Florent Dewez
email: florent.dewez@inria.fr

Stability of propagation features under time-asymptotic 1 approximations for a class of dispersive equations 2

Keywords: wave packet, dispersive equation, oscillatory integral, 6 stationary phase method, frequency band 35B40, 35S10, 35B30, 35Q41, 35Q40

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Introduction

In this paper, we are interested in the time-asymptotic behaviour of wave 10 packets of the form where t ∈ R, x ∈ R and f : R -→ R is a strictly convex symbol. We suppose that the Fourier transform Fu 0 of u 0 ∈ S(R) is supported in a bounded interval [p 1 , p 2 ], where p 1 < p 2 are finite real numbers. In terms of evolution equations, wave packets of the form (1) are solutions of the following type of dispersive equations:

i ∂ t -f D u f (t) = 0 u f (0) = u 0 , (2)
for t ∈ R, where f (D) is the Fourier multiplier associated with f and u 0 ∈ S(R) the initial datum supposed to be in the frequency band [p 1 , p 2 ]. For instance, the solutions of the free Schrödinger equation, of the Klein-Gordon equation or of certain higher-order evolution equations can be described by wave packets of the form (1); we refer to [START_REF] Dewez | Estimates of oscillatory integrals with stationary phase and singular amplitude: Applications to propagation features for dispersive equations[END_REF]Sec. 6] for further details. In the present setting, the frequency band hypothesis prevents the wave packet [START_REF] Ali Mehmeti | Dispersive effects and high frequency behaviour for the Schrödinger equation in star-shaped networks[END_REF] from being too much localised in space according to the uncertainty principle and makes hence challenging the task of describing its spatial propagation. Some approaches solving this challenge time-asymptotically have been developed. In [START_REF] Ali Mehmeti | The Influence of the Tunnel Effect on the L ∞ -time Decay[END_REF], the authors propose to approximate the solution of the Klein-Gordon equation on a star-shaped network by a spatially localised function, the latter tending to the true solution as the time tends to infinity.

This has been achieved by applying precisely the version of the stationary phase method given in [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators I[END_REF]Theorem 7.7.5] to an integral solution formula of the equation. The desired approximation is then given by the first term of the asymptotic expansion from the stationary phase method. The principle of the stationary phase method, which consists in evaluating the integrand of the oscillatory integral of interest at the stationary point of the phase function, combined with the bounded frequency band hypothesis leads to an approximation supported in a space-time cone: this cone describes both the motion and the dispersion of the solution for large times. In particular, the results exhibit in this setting the influence of the tunnel effect on the time-decay rate of the solution.

In [START_REF] Ali Mehmeti | Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation[END_REF], this approach has been adapted to the setting of the free Schrödinger equation on the line. In that paper, the initial data are assumed to have integrable singular frequencies in order to study the effect of such singularities on the time-asymptotic behaviour. The version of the stationary phase method proposed in [18, Sec. 2.9] (also proposed in [START_REF] Erdélyi | Asymptotic Representations of Fourier Integrals and The Method of Stationary Phase[END_REF]) has been used since it covers the case of singular amplitudes; we mention that the authors in [START_REF] Ali Mehmeti | Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation[END_REF] propose modern formulations and detailed proofs of the results from [START_REF] Erdélyi | Asymptotics expansions[END_REF]. The results show that a free particle with a singular frequency tends to travel at the speed associated with this frequency. This is highlighted by the existence of space-time cones, containing the direction given by the singular frequency, in which the time-decay rates are below the rate of the classical decay inherited from the classical dispersion.

However, the expansion provided in [START_REF] Ali Mehmeti | Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation[END_REF] is proved to blow up when approaching the space-time direction associated with the singular frequency:

this prevents the method from approximating uniformly the true solution in regions containing this direction. This is due to the fact that the first term of the expansion inherits the singularity of the initial datum; see [START_REF] Ali Mehmeti | Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation[END_REF]Sec. 3] for more details. To tackle this issue, another approach has been proposed in [START_REF] Dewez | Estimates of oscillatory integrals with stationary phase and singular amplitude: Applications to propagation features for dispersive equations[END_REF]: the precision of asymptotic expansions to one term is removed in favour of less precise but more flexible uniform estimates. In particular, they cover the above critical regions. Further this flexibility has permitted to consider not only the free Schrödinger equation but also equations of type [START_REF] Ali Mehmeti | Dispersive effects for the Schrödinger equation on the tadpole graph[END_REF] with initial data having singular frequencies. The uniform estimates for the so- to the case of singular and integrable amplitudes.

The approach developed in [START_REF] Ali Mehmeti | The Influence of the Tunnel Effect on the L ∞ -time Decay[END_REF] and the subsequent adaptations appearing in [START_REF] Ali Mehmeti | Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation[END_REF][START_REF] Dewez | Estimates of oscillatory integrals with stationary phase and singular amplitude: Applications to propagation features for dispersive equations[END_REF] permit then to describe both the motion and the dispersion of the solutions via the inclination of space-time cones (which depends actually on the frequency support of the initial datum). Nevertheless it is noteworthy that the origin of the space-time cones resulting from the above methods is always put at the space-time point (0, 0), whatever the localisation of the initial datum is: the approximations provided by the method do not have the right spatial positions. As a consequence this method, although asymptotically correct, leads to inaccurate approximations of the solutions on large but finite time intervals [0, T ] for initial data spatially far from the origin.

In view of this, we aim at proposing time-asymptotic approximations whose mean positions and variances are close (even the same) to those of the solutions of equation ( 2) for a better description of the spatial propagation. To this end, we proceed in four steps, each of them containing a new argument:

1. The first step, which is actually independent from the setting of dispersive equations, consists in establishing new uniform and explicit remainder estimates for asymptotic expansions of oscillatory integrals of the form

∀ ω > 0 R U (p) e iωψ(p) dp ,
where the amplitude U : R -→ C is compactly supported and the phase ψ : R -→ R is concave. The new remainder estimate we establish involves the L 2 -norm of the first derivative of the amplitude and not the L ∞ -norm as in the original proofs [START_REF] Ali Mehmeti | Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation[END_REF][START_REF] Erdélyi | Asymptotics expansions[END_REF]. This is achieved by applying Cauchy-Schwarz inequality to the integral representation of the remainder term. We refer to Theorems 3.3 and 3.4 for the precise statements of these results.

2. As a second step, we introduce a space-time shift parametrised by a two-dimensional parameter (t 0 , x 0 ) in the integral formula [START_REF] Ali Mehmeti | Dispersive effects and high frequency behaviour for the Schrödinger equation in star-shaped networks[END_REF]. Broadly speaking, this new argument modifies the initial datum which is then given by the solution at time t 0 spatially translated by x 0 . Inspired by the main lines of the method developed in [START_REF] Ali Mehmeti | The Influence of the Tunnel Effect on the L ∞ -time Decay[END_REF], we apply then carefully a stationary phase method to this shifted solution formula to obtain here a family of time-asymptotic expansions parametrised by (t 0 , x 0 ). This parameter turns out to be the origin of the cone in which lies the support of the first term. This result is given in Theorem 2.2.

3.

Here we use our new version of the stationary phase method described above to expand the formula [START_REF] Ali Mehmeti | Dispersive effects and high frequency behaviour for the Schrödinger equation in star-shaped networks[END_REF]. Since this version makes appear the L 2 -norm of the derivative of Fu 0 , it offers the possibility to apply Plancherel's theorem. This leads to an error bound depending explicitly on the spatial part of the initial datum and on (t 0 , x 0 ). The third step consists then in computing explicitly the parameter (t * , x * ) which minimises the (t 0 , x 0 )-dependent family of error bounds provided in Theorem 2.2. The expansion associated with this optimal parameter, given in Corollary 2.5, is then the one whose error bound is the smallest.

4. In the last step, we prove that the mean position of the first term associated with (t * , x * ) is the same as the one of the solution [START_REF] Ali Mehmeti | Dispersive effects and high frequency behaviour for the Schrödinger equation in star-shaped networks[END_REF] and that the difference between the two variances is constant; this is stated in Theorem 2.6. It is noteworthy that the proof of this theorem shows that the first term associated with (t * , x * ) is actually the only one in the family of first terms we provide which satisfies these two properties.

Contrary to the original method developed in [START_REF] Ali Mehmeti | The Influence of the Tunnel Effect on the L ∞ -time Decay[END_REF] and adapted in [START_REF] Ali Mehmeti | Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation[END_REF][START_REF] Dewez | Estimates of oscillatory integrals with stationary phase and singular amplitude: Applications to propagation features for dispersive equations[END_REF], the present refined approach offers sufficient flexibility to provide time-asymptotic approximations for solutions of equations of type (2) which are more precise and which verify expected propagation properties.

