On semi-classical limit of nonlinear quantum scattering
We consider the nonlinear Schrödinger equation with a short-range external potential, in a semi-classical scaling. We show that for fixed Planck constant, a complete scattering theory is available, showing that both the potential and the nonlinearity are asymptotically negligible for large time. Then, for data under the form of coherent state, we show that a scattering theory is also available for the approximate envelope of the propagated coherent state, which is given by a nonlinear equation. In the semi-classical limit, these two scattering operators can be compared in terms of classical scattering theory, thanks to a uniform in time error estimate. Finally, we infer a large time decoupling phenomenon in the case of finitely many initial coherent states.
We consider the equation
and both semi-classical () and large time () limits. Of course these limits must not be expected to commute, and one of the goals of this paper is to analyze this lack of commutation on specific asymptotic data, under the form of coherent states as described below. Even though our main result (Theorem 1.6) is proven specifically for the above case of a cubic three-dimensional equation, two important intermediate results (Theorems 1.4 and 1.5) are established in a more general setting. Unless specified otherwise, we shall from now on consider , .
1.1. Propagation of initial coherent states
In this subsection, we consider the initial value problem, as opposed to the scattering problem treated throughout this paper. More precisely, we assume here that the wave function is, at time , given by the coherent state
where denote the initial position and velocity, respectively. The function belongs to the Schwartz class, typically. In the case where is a (complex) Gaussian, many explicit computations are available in the linear case (see ). Note that the -norm of is independent of , .
Throughout this subsection, we assume that the external potential is smooth and real-valued, , and at most quadratic, in the sense that
This assumption will be strengthened when large time behavior is analyzed.
1.1.1. Linear case
Resume (1.1) in the absence of nonlinear term:
associated with the initial datum (1.2). To derive an approximate solution, and to describe the propagation of the initial wave packet, introduce the Hamiltonian flow
and prescribe the initial data , . Since the potential is smooth and at most quadratic, the solution is smooth, defined for all time, and grows at most exponentially. The classical action is given by
We observe that if we change the unknown function to by
where the external time-dependent potential is given by
This potential corresponds to the first term of a Taylor expansion of about the point , and we naturally introduce solution to
The obvious candidate to approximate the initial wave function is then:
Therefore, is a good approximation of at least up to time of order (Ehrenfest time).
1.1.2. Nonlinear case
When adding a nonlinear term to (1.3), one has to be cautious about the size of the solution, which rules the importance of the nonlinear term. To simplify the discussions, we restrict our analysis to the case of a gauge invariant, defocusing, power nonlinearity, . We choose to measure the importance of nonlinear effects not directly through the size of the initial data, but through an -dependent coupling factor: we keep the initial datum (1.2) (with an -norm independent of ), and consider
Since the nonlinearity is homogeneous, this approach is equivalent to considering , up to multiplying the initial datum by . We assume , with if : for , defined by
is critical in terms of the effect of the nonlinearity in the semi-classical limit . If , then , given by (1.9)-(1.10), is still a good approximation of at least up to time of order . On the other hand, if , nonlinear effects alter the behavior of at leading order, through its envelope only. Replacing (1.9) by
and keeping the relation (1.10), is now a good approximation of . In  though, the time of validity of the approximation is not always proven to be of order at least , sometimes shorter time scales (of the order ) have to be considered, most likely for technical reasons only. Some of these restrictions have been removed in , by considering decaying external potentials .
1.2. Linear scattering theory and coherent states
We now consider the aspect of large time, and instead of prescribing at (or more generally at some finite time), we impose its behavior at . In the linear case (1.3), there are several results addressing the question mentioned above, considering different forms of asymptotic states at . Before describing them, we recall important facts concerning quantum and classical scattering.
1.2.1. Quantum scattering
Throughout this paper, we assume that the external potential is short-range, and satisfies the following properties:
We suppose that is smooth and real-valued, . In addition, it is short range in the following sense: there exists such that
Our final result is established under the stronger condition (a condition which is needed in several steps of the proof), but some results are established under the mere assumption . Essentially, the analysis of the approximate solution is valid for (see Section 4), while the rest of the analysis requires .
the underlying Hamiltonians. For fixed , the (linear) wave operators are given by
and the (quantum) scattering operator is defined by
See for instance .
1.2.2. Classical scattering
The existence and uniqueness of such a trajectory can be found in e.g. [20, 51], provided that . Moreover, there exists a closed set of Lebesgue measure zero in such that for all , there exists such that
The classical scattering operator is . Choosing implies that the following assumption is satisfied:
The asymptotic center in phase space, is such that the classical scattering operator is well-defined,
and the classical action has limits as :
for some .
1.2.3. Some previous results
It seems that the first mathematical result involving both the semi-classical and large time limits appears in , where the classical field limit of non-relativistic many-boson theories is studied in space dimension .
where denotes the standard Fourier transform (whose definition is independent of ). The main result from  shows that the semi-classical limit for can be expressed in terms of the classical scattering operator, of the classical action, and of the Maslov index associated to each classical trajectory. We refer to  for a precise statement, and to  for the case of long range potentials, requiring modifications of the dynamics.
Then Assumption 1.2 implies that there exists such that
As a corollary, our main result yields another interpretation of the above statement. It turns out that a complete scattering theory is available for (1.9). As a particular case of Theorem 1.5 (which addresses the nonlinear case), given , there exist a unique solution to (1.9) and a unique such that
Then in the above theorem (where is restricted to be a Gaussian), we have
Finally, we mention in passing the paper , where similar issues and results are obtained for
for a short-range potential, and is bounded as well as its derivatives. The special scaling in implies that initially concentrated waves (at scaled ) first undergo the effects of , then exit a time layer of order , through which the main action of corresponds to the above quantum scattering operator (but with due to the new scaling in the equation). Then, the action of becomes negligible, and the propagation of the wave is dictated by the classical dynamics associated to .
