Laurent Thomann

GROWTH OF SOBOLEV NORMS FOR LINEAR SCHR ÖDINGER OPERATORS

Keywords: 2000 Mathematics Subject Classification. 35Q41; 35B08 Linear Schrödinger equation, time-dependent potential, growth of Sobolev norms, reducibility

Growth of Sobolev norms for linear Schrödinger operators

Introduction and main result

The aim of this paper is to present an example of linear, time-dependent, Schrödinger operator which exhibits optimal polynomial growth of Sobolev norms. Moreover, this operator takes the form H + L (t), where H is an elliptic operator with compact resolvent and where the perturbation L (t) is a small, time-dependent, bounded self-adjoint operator. Our construction is actually entirely explicit and it is based on the study of linear Lowest Landau Level equations (LLL) with a timedependent potential.

In Maspero-Robert [START_REF] Maspero | On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms[END_REF], the authors study linear Schrödinger operators, obtain global wellposedness results and prove very precise polynomial bounds on the possible growth of Sobolev norms under general conditions (see Assumption 1.1 below). We show here that these bounds are optimal.

Our setting is the following: consider the 2-dimensional harmonic oscillator

H = -(∂ 2
x + ∂ 2 y ) + (x 2 + y 2 ) = -4∂ z ∂ z + |z| 2 , where z = x + iy, ∂ z = 1 2 (∂ x -i∂ y ). This operator acts on the space E = u(z) = e -|z| 2 2 f (z) , f entire holomorphic ∩ S ′ (C), and if we define the Bargmann-Fock space E by E = u(z) = e -|z| 2 2 f (z) , f entire holomorphic ∩ L 2 (C), then the so-called special Hermite functions (ϕ n ) n≥0 given by

ϕ n (z) = z n √ πn! e -|z| 2 2 ,
form a Hilbertian basis of E, and are eigenfunctions of H, namely

Hϕ n = 2(n + 1)ϕ n , n ≥ 0.

Let 0 ≤ τ < 1 and set ρ(τ ) = 1 2(1-τ ) ∈ [1/2, ∞). We define the operator H τ = (H + 1) ρ(τ ) , which in turn defines the scale of Hilbert spaces H s τ s≥0 by

H s τ = u ∈ L 2 (C), H s/2 τ u ∈ L 2 (C) ∩ E, H 0 τ = E,
and we denote by L the Lebesgue measure on C.

For X a Banach space, we denote by C b R; X the subspace of C R; X composed of bounded functions:

C b R; X = t → u(t) ∈ C R; X : sup t∈R u(t) X < +∞ .
Similarly, for all k ∈ N we define the space

C k b R, X by C k b R; X = u ∈ C b R; X : ∂ j t u ∈ C b R; X , ∀ 0 ≤ j ≤ k .
Let 0 ≤ τ < 1 and s ≥ 0. For a family L (t) t∈R of continuous linear mappings L (t) : H s τ -→ H s τ , we consider the following assumptions : Assumption 1.1. L (t) t∈R is a family of linear operators which satisfies:

(i) One has t -→ L (t) ∈ C b R; L( H s
τ ) for all s ≥ 0. (ii) For every t ∈ R, L (t) is symmetric w.r.t. the scalar product of

H 0 τ , C vL (t)u dL = C uL (t)v dL, ∀u, v ∈ H 0 τ .
(iii) The family L (t) t∈R is H τ τ -bounded in the sense that t -→ [L (t),

H τ ] H -τ τ ∈ C b R, L( H s τ ) for all s ≥ 0. (iv) For all ℓ ∈ N, one has t -→ L (t) ∈ C ℓ b R; L( H s τ ; H s-ℓτ τ
) for all s ≥ 0.

Finally, for s ≥ 0, we consider the initial value problem

i∂ t u = H τ + L (t) u, (t, z) ∈ R × C, u(t, •) |t=t 0 = u 0 ∈ H s τ , (1.1)
and we are able to state our main result :

Theorem 1.2. For any 0 ≤ τ < 1 and any ǫ > 0, there exists a family of linear operators L (t) t∈R which satisfies Assumption 1.1, so that for all s ≥ 0 : (i) There exists C s > 0 such that

sup t∈R L (t) L( H s τ ) ≤ C s ǫ.
(ii) The problem (1.1) is globally well-posed in H s τ : for any u 0 ∈ H s τ , there exists a unique solution

u(t) := U (t, t 0 )u 0 such that u ∈ C R, H s τ to (1.1). Moreover, U has the group property U (t 2 , t 1 )U (t 1 , t 0 ) = U (t 2 , t 0 ), U (t, t) = I d , ∀ t, t 1 , t 2 ∈ R, and U is unitary in H 0 τ U (t, t 0 )u 0 H 0 τ = u 0 H 0 τ , ∀ t ∈ R.
(iii) Any solution to (1.1), with initial condition

u 0 ∈ H s τ , satisfies for all t ∈ R U (t, t 0 )u 0 H s τ ≤ C u 0 H s τ ǫ(t -t 0 ) s 2(1-τ ) .
(iv) There exists a nontrivial initial condition u 0 ∈ ∩ k≥1 H k τ such that the corresponding solution to (1.1)

satisfies for all t ∈ R U (t, t 0 )u 0 H s τ ≥ c u 0 H s τ ǫ(t -t 0 )
Actually, items (ii) and (iii) directly follow from [START_REF] Maspero | On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms[END_REF]Theorem 1.5]. The novelty in our work is item (iv) which shows that the upper bounds obtained in [START_REF] Maspero | On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms[END_REF]Theorem 1.5] are optimal without further assumptions, even for small perturbations L (t), see item (i). It seems that the example of Theorem 1.2 is the first one which covers all the possible values of 0 ≤ τ < 1, and it is noticeable the result of Theorem 1.2 is obtained for any value of 0 ≤ τ < 1, by essentially the same example, written in different scales of Hilbert spaces. An example of such growth was given in [START_REF] Delort | Growth of Sobolev norms for solutions of time dependent Schrödinger operators with harmonic oscillator potential[END_REF] in the case τ = 0 (see also [START_REF] Maspero | Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations[END_REF] for an alternative proof), and in [6, Appendix A] in the case τ = 1/2, but it seems that the other cases were left open. We stress that our example is an operator which takes the form H τ + L (t), where H τ is a (time independent) elliptic operator with compact resolvent and L (t) a bounded self-adjoint operator. Moreover, this perturbation is small and satisfies indeed for all ℓ ≥ 0 sup t∈R

∂ ℓ t L (t) L( H s τ ; H s-ℓτ τ ) ≤ C s,ℓ ǫ.
If one allows unbounded perturbations, it is simpler to obtain growth of Sobolev norms, as it is shown by an elementary example given in Appendix A. In this latter context, growth of Sobolev norms can occur even with time-independent operators.

Observe that the ϕ n are the eigenfunctions of H τ , namely

H τ ϕ n = 2 ρ(τ ) (n + 1) ρ(τ ) ϕ n , n ≥ 0.
Hence in our example, we see an exact correspondence between the asymptotics of the eigenvalues of H τ and the rate of growth for (1.1). If ρ > 1 (which corresponds to τ > 1/2), the operator H τ satisfies a gap condition, but in our example, ∂ ℓ t L (t) is not regular enough (see item (iv) in Assumption 1.1) to meet the hypotheses of [31, Theorems 1.8 and 1.9], in which better upper bounds are obtained.

Let us recall the following characterization of the Sobolev spaces H s τ . By [20, Lemma C.1], for any s ≥ 0, there exist c, C > 0 such that for all u ∈ H s

τ c z sρ(τ ) u L 2 (C) ≤ u H s τ ≤ C z sρ(τ ) u L 2 (C) , z = (1 + |z| 2 ) 1/2 .
As a consequence, in the Bargmann-Fock space, a growth of Sobolev norm corresponds to a transfer of energy in the physical space. In our example, the growth will be induced by a traveling wave. This is in contrast to the previous known examples [START_REF] Delort | Growth of Sobolev norms for solutions of time dependent Schrödinger operators with harmonic oscillator potential[END_REF][START_REF] Bambusi | Reducibility of the quantum harmonic oscillator in ddimensions with polynomial time-dependent perturbation[END_REF][START_REF] Maspero | Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations[END_REF], where the growth was inherited by a time-periodic phenomenon.

We end this section by reviewing some results on the growth of linear Schrödinger equations on manifolds with time-dependent potentials

i∂ t u + ∆u + V (t, x)u = 0. (1.2)
In [START_REF] Bourgain | On growth of Sobolev norms in linear Schrödinger equations with smooth time-dependent potential[END_REF] Bourgain proves a polynomial bound of the Sobolev norm for (1.2), when V (t, x) is a bounded (real analytic) potential. Moreover, when the potential is quasi-periodic in time he obtains in [START_REF] Bourgain | Growth of Sobolev norms in linear Schrödinger equations with quasi-periodic potential[END_REF] a logarithmic bound (see also [START_REF] Delort | Growth of Sobolev norms of solutions of linear Schrödinger equations on some compact manifolds[END_REF][START_REF] Wang | Logarithmic bounds on Sobolev norms for time-dependant linear Schrödinger equations[END_REF][START_REF] Fang | On growth of Sobolev norms in linear Schrödinger equations with time-dependent Gevrey potential[END_REF][START_REF] Grébert | Beating effects in cubic Schrödinger systems and growth of Sobolev norms[END_REF], for more results on norm inflation phenomena in various settings). Delort [START_REF] Delort | Growth of Sobolev norms for solutions of time dependent Schrödinger operators with harmonic oscillator potential[END_REF] constructs an example with polynomial growth for the harmonic oscillator perturbed by a (time-periodic) pseudo-differential operator of order zero. In [START_REF] Bambusi | Reducibility of the quantum harmonic oscillator in ddimensions with polynomial time-dependent perturbation[END_REF], the authors give the example of a time-periodic order one perturbation of the harmonic oscillator which induces polynomial growth. In [START_REF] Arbunich | Stability and instability properties of rotating Bose-Einstein condensates[END_REF], the authors prove exponential growth of the energy norm for a linear (and nonlinear) harmonic oscillator perturbed by the angular momentum operator (see [START_REF] Arbunich | Stability and instability properties of rotating Bose-Einstein condensates[END_REF]Theorem 4.5]). We refer to [START_REF] Maspero | Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations[END_REF][START_REF] Haus | Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators[END_REF] for more examples with growth of norms and to [START_REF] Bambusi | Growth of Sobolev norms for abstract linear Schrödinger equation[END_REF] for bounds on abstract linear Schrödinger equations. Let us mention the article [START_REF] Liang | 1-d quantum harmonic oscillator with time quasi-periodic quadratic perturbation: reducibility and growth of Sobolev norms[END_REF] in which the authors obtain very precise results on the dynamics of a family of perturbations of the harmonic oscillator. Finally, in the recent paper [START_REF] Faou | On weakly turbulent solutions to the perturbed linear harmonic oscillator[END_REF], Faou and Raphaël give examples of solutions to perturbed harmonic oscillators which grow like (log t) α . Interestingly, although being different to ours, one of their approach relies on the study of the so-called continuous resonant equation (CR) which contains the dynamics of the LLL equation.