Let us illustrate our main result in the case of the free Schrödinger equation on the line with initial datum u 0 ∈ S(R), namely

   i ∂ t u S (t) = - 1 2 ∂ xx u S (t) u S (0) = u 0 , (3)
for t ∈ R. We note that equation ( 3) is actually of the form (2) with symbol f (p) = 1 2 p 2 and its solution is given by

∀ (t, x) ∈ R × R u S (t, x) = 1 2π R Fu 0 (p) e -1 2 itp 2 +ixp dp . (4)
In terms of quantum mechanics, the solution of the free Schrödinger equation is the wave function associated with the free particle being in the state u 0 at the initial time. Further the frequency band hypothesis means that the particle has a momentum localised in the interval [p 1 , p 2 ]. According to the physical principle of group velocity, the wave packet (4) will travel in space at different speeds between p 1 and p 2 over time. Hence a free wave packet in the frequency band [p 1 , p 2 ] is expected to be mainly spatially localised in an interval of the form p 1 (t-t 0 )+x 0 , p 2 (t-t 0 )+x 0 , where t 0 and x 0 have to be fixed, illustrating the propagation of the associated particle. For instance, a partial formalisation of this principle is given by Ehrenfest theorem [START_REF] Hall | Quantum Theory for Mathematicians[END_REF]Proposition 3.19] which describes the evolution of the mean position of the particle (but not the dispersion).

In the following result, we apply Corollary 2.5 to the setting of the free Schrödinger equation (3) to provide a time-asymptotic expansion of the solution (4) with explicit error estimates. According to Theorem 2.6, the resulting approximation has the right mean position and its variance is equal to the solution variance plus an explicit constant. This is actually the consequence of the fact that the origin of the space-time cone, in which lies the support of the approximation, is actually put at the mean spatial position of the solution at the time when the variance of the solution is minimal. Hence the resulting approximation describes both position and dispersion of the true solution and the theorem provides a formalisation of the principle of group velocity which seems to be more precise than Ehrenfest theorem in the setting of a free particle.

Theorem 1.1. Consider the free Schrödinger equation on the line (3) with initial datum u 0 ∈ S(R). Let p 1 , p 2 , p1 and p2 be finite real numbers such that

[p 1 , p 2 ] ⊂ (p 1 , p2 ). Suppose u 0 L 2 (R) = 1 and supp Fu 0 ⊆ [p 1 , p 2 ],
and define

• t * = arg min τ ∈R R x 2 u S (τ, x) 2 dx - R x u S (τ, x) 2 dx 2 ; • x * = R x u S (t * , x) 2 dx .
Then for all (t, x)

∈ (t, x) ∈ R\{t * } × R p 1 x -x * t -t * p 2 , we have u S (t, x) - 1 √ 2π e -sgn(t-t * )i π 4 e -it x-x * t-t * 2 +ix x-x * t-t * Fu 0 x -x * t -t * |t -t * | -1 2 C 1 (δ, p1 , p2 ) R x 2 u S (t * , x) 2 dx - R x u S (t * , x) 2 dx 2 |t -t * | -δ ,
where the real number δ is arbitrarily chosen in 1 2 , 3 4 . And for all (t, x) ∈ (t, x) ∈ R\{t * } × R

x -x * t -t * < p 1 or p 2 <

x -x * t -t * , we have

u S (t, x) C 2 (p 1 , p 2 , p1 , p2 ) R x 2 u S (t * , x) 2 dx - R x u S (t * , x) 2 dx 2 + C 3 (p 1 , p 2 , p1 , p2 ) u 0 L 1 (R) |t -t * | -1 .
All the above constants are defined in Theorem 2.2.

See Corollary 2.5 for the general result. Let us now make some comments on this result:

• On one hand, we observe that the first term is spatially well-localised for a solution in a narrow frequency band; on the other hand, the error is bounded by the minimal value of the standard deviation of the solution. Combined with the uncertainty principle, this exhibits a compromise: a frequency well-localised solution (4) can be approximated by a function supported in a narrow space-time cone but a time sufficiently far from t * is required to achieve a good precision; on the other hand, the approximation of the solution (4) with a small minimal standard deviation lies in a larger cone but the error bound is smaller than in the preceding case.

• We remark that the time-decay rate is shifted by t * , which is the time when the variance of the solution (4) is minimal; this corresponds to the fact that the origin of the cone belongs to the vertical space-time line (t, x) ∈ R × R t = t * . Hence if we require an error smaller than a certain threshold ε > 0, then this precision is achieved for all t ∈ R satisfying

|t -t * | > C 1 (δ, p1 , p2 ) 1 δ R x 2 u S (t * , x) 2 dx - R x u S (t * , x) 2 dx 2 1 2δ ε -1 δ =: η(ε) .
In particular if we are interested in the evolution of the solution for positive times and if t * < -η(ε), then the error of the approximation is smaller than ε for all t 0. This has to be compared with the results from the classical approach (as in [START_REF] Ali Mehmeti | Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation[END_REF][START_REF] Ali Mehmeti | The Influence of the Tunnel Effect on the L ∞ -time Decay[END_REF]) which always imply the existence of a small time-interval [0, T ] during which the error is larger than a given threshold. This is due to the lack of flexibility of the classical approach which enforces t * = 0 (the decay rate is then

t -1 2
) and puts automatically the origin of the cone at the origin of space-time.

Let us now comment on some possible improvements or applications of the present results. First of all, an interesting issue would be to apply the approach developed in this paper to more complicated settings. One may consider dispersive equations on certain networks where integral solution formulas are available, as for example the Schrödinger equation on a starshaped network with infinite branches [START_REF] Ali Mehmeti | Dispersive effects and high frequency behaviour for the Schrödinger equation in star-shaped networks[END_REF] or on a tadpole graph [START_REF] Ali Mehmeti | Dispersive effects for the Schrödinger equation on the tadpole graph[END_REF]. In both papers, the time-asymptotic behaviour is studied by a frequency band hypothesis and one may hope a better description of physical phenomena by using our refined method.

We could also consider the Schrödinger equation with a potential. In [START_REF] Dewez | Time-asymptotic propagation of approximate solutions of Schrödinger equations with both potential and initial condition in Fourier-frequency bands[END_REF], As explained in this paper, the notion of frequency band is physically meaningful and permits to describe time-asymptotically the propagation of solutions of certain dispersive equations in a precise way. However it is a restrictive hypothesis: for example, a function in a finite frequency band is necessarily a C ∞ -function. Hence it would be relevant to extend this notion to functions whose Fourier transform is not necessarily compactly supported but still localised in a weaker sense. In this setting, the first term of the expansion is no longer supported in a space-time cone and so one has to quantify the localisation by means of different tools. For instance, we can consider approaches based on weighted norms; such norms have been used

in [START_REF] Egorova | Zero energy scattering for onedimensional Schrödinger operators and applications to dispersive estimates[END_REF], [START_REF] Egorova | Dispersion estimates for one-dimensional Schrödinger and Klein-Gordon equations revisited[END_REF] or in [START_REF] Goldberg | Transport in the One-Dimensional Schrödinger Equation[END_REF] to show that the continuous part of the perturbed Schrödinger evolution transports away from the origin with non-zero velocity.

Our approach makes appear naturally the shifted decay rate |t -

t * | -1 2 ,
where t * minimises the variance of the solution. It would be also interesting to introduce this time-shift in other existing results to obtain greater precision.

For instance, one may consider the important L p -L p estimates for which a simple argument makes apparent the shifted decay; this is proved in the following result:

Proposition 1.2. Consider the free Schrödinger equation on the line (3) with u 0 ∈ S(R) and define t * ∈ R as follows:

t * := arg min τ ∈R R x 2 u S (τ, x) 2 dx - R x u S (τ, x) 2 dx 2 .
Then for all p ∈ [2, ∞] and for all t ∈ R\{t * }, we have

u S (t, .) L p (R) 1 4π -1 2 + 1 p u S (t * , .) L p (R) |t -t * | -1 2 + 1 p ,
where p is the conjugate of p.

Proof. For the sake of clarity, we use the one-parameter group e -it∂xx t∈R which permits to describe the Schrödinger evolution as follows:

∀ t ∈ R u S (t) = e -it∂xx u 0 .
Using the group property, we have for any t ∈ R,

e -it∂xx u 0 = e -i(t-t * )∂xx e -it * ∂xx u 0 ,
and by applying the classical L p -L p estimate [6, Proposition 2.2.3] to the above right-hand side, we obtain for all t = t * ,

u S (t, .) L p (R) = e -it∂xx u 0 L p (R) 1 4π -1 2 + 1 p e -it * ∂xx u 0 L p (R) |t -t * | -1 2 + 1 p = 1 4π -1 2 + 1 p u S (t * , .) L p (R) |t -t * | -1 2 + 1 p .
Note that we are allowed to apply the classical L p -L p estimate since e -it * ∂xx u 0 ∈ S(R) ⊂ L p (R) thanks to the hypothesis u 0 ∈ S(R).

Since the classical L p -L p estimates are exploited to establish Strichartz estimates which are themselves used to study non-linear dispersive phenomena, it is necessary to extend the above shifted L p -L p estimates to spaces larger than the Schwartz space in view of precise applications. In particular, one may examine whether t * defined above still satisfies some optimal conditions; this could be linked with the results established in [START_REF] Cazenave | A note on decay rates for Schrödinger's equation[END_REF].

