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Weakly nonlinear time-adiabatic theory

We revisit the time-adiabatic theorem of quantum mechanics and show that it can be extended to weakly nonlinear situations, that is to nonlinear Schroedinger equations in which either the nonlinear coupling constant or, equivalently, the solution is asymptotically small. To this end, a notion of criticality is introduced at which the linear bound states stay adiabatically stable, but nonlinear effects start to show up at leading order in the form of a slowly varying nonlinear phase modulation. In addition, we prove that in the same regime a class of nonlinear bound states also stays adiabatically stable, at least in terms of spectral projections.


Introduction
The time-adiabatic theorem of quantum mechanics is concerned with systems governed by a slowly varying time-dependent (self-adjoint) Hamiltonian operator H = H(ετ ), where 0 < ε ≪ 1 is a small adiabatic parameter, controlling the time-scales on which H varies. The associated Schrödinger equation, governing the time-evolution of the quantum mechanical wave function Ψ = Ψ(τ, x), with x ∈ R d , reads i∂ τ Ψ = H(ετ )Ψ, Ψ |τ =τ0 = Ψ in (x). In the following, it will be more convenient to rewrite the system using the (slow) macroscopic time variable t = ετ . In this case, the Schrödinger equation becomes a singularly perturbed problem of the form (1.1) iε∂ t Ψ ε = H(t)Ψ ε , Ψ ε |t=t0 = Ψ ε in (x), where Ψ ε (t, x) ≡ Ψ(t/ε, x). A typical example for the time-dependent Hamiltonian H(t), and the one we will be concerned with, is given by where V (t, x) describes some time-dependent (real-valued) potential. It is well-known that in the case where V = V (x) is time-independent, the spectral theorem of self-adjoint operators allows for a precise description of the time-evolution associated to (1.1). In particular, it implies that if the initial data Ψ ε in is concentrated in a given spectral subspace of H, then it will remain so for all times. However, as soon as H = H(t), the spectral subspaces (in general) start to mix during the time-evolution, and thus we do not have any precise information on the solution Ψ(t, ·).
However, one might hope that for small 0 < ε ≪ 1 there is a remedy to the situation. To this end, let us assume that the spectral subspaces of H(t) vary smoothly in time for t ∈ [0, T ], and that the initial wave function Ψ ε in is concentrated in one of these subspaces. Then the classical time-adiabatic theorem of quantum mechanics states that, for sufficiently small ε ≪ 1, the solution Ψ ε (t, ·) approximately (i.e. up to a certain error which vanishes as ε → 0) remains within the same subspace, provided the latter stays isolated from the rest of the spectrum of H(t) for all t ∈ [0, T ], see below. In this situation, the spectral subspace is said to be adiabatically stable under the time-evolution. Note that in the unscaled variable τ this result corresponds to an approximation on time-scales of order τ ∼ O(1/ε). The first adiabatic result for quantum systems appeared as early as 1928, cf. [5]. Since then, many mathematical extensions and developments have taken place, see, e.g., [1,2,11,12,13], and the references therein. For a general introduction to this subject we refer to [26].
A possible way of introducing the slow parameter ε is to think about a quantum mechanical experiment in which the experimentalist is allowed to slowly tune the external potential V = V (ετ, x). With this in mind, it is worth noting that modern quantum mechanical experiments are often performed on ultra-cold quantum gases in the state of their Bose-Einstein condensation [17]. Indeed, ultra-cold quantum gases offer a superb level of control, unprecedented in several respects, which has triggered a vast amount of scientific activity, both theoretical and experimental. It is well-known that within a mean-field approximation the (macroscopic) wave function of the condensate is accurately described by a nonlinear Schrödinger (or, Gross-Pitaevskii) equation, cf. [17] for a general discussion, and [10,15] and the references therein for a rigorous mathematical justification. It therefore seems a natural question to ask, whether one can extend the results of time-adiabatic perturbation theory to the case of nonlinear Schrödinger equations (NLS). This work is a first, modest attempt in this direction, although one should mention that there exist some non-rigorous works in the physics literature, cf. [28]. Moreover, one should distinguish our time-adiabatic setting from the one in [20], which studies solitary wave solutions to nonlinear Schrödinger equations in a space-adiabatic situation, i.e., with a potential of the form V = V (t, εx).
