Strichartz Estimates for Charge Transfer Models

In this note, we prove Strichartz estimates for scattering states of scalar charge transfer models in $\mathbb{R}^{3}$. Based on the idea of the proof of Strichartz estimates which follows \cite{CM,RSS}, we also show the energy of the whole evolution is bounded independently of time without using the phase space method, for example, in \cite{Graf}. One can easily generalize our argument to $\mathbb{R}^{n}$ for $n\geq3$. Finally, in the last section, we discuss the extension of these results to matrix charge transfer models in $\mathbb{R}^{3}$.


Introduction
In this note, following the work of [RSS, Cai], charge transfer models for Schrödinger equations in R 3 will be considered. We study the time-dependent charge transfer Hamiltonian with rapidly decaying smooth potentials V j (x), say, exponentially decaying and a set of mutually non-parallel constant velocities v j . Strichartz estimates for the evolution (1.2) 1 i ∂ t ψ + H(t)ψ = 0 associated with a charge transfer Hamiltonian H(t) will be proved. The starting point is the well-known L p estimates for the free Schrödinger equation (H 0 = − 1 2 ∆) on R n : where 2 ≤ p ≤ ∞, 1 p + 1 p ′ = 1. To analyze the dispersive estimate of linear Schrödinger equations with potentials, we consider the dispersive estimates of the Schrödinger flow (1.4) e itH P c , H = − 1 2 ∆ + V on R n , where P c is the projection onto the continuous spectrum of H. For Schrödinger equations with potentials, there may be bound states, i.e., L 2 eigenfunctions of H. Under the evolution e itH , such bound states are merely multiplied by oscillating factors and thus do not disperse. So we need to project away any bound state. V is a real-valued potential that is assumed to satisfy some decay condition at infinity. This decay is typically expressed in terms of the point-wise decay |V (x)|≤C x −β , for all x ∈ R n and for some β > 0. We use the notation x = 1 + |x| 2 1 2 . Occasionally, we will use an integrability condition V ∈ L p (R n ) (or a weighted variant of it) instead of a point-wise condition. These decay conditions will also be such that H is asymptotically complete: where the spaces on the right-hand side refer to the span of all eigenfunctions, and the absolutely continuous subspace, respectively.
The dispersive estimate for the linear Schrödinger equations with potentials, which we will be most concerned with is of the form Interpolating with the L 2 bound e itH P c f L 2 x ≤ C f L 2 x , we get It is well-known that via a T * T argument the dispersive estimate (1.5) gives rise to the class of Strichartz estimates (1.7) e itH P c f L q t L p x f L 2 for all 2 q + n p = n 2 . The endpoint q = 2 holds for n ≥ 3 but it is not captured by this approach, see [KT].
Roughly speaking, Strichartz estimates can be regarded as smoothing effects in L p x spaces. For example, when we consider the free Schrödinger equation, compared with the trivial conservation of L 2 norm of the solution, in Strichartz estimates one gains space integrability from p = 2 to p > 2, but one loses time integrability from q = ∞ to q < ∞. To be more precise, we can take a function g ∈ L 2 but g / ∈ L p x for p > 2. Then we take f = e 1 2 it0∆ g as the initial data for the free linear Schrödinger equation, then we can see at t = t 0 , e −i 1 2 t0∆ f / ∈ L p x . So without integration or average on time, there is no hope to get L p x estimate for all the time for general L 2 initial data. Strichartz estimates are crucial for the study of long-time behavior of associated nonlinear models.
For the results and historical progress of dispersive estimates and smoothing effects of Schrödinger operators, one can find further details and references in [Sch].
There are extra difficulties for Schrödinger equations with time-dependent potentials. For example, given a general time-dependent potential V (x, t), it is not clear how to introduce an analog of bound states and the spectral projection. And the evolution of equation might not satisfy group properties any more. In this paper, we focus on a particular case of time-dependent potentials, i.e. the charge transfer models in R 3 .
