Gang Pang
email: gangpang@buaa.edu.cn

Songsong Ji

Xavier Antoine

Construction and analysis of discretization schemes for one-dimensional nonlocal Schrödinger equations with exact absorbing boundary conditions

Keywords: Nonlocal Schrödinger equation, exact absorbing boundary condition, stability and error analysis

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Construction and analysis of discretization schemes for one-dimensional nonlocal Schrödinger equations with exact absorbing boundary conditions

Introduction

Two decades ago, Laskin [START_REF] Laskin | Fractional quantum mechanics and Lévy path integrals[END_REF][START_REF] Laskin | Lévy flights over quantum paths[END_REF][START_REF] Laskin | Fractals and quantum mechanics[END_REF][START_REF] Laskin | Fractional quantum mechanics[END_REF] proposed a time-dependent space Fractional Schrödinger Equation (FSE) which can be used e.g. to describe the fractional oscillator of Bohr atom [START_REF] Laskin | Fractional quantum mechanics and Lévy path integrals[END_REF] or longrange dispersive interactions properties [START_REF] Kirkpatrick | On the continuum limit for discrete NLS with longrange lattice interactions[END_REF]. The specificity of the FSE is that it involves a fractional Laplacian term (-∂ 2

x ) s/2 , usually for 1 ≤ s ≤ 2, instead of the standard local Laplacian (s = 2). The FSE led to many developments, from both the applications, theoretical and numerical sides (see e.g. [START_REF] Secchi | Ground state solutions for nonlinear fractional Schrödinger equations in R N[END_REF][START_REF] Guo | Some physical applications of fractional Schrödinger equation[END_REF][START_REF] Antoine | On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions[END_REF][START_REF] Antoine | On the numerical solution and dynamical laws of nonlinear fractional Schrödinger/Gross-Pitaevskii equations[END_REF][START_REF] Bhatti | Fractional Schrödinger wave equation and fractional uncertainty principle[END_REF][START_REF] Bhrawy | A fully spectral collocation approximation for multidimensional fractional Schrödinger equations[END_REF][START_REF] Bhrawy | Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations[END_REF][START_REF] Khaliq | A fourth-order implicit-explicit scheme for the space fractional nonlinear Schrödinger equations[END_REF][START_REF] Kirkpatrick | Fractional Schrödinger dynamics and decoherence[END_REF][START_REF] Liang | The locally extrapolated exponential splitting scheme for multi-dimensional nonlinear space-fractional Schrödinger equations[END_REF][START_REF] Pinsker | Fractional quantum mechanics in polariton condensates with velocity-dependent mass[END_REF]), and has shown to produce specific behaviors compared with the local Schrödinger equation. More generally, fractional PDE models can be embedded into the framework of nonlocal PDEs [START_REF] D'elia | Numerical methods for nonlocal and fractional models[END_REF] to include more advanced long range interactions that are not met in the fractional situation. From this view point, the fractional PDE model can be interpreted as a nonlocal model for a specific kernel. In the present paper, we consider the one-dimensional system with nonlocal Schrödinger equation: find the function ψ solution to    iψ t (x, t) = L δ ψ(x, t), x ∈ R, t ∈ (0, T ], ψ(x, 0) = ψ 0 (x), x ∈ R, lim |x|→+∞ ψ(x, t) = 0, t ∈ (0, T ], [START_REF] Laskin | Fractional quantum mechanics and Lévy path integrals[END_REF] where i = √ -1 and ψ 0 is a given initial data. The maximal time of computation is T . The nonlocal operator L δ appearing in [START_REF] Laskin | Fractional quantum mechanics and Lévy path integrals[END_REF] extends the fractional Laplacian [START_REF] D'elia | Numerical methods for nonlocal and fractional models[END_REF] and is given by

L δ ψ(x) =
• positiveness: γ δ (a, b) ≥ 0;

• symmetry according to the first argument a: γ δ (-a, b) = γ δ (a, b);

• finite horizon: ∃δ > 0 such that γ δ (a, b) = 0 if |a| > δ > 0.

In the above notations, a = xy measures the distance between the two points x and y, and (x + y)/2 controls the local horizon. We also need to precise some additional assumptions that are used later [START_REF] Pang | Artificial boundary conditions for the semi-discretized onedimensional nonlocal Schrödinger equation[END_REF][START_REF] Yan | Numerical computations of nonlocal Schrödinger equations on the real line[END_REF] to solve the nonlocal model set in a finite computational domain ]x -, x + [

• A1: we assume that the initial data ψ 0 is compactly supported in ]x -, x + [;

• A2: γ δ is compactly supported in [-δ, δ] × R for δ ≤ (x + -x -)/2;
• A3: γ δ is homogeneous in the exterior domain, i.e. (3)

Here, we consider that γ δ,± =: γ ∞ to simplify the presentation. Finally, similar problems to (1) could also arise for nonlocal diffusion [START_REF] D'elia | Numerical methods for nonlocal and fractional models[END_REF][START_REF] Lischke | What is the fractional Laplacian? a comparative review with new results[END_REF]. The aim of the paper is to develop truncation techniques for computing the solution to (1) in the finite computational domain ]x -, x + [. Many contributions were proposed to solve this problem for integer order partial differential equations, usually by considering transparent/artificial/absorbing boundary conditions (ABC) at a fictitious boundary, or alternatively absorbing layers or Perfectly Matched Layer (PML). We refer e.g. to [START_REF] Antoine | A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations[END_REF][START_REF] Antoine | A friendly review of absorbing boundary conditions and perfectly matched layers for classical and relativistic quantum waves equations[END_REF] for details about this topic in the case of quantum mechanics equations. Developing ABC or absorbing layers/PML truncation techniques for spatially nonlocal and fractional models remains much less studied than for local PDEs. For ABCs, techniques based on DtN operators have been introduced in [START_REF] Zheng | Numerical solution of the nonlocal diffusion equation on the real line[END_REF][START_REF] Zhang | Absorbing boundary conditions for nonlocal heat equations on unbounded domain[END_REF] in the case of nonlocal heat equations. Furthermore, a full numerical analysis of ABCs is available in [START_REF] Zheng | Stability and error analysis for a second-order fast approximation of the local and nonlocal diffusion equations on the real line[END_REF]. For the two-dimensional nonlocal wave equation, ABCs were constructed in [START_REF] Du | Nonlocal wave propagation in unbounded multiscale media[END_REF][START_REF] Du | Numerical solution of a two-dimensional nonlocal wave equation on unbounded domains[END_REF] while for nonlocal Schrödinger equations, ABCs were proposed and numerically approximated in [START_REF] Yan | Numerical computations of nonlocal Schrödinger equations on the real line[END_REF][START_REF] Wang | Stability and error analysis for a second-order approximation of 1D nonlocal Schrödinger equation under DtN-type boundary conditions[END_REF]. In the framework of the absorbing layers/PML approach, efficient and precised methods were designed in [START_REF] Baeumer | Boundary conditions for fractional diffusion[END_REF][START_REF] Kelly | Boundary conditions for two-sided fractional diffusion[END_REF][START_REF] Antoine | Towards Perfectly Matched Layers for time-dependent space fractional PDEs[END_REF][START_REF] Antoine | Derivation and analysis of computational methods for fractional Laplacian equations with absorbing layers[END_REF] for time-dependent heat and Schrödinger equations involving fractional operators, most particularly the fractional Laplacian. It is nevertheless unclear if these approaches can be extended to more general nonlocal models as the one considered here.

The goal of the present paper is to propose the construction of ABCs for a fully discretized version of [START_REF] Laskin | Fractional quantum mechanics and Lévy path integrals[END_REF] and to develop the error analysis of the scheme. To this end, in section 2.1, we discretize (1) by using a Crank-Nicolson scheme in time and an asymptotically compatible scheme in space. This allows us to derive the exact ABC for the fully discrete scheme in subsection 2.2. In addition, we propose a numerical algorithm for computing the kernel functions involved in the definition of the ABCs in subsection 2.3. After stating some properties of these kernel functions in section 3, we prove the stability of the boundary condition. We next develop in section 4 the error analysis between the exact solution of (1) and the solution of the semi-discretized system associated with (1) in the nontrivial case of a singular interaction kernel γ δ . This shows that the error is second-order both in space and time. In section 5, we study the error analysis for the numerical solution of (1) with given boundary condition, showing the second-order convergence of the scheme. We end the paper by some numerical examples in section 6, a conclusion in section 7 and the technical Appendix A.