To be more concrete, we shall study the following class of NLS: where σ ∈ N, and where λ ∈ R denotes a nonlinear coupling constant, describing either focusing or defocusing behavior, cf. [22] for a broad discussion of these terms. The cubic case σ = 1 corresponds to the classical Gross-Pitaevskii equation.
Clearly, an extension of the time-adiabatic theorem to such nonlinear models is not straightforward, in particular due to the lack of a spectral theory for general nonlinear operators. The basic idea in the present paper is to work in an asymptotic regime for which the nonlinearity can be considered as a small perturbation of the associated linear problem. A possible way to do so is to restrict ourselves to asymptotically small solutions of the form where, as ε → 0, we formally regard ψ ε ∼ O(1), say in L 2 (R d ). Note that the size of the original wave function is then Ψ ε ∼ O(ε 1/(2σ) ) and hence it becomes asymptotically larger the larger σ ∈ N. Rewriting (1.3) in terms of the new unknown ψ ε yields with an effective nonlinear coupling constant λ ε = λε ≪ 1. The equation (1.5) can thus be considered weakly nonlinear.
As we shall see below, a nonlinear coupling constant of order O(ε) will be critical for our analysis, since it corresponds to the threshold for which nonlinear effects are present in the leading order description of ψ ε . In particular, if λ ε were even smaller, the problem would become essentially linearizable (as we will show below). The first main result of this work can now be stated as follows: x) be vanishing as |x| → ∞, for all t ∈ I, and assume that there exists a simple eigenvalue E(t) ∈ spec(H(t)) which stays separated from the rest of the spectrum by some δ > 0, i.e.
Assume that at t = t 0 , the initial data is concentrated in the eigenspace correspond- is a normalized eigenfunction associated to E(t) and γ ε ∈ H k (R d ) is a corrector which is constructed according to (3.4) and such that Then, for any compact time-interval J ⊂ I containing t 0 , there exists ε 0 (J) < 1, and a constant C > 0, such that for any 0 < ε ε 0 (J) the unique solution ψ ε ∈ C(J; H k (R d )) to the nonlinear Schrödinger equation (1.5) exists, and, in addition, where the phase ϕ ε (t) ∈ R is given by with β(t) ∈ iR the classical Berry phase, defined in (2.7).
This theorem shows that the influence of the nonlinearity creates an additional slowly varying phase modulation, similar to the Berry phase, in the leading order approximation of ψ ε . An immediate consequence is the following corollary for the associated spectral projectors (for which we use Dirac's notation): Under the same assumptions as before, we obtain sup t∈J |ψ ε (t, ·) ψ ε (t, ·)| − |χ(t, ·) χ(t, ·)| L 2 →L 2 Cε, In other words, in terms of spectral projections, the linear time-adiabatic theorem is still valid under weakly nonlinear perturbations of the form (1.5).
Unfortunately, due to our method of proof, we require the initial data to be sufficiently well-prepared (in the sense described above), even if one is only interested in the leading order approximation. In the language of, e.g., [26], we require the initial data to be concentrated in a super-adiabatic subspace. This is very similar to the situation encountered in [7], where the semiclassical asymptotics for weakly nonlinear Schrödinger equations with highly oscillatory periodic potentials is studied. In fact, the basic strategy for the proof of Theorem 1.1 is similar to the one used in [7].
