# On scattering for NLS: rigidity properties and numerical simulations via the lens transform

Rémi Carles, Georg Maierhofer

► To cite this version:

Rémi Carles, Georg Maierhofer. On scattering for NLS: rigidity properties and numerical simulations via the lens transform. 2025. hal-05109256

HAL Id: hal-05109256

https://hal.science/hal-05109256v1

Preprint submitted on 12 Jun 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
ON SCATTERING FOR NLS: RIGIDITY PROPERTIES AND NUMERICAL SIMULATIONS VIA THE LENS TRANSFORM

RÉMI CARLES AND GEORG MAIERHOFER

ABSTRACT. We analyse the scattering operator associated with the defocusing nonlinear Schrödinger equation which captures the evolution of solutions over an infinite time-interval under the nonlinear flow of this equation. The asymptotic nature of the scattering operator (involving unbounded time) makes its computation particularly challenging. We overcome this by exploiting the space-time compactification provided by the lens transform, marking the first use of this technique in numerical simulations. This results in a highly efficient and reliable methodology for computing the scattering operator in various regimes. In developing this approach we introduce and prove several new identities and theoretical properties of the scattering operator. We support our construction with several numerical experiments which we show to agree with known analytical properties of the scattering operator, and also address the case of long-range scattering for the one-dimensional cubic Schrödinger equation. Our simulations permit us to further explore regimes beyond current analytical understanding, and lead us to formulate new conjectures concerning fixed and rotating points of the operator, as well as its existence in the long-range setting for both defocusing and focusing cases.

# 1. INTRODUCTION

We consider the defocusing nonlinear Schrödinger equation

$$ (1.1) \qquad i\partial_t u + \frac{1}{2}\Delta u = |u|^{2\sigma}u, \quad (t,x) \in \mathbf{R} \times \mathbf{R}^d, \quad d \ge 1, $$

and assume that the nonlinearity is energy-subcritical, $0 < \sigma < \frac{2}{(d-2)_+}$. As the nonlinearity is defocusing, global existence of solutions in the Sobolev space $H^1(\mathbf{R}^d)$ is classical for such powers, as proven initially in [25]. Once global existence is established, the next standard question is the large time behaviour of the solution. As the Schrödinger equation is a dispersive equation, it is likely that the nonlinearity becomes small as time gets large, and, possibly, eventually negligible. This is the underlying idea of scattering theory, which we now summarise in a more quantitative way.

To understand the large time behaviour of solutions to (1.1), two types of initial data are of special interest:

$$ (1.2) \qquad \text{Initial value problem: } u|_{t=0} = u_0, $$

$$ (1.3) \qquad \text{Final value problem: } U_0(-t)u(t)|_{t=-\infty} = u_-, \quad \text{where } U_0(t) = e^{i\frac{t}{2}\Delta}. $$

Asymptotic states $u_+$ for $t = +\infty$ can also be considered, we address only the case $t = -\infty$ to avoid any ambiguity in the discussion. The scattering operator can be defined after two steps, which we describe below.

<footnote>This work was supported by Centre Henri Lebesgue, program ANR-11-LABX-0020-0. A CC-BY public copyright license has been applied by the author to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.</footnote>
**Existence of wave operators.** For this step, we impose the behaviour of the nonlinear solution as time goes to (plus or minus) infinity. More precisely, for a given asymptotic state $u_-$, we want to find a solution to (1.1) such that

$$u(t) \underset{t \to -\infty}{\sim} U_0(t)u_- = e^{i\frac{t}{2}\Delta}u_-.$$

As the Schrödinger group is dispersive, $U_0(t)u_-$ goes to zero pointwise, this is why the precise requirement is (1.3). If we assume

$$(1.4) \qquad \sigma > 1 \text{ if } d = 1, \quad \sigma > \frac{2}{d+2} \text{ if } d \ge 2,$$

then it is well known that for a given asymptotic state $u_- \in \Sigma = H^1 \cap \mathcal{F}(H^1)$, where

$$(1.5) \qquad \mathcal{F}f(\xi) = \hat{f}(\xi) = \frac{1}{(2\pi)^{d/2}} \int_{\mathbb{R}^d} f(x)e^{-ix\cdot\xi}dx,$$

(1.1)-(1.3) has a unique, global, solution $u \in C(\mathbb{R}; \Sigma)$ ([26], see also [16]). Its initial value $u|_{t=0}$ is the image of the asymptotic state under the action of the wave operator:

$$u|_{t=0} = W_-u_-.$$

The space $\Sigma$ is also characterised as a weighed Sobolev space, since

$$\Sigma = \{f \in L^2(\mathbb{R}^d), \nabla f \in L^2(\mathbb{R}^d), x \mapsto |x|f(x) \in L^2(\mathbb{R}^d)\}.$$

We can define the norm on $\Sigma$ as

$$\|f\|_{\Sigma} = \|f\|_{L^2(\mathbb{R}^d)} + \|\nabla f\|_{L^2(\mathbb{R}^d)} + \|f|f\|_{L^2(\mathbb{R}^d)}.$$

**Asymptotic completeness.** We now turn to the large time behaviour of solutions to the Cauchy problem. If

$$(1.6) \qquad \sigma \ge \frac{2-d+\sqrt{d^2+12d+4}}{4d} =: \sigma_0(d),$$

and $u_0 \in \Sigma$, (1.1)-(1.2) possesses asymptotic states:

$$(1.7) \qquad \exists u_{\pm} \in \Sigma, \quad \|U_0(-t)u(t) - u_{\pm}\|_{\Sigma} \underset{t \to \pm\infty}{\longrightarrow} 0: \quad u_{\pm} = W_{\pm}^{-1}u_0.$$

See [26] for the case $\sigma > \sigma_0(d)$, and [17, 38] for the case $\sigma = \sigma_0(d)$. The same is true under the weaker assumption (1.4), provided that $u_0$ belongs to a sufficiently small ball in $\Sigma$ (whose size is not explicit). We note the inequalities $1/d < \sigma_0(d) < 2/d$, the value $\sigma = 2/d$ corresponding to the $L^2$-critical, or conformally invariant, nonlinearity.

The scattering operator associated to (1.1) is classically defined as

$$(1.8) \qquad S = W_+^{-1} \circ W_-: u_- \mapsto u_+.$$

It maps $\Sigma$ to $\Sigma$ under the assumption (1.6), and a small ball of $\Sigma$ to (a small ball of) $\Sigma$ under the weaker assumption (1.4) (see e.g. [16, Theorem 7.5.6]). Also, for $u_0 \in \Sigma$ and $\sigma > 1/d$, there exist $u_{\pm} \in H^1(\mathbb{R}^d)$ such that

$$(1.9) \qquad \|u(t) - U_0(t)u_{\pm}\|_{H^1} \underset{t \to \pm\infty}{\longrightarrow} 0,$$

as established in [4], extending the initial result of [49].

The assumption $\sigma > 1/d$ is necessary in the sense that if $\sigma \le 1/d$, a long range theory is necessary as the above convergence is possible only in the trivial case $u \equiv 0$. This follows from the stronger result from [1]: if $\sigma \le 1/d$ and

$$\|u(t) - U_0(t)u_+\|_{L^2(\mathbb{R}^d)} \underset{t \to +\infty}{\longrightarrow} 0,$$

then necessarily, $u = u_+ \equiv 0$. Note that in the range $1/d < \sigma < \sigma_0(d)$, it is not known whether scattering holds in $\Sigma$, that is if (1.9) can be improved to (1.7)
for large data $u_0 \in \Sigma$. This point is addressed numerically in Section 4. Our simulations suggest that (1.7) may not be true, at least for some $\sigma \in (1/d, \sigma_0(d))$.

Apart from the existence of the operator $S$, rather few properties are known. The goal of this paper is to prove two identities which appear to be new, and to investigate numerically the behaviour of $S$, by using the lens transform, recalled below.

**Remark 1.1.** The scattering operator can also be defined in some situations for the focusing nonlinear Schrödinger equation,

$$i\partial_t u + \frac{1}{2}\Delta u = -|u|^{2\sigma}u.$$

Statements in such a case require more care, due to the existence of solitary waves, and also of blow-up solutions when $\sigma \ge 2/d$. We choose to consider mostly the defocusing case in order to make the presentation as simple as possible. However in the cubic one-dimensional case (corresponding to long range scattering), we consider numerically the focusing case in Section 5, and explore whether the size of the (explicit) ground states is a threshold for the existence of a (modified) scattering operator.

**1.1. Some known dynamical properties.** It is well-known that the scattering operator $S$, when it is defined like above, is unitary on $L^2$, and on $\dot{H}^1$:

$$\|S(u_-)\|_{L^2(\mathbf{R}^d)} = \|u_-\|_{L^2(\mathbf{R}^d)}; \quad \|\nabla S(u_-)\|_{L^2(\mathbf{R}^d)} = \|\nabla u_-\|_{L^2(\mathbf{R}^d)}.$$

These identities follow from [26, 48, 29], and can be found also in [16].

A natural question is whether the scattering operator is trivial (equal to the identity) or not. It is rather straightforward to compute the first two terms of the asymptotic expansion of $S$ near the origin, which coincide with the first two Picard iterates: for $u_- \in \Sigma$ and $\sigma$ satisfying (1.4),

$$S(\varepsilon u_-) = \varepsilon u_- + \varepsilon^{2\sigma+1} G(u_-) + \mathcal{O}(\varepsilon^{4\sigma+1}) \quad \text{in } L^2(\mathbf{R}^d), \quad \varepsilon \to 0,$$

where

$$G(u_-) = -i \int_{-\infty}^{+\infty} U_0(s) (|U_0(-s)u_-|^{2\sigma} U_0(-s)u_-) ds,$$

see [12] and references therein. By considering for $u_-$ a Gaussian (whose evolution under $U_0$ can be computed explicitly), it is easy to check that the operator $G$ is not identically zero.

We note also that the properties of $S$ near the origin have proven useful in inverse problems (see e.g. [43, 18, 34, 35]); the analyticity of $S$ was addressed in [12] (when $\sigma$ is an integer).