Exact absorbing boundary conditions for the fully discrete nonlocal Schrödinger equation

To design some exact absorbing boundary conditions for the nonlocal Schrödinger equation appearing in system (1), we first fully discretize the equation in subsection 2.1 and next extract the corresponding boundary conditions in subsections 2.2 and 2.3.

Full discretization of the 1D nonlocal Schrödinger equation

For the spatial discretization, we use the Asymptotic Compatibility (AC) discretization scheme derived in [START_REF] Du | Nonlocal wave propagation in unbounded multiscale media[END_REF] to approximate the nonlocal operator L δ given by (2) (see [START_REF] Tian | Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations[END_REF][START_REF] Tian | Asymptotically compatible schemes and applications to robust discretization of nonlocal models[END_REF] for other AC schemes). Let us set discretize [x -, x + ] by using J + 1 equally spaced grid points x j = x -+ jh, 0 ≤ j ≤ J, for the uniform spatial step h := (x +x -)/J > 0, with x 0 = x -and x J = x + . We introduce φ j as the standard hat function of width h centered at point x j . Then, we have the following spatial approximation, for j ∈ Z,

L δ ψ(x j ) ≈ L δ,h ψ j = k∈Z b j,k (ψ j -ψ k ), (4)
where ψ j ≈ ψ(x j , •) and the real-valued coefficients b j,k are given by Let us now consider a Hilbert space H equipped with an inner product (•, •) H and induced norm • H . We introduce the semi-infinite sequence space

b j,k = -c j-k :=    1 (j -k)h R yφ j-k (y)γ δ (y, x j + x k 2 )dy, j = k, 0, j = k,
2 (H) = ß u = {u n } ∞ n=0 : u n ∈ H, u 2 (H) = ( ∞ n=0 u n 2 H ) 1 2 < ∞ ™ , with the inner product: (u, v) 2 (H) ≡ ∞ n=0 (u n , v n ) H , ∀u, v ∈ 2 (H). For a sequence u = {u n } ∞ n=0 ∈ 2 (
H), we define the operator S by: Su = {u n+1 } ∞ n=0 . The average operator E and the forward difference quotient operator D τ with uniform time step τ are given by E = (S + I)/2 and D τ = (S -I)/τ , respectively. We also need the following notations: Su n = (Su) n , Eu n = (Eu) n and D τ u n = (D τ u) n . Let us define the coefficients:

a 0 = -2 K k=1 c k , and a k = c k , k = 1, ..., K. (5)
After some calculations, we obtain the discretized nonlocal Schrödinger equation approximating [START_REF] Laskin | Fractional quantum mechanics and Lévy path integrals[END_REF] and based on the Crank-Nicolson time scheme at time t n = nτ , n ∈ N, and AC discretization

iD τ ψ n j = EL δ,h ψ n j := a 0 Eψ n j + K k=1 a k (Eψ n j+k + Eψ n j-k ), for j ∈ Z, (6)
where we denote by ψ n j the numerical approximation of ψ(x j , t n ).

Exact absorbing boundary condition for the fully discrete nonlocal Schrödinger equation

For ψ 0 j , j = 0, ..., J, compactly supported in [x 0 , x J ], we need to introduce some artificial boundary conditions to complete (6) written for j = 0, ..., J. To this end, we derive in Theorem 1 a relation between the left exterior unknowns ψ n j (red dots in Figure 1), j ≤ -1, and the interior left values ψ n j (blue dots) (0 ≤ j ≤ K -1) in the computational domain, which will play the role of left absorbing boundary condition. A similar relation is given for the right ABC as shown on Figure 1 for the nodes J + 1 ≤ j ≤ J + K.

ψ n -K ψ n -(K-1) ψ n -2 ψ n -1 ψ n K+1 ψ n K ψ n K-1 ψ n K-2 ψ n 2 ψ n 0 ψ n J-K-1 ψ n J-K ψ n J-K+1 ψ n J-K+2 ψ n J-1 ψ n J ψ n J+K ψ n J+K-1 ψ n J+2 ψ n J+1

Computational domain

Left boundary nodes

Left exterior domain

Right boundary nodes

Right exterior domain

Interior nodes Interior connected left nodes

Interior connected right nodes Before giving Theorem 1, we introduce the convolution of two sequences (g n ) n and (f n ) n as: (f g) n := n r=0 g n-r f n . For two sequences of matrices (A n ) n and (B n ) n of size K × K, and for a K components vector sequence (v n ) n , we define the K × K matrix (A B) n := n r=0 A n-r B r , and the vector (of size K) (A v) n := n r=0 A n-r v r . For the tools related to the Z-transform, we introduce:

h x + x -
f (z) := ∞ k=0 f n z -n , A(z) := ∞ n=0 A n z -n and v(z) := ∞ n=0 v n z -n .
Finally, we set: ρ K (z, x) := i z-1 τ -z+1 2 A K (x), with A K (x) := a 0 + 2 K k=1 a k cos(kx). Under these notations, the following result holds.

Theorem 1. Let n ∈ N. The left and right absorbing boundary conditions associated with (6) can be respectively written as

ψ n j = K k=1 (C -j,k ψ k-1 ) n , for -K ≤ j ≤ -1, (7)
ψ n J+j = K k=1 (C j,k ψ J+1-k ) n , for 1 ≤ j ≤ K, (8)
where the sequence of K × K matrices (C n ) n is given by: C(z) = B(z) A -1 (z), z ∈ C. In the above relations, we have:

B n j,k = f n j+k-1 and A n j,k = f n |j-k| , for 1 ≤ j, k ≤ K, with fj (z) = ĝj (z) ĝ0 (z) , ĝj (z) = 1 2π 2π 0 ρ -1 K (z, x)e -ijx dx. (9)
Let us introduce δ 0 j as the Krönecker symbol such that δ 0 0 = 1 and δ 0 j = 0 for j = 0. To prove Theorem 1, we first state the following Lemma.

Lemma 1. The functions f j (z), for j ∈ Z, satisfy

i z -1 τ f j (z) = z + 1 2 a 0 f j (z) + K k=1 a k ( f j+k (z) + f j-k (z)) + δ 0 j g 0 (z) . ( 10
)
Proof of Lemma 1. Let j ∈ Z. A simple calculation based on (9) leads to

a 0 f j + K k=1 a k ( f j+k + f j-k ) = 1 2π g 0 (z) 2π 0 ρ -1 K (z, x)A K (x)e -ijx dx = 2 (z + 1) g 0 (z) - 1 2π 2π 0 e -ijx dx + i z -1 τ g j = 2 (z + 1) g 0 (z) ï i z -1 τ g j (z) -δ 0 j ò ,
which provides [START_REF] Bhatti | Fractional Schrödinger wave equation and fractional uncertainty principle[END_REF].

Let us now prove the following result.

Lemma 2. Let us assume that the kernel functions {f n j } n∈N satisfy (9), for j ∈ Z, j = 0. We consider K given sequences {u n k } n∈N , such that u 0 k = 0, for 0 ≤ k ≤ K -1. If we define

ψ n j = K-1 k=0 (f j-k u k ) n , for j ≤ K -1, (11)
then, for any integer j ≤ -1, we have

iD τ ψ n j = EL δ,h ψ n j , (12)
with ψ 0 j = 0.

Proof of Lemma 2. From [START_REF] Bhrawy | A fully spectral collocation approximation for multidimensional fractional Schrödinger equations[END_REF], we obtain

ψ j (z) = K-1 k=0 f j-k (z) u k (z), for j ≤ K -1. ( 13
)
For j ≤ -1, we have jk < 0, with 0 ≤ k ≤ K -1, and δ 0 j = 0. Hence, from [START_REF] Bhatti | Fractional Schrödinger wave equation and fractional uncertainty principle[END_REF], the functions

ψ j (z) satisfy i z -1 τ ψj (z) = z + 1 2 K-1 k=0 a 0 fj-k (z)û k (z) + K-1 k=0 K p=1 a p ( fj-k-p (z) + fj-k+p (z))û k (z) = z + 1 2 a 0 ψj (z) + z + 1 2 K p=1 a p ( ψj-p (z) + ψj+p (z)),
which leads to [START_REF] Bhrawy | Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations[END_REF] after Z-inversion. Finally, we have ψ 0 j = K-1 k=0 f 0 j-k u 0 k = 0 from (13). Lemma 2 implies that the functions ψ n j , j ≤ -1, satisfy the governing fully discrete nonlocal Schrödinger equation on the left exterior domain. Consequently, if we can compute {u n k } n , for 0 ≤ k ≤ K -1, we obtain the ABC at the left boundary given by [START_REF] Bhrawy | A fully spectral collocation approximation for multidimensional fractional Schrödinger equations[END_REF] and leading to [START_REF] Guo | Some physical applications of fractional Schrödinger equation[END_REF].

