Scattering and inverse scattering for nonlinear quantum walks

We study large time behavior of quantum walks (QWs) with self-dependent (nonlinear) coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QWs such as dispersive estimates and Strichartz estimate. Such argument is standard in the study of nonlinear Schr\"odinger equations and discrete nonlinear Schr\"odinger equations but it seems to be the first time to be applied to QW.

In this paper, we consider the QWs with state dependent (nonlinear) quantum coin. We set H := l 2 (Z; C 2 ) := {u : Z → C 2 | u 2 where u C 2 = (|u 1 | 2 + |u 2 | 2 ) 1/2 for u = t (u 1 , u 2 ) ∈ C 2 . We fix a map C : R × R → U (2), where U (2) is the set of 2 × 2 unitary matrices. We define the (nonlinear) quantum coinĈ : H → H by where u = t (u 1 u 2 ) ∈ H. For (T ± u)(x) = u(x ∓ 1) and S = T − 0 0 T + , we set By definition, S andĈ preserve the l 2 norm, and so does U . Let u 0 ∈ H be an initial state for a walker. Then, the state u(t) of the walker at time t is defined by the recursion relation We define the nonlinear evolution operator U (t) by U (t)u 0 = u(t), t ∈ N 0 := N ∪ {0}.
Notice that u 0 → u = U (·)u 0 is a nonlinear map from H to l ∞ t (N; H). By nonlinear QWs, we mean the nonlinear evolution generated by U (t). If C : R × R → U (2) is a constant function (i.e. C(s 1 , s 2 ) = C 0 ∈ U (2) for all s 1 , s 2 ∈ R), then we will call it linear (or simple) QWs Remark 1.1. One can generalize C by defining it as a function Z × R × R and setting (Ĉu)(x) := C(x, |u 1 (x)| 2 , |u 2 (x)| 2 )u(x). In this paper we will only consider QWs which depend on its state but not explicitly on its position.
To the best of authors knowledge, nonlinear QWs was first proposed by Navarrete, Pérez and Roldán [31] as an nonlinear generalization of optical Galton board.
where g ∈ R.
As pointed out by Navarrete, Pérez and Roldán [31] themselves, this nonlinear evolution does not define a quantum system, but it can be realized in a optical system such as optical Galton board. Notice that this is similar to the relation between (linear) Schrödinger equation which describes quantum system and nonlinear Schrödinger equations which appears in various regions of physics including optics. Moreover, in a way similar to linear QWs, we can define the finding probability p t (x) of a walker at time t at position x through u(t, x) := (U (t)u 0 )(x) as provided that u 0 H = 1. Indeed, p t gives a probability distribution on Z, because U (t) preserves the norm and x∈Z p t (x) = u 0 2 H . From these reasons, it is natural to view the system described by U (t) as a nonlinearization of a linear QWs and thus we simply call it a nonlinear QWs. We also import terminology from QWs and call H and vectors in H the state space and states, respectively. We prove a weak limit theorem for the nonlinear QWs in a companion paper [22].
Other nonlinear QWs have been proposed by several authors as a simulator of nonlinear Dirac equation [21], for studying the nonlinear effect to the topologically protected mode [12], or simply investigating more rich dynamics ( [38]). Example 1.3 (Lee, Kurzyński, and Nha [21]). The following models are proposed by the relation to nonlinear Dirac equations. For Gross-Neveu model (scaler type interaction) and for Thirring model (vector type interaction) where g, θ ∈ R and R(θ) = cos θ − sin θ sin θ cos θ .
In the following, we restrict our nonlinear coin operator to the following type: where g > 0 is a constant, and C N (0, 0) = I 2 . Here I 2 = 1 0 0 1 . We set (1.10) The positive parameter g controls the strength of the nonlinearity. Notice that all models (1.4), (1.6), (1.7) (1.8) given above are included in (1.9).
In this paper, we view nonlinear QWs as space-time discretized nonlinear Schrödinger equations (NLS) and study the dynamical behavior of the walkers. Indeed, we demonstrate that standard estimates such as dispersive estimate and Strichartz estimate hold for QWs (Theorem 2.1, Lemma 2.4). These estimates are fundamental tools for the study of NLS. We show that also for nonlinear QWs, we can prove the scattering by parallel argument as the proof of scattering for NLS. By scattering, we mean the following: Remark 1.6. Scattering is equivalent to Therefore, by scattering, we can conclude that the nonlinear QWs behave similarly to linear QWs after long time. However, u + will be generically different from u 0 .
We use U g=1 (t) to denote the evolution U (t) that has the nonlinear coin defined in (1.9) with g = 1. We observe that for v 0 = √ gu 0 with u 0 l 2 = 1, Hence, instead of changing g, we can fix g = 1 and vary the norm or u 0 l 2 . Small u 0 l 2 will correspond to small g. In the following, we will always fix g = 1.
The first main result in this paper is the following: 1. For the case m = 3, there exists δ > 0 s.t. for any u 0 ∈ l 2 with u 0 l 2 < δ, U (t)u 0 scatters.
2. For the case m = 2, there exists δ > 0 s.t. for any u 0 ∈ l 1 with u 0 l 1 < δ, U (t)u 0 scatters. Theorem 1.7 tells us that if u 0 is sufficiently small (or with fixed u 0 , g is sufficiently small) the dynamics of nonlinear QWs will be similar to the dynamics of linear QWs.
We next consider the inverse scattering problem, which is the problem of identifying unknown nonlinear terms under the assumption that all of the scattering states are known. More precisely, we identify some values concerning to C N using the scattering data (u 0 , W * u 0 ). Here, W * u 0 is the final data u + appearing in Definition 1.5. Such problems naturally arise when one has only partial information of the system. In this case, one would like to reconstruct the parameters governing the system from the data which one can observe. In application, we usually do not have complete information of the system. Therefore it is important to consider inverse scattering problems. As for inverse scattering problems for some nonlinear Schrödinger equations and related equations, there are many papers (see, e.g., [7,30,33,34,35,41,45] and references therein). Using Theorem 1.7 and modifying methods in the above papers, we obtain a reproducing formula for the nonlinear coin.
For simplicity, we consider the case that C N can be expressed as (1 0) and e 2 = t (0 1). Further, for g : R + → C, we define D λ g(λ) = λ −1 (g(2λ) − g(λ)). We define the nonlinear operator W * : u 0 → W * u 0 . It follows from the proof of Theorem 1.7 that W * is well defined on {u 0 ∈ l 1 | u 0 l 1 < δ} and satisfies Theorem 1.8 (Inverse scattering). Assume (A) and that λ > 0 is sufficiently small. Then, we have We note that Theorem 1.8 tells us that we can partially reconstruct the nonlinear coin from the information of the scattering states. For example, the nonlinear coin is given bỹ with some constants g 1 , g 2 , then we can recover which will be the 1st order approximation ofC N (s 1 , s 2 ). Therefore, we can identify the constants g 1 , g 2 in the case. The paper is organized as follows. In section 2, we prove the dispersive estimate and Strichartz estimate for QWs with constant coin. In section 3, we prove Theorem 1.7. In section 4, we prove Theorem 1.8.