As noticed in [17] (see also [16]), if $\mathcal{C}$ denotes the conjugation $f \mapsto \bar{f}$, the time reversibility property associated to (1.1) implies the relation

$$(1.10) \qquad W_{\pm} = \mathcal{C} \circ W_{\mp} \circ \mathcal{C}.$$

In the special $L^2$-critical case $\sigma = 2/d$, two additional properties have been established: in [14], we find the relation

$$\mathcal{F} \circ W_{\pm}^{-1} = W_{\mp} \circ \mathcal{F},$$

which, together with (1.10), implies

$$W_{\pm}^{-1} = (\mathcal{C}\mathcal{F})^{-1} W_{\pm} (\mathcal{C}\mathcal{F}).$$

In [10], still for the case $\sigma = 2/d$, it was proven that for all $\theta \in [0, 2\pi]$, there exists infinitely many $u_- \in \Sigma$ such that $S(u_-) = e^{i\theta}u_-$ (see Proposition 2.4 below). In the next section, we present other algebraic relations which can be viewed as extra rigidity properties of the scattering operator $S$ and the waves operators $W_{\pm}$.
1.2. **New algebraic properties.** The following statement recalls the algebraic properties evoked above, and contains two new identities:

**Theorem 1.2.** Let $\frac{1}{d} < \sigma < \frac{2}{(d-2)_+}$ and $u \in C(\mathbf{R}; \Sigma)$ solution to (1.1). Assume that there exist $u_{\pm} \in \Sigma$ such that

$$ \|U_0(-t)u(t) - u_{\pm}\|_{\Sigma} \underset{t \to \pm\infty}{\longrightarrow} 0. $$

Then, with $u_0 = u_{|t=0}$, the following identities hold:

$$ (1.11) \qquad \|u_{-}\|_{L^2} = \|u_0\|_{L^2} = \|u_{+}\|_{L^2}, $$

$$ (1.12) \qquad \|\nabla u_{-}\|_{L^2}^2 = \|\nabla u_0\|_{L^2}^2 + \frac{2}{\sigma+1} \|u_0\|_{L^{2+2\sigma}}^{2+2\sigma} = \|\nabla u_{+}\|_{L^2}^2, $$

$$ (1.13) \qquad \begin{aligned} \operatorname{Im} \int_{\mathbf{R}^d} \overline{u_{-}}(x) \nabla u_{-}(x) dx &= \operatorname{Im} \int_{\mathbf{R}^d} \overline{u_0}(x) \nabla u_0(x) dx \\ &= \operatorname{Im} \int_{\mathbf{R}^d} \overline{u_{+}}(x) \nabla u_{+}(x) dx, \end{aligned} $$

$$ (1.14) \qquad \int_{\mathbf{R}^d} x|u_{-}(x)|^2 dx = \int_{\mathbf{R}^d} x|u_0(x)|^2 dx = \int_{\mathbf{R}^d} x|u_{+}(x)|^2 dx. $$

In the $L^2$-critical case $\sigma = 2/d$, we have in addition

$$ (1.15) \qquad \|xu_{-}\|_{L^2}^2 + \frac{2d}{2+d} \|\widehat{u_{-}}\|_{L^{2+4/d}}^{2+4/d} = \|xu_0\|_{L^2}^2 = \|xu_{+}\|_{L^2}^2 + \frac{2d}{2+d} \|\widehat{u_{+}}\|_{L^{2+4/d}}^{2+4/d}. $$

**Remark 1.3.** The above result holds (at least) in the following case. Let $u_{-} \in \Sigma$. For $\sigma < \frac{2}{(d-2)_+}$ satisfying (1.6), or more generally (1.4) provided that $\|u_{-}\|_{\Sigma}$ is sufficiently small, we let $u_0 = W_{-}u_{-} \in \Sigma$, and $u_+ = Su_{-} \in \Sigma$.

The identities (1.11), (1.12) and (1.13) are standard. The identities (1.14) and (1.15) are new to the best of our knowledge. The relation (1.15) can be proven by using the pseudo-conformal conservation law discovered in [26]. We give an argument based on the lens transform, which makes the proof very short.

The relation (1.14) implies for instance (together with (1.11)) the fact that the nonlinear scattering map cannot act as a translation operator on a nontrivial asymptotic state. This is obvious in the case of a radially symmetric state, since the dynamics of (1.1) preserves the radial symmetry, and we have more generally:

**Corollary 1.4.** Let $\frac{1}{d} < \sigma < \frac{2}{(d-2)_+}$ satisfying (1.6), and $x_0 \in \mathbf{R}^d$. The identity

$$ S(u_{-})(x) = u_{-}(x - x_0), \quad \forall x \in \mathbf{R}^d, $$

is possible for some $u_{-} \in \Sigma$ only in the trivial case, $u_{-} = 0$ or $x_0 = 0$.

**Remark 1.5.** The Plancherel formula shows that (1.13) can be rewritten, since

$$ \operatorname{Im} \int_{\mathbf{R}^d} \overline{u_{-}}(x) \nabla u_{-}(x) dx = \frac{1}{i} \int_{\mathbf{R}^d} \overline{u_{-}}(x) \nabla u_{-}(x) dx, $$

as

$$ \int_{\mathbf{R}^d} \xi |\widehat{u_{-}}(\xi)|^2 d\xi = \int_{\mathbf{R}^d} \xi |\widehat{u_0}(\xi)|^2 d\xi = \int_{\mathbf{R}^d} \xi |\widehat{u_{+}}(\xi)|^2 d\xi. $$

So the centre in phase space is the same for $u_{-}$, $u_0$, and $u_+$. This property constitutes a precious test for numerical methods.
1.3. **Lens transform.** For $|t| < \pi/2$, introduce $v$, defined by

$$ (1.16) \qquad v(t, x) = \frac{1}{(\cos t)^{d/2}} u \left( \tan t, \frac{x}{\cos t} \right) e^{-i \frac{|x|^2}{2} \tan t}, $$

which has the same value as $u$ at time $t = 0$. As noticed in [32, 42] (see also [8, 45]), $v$ solves (for $|t| < \pi/2$)

$$ (1.17) \qquad i \partial_t v + \frac{1}{2} \Delta v = \frac{|x|^2}{2} v + (\cos t)^{d\sigma-2} |v|^{2\sigma} v, \quad v|_{t=0} = u|_{t=0} = u_0. $$

We note that the nonlinearity in (1.17) is autonomous exactly in the $L^2$-critical case $\sigma = 2/d$. The compactification of time in (1.16) implies the following consequence:

**Lemma 1.6** (See Lemma 2.4 in [10]). Let $u \in C(\mathbf{R}; \Sigma)$ and $v \in C([-\frac{\pi}{2}, \frac{\pi}{2}]; \Sigma)$ solve (1.1) and (2.1), respectively, with $u|_{t=0} = v|_{t=0} = u_0$. Suppose that there exist $u_{\pm} \in \Sigma$ such that

$$ \|U_0(-t)u(t) - u_{\pm}\|_{\Sigma} \underset{t \to \pm\infty}{\longrightarrow} 0. $$

Then

$$ (1.18) \qquad v\left(-\frac{\pi}{2}, x\right) = e^{id\pi/4} \mathcal{F}(u_{-})(-x) \quad ; \quad v\left(\frac{\pi}{2}, x\right) = e^{-id\pi/4} \mathcal{F}(u_{+})\left(x\right). $$

The above property was used in [10] in order to reduce the property $S(u_{-}) = e^{i\theta}u_{-}$ in the $L^2$-critical case to the existence of solutions to an elliptic equation. It has been employed also to prove scattering for low regularity solutions to (1.1) by considering (1.17), in e.g. [5, 6, 40]. It is also present in the argument of [4] for the above mentioned result. The lens transform was used for other questions than scattering theory, see e.g. [21, 45].

1.4. **Numerical approach.** A major difficulty when trying to simulate numerically the scattering map $S$ lies in the dispersive properties of the group $U_0$, since

$$ U_0(t)f(x) \underset{t \to \infty}{\sim} \frac{1}{(it)^{d/2}} \hat{f}\left(\frac{x}{t}\right) e^{i \frac{|x|^2}{2t}}, $$

see Lemma 2.1 for a precise statement. In particular, for large time, the solution leaves any fixed box, and the exiting part cannot be neglected if one wants to take hold of the asymptotic state. In [13, Section 5], a first attempt at computing the scattering operator was made, consisting in computing $U_0(-t)u(t)$ for large values of $t$. The role of the composition with $U_0(-t)$ was to compensate the dispersive effect, in order to get the convergence to the scattering state and to avoid boundary effects. However, since $u(t)$ is computed first, dispersive effects are present, and corrected afterwards, which cannot rule out the possibility of interaction with the boundary. The way out chosen in [13] was to stop the computation at some large (but fixed) time $T$, and choose a computational domain sufficiently large so the interaction of $u$ with the domain remains weak.

In contrast, our approach is based on the lens transform which leads to several advantages: firstly, as pointed out in [45], the lens transform (1.16) compactifies space and time; time is obviously compactified because $t = \infty$ for $u$ corresponds to $t = \pi/2$ for $v$. Space is compactified too because the Laplace operator in (1.1) is replaced by the harmonic oscillator in (1.17), and this operator is confining (it has a compact resolvent). Therefore, it becomes much easier to avoid the interaction of $v$ with the boundary of the computational domain, provided that this domain is chosen sufficiently large. Secondly, as the solution to (1.17) can be computed for $t \in (-\pi/2, \pi/2)$ at least when the cosine factor is locally integrable, we may investigate the large time behaviour of the solution to (1.1) for any $\sigma > 1/d$. Note that the cosine factor ceases to be locally integrable when $\sigma = 1/d$, which
corresponds to the appearance of long range effects in (1.1), that is, the notion of scattering must be modified.

An idea similar to the one described above was used in [15] in order to investigate numerically the dynamics of the Hamiltonian $-\Delta - |x|^2$ perturbed by a logarithmic nonlinearity. The Schrödinger equation in the presence of this repulsive harmonic potential enjoys an exponential dispersive rate (as opposed to the algebraic rate without potential), which causes huge difficulties when it comes to numerics. An analogue of the lens transform was used in [15] in order to neutralise this strong dispersion: that transform removes the potential, and compactifies time (so the dispersion due to the operator $U_0(t)$ is essentially harmless numerically).

**1.5. Contents.** In Section 2, we recall some technical tools and properties useful in the analysis of nonlinear scattering theory, and prove Theorem 1.2. We also recall the argument of the main result from [10], proving the existence of rotating points in the $L^2$-critical case, and sketch some specific aspects of the long range case $\sigma = 1/d$. Section 3 is dedicated to numerical experiments, with emphasis on the existence of rotating points in the $L^2$-critical case and in the $L^2$-supercritical case, and on the long range case in one dimension. In Section 4, we explore numerically the actual threshold value for scattering to hold in $\Sigma$ for large data. In Section 5, we consider the focusing, cubic, one-dimensional Schrödinger equation, and compare numerically the existence of modified scattering with the size of the initial data. Concluding remarks and a few directions for future research are provided in Section 6. In Appendices A & B, we gather auxiliary identities and algorithms for Hermite expansions used in Section 3, and in Appendix C, we show a quantitative continuity argument for the dependence of the scattering operator upon the power $\sigma$, in order to support the discussion on some experiments presented in Section 3.

**Acknowledgements.** The authors are grateful to Valeria Banica and Katharina Schratz for many helpful discussions at the origin of this work. Both authors gratefully acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 850941).

## 2. ANALYTIC BACKGROUND

**2.1. Generalities.** We recall the three standard quantities conserved by the flow of (1.1): mass, energy, and linear momentum, given respectively by

$$ 
\begin{align*}
M &= \|u(t)\|_{L^2(\mathbf{R}^d)}^2, \\
E &= \frac{1}{2} \|\nabla u(t)\|_{L^2(\mathbf{R}^d)}^2 + \frac{1}{\sigma+1} \|u(t)\|_{L^{2\sigma+2}(\mathbf{R}^d)}^{2\sigma+2}, \\
P &= \mathrm{Im} \int_{\mathbf{R}^d} \bar{u}(t, x) \nabla u(t, x) dx.
\end{align*}
$$

The following lemma is standard (see [48] or [41]):

**Lemma 2.1.** Let $f \in L^2(\mathbf{R}^d)$, and recall that $U_0(t) = e^{i \frac{t}{2} \Delta}$.

$$ \|U_0(t)f - A(t)f\|_{L^2(\mathbf{R}^d)} \xrightarrow{t \to \pm\infty} 0, \quad \text{where } A(t)f(x) = \frac{1}{(it)^{d/2}} \hat{f}\left(\frac{x}{t}\right) e^{i \frac{|x|^2}{2t}}, $$

and the Fourier transform is normalised in (1.5).