The linear LLL equation with time-dependent potential

We now present our example more in details. Let W ∈ L ∞ (R × C, R) be a real-valued timedependent potential and consider the linear equation

i∂ t u -δHu = Π W (t, z)u , (t, z) ∈ R × C, δ ∈ R, u(t, •) |t=t 0 = u 0 ∈ E, (2.1)
where Π is the orthogonal projector on the space E (the kernel of Π is very explicit, see (2.13) below). The equation (2.1) is the linearization of the Lowest Landau Level equation

i∂ t u -δHu = Π(|u| 2 u), (2.2)
which is used in the modeling of fast rotating Bose-Einstein condensates. See e.g. the introduction of [START_REF] Gérard | On the cubic lowest Landau level equation[END_REF] for physical motivation, and we refer to [START_REF] Aftalion | Lowest Landau level functional and Bargmann spaces for Bose-Einstein condensates[END_REF][START_REF] Nier | Bose-Einstein condensates in the lowest Landau level: Hamiltonian dynamics[END_REF][START_REF] Gérard | On the cubic lowest Landau level equation[END_REF][START_REF] Biasi | Exact lowest-Landau-level solutions for vortex precession in Bose-Einstein condensates[END_REF][START_REF] Biasi | Solvable cubic resonant systems[END_REF][START_REF] Schwinte | Growth of Sobolev norms for coupled Lowest Landau Level equations[END_REF] for the study of (2.1). Equation (2.1) is a natural mathematical toy model, for which we can try to exhibit some particular dynamics.

The dispersion parameter δ ∈ R does not play a role in the dynamics of equation (2.2). Actually, u solves (2.2) if and only if v = e iδtH u solves (2.2) with δ = 0. This comes from the crucial property e -itH Π e itH a e itH b e itH c = Π a b c , ∀ a, b, c ∈ E, see [21, Lemma 2.4 and Corollary 2.5]. However, the transformation v = e iδtH u does not preserve the left hand side of (2.1), that is why we must keep the parameter δ ∈ R in our study (nonetheless we will see that it does not affect the dynamics of equation (2.1), excepted in the reducibility result stated in Appendix A where we need δ = 0).

In the sequel, by a time translation, we restrict to the case t 0 = 0.

In this section, we state global well-posedness results with optimal bounds on the growth of the Sobolev norms for (2.1). We are also able to obtain reducibility results for (2.1), when W is a small quasi-periodic potential, but these results are direct applications of [START_REF] Grébert | KAM for the quantum harmonic oscillator[END_REF], thus we have postponed the statements to the Appendix A.

2.1. Statement of the results. Our first result concerns the global well-posedness of such an equation under general conditions on W . For s ≥ 0, we denote by

L 2,s = u ∈ L 2 (C), z s u ∈ L 2 (C) , z = (1 + |z| 2 ) 1/2 (2.3)
the weighted Lebesgue space and L 2,s E = L 2,s ∩ E. Then our well-posedness result reads:

Theorem 2.1. Let δ ∈ R and W ∈ L ∞ (R × C, R). For all u 0 ∈ E, there exists a unique solution u ∈ C(R, E) to equation (2.1). Moreover, for every t ∈ R, C |u(t, z)| 2 dL(z) = C |u 0 (z)| 2 dL(z).
Furthermore, if for some s > 0, u 0 ∈ L 2,s E , then u(t) ∈ L 2,s E for every t ∈ R. A natural question is the control of higher order Sobolev norms of the solution for large times and this will be achieved, under some additional conditions on W . The notation W ∈ C ∞ bt R × R 2 , R means that W is continuous and bounded in t and smooth in the variables (x, y). We stress that derivation in the time variable is not needed. For notational convenience, we sometimes identify (x, y) ∈ R 2 and z = x + iy ∈ C. In particular, for a function of the variables (x, y) we use the notations

∂ z = 1 2 (∂ x -i∂ y ), ∂ z = 1 2 (∂ x + i∂ y ). (2.4) Theorem 2.2. Let δ ∈ R and s ≥ 0. Assume that W ∈ C ∞ bt R × R 2 , R is such that sup 0≤k≤ s ∂ k z W (t, •) L ∞ (C) ≤ C 0 , t ∈ R, (2.5)
then any solution to (2.1), with initial condition

u 0 ∈ L 2,s E , satisfies for all t ∈ R z s u(t) L 2 (C) ≤ C z s u 0 L 2 (C) C 0 t s , (2.6)
where the constant C > 0 only depends on s ≥ 0.

Condition (2.5) is rather natural in the space E. For instance, it is satisfied by the following class of potentials: assume that V (t, •) ∈ E, uniformly in t ∈ R, then W = |V | 2 satisfies (2.5) for all k ≥ 0, by Lemma C.2 and (2.15). We stress that in (2.5) one needs the operator

∂ z (not ∂ x or ∂ y ).
This bound is indeed optimal as shown by the next result:

Theorem 2.3. Let δ ∈ R. For all ǫ > 0, there exists W ǫ ∈ S (R × R 2 , R) such that for all 1 ≤ p ≤ ∞, W ǫ (t, •) L p (C) ≤ ǫ,
and such that for all k, j ≥ 0 and uniformly in time

∂ j z ∂ k z W ǫ (t, •) L ∞ (C) ≤ ǫC jk , t ∈ R, (2.7)
and there exists a nontrivial initial condition u 0 ∈ k≥0 L 2,k E such that the corresponding solution to (2.1) satisfies for all s ≥ 0 and t ∈ R

z s u(t) L 2 (C) ≥ c s z s u 0 L 2 (C) ǫt s .
(2.8)

Moreover, we have the following equivalence, when t -→ ±∞

z s u(t) L 2 (C) ∼ c s ǫ s |t| s u 0 L 2 (C) .
(2.9)

The inequality (2.8) is stated like that in order to give a counterpart to (2.6). Actually, as shown in (2.9), the coefficient of the leading order of the lower term depends on u 0 L 2 (C) (and not on z s u 0 L 2 (C) ). The inequality (2.8) holds true since u 0 ∈ k≥0 L 2,k E is here fixed, and the constant c s > 0 depends on u 0 .

Observe that using the notations (2.4), the condition (2.7) can be rephrased as

∂ j x ∂ k y W ǫ (t, •) L ∞ (C) ≤ ǫC jk , t ∈ R.
The result of Theorem 2.3 is a direct consequence of [35, Theorem 1.5 and Corollary 1.6] and we can make the explicit choices

u 0 = √ ǫ 1 2 ϕ 0 + i √ 3 2 ϕ 1 , α = √ 3 32π ǫ,
and

W ǫ (t, z) = ǫ 4π 1 -i √ 3(e -2iδt z + αt) 2 e -|e -2iδt z+αt| 2 . (2.10)
Actually, in [35, Theorem 1.5 and Corollary 1.6] (see also [35, equation (2.4)]), unbounded trajectories where constructed for the system

     i∂ t u -δHu = Π(|v| 2 u), (t, z) ∈ R × C, i∂ t v -δHv = -Π(|u| 2 v), u(0, •) = u 0 , v(0, •) = v 0 , (2.11)
and the idea is here to consider the second equation in (2.11) as given, and to interpret the term |v| 2 in the first line as a given time-dependent potential.

Notice that the growth of Sobolev norms is not obtained by a periodic potential as in [START_REF] Delort | Growth of Sobolev norms of solutions of linear Schrödinger equations on some compact manifolds[END_REF][START_REF] Bambusi | Reducibility of the quantum harmonic oscillator in ddimensions with polynomial time-dependent perturbation[END_REF]. Here, as it is shown in (2.10), the growth is exhibited by a time translation (more precisely, by a magnetic translation in the Bargmann-Fock space).

In general, growth of Sobolev norms is a phenomenon which happens due to resonances of the equation. Recall that the dynamics of the cubic LLL equation i∂ t u = Π(|u| 2 u) is included in the so-called cubic resonant (CR) equation, which was derived in [START_REF] Faou | The weakly nonlinear large box limit of the 2D cubic NLS[END_REF] as a resonant approximation of NLS (we also refer to [START_REF] Germain | On the continuous resonant equation for NLS. I. Deterministic analysis[END_REF] for a comprehensive study of the (CR) equation).

In the last result of this section we show that if W has additional spacial decay, then the possible growth of the solution of (2.1) enjoys better controls :

Theorem 2.4. Let δ ∈ R and s ≥ 0 and let W ∈ C ∞ bt R × R 2 , R . (i) Assume that sup 1≤j≤k≤ s z 2k-j ∂ j z W (t, •) L ∞ (C) ≤ C 1 , t ∈ R,
then any solution to (2.1), with initial condition

u 0 ∈ L 2,s E , satisfies for all t ∈ R z s u(t) L 2 (C) ≤ C z s u 0 L 2 (C) C 1 t 1/2 ,
where the constant C > 0 only depends on s ≥ 0. (ii) Let ǫ > 0. Assume that for all k ≥ 1 and uniformly in time

z 2k ∂ k z W (t, •) L ∞ (C) ≤ C k , t ∈ R, (2.12)
then any solution to (2.1), with initial condition

u 0 ∈ L 2,s E , satisfies for all t ∈ R z s u(t) L 2 (C) ≤ C z s u 0 L 2 (C) t ǫ ,
where the constant C > 0 depends on W , s ≥ 0 and ǫ > 0.

This result is the analogous to [START_REF] Bourgain | On growth of Sobolev norms in linear Schrödinger equations with smooth time-dependent potential[END_REF][START_REF] Delort | Growth of Sobolev norms of solutions of linear Schrödinger equations on some compact manifolds[END_REF] in which similar bounds are obtained for the linear Schrödinger equation with time-dependent potential, but in our case the proof is much simpler.

The result of Theorem 2.4 shows that growth of Sobolev norms can occur only if W is concentrated in the region |z| ≫ 1 when t -→ ±∞. This is typically the case with the example of the traveling wave exhibited in Theorem 2.3 (see (2.10)).

Under additional conditions on W (analyticity in time and quasi-periodicity) one can show the solutions are indeed bounded, see Theorem B.1.

2.2.

Plan of the paper. The rest of the paper is organized as follows. We end this section by giving some notations. In Section 3 we study the linear LLL equation (2.1). We are then able to apply these results to prove Theorem 1.2 in Section 4. In Appendix A we give an another example of Schrödinger operator with unbounded orbits and in Appendix B we state a reducibility result for (2.1).

2.3. Some recalls and notations. The harmonic oscillator H is defined by

H = -4∂ z ∂ z + |z| 2 ,
with the classical notations z = x + iy and

∂ z = 1 2 (∂ x -i∂ y ), ∂ z = 1 2 (∂ x + i∂ y ).
Recall that the family of the special Hermite functions (ϕ n ) n≥0 is given by

ϕ n (z) = z n √ πn! e -|z| 2 2 .
The family (ϕ n ) n≥0 forms a Hilbertian basis of E (see [39, Proposition 2.1]), and the ϕ n are the eigenfunctions of H, Hϕ n = 2(n + 1)ϕ n , n ≥ 0.