Let us now admit the following Lemma. Its proof is available at the end of this section since it requires some preliminary analytical results. Lemma 3. For K given sequences {ψ n j } n , with ψ 0 j = 0 and 0 ≤ j ≤ K -1, we can find K sequences {u n k } n , with 0 ≤ k ≤ K -1, such that,

ψ n j = K-1 k=0 (f j-k u k ) n , 0 ≤ j ≤ K -1. ( 14
)
Lemma 3 shows that, being given the interior values {ψ n j } n , 0 ≤ j ≤ K -1, we can compute {u n j } n through [START_REF] Kirkpatrick | Fractional Schrödinger dynamics and decoherence[END_REF] and then obtain {ψ n j } n , j ≤ K -1, satisfying [START_REF] Bhrawy | A fully spectral collocation approximation for multidimensional fractional Schrödinger equations[END_REF] and the exterior left problem [START_REF] Bhrawy | Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations[END_REF], for j ≤ -1. Based on the two previous Lemmas, we prove Theorem 1.

Proof of Theorem 1. By using Lemma 3, we can find K sequences {u n k } n , 0 ≤ k ≤ K -1, such that [START_REF] Kirkpatrick | Fractional Schrödinger dynamics and decoherence[END_REF] holds for the boundary values {ψ n j } n , 0 ≤ j ≤ K -1. Now, for the same sequences {u n k } n , we can define {ψ n j } n based on [START_REF] Bhrawy | A fully spectral collocation approximation for multidimensional fractional Schrödinger equations[END_REF], for j ≤ -1. Therefore, {ψ n j } n satisfy (12), -K ≤ j ≤ -1, by Lemma 2. Combining [START_REF] Bhrawy | A fully spectral collocation approximation for multidimensional fractional Schrödinger equations[END_REF] and [START_REF] Kirkpatrick | Fractional Schrödinger dynamics and decoherence[END_REF] provides the left boundary condition. More precisely, we have

           ψ n j = K k=1 (f -j+k-1 u k-1 ) n , -K ≤ j ≤ -1, ψ n k-1 = K =1 (f k-u -1 ) n , 1 ≤ k ≤ K, (15)
since f n -j+k-1 = f n 1-k+j . Now, by applying the Z-transform to (15) and eliminating u(z), we directly obtain the equivalent expression

ψ n j = K k=1 (C -j,k ψ k-1 ) n , -K ≤ j ≤ -1, where A, B and C are K × K matrices such that C(z) = B(z) A -1 (z), (B n ) j,k = f n j+k-1 and (A n ) j,k = f n j-k , 1 ≤ j, k ≤ K.

Similar calculations on the right domain lead to (8).

Let us note that, from the definition of A and B, and since the index of

f n j+k-1 is such that 0 ≤ j + k -1 ≤ 2K -1, from f n -j = f n j and (B n ) j,k = f n j+k-1 , we only need to compute f n j for 0 ≤ j ≤ 2K -1 to determine C(z) = B(z) A -1 (z) in subsection 2.3. In addition, we remark that considering g n (z) instead of f n (z) is sufficient to compute C. Indeed, since B j,k (z) = g j+k-1 (z)/ g 0 (z) and A j,k (z) = g |j-k| (z)/ g 0 (z), for 1 ≤ j, k ≤ K, from the relation C = B A -1
, we can modify the two matrices to be B j,k (z) = g j+k-1 (z) and A j,k (z) = g |j-k| (z), with g j (z) given by (9).

Computation of the functions {g n

j } n , for 0 ≤ j ≤ 2K -1 Let us define {g n j } n as the inverse discrete Z-transform of g j (z). Since g n j = g n -j , from (9), we only need to compute g n j , for 0 ≤ j ≤ 2K -1, to derive the absorbing boundary conditions ( 7) and [START_REF] Antoine | On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions[END_REF]. From Lemma 1, the functions g j (z), for j ∈ Z,

satisfy i z -1 τ g j (z) = z + 1 2 a 0 g j (z) + K k=1 a k ( g j+k (z) + g j-k (z)) + δ 0 j , (16)
leading to iD τ g n j = EL δ,h g n j + δ 0 j , with g 0 j = 0. Let us now state the following Lemma.

Lemma 4. The sequences {g n j } n , for any integer j ∈ Z and n ≥ 2, are such that

g n j = g n j-2K + 1 Ka K K-1 k=1 ka k (g n j-K-k -g n j-K+k ) + (2n -3) (n -1)Ka K K k=1 ka k (g n-1 j-K-k -g n-1 j-K+k ) + (n -2) (n -1)Ka K K k=1 ka k (g n-2 j-K-k -g n-2 j-K+k ) - 4i(j -K) (n -1)Ka K 1 τ g n-1 j-K . ( 17
)
Proof of Lemma 4. For any integer j ∈ Z, we define ĥj (z) = (z + 1)ĝ j (z). Then, we have

d dz K k=1 ka k ( ĥj-k -ĥj+k ) = 1 2π d dz 2π 0 2i(1 + z)ρ -1 K (z, x)A K (x)e -ijx dx = - 4ij (z + 1)τ ĝj , which leads to K k=1 ka k (ĝ j-k (z) -ĝj+k (z)) + (z + 1) K k=1 ka k ( d dz ĝj-k (z) - d dz ĝj+k (z)) = - 4ij (z + 1)τ ĝj (z). ( 18
)
After multiplying ( 18) by (1 + z), we obtain

(z + 1) K k=1 ka k ∞ n=0 (g n j-k -g n j+k )z -n + (z + 1) 2 K k=1 ka k ∞ n=0 k(g n j+k -g n j-k )z -(n+1) = -4ij 1 τ ∞ n=0 g n j z -n .
After some manipulations, we deduce

g n+1 j+K = g n+1 j-K + 1 Ka K K-1 k=1 ka k (g n+1 j-k -g n+1 j+k ) + (2n -1) nKa K K k=1 ka k (g n j-k -g n j+k ) + (n -1) nKa K K k=1 ka k (g n-1 j-k -g n-1 j+k ) - 4ij nKa K 1 τ g n j .
Finally, we can rewrite the above expression for g n j as [START_REF] D'elia | Numerical methods for nonlocal and fractional models[END_REF]. We still need to compute g 1 j , j ∈ Z. To this end, we write

g 1 j = 1 2π 2π 0 e ijx i/τ -a 0 /2 -K k=1 a k (e ikx + e -ikx )/2 de ix ie ix = 1 2πi |z|=1 z K+j-1 q(z) dz, defining: q(z) = (i/τ -a 0 /2)z K -K k=1 a k (z K+k + z K-k )/2, which has 2K different roots r k , 1 ≤ k ≤ 2K. We suppose that K 1 zeroes are such that: |r k | < 1, 1 ≤ k ≤ K 1 ≤ K. Then, we have g 1 j = 1 2πi |z|=1 z K+j-1 q(z) dz = K 1 k=1 Res z K+j-1 q(z) z=r k = K 1 k=1 r K+j-1 k q (r k ) .
Relation ( 16) can be used to compute the 2K sequences {g n j } n , for 0 ≤ j ≤ 2K -1. To this end, the sequences {g n j } n , for 2K ≤ j ≤ 3K -1 and -K ≤ j ≤ -1, need to be determined to close [START_REF] Pinsker | Fractional quantum mechanics in polariton condensates with velocity-dependent mass[END_REF]. Furthermore, {g n j } n , for -K ≤ j ≤ -1, can be obtained by the property: g n -j = g n j , and the sequences {g n j } n , for 2K ≤ j ≤ 3K -1, can be computed based on [START_REF] D'elia | Numerical methods for nonlocal and fractional models[END_REF]. Therefore, we determine the discrete system for {g n j } n , with 0 ≤ j ≤ 2K -1, and obtain B = C A. Finally, we prove Lemma 3. Proof of Lemma 3. To solve [START_REF] Kirkpatrick | Fractional Schrödinger dynamics and decoherence[END_REF] with ψ 0 j = 0, 0 ≤ j ≤ K -1, we can equivalently consider