The lemma is an easy consequence of the factorisation

$$ U_0(t) = M_t D_t \mathcal{F} M_t, $$
where $M_t$ stands for the multiplication by the function $e^{i\frac{|x|^2}{2t}}$, $\mathcal{F}$ is the Fourier transform defined in (1.5), and $D_t$ is the dilation operator

$$ (D_t f)(x) = \frac{1}{(it)^{d/2}} f\left(\frac{x}{t}\right). $$

These three operators are unitary on $L^2$, and the lemma follows from the Dominated Convergence Theorem, noticing that $A(t) = M_t D_t \mathcal{F}$.

The sharp decay in time of linear solutions is classical and recalled in the following lemma:

**Lemma 2.2.** There exists $C$ such that for all $f \in \Sigma$, and all $2 \le p < \frac{2d}{(d-2)_+}$,

$$ \|U_0(t)f\|_{L^p(\mathbf{R}^d)} \le \frac{C}{\langle t \rangle^{d(\frac{1}{2}-\frac{1}{p})}} \|f\|_{\Sigma}, \quad \forall t \in \mathbf{R}. $$

*Proof.* The tools and properties evoked in this proof can be found in e.g. [16] or [24]. Gagliardo-Nirenberg inequality yields

$$ \|U_0(t)f\|_{L^p(\mathbf{R}^d)} \lesssim \|U_0(t)f\|_{L^2(\mathbf{R}^d)}^{1-\delta(p)} \|\nabla U_0(t)f\|_{L^2(\mathbf{R}^d)}^{\delta(p)}, \quad \delta(p) := d\left(\frac{1}{2} - \frac{1}{p}\right), $$

and since $U_0(t)$ is unitary on $L^2$ and commutes with $\nabla$,

$$ \|U_0(t)f\|_{L^p(\mathbf{R}^d)} \lesssim \|f\|_{L^2(\mathbf{R}^d)}^{1-\delta(p)} \|\nabla f\|_{L^2(\mathbf{R}^d)}^{\delta(p)}, \quad \forall t \in \mathbf{R}. $$

For $|t| > 1$, we recall that the operator $J(t) = x + it\nabla$ can be factorised as

$$ J(t)f = it e^{i\frac{|x|^2}{2t}} \nabla \left( f e^{-i\frac{|x|^2}{2t}} \right), $$

so Gagliardo-Nirenberg inequality also yields

$$ \|f\|_{L^p(\mathbf{R}^d)} \lesssim \frac{1}{|t|^{\delta(p)}} \|f\|_{L^2(\mathbf{R}^d)}^{1-\delta(p)} \|J(t)f\|_{L^2(\mathbf{R}^d)}^{\delta(p)}. $$

Moreover, $J(t) = U_0(t)xU_0(-t)$, so, since $U_0(t)$ is unitary on $L^2$,

$$ \|U_0(t)f\|_{L^p(\mathbf{R}^d)} \lesssim \frac{1}{|t|^{\delta(p)}} \|f\|_{L^2(\mathbf{R}^d)}^{1-\delta(p)} \|xf\|_{L^2(\mathbf{R}^d)}^{\delta(p)}, $$

hence the lemma.

□

**2.2. Proof of Theorem 1.2.** The property

$$ \|U_0(-t)u(t) - u_{\pm}\|_{\Sigma} \xrightarrow{t \to \pm\infty} 0, $$

together with Lemma 2.1, Sobolev embedding, and Lemma 2.2, then easily yield (1.11), (1.12) and (1.13).

Formally, the centre of mass

$$ \bar{x}(t) = \int_{\mathbf{R}^d} x|u(t, x)|^2 dx $$

evolves according to

$$ \begin{aligned} \frac{d}{dt} \bar{x}(t) &= 2 \operatorname{Re} \int_{\mathbf{R}^d} x \bar{u}(t, x) \partial_t u(t, x) dx = - \operatorname{Im} \int_{\mathbf{R}^d} x \bar{u}(t, x) \Delta u(t, x) dx \\ &= \operatorname{Im} \int_{\mathbf{R}^d} \bar{u}(t, x) \nabla u(t, x) dx = P, \end{aligned} $$
and so its derivative is constant. This identity can be made rigorous by resuming
the argument given in the proof of [16, Lemma 6.5.2], which consists, in the present
case, in considering as an intermediate quantity

$$
\bar{x}_{\varepsilon}(t) = \int_{\mathbb{R}^d} xe^{-\varepsilon|x|^2} |u(t, x)|^2 dx.
$$

Let

$$
z(t) = \operatorname{Re} \int_{\mathbb{R}^d} \bar{u}(t, x) J(t) u(t, x) dx = \int_{\mathbb{R}^d} \bar{u}(t, x) J(t) u(t, x) dx,
$$

where $J(t) = x + it\nabla$ was introduced in the proof of Lemma 2.2. The reason why
the imaginary part of the above integral is zero is straightforward, by writing

$$
\int_{\mathbb{R}^d} \bar{u}(t, x) J(t) u(t, x) dx = \int_{\mathbb{R}^d} x \bar{u}(t, x) u(t, x) dx + it \int_{\mathbb{R}^d} \bar{u}(t, x) \nabla u(t, x) dx.
$$

More precisely, expanding,

$$
z(t) = \bar{x}(t) - tP = \bar{x}(0) = \int_{\mathbb{R}^d} x|u_0(x)|^2 dx.
$$

On the other hand, the factorisation $J(t) = U_0(t)xU_0(-t)$ and the assumption
$\|U_0(-t)u(t) - u_\pm\|_{\Sigma} \to 0$ yield, with $\langle f, g \rangle = \int f \bar{g}$, and since $U_0(t)^* = U_0(-t)$,

$$
\begin{align*}
z(t) &= \int_{\mathbb{R}^d} \overline{u(t,x)} U_0(t) x U_0(-t) u(t,x) dx = \langle U_0(t) x U_0(-t) u(t), u(t) \rangle \\
&= \langle x U_0(-t) u(t), U_0(-t) u(t) \rangle \xrightarrow{t \to \pm\infty} \langle x u_\pm, u_\pm \rangle = \int_{\mathbb{R}^d} x |u_\pm(x)|^2 dx,
\end{align*}
$$

hence (1.14).

To prove (1.15), we use the lens transform (1.16). We first emphasise the fact
that the lens transform yields an alternative proof of (1.13)-(1.14), since, like in the
linear case (discovered in [22]), $v$, solution to (1.17), satisfies

$$
\begin{align*}
\frac{d}{dt} \operatorname{Im} \int_{\mathbb{R}^d} \bar{v}(t, x) \nabla v(t, x) dx &= \int_{\mathbb{R}^d} x |v(t, x)|^2 dx, \\
\frac{d}{dt} \int_{\mathbb{R}^d} x |v(t, x)|^2 dx &= \operatorname{Im} \int_{\mathbb{R}^d} \bar{v}(t, x) \nabla v(t, x) dx.
\end{align*}
$$

This implies

$$
\begin{align*}
\operatorname{Im} \int_{\mathbb{R}^d} \bar{v}(t, x) \nabla v(t, x) dx &= \cos t \times \operatorname{Im} \int_{\mathbb{R}^d} \bar{u}_0(x) \nabla u_0(x) dx + \sin t \times \int_{\mathbb{R}^d} x |u_0(x)|^2 dx, \\
\int_{\mathbb{R}^d} x |v(t, x)|^2 dx &= \cos t \times \int_{\mathbb{R}^d} x |u_0(x)|^2 dx + \sin t \times \operatorname{Im} \int_{\mathbb{R}^d} \bar{u}_0(x) \nabla u_0(x) dx,
\end{align*}
$$

so (1.13)-(1.14) follow from Plancherel equality and Lemma 1.6, since

$$
\operatorname{Im} \int_{\mathbb{R}^d} \bar{v}(t, x) \nabla v(t, x) dx = \frac{1}{i} \int_{\mathbb{R}^d} \bar{v}(t, x) \nabla v(t, x) dx.
$$

In the case $\sigma = 2/d$, (1.17) is autonomous,

$$
(2.1) \qquad i \partial_t v + \frac{1}{2} \Delta v = \frac{|x|^2}{2} v + |v|^{4/d} v, \quad v|_{t=0} = u|_{t=0} = u_0,
$$

and we know that if $v|_{t=0} \in \Sigma$, then (2.1) has a unique, global solution $v \in C(\mathbb{R}; \Sigma) \cap C^1(\mathbb{R}; \Sigma^*)$ (see [16, Section 9.2] or [9]).
**Lemma 2.3** (From Lemma 3.1 in [9]). Let $u_0 \in \Sigma$: the solution $v \in C(\mathbf{R}; \Sigma)$ to (2.1) satisfies

$$ (2.2) \qquad \frac{d}{dt} \left( \|xv(t) \sin t - i \cos t \nabla v(t)\|_{L^2}^2 + \frac{2d}{2+d} \cos^2 t \|v(t)\|_{L^{2+4/d}}^{2+4/d} \right) = 0, $$

$$ (2.3) \qquad \frac{d}{dt} \left( \|xv(t) \cos t + i \sin t \nabla v(t)\|_{L^2}^2 + \frac{2d}{2+d} \sin^2 t \|v(t)\|_{L^{2+4/d}}^{2+4/d} \right) = 0. $$

We can now complete the proof of Theorem 1.2. For $u_- \in \Sigma$, let $u \in C(\mathbf{R}; \Sigma)$ be given by the wave operator, with $u|_{t=0} = u_0$. We define $v$ as the solution to (2.1). Considering the constant of motion appearing in (2.2) at times $t = -\pi/2, 0$ and $\pi/2$, we get

$$ \|xv\left(-\frac{\pi}{2}\right)\|_{L^2}^2 = \|\nabla u_0\|_{L^2}^2 + \frac{2d}{2+d} \|u_0\|_{L^{2+4/d}}^{2+4/d} = \|xv\left(\frac{\pi}{2}\right)\|_{L^2}^2. $$

Using Lemma 1.6, this yields (1.12).

Similarly, considering the constant of motion appearing in (2.3) at times $t = -\pi/2, 0$ and $\pi/2$,

$$ \begin{aligned} \|\nabla v\left(-\frac{\pi}{2}\right)\|_{L^2}^2 + \frac{2d}{2+d} \|v\left(-\frac{\pi}{2}\right)\|_{L^{2+4/d}}^{2+4/d} &= \|xu_0\|_{L^2}^2 \\ &= \|\nabla v\left(\frac{\pi}{2}\right)\|_{L^2}^2 + \frac{2d}{2+d} \|v\left(\frac{\pi}{2}\right)\|_{L^{2+4/d}}^{2+4/d}, \end{aligned} $$

and Lemma 1.6 yields (1.15).

## 2.3. Rotating points for the scattering operator in the $L^2$-critical case.

We recall the main result from [10], and sketch the main arguments of the proof:

**Proposition 2.4** (From [10]). Let $d \ge 1$ and $\sigma = 2/d$ in (1.1). For any $\theta \in [0, 2\pi)$, there exist infinitely many functions $u_- \in \Sigma$ such that $S(u_-) = e^{i\theta}u_-$.

As (1.1) is invariant under the gauge transform $u \mapsto e^{i\omega}u$ for any constant $\omega \in \mathbf{R}$, we emphasise that the above statement holds even modulo this equivalence relation. The proof actually allows to consider $u_0 = W_-u_-(= \psi$ below) real-valued.