1.3. Main results
We now consider the nonlinear equation
along with asymptotic data (1.14). We first prove that for fixed , a scattering theory is available for (1.15): at this stage, the value of is naturally irrelevant, as well as the form (1.14). To establish a large data scattering theory for (3.1), we assume that the attractive part of the potential,
is not too large, where for any real number .
We emphasize the fact that several recent results address the same issue, under various assumptions on the external potential :  treats the case where is an inverse square (a framework which is ruled out in our contribution), while in , the potential is more general than merely inverse square. In , a magnetic field is also included, and the Laplacian is perturbed with variable coefficients. We make more comparisons with  in Section 3.
The second result of this paper concerns the scattering theory for the envelope equation:
As mentioned above, the proof includes the construction of a linear scattering operator, comparing the dynamics associated to (1.9) to the free dynamics . In the above formula, we have incorporated the information that is unitary on , but not on (see e.g. ).
We can now state the nonlinear analogue to Theorem 1.3. Since Theorem 1.4 requires , we naturally have to make this assumption. On the other hand, we will need the approximate envelope to be rather smooth, which requires a smooth nonlinearity, . Intersecting this property with the assumptions of Theorem 1.4 leaves only one case: and , that is (1.1), up to the scaling. We will see in Section 5 that considering is also crucial, since the argument uses dispersive estimates which are known only in the three-dimensional case for satisfying Assumption 1.1 with (larger values for could be considered in higher dimensions, though). Introduce the notation
As a corollary of the proof of the above result, and of the analysis from , we infer:
Corollary 1.8 (Asymptotic decoupling).
with initial datum
where , , so that scattering is available as for , in the sense of Assumption 1.2, and . We suppose for . Then we have the uniform estimate:
where is the approximate solution with the -th wave packet as an initial datum. As a consequence, the asymptotic expansion holds in , as :
where the inverse wave operators stem from Theorem 1.4, the ’s are the asymptotic states emanating from , and
In the case , the approximation by wave packets is actually exact, since then , hence . For one wave packet, Theorem 1.6 becomes empty, since it is merely a rescaling. On the other hand, for two initial wave packets, even in the case , Corollary 1.8 brings some information, reminiscent of profile decomposition. More precisely, define by (1.6), and choose (arbitrarily) to privilege the trajectory . The Cauchy problem is then equivalent to
where we have set and . Note however that the initial datum is uniformly bounded in , but in no for (if ), while the equation is -critical, Therefore, even in the case , Corollary 1.8 does not seem to be a consequence of profile decompositions like in e.g. [21, 42, 45]. In view of (1.4), the approximation provided by Corollary 1.8 reads, in that case:
where the phase shift is given by
Notation. We write whenever there exists independent of and such that .
2. Spectral properties and consequences
In this section, we derive some useful properties for the Hamiltonian
Since the dependence upon is not addressed in this section, we assume .
First, it follows for instance from  that Assumption 1.1 implies that has no singular spectrum. Based on Morawetz estimates, we show that has no eigenvalue, provided that the attractive part of is sufficiently small. Therefore, the spectrum of is purely absolutely continuous. Finally, again if the attractive part of is sufficiently small, zero is not a resonance of , so Strichartz estimates are available for .
2.1. Morawetz estimates and a first consequence
In this section, we want to treat both linear and nonlinear equations, so we consider
Morawetz estimate in the linear case will show the absence of eigenvalues. In the nonlinear case , these estimates will be a crucial tool for prove scattering in the quantum case. The following lemma and its proof are essentially a rewriting of the presentation from .
Proposition 2.1 (Morawetz inequality).
In other words, the main obstruction to global dispersion for comes from , which is the attractive contribution of in classical trajectories, while is the repulsive part, which does not ruin the dispersion associated to (it may reinforce it, see e.g. , but repulsive potentials do not necessarily improve the dispersion, see ).
The proof follows standard arguments, based on virial identities with a suitable weight. We resume the main steps of the computations, and give more details on the choice of the weight in our context. For a real-valued function , we compute, for solution to (3.1),
In the case , the standard choice is , for which
This readily yields Proposition 2.1 in the repulsive case , since .
In the same spirit as in , we proceed by perturbation to construct a suitable weight when the attractive part of the potential is not too large. We seek a priori a radial weight, , so we have
We construct a function such that , so the condition will remain. The goal is then to construct a radial function such that the second line in (2.3) is non-negative, along with for some .
Case . In this case, the expression for is simpler, and the above conditions read
Since we do not suppose a priori that is a radial potential, the first condition is not rigorous. We actually use the fact that for , Assumption 1.1 implies
To achieve our goal, it is therefore sufficient to require:
In view of (2.5), we seek
We readily have
along with the properties ,
It is now natural to set
so we have and
This function is indeed in if and only if . We define by ,
where is proportional to
We infer that (2.6) is satisfied for , provided that . Note then that by construction, we may also require
for morally very small.
Case . Resume the above reductions, pretending that the last two terms in are not present: (2.6) just becomes
and we see that with and defined like before, we have
Since this term is negative at and has a non-positive derivative, we have , so finally . ∎
2.2. Strichartz estimates
In [3, Proposition 3.1], it is proved that zero is not a resonance of , but with a definition of resonance which is not quite the definition in , which contains a result that we want to use. So we shall resume the argument.
By definition (as in ), zero is a resonance of , if there is a distributional solution , such that for all , to .
Under the assumptions of Proposition 2.1, zero is not a resonance of .
This implies that , by taking for instance in
By definition, for all test function ,
Let be the weight constructed in the proof of Proposition 2.1, and consider
By construction of , this implies
hence . ∎