We can show (see [START_REF] Gérard | On the cubic lowest Landau level equation[END_REF]) that Π, the orthogonal projection on E, is given by the formula

(Πu)(z) = 1 π e -|z| 2 2 C e wz-|w| 2 2 u(w) dL(w), (2.13)
where L stands for Lebesgue measure on C. Recall (see (2.3)) that for s ≥ 0, the weighted Lebesgue space L 2,s is defined by

L 2,s = u ∈ L 2 (C), z s u ∈ L 2 (C) , z = (1 + |z| 2 ) 1/2
and L 2,s E = L 2,s ∩ E. For s ≥ 0, we define the harmonic Sobolev spaces by

H s = u ∈ L 2 (C), H s/2 u ∈ L 2 (C) ∩ E, equipped with the natural norm u H s = H s/2 u L 2 (C) .
Then by [20, Lemma C.1], we have

H s = L 2,s E with the equivalence of norms c z s u L 2 (C) ≤ u H s ≤ C z s u L 2 (C) , ∀ u ∈ L 2,s E . (2.14)
Recall the hypercontractivity estimates (see [START_REF] Carlen | Some integral identities and inequalities for entire functions and their application to the coherent state transform[END_REF] or [START_REF] Schwinte | Growth of Sobolev norms for coupled Lowest Landau Level equations[END_REF]Lemma A.2] for the bounds without the optimal constants which will be enough for our purpose) : for all 1 ≤ p ≤ q ≤ +∞ and u ∈ E

q 2π 1/q u L q (C) ≤ p 2π 1/p u L p (C) . (2.15)
In this paper c, C > 0 denote universal constants the value of which may change from line to line.

Study of the linear LLL equation

3.1. Global existence. To solve equation (2.1) we find a fixed point in a ball of E to

F : u -→ e -iδtH u 0 -i t 0 e -iδ(t-s)H Π(W u)(s) ds.
Let us sketch the proof: since e iτ H is unitary in L 2 , we have

F (u)(t) L 2 ≤ u 0 L 2 + t 0 Π(W u)(s) L 2 ds ≤ u 0 L 2 + Ct sup s∈[0,t] u(s) L 2 W L ∞ ,
where we used the continuity of Π in L 2 in the last line (for continuity results for Π we refer to [START_REF] Gérard | On the cubic lowest Landau level equation[END_REF]Proposition 3.1]). Contraction estimates are obtained similarly, and this gives a local in time solution. Globalization can be obtained by the Grönwall inequality since the equation is linear. The L 2 norm of a solution is a conserved quantity, since the potential W is real valued.

If moreover u 0 ∈ L 2,s E , we can prove the wellposedness in L 2,s E , thanks to the following lemma, which we quote for future reference:

Lemma 3.1. Let W ∈ L ∞ (C) and v ∈ L 2,s E , then z s Π W v L 2 (C) ≤ C W L ∞ (C) z s v L 2 (C) , (3.1)
and

z s e iτ H v L 2 = z s v L 2 . (3.2)
Proof. The bound (3.1) is a consequence of [START_REF] Gérard | On the cubic lowest Landau level equation[END_REF]Proposition 3.1]. For (3.2), we first observe that for all u ∈ E = L 2,0 E , we have e iτ H u(z) = e 2iτ u(e 2iτ z), as can be seen by testing on the complete family (ϕ n ) n≥0 . Then (3.2) follows from the change of variables z → e -2iτ z.

3.2.

Bounds on Sobolev norms: proof of Theorem 2.2. Now that equation (2.1) is wellposed, let us inspect the behaviour of the norms of the solutions. For this we need a result, which is an consequence of [35, Lemma 2.1] : We are now able to prove Theorem 2.2. By linearity, it is enough to consider the case z k u 0 L 2 (C) = 1. We use the identity (3.3). Then, since

Lemma 3.2. Let k ∈ N and let W ∈ C k (R × R 2 , R) be a real valued function. Assume that u ∈ L 2,k E satisfies i∂ t u -δHu = Π W u . Then d dt C |z| 2k |u(t, z)| 2 dL(z) = -2 k j=1 (-1) j k j Im C z k z k-j |u(t, z)| 2 ∂ j z W (t, z) dL(z). (3.3) Proof. We compute d dt C |z| 2k |u| 2 dL = 2Re C |z| 2k u∂ t udL = 2Im C |z| 2k uΠ(W u)dL + 2δIm C |z| 2k uHudL.

Let us first show that

Im C |z| 2k uHudL = 0. Since H = -4∂ z ∂ z + |z| 2 , it remains to show that Im C |z| 2k u∂ z ∂ z udL = 0. Write u(z) = f (z)e -1 2 |z| 2 , then Im C |z| 2k u∂ z ∂ z udL = Im C |z| 2k f e -1 2 |z| 2 ∂ z ∂ z f e -1 2 |z| 2 )dL = - 1 2 Im C |z| 2k f f + z∂ z f - 1 2 |z| 2 f )e -|z| 2 dL = - 1 2 Im C z k+1 z k f (∂ z f )e -|z|
∂ j z W L ∞ (C) ≤ C 0 for all 1 ≤ j ≤ k, we deduce by Hölder d dt C |z| 2k |u| 2 dL ≤ CC 0 C z 2k-1 |u| 2 dL ≤ CC 0 C z 2k |u| 2 dL 1-1 2k C |u| 2 dL 1 2k
, therefore, using the conservation of the mass,

d dt z k u 2 L 2 (C) ≤ CC 0 z k u 2-1 k L 2 (C) ,
which in turn implies, by time integration,

z k u(t) L 2 (C) ≤ z k u 0 1/k L 2 (C) + CC 0 |t| k ≤ C(1 + C 0 |t|) k ,
hence the result when k is an integer. The general case follows by interpolation.

z k u 2 L 2 (C) ≤ z k u 0 2 L 2 (C) + CC 1 |t| u 0 2 L 2 (C) . (ii)
We assume that the stronger condition (2.12) holds. We use here interpolation theory for linear operators. Fix s > 0, ǫ > 0 and set k ∈ N such that s/k < ǫ. Then from (i) we have

z k u(t) L 2 (C) ≤ C t 1/2 z k u 0 L 2 (C) .
Next, the L 2 -conservation yields u(t) L 2 (C) ≤ u 0 L 2 (C) . Then by interpolation, we get that for all 0 ≤ θ ≤ 1

z θk u(t) L 2 (C) ≤ C θ t θ/2 z θk u 0 L 2 (C) .
The result then follows by taking θ = s/k. As it can be checked on the (ϕ n ) n≥0 , we have e itH = e 2it L 2t for all t ∈ R. Now we refer to [35, Section 1.7.2]. The system

     i∂ t u -δHu = Π(|v| 2 u), (t, z) ∈ R × C, i∂ t v -δHv = -Π(|u| 2 v), u(0, z) = u 0 (z), v(0, z) = v 0 (z),
admits the following explicit solutions:

(u, v) = e -iλt e -iδtH R αt U, e -iµt e -iδtH R αt V = e -i(λ+2δ)t L -2δt R αt U, e -i(µ+2δ)t L -2δt R αt V , with U = √ ǫ 1 2 ϕ 0 + i √ 3 2 ϕ 1 , V = √ ǫ 1 2 ϕ 0 -i √ 3 2 ϕ 1 , and
λ = 7ǫ 32π , µ = - 7ǫ 32π , α = √ 3 32π ǫ.
It remains to check that W := |v| 2 and u satisfy the assumptions and the conclusions of Theorem 2.3. On the one hand, for all t ∈ R, u(t) L 2 = √ ǫ, and for s > 0,

z s u 0 L 2 = z s U L 2 ≤ c s √ ǫ. (3.4)
Let us prove that there exists c s > 0 such that for all t ∈ R

z s u(t) L 2 ≥ c s √ ǫ ǫt s . (3.5)
We have

z s u(t) 2 L 2 = z s R αt U 2 L 2 = z -αt s U 2 L 2 = C 1 + |z -αt| 2 s |U (z)| 2 dL(z).
Therefore,

z s u(t) 2 L 2 ≥ 1 2 C 1 + |z -αt| 2s |U (z)| 2 dL(z). (3.6)
• By the triangle inequality we have

|αt| 2s ≤ (|z -αt| + |z|) 2s ≤ 4 s (|z -αt| 2s + |z| 2s ),
which in turn implies

|z -αt| 2s ≥ 4 -s |αt| 2s -|z| 2s .
As a consequence, by (3.4) and (3.6)

z s u(t) 2 L 2 ≥ c|αt| 2s C |U (z)| 2 dL(z) -C C |z| 2s |U (z)| 2 dL(z) ≥ (c|αt| 2s -C)ǫ ≥ (c|ǫt| 2s -C)ǫ, (3.7)
where the constants c, C > 0 have varied from line to line. From (3.7) we deduce that there exists C 0 > 0 such that if |ǫt| > C 0 we have

z s u(t) L 2 ≥ c ǫt s √ ǫ.
• In the regime |ǫt| ≤ C 0 , we use the inequality (3.6) to write

z s u(t) L 2 ≥ c U L 2 = c √ ǫ ≥ c ǫt s √ ǫ.
As a consequence, we have proven (3.5).

On the other hand, we have the explicit expression

W (t, z) = ǫ 4π 1 -i √ 3(e -2iδt z + αt) 2 e -|e -2iδt z+αt| 2 = L -2δt R αt V (z) 2 .
Therefore we have

W L 1 = L -2δt R αt V 2 L 2 = V 2 L 2 = ǫ and from (2.15) we have W L ∞ = L -2δt R αt V 2 L ∞ = V 2 L ∞ ≤ ǫ.
Moreover, from Lemma C.2, we deduce

∂ j z ∂ k z W L ∞ (C) = ∂ j z ∂ k z |L -2δt R αt V | 2 L ∞ (C) ≤ C jk L -2δt R αt V 2 L ∞ ≤ ǫC jk ,
which was the claim.

4. Proof of Theorem 1.2 4.1. Some notations. For 0 ≤ τ < 1 we set ρ(τ ) = 1 2(1-τ ) > 0 and we define the operator H τ = (H + 1) ρ(τ ) , where H is the harmonic oscillator defined by

H = -4∂ z ∂ z + |z| 2 .
We then define the family of Hilbert spaces H s τ s≥0 by

H s τ = u ∈ L 2 (C), H s/2 u ∈ L 2 (C) ∩ E, H 0 τ = E. Recall that, by [20, Lemma C.1], we have c z sρ(τ ) u L 2 (C) ≤ u H s τ ≤ C z sρ(τ ) u L 2 (C) , z = (1 + |z| 2 ) 1/2 .
Observe also that H s τ = H sρ(τ ) (4.1) where H σ stands for the harmonic Sobolev space based on the harmonic oscillator H, and we have

c z s u L 2 (C) ≤ u H s ≤ C z s u L 2 (C) . (4.2)
In the sequel, in order to alleviate notations, we simply write H = H τ and H s = H s τ . 4.2. Definition of the operator L (t). Define the potential W 0 (t, z) as follows:

V = √ ǫ 1 2 ϕ 0 -i √ 3 2 ϕ 1 , W 0 (t, z) = |R αt V (z)| 2 = ǫ 4π 1 -i √ 3(z + αt) 2 e -|z+αt| 2 , α = √ 3 32π ǫ,
and with Lemma C.2, we show that all the derivatives of W 0 are bounded uniformly in t ∈ R:

∂ j z ∂ k z W 0 (t) L ∞ (C) = ∂ j z ∂ k z |R αt V | 2 L ∞ (C) ≤ C jk R αt V 2 L ∞ = C jk V 2 L ∞ ≤ ǫC jk . (4.3)
Now we define the mapping

L (t) : H s -→ H s u -→ e -it H Π W 0 (t)e it H u = e -it(H+1) ρ Π W 0 (t)e it(H+1) ρ u , (4.4)
and we consider the initial value problem

i∂ t u = H + L (t) u, (t, z) ∈ R × C, u(t) |t=t 0 = u 0 ∈ H s . (4.5)
4.3. Verification of Assumption 1.1. We now prove that L (t) satisfies the required properties.