*Sketch of the proof.* Using the lens transform (1.16), Lemma 1.6 shows that it suffices to construct a solution $v$ to (1.17) such that

$$ v\left(\frac{\pi}{2}, x\right) = e^{i\theta} e^{-id\pi/2} v\left(-\frac{\pi}{2}, -x\right), \quad \forall x \in \mathbf{R}^d. $$

Since (1.17) is autonomous in the $L^2$-critical case $\sigma = 2/d$, we can seek such a special solution of the form $v(t, x) = e^{-i\nu t}\psi(x)$, where $\psi$ is a radial solution to the elliptic problem

$$ (2.4) \qquad \nu\psi = H\psi + |\psi|^{4/d}\psi, \quad H = -\frac{1}{2}\Delta + \frac{|x|^2}{2}, $$

and $\nu$ belongs to the set

$$ \left\{ \frac{d}{2} - \frac{\theta}{\pi} + 2j, \quad j \in \mathbb{N} \setminus \{0\} \right\}. $$

It is then easy to see that for any $\nu > d/2$ and any $0 < \sigma < \frac{2}{(d-2)_+}$, there exists a radial solution $\psi \in \Sigma$ to

$$ (2.5) \qquad \nu\psi = H\psi + |\psi|^{2\sigma}\psi, $$

by minimising the action of the linear case

$$ I(\psi) = \langle H\psi, \psi \rangle - \nu \langle \psi, \psi \rangle, $$
under the nonlinear constraint

$$ \psi \in M = \left\{ \psi \in \Sigma, \psi(x) = \psi(|x|); \frac{1}{1+\sigma} \int_{\mathbb{R}^d} |\psi(x)|^{2+2\sigma} = 1 \right\}. $$

The uncertainty principle and the compactness of the resolvent of H imply that the infimum is negative, and achieved by a function $\tilde{\psi} \in M$. One concludes by examining the sign of the Lagrange multiplier and using the homogeneity of the nonlinearity. □

We emphasise that the elliptic argument does not use the fact that the nonlinearity is $L^2$-critical, as we solve (2.5). This property is crucial however to reduce the proof of Proposition 2.4 to the construction of a (radial) solution to (2.5), since if $\sigma \neq 2/d$, the nonlinearity in the equation obtained after the lens transform, (1.17), depends on time, and it is hopeless to seek a special solution in the form of a solitary wave $v(t, x) = e^{-i\nu t}\psi(x)$. It is therefore unclear whether the statement of Proposition 2.4 can be extended to other powers than $\sigma = 2/d$ or not. One might think for instance of an argument based on Leray-Schauder degree theory (see e.g. [19, 37]), to change the power of $\cos t$ from zero (like in the proof of Proposition 2.4) to $d\sigma - 2$ like in (1.17). The numerical simulations that we present in Section 3 suggest that this strategy may not be successful, and that Proposition 2.4 may be bound to the $L^2$-critical case.

2.4. **Long range scattering.** In this section, we discuss only the one-dimensional case $d=1$. As recalled in the introduction, when $\sigma = 1$ (cubic nonlinearity),

$$ (2.6) \qquad i\partial_t u + \frac{1}{2}\partial_x^2 u = |u|^2 u, \quad x \in \mathbb{R}, $$

the solution $u$ cannot behave asymptotically like the linear evolution of a scattering state $u_+$, unless we deal with the zero solution. The large time dynamics of $u$ becomes more complicated to describe than with the mere linear group $U_0(t)$. Without entering into details, we summarise the main results from e.g. [27, 28, 31, 39].

Given asymptotic states $u_-$ and $u_+$, the long range phase corrections (different whether $t \to -\infty$ of $t \to +\infty$) are given by

$$ (2.7) \qquad \Phi_{\pm}(t, x) := \mp \left|\hat{u}_{\pm}\left(\frac{x}{t}\right)\right|^2 \log|t|. $$

As proven initially in [39], given $u_-$ (sufficiently smooth, localised, and small), there exists a unique solution $u$ to (1.1) such that

$$ u(t, x) \underset{t \to -\infty}{\sim} e^{i\Phi_{-}(t,x)} U_0(t) u_{-}(x) \underset{t \to -\infty}{\sim} e^{i\Phi_{-}(t,x)+i\frac{x^2}{2t}} \frac{1}{(it)^{1/2}} \hat{u}_{-}\left(\frac{x}{t}\right). $$

Note that the phase modification $\Phi_-$ is by no means negligible as $t \to -\infty$: it encodes long range effects. The above mentioned smallness assumption was removed very recently, in [33], under various (limited) smoothness assumptions.

The modified asymptotic completeness is similar. As proven initially in [27], given $u_0$ (sufficiently smooth, localised, and small), there exists $u_+$ such that the solution $u$ to (2.6) with $u|_{t=0} = u_0$ satisfies

$$ u(t) \underset{t \to +\infty}{\sim} e^{i\Phi_{+}(t)} U_0(t) u_+. $$

We note that the argument in the proof of [27, Theorem 1.2] (see also [31]) goes as follows:

* First, it is proven that, setting $f(t) = U_0(-t)u(t)$, there exists $W \in L^\infty \cap L^2$ and $\beta > 0$ such that

$$ \hat{f}(t) \exp \left( -i \int_1^t |\hat{u}(s)|^2 \frac{ds}{s} \right) - W = \mathcal{O}(t^{-\beta}) \quad \text{as } t \to \infty, \text{ in } L^2 \cap L^\infty. $$
• Then, it is established that there exists $\varphi \in L^\infty$ such that

$$ \int_1^t |\hat{u}(s)|^2 \frac{ds}{s} - |W|^2 \log t - \varphi = \mathcal{O}(t^{-\beta}), \quad \text{in } L^\infty. $$

• In view of these two results, one infers (using a refined version of Lemma 2.1),

$$ u(t,x) = \frac{1}{(it)^{1/2}} W\left(\frac{x}{t}\right) \exp\left(i\frac{x^2}{2t} - i\left|W\left(\frac{x}{t}\right)\right|^2 \log t - i\varphi\left(\frac{x}{t}\right)\right) + \rho(t,x), $$

with

$$ \|\rho(t)\|_{L^2} = \mathcal{O}(t^{-\beta}), \quad \|\rho(t)\|_{L^\infty} = \mathcal{O}(t^{-1/2-\beta}). $$

In particular, Lemma 2.1 shows that the modified asymptotic state $u_+$ is given by $\hat{u}_+ = We^{-i\varphi}$. We emphasise the fact that in the argument from [27] (or [31]), the rate $\beta$ may depend on the size of $u_0$.

When using the lens transform, a feature of the cubic one-dimensional case is that the cosine factor in the nonlinearity in (1.17) ceases to be integrable near $t = \pm\pi/2$. Indeed, as the result by Barab [1] evoked in the introduction shows that one cannot have $u(t) \sim U_0(t)u_-$ as $t \to -\infty$, unless $u = u_- \equiv 0$, Lemma 1.6 suggests that $v$ may indeed have some singularity as $t \to \pm\pi/2$. This is consistent with the fact that a nontrivial phase shift $\Phi_\pm$ must be introduced in order to define the modified scattering theory in this case.

In view of the construction of $u_+$ described above, and of Lemma 1.6, when dealing with (1.17) in the case $d = \sigma = 1$, it is rather natural to first consider

$$ w(t,x) = v(t,x) \exp \left( i \int_0^t |v(s,x)|^2 \frac{ds}{\cos s} \right). $$

Indeed, in view of (1.16), Lemma 2.1, and the construction of the modified scattering operator, we have

$$ \int_0^t |v(s,x)|^2 \frac{ds}{\cos s} \underset{t \to \frac{\pi}{2}}{\sim} \int_0^t |\hat{u}_+(x)|^2 \frac{ds}{\cos s} \underset{t \to \frac{\pi}{2}}{\sim} |\hat{u}_+(x)|^2 \log \left(\frac{\pi}{2} - t\right) \underset{t \to \frac{\pi}{2}}{\sim} |\hat{u}_+(x)|^2 \log \tan t, $$

as well as

$$ (2.8) \qquad v(t,x) \underset{t \to \frac{\pi}{2}}{\sim} \hat{u}_+(x) e^{-i|\hat{u}_+(x)|^2 \log \tan t}, $$

which confirms the fact that $v$ is singular as $t \to \pi/2$, but $w$ is not,

$$ w(t,x) \underset{t \to \frac{\pi}{2}}{\longrightarrow} \hat{u}_+(x). $$

We also check that $w$ solves

$$ i\partial_t w = (Hv) \exp \left( i \int_0^t |v(s,x)|^2 \frac{ds}{\cos s} \right) = (Hv) \exp \left( i \int_0^t |w(s,x)|^2 \frac{ds}{\cos s} \right), $$

where $H = -\frac{1}{2}\partial_x^2 + \frac{x^2}{2}$. As $w$ is continuous at $t = \pm\pi/2$, this formulation could be expected to be more tractable numerically, and compatible with iterative process. However, due to the interaction of $w$ with $v$ in the above formulation, our current numerical approach does not directly benefit from this observation. Instead, in order to get stable numerical simulations, we have kept $v$ as the main object of study.
### 3. NUMERICAL COMPUTATION OF THE SCATTERING OPERATOR

3.1. **Methodology.** As mentioned in Section 1.4 we can exploit the lens transform (1.16) to compactify time and localise solution values in space, thus providing the basis for efficient numerical computation of the scattering operator S (cf. (1.8)). For this we can use the following procedure:

(I) For a given initial condition $u_-$ we compute the initial state $v(-\pi/2, x)$ using (1.18);

(II) We propagate this initial state forward to $t_{\text{lens}} = \pi/2$ using (1.17);

(III) Finally, we compute $S(u_-) := u_+$ using (1.18).

A natural spatial discretisation for (1.17) is a Hermite spectral method (cf. [46]) with the following $L^2(\mathbf{R}^d)$-orthonormal basis:

$$ \mathcal{H}_m(x) = \prod_{j=1}^{d} \left( H_{m_j}(x_j) e^{-\frac{1}{2} x_j^2} \right), \quad m \in \mathbf{N}^d, $$

where $H_{m_j}$ denotes the Hermite polynomial of degree $m_j \in \mathbf{N}$ normalised with respect to the weight $w(x) = \exp(-x^2)$. The $\mathcal{H}_m$ are eigenfunctions of the Schrödinger operator with Harmonic potential, $H$, with

$$ (3.1) \qquad H(\mathcal{H}_m) = \left( \frac{d}{2} + \sum_{j=1}^{d} m_j \right) \mathcal{H}_m, $$

thus forming a convenient basis for a splitting approach to (II). Moreover, this basis allows for the direct computation of $\| \cdot \|_{\Sigma}$: introducing for $k \in \mathbf{N}$,

$$ \Sigma^k = \{ f \in L^2(\mathbf{R}^d), \|f\|_{\Sigma^k} := \|f\|_{H^k(\mathbf{R}^d)} + \|x\|^k f\|_{L^2(\mathbf{R}^d)} < \infty \}, $$

we have the following equivalence of norms (see e.g. [2, 30])

$$ (3.2) \qquad \|f\|_{\Sigma^k}^2 \sim \sum_{m \in \mathbf{N}^d} \lambda_m^k |\alpha_m|^2, \quad \lambda_m = \frac{d}{2} + \sum_{j=1}^{d} m_j, \quad \text{for } f(x) = \sum_{m \in \mathbf{N}^d} \alpha_m \mathcal{H}_m(x). $$

We thus expand our initial conditions in this basis

$$ (3.3) \qquad u_-(x) \approx \sum_{\substack{m \in \mathbf{N}^d \\ 0 \le m \le M-1}} \alpha_m^- \mathcal{H}_m(x), $$

and we note that the basis $\mathcal{H}_m$ satisfies a range of useful identities which facilitate the quick computation of Fourier transforms and derivatives of functions of the form (3.3). In particular, there is a numerically stable way of transforming between Hermite coefficients and function values introduced by [3] and explained in further detail in [36] (cf. also the implementation in Julia in [50, 47, 44]) which is briefly recalled for completeness in Appendix A. Some further identities relevant to the computations conducted in this work are listed in Appendix B. As a result of these identities and of (3.1), this basis lends itself naturally to the integration of (1.17). In order to perform step (II) above, we will thus resort to Lie-splitting of (1.17), by considering the following two subproblems

$$ (3.4) \qquad i \partial_t v = H v, \quad i \partial_t v = (\cos t)^{d\sigma-2} |v|^{2\sigma} v. $$

Note that there is a second natural choice for the splitting in (3.4), which is to combine the nonlinearity and the harmonic potential in one subproblem and keep the Laplacian part of the linear term separate. This second approach lends itself naturally to a Fourier pseudospectral spatial discretisation, whereby one has to choose an appropriate bounding box for the domain $\mathbf{R}^d$. In principle, it was observed by [46] that the numerical performances of both methods are comparable, but we note
that in our case it is crucial to localise dispersive effects for the reliable numerical computation of solutions as $t \to \pm\infty$ and thus it is natural to use (3.4).