Regarding long-term perspectives of our work, one could consider the full soliton resolution for non-linear dispersive equations [START_REF] Duyckaerts | Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation[END_REF][START_REF] Duyckaerts | Profiles of bounded radial solutions of the focusing, energy-critical wave equation[END_REF][START_REF] Duyckaerts | Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case[END_REF][START_REF] Duyckaerts | Classification of radial solutions of the focusing, energy-critical wave equation[END_REF][START_REF] Duyckaerts | Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations[END_REF], which aims at classifying the asymptotic behaviour of the non-linear solutions. A key argument for the results contained in this series of papers is the channel energy method [START_REF] Kenig | Channels of energy for the linear radial wave equation[END_REF], which consists in estimating the associated free solution outside a space-time cone or channel; this estimate is then used to prove that a dispersive term appearing in the decomposition of the non-linear solution tends to 0 in the energy-space. In particular, we mention that the authors of [START_REF] Côte | Energy partition for the linear radial wave equation[END_REF] have to shift in time the cones and channels to derive the desired estimates.

Hence one might hope that the ideas proposed in the present paper could help to understand the requirement for this shift and more generally to refine the channel energy method.

Finally we could also think about minimal escape velocities [START_REF] Hunziker | Minimal Escape Velocities[END_REF][START_REF] Sigal | Local decay and velocity bounds for timeindependent and time-dependent Hamiltonians[END_REF] which aim at exhibiting propagation features for evolution operators of type e -itH , where H is a general Hamiltonian; for instance, one may consider the opera-

tor H = -1 2 ∂ xx + V where V is real-valued potential.
As explained in [START_REF] Huang | Uncertainty principle, minimal escape velocities and observability inequalities for schrödinger equations[END_REF],

the method to establish these estimates generalises the integration by parts which is actually crucial to describe the time-asymptotic behaviour of wave packets, as illustrated in the present paper. Our approach could bring more precision to the abstract setting and hence lead to estimates containing more information on the propagation of general wave packets.

The paper is organised as follows: in the following section, we state the main results on time-asymptotic expansions for dispersive equations of the form [START_REF] Ali Mehmeti | Dispersive effects for the Schrödinger equation on the tadpole graph[END_REF]. Since the proofs of the main results are substantially based on a careful application of the stationary phase method, Section 3 is devoted to our new version of this method. Finally Section 4 contains the proofs of each result presented in Section 2.

Main results on time-asymptotic approximations for dispersive equations

In this section, we present the main results of our paper. Prior to this, we introduce the mathematical setting as well as some notations.

The Fourier transform Fu : R -→ C of a function u : R -→ C belonging to the Schwartz space S(R) is defined by

∀ p ∈ R Fu(p) := R u(x) e -ixp dx .
The Fourier transform defines an invertible operator from S(R) onto itself, and can be extended to the space of square-integrable functions L 2 (R) and to the tempered distributions S (R). Moreover, for u ∈ L 2 (R), Plancherel theorem assures the following equality:

∀ u ∈ L 2 (R) u L 2 (R) = 1 √ 2π Fu L 2 (R) ;
see [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators I[END_REF]Theorem 7.1.6].

Consider now a C ∞ -function f : R -→ R such that all its derivatives grow at most as a polynomial at infinity and consider the associated operator f (D) : S(R) -→ S(R) defined by

∀ x ∈ R f (D)u(x) := 1 2π R f (p) Fu(p) e ixp dp = F -1 f Fu (x) ,
which can be extended to the tempered distributions S (R). The operator

f (D) : S (R) -→ S (R) is called a Fourier multiplier associated with the symbol f .
Given such an operator, we introduce the following evolution equation on the line,

i ∂ t -f D u f (t) = 0 u f (0) = u 0 , (5)
for t ∈ R. If we suppose u 0 ∈ S (R) then the equation ( 5) has a unique solution in C 1 R, S (R) given by the following solution formula,

u f (t) = F -1 e -itf Fu 0 .
We refer to [START_REF] Ben Artzi | Uniform Estimates for a Class of Dispersive Equations[END_REF] for a detailed study of this family of equations. In this paper, we suppose that the symbol f is strictly convex; an important example of such an equation is given by the free Schrödinger equation whose symbol is

f S (p) = 1 2 p 2 .
For the sake of better presentation of the results, we consider initial data u 0 belonging only to the Schwartz space S(R) to focus on the approach we propose. We mention that it is possible to extend our results to the case of initial data in L 2 (R) with additional assumptions on regularity and decay; but this falls out of the scope of the paper.

Further the initial data are assumed to be in bounded frequency bands, meaning that their Fourier transforms are supported on bounded intervals [p 1 , p 2 ], where p 1 < p 2 are finite real numbers. Under such hypotheses, the solution formula for the equation ( 5) defines a function u f : R × R -→ C

given by

u f (t, x) = 1 2π p 2 p 1 Fu 0 (p) e -itf (p)+ixp dp . (6)
We define now the space-time cone related to the symbol f and to the frequency band [p 1 , p2 ] with origin (t 0 , x 0 ) ∈ R 2 .

Definition 2.1. Let t 0 , x 0 , p1 and p2 be finite real numbers such that p1 < p2 and let f : R -→ R be a symbol.

1. We define the space-time cone C f [p 1 , p2 ], (t 0 , x 0 ) as follows:

C f [p 1 , p2 ], (t 0 , x 0 ) := (t, x) ∈ R\{t 0 } × R f (p 1 ) x -x 0 t -t 0 f (p 2 ) . 2. Let C f [p 1 , p2 ], (t 0 , x 0 ) c be the complement of the space-time cone C f [p 1 , p2 ], (t 0 , x 0 ) in R\{t 0 } × R .
We present now our first main result. and fix (t 0 , x 0 ) ∈ R 2 . Then

1. for all (t, x) ∈ C f [p 1 , p2 ], (t 0 , x 0 ) , we have u f (t, x) - 1 √ 2π e -sgn(t-t 0 )i π 4 e -itf (p 0 (t,x))+ixp 0 (t,x) Fu 0 p 0 (t, x) f p 0 (t, x) |t -t 0 | -1 2 C 1 (f, δ, p1 , p2 ) (. -x 0 ) u f (t 0 , .) L 2 (R) + C 2 (f, δ, p1 , p2 ) u 0 L 1 (R) |t -t 0 | -δ , (7)
where the real number δ is arbitrarily chosen in 1 2 , 3 4 and

• p 0 (t, x) := (f ) -1 x -x 0 t -t 0 ; • C 1 (f, δ, p1 , p2 ) := 2 δ+ 1 2 L(δ) √ π √ 3 -4δ p2 -p1 3-4δ 2 c 1 (f, δ, p1 , p2 ) ; • C 2 (f, δ, p1 , p2 ) := 2 δ-2 L(δ) π(1 -δ) p2 -p1 2-2δ c 2 (f, δ, p1 , p2 ) ; • c 1 (f, δ, p1 , p2 ) := f 3 2 -δ L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ] f -3 2 ; • c 2 (f, δ, p1 , p2 ) := f 5 2 -δ L ∞ (p 1 ,p 2 ) f (3) L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ] f -7 2 + 1 3 f 7 2 -δ L ∞ (p 1 ,p 2 ) f (3) L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ] f -9 2 .
The constant L(δ) > 0 is defined in (13);

2. for all

(t, x) ∈ C f [p 1 , p2 ], (t 0 , x 0 ) c , we have u f (t, x) C 3 (f, p 1 , p 2 , p1 , p2 ) (. -x 0 ) u f (t 0 , .) L 2 (R) + C 4 (f, p 1 , p 2 , p1 , p2 ) u 0 L 1 (R) |t -t 0 | -1 , (8)
where

• C 3 (f, p 1 , p 2 , p1 , p2 ) := 1 √ 2π (p 2 -p 1 ) 1 2 min f (p 1 ) -f (p 1 ), f (p 2 ) -f (p 2 ) -1 ; • C 4 (f, p 1 , p 2 , p1 , p2 ) := 1 2π min f (p 1 ) -f (p 1 ), f (p 2 ) -f (p 2 ) -1 .
To state the two other main results, we define the following moment-type and variance-type quantities for normalized u ∈ L 2 (R).

Definition 2.3. Let u ∈ L 2 (R) such that u L 2 (R) = 1
and let f : R -→ R be a symbol. If they exist, we define the real numbers M f (u) and V f (u) as follows,

M f (u) := R f (x) u(x) 2 dx , V f (u) := M f 2 (u) -M f (u) 2 .
If f (x) = x, then we note for simplicity

M 1 (u) := M f (u) , M 2 (u) := M f 2 (u) , V(u) := V f (u) .
Remark 2.4. The above quantities M 1 (u), M 2 (u) and V(u) are respectively the mean, the second moment and the variance of |u| 2 .

The explicitness of the family of error bounds given in Theorem 2.2 allows to compute the parameter (t * , x * ) giving the smallest bound; this computation is carried out in Lemma 4.2.1. In the following result, we provide the values of t * and x * and we provide the time-asymptotic expansion for the solution of equation ( 5) associated with this optimal parameter.