(i) Let us check that L ∈ C b R, L( H s ) , with norm L (t) L( H s ) ≤ C s ǫ. First, by (4.1) it is equivalent to show that L ∈ C b R, L(H s ) .
Then, since e it(H+1) ρ is unitary in H s , and by (4.

2)

e -it(H+1) ρ Π W 0 (t)e it(H+1) ρ u H s = Π W 0 (t)e it(H+1) ρ u H s ≤ C z s Π W 0 (t)e it(H+1) ρ u L 2 (C) .
Next, by (3.1) and (3.2)

z s Π W 0 (t)e it(H+1) ρ u L 2 (C) ≤ C W 0 (t) L ∞ (C) z s e it(H+1) ρ u L 2 (C) = C W 0 (t) L ∞ (C) z s u L 2 (C) ≤ C W 0 (t) L ∞ (C) u H s . Recall that W 0 (t) = |R αt V | 2 , where V ∈ E, then W 0 (t) L ∞ (C) = V 2 L ∞ ≤ Cǫ.
Putting all the previous estimates toghether, we obtain e -it(H+1) ρ Π W 0 (t)e it(H+1) ρ u H s ≤ C s ǫ u H s , hence the announced bound. The time-continuity of L follows from the previous estimates together with the continuity of the translations for the Lebesgue measure and the fact that e it H ∈ C b R, L( H s ) .

(ii) The symmetry of L , w.r.t. the scalar product of H 0 = E, is a consequence of the symmetry of Π, the conjugation by the unitary operator e it(H+1) ρ and the fact that W 0 is a real valued function.

(iii) Let us check that H, L (t) is H τ -bounded. By Lemma C.1, the operators H and Π commute, thus

H, L (t) H -τ = (H + 1) ρ , L (t) (H + 1) -ρτ = e -it(H+1) ρ Π (H + 1) ρ , W 0 (t) (H + 1) -ρτ e it(H+1) ρ .
Recall that Π is bounded in all the H s spaces, as well as the operators e -it(H+1) ρ .

• Case s = 0. Let us first prove that Π (H + 1) ρ , W 0 (t) (H + 1) -ρτ : E -→ E is bounded, uniformly in t ∈ R. For that, we use the Weyl-Hörmander pseudo-differential calculus (we refer to [START_REF] Robert | Autour de l'approximation semi-classique[END_REF][START_REF] Helffer | Théorie spectrale pour des opérateurs globalement elliptiques[END_REF] or to [START_REF] Parmeggiani | Spectral theory of non-commutative harmonic oscillators: an introduction[END_REF]Chapter 3] for a review of this theory). Denote by z = x 1 + ix 2 , ξ = ξ 1 + ξ 2 , and consider the metric

dx 2 1 + dx 2 2 + dξ 2 1 + dξ 2 2 1 + |z| 2 + |ξ| 2 . (4.6)
The Planck function associated to the metric (4.6) is given by h

(x 1 , x 2 , ξ 1 , ξ 2 ) = (1 + |z| 2 + |ξ| 2 ) -1/2 and for m ∈ R, the symbol class S m is S m = a ∈ C ∞ (R 4 ; C) : ∂ α 1 x 1 ∂ α 2 x 2 ∂ β 1 ξ 1 ∂ β 2 ξ 2 a(x 1 , x 2 , ξ 1 , ξ 2 ) ≤ C α,β |z| + |ξ| m-β 1 -β 2 , ∀ α, β ∈ N 2 .
For a ∈ S m , we define its Weyl-quantization by the formula

a w (x, D)u(x) = 1 (2π) 2 R 2 R 2 e i(x-y)•ξ a( x + y 2 , ξ)u(y)dydξ, u ∈ S (R 2 ).
First, using the functional calculus associated to the operator H, we obtain that (H + 1) ρ is a pseudo-differential operator with symbol in S 2ρ . By (4.3), W 0 (t) ∈ S 0 uniformly in t ∈ R, and therefore the commutator (H + 1) ρ , W 0 (t) is a pseudo-differential operator with symbol in S 2ρ-1 , which in turn implies that (H +1) ρ , W 0 (t) (H +1) -ρτ is a pseudo-differential operator with symbol in S 0 (because 2ρ -1 -2ρτ = 0), hence it is bounded.

• The proof in the general case s ≥ 0 is similar.

(iv) From (4.4), a direct computation gives

∂ t L (t) = e -it H Π(∂ t W 0 (t))e it H -ie -it H Π H, W 0 (t) e it H = e -it H Π(∂ t W 0 (t))e it H -i H, L (t) . (4.7)
From the expression of W 0 , we deduce that sup t∈R ∂ ℓ t W 0 (t) L ∞ (C) ≤ C ℓ for all ℓ ≥ 0. In particular, the first term in the right hand side of (4.7) is bounded H s -→ H s . In item (iii) we have shown that [ H, L (t) : H s -→ H s-τ is bounded, uniformly in t ∈ R. As a consequence, for all s ≥ 0, L ∈ C 1 b R; L( H s ; H s-τ ) . The general case ℓ ≥ 0 is obtained by induction.

4.4. Proof of Theorem 1.2. We are now ready to complete the proof of Theorem 1.2. Consider the problem (4.5) and for convenience, assume that t 0 = 0. By (4.4), the equation (4.5) is equivalent to

i∂ t v = Π W 0 (t)v , (t, z) ∈ R × C, v(0, •) = u 0 ∈ H s = H ρs ,
with the change of unknown v = e it(H+1) ρ u. As a consequence we can directly apply the results of Section 2 (case δ = 0) to this model.

(i) The fact that L (t) L( H s ) ≤ C s ǫ has already been shown in the previous paragraph.

(ii) For all s ≥ 0, the problem (4.5) is globally well-posed, in H s by Theorem 2.1. The group property of U is a consequence of uniqueness, and its unitarity follows from the conservation of the L 2 norm.

(iii) The upper bound is given by Theorem 2.2, namely, for all t ∈ R

u(t) H s ≤ C z ρs u(t) L 2 (C) ≤ C z ρs u 0 L 2 (C) ǫt ρs ≤ C u 0 H s ǫt ρs , where ρ = 1 2(1-τ ) .
(iv) Consider the function

u 0 ∈ ∩ k≥1 L 2,k E = ∩ k≥1 H k given by Theorem 2.3, (see paragraph 3.4) then u 0 H s ≤ C √ ǫ and u(t) H s ≥ c z ρs u(t) L 2 (C) ≥ c √ ǫ ǫt ρs ≥ c u 0 H s ǫt ρs ,
hence the result.

Notice that the items (ii) and (iii) also directly follow from the general result [START_REF] Maspero | On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms[END_REF]Theorem 1.5].

Now, for ǫ > 0, we consider the problem,

i∂ t u = -iǫ∂ x u, (t, x) ∈ R × R, u(0, •) = u 0 ∈ H s . (A.2)
In this framework, we are able to prove the following result for the operator iǫ∂ x in the spaces H s :

Lemma A.1. (i) One has iǫ∂ x ∈ L( H s+1/ρ , H s ) for all s ≥ 0, and

iǫ∂ x L( H s+1/ρ , H s ) ≤ C s ǫ.
(ii) The operator iǫ∂ x is symmetric on H 1/ρ w.r.t. the scalar product of

H 0 , R v(iǫ∂ x u) dx = R u(iǫ∂ x v) dx, ∀u, v ∈ H 1/ρ .
(iii) The operator iǫ∂ x is H τ -bounded in the sense that [iǫ∂ x , H] H -τ ∈ L( H s ) for all s ≥ 0.

Therefore the operator i∂ x satisfies the assumptions (H0)-(H3) of [START_REF] Maspero | On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms[END_REF].

On the other hand, we have the following elementary result:

Proposition A.2. Let s ≥ 0, then (i) The problem (A.
2) is globally well-posed in H s , and the solution is explicitly given by u(t, x) = u 0 (x -ǫt).

(ii) The following bounds hold true : for all u 0 ∈ H s and for all t ∈ R c ǫt

s 2(1-τ ) u 0 H s ≤ u(t) H s ≤ C ǫt s 2(1-τ ) u 0 H s . (A.3)
This result is directly obtained using (A.1) and the expression ρ(τ ) = 1 2(1-τ ) .

Remark A.3. Notice that the function v(t) = e -it H u(t) is solution to

i∂ t v -Hv = -iǫ e -it H ∂ x e it H v, (t, x) ∈ R × R, v(0, •) = v 0 = u 0 ∈ H s ,
and satisfies the conclusions of Lemma A.1 and the bounds (A.3). This yields an example in the spirit of the one exhibited in Theorem 1.2, but in the present case, the perturbation is of order 1/ρ ∈ (0, 2] instead of being of order 0.

Proof of Lemma A.1. Item (i) is a direct consequence of (A.1) and (ii) is elementary.

(iii) It is convenient to introduce the Sobolev space based on the harmonic oscillator (s ≥ 0)

H s = u ∈ L 2 (R), H s/2 u ∈ L 2 (R) , H 0 = L 2 (R).
Thanks to the pseudo-differential calculus associated to H (see also paragraph 4.3), we first prove that for all r > 0, [i∂ x , (H + 1) r ](H + 1) -r+ 1 2 ∈ L L 2 (R) .

(A.4) Here the symbol class S m reads

S m = a ∈ C ∞ (R 2 ; C) : ∂ α x ∂ β ξ a(x, ξ) ≤ C α,β |x| + |ξ| m-β , ∀ α, β ∈ N .
The symbol of [i∂ x , (H + 1) r ], modulo terms in S 2r-1 , is given by the formula

-i ξ, (x 2 + ξ 2 + 1) r = -i∂ ξ (ξ)∂ x (x 2 + ξ 2 + 1) r + i∂ x (ξ)∂ ξ (x 2 + ξ 2 + 1) r = -i2rx(x 2 + ξ 2 + 1) r-1 ∈ S 2r-1 .
Since the symbol of (H + 1) -r+ 1 2 belongs to S -2r+1 , we deduce (A.4).

Theorem B.1. Assume that W satisfies (B.2). Then there exists ǫ 0 such that for all 0 ≤ ǫ < ǫ 0 there exists a set Λ ǫ ⊂ [0, 2π) n of positive measure and asymptotically full measure: Meas(Λ ǫ ) → (2π) n as ǫ → 0, such that for all ω ∈ Λ ǫ , the linear equation (B.1) reduces, in E, to a linear equation with constant coefficients.

We refer to [START_REF] Grébert | KAM for the quantum harmonic oscillator[END_REF]Theorem 7.1] for a more precise statement, giving in particular more information on the transformation.