Clearly, Theorem 1.1 can be reformulated in terms of Ψ ε , yielding a timeadiabatic result for asymptotically small solutions. In this case, a connection to the theory of nonlinear bound states for NLS equations becomes apparent. To this end, consider the "stationary" Schrödinger equation associated to (1.3), i.e., where E * ≡ E * (t) ∈ R is a nonlinear energy-eigenvalue. Now, let t ∈ R be fixed. Then, if the potential V (t, x) is such that H(t) has a discrete (linear) eigenvalue/eigenfunction pair (E, χ), classical bifurcation theory (see, e.g., [14]) implies that for E * ≈ E small amplitude nonlinear bound states Φ exist, and are, in leading order, given by small multiples of χ. In the context of NLS, this has been rigorously proved in a number of papers, cf. [16,19,27] (see also, [8]). Combining this fact with the result in Theorem 1.1 will allow us to prove that, under certain circumstances, these nonlinear bound states are also adiabatically stable. More precisely, we have the following result: Assume that for all t ∈ I, H(t) has exactly one simple eigenvalue E(t), which stays separated from the rest of the spectrum by some δ > 0.
Denote by Ψ ε the solution of (1.3) with initial data Ψ ε in = √ εψ ε in , where ψ ε in is as in Theorem 1.1. Then there exists a nonlinear bound state Φ ∈ C 1 b (I; H 2 (R 3 )) with its associated energy-eigenvalue E * ∈ C 1 b (I), a constant K > 0, and anε 0 ǫ 0 (J) such that for all ε ∈ (0,ε 0 ] it holds where ϕ ε is as before and J * ⊆ J is the maximal time-interval for which it holds The restrictions on d and σ are due to the fact that for the proof of Theorem 1.3 we will quote a result from [8] in which only cubic interactions in three dimensions are considered (however, a generalization to higher dimensions and/or nonlinearities is certainly possible). Similarly, the restrictions on V are chosen such that both Theorem 1.1 and the aforementioned result of [8] can be easily applied. One should mention that in [8] a similar theorem is proven but the precise form of the nonlinear phase modulation present in ϕ ε is not given. More importantly, the result in [8] requires two small parameters, namely ε ≪ 1 and M := Ψ ε (t, ·) 2 L 2 ≪ 1, sufficiently small, but it does not give a quantitative estimate on either of them. In comparison, Theorem 1.3 holds for (small) solutions Ψ ε of order O( √ ε). The main drawback of our theorem is the fact that it is very hard to give a precise estimate on the time-interval J * defined by the condition above. The reason for this is two-fold: First, the proof of Theorem 1.1 shows that it is rather difficult to give a precise estimate on ε 0 . Second, the existence of a nonlinear eigenvalue E * (t) near to E(t) is obtained through bifurcation theory, which in itself relies on the implicit function theorem. The later usually does not yield a precise estimate on the size of the neighborhood on which the implicit function exists. Thus one usually does not know how close E and E * are, as time evolves. As a final remark, we note that one can obviously reformulate Theorem 1.3 in terms of spectral projections to obtain The paper is now organized as follows: In Section 2 we shall show how to obtain the leading order approximation by means of formal asymptotic expansions. These expansions will then be made mathematically rigorous in Section 3. The nonlinear stability of our approximation is proved in (4), yielding the proof of Theorem 1.1. Possible extensions and variations of our results, in particular, the proof of Theorem 1.3 are then discussed in Section 5.