Corollary 2.5. Assume that the hypotheses of Theorem 2.2 are satisfied and assume in addition that u 0 L 2 (R) = 1. Then the (t 0 , x 0 )-dependent righthand sides of inequalities ( 7) and (8) in Theorem 2.2 have a global minimum at the point t * , x * ∈ R 2 with

• t * = arg min τ ∈R V u f (τ, .) ; • x * = M 1 u f (t * , .) .
In this case, for all (t, x) ∈ C f [p 1 , p2 ], (t * , x * ) , we have

u f (t, x) - 1 √ 2π
e -sgn(t-t * )i π 4 e -itf (p * (t,x))+ixp * (t,x) Fu 0 p * (t, x) f p * (t, x)

|t -t * | -1 2 C 1 (f, δ, p1 , p2 ) min τ ∈R V u f (τ, .) + C 2 (f, δ, p1 , p2 ) u 0 L 1 (R) |t -t * | -δ ,
where the real number δ is arbitrarily chosen in 1 2 , 3 4 and

p * (t, x) := (f ) -1 x -x * t -t * .
And for all (t, x)

∈ C f [p 1 , p2 ], (t * , x * ) c , we have u f (t, x) C 3 (f, p 1 , p 2 , p1 , p2 ) min τ ∈R V u f (τ, .) + C 4 (f, p 1 , p 2 , p1 , p2 ) u 0 L 1 (R) |t -t * | -1 .
All the constants are defined in Theorem 2.2.

In the last result of this section, we consider mean position and variance of the time-asymptotic approximation of the solution (6) associated with the optimal parameter (t * , x * ). Theorem 2.6 claims that the mean position of this approximation is equal to the one of the true solution and the difference between the two variances is constant; the value for this constant is provided.

Theorem 2.6. Assume that the hypotheses of Theorem 2.2 are satisfied and assume in addition that u 0 L 2 (R) = 1. For all t ∈ R\{t * } and x ∈ R, we define

H f (t, x, u 0 , t * , x * ) := 1 √ 2π e -sgn(t-t * )i π 4 e -itf (p * (t,x))+ixp * (t,x) Fu 0 p * (t, x) f p * (t, x) |t -t * | -1 2 ,
where t * and x * are introduced in Corollary 2.5 and p * (t, x) := (f ) -1 x-x * t-t * . Then for all t ∈ R\{t * }, we have

M 1 u f (t, .) = M 1 H f (t, ., u 0 , t * , x * ) V u f (t, .) -V H f (t, ., u 0 , t * , x * ) = min τ ∈R V u f (τ, .
) .

Explicit error estimates for a stationary phase method via Cauchy-Schwarz inequality

The proof of Theorem 2.2 is substantially based on a careful application of a new version of the stationary phase method for oscillatory integrals of the form

∀ ω > 0 R U (p) e iωψ(p) dp , (9)
where the amplitude U : R -→ C is a continuously differentiable function supported on a bounded interval and the phase ψ : R -→ R is a strictly concave C 3 -function having a unique stationary point p 0 . The remainder estimates we provide here are explicit, uniform with respect to p 0 and involve the L 2 -norm of the first derivative of the amplitude. This plays actually a key role in the proof of Theorem 2.2.

This section is fully devoted to this new version since it is generic and independent from the setting of Section 2. The asymptotic expansions together with the uniform and explicit remainder estimates are established in Theorems 3.3 and 3.4.

We start by stating two technical lemmas which will be substantially used in the proof of Theorem 3.3.

The first step to expand ω-asymptotically integrals of type (9) consists in making simpler the phase function in order to integrate then by parts. To do so, we use the diffeomorphisms ϕ j (j = 1, 2) defined and studied in the following lemma. The values of these diffeomorphisms at the stationary point p 0 are provided in order to compute explicitly the first term of the expansions and two inequalities for ϕ j are established to bound the remainders of these expansions.

The proof of the following result lies mainly on an integral representation of ϕ j .

Lemma 3.1. Let p 0 , p1 and p2 be finite real numbers such that p 0 ∈ (p 1 , p2 ). Suppose that ψ ∈ C 3 (R, R) is a strictly concave function which has a unique stationary point at p 0 . Then, for j = 1, 2, the function

ϕ j : I j -→ [0, s j ] p -→ ψ(p 0 ) -ψ(p) 1 2
where I 1 := [p 1 , p 0 ], I 2 := [p 0 , p2 ] and s j := ϕ j (p j ), satisfies the following properties:

1. the function ϕ j is a C 2 -diffeomorphism between I j and [0, s j ] ;

2. we have ϕ j (p 0 ) = (-1) jψ (p 0 ) 2 ;

3. for all p ∈ I j , the absolute value of ϕ j (p) is lower bounded as follows:

ϕ j (p) 1 √ 2 min [p 1 ,p 2 ] -ψ ψ -1 2 L ∞ (p 1 ,p 2 ) ;
4. we have the following L ∞ -norm estimate for ϕ -1 j :

ϕ -1 j L ∞ (0,s j ) ψ 3 2 L ∞ (p 1 ,p 2 ) ψ (3) L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ]
-ψ

-7 2 + 1 3 ψ 5 2 L ∞ (p 1 ,p 2 ) ψ (3) L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ] -ψ -9 2 .
Proof. Let j ∈ {1, 2} and fix p 0 ∈ (p 1 , p2 ). The proof of the present lemma is mainly based on the following integral representation of the function ϕ j :

ϕ j (p) = (-1) j (p -p 0 ) 1 0 1 0 ψ(p, τ, ν) dνdτ 1 2 , (10)
for all p ∈ I j , where

ψ(p, τ, ν) := -ψ (1 -τ )(1 -ν)p + (ν -ντ + τ )p 0 (1 -τ ) .
This representation can be derived by noting firstly that

ψ(p 0 ) -ψ(p) = - p 0 p ψ (p 0 ) -ψ (t) dt = p 0 p p 0 t -ψ (v) dv dt ,
where we have used the hypothesis ψ (p 0 ) = 0 to obtain the first equality.

Then we make the change of variable (ν, τ ) = v-t p 0 -t , t-p p 0 -p , leading to

ψ(p 0 ) -ψ(p) = (p -p 0 ) 2 1 0 1 0 ψ(p, τ, ν) dνdτ ,
and we take finally the square root of the preceding equality to obtain the desired representation [START_REF] Dewez | Time-asymptotic propagation of approximate solutions of Schrödinger equations with both potential and initial condition in Fourier-frequency bands[END_REF].

1. Since ψ is a strictly concave function on R, the function ϕ j is actually the square root of the non-negative

C 3 -function p -→ ψ(p 0 ) -ψ(p),
showing that ϕ j is twice continuously differentiable on I j \{p 0 } (ϕ j is actually a C 3 -function on this domain). Let us prove that it is also twice differentiable on the whole I j . To do so, note that we have for p ∈ I j \{p 0 },

ϕ j (p) = - 1 2 ψ (p) ψ(p 0 ) -ψ(p) -1 2 = - 1 2 p 0 p -ψ (q) dq ϕ j (p) -1 = 1 2 (p -p 0 ) 1 0 -ψ (1 -t)p + tp 0 dt ϕ j (p) -1 = (-1) j 2 1 0 -ψ (1 -t)p + tp 0 dt × 1 0 1 0 ψ(p, τ, ν) dνdτ -1 2 . ( 11
)
The preceding equality combined with the positivity of the C 1 -function -ψ shows that ϕ j is continuously differentiable on I j whose derivative is given by

ϕ j (p) = (-1) j 2 1 0 -ψ (3) (p -p 0 )t + p 0 (1 -t) dt × 1 0 1 0 ψ(p, τ, ν) dνdτ -1 2 + (-1) j 2 1 0 -ψ (1 -t)p + tp 0 dt × - 1 2 1 0 1 0 ψ(p, τ, ν) dνdτ 1 0 1 0 ψ(p, τ, ν) dνdτ 3 2
, for all p ∈ I j , where

ψ(p, τ, ν) := -ψ (3) (1 -τ )(1 -ν)p + (ν -ντ + τ )p 0 (1 -τ ) 2 (1 -ν) .
Now, according to equality [START_REF] Duyckaerts | Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation[END_REF], we observe that ϕ j is negative for j = 1 and positive for j = 2 since -ψ > 0. By the inverse function theorem, we deduce that ϕ j is a C 2 -diffeomorphism.

2. Thanks to the integral representation [START_REF] Dewez | Time-asymptotic propagation of approximate solutions of Schrödinger equations with both potential and initial condition in Fourier-frequency bands[END_REF], we have

ϕ j (p 0 ) = lim p→p 0 ϕ j (p) -ϕ j (p 0 ) p -p 0 = (-1) j lim p→p 0 1 0 1 0 ψ(p, τ, ν) dνdτ 1 2 = (-1) j - ψ (p 0 ) 2 .
3. From equality [START_REF] Duyckaerts | Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation[END_REF] (which holds actually for all p ∈ I j ), we deduce the following lower estimate for ϕ j :

∀ p ∈ I j ϕ j (p) 1 √ 2 min [p 1 ,p 2 ] -ψ ψ -1 2 L ∞ (p 1 ,p 2 ) . ( 12
)
4. From the expression of ϕ j computed above, we obtain the following upper estimate:

ϕ j (p) 1 2 √ 2 ψ (3) L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ]
-ψ

-1 2 + 1 6 √ 2 ψ L ∞ (p 1 ,p 2 ) ψ (3) L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ] -ψ -3 2 ,
for all p ∈ I j . By combining the preceding inequality with estimate ( 12) and the following relation,

∀ s ∈ [0, s j ] ϕ -1 j (s) = - ϕ j ϕ -1 j (s) ϕ j ϕ -1 j (s) 3 ,
we obtain finally for all s ∈ [0, s j ],

ϕ -1 j (s) ψ 3 2 L ∞ (p 1 ,p 2 ) ψ (3) L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ]
-ψ

-7 2 + 1 3 ψ 5 2 L ∞ (p 1 ,p 2 ) ψ (3) L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ] -ψ -9 2 .
After having applied the above diffeomorphism to the integral (9) (previously split at p 0 ), the phase becomes the quadratic function s -→ -s 2 .