Firstly, we consider the scalar model in the following sense: By a charge transfer model we mean a Schrödinger equation where v j 's are distinct vectors in R 3 , and the real potentials V k are such that for every 1 ≤ k ≤ m 1) V k is time-independent and decays exponentially (or has compact support) 2) 0 is neither an eigenvalue nor a resonance of the operators Recall that ψ is a resonance at 0 if it is a distributional solution of the equation H k ψ = 0 which belongs to the space f : x −σ f ∈ L 2 for any σ > 1 2 , but not for σ = 0.
To simplify our argument, we discuss when m = 2 case with V 1 is stationary and V 2 moves along − → e 1 with the unit speed. It is easy to see our arguments work for general cases.
Remark. The assumptions are always assumed when we want to prove dispersive estimate and Strichartz estimates, e.g, [JSS, Sch, Ya, RSS, Cai]. The decay required of the potentials is not optimal but merely for convenience.
An indispensable tool in the study of charge transfer models are the Galilei transformations (1.10) g v,y (t) = e i | v| 2 2 t e ix· v e −i(y+ vt)· p , cf. [Graf, Cai, RSS], where p = −i ∇. They are the quantum analogues of the classical Galilei transforms To see this, we take a Schwartz function f such that f andf are centered around the origin, then g v,y (t)f is centered around t v + y, and g v,y (t)f is centered around v. The Galilei transformations have a very important conjugacy property: Another important property of the Galilei transformations is that g v,y (t) are isometries in all L p spaces. Finally, in our case, as discussed above, we always assume y = 0. To simplify our notations, we write g v (t) := g v,0 (t) and notice that g e1 (t) −1 = g − e1 (t).
We recall some consequences from [RSS, Cai]. Again, we consider with V 1 and V 2 decaying rapidly. Let w 1 , . . . , w m and u 1 , . . . , u ℓ be the normalized bound states of H 1 and H 2 associated to the negative eigenvalues λ 1 , . . . , λ m and µ 1 , . . . , µ ℓ respectively (notice that by our assumptions, 0 is not an eigenvalue). Following the notations in [RSS], we denote by P b (H 1 ) and P b (H 2 ) the projections onto the the bound states of H 1 and H 2 , respectively, and let P c To be more explicit, we have It is well-known, from the standard case with stationary potentials that we need to project away from bound states as we discussed at the very beginning. Here following [RSS], we recall the analogous condition in our case.
x) be the solution of equation (1.15). We say that ψ 0 or ψ (x, t) is asymptotically orthogonal to the bound states of H 1 and Here It is clear that all ψ 0 that satisfy (1.17) form a closed subspace of L 2 (R n ). We call elements in this subspace scattering states at t = 0 and denote the subspace by H s (0). We name H s (0) as scattering space at t = 0. With H s (0), we define P s (0) to be the projection onto H s (0).
Remark. The subspace above coincides with the space of scattering states for the charge transfer problem which appears in Graf's asymptotic completeness result [Graf]. We will see more details in Section 2.
We now formulate our main results.

Theorem 1.3 (Strichartz estimates).
Consider the charge transfer model as in Definition 1.1 with two potentials in R 3 as above. Suppose the initial data ψ 0 ∈ L 2 R 3 is asymptotically orthogonal to the bound states of H 1 and H 2 in the sense of Definition 1.2. Then for ψ(t, x) = U (t, 0)ψ 0 and a Schrödinger admissible pair (p, q) in R 3 , i.e., We also have the boundedness of the energy.
Theorem 1.4. Let ψ 0 ∈ H 1 R 3 and ψ(t, x) = U (t, 0)ψ 0 be a solution to (1.15) with the initial data ψ 0 . Then The paper is organized as follows: In Section 2, we will recall some results from [RSS, Cai]. Then in Section 3, we establish Strichartz estimates for the evolution that is not associated to the bound states of H j for the scalar charge transfer model. In Section 4, we will show the energy of the whole evolution is bounded independently of time. Finally, we will generalize our arguments to non-selfadjoint matrix cases in Section 5.

Preliminaries
In this section, we formulate the important results from [RSS, Cai] which are crucial for later sections.
First of all, if the evolution is asymptotically orthogonal to the bound states of H 1 and H 2 , we can actually get a decay rate for RSS], Proposition 3.1). Let ψ(t, x) be a solution to (1.15) which is asymptotically orthogonal to the bound states of H 1 and H 2 in the sense of Definition 1.2. Then we have the decay rate that for some α > 0.