Formal construction of the approximate solution
2.1. The linear case. In this section we shall perform a formal multiple scales expansion of the solution of (1.5) in the linear case λ = 0. To this end, we make the following ansatz , where ϕ(t) ∈ R is some sufficiently smooth phase function, and the complex-valued amplitude U ε is assumed to be of the form in the sense of formal asymptotic expansions. Plugging this into (1.5) yields where H(t) is given by (1.2). Next, we plug in (2.1) and equate powers in ε. In leading order, i.e. by equating terms of order O(1), we find: This can be seen as an eigenvalue problem for the operator H(t) with eigenvalue E(t) =φ(t) and we consequently conclude that the so-called dynamic phase. Assuming for the moment that E(t) is a simple eigenvalue for all t ∈ I ⊆ R, with associated normalized eigenfunction for some yet to be determined coefficient Next, by equating terms of order O(ε), we find the following inhomogeneous equation Using the information form the step before, this can be rewritten as where from now on, we shall denote The kernel of L E (t) is given by span(χ(t, ·)) and we consequently decompose where v 1 (t, ·), χ(t, ·) L 2 = 0, for all t ∈ I. In order to guarantee that (2.5) has a solution, Fredholm's alternative asserts that the right hand side of (2.5) has to be orthogonal to χ(t, ·), for all t ∈ I. Taking the L 2 (R d ) inner product of (2.5) with χ gives and thus (up to a multiplicative constant which we shall choose equal to 1 for simplicity), we find denotes the famous Berry phase term [2,3]. Note that β(t) ∈ iR, for all t ∈ I, as one can easily see from differentiating the normalization condition χ(t, ·), χ(t, ·) L 2 = 1.
In leading order, we therefore find the well known approximation of linear timeadiabatic theory. Namely, that for ε → 0 the solution ψ ε behaves like Remark 2.1. In the case where the time-dependence of H(t) is not periodic, it is usually possible to choose a gauge such that β(t) = 0, see [2,3]. For a general discussion of the physical significance of the Berry phase (and similar geometric phases), we refer to [4].
With this in hand, it is possible to determine v 1 through (2.5). At least formally, x) , where we denote the partial inverse (or, partial resolvent) of L E (t) by , with P (t) = |χ(t, ·) χ(t, ·)| being the projection onto the eigenspace corresponding to E(t) ∈ R. Note that this also shows that initially v 1 (t 0 , x) = 0, in general.
The remaining unknown u 1 appearing in (2.6) can then be obtained by equating terms of order O(ε 2 ). Indeed, by looking at the solvability condition for one finds that u 1 (t) solves the following equatioṅ Choosing, for simplicity, u 1 (t 0 ) = 0, we get By repeating these steps, one easily finds that all amplitudes U n (t, x), n 1, appearing in (2.1), are of the form where every u n (t) is determined through an ordinary differential equation obtained from the solvability condition at order O(ε n+1 ), together with the initial data u n (t 0 ) = 0.

2.2.
Adding a nonlinearity. Next, we want to understand how to take into account a (sub-)critical nonlinearity in our asymptotic expansion. To this end, we first note that (2.1) yields . Thus, the leading order eigenvalue problem (2.2) does not change. The nonlinearity enters only in the expressions of order O(ε) or higher. For the former we find the following analog of (2.4): Here, we can use our knowledge from before to make the following ansatz for U 0 : where β(t) is the Berry phase defined in (2.7) and θ(t) ∈ R is some other phase yet to be determined. By doing so, the solvability condition requiring that the right hand side of (2.10) has to be orthogonal to ker L E (t) yields dθ dt = λ |χ(t, ·)| 2σ χ(t, ·), where we have used the fact that β(t) ∈ iR. Assuming, for the moment, that χ(t, ·) ∈ L 2σ+2 (R d ), we thus find In view of (2.11), we see that the nonlinearity contributes in leading order by adding an additional nonlinear phase modulation to the classical Berry phase, i.e., then instead of (1.5) we would obtain Performing the same asymptotic expansion as before, we see that if α 2, then no nonlinear effects are present in the leading order asymptotics. The problem thus becomes essentially linearizable, and can be considered sub-critical with respect to our asymptotic analysis. A somewhat intermediate regime is obtained in the case where α is no longer a natural number and such that 1 < α < 2. This situation will be discussed in more detail in Section 5.1.3. Finally, if 0 α < 1, the problem can be considered super-critical with respect to our asymptotic expansion. The case α = 0 is probably the most relevant from the physics point of view, but clearly also mathematically much more challenging and thus beyond the scope of the present work. It seems clear, though, that this question is intimately related to the modulation stability of nonlinear ground states studied in [27] (see also [23]).