ψ j = K-1 k=0 f j-k (z) u k (z), 0 ≤ j ≤ K -1. ( 19
)
Since f j-k (z) = g j-k / g 0 , then one gets the new form of ( 19)

g 0 ψ j = K-1 k=0 g j-k (z) u k , 0 ≤ j ≤ K -1. ( 20
)
If we define the two vectors " ψ K = ( g 0 ψ 0 , g 0 ψ 1 , ... g 0 ψ K-1 ) T and u K = ( u 0 , u 1 , ... u K-1 ) T , then (20) can be rewritten under the matrix form: " ψ K = A u K , where A is the K ×K matrix with coefficients: A j,k (s) = g |j-k| (z), 0 ≤ j, k ≤ K -1, since g j-k = g k-j . In addition, we obtain

g j (z) = 1 2π 2π 0 ρ -1 K (z, x)e -ijx dx = 1 2π ∞ =0 1 (z) +1 2π 0 (i/τ + A K (x)/2) (i/τ -A K (x)/2) +1 e -ijx dx = 1 z 2π 0 1 2π(i/τ -A K (x)/2) e -ijx dx + O( τ z 2 ) = γ j z + O( τ z 2 ),
for large values of |z|. Moreover, we have γ j = -iτ δ 0 j + O(τ 2 ). Therefore, we deduce that A = -iτ z -1 (I + O(τ ) + O(z -1 )), where I is the K × K identity matrix. For z large enough and small τ , we have: " ψ K ≈ A u K , where A := -iτ z -1 I, and (20) has a unique sequence solution u K , ending hence the proof of Lemma 3.

3. Properties of the kernel {C n } n and stability analysis

Properties of the kernel {C n } n

Let us now prove some useful properties of {C n } n for the stability and error analysis. From now on, z is the complex conjugate of z ∈ C, and M T is the transpose of any complex-valued matrix M.

Proposition 1. For any sequence v = {v n } n of elements v n ∈ C K such that v 0 = 0, the following inequality holds, for N ≥ 1, N -1 n=0 (E(C v)) n T • S(Ev) n ≥ 0, ( 21
)
with v n = [v n 0 , v n 1 , ...v n K-1 ]
T and S is the K × K Hankel matrix based on {a k } k=1,...,K defined by (5) (on the first row), with zeroes under the anti-diagonal.

Proof of Proposition 1. Let us first prove [START_REF] Antoine | A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations[END_REF]. By Lemma 3, for any vector sequence v = {v n } n such that v 0 = 0, we can find some sequences

{u n k } n , 1 ≤ k ≤ K, such that: v n j = K-1 k=0 (f j-k u k ) n , 0 ≤ j ≤ K -1. Hence, one can directly define: v n j = K-1 k=0 (f j-k u k ) n , for j ≤ -1. Since a 0 = -2 K k=1 a k , then {v n j } n , j ≤ -1
, satisfy the fully discrete nonlocal Schrödinger equation ( 12)

iD τ v n j = a 0 Ev n j + K k=1 a k (Ev n j-k + Ev n j+k ) = K k=1 a k (Ev n j-k -Ev n j ) + K k=1 a k (Ev n j+k -Ev n j ).
Multiplying the above equality by Ev n j , and summing from j = -J to j = -1 yields

-1 j=-J i v n+1 j -v n j τ Ev n j = -1 j=-J K k=1 a k (Ev n j+k -Ev n j )(Ev n j -Ev n j+k ) + -1 j=-J K k=1 a k (Ev n j+k -Ev n j )Ev n j+k + -1 j=-J K k=1 a k (Ev n j-k -Ev n k )(Ev n j -Ev n j-k ) + -1 j=-J K k=1 a k (Ev n j-k -Ev n j )Ev n j-k .( 22
)
Computing ( 22)-( 22) leads to

-1 j=-J i |v n+1 j | 2 -|v n j | 2 τ = - K k=1 -1 j=-k a k Ev n j Ev n j+k + K k=1 -1 j=-k a k Ev n j Ev n j+k - K k=1 -J-1 j=-J-k a k Ev n j+k Ev n j + K k=1 -J-1 j=-J-k a k Ev n j+k Ev n j . ( 23
)
Taking J → +∞ in [START_REF] Zheng | Numerical solution of the nonlocal diffusion equation on the real line[END_REF], and since v n j → 0 when |j| → +∞, we obtain

0 j=-∞ i |v n+1 j | 2 -|v n j | 2 τ = - K k=1 -1 j=-k a k Ev n j Ev n j+k + K k=1 -1 j=-k a k Ev n j Ev n j+k . (24)
Let us define

u n = [u n 0 , u n 1 , • • • , u n K-1 ] T , v n = [v n 0 , v n 1 , • • • , v n K-1 ] T and w n = [v n -1 , v n -2 , • • • , v n -K ] T . By Lemma 3 we have v n = (A u) n and w n = (B u) n , implying that: w n = (C v) n . Summing up (24) from n = 0 to N -1, one finally gets (21) since i -1 j=-∞ |v N j | 2 τ = 2i ( N -1 n=0 (E(C v)) n T • S(Ev) n ). ( 25
)

Stability analysis

For ψ n := (ψ n j ) j=0,...,J , n ∈ N, we introduce the norm ||ψ n || 2

J := h 1/2 ( J j=0 |ψ n j | 2 ) 1/2
. We now prove the 2 J -stability of the fully-discrete scheme with ABC.

Theorem 2. Let N ∈ N. The solution of the fully discrete scheme [START_REF] Secchi | Ground state solutions for nonlinear fractional Schrödinger equations in R N[END_REF] with ABCs ( 7)-( 8) satisfies the stability inequality

||ψ N || 2 J ≤ ||ψ 0 || 2 J . ( 26
)
Proof of Theorem 2. Repeating the same procedure as in the proof of Proposition 1 by multiplying (6) by Eψ n j and summing up all the terms from j = 0 to j = J, we obtain

J j=0 i ψ n+1 j -ψ n j τ Eψ n j = J j=0 K k=1 a k (Eψ n j+k -Eψ n j )(Eψ n j -Eψ n j+k ) + J j=0 K k=1 a k (Eψ n j+k -Eψ n j )Eψ n j+k + J j=0 K k=1 a k (Eψ n j-k -Eψ n j )(Eψ n j -Eψ n j-k ) + J j=0 K k=1 a k (Eψ n j-k -Eψ n j )Eψ n j-k .( 27
)
Let us define (v 27)-( 27) yields ( 26)

+ [ψ]) n = [ψ n J , ψ n J-1 , • • • , ψ n J-K+1 ] T , (w + [ψ]) n = [ψ n J+1 , ψ n J+2 , • • • , ψ n J+K ] T and (v -[ψ]) n = [ψ n 0 , ψ n 1 , • • • , ψ n K-1 ] T , (w -[ψ]) n = [ψ n -1 , ψ n -2 , • • • , ψ n -K ] T . The boundary conditions read (w + [ψ]) n = (C v + [ψ]) n , (w -[ψ]) n = (C v -[ψ]) n . Computing (
J j=0 i |ψ N j | 2 -|ψ 0 j | 2 τ = -2i N -1 n=0 (E(C v -[ψ])) n T • SEv -[ψ] n + N -1 n=0 (E(C v + [ψ])) n T • SEv + [ψ] n .

Truncated and boundary errors analysis

Let us introduce the following norms

||f || L ∞ = max x∈R |f (x)|, ||f || L 2 = ( R |f (x)| 2 dx) 1/2 , ||f || L 1 δ = δ -δ |f (x)|dx, ||f || L ∞ x,δ = max y∈[x-δ;x+δ] |f (y)|,
for δ > 0. We recall that γ δ is assumed to be homogeneous in the exterior domain according to assumption A3 (see [START_REF] Laskin | Fractals and quantum mechanics[END_REF]). In the computational domain, it may be a priori inhomogeneous. However, to simplify the presentation of the proofs, we suppose from now on that γ δ is also homogeneous inside the computational domain, has a singularity behaviour γ δ (a) ∼ |a| -α , α ≤ 1/2, and that

||γ δ || L 1 δ < +∞.
Nevertheless, if γ δ is non homogeneous in the domain of computation, the proofs can be adapted for similar singularity types but at the price of more complexity.