As pointed out above, ∀ψ 0 ∈ L 2 such that the asymptotically orthogonal condition 1.17 are satisfied, they form a subspace H s (0)⊂L 2 . We can do a more general time-dependent construction. Denote the evolution starting from τ to t by U (t, τ ). Similar as our original construction there is a subspace H s (τ ) ⊂ L 2 such that for ψ ∈ H s (s), Similarly as above, we can also obtain a decay rate that for some α > 0, It is crucial to notice an important property of H s (τ ).
If the evolution is asymptotically orthogonal to the bound states, one also have the usual L 1 → L ∞ dispersive estimate. [Cai]). Consider the charge transfer model as in Definition 1.1 with two potentials as above. Assume V 1 , V 2 ∈ L 1 (R n ). Then for any initial data ψ 0 ∈ L 1 (R n ), which is asymptotically orthogonal to the bound states of H 1 and H 2 in the sense of Definition 1.2, one has the decay estimate A similar estimate holds for any number of potentials.
Note that since the potentials depend on time, Strichartz estimates do not follow from the dispersive estimate and T T * argument.
With the decay estimate (2.4), we obtain the asymptotic completeness of the charge transfer Hamiltonian: RSS, Cai]). Let w 1 , . . . , w m and u 1 , . . . , u ℓ be the normalized bound states of H 1 and H 2 associated to the negative eigenvalues λ 1 , . . . , λ m and µ 1 , . . . , µ ℓ . Then for any initial data ψ 0 ∈ L 2 (R n ), the solution ψ(t, x) = U (t, 0)ψ 0 of the charge transfer model, equation (1.15), can be written in the form (2.5) with some choice of the constants A r , B s and the function φ 0 . The remainder term R(t) satisfies the estimate, With the asymptotic completeness of the charge transfer Hamiltonian, we can construct a time-dependent decomposition of L 2 with scattering states and analogous of bound states associated with H 1 and H 2 . The construction should be similar as [Graf,Ya1]. Following the notations in [RSS, Graf], with the proof of Theorem 2.4, we know the existence of the following wave operators in L 2 : for s ∈ R, where limits are taken as strong operator topology. From [RSS], the ranges of the above operators has the following relation: . Naturally the above constructions will introduce a time-dependent decomposition of L 2 and one can observe that and RanΩ − 2 (τ ) are analogous as the spans bound states associated with H 1 and H 2 respectively. Notice that by construction, one can find a basis for Ω − i (τ ), i = 1, 2. With our notations above, w 1 , . . . , w m and u 1 , . . . , u ℓ be the normalized bound states of H 1 and H 2 associated to the negative eigenvalues λ 1 , . . . , λ m and µ 1 , . . . , µ ℓ respectively. Then is a basis for RanΩ − 2 (τ ). By asymptotic completeness and intuition, as τ → ∞, [RSS], one can actually extract a convergent rate. We focus onΩ − 1 (τ )w i → w i and for the other case, we just need to apply the same argument after applying a Galilei transformation.
It suffices to estimate the L 2 norm of

Strichartz Estimates
In this section, we prove Strichartz estimates for charge transfer models. The ideas will be based on methods in [CM, RSS]. Certainly, we need to project away from the bound states of H 1 and the moving bound states associated to H 2 (t). We will show certain weighted estimates for the evolution of states in the scattering space defined in [RSS] and in the sense of Definition 1.2. Now we formulate the following two estimates when our initial state is in the scattering space. The first one is: for all x 0 and x 1 .
Here 2 → 2 means the norm as an operator from L 2 to L 2 and P s defined as the projection onto the scattering space as above in sense of Definition 1.2. Also as usual, x = |x| 2 + 1 1 2 . The second weighted estimate we want to show is the following: Heuristically, we can see the above two estimates hold for the evolution of a free Schrödinger equation since a free particle moves towards infinity. The weights just play roles like indicator functions of certain finite regions. Then surely, as time evolves, the particle will leave any of those regions. So we have the decay of the wave function. In our case, the state in the scattering space will just move asymptotically like a free particle, so we should expect the above result. The second estimate is a variant of the above heuristics adjusted to our model since we have moving potentials.