A mathematical framework for asymptotic expansions
We wish to make the formal multiple scales computations of the foregoing section mathematically rigorous. To this end, we shall impose the following basic assumptions on the time-dependent potential: Assumption 1. Let I ⊆ R be some open interval containing t 0 ∈ R. We assume that, for some k ∈ N, the potential V (t, x) satisfies V ∈ C 1 b (I; C k b (R d )), and that lim |x|→∞ |V (t, x)| = 0, ∀ t ∈ I.
Fix t ∈ I. Then it is well known (see, e.g., [25,Chapter 10.1]) that for V (t, ·) bounded, i.e., k = 0, and decaying at infinity, the Hamiltonian H(t) is a self-adjoint operator with dom(H(t)) = H 2 (R d ) ⊂ L 2 (R d ). Moreover, for any fixed t ∈ I the spectrum of H(t) is of the standard form, i.e., see, e.g., [9]. Of course as these eigenvalues vary in time, they might cross each other, or disappear into the continuous spectrum. Our main assumption, necessary for the validity of an adiabatic approximation, is that the eigenvalue E(t) we are interested in stays separated from the rest of the spectrum by a spectral gap.
Assumption 2. We further assume that there exists a simple eigenvalue E(t) ∈ spec(H(t)) and a constant δ > 0, satisfying Note that this implies E(t) −δ, for all t ∈ I. Denoting by χ(t, ·) ∈ L 2 (R d ) a normalized eigenfunction corresponding to such a well separated eigenvalue E(t), we have the following regularity result.
Lemma 3.1. Let Assumptions 1 and 2 hold, then χ ∈ C 1 b (I, H k+2 (R d )). Proof. The proof follows from standard arguments. Indeed, we first notice that, for any fixed t ∈ R, χ(t, ·) satisfies the Schrödinger eigenvalue problem which, in view of Assumption 1 and [9, Proposition 1.2], implies the asserted regularity in H k+2 (R d ). Thus it only remains to prove the differentiability property in time. This follows from the fact that as long as E(t) stays separated from the rest of the spectrum, the associated orthogonal projector P (t) = |χ(t, ·) χ(t, ·)| can be expressed via Riesz' formula as where Γ(t) ⊂ C is a continuous (positively oriented) curve encircling E(t) once, such that inf t∈I dist(E(t), spec H(t)) = δ/2, i.e., no other points within spec(H(t)) are enclosed by Γ(t). Using this, we see that ). The latter is proved for example in [26, Lemma 2.4].
By Sobolev imbedding we also have H s (R d ) ֒→ L ∞ (R d ), provided s > d 2 , in which case χ ∈ L q (R d ) for any q ∈ [2, ∞]. In particular, for the expression for the nonlinear phase modulation θ(t) given by (2.12) is welldefined in this situation. Moreover, for s > d 2 , the Sobolev space H s (R d ) is in fact an algebra, i.e., if f, g ∈ H s (R d ) then f g ∈ H s (R d ). This can be used to prove the following regularity result: Lemma 3.2. Let σ ∈ N, λ ∈ R, and Assumptions 1 and 2 hold for some k > d− 4 2 . Then the expressions appearing in the asymptotic expansion (2.1) satisfy U n ∈ C 1 b (I; H k+2 (R d )), for any n ∈ N. Proof. Each U n is of the form given in (2.9), i.e., U n (t, x) = u n (t)χ(t, x) + v n (t, x), with v 0 ≡ 0. In view of our assumption on the potential it is clear that u n ∈ C 1 b (I). Together with Lemma 3.1 we thus have u n χ ∈ C 1 b (I; H k+2 (R d )). Moreover, we know that all v n , for n 1 are determined by inverting an elliptic equation for any fixed t ∈ I, i.e., where we denote the nonlinearity by F (z) = |z| 2σ z, which for σ ∈ N is smooth. Note that the derivative of F appearing on the right hand side, is in fact a sum of products of U ℓ 's (see also the proof of Proposition 3.3 below). The fact that H k+2 (R d ) for k > d−4 2 forms an algebra, implies that the right hand side is indeed , in view of (2.8), the assertion follows by induction over n.