In order to make an integration by parts, creating then the first and the remainder terms of the integral, one needs an expression for a primitive of the function s ∈ [0, s 0 ] -→ e -iωs 2 ∈ C, for fixed s 0 , ω > 0. In the following lemma, a useful integral representation of such a primitive is given. As in the preceding result, its value at the origin and an inequality are also provided to compute respectively the first term of the expansion and an upper bound for the remainder term.

To prove Lemma 3.2, we refer to the paper [START_REF] Ali Mehmeti | Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation[END_REF] which gives actually the successive primitives of more general functions by using essentially complex analysis; see [3, Theorems 6.4, 6.5 and Corollary 6.6].

Lemma 3.2. Let ω, s 0 > 0 be real numbers and let φ(., ω) : [0, s 0 ] -→ C be the function defined by φ(s, ω) := -

Λ(s)
e -iωz 2 dz , where Λ(s) is the half-line in the complex plane given by Λ(s) := s + t e -i π 4 ∈ C t 0 .

Then

1. the function φ(., ω) is a primitive of s ∈ [0, s 0 ] -→ e -iωs 2 ∈ C ; 2. we have φ(0, ω) = - 1 2
√ π e -i π 4 ω -1 2 ;

3. the function φ(., ω) satisfies

∀ s ∈ (0, s 0 ] φ(s, ω) L(δ) s 1-2δ ω -δ ,
where the real number δ is arbitrarily chosen in 1 2 , 1 and the constant

L(δ) > 0 is defined by L(δ) := √ π 2 1 2 √ π + 1 4π + 1 2 2δ-1 . ( 13
)
Proof. The function φ(., ω) of the present paper corresponds actually to the function φ

(2) 1 (., ω, 2, 1) defined in [START_REF] Ali Mehmeti | Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation[END_REF]Theorem 2.3]. Hence we apply the results

established in [START_REF] Ali Mehmeti | Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation[END_REF] to the present situation:

1. One proves this first point by applying [3, Corollary 6.6], which is a consequence of Theorems 6.4 and 6.5 of [START_REF] Ali Mehmeti | Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation[END_REF], in the case n = 1, j = 2, ρ j = 2 and µ j = 1.

2. The proof of this point lies only on basic computations which are carried out in the fourth step of the proof of [3, Theorem 2.3].

3. The combination of Lemmas 2.4 and 2.6 of [START_REF] Ali Mehmeti | Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation[END_REF] assures this last point.

Thanks to the two preceding lemmas, we are now in position to establish the desired asymptotic expansions with respect to the parameter ω for oscillatory integrals of type [START_REF] Dewez | Estimates of oscillatory integrals with stationary phase and singular amplitude: Applications to propagation features for dispersive equations[END_REF]. In the following theorem, we are interested in the case where the stationary point p 0 of the phase belongs to a neighbourhood of the support of the amplitude. We emphasise that the remainder estimate we provide is different from those appearing in the original paper [START_REF] Erdélyi | Asymptotics expansions[END_REF] and in [START_REF] Ali Mehmeti | Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation[END_REF].

Technically speaking, we split the integral at the stationary point p 0 and we study separately the two resulting integrals. In each situation, the method consists firstly in using the diffeomorphism introduced in

U ∈ C 1 (R, C) is a function satisfying supp U ⊆ [p 1 , p 2 ] .
Then we have for all ω > 0,

R U (p) e iωψ(p) dp - √ 2π e -i π 4 e iωψ(p 0 ) U (p 0 ) -ψ (p 0 ) ω -1 2 C 5 (ψ, δ, p1 , p2 ) U L 2 (R) + C 6 (ψ, δ, p1 , p2 ) U L ∞ (R) ω -δ ,
where the real number δ is arbitrarily chosen in 1 2 , 3 4 and

• C 5 (ψ, δ, p1 , p2 ) := 2 δ+1 L(δ) √ 3 -4δ p2 -p1 3-4δ 2 c 5 (ψ, δ, p1 , p2 ) ; • C 6 (ψ, δ, p1 , p2 ) := 2 δ-1 L(δ) 1 -δ p2 -p1 2-2δ c 6 (ψ, δ, p1 , p2 ) ; • c 5 (ψ, δ, p1 , p2 ) := ψ 3 2 -δ L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ] -ψ -3 2 ; • c 6 (ψ, δ, p1 , p2 ) := ψ 5 2 -δ L ∞ (p 1 ,p 2 ) ψ (3) L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ]
-ψ

-7 2 + 1 3 ψ 7 2 -δ L ∞ (p 1 ,p 2 ) ψ (3) L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ] -ψ -9 2 .
The constant L(δ) > 0 is defined in [START_REF] Duyckaerts | Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case[END_REF]. Splitting the above integral at the point p 0 and using the two C 2 -diffeomorphisms defined in Lemma 3.1, we obtain

I(ω) = - s 1 0 U • ϕ -1 1 (p) ϕ -1 1 (p) e -iωs 2 ds e iωψ(p 0 ) + s 2 0 U • ϕ -1 2 (p) ϕ -1 2
(p) e -iωs 2 ds e iωψ(p 0 ) ; note that we have used the fact that ϕ 1 and ϕ 2 are respectively decreasing and increasing. We integrate now by parts by using the primitive s -→ φ(s, ω) given in Lemma 3.2 and the regularity of ϕ j :

(-1) j s j 0 U • ϕ -1 j (s) ϕ -1 j (s) e -iωs 2 ds = (-1) j U • ϕ -1 j (s) ϕ -1 j (s) φ(s, ω) s j 0 + (-1) j+1 s j 0 U • ϕ -1 j ϕ -1 j (s) φ(s, ω) ds = (-1) j+1 U • ϕ -1 j (0) ϕ -1 j (0) φ(0, ω) + (-1) j+1 s j 0 U • ϕ -1 j ϕ -1 j (s) φ(s, ω) ds = 1 2 √ 2π e -i π 4 U (p 0 ) -ψ (p 0 ) ω -1 2 + (-1) j+1 s j 0 U • ϕ -1 j ϕ -1 j (s) φ(s, ω) ds ;
the second equality has been obtained by using the fact that U (p j ) = 0 and the last one by applying Lemma 3.1 point 2 and Lemma 3.2 point 2. Hence it follows

I(ω) = √ 2π e -i π 4 e iωψ(p 0 ) U (p 0 ) -ψ (p 0 ) ω -1 2 + 2 j=1 (-1) j+1 s j 0 U • ϕ -1 j ϕ -1 j (s) φ(s, ω) ds e iωψ(p 0 ) .
To estimate each term of the remainder, we proceed as follows:

(-1) j+1 s j 0 U • ϕ -1 j ϕ -1 j (s) φ(s, ω) ds s j 0 U • ϕ -1 j (s) ϕ -1 j (s) 2 φ(s, ω) ds + s j 0 U • ϕ -1 j (s) ϕ -1 j (s) φ(s, ω) ds s j 0 U • ϕ -1 j (s) ϕ -1 j (s) 2 2 ds 1 2 s j 0 φ(s, ω) 2 ds 1 2 + s j 0 φ(s, ω) ds U L ∞ (R) ϕ -1 j L ∞ (0,s j )
; let us remark that we have applied Cauchy-Schwarz inequality to the first integral. We continue the proof by estimating each resulting term; first of all, by making the change of variable p = ϕ -1 j (s) and by using Lemma 3.1 point 3, we obtain

s j 0 U • ϕ -1 j (s) ϕ -1 j (s) 2 2 ds 1 2 2 3 4 ψ 3 4 L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ] -ψ -3 2 U L 2 (R) .
Then we use the point 3 of Lemma 3.2 to derive the two following inequalities:

• s j 0 φ(s, ω) ds L(δ) s j 0 s 1-2δ ds ω -δ L(δ) 2 -2δ ϕ j (p j ) 2-2δ ω -δ ; • s j 0 φ(s, ω) 2 ds 1 2 L(δ) √ 3 -4δ ϕ j (p j ) 3-4δ 2 ω -δ .
By using the integral representation (10) of ϕ j , we obtain

ϕ j (p j ) 1 √ 2 ψ 1 2 L ∞ (p 1 ,p 2 ) ( p2 -p1 ) ,
which permits to deduce

• s j 0 φ(s, ω) ds 1 2 1-δ L(δ) 2 -2δ p2 -p1 2-2δ ψ 1-δ L ∞ (p 1 ,p 2 ) ω -δ ; • s j 0 φ(s, ω) 2 ds 1 2 1 2 3 4 -δ L(δ) √ 3 -4δ p2 -p1 3-4δ 2 ψ 3 4 -δ L ∞ (p 1 ,p 2 ) ω -δ .
And, from Lemma 3.1 point 4, we recall that

ϕ -1 j L ∞ (0,s j ) ψ 3 2 L ∞ (p 1 ,p 2 ) ψ (3) L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ]
-ψ

-7 2 + 1 3 ψ 5 2 L ∞ (p 1 ,p 2 ) ψ (3) L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ] -ψ -9 2 .
Putting everything together provides the desired estimate, namely,

I(ω) - √ 2π e -i π 4 e iωψ(p 0 ) U (p 0 ) -ψ (p 0 ) ω -1 2 2 j=1 (-1) j+1 s j 0 U • ϕ -1 j ϕ -1 j (s) φ(s, ω) ds e iωψ(p 0 ) 2 δ+1 L(δ) √ 3 -4δ p2 -p1 3-4δ 2 ψ 3 2 -δ L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ] -ψ -3 2 U L 2 (R) ω -δ + 2 δ-1 L(δ) 1 -δ p2 -p1 2-2δ U L ∞ (R) × ψ 5 2 -δ L ∞ (p 1 ,p 2 ) ψ (3) L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ]
-ψ

-7 2 + 1 3 ψ 7 2 -δ L ∞ (p 1 ,p 2 ) ψ (3) L ∞ (p 1 ,p 2 ) min [p 1 ,p 2 ] -ψ -9 2 ω -δ .
We end this section by providing an explicit and uniform bound for oscillatory integrals of type [START_REF] Dewez | Estimates of oscillatory integrals with stationary phase and singular amplitude: Applications to propagation features for dispersive equations[END_REF] in the case where there is no stationary point inside the support of the amplitude, making the decay with respect to ω faster. As above, the estimate involves the L 2 -norm of the first derivative of the amplitude.