### 3.1.1. Error analysis.
Using the identities in Appendix B, the Fourier transforms in steps (I) & (III) can be performed exactly, so the only error incurred in our method for computing $S$ is through the spatial discretisation and the splitting method. For the splitting part, corresponding convergence results are already available in the literature. In particular, for high-regularity solutions the convergence of the aforementioned splitting method was proved by Gauckler [23]. In later work, discrete Strichartz estimates were used in [11] to prove convergence of the Lie splitting method in the low-regularity régime. For completeness we recall the main theorem from [23], adapted to our current setting. Note that [23] is focused on the Strang splitting, but the corresponding result for the Lie splitting employed in the present work follows straightforwardly from a similar analysis. In the following we denote the splitting method approximation of $v_+$ by $v_+^{M,\tau}$, where we used $M \in \mathbb{N}$ Hermite modes, and where $\tau$ is the timestep.

**Theorem 3.1** (From Theorem 3.4 in [23]). Suppose for a given $v(-\pi/2, x)$ the corresponding exact solution of (1.17) is given by $v(\pi/2, x)$ and that $s > \lceil \frac{d+1}{2} \rceil + 2 + \frac{2d}{3}$ is an integer. Let $R_{s+2} = \sup_{-\pi/2 \le t \le \pi/2} \|v(t, \cdot)\|_{\Sigma^{s+2}}$. Then there exist $h_0, M_0 > 0$ such that the following error bound holds for step sizes $0 < h \le h_0$ and number of Hermite modes $M \ge M_0$:

$$ \|v(\pi/2, \cdot) - v_{+}^{M,\tau}\|_{L^2} \le C \left(M^{1+\frac{d}{3}-\frac{1}{2}s} + \tau\right), $$

where $C$ only depends on $d, s$, and $R_{s+2}$.

**Remark 3.2.** In view of [11], the restriction on $s$ in the above results can be significantly weakened. By using a spectral cut-off, which amounts to imposing a CFL condition in the above method (relating $M$ with $\tau$ – typically $M = 1/\tau$ when $d=1$), it is shown in [11] that for $v(-\pi/2, \cdot) \in \Sigma^2$, the error is again $O(\tau)$ at time $\pi/2$.

Let us now denote the numerical solution of the scattering operator following the above process by $u_+^{N,\tau}$. Then we have the following immediate corollary:

**Corollary 3.3.** Suppose $s$ is as in Theorem 3.1, then there are $\tilde{M}_0, \tilde{h}_0, \tilde{C} > 0$ depending only on $d, s$, and $R_{s+2}$ such that for step sizes $0 < h \le h_0$ and number of Hermite modes $M \ge M_0$,

$$ \|u_+(\cdot) - u_+^{M,\tau}\|_{L^2} \le C \left(M^{1+\frac{d}{3}-\frac{1}{2}s} + \tau\right). $$

### 3.2. Visualisation of the scattering operator.
Having described our methodology above, we can now provide a first visualisation of the scattering operator for a given initial condition. In Figure 1 we can see an example of the action of the scattering operator on the function $u_-(x) = e^{ix-(x-1)^2/2} + e^{-(x+2)^2/4}$.

**Error in conservation laws.** We note that the conservation laws (1.11)-(1.14) provide a useful way of validating our numerical method. Indeed, we can perform the following test to see if our numerical scheme (as described in Section 3.1) preserves these quantities: We begin by initialising $u_-$ in the following form (for us $M = 200, d = 1$)

$$ u_-(x) = \sum_{m=0}^{M-1} U_m \mathcal{H}_m(x), $$

where $U_m \sim U([0, 1] + i[0, 1])$ are drawn uniformly at random in the complex unit square. For each such sample $u_-$ we compute $u_+ := S(u_-)$ and compare the value
<figure><img src="image_2.png" /><figcaption>FIGURE 1. The action of the scattering operator on different initial conditions.</figcaption></figure>

of each conserved quantity in $u_+$ with that of $u_-$, where we use the following notation:

$$ \begin{aligned} \mathcal{I}_1(u) &= \|u\|_{L^2}, \quad \mathcal{I}_2(u) = \|\nabla u\|_{L^2}^2, \quad \mathcal{I}_3(u) = \operatorname{Im} \int_{\mathbb{R}^d} \bar{u}(x) \nabla u(x) dx, \\ \mathcal{I}_4(u) &= \int_{\mathbb{R}^d} x |u(x)|^2 dx. \end{aligned} $$

The results are shown in Figure 2 for two different values of the timestep $\tau_{\text{lens}}$ taken in step (II) of our algorithm (cf. Section 3.1). The figure shows that these quantities are indeed preserved to reasonable accuracy, thus confirming that our methodology for approximating $S$ is consistent with theoretical properties of the scattering operator.

<figure><img src="image_3.png" /><figcaption>FIGURE 2. The error in conservation laws (1.11)-(1.14) for a range of randomly chosen initial conditions.</figcaption></figure>

### 3.3. Rotating-points in the $L^2$-critical case.
As discussed in Section 2.3, the scattering operator $S$ has a selection of fixed/rotating points which are given as solutions to (2.4). We can solve this equation numerically by expanding the unknown
rotating point $\psi$ in the Hermite basis $\{\mathcal{H}_m\}_{m \in \mathbb{N}^d, 0 \le m \le M-1}$ in the form

$$ \psi \approx \sum_{\substack{m \in \mathbb{N}^d \\ 0 \le m \le M-1}} \beta_m \mathcal{H}_m, $$

which transforms (2.4) into a finite dimensional, nonlinear system for the coefficients $\beta$. This system can be solved directly using Matlab's `fsolve` command which uses a trust-region algorithm to solve the nonlinear system. With appropriate initialization (taking $\psi_0$ non-negative), we observed that this solver tends to converge within a few dozen iterations. In practice we found that for higher energy states $j > 2$ it is helpful to initialise this nonlinear solver with corresponding lower energy states, i.e. $\psi_0^{(j)} = \psi^{(j-1)}$. The shape of two rotating points and the corresponding action of the scattering operator can be observed in Figure 3.

<figure><img src="image_4.png" /><figcaption>FIGURE 3. The action of the scattering operator on rotating point initial conditions.</figcaption></figure>

Note since rotating points provide a known reference value for $Su$, we can use this information to verify the convergence properties of our methodology. In Figure 4 we observe the error committed in our approximation of $S$ for both of the aforementioned fixed points as a function of the timestep in the lens-transformed system (1.17) (cf. step (2) in Section 3.1).

3.4. **Finding rotating points in supercritical cases.** Having gained confidence in the accuracy and performance of our numerical approximation of $S$, we can use this technique to numerically test whether we can find rotating points in the
<figure><img src="image_5.png" /><figcaption>FIGURE 4. Convergence behaviour of our method for known reference values in the rotating point case.</figcaption></figure>

supercritical case $\sigma > 2/d$. To test this we performed the following steps for a given $\sigma > 2/d$.

(1) Based on the continuity arguments outlined in Appendix C we solve (2.4) for the specific value of $\sigma > 2/d$, and use this to initialise $u_-$;

(2) For this value of $u_-$ we try to find the optimal $\theta$ such that

$$ \theta_0 = \operatorname{argmin}_{\theta \in [0, 2\pi)} |u_{+}^{N, \tau_{\text{lens}}} - e^{i\theta} u_{-}| $$

(3) We optimise both $u_-$ and $\theta$ simultaneously to match the rotating point condition from the initialisation $u_-$ using Matlab's `fsolve`: Let us denote the Hermite coefficients of $u_-$ by $\alpha_m$ such that $u_-= \sum_{m=0}^{M-1} \alpha_m \mathcal{H}_m(x)$, and find

$$ \tilde{\alpha}, \tilde{\theta} = \operatorname{argmin}_{\alpha, \theta} |u_{+}^{N, \tau_{\text{lens}}} - e^{i\theta} u_{-}| $$

The outcomes of all of those steps are shown in Figure 5, and we observe that while in the critical case (blue solid line with $\sigma = 2.0, \theta = 0.5$) our scattering operator recovers $u_-$ to within an error of size roughly $10^{-6}$, none of our approaches lead to a rotating point for the supercritical régime (noting that all errors are on the order of $10^{-2}$, and this error could not be improved in our experiments, even with finer temporal or spatial resolution in the solver). While we recognise this is not definite conclusive evidence (see Appendix C for more precise comments), this experiment provides a numerical indication that if rotating points exist in the supercritical régime, they may not be located in the proximity of rotating points from the critical régime.

### 3.5. Long-range case.
Finally, we note that we can use our approach also to simulate the long-range case. In this case, we can proceed similarly to the algorithm described in Section 3.1 with the small exception that the Lie splitting of (II) involves the integration of

$$ i\partial_t v = (\cos t)^{-1} |v|^2 v, $$

which has a singularity at $t = \pm \pi/2$.

We overcome this by starting our time-evolution at $-\pi/2 + \tau_{\text{lens}}$ and ending it at $\pi/2 - \tau_{\text{lens}}$, i.e. we omit the first and the final timestep, and in addition we incorporate the phase shift (2.7) in steps (I) & (III). Figure 6c provides numerical evidence that our approach seems to recover the convergence of the lens-transformed variable
<figure><img src="image_6.png" /><figcaption>FIGURE 5. The error observed in pseudo-rotating points in the supercritical regime.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>FIGURE 6. The action of the scattering operator and convergence of the numerical method in long-range case.</figcaption></figure>
in modulus, and thus indicates that this approach leads to accurate computation of the scattering operator also in the long-range case. An example of the action of the scattering operator in this régime can be seen in figures 6a & 6b. The numerical study of the long range case is mainly motivated by the question addressed in Section 5.

## 4. APPROACHING THE LONG RANGE THRESHOLD

As recalled in the introduction, a complete (short range) scattering theory is available in the space $\Sigma$ provided that the power $\sigma$ satisfies (1.6), that is

$$ \sigma \geq \frac{2 - d + \sqrt{d^2 + 12d + 4}}{4d} =: \sigma_0(d). $$

We can check that $\sigma_0(d) > 1/d$, and the gap between this value and the threshold for long range scattering, $1/d$, is described as follows, in dimensions $d = 1$ and $2$ (for $d \geq 3$, the condition (1.4) for the existence of wave operators shows that the picture is less complete): if ($d \leq 2$ and) $\sigma > 1/d$,

* The wave operators (mapping $u_-$ to $u_0$) are well-defined from $\Sigma$ to $\Sigma$.