With this result in hand, we set where ϕ(t) is the dynamic phase given by (2.3). Note that at t = t 0 the U n can in general not be chosen arbitrarily, since parts of it need to be determined recursively as given in (3.2). In particular, we have with v n (t 0 , x) as above. This definition of ψ ε N then yields an approximate solution of the nonlinear Schrödinger equation (1.5) in the following sense: Proposition 3.3. Let σ ∈ N, λ ∈ R, and Assumptions 1 and 2 hold for some Proof. By plugging ψ ε N into the nonlinear Schrödinger equations, the asymptotic expansion above shows that In view of the regularity result established in Lemma 3.2, and the algebra property of H k+2 (R d ), for k > d− 4 2 , we directly obtain the estimate on the remainder stated above.
This result, however, is not sufficient to conclude that the exact solution ψ ε will stay close to the approximate solution ψ ε N for times of order O(1). We shall show in the next section that this is indeed the case. 4. Nonlinear stability of the approximation 4.1. Preliminaries. Before we can prove stability of our asymptotic expansion, we need a basic existence result for solutions to nonlinear Schrödinger equations of the form (1.5).
Lemma 4.1. Let σ ∈ N, λ ∈ R, and N ∋ k > d 2 . Moreover, let ψ ε in ∈ H k (R d ) and the potential satisfy V ∈ L ∞ loc (I; C k b (R d )). Then there exist T ε 1 , T ε 2 > 0, and a unique solution Proof. The proof is a straightforward extension of the one given in, e.g., [24,Proposition 3.8] for the case without potential. We rewrite the NLS using Duhamel's principle Clearly, the free Schrödinger group e −it ∆ 2ε is an isometry on H k (R d ) for any k ∈ R, and our assumptions on V guarantee that there is a constant C = C(k, d) > 0 such that Moreover, for σ ∈ N, the nonlinearity F (z) = |z| 2σ z is smooth which, together with the fact that H k (R d ) for k > d 2 forms an algebra, allows us to estimate

R.
Then, we can show that the u → Ξ(u) maps the ball B 2R (0) ⊂ X into itself. Indeed, the estimate (4.1) implies Hence, we can choose T ε 1 , T ε 2 R −2σ 2 2σ+2 C ε > 0 and such that [t 0 − T ε 1 , t 0 + T ε 2 ] ⊂ I. The same type of estimate shows that u → Ξ(u) is a contraction on B 2R (0) ⊂ X and hence there exists a unique fixed point u = ψ ε ∈ X. The conservation of the L 2 -norm of the solution then follows from the fact that H(t) is self-adjoint.
Remark 4.2. By carefully tracking the ε-dependence of T ε 1,2 , one finds that, in general, T ε 1,2 will go to zero, as ε → 0. However, the stability proof below actually shows that for our choice of initial data, one can find T ε 1,2 > 0 independent of ε. We will also need the following Moser type result, proved in, e.g., [18,Lemma 8.1.1].

Nonlinear stability.