Error estimates: main result

For u n := (u n j ) j∈Z , n = 0, • • • , N , and

U N = (u n ) 0≤n≤N -1 , with N ≥ 1, we define the norms ||u n || ∞ := max j∈Z |u n j |, ||u n || 2 := h 1/2 ( j∈Z |u n j | 2 ) 1/2 , ||u n || 2 K := h 1/2 ( K k=1 |u n k | 2 ) 1/2 , ||U N || ∞,N := max 0≤n≤N -1 ||u n || ∞ ||U N || 2,N := τ 1/2 ( N -1 n=0 ||u n || 2 2 ) 1/2 .
For ψ solution to (1), we set ψ j (t) := ψ(x j , t), for j ∈ Z. The error e n j for ( 6) is then

e n j = i 2 τ (ψ j (t n+1 ) -ψ j (t n )) -L δ,h (ψ j (t n+1 ) + ψ j (t n )),
where 0 ≤ n ≤ N -1 and j ∈ Z. We set e n := (e n j ) j∈Z , n = 0, • • • , N , and E N = (e n ) 0≤n≤N -1 . We define the backward derivative operator ∇ -e n j := (e n j -e n j-1 )/h, and ∇ -E N := (∇ -e n ) 0≤n≤N -1 , for e n := (e n j ) j∈Z . Let us introduce the time-dependent K components vectors

V -ψ = ((ψ j ) j=0,••• ,K-1 ) T , W -ψ = ((ψ j ) j=-1,••• ,-K ) T , V + ψ = ((ψ J-j ) j=0,...,K-1 ) T and W + ψ = ((ψ j ) j=J+1,••• ,J+K ) T .
We also need to define the boundary operators

G ± = C V ± -W ± such that: C V ± (t) = ∞ m=0 C m V ± (t- t m ), setting V ± (t) = 0 for t < 0.
The main result of this section concerns the following Theorem which shows that both the truncated and boundary errors of the interior scheme are of order O(h 2 + τ 2 ). Theorem 3. For N ≥ 1, we have the following error bounds

||E N || ∞,N ≤ C(τ 2 + h 2 ), ( 28
) ||E N || 2,N ≤ C(τ 2 + h 2 ), ( 29
) ||∇ -E N || 2,N ≤ C(τ 2 + h 2 ), ( 30
) ||G n ± || 2 K ≤ C(τ 2 + h 2 ), 0 ≤ n ≤ N, (31)
where C > 0 are some constants that only depend on γ δ , δ, T and the initial data ψ 0 .

Estimates for the solution ψ of system (1)

We first derive some useful estimates for the solution ψ of the continuous problem (1).

Lemma 5. Let us assume that ∂ x ψ 0 is in L 2 , for a positive integer ≥ 0. Then, we have:

||∂ x ψ|| L 2 = ||∂ x ψ 0 || L 2 . In addition, if ∂ +1
x ψ 0 is in L ∞ , then the following bound holds

||∂ x ψ|| L ∞ ≤ C(||∂ +1 x ψ 0 || L 2 + ||∂ x ψ 0 || L 2 ),
where C > 0 is a constant.

Proof of Lemma 5. Let us consider the governing equation

i∂ t ψ(x, t) = +δ -δ γ δ (y)(ψ(x, t) -ψ(x -y, t))dy. ( 32
)
Summing up [START_REF] Antoine | Derivation and analysis of computational methods for fractional Laplacian equations with absorbing layers[END_REF] multiplied by ψ and the conjugate of (32) multiplied by -ψ, and integrating the equality from -L to L(> 0), we obtain

i∂ t L -L |ψ| 2 dx = δ -δ γ δ (y) - L+y L + -L+y -L ψ(x, t)ψ(x -y, t)dxdy. Since |ψ(x, t)| → 0 as |x| → ∞, taking L → +∞ in the above expression leads to ∂ t ||ψ|| 2 L 2 = 0, providing ||ψ|| 2 L 2 = ||ψ 0 || 2 L 2 . It is easy to see that ||∂ x ψ|| 2 L 2 = ||∂ x ψ 0 || 2 L 2 also holds.
In addition, we have the following inequality which proves the result for = 0 [START_REF] Antoine | Derivation and analysis of computational methods for fractional Laplacian equations with absorbing layers[END_REF]. A consequence of Lemma 5 is given in the next result. Lemma 6. Let k ≥ 1 and ∈ N. For ψ 0 and ψ sufficiently smooth, and for ψ 0 compactly supported in ]x -; x + [, we have, for x ∈ R and t > 0,

||ψ|| L ∞ ≤ ∞ -∞ (1 + |ξ|) -1 (1 + |ξ|)| ψ(ξ)|dξ ≤ C(||ψ|| L 2 + ||∂ x ψ|| L 2 ). Similarly, we obtain ||∂ x ψ|| 2 L 2 = ||∂ x ψ 0 || 2 L 2 by considering ∂ t ∂ x ψ instead of ∂ t ψ in
|∂ k t ∂ x ψ(x, t)| ≤ C2 k ||γ δ || k L 1 δ (||∂ +1 x ψ 0 || 2 L 2 + ||∂ x ψ 0 || 2 L 2 ). ( 33
)
Proof of Lemma 6. From (32), we have:

|∂ t ψ(x, t)| ≤ ||γ δ || L 1 δ ||ψ|| L ∞ ,

and by iteration

|∂ k t ψ(x, t)| ≤ 2||γ δ || L 1 δ ||∂ k-1 t ψ|| L ∞ ≤ 2 k ||γ δ || k L 1 δ ||ψ|| L ∞ ≤ C2 k ||γ δ || k L 1 δ (||∂ x ψ 0 || 2 L 2 + ||ψ 0 || 2 L 2 ).
Similarly, we prove (33)

|∂ k t ∂ x ψ(x, t)| ≤ 2||γ δ || L 1 δ ||∂ k-1 t ∂ x ψ|| L ∞ ≤ 2 k ||γ δ || k L 1 δ ||∂ x ψ|| L ∞ ≤ C2 k ||γ δ || k L 1 δ (||∂ +1 x ψ 0 || 2 L 2 + ||∂ x ψ 0 || 2 L 2 ).

Boundary error estimates

Let us introduce j such that

j (t) = a 0 ψ j (t) + K k=1 a k (ψ j+k (t) + ψ j-k (t)) -L δ ψ j (t), (34)
for j ∈ Z, where ψ designates the solution to [START_REF] Laskin | Fractional quantum mechanics and Lévy path integrals[END_REF]. We rewrite j as,

j = K-1 k=1 ψ j-k -2ψ j + ψ j+k kh (k+1)h (k-1)h φ k (ς)ςγ δ (ς)dς + ψ j-K -2ψ j + ψ j+K Kh Kh (K-1)h φ K (ς)ςγ δ (ς)dς - δ 0 (ψ(x j -ς, t) -2ψ(x j , t) + ψ(x j + ς, t))γ δ (ς)dς, (35)
where we recall that φ k is the hat function of width h centered at kh. Then, the following technical Lemma holds (see Appendix A for the proof).

Lemma 7. Let us assume that ψ(x, t) ∈ C 5 ([-∞, ∞] × [0, T ]). Then, we have: j (t) = h 2 Q(x j , t), for j ∈ Z and t > 0, with

h j∈Z |Q(x j , t)| 2 ≤ C R 5 p=2 |∂ p x ψ(y, 0)| 2 dy, (36) setting
Q(x, t) = h K-1 k=1 1 -1 φ 0 (ςh)ς 2 dς 1 0 qk (kh + ςyh, x, t)(1 -y)dy + 0 -1 φ 0 (ςh)ςdς 1 0 qK (Kh + ςyh, x, t)dy -h 1 0 ς 2 φ 0 (ςh)γ δ (ςh)dς 1 0 ξ 1 0 (∂ 2 x ψ(x + ςξ 2 h, t) + ∂ 2 x ψ(x -ςξ 2 h, t))dξ 2 dξ 1 . (37)
In the above relation, we defined

q k (ς, x, t) = ςγ δ (ς) 1 0 khξ 1 ςξ 1 (∂ 2 x ψ(x + ξ 2 , t) + ∂ 2 x ψ(x -ξ 2 , t))dξ 2 dξ 1 , (38)
and q(ς, x, t) designates the second-order derivative of q p (ς, x, t) with respect to ς.