* In the small data case, scattering is known in $\Sigma$: if $u_0 \in \Sigma$ is such that $\|u_0\|_{\Sigma}$ is sufficiently small, then there exists $u_+ \in \Sigma$ such that

$$ (4.1) \qquad \|U_0(-t)u(t) - u_+\|_{\Sigma} \xrightarrow{t \to +\infty} 0. $$

* Some partial asymptotic completeness holds ([4]): for any $u_0 \in \Sigma$, there exists a unique asymptotic state $u_+ \in H^1(\mathbf{R}^d)$, such that

$$ (4.2) \qquad \|U_0(-t)u(t) - u_+\|_{H^1(\mathbf{R}^d)} = \|u(t) - U_0(t)u_+\|_{H^1(\mathbf{R}^d)} \xrightarrow{t \to +\infty} 0. $$

A natural question in view of these two properties is: if $u_0 \in \Sigma$ is large, does $u_+$ belong to $\Sigma$? If yes, does the convergence hold in $\Sigma$, in the sense that (4.1) holds? A first issue when addressing such questions numerically is that the above notion of smallness is implicit. So we have to gradually increase the size of $u_0$ and check whether the convergence still holds in $\Sigma$ up to the final time $\pi/2$ (in terms of $v$).

For a more quantitative statement, resume the operator present in (2.2) and (2.3), and apply them to the identity (1.16):

$$ \begin{aligned} xv(t, x) \sin t - i \cos t \nabla v(t, x) &= \frac{-ie^{-i \frac{|x|}{2}^2} \tan t}{(\cos t)^{d/2}} \nabla u \left( \tan t, \frac{x}{\cos t} \right), \\ xv(t, x) \cos t + i \sin t \nabla v(t, x) &= \frac{e^{-i \frac{|x|}{2}^2} \tan t}{(\cos t)^{d/2}} \left( \frac{x}{\cos t} u \left( \tan t, \frac{x}{\cos t} \right) \right. \\ & \qquad \left. + i \tan t \nabla u \left( \tan t, \frac{x}{\cos t} \right) \right). \end{aligned} $$

Taking the $L^2$-norms yields

$$ \|xv(t) \sin t - i \cos t \nabla v(t)\|_{L^2} = \|\nabla u(\tan t)\|_{L^2}, $$

$$ \|xv(t) \cos t + i \sin t \nabla v(t)\|_{L^2} = \|yu(\tan t) + i \tan t \nabla u(\tan t)\|_{L^2}. $$

In the last term, we recognise the operator $J(\tan t)$, and the identities listed in Lemma 2.2 imply

$$ \|xv(t) \cos t + i \sin t \nabla v(t)\|_{L^2} = \|J(\tan t)u(\tan t)\|_{L^2} = \|xU_0(-\tan t)u(\tan t)\|_{L^2}. $$
In view of (4.2),

$$ \begin{aligned} \|xv(t) \sin t - i \cos t \nabla v(t)\|_{L^2} &= \|\nabla u(\tan t)\|_{L^2} = \|U_0(-\tan t)\nabla u(\tan t)\|_{L^2} \\ &= \|\nabla U_0(-\tan t)u(\tan t)\|_{L^2} \underset{t \to \frac{\pi}{2}}{\longrightarrow} \|\nabla u_+(t)\|_{L^2}. \end{aligned} $$

Therefore, if $\|xv(t) \sin t - i \cos t \nabla v(t)\|_{L^2}$ is unbounded as $t \to \frac{\pi}{2}$, then (4.1) cannot hold. Finally, in view of the identity

$$ \begin{pmatrix} xv(t) \sin t - i \cos t \nabla v(t) \\ xv(t) \cos t + i \sin t \nabla v(t) \end{pmatrix} = \begin{pmatrix} \sin t & -\cos t \\ \cos t & \sin t \end{pmatrix} \begin{pmatrix} xv(t) \\ i \nabla v(t) \end{pmatrix}, $$

and since the determinant of the matrix on the right hand side is one, the unbound- edness of $xv(t) \sin t - i \cos t \nabla v(t)$ in $L^2$ is equivalent to the unboundedness of $v(t)$ in $\Sigma$.

<figure><img src="image_8.png" /><figcaption>FIGURE 7. The time-evolution of the $\Sigma$-norm of the lens-transformed variable $v$ with $\tau_{\text{lens}} = 2^{-14}$, $\|v_0\|_{\infty} = 10$.</figcaption></figure>
We consider the one-dimensional case $d = 1$, where

$$ \sigma_0(1) = \frac{1 + \sqrt{17}}{4} \approx 1.28. $$

Similarly to Section 3.2, the following numerical experiment is performed with randomised initial data of the form

$$ v(t_{\text{lens}} = 0) = \sum_{m=0}^{M-1} \lambda_m^{-1} U_m \mathcal{H}_m(x), $$

where $U_m \sim U([0, 1] + i[0, 1])$ and $\lambda_m$ are as in (3.2). We then evolve this initial data according to (1.17) using (II).

As a reference case, we consider $\sigma = 1.5$: we know that scattering holds in $\Sigma$ for arbitrary large data. For this value, the cosine factor is nontrivial in (1.17), so checking that indeed $v(t, \cdot)$ always converges in $\Sigma$ as $t \to \pi/2$ is a way to verify that the numerical scheme is robust, as illustrated by the top picture in Figure 7.

On the other hand, the value $\sigma = 1.15$ belongs to the “unknown range” $(1, \sigma_0(1))$. The numerical simulation for this value of $\sigma$ is reported in the second picture in Figure 7. We observe that the $\Sigma$-norm of $v$ becomes unbounded as $t \to \pi/2$ (this is to be compared with the top picture, where the norm remains bounded). This suggests that for this value of $\sigma$, asymptotic completeness in $\Sigma$ (like stated in (4.1)) does not hold for large data. In view of the result from [4], this means that the failure in (4.1) comes from the absence of convergence of the moment.

## 5. FOCUSING LONG RANGE CASE

The results recall in Section 2.4 turn out to be essentially independent of the sign in front of the nonlinearity: the nonlinearity may be defocusing or focusing, the main difference being that the signs on the right hand side of (2.7) are swapped in the focusing case. A feature of the focusing one-dimensional cubic case,

$$ (5.1) \qquad i \partial_t u + \frac{1}{2} \partial_x^2 u = -|u|^2 u, $$

is that there exist solitary waves of the form $e^{-i\nu t}\phi_\nu(x)$. The function $\phi_\nu$ must solve

$$ \nu \phi_{\nu} + \frac{1}{2} \phi_{\nu}'' = -|\phi_{\nu}|^2 \phi_{\nu}. $$

The ground state is the only real, positive, and even solution. It is given by the explicit formula

$$ \phi_{\nu}(x) = \sqrt{2\nu} \operatorname{sech} \left( x\sqrt{2\nu} \right). $$

Two “natural” Lebesgue norms are independent of $\nu > 0$: $\|\phi_\nu\|_{L^1}$ and $\|\hat{\phi}_\nu\|_{L^\infty}$. In [7], it is shown (after making the normalisation for the Fourier transform consistent with the present convention) that if

$$ \|\hat{u}_-\|_{L^\infty} < \frac{1}{\sqrt{2}}, $$

then the modified wave operator is well-defined (the smallness assumption for the existence of modified scattering is not explicit). This assumption was relaxed in [33], $\|\hat{u}_-\|_{L^\infty} < 1$. As

$$ \int_{-\infty}^{x} \sech(y) dy = 2 \arctan (e^x), $$

we have $\|\phi_\nu\|_{L^1} = \pi$. We also infer, since $\hat{\phi}_\nu(0) = \frac{1}{\sqrt{2\pi}} \int \phi_\nu = \frac{1}{\sqrt{2\pi}} \|\phi_\nu\|_{L^1}$,

$$ \|\hat{\phi}_\nu\|_{L^\infty} = \sqrt{\frac{\pi}{2}}. $$
It is not known whether either of these two values is sharp for the existence of the
modified wave operator (mapping $u_-$ to $u_0$ in Section 2.4) or its inverse (mapping
$u_0$ to $u_+$ in Section 2.4). The numerical tests presented below suggest that the
$L^\infty$-norm of $\hat{\phi}_\nu$ is not a threshold for the existence of the inverse of the modified
wave operator, in the sense that apparently, this operator is not well-defined for
initial data $\phi$ such that $\|\hat{\phi}\|_{L^\infty} < \|\hat{\phi}_\nu\|_{L^\infty}$.

The parameter $\nu$ in the family of solitary waves must be chosen carefully for the
numerical simulations. If $\nu > 0$ is too small, then $\phi_\nu$ is small and diffuse. The smallness is delicate for precise computation, the diffusion is delicate to avoid boundary
effects in the computational domain. On the other hand, is $\nu$ is too large, then
$\phi_\nu$ is large and peaked $(\phi_\nu \to \pi\delta_{x=0}$ as $\nu \to \infty$), which is also delicate to manage
numerically. We have observed that the value $\nu = 1/2$ is a good compromise, and
report the simulations in that case, in Figure 8.

<figure><img src="image_9.png" /><figcaption>FIGURE 8. The $L^\infty$-norm evolution for $v_0 = \alpha\phi_{1/2}$ with $\tau_{\text{lens}} = 2^{-14}$, $M = 8096$.</figcaption></figure>
The experiments then consist in considering

$$u_0(x) = v_0(x) = \alpha\phi_{1/2}(x), \quad 0 < \alpha \le 1.$$

In order to provide a good confidence level in the observations made in these graphs, we use the highest numerical resolution employed in the previous section, i.e. $\tau_{\text{lens}} = 2^{-14}$, $M = 8096$. We know that for $\alpha = 1$, there is no modified scattering, since the exact solution is $u(t, x) = e^{-it/2}\phi_{1/2}(x)$, and we ask whether or not for small values of $\alpha$, modified scattering may be ruled out.

We do not measure the $\Sigma$-norm of $v$, since when there is modified scattering, the phase modification $\Phi_+$ causes the $\Sigma$-norm of $v$ to be unbounded as $t \to \pi/2$, see (2.8), and stick to the observation of the $L^\infty$-norm. When $u$ disperses like in the linear case, the singular cosine factor $(\cos t)^{-d/2}$ in (1.16) is compensated by $(\tan t)^{-d/2}$ in (1.16). The top picture in Figure 8 shows the case $u_0 = \phi_{1/2}$, for which we know that there is no modified scattering, and not even dispersion. We compare the behaviour of $v$ with the function $(\cos t)^{-1/2}$, which is the factor in front of $u$ in (1.16). The absence of dispersion of the solitary wave is observed by the coincidence of the two curves, up to some time relatively close to $\pi/2$, after which the numerical scheme becomes unstable. The second picture illustrates, from different values of $\alpha$, the convergence or divergence of $|v|$ as time approaches $\pi/2$, with the upper bound in time for stability observed in the case $\alpha = 1$. The growth in the case $\alpha = 0.75$ suggests that the value $\alpha = 1$ may not be the threshold to decide whether there is modified scattering or not, and modified scattering may be ruled out for initial data smaller than the ground state.

## 6. CONCLUSIONS

We have recalled several known features of the scattering operator associated to the nonlinear Schrödinger equation (1.1), and established two new conservation laws. These analytical properties have served as tests to validate the numerical approach presented here, which relies on the lens transform. This transform avoids spatial boundary effects since the harmonic oscillator is confining. It also avoids the issue of computing over large time and accumulating numerical errors, as time is compactified thus allowing for efficient, reliable simulations. By using Hermite functions (which form an eigenbasis of the harmonic oscillator) and Lie time splitting, our simulations are in very good agreement with known properties of the scattering operator, even when this operator has to be modified, due to long range effects (only the one-dimensional cubic case is considered here).