We are now in the position to prove the desired stability result for the asymptotic expansion obtained above. Given an approximate solution ψ ε N ∈ C 1 b (I, H k (R d )) of the form (3.3), with N > k, we assume that, at t = t 0 , the initial data ψ ε Then, for any compact time-interval J ⊂ I containing t 0 , there exists an ε 0 (J) > 0, and a constant C > 0, such that for any 0 < ε ε 0 (J) the unique solution ψ ε ∈ C(J; H k (R d )) to (1.5) exists and, in addition, Note that this result in particular implies that the solution ψ ε to (1.5) cannot exhibit blow-up on any finite time-interval J ⊂ I ⊆ R. (1.5). We denote the difference between the exact and the approximate solution by , where τ ε j = min(T ε j , T j ), with j = 1, 2. We prove that for ε sufficiently small, η ε may be extended up to the time-interval J ⊂ I, with η ε ∈ C(J, H k (R d )). For simplicity, we shall only show the argument for the times bigger than t 0 . A similar argument applies on the time Take ε 0 > 0 so that C 0 ε 0 ≤ 1 2 , and for ε ∈]0, ε 0 ], let We already know that t ε > 0 by the local existence result for ψ ε . By possibly reducing ε 0 > 0 even further, we shall show that t ε t 0 + T 2 . The error satisfies ) by assumption. Next, we multiply (4.2) by η ε , integrate over R d , and take the real part of the resulting expression. Since H(t) is self-adjoint, this yields In view of Proposition 3.3, we have r ε N H k = O(ε N +1 ). On the other hand, for k > d 2 the Gagliardo-Nirenberg inequality implies η ε Applying Lemma 4.3 with k = 0, we consequently obtain and, by using Grownwall's lemma, we thus find . The idea is now to obtain a similar estimate for (weak) derivatives of η ε , in order to close the argument in H k (R d ). To this end, we first note that ) + ∇r ε N , and the same type of argument as before, together with the Cauchy Schwartz inequality, yields x), which is bounded by assumption, and so Invoking again Lemma 4.3 with k = 1, and the bound (4.3), we infer that ∀t ∈ [t 0 , t ε ] it holds ∂ t ∇η ε L 2 ∇η ε L 2 + ε N −1 , and Grownwall's lemma, together with (4.3), then yields By induction over k ∈ N, we obtain, more generally and the Gagliardo-Nirenberg inequality consquently implies For N − k > 0, this shows that the assumptions needed to apply Lemma 4.3 are fulfilled for all ε ∈ (0, ε 0 ] and all t t ε . Continuity of η ε (·, t) H k implies that t ε t 0 + T 2 , for ε ε 0 (T 2 ) sufficiently small. In particular, we obtain that η ε , and hence ψ ε , is well defined for all t ∈ [t 0 , t 0 + T 2 ], thus showing T ε 2 T 2 . Since the same argument can be applied for times smaller than t 0 ,we finally conclude that ψ ε is well defined for all t ∈ J = [t 0 − T 1 , t 0 + T 2 ] ⊂ I and 0 < ε ε 0 (J).
To complete the proof of the theorem, we first note that (4.4) implies and since N > k, we also have Thus, we can use the triangle inequality to replace ψ ε N with ψ ε N −1 in our estimate, which yields the desired result.

Possible extensions and variations
5.1. Remarks on closely related cases. In this section we collect several remarks on how to extend Theorem 1.1 to other, closely related, situations. 5.1.1. Degenerate Eigenvalues. The results above readily generalize to the case of an M -fold degenerate eigenvalue E(t), provided that there exists a smooth basis χ ℓ (t, ·) ∈ L 2 (R d ), where ℓ = 1, . . . , L, of the associated eigenspace. The associated projection onto the eigenspace corresponding to E(t) then becomes Using this, one can proceed along the same lines as above to obtain that However, the formulas in general become more complicated, since the coefficient functions, u ℓ,0 (t) are now determined by an L × L system of ordinary equation (leading to matrix-valued Berry phases and analogous nonlinear phase modulations). This consequently leads to rather tedious computations in the subsequent steps of our asymptotic expansion, which is why we shall not go into further details here.

Quadratic potentials.