Let us now prove Theorem 3. Proof of Theorem 3. Let us start with ( 28) and [START_REF] Baeumer | Boundary conditions for fractional diffusion[END_REF]. Using Taylor's expansions of ψ(x, t n ) and ψ(x, t n+1 ) and after some computations, we deduce

e n (x) := i 2 τ (ψ(x, t n+1 ) -ψ(x, t n )) -L δ,h (ψ(x, t n+1 ) + ψ(x, t n )) = -2h 2 Q(x, t n+1/2 ) - τ 2 h 2 12 ∂ 2 t Q(x, t n+1/2 ) -L δ ∂ 2 t ψ(x, t n+1/2 ) + h 2 ∂ 2 t Q(x, t n+1/2 ) τ 2 /6 + i 12τ τ /2 0 (∂ 5 t ψ(x, t n+1/2 + ς) -∂ 5 t ψ(x, t n+1/2 -ς))(τ /2 -ς) 4 dς - 1 6 τ /2 0 (L δ,h ∂ 4 t ψ(x, t n+1/2 + ς) + L δ,h ∂ 4 t ψ(x, t n+1/2 -ς) (τ /2 -ς) 3 dς. By using: L δ,h ∂ k t ψ(x j , t) = L δ ∂ k t ψ(x j , t) + h 2 ∂ k t Q(x j , t), and since τ /2 0 (L δ,h ∂ 4 t ψ(x, t n+1/2 + ς) + L δ,h ∂ 4 t ψ(x, t n+1/2 -ς))(τ /2 -ς) 3 dς 2 ≤ Cτ 7 τ /2 -τ /2 L δ,h ∂ 4 t ψ(x, t n+1/2 + ς) 2 dς,
one gets

|e n j | 2 = |e n (x j )| 2 ≤ Ch 4 |Q(x j , t n+1/2 )| 2 + τ 4 h 4 |∂ 2 t Q(x j , x n+1/2 )| 2 + Cτ 4 |L δ ∂ 2 t ψ(x j , t n+1/2 )| 2 +Cτ 7 τ /2 -τ /2 |L δ ∂ 4 t ψ(x j , t n+1/2 + ς)| 2 + h 4 |∂ 4 t Q(x j , t n+1/2 + ς)| 2 + |∂ 5 t ψ(x j , t n+1/2 + ς)| 2 dς.
From Lemmas 5-7 and the above equality, we deduce (28) since from (A.10) (see page 23) we have

|∂ t Q(x, t)| ≤ C ||∂ 2 y ∂ t ψ(y, t)|| L ∞ x,δ + ||∂ 3 y ∂ t ψ(y, t)|| L ∞ x,δ , = 0, 2, 4,
with L δ ∂ 2 t ψ(x, t) = i∂ 3 t ψ(x, t) and L δ ∂ 4 t ψ(x, t) = i∂ 5 t ψ(x, t). To prove (29), we remark for example that, for N τ ≤ T , we have

τ N -1 n=0 |Q(x, t n+1/2 )| 2 ≤ C T 0 |Q(x, t)| 2 dt.
All the other terms can be treated similarly. This yields

N -1 n=0 -1 j=-∞ |e n j | 2 hτ ≤ C(h 4 + τ 4 ) T 0 R 0≤p, ≤5 |∂ p x ∂ t ψ(x, t)| 2 dxdt ≤ C(h 4 + τ 4 ),
which provides [START_REF] Baeumer | Boundary conditions for fractional diffusion[END_REF]. We obtain [START_REF] Kelly | Boundary conditions for two-sided fractional diffusion[END_REF] similarly.

Let us now prove [START_REF] Antoine | Towards Perfectly Matched Layers for time-dependent space fractional PDEs[END_REF]. For the sequence {ψ n j } n = {ψ j (t n )} n based on the solution of (1), with ψ 0 j = 0, for 0 ≤ j ≤ K -1, from Lemma 2 we can find K sequences {u n k } n , with 0 ≤ k ≤ K -1, such that

ψ n j = K-1 k=0 (f j-k u k ) n , 0 ≤ j ≤ K -1. ( 39
)
Then, the values ψ n j , for j ≤ -1, can be defined by

ψ n j = K k=1 (f j-k u k ) n , j ≤ -1. ( 40
)
In this case, the solution satisfies: i(ψ n+1 j ψ n j )/τ = L δ,h Eψ n j , j ≤ -1, with |ψ n j | → 0 as j → -∞. Now, let us consider w n j = ψ n jψ(x j , t n ). For j ≤ -1, we have

iD τ w n j = a 0 Ew n j + K k=1 a k (Ew n j-k + Ew n j+k ) - 1 2 e n j .
Multiplying the above equality by Ew n j , and summing up all the term from j = -J to -1 we get

-1 j=-J i w n+1 j -w n j τ Ew n j = -1 j=-J K k=1 a k (Ew n j+k -Ew n j )Ew n j + K k=1 a k (Ew n j-k -Ew n j )Ew n j - 1 2 -1 j=-J e n j Ew n j . ( 41
)
Repeating the same procedures, we compute (41)-( 41). Taking J → ∞, since ψ n j → 0 as |j| → ∞ we have

0 j=-∞ i |w n+1 j | 2 -|w n j | 2 τ = 2i N -1 n=0 (E(w -[w])) n T • S(E(v -[w])) n - 1 2 -1 j=-∞ e n j Ew n j + 1 2 -1 j=-∞ e n j Ew n j . ( 42
)
From the choice of {ψ n j } n for 0 ≤ j ≤ K -1, it is clear that:

w n j = ψ n j -ψ(x j , t n ) = 0 for 0 ≤ j ≤ K -1, and hence v -[w] = 0. Summing up (42) from n = 0 to N -1, we obtain -1 j=-∞ |w N j | 2 τ ≤ C -1 j=-∞ N -1 n=0 |e n j | 2 + -1 j=-∞ N -1 n=0 |w n j | 2 .
Then, using ( 29) and Gronwall's inequality [START_REF] Chandra | On a generalization of the Gronwall-Bellman lemma in partially ordered Banach spaces[END_REF] leads to

h -1 j=-∞ |w N j | 2 ≤ Cτ h -1 j=-∞ N -1 n=0 |e n j | 2 ≤ (h 4 + τ 4 )C(ψ(x, 0), γ, δ, T ).
From ( 39) and (40), we have (w

-[ψ]) n = (C v -[ψ]) n = (C V -[ψ]) n . Therefore, we deduce: [w n -1 , w n -2 , ...w n -K ] T = (w -[ψ]) n -W -(t) = (C V -[ψ]) n -W -(t) = G -(t).

Error estimates of the scheme

Let us recall that w n j = ψ n j -ψ(x j , t n ), where ψ(x j , t n ) is the solution of the nonlocal Schrödinger equation (1) evaluated at (x j , t n ), and ψ n j solves the fully discrete version [START_REF] Secchi | Ground state solutions for nonlinear fractional Schrödinger equations in R N[END_REF]. In addition, let us define the norm:

||w n || ∞ J := max 0≤j≤J |w n j |.
We set the discrete forward derivatives as:

∇ + ψ(x j , t n ) = (ψ(x j+1 , t n ) -ψ(x j , t n ))/h and ∇ + ψ n j = (ψ n j+1 -ψ n j )/h.
Let us give the main result concerning the error estimate. Theorem 4. For τ > 0 and N τ ≤ T , we have the error bounds

w N 2 J ≤ Cτ 2 , ( 43
)
∇ + w N 2 J-1 ≤ Cτ 2 , ( 44
)
||w N || ∞ J ≤ Cτ 2 , ( 45
)
for some constants C > that only depend on γ δ , δ, T and ψ 0 .

It is straightforward to check that the error w n j satisfies for n ≥ 0

i 1 τ (w n+1 j -w n j ) = 1 2 L δ,h (w n+1 j + w n j ) - 1 2 e n j , (C v ± [w]) n -(w ± [w]) n = G ± (t n ),
with w 0 n = 0, where e n j (respectively G ± (t n )) is the interior (respectively boundary truncation) error according to the time and space discretizations defined in Theorem 3.

The proof of Theorem 4 is a consequence of Theorem 3 and the following Lemma.