Before we prove Lemma 3.1 and Lemma 3.2, we show how to derive Strichartz estimates for the charge transfer model based on them.
Proof of Theorem 1.3. Let ψ(t) = U (t, 0)ψ 0 and by our assumption we have P s (0)ψ 0 = ψ 0 . Rewrite the charge transfer model as Now we apply the endpoint Strichartz estimate [KT] for the free Schrödinger equation, we get for a Schrödinger admissible pair (p, q) in R 3 , one has Since our potentials decay fast, we can pick m large (in particular m > 3 2 ) such that by Hölder's inequality we have, x Now by our above two claimed estimates Lemma 3.1 and Lemma 3.2, we have Then combine all estimates above, we get Therefore, we have the desired Strichartz estimate In the next section, we will show as a byproduct of Strichartz estimates, (1.19), we can get the energy boundedness of the whole evolution of the charge transfer model.
3.1. Proof of Lemmas 3.1 and 3.2. To rigorously show Lemmas 3.1 and 3.2 are consistent with our heuristics, we consider the free evolution first. We claim that the first estimate (3.1) holds for the free Schrödinger equation.
If |s| ≥ 1, we apply the dispersive estimate for the free evolution. Then by Young's inequality we get So the desired estimate holds.
Also the second estimate (3.2) holds for the free Schrödinger evolution by the endpoint Strichartz estimate, estimate (1.7).
Proof. By Hölder's inequality, we have Then by the endpoint Strichartz estimate in R 3 , [KT], we have Therefore, we can concludê The Lemma is proved. Now we show Lemmas 3.1 and 3.2 by a bootstrap argument similar to the one in [RSS]. As usual, the constant C varies from line to line.
First of all, we note the following simple facts: Since P s (t 0 )u satisfies following estimates, for p ≥ 2, Then surely, For the second weighted estimate, with some p ≥ 2, x . By the Duhamel formula, we write Surely, there is no problem with the free piece F as we discussed above by Lemmas 3.3 and 3.4. Now fix T large enough and apply Gronwall's equality. Then we can find a large constant C(T ) such that (3.7) x hold for t ≤ T . Next we imitate the bootstrap process in [RSS] and [CM]. Fix a large constant A to be determined later. We also assume T − t 0 ≫ A. As in [RSS], for t ≤ T we First, we bound L 1 . With Lemma 3.3, we have Here we just emphasize that the constant in above estimate does not depend on T . Notice that G 1 can be bounded similarly as above.
For the second part, with Lemma 3.4, Also G 1 can be bounded similarly.
Next, we analyze L 2 . With Lemma 3.3 and the bootstrap assumption (3.7), for an absolute constant C.
For the other estimate, with Lemma 3.4 and bootstrap assumption (3.8), we conclude that So when A is large, we recapture our bootstrap argument conditions, i.e., h(A)C 2 (T ) will be a small portion of C(T ) provided A is large enough. Similar estimates hold for G 2 . It remains to analyze L 3 and G 3 . We will expand U again. And the following two versions of weighted estimates for Schrödinger equations with rapidly decaying potentials will be used.
Lemma 3.5. For σ > 3 2 , and H j = − 1 2 ∆ + V j , where V j satisfies the decay assumption for our charge transfer Hamiltonian, then we have where P c (H j ) is the projection onto the continuous spectrum of H j .
Proof. These two estimates follow from the boundedness of wave operators [Ya] and Lemma 3.3 and Lemma 3.4. Or one can apply the dispersive estimate and Strichartz estimates for perturbed Schrödinger equations.

Now we analyze
Splitting L 3 with respect to the spectrum of H 1 , one has Surely, there is no problem with L 3,b by the discussion at the very beginning of this sectionP b (H 1 ) U (s, t 0 )P s (t 0 ) decays exponentially. For L 3,c , we use the ideas from [RSS] to decompose our evolution into low velocity and high velocity pieces. For the low velocity piece, we directly use a commutator argument, non-stationary phase and the fact the supports of V 1 and V 2 become almost disjoint. For the high velocity part, we use a version of the Kato smoothing estimate.