Assume that (θ, z) → V (θ, z) is analytic in θ on |Im θ| < τ , that V (θ, •) ∈ E for all θ ∈ T n , and satisfies, for some γ > 0, the bound

|V (θ, z)| ≤ C z -γ uniformly in θ ∈ T n . Then, by Lemma C.2, W = |V | 2 satisfies (B.2). Such a potential even satisfies |∂ j z ∂ ℓ z W (θ, z)| ≤ C for all k, ℓ ∈ N (without additional assumptions on V ∈ E).
We also have the following result on the dynamics of the solutions of (B.1) :

Corollary B.2. Assume that W is C ∞ in x,
y with all its derivatives bounded and satisfying (B.2). Let s ≥ 0 and u 0 ∈ H s . Then there exists ǫ 0 > 0 so that for all 0 < ǫ < ǫ 0 and ω ∈ Λ ǫ , there exists a unique solution u ∈ C R ; H s of (B.1) so that u(0) = u 0 . Moreover, u is almost-periodic in time and we have the bounds

(1 -ǫC) u 0 H s ≤ u(t) H s ≤ (1 + ǫC) u 0 H s , ∀ t ∈ R, for some C = C(s, ω).
The result of Theorem B.1 can also be formulated in term of the Floquet operator. Consider the Floquet Hamiltonian operator, defined on E ⊗ L 2 (T n ) by

K := i n k=1 ω k ∂ θ k + H + ǫΠ W (θ, z) • ,
then we can state Corollary B.3. Assume that W satisfies (B.2). There exists ǫ 0 > 0 so that for all 0 < ǫ < ǫ 0 and ω ∈ Λ ǫ , the spectrum of the Floquet operator K is pure point.

We refer to [START_REF] Grébert | KAM for the quantum harmonic oscillator[END_REF]Section 7], where similar results are proven for the 1D quantum harmonic oscillator.

For the reducibility of the periodic Schrödinger equation, we refer to [START_REF] Eliasson | On reducibility of Schrödinger equations with quasiperiodic in time potentials[END_REF] and for the reducibility of the quantum harmonic oscillator in any dimension to [START_REF] Grébert | On reducibility of quantum harmonic oscillator on R d with quasiperiodic in time potential[END_REF][START_REF] Liang | Reducibility of quantum harmonic oscillator on R d with differential and quasi-periodic in time potential[END_REF][START_REF] Bambusi | Reducibility of the quantum harmonic oscillator in ddimensions with polynomial time-dependent perturbation[END_REF] and we refer to [START_REF] Bambusi | Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations[END_REF][START_REF] Bambusi | Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations[END_REF][START_REF] Bambusi | Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations[END_REF] for the reducibility for 1-d operators with unbounded perturbations. For references on the theory of Floquet operators, see [START_REF] Eliasson | Almost reducibility of linear quasi-periodic systems. Smooth ergodic theory and its applications[END_REF][START_REF] Wang | Pure point spectrum of the Floquet Hamiltonian for the quantum harmonic oscillator under time quasi-periodic perturbations[END_REF].

Finally, let us mention that, concerning the nonlinear cubic LLL equation, the abstract KAM result of [START_REF] Grébert | KAM for the quantum harmonic oscillator[END_REF] was applied in [START_REF] Gérard | On the cubic lowest Landau level equation[END_REF]Theorem 4.3] in order to show the existence of invariant torii, and that the result of [START_REF] Grébert | Normal forms for semilinear quantum harmonic oscillators[END_REF] was applied to show an almost global existence result for the cubic LLL equation. We refer to [START_REF] Gérard | On the cubic lowest Landau level equation[END_REF]Section 4.2] for more details.

The arguments of [START_REF] Grébert | KAM for the quantum harmonic oscillator[END_REF]Section 7] can be directly applied to the equation (B.1), and we address the reader to this latter paper for the proofs of the previous results. Let us just sketch the idea : we expand u and ū on the basis given by the special Hermite functions Lemma C.2. For all j, k ≥ 0 there exists C > 0 such that for all 1 ≤ p ≤ ∞ and v ∈ E,

∂ j z ∂ k z |v| 2 L p (C) ≤ C v 2 L 2p (C) .

  parts, hence the result. To complete the proof, we apply[START_REF] Schwinte | Growth of Sobolev norms for coupled Lowest Landau Level equations[END_REF] Lemma 2.1].

3. 4 . 1 2 1 2

 411 Growth of Sobolev norms: proof of Theorem 2.3. Let us define the magnetic translations by the formula R α : (u, v)(z) → u(z + α)e (zα-zα) , v(z + α)e (zα-zα) , α ∈ C, as well as the space rotations L θ : (u, v)(z) → u(e iθ z), v(e iθ z) , θ ∈ T.

-|z| 2 2 Ceπ e -|z| 2 2 C e wz-|w| 2 2 ∂-|z| 2 2 C 2 u

 22222 u = j≥0 c j ϕ j , u = j≥0 c j ϕ j .Then equation (B.1) reads as an autonomous Hamiltonian system in an extended phase space           ċj = -2i(j + 1)c j -iǫ∂ cj Q(θ, c, c) j ≥ 0 ċj = 2i(j + 1)c j + iǫ∂ c j Q(θ, c, c) j ≥ 0 θj = ω j j = 0, . . . , n Ẏj = -ǫ∂ θ j Q(θ, z, z) j = 0, . . . , n (B.3)where Q is a quadratic functional in (c, c) given byQ(θ, c, c) = C W (θ, z) j≥0 c k ϕ k (z) j≥0 c j ϕ j (z) dL(z), (B.4) and the Hamiltonian of the system (B.3) is n j=1 ω j Y j + 2 j≥0 (j + 1)c j c j + Q(θ, c, c). Then, we can check that (B.4) satisfies the assumptions of [25, Theorem 7.1]. The dispersive estimate ϕ n L ∞ (C) ≤ Cn -1/4 satisfied by the (ϕ n ) n≥0 , is the key ingredient which allows to follow the lines of [25, Section 7]. Appendix C. Some technical results Lemma C.1. The operators H and Π commute. wz-|w| 2 2 u(w) dL(w), and that H = -4∂ z ∂ z + |z| 2 . On the one hand, by integration by parts Π∂ z ∂ z u (z) = 1 w ∂ w u(w) dL(w) ∂ w ∂ w e wz-|w| 2 2 u)(z), thus ΠHu(z) = 2Πu(z) + 2zΠ(wu)(z). On the other hand ∂ z Πu(z) =that HΠu(z) = 2Πu(z) + 2zΠ(wu)(z) = ΠHu(z), which was the claim. We recall a short version of [35, Lemma A.2] :

The author is supported by the grants "BEKAM" ANR-15-CE40-0001 and "ISDEEC" ANR-16-CE40-0013. 1 s 2(1-τ ) .

Appendix A. A non-perturbative & time-independent example

In this section, we give another example of linear Schrödinger operator which yields unbounded dynamics, and which meets the assumptions (H0)-(H3) of [START_REF] Maspero | On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms[END_REF]. This example differs from the one exhibited in Theorem 1.2 in two main aspects :

• it is a non-perturbative example : it is not a lower order perturbation of a time-independent elliptic differential operator ; • it is time-independent. However, with a change of unknown, we can obtain a time-dependent perturbation of a constant coefficient self-adjoint elliptic operator, see Remark A.3 below. This example is very simple, and that is why we decided to develop it here. Actually the mechanism involved in the norm inflation is the same as in Theorem 1.2: it is a traveling wave measured in a weighted L 2 space. Actually, our example is close to the one developed in [6, Appendix A], after change of variables.

On L 2 (R) we define the usual harmonic oscillator

We define the operator H = (H + 1) ρ(τ ) and the scale of Hilbert spaces H s s≥0 by

. By [38, Lemma 2.4], we have the following equivalence of norms

Next, for ρ > 0 and s ≥ 0

and by applying (A.4) twice, we deduce that for all s ≥ 0

Finally recall that H = (H + 1) ρ , thus (A.5) is equivalent to

, which was the claim.

Appendix B. On the reducibility of the linear LLL equation

We state here a reducibility result for the linear LLL equation. It turns out that the abstract reducibility result obtained in [START_REF] Grébert | KAM for the quantum harmonic oscillator[END_REF] can be applied to this model, which is close in many aspects to the usual 1D cubic quantum harmonic oscillator with time-dependent potential. We consider the linear equation

where δ = 0, where ǫ > 0 is small and where the parameter ω ∈ [0, 2π) n is the frequency vector, for some given n ≥ 1. Up to a rescaling, we can assume that δ = 1. We assume in the sequel that the potential W :

is analytic in θ on |Im θ| < τ for some τ > 0, and C 2 in x, y (where z = x + iy), and we suppose moreover that there exists γ > 0 and C > 0 so that for all θ ∈ T n and z ∈ C

for any 0 ≤ j, ℓ ≤ 1.

When ω = 0, all the solutions to (B.1) are almost periodic in time. This can be proved by constructing a Hilbertian basis 1 (ψ k ) k≥0 of E composed of eigenfunctions of the operator u → Hu + ǫΠ(W (0, z)u), such that

Then (B.1) can be solved by

which shows that any solution to (B.1) is an infinite superposition of periodic functions, hence it is an almost-periodic function in time.

For ω = 0, the reducibility theory adresses the question if, by the means of a time quasi-periodic transformation, one can reduce to the previous case. It turns out that for (B.1), it is the case for a large set of values ω ∈ Λ ǫ :

1 Such a Hilbertian basis exists, since u → Hu + ǫΠ(W (0, z)u) is a self-adjoint operator with compact resolvent in E .


Alvaro A Grandi

Suzie Protière
email: suzie.protiere@sorbonne-universite.fr

Arnaud Lazarus
email: arnaud.lazarus@sorbonne-universite.fr

Enhancing and controlling parametric instabilities in mechanical systems

Keywords:

We gain new fundamental insights on parametric instabilities that are at the heart of many physical phenomena from the dynamic buckling of slender structures in periodic compression to the emergence of Faraday waves or the spontaneous symmetry breaking in Floquet time crystals. Combining theoretical models and precision desktop experiments, we explain how to periodically vary the evolution function of a dynamical system to enhance and get control on parametric instabilities. We show on a proof of concept that is an electromagnetic pendulum: i) how to observe extremely high orders of parametric resonance, even in the presence of dissipation, ii) how to trigger and efficiently sustain the natural vibrations of an oscillator. The presented concepts being universal, they could offer new dynamical functionalities in various fields and at any scale, from actuation in soft robotics to vibrational motions in microelectromechanical resonators.