In view of a possible application to Bose-Einstein condensates, the assumption that V (t, x) vanishes as |x| → ∞ seems unrealistic, since one typically considers trapping potentials of the form i.e., a time-dependent harmonic oscillator. There is, however, no fundamental difficulty in extending our result to such a situation. Indeed, as long as Ω j (t) > 0, the existence of eigenvalues E(t) together with their associated smooth (and rapidly decaying) eigenfunctions is guaranteed (see, e.g., [25]), and the asymptotic expansion, stays (at least formally) exactly the same as before. Only from the point of view of estimates, one needs to shift from the usual Sobolev space setting H k (R d ), to weighted spaces of the form The basic existence and well-posedness theory for NLS in such weighted spaces has been established in [6], yielding a unique solution ψ ε ∈ C(I; Σ k ), provided ψ in ∈ Σ k . Moreover, an extension of the Moser type Lemma 4.3 to Σ k is straightforward. The only extra work needed is in the proof of the nonlinear stability, where now [∇, H(t)] = ∇V (t, x) is no longer bounded. However, since ∇V η ε L 2 ≃ xη ε L 2 , and since [x, H(t)] = ∇, a closed set of estimates for the combined L 2 -norms of xη ε and ∇η ε (and thus for the Σ 1 -norm of η ε ) can be obtained, cf. [7] for more details. Iterating this then yields a stability result in Σ k . 5.1.3. The intermediate regime 1 < α < 2. We go back to the discussion started in Remark 2.2 and consider the slightly more general situation of We already know that if α = 1 the problem is critical, and that if α 2, the problem is sub-critical (i.e., linearizable). The intermediate regime 1 < α < 2, however, is slightly more complicated, since the asymptotic expansion used before fails to match the size of the nonlinearity. One way to overcome this problem is to include the nonlinearity in the equation of order O(ε), which yields instead of (2.10). For ε α−1 ≪ 1 this can be seen as a regular perturbation problem of the associated linear situation. The corresponding solvability condition now yields an ε-dependent leading order amplitude of the form , with θ(t) given by (2.11). The nonlinear phase modulation appearing in this expression is obviously rather weak, due to the small ε α−1 ≪ 1 factor in front. The price to pay is that now all the U ε n become ε-dependent, and in order to state a rigorous result one would need to carefully track the ε-dependence through all the required estimates. This is doable, in principle, but again rather cumbersome and we shall leave the details to the reader.

5.2.
Connection to nonlinear bound states. We finally turn to the proof of Theorem 1.3. To this end, we first recall that for σ = 1 nonlinear bound states are solutions (at any fixed time t ∈ R) of the stationary Schrödinger equation where now x ∈ R 3 and the potential V is assumed to satisfy the following conditions: Assumption 3. The potential V ∈ C 1 b (I, S(R 3 )) is such that for all t ∈ I the operator H(t) admits only one simple eigenvalue E(t) ∈ spec(H(t)) satisfying inf t∈I dist E(t), spec(H(t)) \ {E(t)} = δ > 0.
Remark 5.1. Similar assumptions are imposed in [8]. In comparison, we strengthen the regularity and decay properties of V considerably in order to be able to impose a single, simple condition which ensures applicability of both Theorem 1.1 and [8, Propositon 5.2]. One can certainly generalize all of our results to potentials which satisfy much weaker conditions. In addition, one should note that [8] also imposes the assumption that for all t ∈ I, the potential V is such that H(t) does not admit a zero energy resonance. It seems, however, that this condition is only needed for the applicability of certain dispersive estimates which are not used for the proof of the result quoted below.
The existence of a unique family of nonlinear bound states t → Φ(t, ·) satisfying (5.1) is proven in [8,Proposition 5.2], whose main assertions we recall below: Proposition 5.2. Let Assumption 3 hold and t ∈ I be fixed. Then there exist 0 < ε 1 , ε 2 ≪ 1 such that (5.1) admits a solution (Φ, E * ) with Φ 2 where χ(t, ·) ∈ H 2 (R 3 ) is the normalized eigenfunction associated to the linear eigenvalue E(t). In addition, for any M ε 1 , there is a unique positive family of bound states t → Φ(t, ·) ∈ C 1 b (I; H 2 (R 3 )) with constant mass Φ(t, ·) 2 L 2 = M . With this assertion in hand, we can now give the proof of our second main result.
Remark 5.3. Interestingly, the proof shows that in the asymptotic regime being considered here, the linear subspace satisfies a better approximation estimate than the nonlinear one. At this point, it is not clear if this is only due to our method of proof, or indeed to a real phenomenon (which might be related to the particular choice of the nonlinear bound states with constant mass).