The proof of the following result lies on classical arguments (as those in [28, Chap. VIII, Sec. Then we have for all ω > 0,

R U (p) e iωψ(p) dp C 7 (ψ, p 1 , p 2 ) U L 2 (R) + C 8 (ψ, p 1 , p 2 ) U L ∞ (R) ω -1 ,
where Moreover we have -i

• C 7 (ψ, p 1 , p 2 ) := (p 2 -p 1 )
p 2 p 1 U ψ (p) e iωψ(p) dp p 2 p 1 U (p) ψ (p) -1 e iωψ(p) dp + p 2 p 1 U (p) ψ (p) ψ (p) -2 dp U L 2 (R) (ψ ) -1 L 2 (p 1 ,p 2 ) + U L ∞ (R) p 2 p 1 ψ (p) ψ (p) -2 dp ;
as in the preceding proof, we have applied Cauchy-Schwarz inequality to the first integral. Now the hypotheses ψ 0 and ψ is monotonic with a constant sign allow to carry the following computations out:

p 2 p 1 ψ (p) ψ (p) -2 dp = - p 2 p 1 ψ (p) ψ (p) -2 dp = ψ (p 2 ) -1 -ψ (p 1 ) -1 m p 1 ,p 2 (ψ ) -1 .
Furthermore, we have

(ψ ) -1 L 2 (p 1 ,p 2 ) m p 1 ,p 2 (ψ ) -1 (p 2 -p 1 ) 1 2
.

Consequently we obtain

R U (p) e iωψ(p) dp m p 1 ,p 2 (ψ ) -1 (p 2 -p 1 ) 1 2 U L 2 (R) + m p 1 ,p 2 (ψ ) -1 U L ∞ (R) ω -1 .

Proofs of the main results

This section is devoted to the proofs of the results stated in Section 2.

Proof of Theorem 2.2

The proof of Theorem 2.2 is inspired by the one of [3, Theorem 5.2]: it consists mainly in rewriting wisely the solution formula (6) as an oscillatory integral with respect to time and in applying then a version of the stationary phase method.

In the present paper, the expansions in cones with arbitrary origin are obtained thanks to a space-time shift in the integral defining [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]. And the explicitness of the error bounds with respect to the origin of the cones is achieved thanks to the new generic remainder estimates given in Theorems 3.3 and 3.4; this allows the application of Plancherel theorem in the present setting.

Proof of Theorem 2.2. The cases t > t 0 and t < t 0 are distinguished for the sake of readability.

Case 1: t > t 0 We rewrite the solution formula as an oscillatory integral by proceeding as follows1

u f (t, x) = 1 2π R Fu 0 (p) e -itf (p)+ixp dp = R 1 2π Fu 0 (p) e -it 0 f (p)+ix 0 p e i(t-t 0 ) x-x 0 t-t 0 p -f (p) dp = R U f (p, t 0 , x 0 ) e i(t-t 0 )Ψ f (p,t,x,t 0 ,x 0 ) dp =: I f (t, x, u 0 , t 0 , x 0 ) .
We note that the amplitude

U f (p, t 0 , x 0 ) := 1 2π Fu 0 (p) e -it 0 f (p)+ix 0 p ,
which is actually the Fourier transform of 1 2π u f (t 0 , . + x 0 ), is a C ∞ -function (with respect to the variable p) whose support is included in [p 1 , p 2 ]. The phase function Ψ f (p, t, x, t 0 , x 0 ) :=

x -

x 0 t -t 0 p -f (p)
is a C ∞ -function on R which is strictly concave since we have supposed f > 0 in this section. Now we remark that the existence of a stationary point for the phase inside the interval Ĩ := (p 1 , p2 ) depends on the value of x-x 0 t-t 0 : it exists and is unique if and only if x-x 0 t-t 0 ∈ f Ĩ . In this case, the stationary point p 0 (t, x) is given by

p 0 (t, x) = (f ) -1 x -x 0 t -t 0 .
Let us now distinguish two sub-cases to apply Theorem 3.3 and Theorem 3.4.

405

• Case x-x 0 t-t 0 ∈ f Ĩ . In this case, the stationary point belongs to Ĩ. Hence we are allowed to apply Theorem 3.3 to the oscillatory integral I f (t, x, u 0 , t 0 , x 0 ) with ω = t -t 0 :

I f (t, x, u 0 , t 0 , x 0 ) - 1 √ 2π e -i π 4 e -itf (p 0 (t,x))+ixp 0 (t,x) Fu 0 p 0 (t, x) f p 0 (t, x) (t -t 0 ) -1 2 1 2π C 5 (Ψ f , δ, p1 , p2 ) ∂ p Fu 0 (•) e -it 0 f (•)+ix 0 • L 2 (R) + C 6 (Ψ f , δ, p1 , p2 ) Fu 0 L ∞ (R) (t -t 0 ) -δ , with δ ∈ 1 2 , 3 4 and the constants C 5 (Ψ f , δ, p1 , p2 ), C 6 (Ψ f , δ, p1 , p2 ) > 0 are defined in Theorem 3.3. Since we have ∂ 2 p Ψ f (p, t, x, t 0 , x 0 ) = -f (p) , ∂ 3 p Ψ f (p, t, x, t 0 , x 0 ) = -f (3) (p) ,
and since the constants C 5 (Ψ f , δ, p1 , p2 ) and C 6 (Ψ f , δ, p1 , p2 ) depend only on the second and third derivatives (with respect to p) of the phase, we can claim that these constants depend on f rather than Ψ f . Furthermore, Plancherel theorem and standard properties of the Fourier transform provide

∂ p Fu 0 (.)e -itf (•)+ix 0 • L 2 (R) = √ 2π x -→ x F -1 Fu 0 (.)e -it 0 f (•)+ix 0 • (x) L 2 (R) = √ 2π x -→ x F -1 Fu 0 (.) e -it 0 f (•) (x + x 0 ) L 2 (R) = √ 2π x -→ (x -x 0 ) F -1 Fu 0 (.) e -it 0 f (•) (x) L 2 (R) = √ 2π x -→ (x -x 0 ) u f (t 0 , x) L 2 (R)
.

Hence we obtain finally

u f (t, x) - 1 √ 2π e -i π 4 e -itf (p 0 (t,x))+ixp 0 (t,x) Fu 0 p 0 (t, x) f p 0 (t, x) (t -t 0 ) -1 2 1 √ 2π C 5 (-f, δ, p1 , p2 ) (. -x 0 ) u f (t 0 , .) L 2 (R) + 1 2π C 6 (-f, δ, p1 , p2 ) u 0 L 1 (R) (t -t 0 ) -δ ,
where we have used the classical estimate Fu 0 L ∞ (R) u 0 L 1 (R) .

• Case x-x 0 t-t 0 / ∈ f Ĩ . As previously, we rewrite the solution formula as the oscillatory integral I f (t, x, u 0 , t 0 , x 0 ). Here the phase Ψ f (., t, x, t 0 , x 0 ) has no stationary point inside the interval Ĩ = (p 1 , p2 ) and one has

∀ p ∈ [p 1 , p 2 ] ∂ p Ψ f (p, t, x, t 0 , x 0 ) = x -x 0 t -t 0 -f (p) m Ĩ (f ) > 0 ,
where m Ĩ (f ) := min f (p 1 ) -f (p 1 ), f (p 2 ) -f (p 2 ) . Consequently we can apply Theorem 3.4 which provides

u f (t, x) 1 √ 2π C 7 (-f, p1 , p2 ) (. -x 0 ) u f (t 0 , .) L 2 (R) + 1 2π C 8 (-f, p1 , p2 ) u 0 L 1 (R) (t -t 0 ) -1 ,
where the constants C 7 (-f, p1 , p2 ), C 8 (-f, p1 , p2 ) > 0 are defined in Theorem 3.4.