Introduction

Parametric instabilities can develop in any dynamical system whose local evolution function is periodically varied in synchrony with one of its natural time scale [START_REF] Richards | Analysis of periodically time-varying systems[END_REF]. In practice, the evolution rule is varied by modulating a physical parameter of the system: a child can pump a playground swing by periodically squatting and standing, thus slightly varying the moment of inertia of the swing as a pendulum [START_REF] Juan R Sanmartin | O botafumeiro: Parametric pumping in the middle ages[END_REF]; Faraday waves can emerge by vertically oscillating a layer of fluid, thus modulating the effective gravity felt by the perturbed surface waves [START_REF] Faraday | On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces[END_REF][START_REF] Brooke | The stability of the plane free surface of a liquid in vertical periodic motion[END_REF][START_REF] Douady | Experimental study of the faraday instability[END_REF][START_REF] Raymond E Goldstein | Coffee stains, cell receptors, and time crystals: Lessons from the old literature[END_REF] and a slender structure under periodic compressive loads can dynamically buckle due to the small variations of its apparent flexural rigidity [START_REF] Vv Bolotin | The dynamic stability of elastic systems[END_REF][START_REF] Vafai | Parametric instability of edge cracked plates[END_REF]. This concept being universal, parametric instabilities are encountered in various scientific areas from gravitational waves detectors [START_REF] Evans | Observation of parametric instability in advanced ligo[END_REF] to plasma [START_REF] Alexey V Arefiev | Parametric amplification of laser-driven electron acceleration in underdense plasma[END_REF] or Micro Electro Mechanical Systems (MEMS) [START_REF] Turner | Five parametric resonances in a microelectromechanical system[END_REF][START_REF] Szorkovszky | Mechanical squeezing via parametric amplification and weak measurement[END_REF]. By nature, parametric oscillators should allow for the design of complex stability diagrams in the modulation parameters space with numerous parametric instability tongues that could be exploited for promising dynamical functionalities. But apart in MEMS thanks to their extremely low damping [START_REF] Turner | Five parametric resonances in a microelectromechanical system[END_REF][START_REF] Jia | Twenty-eight orders of parametric resonance in a microelectromechanical device for multi-band vibration energy harvesting[END_REF] and Floquet time crystals [START_REF] Wilczek | Quantum time crystals[END_REF][START_REF] Dominic V Else | Floquet time crystals[END_REF], because they are many-body coupled oscillators with low dissipation [START_REF] Heugel | Classical many-body time crystals[END_REF], classical parametric instability regions are actually rather rare and barely controllable.

In this work, we investigate what features of parametric systems are essential to practically tailor the shape and number of their parametric instability tongues. We focus on a single linear oscillator and study the instability thresholds and dynamical responses of an electromagnetic pendulum to experimentally illustrate our theoretical concepts. We find that by carefully choosing the physical parameter to greatly ORCID(s): vary the evolution function of a linear dynamical system, it is possible to enhance and fully control parametric instabilities. For instance, it becomes simple to trigger extremely high super-harmonic instability regions or to practically use any periodic modulation functions to achieve interesting cyclic a) b) motions. This work brings new physical insights on parametric instabilities and show that new dynamical functionalities can emerge when one periodically change the local evolution function about the equilibrium point of a dynamical system. The presented energetic concepts being universal, they could apply at any scale and in various fields, from electronics to physics or biology. In mechanics, possible applications range from the original realization of cyclic motions such as frequency dividers or clocks in MEMS to efficient dynamical actuation between equilibrium positions in soft robotics.

 = 1 2 ( 2 -2 ) 2 . A: 2 < 0, 2 < -2 . B: 2 < 0, 2 > -2 . C: 2 > 0, 2 > 2 . D: 2 > 0, 2

From classical to extreme parametric systems

The simplest linear model to get physical insights in the stability behavior of parametric systems is a mass locally moving in a harmonic potential whose curvature is periodically varying with time as illustrated in Fig. 1a). Denoting ( ) and ̇ ( ) the dimensionless position and velocity of the mass about the equilibrium ( ) = ̇ ( ) = 0, the total energy of the system can be expressed as the sum of a kinetic and potential part, respectively reading  = 1 2 ̇ 2 and  = 1 2 ( 2 + 2 ( )) 2 where ( ) = ( +  ) is a  -periodic modulation function between +1 and -1 and 2 and 2 are the local curvatures of the potential. The linear equation of motion deriving from this energetic model reads

̈ ( ) + 2 + 2 ( ) ( ) = 0 (1)
which is an Initial Value Problem in the form of a linear ODE with periodic coefficient. We recall the solution of Eq.( 1) can be sought in the Floquet form ( ) = ( ) + ̄ ( ) - where ( ) = ( +  ) is a  -periodic complex eigenfunction and is a complex eigenvalue called the Floquet exponent [START_REF] Bentvelsen | Modal and stability analysis of structures in periodic elastic states: application to the ziegler column[END_REF][START_REF] Lazarus | Discrete dynamical stabilization of a naturally di-verging mass in a harmonically time-varying potential[END_REF]. Fig. 1b) shows the numerical evolution of = (ℜ(± )) as a function of modulation parameters 2 and 2 in the classic "Mathieu" case where ( ) = cos( ) and  = 2 . The color dots > 0 indicate solutions ( ) diverging from ( ) = 0, i.e. a particle that is locally unstable, when the white regions = 0 indicate quasi-periodic oscillating solutions ( ) about ( ) = 0, i.e. a marginally stable particle.

The particular case 2 = 0 corresponds to the classic harmonic oscillator where the mass is either marginally stable when 2 > 0 or locally unstable when 2 < 0. The generalized case 2 > 0 models parametric oscillators and details on the rationalization of classic experiments by Eq.( 1) are given in Appendix A. For 2 < 0, it is eventually possible to dynamically stabilize the naturally diverging mass [START_REF] Lazarus | Discrete dynamical stabilization of a naturally di-verging mass in a harmonically time-varying potential[END_REF][START_REF] Apffel | Floating under a levitating liquid[END_REF], but in a very small region of the modulation parameter space and only if 2 > | 2 |, i.e. if the curvature of the potential is at least shortly positive (see cases A and B in Fig. 1). The classic "Kapitza's pendulum" experiment [START_REF] Stephenson | Xx. on induced stability[END_REF][START_REF] Dj Acheson | Upside-down pendulums[END_REF] that consists in stabilizing an inverted pendulum by modulating the gravitational energy potential through the vertical oscillation of its pivot point is indicated by a black square in the stability diagram of Fig. 1b) [START_REF] Smith | Experimental study of an inverted pendulum[END_REF]. The case 2 > 0 and 2 > 2 (case D in Fig. 1) corresponds to classic parametric instabilities characterized by disconnected instability tongues whose tips originate, for → 0, at = 0.5 with a positive integer. The ℎ instability region indicates the emergence of 2 or  -periodic motions depending if is odd or even, respectively. Classic experiments on the emergence of Faraday waves [START_REF] Protiere | Gouttes rebondissantes: une association ondeparticule à échelle macroscopique[END_REF] or the parametric pumping of "O Botafumeiro" from Santiago de Compostella [START_REF] Juan R Sanmartin | O botafumeiro: Parametric pumping in the middle ages[END_REF] are represented by green circles and blue triangles, respectively, in Fig. 1b). Super-harmonic parametric instabilities in MEMS [START_REF] Jia | Twenty-eight orders of parametric resonance in a microelectromechanical device for multi-band vibration energy harvesting[END_REF] are also represented in the same figure by red crosses. Finally, case C corresponds to periodic modulations that are so large, that the curvature of the potential in Fig. 1a) is shortly negative. This is the case of few Faraday instability experiments when the acceleration of the shaker is such, than the effective gravity is negative for a short time, i.e. the fluid tends to vertically escape the recipient against gravity [START_REF] Protiere | Gouttes rebondissantes: une association ondeparticule à échelle macroscopique[END_REF]. The experimental Floquet oscillator under study is a planar pendulum with a metallic marble that is symmetrically placed between two identical attracting electromagnets whose attracting force depends on the imposed electrical current (see sketch in inset). The scalar 2 ( ), characterizing the local evolution function of the electromagnetic pendulum, drastically depends on the constant control parameter .

It is clear from Fig. 1b) that parametric instabilities usually exploited in mechanical Floquet systems are the ones close to the tips of the instability tongues. In fact, when periodically varying the length of a pendulum, the acceleration of a shaker or the membrane stiffening of a micrometric plate, it is either impractical or energetically too costly to maintain a large variation of the evolution function of a system over relatively long periods of time, i.e. having both large 2 and 2 . And since because of inherent damping, there is an exponential narrowing of the tips of the tongues with increasing instability region number [START_REF] Turner | Five parametric resonances in a microelectromechanical system[END_REF], macroscopic systems are typically limited around the first instability region and higher orders parametric resonances have only been observed in MEMS thanks to their extremely large quality factors [START_REF] Turner | Five parametric resonances in a microelectromechanical system[END_REF][START_REF] Jia | Twenty-eight orders of parametric resonance in a microelectromechanical device for multi-band vibration energy harvesting[END_REF]. In Fig. 2, we propose an experiment where parametric instabilities are enhanced. The setup consists of a 2 pendulum: a metallic marble with radius 0.6 cm is attached to a string of length = 4.3cm. The marble is symmetrically placed between two electromagnets (with typical holding force of 1000 N) that are separated by a distance = 4 cm. Upon a given electrical current , the magnets are turned ON and both attract the marble. The local evolution function of this electromagnetic pendulum is characterized, for a given control parameter , by locally perturbing the pendulum from its trivial vertical equilibrium position with a small initial angle (0) at time = 0. We find linear angular responses can be written in the form ( ) = (0)ℜ( - √2 ( ) ) where 2 ( ) is shown in Fig. 2. Two qualitative responses are observed: below < , the pendulum is harmonically oscillating in the form ( ) = (0) cos( ( ) ) with a natural frequency ( ) that decreases as increases. Above a critical current , the pendulum locally diverges in the form (0) ( ) , eventually choosing one or the other side depending of initial symmetry imperfections. Because the bifurcation at = is sub-critical, there is always a "jump" of 2 ( ) around where our experiment is imperfection sensitive. a)

ω 2 (0) ω 2 (I)
Figure 3: Sketch of the input-output synchronization that would be required for parametric pumping in the case of a time-periodic system with a square wave modulation function 1 = 2 = ∕2. To amplify any small perturbation about the equilibrium ( ) = 0, one needs to turn the magnets ON (decrease effective gravity) when the pendulum height is minimal, just before the mass goes up, and turn them OFF (put back strong gravity) when the pendulum height is maximum, just before the mass goes down.

Beyond the tip of the parametric instability tongues

Unlike classical parametric systems, it is straightforward with the system of Fig. 2 to fully explore regionsof Fig. 1b) by periodically varying the electrical current below or above . In this work, we modulate 2 ( ( )) in a square wave fashion as shown in Fig. 3 (case D) and 5a) (case C). Precisely, we turn the electromagnets OFF ( = 0) and ON ( ≠ 0) during 1 and 2 seconds, respectively, so that the modulation period is = 2 ∕Ω = 1 + 2 . The linear equation of motion of this two-state oscillator reads simply

̈ ( ) + 2 (0) ( ) = 0 during 1 , ̈ ( ) + 2 ( ) ( ) = 0 during 2 . ( 2
)
Introducing the dimensionless time = Ω , the square wave modulation function ( ) = +1 during  1 = 2 1 ∕ and ( ) = -1 during  2 = 2 2 ∕ and the dimensionless modulation parameters

2 = 2 (0) + 2 ( ) 2Ω 2 , 2 = 2 (0) -2 ( ) 2Ω 2 , ( 3
)
Eq.( 2) can be reduced in the dimensionless form of Eq.( 1) which, in the case of a piecewise modulation function ( ), is called the Meissner equation [START_REF] Richards | Analysis of periodically time-varying systems[END_REF]. On top of being easy to experimentally implement, a square wave parametric oscillator can be analytically solved [START_REF] Van Der Pol | Ii. on the stability of the solutions of mathieu's equation[END_REF][START_REF] Sato | Correction of stability curves in hill-meissner's equation[END_REF]. Notably, introducing = √ 2 + 2 and = √ 2 -2 , the analytical growth rate = (ℜ(± )) of the Floquet forms of Eq.( 1) reads