Lemma 8. Let us assume that the singular behavior of

γ δ around 0 is γ δ (a) ∼ |a| -α , α ≤ 1/2.
Then, the coefficients a k in L δ,h given by ( 5)-( 6) satisfy the following inequality:

|a k | ≤ Ch 1-α , for k = 1, ..., K.
Proof of Lemma 8. Since a k = a -k , we only need to estimate a k for k ≥ 0. For k = 1, one gets

|a k | = 2h 0 φ k (ς)ςγ δ (ς) kh dς ≤ C 2h 0 ς 1-α h dς ≤ Ch 1-α ,
for a constant C > 0. In addition, for k ≥ 2, we obtain

|a k | = (k+1)h (k-1)h φ k (ς)ςγ δ (ς) kh dς ≤ C (k+1)h (k-1)h ς 1-α kh dς ≤ C h α (k+1)h (k-1)h ς kh dς ≤ Ch 1-α .
We can now prove Theorem 4. Proof of Theorem 4. Let us start with (43). For a time step τ , we have

i 1 τ (w n+1 j -w n j ) = 1 2 L δ,h (w n+1 j + w n j ) - 1 2 e n j .
By a procedure similar to the one given in Proposition 1, we obtain

i τ J j=0 (|w N j | 2 -|w 0 j | 2 ) = -2i N -1 n=0 (E(w -[w])) n T • S(E(v -[w])) n (46) + N -1 n=0 (E(w + [w])) n T • S(E(v + [w])) n + N -1 n=0 J j=0 1 2 -e n j Ew n j + e n j Ee n j ,
which leads to

J j=0 h(|w N j | 2 -|w 0 j | 2 ) ≤ -2τ h N -1 n=0 (E(C v + [w])) n T • S(E(v + [w])) n - N -1 n=0 E(G + (t n )) T • S(E(v + [w])) n -2τ h N -1 n=0 (E(C v -[w])) n T • S(E(v -[e])) n - N -1 n=0 E(G -(t n )) T • S • (E(v -[e])) n + τ h 2 N -1 n=0 J j=0 |e n j ||w n j | + |e n j ||w n+1 j | . (47)
In addition, from a k = O(h 1-α ), for 1 ≤ k ≤ K, and the expression of S, we derive

S(E(v ± [w])) n k ≤ Ch 1-α K q=1 |w n J+1-q | + K q=1 |w n+1 J+1-q | . (48)
Thus, for α ≤ 1/2, from (48) and ( 31), (47) can be written as

-2τ h N -1 n=0 (E(C v ± [w])) n T • S(E(v ± [w])) n - N -1 n=0 E(G ± (t n )) T • S(E(v ± [w])) n ≤ C(h 2 + τ 2 ) 2 + Cτ N n=0 ||w n || 2 2 J . (49)
Finally, ( 29), ( 46) and (49) lead to

||w N || 2 2 J -||w 0 || 2 2 J ≤ C(τ 2 + h 2 ) 2 + Cτ N n=0 ||w n || 2 2 J
, which proves (43):

w N 2 J ≤ Ce CN τ (τ 2 + h 2 ) ≤ Ce CT (τ 2 + h 2 ).
Let us now focus on (44). For {ψ n j } n solution to [START_REF] Secchi | Ground state solutions for nonlinear fractional Schrödinger equations in R N[END_REF], with ψ 0 j = 0 for 0

≤ j ≤ K -1, from Lemma 3 we can find K sequences {u n k } n , with 0 ≤ k ≤ K -1, such that: ψ n j = K-1 k=0 (f j-k u k ) n , 0 ≤ n ≤ K -1.
Then, ψ n j , for j ≤ -1, can be defined by:

ψ n j = K k=1 (f j-k u k ) n . In this case, the solution satisfies: iτ -1 (ψ n+1 j -ψ n j ) = L δ,h Eψ n j , j ≤ 0, and |ψ n j | → 0 as j → -∞. This leads to i ∇ + ψ n+1 j -∇ + ψ n j τ = L δ,h E∇ + ψ n j , j ≤ -1.
Introducing ϕ n j = ∇ + ψ n j , for 0 ≤ j ≤ K -1, from Lemma 2 we can find K sequences {v n k } n , with 1 ≤ k ≤ K, such that:

ϕ n j = K-1 k=0 (f j-k v k ) n , 0 ≤ j ≤ K -1.
The terms ϕ n j , for j ≤ -1, can be defined by: ϕ

n j = K k=1 (f j-k v k ) n , j ≤ -1. In this case, we have i ϕ n+1 j -ϕ n j τ = L δ,h Eϕ n j , j ≤ -1,
and

|ϕ n j | → 0 as j → -∞. By defining ω n j = ∇ + ψ n j -ϕ n j , we obtain i ω n+1 j -ω n j τ = L δ,h Eω n j , j ≤ -1,
and (v -[ω]) n = [∇ + ψ n k -ϕ n k ] T k=0,...,K-1 = 0.
Repeating the procedure as for ( 25), we deduce

-1 j=-∞ |ω N j | 2 τ = 2 N -1 n=0 (E(w -[ω])) n T • S(Ev -[ω]) n = 0.
Then, we have

∇ + ψ n j -ϕ n j = ω n j = 0 for j ≤ -1. By defining (v -[∇ + ψ]) n = [∇ + ψ n k ] T k=0,...,K-1 and (w -[∇ + ψ]) n = [∇ + ψ n -k ] T k=1,...K , we obtain: (w -[ω]) n = (w -[∇ + ψ]) n -(v -[ϕ]) n = 0. From (w -[ϕ]) n = (C v -[ϕ]) n , one gets (w -[∇ + ψ]) n = (w -[ϕ]) n = (C v -[ϕ]) n = (C v -[∇ + ψ]) n . (50)
Similarly, we derive: (w

+ [∇ -ψ]) n = (C v + [∇ -ψ]) n , with (v + [∇ -ψ]) n = [∇ -ψ n L-k ] T k=0,...,K-1 and (w + [∇ -ψ]) n = [∇ -ψ n
L+k+1 ] T k=0,...,K-1 . From (34) and Lemma 7, we have

∇ + j (t) = a 0 ∇ + ψ j (t) + K k=1 a k (∇ + ψ j+k (t) + ∇ + ψ j-k (t)) -L δ ∇ + ψ j (t) = h 2 ∇ + Q(x j , t),
for ∇ + u j (t) = (u j+1 (t)u j (t))/h. From Lemma 7, we deduce

h j∈Z |∇ + Q(x j , t)| 2 ≤ C R 6 p=3 |∂ p y ψ(y, 0)| 2 dy.
Therefore, we obtain

N -1 n=0 -1 j=-∞ |∇ + e n j | 2 hτ ≤ C(h 4 + τ 4 ) T 0 R 0≤p≤6,0≤k≤5 |∂ p x ∂ k t ψ(x, t)| 2 dxdt ≤ C(h 4 + τ 4 ).
For the sequence {ψ j (t n )} n solution of (1), with ψ 0 j = 0, for 0

≤ j ≤ K -1, from Lemma 2 we can find K sequences {u n k } n , with 0 ≤ k ≤ K -1, such that χ n j = ∇ + ψ j (t n ) = K-1 k=0 (f j-k u k ) n , 0 ≤ j ≤ K -1. (51)
Then, the values χ n j , for j ≤ -1, can be defined by

χ n j = K k=1 (f j-k u k ) n , j ≤ -1. (52)
In this case, the solution satisfies

i χ n+1 j -χ n j τ = L δ,h Eχ n j , j ≤ -1,
with |χ n j | → 0 as j → -∞. Now, let us consider d n j = χ n j -∇ + ψ(x j , t n ). For j ≤ -1, we have

iD τ d n j = a 0 Ed n j + K k=1 a k (Ed n j-k + Ed n j+k ) - 1 2 ∇ + e n j .
Multiplying the above equality by Ed n j , and summing from j = -J to -1 we get

-1 j=-J i d n+1 j -d n j τ Ed n j = -1 j=-J K k=1 a k (Ed n j+k -Ed n j )Ed n j + K k=1 a k (Ed n j-k -Ed n j )Ed n j - 1 2 -1 j=-J ∇ + e n j Ed n j . (53)
Computing ( 53)-(53), taking J → ∞ in (53), since ϕ n j → 0 when |j| → ∞ we have

0 j=-∞ i |d n+1 j | 2 -|d n j | 2 τ = 2i N -1 n=0 (E(w -[d])) n T • S(E(v -[d])) n - 1 2 -1 j=-∞ ∇ + e n j Ed n j + 1 2 -1 j=-∞ ∇ + e n j Ed n j . (54)
From the choice of {χ n j } n for 0 ≤ j ≤ K -1, we have: d n j = χ n j -∇ + ψ(x j , t n ) = 0 for 0 ≤ j ≤ K -1, and then v -[d] = 0. Next summing up (54) from n = 0 to N -1 provides

-1 j=-∞ |d N j | 2 τ = N -1 n=0 0 j=-∞ |d n+1 j | 2 -|d n j | 2 τ = 1 2i - N -1 n=0 -1 j=-∞ ∇ + e n j Ed n j + N -1 n=0 -1 j=-∞ ∇ + e n j Ed n j ≤ C -1 j=-∞ N -1 n=0 |∇ + e n j | 2 + -1 j=-∞ N -1 n=0 |d n j | 2 .
Then, combining (51) and a Gronwall inequality [START_REF] Chandra | On a generalization of the Gronwall-Bellman lemma in partially ordered Banach spaces[END_REF] yields

h -1 j=-∞ |d N j | 2 ≤ Cτ h -1 j=-∞ N -1 n=0 |∇ + e n j | 2 ≤ (h 4 + τ 4 )C(ψ 0 , γ δ , δ, T ).
From ( 51) and ( 52), we have (w