Case 2: t < t 0 Here we have

u f (t, x) = R 1 2π Fu 0 (p) e -it 0 f (p)+ix 0 p e i(t-t 0 ) x-x 0 t-t 0 p -f (p) dp = R 1 2π Fu 0 (p) e -it 0 f (p)+ix 0 p e i(t 0 -t) x-x 0 t-t 0 p -f (p) dp = R U f (p, t 0 , x 0 ) e i(t 0 -t)Ψ f (p,t,x,t 0 ,x 0 ) dp .
Following the arguments and computations of the preceding case t > t 0 , we obtain

u f (t, x) - 1 √ 2π
e +i π 4 e -itf (p 0 (t,x))+ixp 0 (t,x) Fu 0 p 0 (t, x) f p 0 (t, x)

(t 0 -t) -1 2 1 √ 2π C 5 (-f, δ, p1 , p2 ) (. -x 0 ) u f (t 0 , .) L 2 (R) + 1 2π C 6 (-f, δ, p1 , p2 ) u 0 L 1 (R) (t 0 -t) -δ , for all (t, x) ∈ (-∞, t 0 ) × R such that x-x 0 t-t 0 ∈ f Ĩ , and
u f (t, x) 1 √ 2π C 7 (-f, p1 , p2 ) (. -x 0 ) u f (t 0 , .) L 2 (R) + 1 2π C 8 (-f, p1 , p2 ) u 0 L 1 (R) (t 0 -t) -1 ,
for all (t, x) ∈ (-∞, t 0 ) × R such that x-x 0 t-t 0 / ∈ f Ĩ , leading to the desired estimates.

Proof of Corollary 2.5

This subsection is devoted to the proof of Corollary 2.5. It is based on a combination between Theorem 2.2 and the following Lemma 4.2.1. This lemma gives the minimum of the function (t 0 , x 0 ) -→ (. -x 0 )u f (t 0 , .)

2 L 2 (R)
which is involved in the error estimates in Theorem 2.2. This function is actually the moment of order 2 of the function x -→ u f (t 0 , x + x 0 ) 2 for fixed (t 0 , x 0 ) ∈ R 2 .

Lemma 4.2.1. Assume that the hypotheses of Theorem 2.2 are satisfied and assume in addition that u 0 L 2 (R) = 1. Then the function g : R 2 -→ R + defined by

g(t 0 , x 0 ) = (. -x 0 ) u f (t 0 , .) 2 L 2 (R) has a global minimum at (t * , x * ) ∈ R 2 with • t * = arg min τ ∈R V u f (τ, .) ; • x * = M 1 u f (t * , .) .
Proof. For fixed t 0 ∈ R, differentiating twice the function g(t 0 , .) with respect to its second argument shows that

∂ 2 x 0 g(t 0 , x 0 ) = 2 > 0 .
Hence g(t 0 , .) is a polynomial function of degree 2 whose unique global minimum is

x(t 0 ) = R x u f (t 0 , x) 2 dx = M 1 u f (t 0 , .) .
It follows that

g t 0 , x(t 0 ) = R x -M 1 u f (t 0 , .) 2 u f (t 0 , x) 2 dx = V u f (t 0 , .) . Lemma Appendix A.3 assures that t 0 ∈ R -→ V u f (t 0 , .) ∈ R + is a poly- nomial of degree 2 whose leading coefficient is V f 1 √
2π Fu 0 . Since we have by simple calculations,

V f 1 √ 2π Fu 0 = M f 2 1 √ 2π Fu 0 -M f 1 √ 2π Fu 0 2 = 1 2π R f (p) -M f 1 √ 2π Fu 0 2 Fu 0 (p) 2 dp ,
and since f is supposed to be strictly convex in this paper, the leading coefficient V f 1 √ 2π Fu 0 is necessarily positive. Thus t 0 ∈ R -→ g t 0 , x(t 0 ) ∈ R + has a global minimum at a certain t * ∈ R, i.e., t * = arg min τ ∈R V u f (τ, .) .

Finally we define

x * := x(t * ) = M 1 u f (t * , .) .

418

Remark 4.2.2. The polynomial nature of t 0 ∈ R -→ g t 0 , x(t 0 ) ∈ R + permits to derive the following formula for t * :

t * = 1 V f 1 √ 2π Fu 0 - 1 2π R f (p) Fu 0 (p) Fu 0 (p) dp + M f 1 √ 2π Fu 0 M 1 (u 0 ) . ( 14
)
Proposition 4.3.1. Let (t 0 , x 0 ) ∈ R 2 . Assume that the hypotheses of Theorem 2.2 are satisfied and assume in addition that u 0 L 2 (R) = 1. Then for all t ∈ R\{t 0 }, we have M 1 H f (t, ., u 0 , t 0 , x 0 ) = x 0 + M f 1 √ 2π Fu 0 (t -t 0 ) .

Proof. Let t ∈ R\{t 0 }. First of all, we note that

x 0 + f (p 1 )(t -t 0 ) < x 0 + f (p 2 )(t -t 0 ) ⇐⇒ t > t 0 .

Hence using the definition of H f (t, x, u 0 , t 0 , x 0 ) given in Theorem 2.6, we have x H f (t, x, u 0 , t 0 , x 0 ) 2 dx = sgn(t -t 0 ) 2π

x 0 +f (p 2 )(t-t 0 )

x 0 +f (p 1 )(t-t 0 )

x Fu 0 p 0 (t, x) x 0 +f (p 2 )(t-t 0 )

x 0 +f (p 1 )(t-t 0 )

x Fu 0 p 0 (t, x) 2 f p 0 (t, x) dx (t -t 0 ) -1 .

We make now the change of variable x = x 0 + f (p)(t -t 0 ) to obtain the desired result: x 0 + f (p)(t -t 0 ) Fu 0 (p)

2 dp = x 0 + M f 1 √ 2π Fu 0 t -M f 1 √ 2π Fu 0 t 0 ;
note that we have used the fact that 1 2π Fu 0 2 L 2 (p 1 ,p 2 ) = 1, which is a direct consequence of the assumption u 0 L 2 (R) = 1.

In the following result, we prove that the mean positions of the solution of equation ( 5) and of the term H f (t, ., u 0 , t 0 , x 0 ) are equal if and only if x 0 is equal to the mean position of u f (t 0 , .).

Proposition 4.3.2. Let (t 0 , x 0 ) ∈ R 2 . Assume that the hypotheses of Theorem 2.2 are satisfied and assume in addition that u 0 L 2 (R) = 1. Then for all t ∈ R\{t 0 }, we have M 1 u f (t, .) = M 1 H f (t, ., u 0 , t 0 , x 0 ) ⇐⇒ x 0 = M 1 u f (t 0 , .) . x 0 +f (p 2 )(t-t 0 )

x 0 +f (p 1 )(t-t 0 )

x 2 Fu 0 p 0 (t, x)

It follows:

V H f (t, ., u 0 , t 0 , x 0

) = M f 2 1 √ 2π Fu 0 -M f 1 √ 2π Fu 0 2 (t -t 0 ) 2 = V f 1 √ 2π Fu 0 (t -t 0 ) 2 .
Using the formula for V u f (t, .) from Proposition Appendix A.3, we obtain finally V u f (t, .) -V H f (t, ., u 0 , t 0 , x 0 )

= V f 1 √ 2π Fu 0 t 2 + 2 1 2π R f (p) Fu 0 (p) Fu 0 (p) dp -M f 1 √ 2π Fu 0 M 1 (u 0 ) t + V(u 0 ) -V f 1 √ 2π Fu 0 (t -t 0 ) 2 = 2 1 2π R f (p) Fu 0 (p) Fu 0 (p) dp -M f 1 √ 2π Fu 0 M 1 (u 0 ) + V f 1 √ 2π Fu 0 t 0 t + V(u 0 ) -V f 1 √ 2π Fu 0 t 2 0 .
446 Proposition Appendix A.1. Suppose that u 0 ∈ S(R). Then for all t ∈ R, we have

M 1 u f (t, .) = M f 1 √ 2π Fu 0 t + M 1 (u 0 ) .
Proof. For t ∈ R, we have

R x u f (t, x) 2 dx = R x u f (t, x) u f (t, x) dx = 1 2π R F x → x u f (t, x) (p) F x → u f (t, x) (p) dp = i 2π R ∂ p F x → u f (t, x) (p) F x → u f (t, x) (p) dp ; (A.1)
the second and third equalities have been obtained by applying Plancherel theorem and basic properties of the Fourier transform. Using now the formula (1), we obtain for all p ∈ R,

• ∂ p F x → u f (t, x) (p) = e -itf (p) -itf (p) Fu 0 (p) + (Fu 0 ) (p) ;

• F x → u f (t, x) (p) = e itf (p) Fu 0 (p) .

By combing the two last equalities with (A.1) and by using again basic properties of the Fourier transform, it follows

lutions have been achieved by applying a generalisation [ 9 ,

 9 Theorem 4.8] of the classical van der Corput Lemma [28, Chap. VIII, Sec. 1, Proposition 2]

  the time-asymptotic behaviour of the two first terms of the Dyson-Phillips series [16, Chap. III, Theorem 1.10] representing the perturbed solution is studied by means of asymptotic expansions. The results concerning the second term of the series are interpreted as follows: if the initial state travels from left to right in space, then the positive frequencies of the potential tend to accelerate the motion of the second term while the negative frequencies tend to slow down or even reverse it, exhibiting advanced and retarded transmissions as well as reflections. The application of the present results could bring more information on these phenomena, in particular precise spatial information on the transmitted and reflected wave packets.