= |ℜ( 1 2 ln( Δ± √ Δ 2 -4 2
))| where

Δ=2cosh(  1 ) cosh(  2 )+( + ) sinh(  1 ) sinh(  2 ). ( 4
)
Evolution of is shown in Fig. 4a) for 1 = 2 = ∕2 in an extremely extended parameter space ( 2 , 2 ) (two order of magnitude more as compared to the stability chart of Fig. 1b)). We recognize the ℎ instability tongues originating at = 0.5 for → 0 like for the Mathieu equation, although the tongues are here tailored in discrete pockets. We found that the discrete location of those instability pockets can be rationalized by the input-output synchronization sketched in Fig. 3. Parametric pumping is achieved by dropping the effective gravity (magnets ON) when the pendulum has minimal potential energy, i.e. at ( ) = 0, and putting back strong gravity (magnets OFF) when potential energy is maximum, i.e. when ( ) is an extremum. For example, Fig. 3 illustrates the most efficient pumping at the origin of the classic primary instability tongue, that consists in turning the magnets ON and OFF every quarter of a cycle of the pendulum's response. But many higher super-harmonic modes of parametric amplification are theoretically possible that can verify the aforementioned input-output synchronization. Knowing the electromagnetic pendulum has a natural period 2 ∕ √ 2 + 2 when OFF during  1 = and

2 ∕ √ 2 -2 when ON during  2 = , those modes need to fulfill (2 -1)2 4 √ 2 + 2 = and (2 -1)2 4 √ 2 -2 = ( 5
)
where and are positive integers and > , i.e. 2 > 2 . As the mode numbers ( , ) are varied, Eq.( 5) gives us a new discrete design rule in the ( 2 , 2 ) space to easily locate the center of the instability pockets as shown by the grey dots in Fig. 4a). Physically, (2 -1) and (2 -1) represent the number of quarters of a cycle of the pendulum's response one can fit in the modulation periods 1 and pumpings at the origin of the observed limit cycles are in good qualitative agreement with the synchronization principle illustrated in Fig. 3. Expressing the experimental modulation parameters ( ) and Ω associated with the responses of Fig. 4b) in the dimensionless form 2 and 2 thanks to Eq.( 3), we report experimental data (green crosses) in the stability diagram of Fig. 4a) and find excellent agreement with the expected ( , ) parametric pumping mode from Eq.( 5). The parametric instability of Fig. 4b1), found in the classic primary instability region = 1, leads to a sub-harmonic 2 -periodic limit cycle. Fig. 4b2) corresponds to a ( , ) = (2, 1) unstable parametric mode, located in the instability region = 2, and leads to a -periodic limit cycle as expected since is even (see movie 1 in [START_REF] Protière | Movie 1 showing the experimental response of the electromagnetic pendulum periodically turned in an oscillating state (0) = √ ∕ = 15.24 rad/s during 1 = 0.31 s and ( ) = 6.8 rad/s during 2 = 0.31s. Movie 2 showing the experimental response of the electromagnetic pendulum periodically turned in an oscillating state (0) = √ ∕ = 15.24 rad/s during 1 = 5.08 s and ( )[END_REF]). Fig. 4b3) and Fig. 4b4) display a (15, 7) and (25, 12) parametric pumping mode respectively, leading to extremely high super-harmonic 2periodic motions since is odd in both cases. Since the instability region number follows = + -1, the response of Fig. 4b4) (see movie 2 in [START_REF] Protière | Movie 1 showing the experimental response of the electromagnetic pendulum periodically turned in an oscillating state (0) = √ ∕ = 15.24 rad/s during 1 = 0.31 s and ( ) = 6.8 rad/s during 2 = 0.31s. Movie 2 showing the experimental response of the electromagnetic pendulum periodically turned in an oscillating state (0) = √ ∕ = 15.24 rad/s during 1 = 5.08 s and ( )[END_REF]) is actually located in the 36 ℎ instability tongue as shown in Fig. 4a); an achievement since, to our knowledge, the record = 28 was observed in a MEMS in 2016 [START_REF] Jia | Twenty-eight orders of parametric resonance in a microelectromechanical device for multi-band vibration energy harvesting[END_REF]. It is beyond the scope of this paper to report the plethora of parametric instabilities existing in this fundamental electromagnetic pendulum, as suggested by the impressive number of instability pockets in Fig. 4a), but the various motions displayed in Fig. 4b) already highlights the potential of extreme parametric instabilities for new dynamical control in mechanical systems.

Triggering and sustaining a natural oscillation

Another overlooked limit that could be useful in mechanical Floquet systems is the almost "impulse train" case illustrated in the sketch of Fig. 5a). Here, the magnets are almost continuously OFF, except for every period 1 ≈ where they are turned shortly ON during 2 with a current . Our system is a locally stable pendulum, characterized by a harmonically damped oscillating perturbation with natural period 0 = 2 ∕ (0), except every period 1 where the local evolution function of the pendulum is shortly but drastically changed. This original scenario is still theoretically described by the governing equations Eqs.( 1)-( 4). Notably, the analytical growth rate as a function of ( 2 , 2 ) is given in Fig. 5a) for 1 = 0.98 and 2 = 0.02 . Now the parametric instability tongues are thin parallel regions whose "left" frontiers correspond to the black lines √ 2 + 2 = 0.5 , that have been obtained by introducing the limits  2 → 0 and  1 → 2 in Eq.( 4). This stability diagram actually means one should be able to parametrically amplify the harmonic perturbation with natural period 0 as soon as ≈ 0.5 0 , where is the positive integer representing the instability tongue number.

As a proof of concept, we show in Fig. 5b) the experimental response of the electromagnetic pendulum (characterized by a natural period 0 = 2 ∕ √ ∕ = 412 ms and a natural frequency 0 = 1∕ 0 = 2.43 Hz) when the magnets are continuously OFF except every ≈ 1 ≈ 1220 ms where ≈ 1.2 A during 2 ≈ 25 ms (see movie 3 in [START_REF] Protière | Movie 1 showing the experimental response of the electromagnetic pendulum periodically turned in an oscillating state (0) = √ ∕ = 15.24 rad/s during 1 = 0.31 s and ( ) = 6.8 rad/s during 2 = 0.31s. Movie 2 showing the experimental response of the electromagnetic pendulum periodically turned in an oscillating state (0) = √ ∕ = 15.24 rad/s during 1 = 5.08 s and ( )[END_REF]). Expressing the experimental parameters (0) = 2 0 , ( ) and Ω = 2 ∕ in the dimensionless form 2 and 2 thanks to Eq.( 3), we report experimental data (green cross) in the stability diagram of Fig. 5a). We find, as expected since the applied modulation period is 6 th time slower than half the natural period of the pendulum, that the limit cycle of Fig. 5b) results from a parametric resonance of order = 6. Since is even, we expect to trigger a -periodic limit cycle which is confirmed by the FFT of the experimental permanent regime in Fig. 5b) that shows spectral rays located every 0 ± ℎ with ℎ an integer and = 1∕ . Interestingly, the limit cycle is very close to a purely sinusoidal motion with a fundamental frequency 0 since the amplitudes of the secondary harmonics weight no more than 3% of the fundamental one. Actually, as illustrated by the inset of Fig. 5b) that displays the last three periods of the recorded experimental response, the triggered limit cycle is the free damped oscillation of the pendulum with natural period 0 that is "reset" every 3 0 thanks to the synchronized impulse of input energy represented by red vertical lines. Every ≈ 0.5 0 , the same periodically reseted natural sinusoidal motion could be triggered and sustained with the difference that the growth rate of the transient oscillations decreases with until eventually no instability occurs for large = because of insufficient energy exchanges (see Appendix B for the case ≈ 6 0 ). To increase the maximum order of observed parametric resonance , one could increase the factor of the pendulum or the current intensity . Interestingly, unlike the classic resonance phenomenon, the bandwidth of this parametric resonance, i.e. the width of the instability tongues of Fig. 5a), depends on the "width" of the imposed impulse train and not on the factor: the shorter the impulse train, the shorter the periodic energy exchange and the thinner the instability regions. Another useful property is the tuning capabilities offered by this system: by maintaining the magnets almost continuously ON with < instead of OFF with = 0 as in Fig. 5a), one could efficiently trigger and sustain any sinusoidal oscillations with angular frequency ( ) between (0) and almost 0, as suggested by Fig. 2. For systems like MEMS that are tiny beam or plate resonators with enormous factors (because they operate in Ultra High Vacuum) driven by periodic electrostatic fields, an application of this particular parametric pumping could allow to efficiently generate highly super-harmonic sinusoidal signals with extremely tunable fundamental frequencies.

Conclusions and perspectives

Based on a better fundamental understanding of parametric oscillators, we have shown through theoretical and experimental proofs of concepts, how to extremely enhance and better control parametric instabilities. The presented concepts being universal, they could apply to any dynamical systems with a natural cycle which time scale could be periodically varied: from resonant MEMS to purely electronic circuits or lasers where researchers are often seeking for new dynamical functionalities to exploit, but also non engineered systems such as business cycles or circadian rhythms.

Quasi-static symmetry breaking like buckling allows, through the slow variation of a single parameter, to switch from one unstable to a new stable equilibrium configuration, albeit with a unique time scale mainly determined by the growth rate of the diverging instability. Parametric instability is a dynamic symmetry breaking that, upon the periodic variation of a parameter, should allow for even more control on switchings between equilibrium or time periodic configurations, as soon as one modulates the evolution function of a system close to its diverging instability. Since elastic buckling has been shown to enable actuations of soft machines [START_REF] Yang | Buckling of elastomeric beams enables actuation of soft machines[END_REF] or structures [START_REF] Douglas | Snapping surfaces[END_REF], the application of extreme parametric instabilities to those fields should improved actuation, especially in time.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

A. Rationalization of the instability thresholds of classical parametric systems

We rationalize various experiments from the literature with the linear parametric oscillator governed by Eq.( 1) of the manuscript. All experimental data can then be represented in the same stability diagram.

A.1. Faraday instability

The first experimental demonstration of a parametric behavior was made by Faraday in 1831. His experiment consists in creating a wave motion in a fluid by vertically oscillating it. A fluid is placed over a vibrating pot with a frequency of excitation and an acceleration of the excitation Γ. The fluid is characterized by : ( ) the amplitude of displacement of the surface, ℎ the depth of the fluid, the surface tension, the density of the fluid, the wavelength and the gravitational acceleration.

! "# ! cos #' Γ Vibrating pot h )(#) , -, / !(#)
Based on the work of Benjamin and Ursell [START_REF] Brooke | The stability of the plane free surface of a liquid in vertical periodic motion[END_REF] and considering only the first the evolution of the amplitude of displacement of the surface of the fluid for the Faraday instability reads

2 ( ) 2 + tanh( ℎ) 2 + -Γ cos( ) ( ) = 0 (6)
where = 2 ∕ is the fluid wavenumber and the dispersion equation for capillaro-gravitational waves is

= √ tanh ( ℎ) + 2 . ( 7
)
By implementing a change of variable = and using a first approximation for small angles ( ) = ( )∕ , we are able to write Eq.(1) in the standard dimensionless form of the manuscript: . -2 0.250 0.267

Table 1

Experimental parameters for the Faraday instability used by Protière [START_REF] Protiere | Gouttes rebondissantes: une association ondeparticule à échelle macroscopique[END_REF] and corresponding values of the dimensionless modulation parameters 2 and 2 . Data points presented in Fig. 1 of the manuscript.

