this section, we briefly discuss how to obtain the limiting model in the case of strong two-dimensional confinement within the original (three-dimensional) Gross-Pitaevskii equation. To this end, we we start with the analog of (1.1), given by Note that in comparison to (1.1) the roles of x and z have been reversed. Thus, in (4.1), the x variables are now the ones which represent the strongly confined directions, and we consequently aim to derive an effective model depending on the z-variable only. To this end, we rescale and assume that β ε = λε 2σ , i.e., an even weaker interaction regime as before. The rescaled NLS equation then becomes with ψ ε (t = 0, x, z) = ψ 0 (x, z). In order to get rid of the singular perturbation we invoke the same change of variables (up to a sign) as in the case of a strong one-directional confinement, i.e., After a somewhat lengthy computation, this yields the following analog of (1.4): In order to average out the fast oscillations stemming from H x , we introduce which satisfies G ∈ C(R × Σ s ; Σ s ). Moreover, G is easily seen to be a 2π-periodic function in θ, since the spectrum of the two-dimensional harmonic oscillator H x is given by {λ n = n + 1 , n ∈ N 0 }. Note however, that the eigenspace corresponding to λ n is (n + 1)-fold degenerate. We consequently denote the associated averaged nonlinearity by G av (u) := 1 2π 2π 0 e iθHx e −iθHx u 2σ e −iθHx u dθ, and find, that, as ε → 0, the new limiting model becomes Again, we note the appearance of an additional negative (repulsive) quadratic potential, provided Ω 1 , Ω 2 = 0.
Proof. The approximation proof follows along the same lines as the one for Theorem 1.1, up to adjusting the notation (i.e., switching the roles of x and z). The main difference concerns the proof of well-posedness for the limiting equation (4.4). In contrast to the case of one-dimensional confinement, equation (4.4) only admits dispersive properties only in one direction, which might not be sufficient for the use of Strichartz estimates. However, since we are working in Σ 2 ֒→ L ∞ (R 3 x,z ), local in-time well-posedness follows from standard arguments, see [9].
In comparison to initial data polarized along the ground state λ 0 , the situation for initial data polarized along some higher energy eigenvalue λ n , n ≥ 1, is much more complicated, due to their (n+1)-fold degeneracy. The corresponding solutions are then of the form φ(t, x, z) = e −itλn/ε 2 where the coefficients ϕ α ≡ ϕ α 1 ,α 2 solve a system of n + 1 coupled NLS. The latter mixes the ϕ α through the nonlinearity and describes the dynamics within the n-th eigenspace. The precise form of the NLS system is rather complicated and hence, we shall leave its details to the reader, in particular, since one anyway might prefer the description of the dynamics via the effective model (4.4) instead.