Theorem 3 . 3 .

 33 Lemma 3.1 to make the phase function simpler, secondly in integrating by parts to create the expansion by using Lemma 3.1 point 2, Lemma 3.2 point 1 and point 2, and finally in bounding the remainder term by combining Lemma 3.1 point 3, point 4 and Lemma 3.2 point 3 with Cauchy-Schwarz inequality. Let p 1 , p 2 , p1 and p2 be finite real numbers such that [p 1 , p 2 ] ⊂ (p 1 , p2 ). Suppose that ψ ∈ C 3 (R, R) : R -→ R is a strictly concave function which has a unique stationary point at p 0 ∈ (p 1 , p2 ). And assume that

Proof.U

  Let ω > 0 and choose p 0 ∈ (p 1 , p2 ). First of all, since the support of the amplitude is included in [p 1 , p 2 ] ⊂ (p 1 , p2 ), we have clearly R U (p) e iωψ(p) dp = p2 p1 (p) e iωψ(p) dp =: I(ω) .

1 2 1 ;• C 8 1 . 2 p 1 U

 118121 min ψ (p 1 ) , ψ (p 2 ) -(ψ, p 1 , p 2 ) := min ψ (p 1 ) , ψ (p 2 ) -Proof. Let ω > 0. Since ψ is monotonic and has a constant sign on [p 1 , p 2 ], we have ∀ p ∈ [p 1 , p 2 ] ψ (p) min ψ (p 1 ) , ψ (p 2 ) =: m p 1 ,p 2 (ψ ) > 0 . Hence we are allowed to integrate by parts as follows: R U (p) e iωψ(p) dp = p (p) e iωψ(p) dp = -i e iωψ(p) dp ω -1 .

R

2 f p 0

 20 (t, x) dx |t -t 0 | -1 = 1 2π

Proposition 4 .

 4 3.1, we have

M 2 H

 2 f (t, ., u 0 , t 0 , x 0 ) = 1 2π

R

  x u f (t, x)2 dx = i 2π R -itf (p) Fu 0 (p) + (Fu 0 ) (p) Fu 0 (p) dp = 1 2π R f (p) Fu 0 (p) 2 dp t + i 2π R (Fu 0 ) (p) Fu 0 (p) dp = 1 2π R f (p) Fu 0 (p) 2 dp t + 1 2π R F x → x u 0 (x) (p) Fu 0 (p) dp = 1 2π R f (p) Fu 0 (p) 2 dp t + R x u 0 (x) 2 dx ,leading finally to the desired equality.

  465

  Let p 1 , p 2 , p1 and p2 be finite real numbers such that[p 1 , p 2 ] ⊂ (p 1 , p2 ). Suppose that ψ ∈ C 2 (R, R) : R -→ R is a concave function such that |ψ | > 0 on [p 1 , p 2 ]. And assume that U ∈ C 1 (R, C) is a function satisfying supp U ⊆ [p 1 , p 2 ] .

	Theorem 3.4.
	1, Proposition 2]) combined with Cauchy-Schwarz inequali-
	ty.

  (t -t 0 ) 2 + 2 x 0 M f 1 √ 2π Fu 0 (t -t 0 ) .Moreover, applying Proposition 4.3.1 givesM 1 H f (t, ., u 0 , t 0 , x 0 )

								2
								f p 0 (t, x)	dx (t -t 0 ) -1
	=	1 2π	p 1 p 2	x 0 + f (p)(t -t 0 )	2 Fu 0 (p)	2 dp
	= x 2 0 + M f 2	1 √ 2π Fu 0 2
	= x 2 0 + M f		1 √ 2π	Fu 0	2	(t -t 0 ) 2 + 2 x 0 M f	1 √ 2π	Fu

0 (t -t 0 ) .

In[START_REF] Ali Mehmeti | Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation[END_REF][START_REF] Dewez | Estimates of oscillatory integrals with stationary phase and singular amplitude: Applications to propagation features for dispersive equations[END_REF], the parameters t 0 and x 0 are implicitly equal to 0. Allowing these parameters to be arbitrary produces a space-time shift in the solution formula and permits to consider cones with arbitrary origin.

Furthermore, from Lemma Appendix A.1, we have

inserting formula [START_REF] Duyckaerts | Classification of radial solutions of the focusing, energy-critical wave equation[END_REF] into the preceding equality provides

We are now in position to prove Corollary 2.5.

Proof of Corollary 2.5. We apply Theorem 2.2 with t 0 = t * and x 0 = x * , where t * and x * are defined in Lemma 4.2.1, to the solution formula [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]. This provides the time-asymptotic expansions appearing in [START_REF] Cazenave | A note on decay rates for Schrödinger's equation[END_REF] and [START_REF] Côte | Energy partition for the linear radial wave equation[END_REF] together with the error bounds in which the following term appears:

where g : R 2 -→ R is defined in Lemma 4.2.1. By the definitions of t * and x * , we obtain

Further it is clear that minimising the two families of (t 0 , x 0 )-dependent error bounds given in inequalities [START_REF] Cazenave | A note on decay rates for Schrödinger's equation[END_REF] and ( 8) is equivalent to minimising the function g. By Lemma 4.2.1, the parameter (t * , x * ) minimises these two families, which ends the proof.

Proof of Theorem 2.6

We prove Theorem 2.6 in this last subsection. This will be done in two main steps: for arbitrary (t 0 , x 0 ) ∈ R 2 , we first compute the mean position of the term H f (t, ., u 0 , t 0 , x 0 ) defined in the statement of Theorem 2.6 and compare it to the mean position of the solution (6); a similar study for the variances is carried out. Theorem 2.6 is finally a direct consequence of these two studies.

In the first proposition, we compute the mean position of H f (t, ., u 0 , t 0 , x 0 ) for all t = t 0 .

Proof. Let t ∈ R\{t 0 }. According to Propositions 4.3.1 and Appendix A.1, we have

Hence these two mean positions are equal if and only if

The definitions of t * and x * from Corollary 2.5 and the preceding proposition assure that the mean positions of u f (t, .) and the term

In the two following results, we focus on the variances of the solution u f (t, .) and of the term H f (t, ., u 0 , t 0 , x 0 ) for all t = t 0 . We give firstly a formula for the difference between the two variances for arbitrary (t 0 , x 0 ).

Assume that the hypotheses of Theorem 2.2 are satisfied and assume in addition that u 0 L 2 (R) = 1. Then for all t ∈ R\{t 0 }, we have

Proof. Let t ∈ R\{t 0 }. Similarly to the arguments employed in the proof of In view of the preceding result, the difference between the variances of u f (t, .) and H f (t, ., u 0 , t 0 , x 0 ) is an affine function with respect to t. Consequently the unique way to make this difference constant is to choose t 0 in such way that the leading coefficient is equal to 0. It turns out that the unique t 0 satisfying this property is the one minimising the variance of the solution (6), namely t * introduced in Corollary 2.5.

Assume that the hypotheses of Theorem 2.2 are satisfied and assume in addition that u 0 L 2 (R) = 1. Then we have the following equivalence:

In particular, we have

Proof. According to Proposition 4.3.4, the difference between the variances of u f (t, .) and H f (t, ., u 0 , t 0 , x 0 ) is constant if and only if

which is equal to t * according to Remark 4.2.2. By evaluating equality [START_REF] Duyckaerts | Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations[END_REF] at t * , we obtain for all t ∈ R\{t * },

We note now that

From equalities ( 16) and ( 17), it follows finally

and Proposition 4.3.5 assures that

which ends the proof.

Appendix A. Mean position and variance of the free wave packet

In this appendix, we give the formulas for the mean position and the variance of the wave packet defined in [START_REF] Ali Mehmeti | Dispersive effects and high frequency behaviour for the Schrödinger equation in star-shaped networks[END_REF]. The proofs we propose here are substantially based on the fact that the wave packet is defined via the Fourier transform, permitting to apply some properties of this transform.

We begin with the formula for the mean position.

Remark Appendix A.2. The preceding formula is actually an extension of the well-known Ehrenfest theorem [START_REF] Hall | Quantum Theory for Mathematicians[END_REF]Proposition 3.19] to the family of dispersive equations of type [START_REF] Ben Artzi | Uniform Estimates for a Class of Dispersive Equations[END_REF]. We recall that Ehrenfest theorem in the setting of the free Schrödinger equation (3) gives the following formula for the mean position of a free particle:

The formula for the variance is provided in the following result.

466 Proposition Appendix A.3. Suppose that u 0 ∈ S(R). Then for all t ∈ R, we have

Proof. Following the computational arguments of the proof of Proposition Appendix A.1, we have for all t ∈ R, R

x 2 u f (t, x)

Inserting the following relation

into (A.2) and expanding then the square of the absolute value provides

Now by using Proposition Appendix A.1, we have

which leads finally to V u f (t, .)

= M 2 u f (t, .) -M 1 u f (t, .)

467

Appendix B. Acknowledgements

The author gratefully thanks Prof. Felix Ali Mehmeti whose numerous and insightful comments helped to improve the present paper. The author is also grateful to the referee for valuable remarks which helped to improve significantly the presentation of the paper.