(

) 8
An experimental study of Faraday instability is presented in the work of Protière [START_REF] Protiere | Gouttes rebondissantes: une association ondeparticule à échelle macroscopique[END_REF]. The properties of the fluid are: surface tension = 0.0209 ∕ , density = 0.965 × 10 3

. -3 , the depth of the fluid is ℎ = 4×10 -3 and the gravitational acceleration is = 9.81 . -2 . Using the dispersion equation of capillary-gravitational waves Eq.( 7) with the explicit form of parametric parameters of Eq.( 8), we are able to calculate in Table 1 the dimensionless parameters ( 2 , 2 ) for each experimental point that are then reported in the Fig. 1 of the manuscript.

A.2. Parametric pendulum

The system is a pendulum where the pivot point is vertically oscillating as shown in Fig. The undamped equation of motion for this system is the following where is the moment of inertia and the rest of the parameters are defined on Fig. 2. In this case parametric instabilities are used to trigger the motion of the pendulum in the equilibrium position = 0. By linearizing Eq.( 9) about = 0 (approximation of small angles for the angular displacement ( )) and upon the change of variable = , we are able

Figure 1 : 1 22 2

 112 Figure 1: Linear stability of a mass locally moving in a harmonically varying potential  = 1 2 ( 2 + 2 cos( )) 2 . a) Four qualitative scenarios depending on 2 and 2 . Black lines show  = 1 2 2 2 , red dashed lines  = 1 2 ( 2 + 2 ) 2 and blue dashed lines  = 1 2 ( 2 -2 ) 2 . A: 2 < 0, 2 < -2 . B: 2 < 0, 2 > -2 . C: 2 > 0, 2 > 2 . D: 2 > 0, 2 < 2 . b) Linear stability diagram of the Mathieu equation Eq.(1) in the modulation parameter space ( 2 , 2 ). Triangle, circle, cross, square represent classic experiments on parametric instability.

  < 2 . b) Linear stability diagram of the Mathieu equation Eq.(1) in the modulation parameter space ( 2 , 2 ). Triangle, circle, cross, square represent classic experiments on parametric instability.

Figure 2 :

 2 Figure 2:The experimental Floquet oscillator under study is a planar pendulum with a metallic marble that is symmetrically placed between two identical attracting electromagnets whose attracting force depends on the imposed electrical current (see sketch in inset). The scalar 2 ( ), characterizing the local evolution function of the electromagnetic pendulum, drastically depends on the constant control parameter .

Figure 4 :

 4 Figure 4: Triggering extremely high orders of parametric resonances. a) Stability diagram of the Meissner equation Eq.(1) showing , up to 0.4, in the extremely extended ( 2 , 2 ) space. Grey dots ( , ) represent the discrete solutions of Eq.(5). Green crosses represent the experimental parameters expressed in the ( 2 , 2 ) space. Inset zooms on the classic first instability regions. b) Dynamical responses of the electromagnetic pendulum of Fig.2 when 2 ( ( )) is modulated in a square wave fashion with 1 = 2 = ∕2. b1) A classic "(1, 1)" parametric amplification obtained for (0) = √ ∕ ≈ 15.15 rad/s, ( ) ≈ 6.8 rad/s and ≈ 0.33s. b2) A (2, 1) parametric instability obtained for (0) ≈ 15.15 rad/s, ( ) ≈ 6.8 rad/s and ≈ 0.62s. b3) A (15, 7) parametric pumping obtained for (0) ≈ 15.15 rad/s, ( ) ≈ 6.5 rad/s and ≈ 6.02s. b4) A (25, 12) parametric amplification obtained for (0) ≈ 15.15 rad/s, ( ) ≈ 6.5 rad/s and ≈ 10.16s.

Figure 5 :

 5 Figure 5: Triggering a "natural" limit cycle. a) Stability diagram of the Meissner equation Eq.(1) in the "impulse train" case when 1 = 0.98 and 2 = 0.02 . Black lines show the limit 1 → and 2 → 0. The green cross represents the experimental modulation parameters associated with the response shown in b) Left: experimental response observed when the electromagnetic pendulum is OFF during 1 ≈ 1.22 s and ON, with ≈ 1.2 A ( > ) during 2 ≈ 0.025 s (≈ 0.02 1 ). Inset zooms on the -periodic limit cycle. Red lines represent the very short time the magnets are ON. Right: Fast Fourier Transform of the permanent regime of the response. The black dashed-line represents the value of the natural frequency of the pendulum when the red ones represent the secondary harmonics 0 ± ℎ , where = 1∕ and ℎ is an integer.

Figure 6 :

 6 Figure 6: Experimental setup to observe Faraday instability.A fluid is placed over a vibrating pot with a frequency of excitation and an acceleration of the excitation Γ. The fluid is characterized by : ( ) the amplitude of displacement of the surface, ℎ the depth of the fluid, the surface tension, the density of the fluid, the wavelength and the gravitational acceleration.

̈

  ( ) + 2 -2 cos( ) ( ) = 0 with 2 = tanh ( ℎ)

Figure 7 :

 7 Figure 7: Experimental setup and parameters for a parametric pendulum. The frequency of excitation is and the amplitude of excitation is . The pendulum is characterized by its length , mass and ( ) the angular displacement measured from the vertical position. The gravitational acceleration is .

, respectively. Fig.4b) shows four experimental dynamical responses whose transient regimes are representative of the many parametric instabilities observed in the electromagnetic pendulum of Fig.2. As expected, the parametric

Acknowledgments

A. Lazarus and S. Protière are grateful for financial support from Sorbonne University (EMERGENCES grants).

Experimental parametric parameters

= 0.22 Hz ; = 2.9 0.244 0.141

Table 2

Experimental parameters used by San Martin [START_REF] Juan R Sanmartin | O botafumeiro: Parametric pumping in the middle ages[END_REF] and corresponding values of the dimensionless modulation parameters 2 and 2 . Data point presented in Fig. 1 of the manuscript.

to write Eq.( 9) in the form of the Mathieu equation of the manuscript:

̈ ( ) + 2 -2 cos( ) ( ) = 0

An experimental example of this system is the giant censer ("O Botafumeiro") from Santiago de Compostella presented in the work of San Martin [START_REF] Juan R Sanmartin | O botafumeiro: Parametric pumping in the middle ages[END_REF]. The length of the pendulum is = 20.6 , the mass is = 56.5 and the gravitational acceleration is = 9.81 . -2 . The dimensionless experimental parameters are calculated using the experimental values of San Martin [START_REF] Juan R Sanmartin | O botafumeiro: Parametric pumping in the middle ages[END_REF] and Eq.( 10) and are reported in Table 2.

A.3. Inverted pendulum

The system is an inverted pendulum where the pivot point is vertically oscillating (the displacement is often imposed by a shaker). In this case parametric instabilities are used to stabilize the naturally diverging equilibrium.

Vibrating pot

Figure 8: Experimental setup and parameters for stabilizing an inverted pendulum. The inverted pendulum is placed over a shaker with a frequency of excitation , an amplitude of excitation and an acceleration of excitation Γ = 2 . The pendulum is characterized by its length , mass and ( ) is the angular displacement measured from the vertical position. The gravitational acceleration is .

The undamped equation of motion for this system is the same presented for the parametric pendulum Eq.( 9) but in this case the considered equilibrium position is = . By linearizing Eq.( 9) about this equilibrium position (approximation of small angles for the angular displacement ( )) and upon the change of variable = , we are able to write = 157.17 Hz ; Γ = 61.3 . -2 0.050 0.31

Table 3

Experimental parameters used by Smith [START_REF] Smith | Experimental study of an inverted pendulum[END_REF] and values of the corresponding dimensionless parameters 2 and 2 . Data point presented in Fig. 1 of the manuscript.

Eq.( 9) in the form:

For this system, an experimental example is presented in the work of Smith [START_REF] Smith | Experimental study of an inverted pendulum[END_REF]. The length of the pendulum is = 0.008 , the mass is = 0.19 × 10 -3 and the gravitational acceleration is = 9.81 . -2 . The dimensionless experimental parameters are calculated using the experimental values of Smith [START_REF] Smith | Experimental study of an inverted pendulum[END_REF] and Eq.( 11) and are shown in Table 3.

A.4. Microelectromechanical systems: MEMS

In this case parametric instabilities are used to amplify the movement of microelectromechanical membranes (MEMS). Jia et al. [START_REF] Jia | Twenty-eight orders of parametric resonance in a microelectromechanical device for multi-band vibration energy harvesting[END_REF] used a Duffing oscillator to describe the motion of the membrane:

where is the displacement of the membrane, is the natural frequency of the mode , 1 is the viscous damping ratio, 2 is the nonlinear quadratic damping coefficient, is the mass normalised Duffing coefficient, is a standard coefficient relating the external excitation to the parametric excitation, is the frequency of excitation and Γ is the acceleration of the excitation.

To compare this equation with our model we make the assumption that there is no damping and we neglect non linear terms, i.e.: we consider only small oscillations about the equilibrium position. Upon the change of variable = , equation Eq.( 12) can be expressed in the standard dimensionless form of the Mathieu equation of the manuscript:

The experimental parameters are the one used in the work of Jia et al. [START_REF] Jia | Twenty-eight orders of parametric resonance in a microelectromechanical device for multi-band vibration energy harvesting[END_REF]. The natural frequency of the membrane is 0 = 980 Hz and the value of the standard coefficient is = 1.7744 × 10 6 . The dimensionless experimental parameters are calculated using the experimental values of Jia et al. [START_REF] Jia | Twenty-eight orders of parametric resonance in a microelectromechanical device for multi-band vibration energy harvesting[END_REF] and Eq.( 13) and are reported in Table 4. 4.17 1.56

Experimental parametric parameters

Table 4

Experimental parameters used by Jia et al. [START_REF] Jia | Twenty-eight orders of parametric resonance in a microelectromechanical device for multi-band vibration energy harvesting[END_REF] and values of the corresponding dimensionless modulation parameters 2 and 2 . Data points presented in Fig. 1 of the manuscript.

B. Triggering and sustaining a natural oscillation

We present the experimental observation of a 12 th order parametric resonance using the "impulse train" scenario described in Section 4. The period of modulation is = 6 0 and current intensity = 1.25A. To assure a minimal exchange of energy we set 1 = 0.99 0 and 2 = 0.01 0 . Expressing the experimental parameters in the ( 2 , 2 ) dimensionless space, we report experimental data (green cross) in the stability diagram of Fig. 9a). We find, as expected since the applied modulation period is 12 th time slower than half the natural period of the pendulum, that the limit cycle of Fig. 9b) results from a parametric resonance of order = 12. Since is even, we expect to trigger a -periodic limit cycle which is confirmed by the FFT of the experimental permanent regime in Fig. 9b). Interestingly, the limit cycle is very close to a purely sinusoidal motion with a fundamental frequency 0 since the amplitudes of the secondary harmonics weight no more than 3% of the fundamental one. Actually, as illustrated by the inset of Fig. 9b) that displays the last two periods of the recorded experimental response, the triggered limit cycle is the free damped oscillation of the pendulum with natural period 0 that is "reset" every 6 0 thanks to the synchronized impulse of input energy represented by red vertical lines.