This confidence in the numerical approach led us to consider some questions remaining open on the analytical level at the present time. Our simulations suggest the following conjectures:

* In the $L^2$-supercritical régime $\sigma > 2/d$, there may be no rotating point for the scattering operator.

* In the intermediate range $1/d < \sigma < \sigma_0(d)$, scattering may not hold in $\Sigma$ for large data.

* In the focusing cubic one-dimensional case, the ground state(s) may not be the minimal object to rule out the existence of modified scattering.

Apart from securing these conjectures with more numerical experiments, possibly with different approaches, some next natural questions could be:

* In the intermediate range $1/d < \sigma < \sigma_0(d)$, for large data, is there convergence of intermediate momenta in (4.1), that is, is there $s \in (0, 1)$ such that $x \mapsto \langle x \rangle^s u_+(x) \in L^2(\mathbf{R}^d)$ and

$$\|\langle x \rangle^s (U_0(-t)u(t, x) - u_+(x))\|_{L^2} \underset{t \to +\infty}{\longrightarrow} 0?$$
* In the focusing cubic one-dimensional case, which criterion regarding the initial state $u_0$ makes it possible to decide whether there is modified scattering or not?

## APPENDIX A. FAST, STABLE HERMITE TRANSFORMS

As part of the splitting method to solve (1.17), we have to be able to efficiently transform between function values and Hermite coefficients. In particular, let $\mathbf{x} = (x_0, \dots, x_{M-1})$ be the Gauss-Hermite quadrature nodes on $\mathbf{R}$ of degree $M$. For a function $f(x)$ we can thus approximate its Hermite expansion coefficients as follows: Let $f(x) = \sum_{m \ge 0} c_m \mathcal{H}_m(x)$, then

$$c_m = \int_{\mathbf{R}} f(x) \mathcal{H}_m(x) dx = \int_{\mathbf{R}} f(x) \mathbf{H}_m(x) e^{\frac{x^2}{2}} e^{-x^2} dx \approx \sum_{k=0}^{M-1} w_k f(x_k) \mathbf{H}_m(x_k) e^{\frac{x_k^2}{2}},$$

where $w_k$ are the Gauss-Hermite quadrature weights. In other words the transformation from function values to Hermite coefficients can be expressed as the following matrix multiplication

$$\mathbf{c} = \mathbf{T} f(\mathbf{x}),$$

where $\mathbf{T}_{km} = w_m \mathbf{H}_k(x_m) \exp(x_m^2/2)$. The entries of $\mathbf{T}$ grow super-exponentially in the number of Hermite modes $M$, and as a result both the assembly of the matrix and the computation with this matrix become unstable even for moderate values of $M$. This was realised in [3], and further developed in [36] (see also the Julia implementations in [50, 47, 44]), who proposed a stable way of computing the entries of $\mathbf{T}$ (see [3, Appendix]), and a stable way of computing with $\mathbf{T}$ (see [36]). The basis for this stable construction is the following recurrence for the normalised Hermite polynomials

$$\mathbf{H}_{k+1}(\mathbf{x}) = \sqrt{\frac{2}{k+1}} \mathbf{x} \mathbf{H}_k(\mathbf{x}) - \sqrt{\frac{k}{k+1}} \mathbf{H}_{k-1}(\mathbf{x}),$$

and the reader is referred to [36] for full details on the algorithm used in our implementation.

## APPENDIX B. AUXILIARY IDENTITIES FOR HERMITE EXPANSIONS

Here we recall a few identities for Hermite expansions which are used in the efficient implementation of our algorithm for computing the lens transform (cf. Section 3.1). For the following identities, we suppose that we have a representation of a function $f$ in the form

$$f(x) = \sum_{\substack{m \in \mathbf{N}^d \\ 0 \le m \le M-1}} \alpha_m \mathcal{H}_m(x).$$

B.1. **Fourier transform.** The Fourier transform of $f$ can then be written as

$$\begin{aligned} \mathcal{F} f(\xi) &= \sum_{\substack{m \in \mathbf{N}^d \\ 0 \le m \le M-1}} \alpha_m \mathcal{H}_m(\xi) \\ &= \sum_{\substack{m \in \mathbf{N}^d \\ 0 \le m \le M-1}} \alpha_m (-i)^m \mathcal{H}_m(\xi), \end{aligned}$$

where we used the well-known expression for the Fourier transform of Hermite functions (cf. [20, 18.17.21_3]).
### B.2. Differentiation.
In the interest of notational simplicity, we restrict the following to the one dimensional case. The case $d \ge 2$ can be treated similarly by tensor product extension. In order to compute the derivative of a Hermite expansion, we note that the normalised Hermite polynomials are of the form

$$H_m = \frac{1}{\sqrt{2^m \sqrt{\pi m}!}} \tilde{H}_m,$$

where $\tilde{H}_m$ satisfies the identity

$$\frac{d}{dx} (\tilde{H}_m e^{-x^2/2}) = x \tilde{H}_m e^{-x^2/2} - \tilde{H}_{m+1} e^{-x^2/2}, \quad m \ge 0,$$

cf. [20, 18.9.26]. This leads to the identity

$$f'(x) = \sum_{m \ge 0} \beta_m H_m e^{-x^2/2},$$

where $\beta_m = \gamma_m - \sqrt{2(m+1)}\alpha_{m+1}$ and $\gamma_m$ are the Hermite coefficients of $xf(x)$, i.e.

$$xf(x) = \sum_{m \ge 0} \gamma_m H_m e^{-x^2/2}.$$

## APPENDIX C. CONTINUITY OF SCATTERING OPERATOR WITH RESPECT TO $\sigma$

In this section, we sketch an analytic explanation for Figure 5, in the case $d=1$, where we consider a slightly $L^2$-supercritical nonlinearity, $\sigma = 2+\varepsilon$, $0 < \varepsilon \ll 1$. Consider the solutions $v$ and $v^\varepsilon$ corresponding to the $L^2$-critical case and this slightly $L^2$-supercritical case, respectively, with possibly different data at $-\frac{\pi}{2}$,

$$(C.1) \quad \begin{cases} i \partial_t v + \frac{1}{2} \partial_x^2 v = \frac{x^2}{2} v + |v|^4 v, & v|_{t=-\pi/2} = v_- \in \Sigma, \\ i \partial_t v^\varepsilon + \frac{1}{2} \partial_x^2 v^\varepsilon = \frac{x^2}{2} v^\varepsilon + (\cos t)^\varepsilon |v^\varepsilon|^{4+2\varepsilon} v^\varepsilon, & v|_{t=-\pi/2} = v_- \in \Sigma. \end{cases}$$

Since $v^\varepsilon$ will eventually be close to $v_-$ ($\ne 0$), we assume

$$\sup_{\varepsilon \in (0,1)} \|v^\varepsilon\|_{\Sigma} \le 2 \|v_-\|_{\Sigma}.$$

As the nonlinearity is defocusing, we infer a uniform bound of the form

$$\sup_{-\pi/2 \le t \le \pi/2} \|v(t)\|_{\Sigma} + \sup_{\varepsilon \in (0,1)} \sup_{-\pi/2 \le t \le \pi/2} \|v^\varepsilon(t)\|_{\Sigma} \le C (\|v_-\|_{\Sigma}).$$

Since we are in the one-dimensional case, $H^1(\mathbf{R}) \hookrightarrow L^\infty(\mathbf{R})$, and we deduce that there exists $M \ge 1$ (this lower bound is imposed to simplify the formulas below) such that

$$(C.2) \quad \sup_{-\pi/2 \le t \le \pi/2} \|v(t)\|_{L^\infty} + \sup_{\varepsilon \in (0,1)} \sup_{-\pi/2 \le t \le \pi/2} \|v^\varepsilon(t)\|_{L^\infty} \le M.$$

We readily observe that the difference $w = v - v^\varepsilon$ solves

$$i \partial_t w + \frac{1}{2} \partial_x^2 w = \frac{x^2}{2} w + |v|^4 v - (\cos t)^\varepsilon |v^\varepsilon|^{4+2\varepsilon} v^\varepsilon \quad ; \quad w|_{t=-\pi/2} = v_- - v_-^\varepsilon.$$

Inserting the term $\pm (\cos t)^\varepsilon |v|^{4+2\varepsilon} v$, energy estimate yields, for $t \in [-\frac{\pi}{2}, \frac{\pi}{2}]$,

$$\|w(t)\|_{L^2} \le \|w(-\pi/2)\|_{L^2} + 2 \int_{-\pi/2}^{t} (\cos s)^\varepsilon \| |v|^{4+2\varepsilon} v - |v^\varepsilon|^{4+2\varepsilon} v^\varepsilon \|_{L^2} ds \\ + 2 \int_{-\pi/2}^{t} \| |v|^4 v - (\cos s)^\varepsilon |v|^{4+2\varepsilon} v \|_{L^2} ds,$$
where the implicit time variable for $v$ and $v^\epsilon$ is $s$. In view of (C.2), for all $s \in [-\frac{\pi}{2}, \frac{\pi}{2}]$,

$$(\cos s)^\epsilon \|v|^{4+2\epsilon}v - |v^\epsilon|^{4+2\epsilon}v^\epsilon\|_{L^2} \le CM^{4+2\epsilon}\|w(s)\|_{L^2} \le CM^6\|w(s)\|_{L^2},$$

for some constant $C$ uniform in $\epsilon \in (0, 1)$. In order to apply Grönwall's lemma, we therefore focus on the source term

$$|v|^4 v - (\cos s)^\epsilon |v|^{4+2\epsilon} v = |v|^4 v - |v|^{4+2\epsilon} v + (1 - (\cos s)^\epsilon) |v|^{4+2\epsilon} v.$$

With $y \ge 0$ a placeholder for $|v|^4$, and up to changing $\epsilon$ with $2\epsilon$ to lighten notations, consider the function

$$f^\epsilon(y) = y - y^{1+\epsilon}.$$

This function is increasing on $[0, \frac{1}{(1+\epsilon)^{1/\epsilon}}]$, decreasing on $(\frac{1}{(1+\epsilon)^{1/\epsilon}}, +\infty)$, and so

$$\sup_{0 \le y \le M^4} |f^\epsilon(y)| \le \max \left( f^\epsilon \left( \frac{1}{(1+\epsilon)^{1/\epsilon}} \right), |f^\epsilon(M^4)| \right),$$

where we have chosen for $M$ the constant from (C.2). We compute

$$f^\epsilon \left( \frac{1}{(1+\epsilon)^{1/\epsilon}} \right) = \frac{1}{(1+\epsilon)^{1/\epsilon}} \left( 1 - \frac{1}{1+\epsilon} \right) = \mathcal{O}(\epsilon),$$

and (since $M \ge 1$)

$$|f^\epsilon(M^4)| = M^{4+4\epsilon} - M^4 = M^4 (e^{4\epsilon \log M} - 1) = \mathcal{O}(\epsilon).$$

We infer from (C.2) and the conservation of the $L^2$-norm for $v$,

$$\int_{-\pi/2}^{\pi/2} \|v|^4 v - |v|^{4+2\epsilon} v\|_{L^2} ds \le \int_{-\pi/2}^{\pi/2} \| |v|^4 - |v|^{4+2\epsilon} \|_{L^\infty} \|v(s)\|_{L^2} ds \\ \le \pi \|v_\perp\|_{L^2} \sup_{0 \le y \le M^4} |f^\epsilon/2(y)| \le C\epsilon.$$