Dispersive and Strichartz estimates
We first derive the dispersive estimate for the linear evolution U 0 = SĈ 0 by using stationary phase method. We note that this dispersive estimate was first obtained by Sunada and Tate [43] in a slightly different form.
We define the (discrete) Fourier transform by and the inverse Fourier transform by Notice thatÛ is also unitary and the eigenvalues are given by Thus diagonalizingÛ (ξ), we have where We set a = |a|e iθa . Then, since we have 0 <p < π by (2.4), setting we havep(ξ) = p(ξ + θ a ). Differentiating (2.7), we obtain (2.10) We set the projections P ± by and P − = 1 − P + . We define where Then, we can express the generator by The following is the dispersive estimate for QW.
From Lemma 2.3, we obtain the claim of Theorem 2.1.
As the case of Schrödinger equations and discrete Schödinger equaiton (or continuous time QWs), we can derive the Strichartz estimate from dispersive estimate. We define where u l p t l q We further define the weak l p space l p,∞ by its norm where # is the counting measure. It is well known that f l p,∞ ≤ f l p and moreover we have · −1/p l p,∞ < ∞ ( x := (1 + |x| 2 ) 1/2 ) and the Young's inequality for weak type spaces where C > 0 is a constant.

Scattering
We now prove Theorem 1.7 Proof of Theorem 1.7 1. We first estimate the Strichartz norm. Set Φ : Notice that U (t)u 0 is the unique solution of (1.3) if and only if it is a fixed point of Φ. We show that if δ := u 0 l 2 ≪ 1, then Φ has an fixed point. Indeed, by lemma 2.4 Thus, if we set B := {u ∈ Stz | u Stz ≤ 2 u 0 l 2 = 2δ}, we see that Φ : B → B is a contraction mapping, provided δ > 0 sufficiently small. Therefore, there exists a unique u s.t. Φ(u) = u, which is actually U (t)u 0 . Further, since Thus, Therefore, we have the conclusion.
We can show scattering by using decay.
Let u 1 = u 11 u 12 ∈ l 1 and u 2 = u 21 u 22 ∈ l 1 . Then we have for any x ∈ Z, Using (1.11), we obtain for any x ∈ Z, Hence it follows that where we have used (4.1) and Lemma 3.1 in the last line. This completes the proof of (4.2).