Expanding U with respect to H 1 , we can write Then we can write Consider the decomposition L 3,c = I + iK, There is no problem with I by similar arguments for the free case with Lemma 3.5. Next, we decompose K further as follows: For S, a similar argument as for L 1 implies As usual, the constant C does not depend on T .
For the second piece, we also havê Next, for Z, following a similar argument to L 2 , we obtain For the second estimate, where as before, Therefore, when we pick B large enough, we have satisfied all the conditions for the bootstrap argument.
Finally, we analyze J: We decompose the integral into low and high frequency parts: where F (| p| ≤ M ) and F (| p| ≥ M ) denote smooth projections onto frequencies | p| ≤ M and | p| ≥ M respectively. To analyze the low frequency part, we observe that for arbitrary ǫ > 0, x σ , then we look at the following quantity, This decay result follows from the following two facts: Integration by parts with and the decay estimate: So we can conclude that for any N > 0, By some similar calculations in [Graf], we conclude But in our particular situation, one can do easy calculations based on Duhamel formula, provided M is large enough. Therefore, for J L , So when A, B is large, we conclude that the coefficient satisfies the bootstrap conditions. For the second part, Again, we know when ǫ is small, we recapture the bootstrap conditions. Finally, we need to check J c,H . We will use the following version of the Kato smoothing estimate, or we can apply a variant of Kato's smoothing estimate from [RSS].
we also haveR We will use Lemma 3.6, but for the sake of completeness, we formulate the result from [RSS].

1.
So for the first estimate, with bootstrap assumption (3.7), we get For the other estimate, So we can pick M large, then the coefficient satisfies the bootstrap condition again.
To sum up, when we pick A, B and M large enough independent of T , if we have for t ∈ [t 0 , T ] we can improve it to Therefore, we can make for t ∈ [t 0 , T ], ≤ C holds for arbitrary t which shows Lemma 3.1.
For the second part we proceed analogously. Indeed, if we supposê then we can improve the estimate tô So we can obtain a bound for which is independent of T . Therefore, we can send T to ∞ above. Finally, we obtainˆ∞ Remark 3.7. With Theorem 2.3, one can show Lemma 3.1 easily as the free case. Set s = t − t 0 , first, if |s| ≤ 1, clearly by U (t, t 0 ) 2→2 ≤ 1 and the integrability condition in R 3 , i.e. σ > 3 2 , we can get the desired result. If |s| ≥ 1, we apply the dispersive estimate for the free motion, by Young's inequality we get x −σ 2 L 2 U (t, t 0 )P s (t 0 ) 1→∞ , and from Theorem 2.3, But we proved Lemma 3.1 together with Lemma 3.2, since the dispersive estimate might not be available in other contexts.

Boundedness of The Energy
In this section, we use Strichartz estimates to show that the energy of the whole evolution of the charge transfer model is bounded independently of time. The asymptotic completeness of the Hamiltonian shown in [RSS] will be used. We will still consider the model with two potentials as in the previous section.
Proof of Theorem 1.4. From Theorem 2.4, we can write the evolution as: for some φ 0 ∈ L 2 R 3 , where g is the Galilei transformation. It is trivial to see the part associated with bound states and moving bound states, A r e −iλr t w r + ℓ s=1 B s e −iµst g − − → e1 (t)u s has bounded energy. Indeed, to be more precise, we have So it suffices to consider In other words, we might assume P s (t)ψ(t) = ψ(t).
We can differentiate the equation (4.4) and set v = ∂ x1 ψ =: ∂ 1 ψ, Again, it suffices to consider ψ is in the scattering space. Since other components are easily to be bounded. To see this, we look at v, w r L 2 = − e −it ∆ 2 φ 0 + R(t), ∂ x1 w r L 2 . By the asymptotic completeness result, we know In particular, we know since from Agmon's estimate, ∂ x1 w r is still exponentially decaying.
Notice that since we can approximate φ 0 by φ n ∈ L 2 ∩ L 1 in L 2 and then by the dispersive estimate for the free equation A similar discussion holds for u s , we can conclude that By the above argument, we can actually conclude that v is asymptotically orthogonal to the bound states of H 1 and moving bound states associated to H 2 (t). We can in fact obtain an explicit rate of decay for the term goes to 0, but it is enough for our purposes to know that it is just bounded by ψ 0 H 1 . Then by Proposition 2.5, Therefore, it is sufficient to estimate P s (t)v(t) L 2 and hence, without loss of generality, we assume We do a similar argument as the proof for Strichartz estimates, Theorem 1.3.