The last part of the source term is

$$\int_{-\pi/2}^{t} (1 - (\cos s)^{\epsilon}) \|v|^{4+2\epsilon} v\|_{L^2} ds \le M^6 \|v_{-}\|_{L^2} \int_{-\pi/2}^{\pi/2} (1 - (\cos s)^{\epsilon}) ds.$$

Taylor formula for the function $g(\epsilon) = (\cos s)^\epsilon$ yields, for $s \in (-\frac{\pi}{2}, \frac{\pi}{2})$,

$$1 - (\cos s)^{\epsilon} = g(0) - g(\epsilon) = -\epsilon \log \cos s \int_{0}^{1} (\cos s)^{\theta\epsilon} d\theta \le \epsilon \log \frac{1}{\cos s}.$$

Since the right hand side is integrable on $(-\frac{\pi}{2}, \frac{\pi}{2})$, we conclude

$$\int_{-\pi/2}^{\pi/2} \|v|^4 v - (\cos s)^\epsilon |v|^{4+2\epsilon} v\|_{L^2} ds \le C\epsilon,$$

for some constant $C$ independent of $\epsilon$. Invoking Grönwall's lemma, we have proved:

**Proposition C.1.** There exists $C$ independent of $\epsilon \in (0, 1)$ such that the solutions to (C.1) satisfy

$$\sup_{-\pi/2 \le t \le \pi/2} \|v(t) - v^\epsilon(t)\|_{L^2} \le C \|v_\perp - v^\epsilon\|_{L^2} + C\epsilon.$$

This result explains why the existence of rotating points for the scattering operator when the nonlinearity is not $L^2$-critical is doubtful in view of our numerical experiments. More precisely, in Figure 5, we consider $\sigma = 2.01$, hence $\epsilon = 10^{-2}$, which corresponds to the order of magnitude in the error observed, compared to the case $\sigma = 2$ where the existence of rotating points is granted analytically and validated numerically.
REFERENCES

[1] J. E. Barab. Nonexistence of asymptotically free solutions for nonlinear Schrödinger equation. *J. Math. Phys.*, 25:3270–3273, 1984.

[2] N. Ben Abdallah, F. Castella, and F. Méhats. Time averaging for the strongly confined nonlinear Schrödinger equation, using almost periodicity. *J. Differential Equations*, 245(1):154–200, 2008.

[3] B. F. Bunck. A fast algorithm for evaluation of normalized hermite functions. *BIT Numerical Mathematics*, 49(2):281–295, 2009.

[4] N. Burq, V. Georgiev, N. Tzvetkov, and N. Visciglia. $H^1$ scattering for mass-subcritical NLS with short-range nonlinearity and initial data in $\Sigma$. *Ann. Henri Poincaré*, 24(4):1355–1376, 2023.

[5] N. Burq and L. Thomann. *Almost sure scattering for the one dimensional nonlinear Schrödinger equation*, volume 1480 of *Mem. Am. Math. Soc*. Providence, RI: American Mathematical Society (AMS), 2024.

[6] N. Burq, L. Thomann, and N. Tzvetkov. Long time dynamics for the one dimensional non linear Schrödinger equation. *Ann. Inst. Fourier*, 63(6):2137–2198, 2013.

[7] R. Carles. Geometric optics and long range scattering for one-dimensional nonlinear Schrödinger equations. *Comm. Math. Phys.*, 220(1):41–67, 2001.

[8] R. Carles. Critical nonlinear Schrödinger equations with and without harmonic potential. *Math. Models Methods Appl. Sci.*, 12(10):1513–1523, 2002.

[9] R. Carles. Remarks on nonlinear Schrödinger equations with harmonic potential. *Ann. Henri Poincaré*, 3(4):757–772, 2002.

[10] R. Carles. Rotating points for the conformal NLS scattering operator. *Dyn. Partial Differ. Equ.*, 6(1):35–51, 2009.

[11] R. Carles. Time splitting and error estimates for nonlinear Schrödinger equations with a potential. Preprint, archived at https://hal.science/hal-04677959, 2024.

[12] R. Carles and I. Gallagher. Analyticity of the scattering operator for semilinear dispersive equations. *Comm. Math. Phys.*, 286(3):1181–1209, 2009.

[13] R. Carles and L. Gosse. Numerical aspects of nonlinear Schrödinger equations in the presence of caustics. *Math. Models Methods Appl. Sci.*, 17(10):1531–1553, 2007.

[14] R. Carles and T. Ozawa. On the wave operators for the critical nonlinear Schrödinger equation. *Math. Res. Lett.*, 15(1):185–195, 2008.

[15] R. Carles and C. Su. Numerical study of the logarithmic Schrödinger equation with repulsive harmonic potential. *Discrete Contin. Dyn. Syst.*, Ser. B, 28(5):3136–3159, 2023.

[16] T. Cazenave. *Semilinear Schrödinger equations*, volume 10 of *Courant Lecture Notes in Mathematics*. New York University Courant Institute of Mathematical Sciences, New York, 2003.

[17] T. Cazenave and F. Weissler. Rapidly decaying solutions of the nonlinear Schrödinger equation. *Comm. Math. Phys.*, 147:75–100, 1992.

[18] G. Chen and J. Murphy. Recovery of the nonlinearity from the modified scattering map. *Int. Math. Res. Not.*, 2024(8):6632–6655, 2024.

[19] J. Cronin. *Fixed points and topological degree in nonlinear analysis*. Mathematical Surveys, No. 11. American Mathematical Society, Providence, RI, 1964.

[20] NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, Release 1.2.4 of 2025-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.

[21] T. Duyckaerts, F. Merle, and S. Roudenko. Maximizers for the Strichartz norm for small solutions of mass-critical NLS. *Ann. Sc. Norm. Super. Pisa Cl. Sci.* (5), 10(2):427–476, 2011.

[22] P. Ehrenfest. Bemerkung über die angenaherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik. *Zeitschrift für Physik*, 45(7-8):455–457, 1927.

[23] L. Gauckler. Convergence of a split-step Hermite method for the Gross-Pitaevskii equation. *IMA J. Numer. Anal.*, 31(2):396–415, 2011.

[24] J. Ginibre. Introduction aux équations de Schrödinger non linéaires. In French. Cours de DEA, 1995. Paris Onze Édition. Archived at http://sites.mathdoc.fr/PMO/PDF/G_GINIBRE-48.pdf.

[25] J. Ginibre and G. Velo. On a class of nonlinear Schrödinger equations. I The Cauchy problem, general case. *J. Funct. Anal.*, 32:1–32, 1979.

[26] J. Ginibre and G. Velo. On a class of nonlinear Schrödinger equations. II Scattering theory, general case. *J. Funct. Anal.*, 32:33–71, 1979.

[27] N. Hayashi and P. Naumkin. Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations. *Amer. J. Math.*, 120(2):369–389, 1998.
[28] N. Hayashi and P. Naumkin. Domain and range of the modified wave operator for Schrödinger equations with a critical nonlinearity. *Comm. Math. Phys.*, 267(2):477–492, 2006.

[29] N. Hayashi and Y. Tsutsumi. Remarks on the scattering problem for nonlinear Schrödinger equations. In *Differential equations and mathematical physics (Birmingham, Ala., 1986)*, volume 1285 of *Lectures Notes in Math.*, page 162–168. Springer, Berlin, 1987.

[30] B. Hellffer. *Théorie spectrale pour des opérateurs globalement elliptiques*, volume 112 of *Astérisque*. Société Mathématique de France, Paris, 1984. With an English summary.

[31] J. Kato and F. Pusateri. A new proof of long-range scattering for critical nonlinear Schrödinger equations. *Differ. Integral Equ.*, 24(9-10):923–940, 2011.

[32] O. Kavian and F. Weissler. Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation. *Michigan Math. J.*, 41(1):151–173, 1994.

[33] M. Kawamoto and H. Mizutani. Modified wave operators for the defocusing cubic nonlinear Schrödinger equation in one space dimension with large scattering data. Preprint, archived at https://arxiv.org/abs/2506.01871, 2025.

[34] R. Killip, J. Murphy, and M. Visan. The scattering map determines the nonlinearity. *Proc. Amer. Math. Soc.*, 151(6):2543–2557, 2023.

[35] R. Killip, J. Murphy, and M. Visan. Determination of Schrödinger nonlinearities from the scattering map. *Nonlinearity*, 38(1):015021, 2025.

[36] G. Maierhofer and M. Webb. A note on stable Hermite transforms, 2025. In preparation.

[37] J. Mawhin. *Topological degree methods in nonlinear boundary value problems*, volume 40 of *Reg. Conf. Ser. Math*. American Mathematical Society (AMS), Providence, RI, 1979.

[38] K. Nakanishi and T. Ozawa. Remarks on scattering for nonlinear Schrödinger equations. *NoDEA Nonlinear Differential Equations Appl.*, 9(1):45–68, 2002.

[39] T. Ozawa. Long range scattering for nonlinear Schrödinger equations in one space dimension. *Comm. Math. Phys.*, 139:479–493, 1991.

[40] A. Poiret, D. Robert, and L. Thomann. Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator. *Anal. PDE*, 7(4):997–1026, 2014.

[41] J. Rauch. *Partial Differential Equations*, volume 128 of *Graduate Texts in Math*. Springer-Verlag, New York, 1991.

[42] A. V. Rybin, G. G. Varzugin, M. Lindberg, J. Timonen, and R. K. Bullough. Similarity solutions and collapse in the attractive Gross-Pitaevskii equation. *Phys. Rev. E (3)*, 62(5, part A):6224–6228, 2000.

[43] H. Sasaki. The inverse scattering problem for Schrödinger and Klein-Gordon equations with a nonlocal nonlinearity. *Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods*, 66(8):1770–1781, 2007.

[44] R. M. Slevinsky, S. Olver, et al. Fast transforms.jl. https://github.com/JuliaApproximation/FastTransforms.jl, 2025. Version 0.17.0.

[45] T. Tao. A pseudoconformal compactification of the nonlinear Schrödinger equation and applications. *New York J. Math.*, 15:265–282, 2009.

[46] M. Thalhammer, M. Caliari, and C. Neuhauser. High-order time-splitting Hermite and Fourier spectral methods. *J. Comput. Phys.*, 228(3):822–832, 2009.

[47] A. Townsend, S. Olver, et al. Fastgaussquadrature.jl. https://github.com/JuliaApproximation/FastGaussQuadrature.jl, 2024. Version 1.0.2.

[48] Y. Tsutsumi. Scattering problem for nonlinear Schrödinger equations. *Ann. Inst. H. Poincaré Phys. Théor.*, 43(3):321–347, 1985.

[49] Y. Tsutsumi and K. Yajima. The asymptotic behavior of nonlinear Schrödinger equations. *Bull. Amer. Math. Soc. (N.S.)*, 11(1):186–188, 1984.

[50] M. Webb. Quantumtimesteppers.jl. https://github.com/marcusdavidwebb/QuantumTimeSteppers.jl, 2025. Accessed on 07/05/2025.

(R. Carles) CNRS, IRMA - UMR 6625, RENNES, FRANCE

Email address: Remi.Carles@math.cnrs.fr

(G. Maierhofer) DEPARTMENT OF APPLIED MATHEMATICS AND THEORETICAL PHYSICS, UNIVERSITY OF CAMBRIDGE, UNITED KINGDOM

Email address: gam37@cam.ac.uk