Next we bound U 3 . We again apply Hölder's inequality and the endpoint Strichartz estimate, It remains to bound ˆt By Lemma 3.1, Lemma 3.2, we have Also, we can get for any Schrödinger admissible pair (p, q).
For the second piece, we use For the first piece, by Hölder's inequality, we have x .
Then applying the endpoint Strichart estimate to ψ by Theorem 1.3, we get The same argument applies to all other partial derivatives of ψ. So we can conclude that

The theorem is proved
By a simple inductive argument, we obtain the following corollary: where k is a non-negative integer, then Remark. As a concluding remark, we notice that we proved the boundedness of the energy based on Strichartz estimates and the asymptotic completeness of the Hamiltonian. In [Graf], Graf proved the asymptotic completeness based on the boundedness of the energy. So, we can see, modulo some technical assumptions on the spectrum of the Schödinger operator, the boundedness of the energy is equivalent to the asymptotic completeness of the Hamiltonian. Also note that the asymptotic completeness can be also proved by the dispersive estimate as in [RSS].

Matrix Charge Transfer Models
In this section, we extend our above results to matrix charge transfer models in R 3 similarly as the work in [RSS]. For the sake of completeness, we start from the basic definitions following [RSS].
Definition 5.1. By a matrix charge transfer model we mean a system where v j are distinct vectors in R 3 , and V j are matrix potentials of the form · v j + γ j with α j , γ j ∈ R and α j = 0. Furthermore, we require that each  where ω(t) = α 2 t + γ. Then we have the following relation For matrix charge transfer models, the analysis should be similar to the scalar case except that we have to modify the asymptotic orthogonality condition. Recall that as we remarked above, it is not necessary to use the asymptotic completeness results. In the scalar case, the asymptotic orthogonality condition is sufficient for us. In the matrix case, the asymptotic orthogonality condition is replaced by the definition of "scattering states" in Definition 8.3 in [RSS] which is similar to the scattering space in the sense of Definition 1.2 for the scalar case.
Definition 5.4. Let U (t) ψ 0 = ψ(t, ·), we call that ψ 0 a scattering state relative to H j if By the discussion in Section 8.3 in [RSS], if ψ 0 a scattering state relative to each H j , we have the rate of convergence similar to the scalar case, for some α > 0. With all the preparations above, we now can formulate our Strichartz estimates for matrix charge transfer models.
Theorem 5.5. Consider the matrix charge transfer model as in Definition 5.1. We denote ψ(t) = U (t, 0) ψ 0 and assume ψ 0 is a scattering state relative to each H j in sense of Definition 5.4. Then for a Schrödinger admissible pair (p, q) in R 3 , i.e., for some finite constant C.
As in the scalar case, the proof Theorem 5.5 is based on certain weighted estimates which rely on a bootstrap argument. Since the proof is basically identical as with the scalar case, we do not carry out the details. We only discuss it briefly. Recall that in our proof, there are several important ingredients: dispersive estimates for stationary potentials, the boundedness of wave operators, the Kato smoothing estimate. All of them hold for the matrix case. For the dispersive estimates for stationary potentials, one can find details in [Cu, RSS, ES]; for the boundedness of wave operators, the results are discussed in [Cu]; the Kato smoothing estimates can be obtained as for the scalar case in [RSS]. Hence with the remark at the beginning of the second section, and all the proofs above, we can conclude that Strichartz estimates hold for the matrix case.
Remark. With the dispersive estimate for matrix transfer models and the results on scattering states, we can follow the proof in [RSS] to prove the asymptotic completeness for matrix charge transfer Hamiltonians.
Similar to the scalar case, we also have the energy estimate.
Theorem 5.6. For ψ 0 ∈ H 1 := H 1 R 3 × H 1 R 3 , we have Corollary 5.7. For ψ 0 ∈ H k := H k R 3 × H k R 3 where k is a non-negative integer, then we have