-[χ]) n = (C v -[χ]) n = (C V -[∇ + ψ j (t n )]) n . We deduce [d n -1 , d n -2 , ...d n -K ] T = [χ n -1 -∇ + ψ -1 (t n ), χ n -2 -∇ + ψ -2 (t n ), ...χ n -K -∇ + ψ -K (t n )] T = (w -[χ]) n -∇ + W -(t n ) = (C V -[∇ + ψ j (t n )]) n -∇ + W -(t n ) = ∇ + G -(t n ),
and similarly to the previous calculations, by (51),

||∇ + e n || 2 J-1 ≤ C(τ 2 + h 2 ), ||∇ ∓ G n ± || 2 K ≤ C(τ 2 + h 2 ), 0 ≤ n ≤ N.
From w n j = ψ n jψ(x j , t n ), we have

i 1 τ (∇ + w n+1 j -∇ + w n j ) = L δ,h (∇ + w n+1 j + ∇ + w n j ) - 1 2 ∇ + e n j .
By (w

∓ [∇ ± ψ]) n = (C v ∓ [∇ ± ψ]) n in (50), we have (w -[∇ + w]) n = (w -[∇ + ψ]) n -(w -[∇ + ψ(t n )]) = (C v -[∇ + ψ]) n -(w -[∇ + ψ(t n )]) which leads to: (w -[∇ + w]) n -(C v -[∇ + w]) n = (C v -[∇ + ψ(t n )])- (w -[∇ + ψ(t n )]) = ∇ + G -(t n ).
Thus repeating a procedure similar as in Proposition 1, we obtain i τ

J-1 j=0 (|∇ + w N j | 2 -|∇ + w 0 j | 2 ) = -2i N -1 n=0 (E(w -[∇ + w])) n T • S(E(v -[∇ + w])) n + N -1 n=0 (E(w + [∇ -w])) n T • S(E(v + [∇ -w])) n + N -1 n=0 J-1 j=0 1 2 -∇ + e n j E∇ + w n j + ∇ + e n j E∇ + w n j ,
which leads to

J-1 j=0 h(|∇ + w N j | 2 -|∇ + w 0 j | 2 ) = -2τ h N -1 n=0 (E(C v + [∇ -w])) n T • S(E(v + [∇ -w])) n - N -1 n=0 E(G + (t n )) T • S(E(v + [∇ -w])) n -2τ h N -1 n=0 (E(C v -[∇ + w])) n T • S(E(v -[∇ + w])) n - N -1 n=0 E(G -(t n )) T • S • (E(v -[∇ + w])) n + τ h 2 N -1 n=0 J-1 j=0 |∇ + e n j ||∇ + w n j | + |∇ + e n j ||∇ + w n+1 j | .
We have

||∇ + w N || 2 2 J-1 -||∇ + w 0 || 2 2 J-1 ≤ C(τ 2 + h 2 ) 2 + Cτ N n=0 ||∇ + w n || 2 2 J-1 + τ h 2 N -1 n=0 J-1 j=0 |∇ + e n j ||∇ + w n j | + |∇ + e n j ||∇ + w n+1 j | ≤ C(τ 2 + h 2 ) 2 + Cτ N n=0 ||∇ + w n || 2 2 J-1
,

which implies: ||∇ + w N || 2 J-1
≤ Ce CT (τ 2 + h 2 ), and proves (44). Finally, by the discrete Sobolev imbedding theorem, we obtain (45). ∞,T at time T = 2 in Table 1. This example shows that the scheme is second-order in time as expected. Let us now fix the time step to τ = 1 × 10 -3 . The reference solution ψ n,ref is obtained for h = 2.5 × 10 -3 and τ = 1 × 10 -3 . We give in Table 2 the error e h,τ ∞,T at T = 2 with respect to the mesh refinement h where we observe that the convergence rate is equal to 2. . We observe that the reflection at the boundary related to the ABC is small and has an amplitude similar to the numerical error of the interior scheme.

Example 2

We consider now the singular kernel γ(a, b) = 3 for various values of δ. The reference solution ψ n,ref is evaluated for the parameters h = 5 × 10 -2 and τ = 1 × 10 -4 . We again observe that the scheme is second-order accurate in time. Let us now fix the time step to τ = 1 × 10 -3 . The reference solution ψ n,ref is obtained for h = 2.5 × 10 -3 and τ = 1 × 10 -3 . We report in Table 4 the error e h,τ ∞,T at T = 2 with respect to the mesh refinement h. For the various values of δ, we observe that the convergence rate is equal to 2 according to h in Table 4. For the same parameters values, we finally plot in Figure 3 (left) the evolution of the solution and the error (right) between the numerical and the reference solutions which again attest of the accuracy of both the scheme and discrete ABC.

e h,τ ∞,T τ = 1 × 10 -2 τ = 5 × 10 -3 τ = 4 × 10 -3 τ = 2.5 × 10 -3 rate δ = 0.4 6

Conclusion

We provided the construction of fully discrete ABCs for the one-dimensional nonlocal Schrödinger equation discretized by the Crank-Nicolson scheme in time and an asymptotically compatible scheme in space. An algorithm is given for evaluating these ABCs. In addition, the stability and error analysis of the scheme are developed for a singular kernel. Numerical examples confirm the accuracy of the ABCs. Further works include the extension to the singular cases for α ≥ 1/2, higher dimensional problems as well as considering nonlinearities into the equation.

Appendix A. Proof of Lemma 7

From the identity u(xς) -2u(x) + u(x + ς) = ς 2 1 0 ξ 1 0 (u (x + ςξ 2 ) + u (xςξ 2 ))dξ 2 dξ 1 , the third term on the right hand side of (35) can be written as δ 0 (ψ(x jς, t) -2ψ(x j , t) + ψ(x j + ς, t))γ δ (ς)dς = h 3 1 0 ς 2 φ 0 (ςh)γ δ (ςh)dς In the same way, the two first terms on the right hand side can be recast as

K-1 k=1
ψ j-k -2ψ j + ψ j+k kh

x k+1

x k-1 φ k (ς)ςγ δ (ς)dς + ψ j-K -2ψ j + ψ j+K Kh where q k (ς, x, t) is defined by (38). It is easy to see that q k (ς, x, t) = q k (kh, x, t) + qk (kh, x, t)(ςkh) + ς kh qk (y, x, t)(ςy)dy, q k (ς, x, t) = q k (kh, x, t) + ς kh qk (y, x, t)dy,

(A.4)
where we use the dots to represent the derivative of q k (ς, x, t) with respect to the first variable ς. By using (A.4) and q k (kh, x, t) = 0, we deduce after some calculations Therefore from (A.3) and (A.5), we obtain: j (t) = h 2 Q(x j , t), with Q(x, t) defined by (37). Now, the last term of (37) satisfies |h 1 0 ς 2 φ 0 (ςh)γ δ (ςh)dς

  γ δ (a, b) = γ δ,-(a), for b ∈ (-∞, x -+ δ], γ δ (a, b) = γ δ,+ (a), for b ∈ [x +δ, +∞).

  and b k,j = b j,k . Since the kernel function γ δ is compactly supported, then one gets: b j,k = 0 for |j -k| > K := [δ/h] + 1, where [•] is the floor function. Therefore, the sum in (4) is finite with index k = -K, ..., K.

Figure 1 :

 1 Figure 1: Discretization of the domain and unknowns.

  Let us start by considering the nonlocal Schrödinger equation (1) with the constant kernel function γ δ (a, b) = 2δ -3 , for a ∈ [-δ, δ]. The computational domain is [x -, x + ] := [-3, 3], and the final time is set to T = 2, for the initial gaussian data ψ(x, 0) = 5e 2ix-25x 2 /4 . Let us fix the spatial discretization step to h = 5×10 -2 and analyze the ∞ J -norm error defined by: e h,τ ∞,T := ||ψ N jψ N,ref || ∞ J thanks to τ , with N τ = T . Here, we introduced ||u j || ∞ J := max 0≤j≤J |u j |. The reference solution ψ n,ref is computed for the discretization parameters h = 5 × 10 -2 and τ = 1 × 10 -4 . For different values of δ, we report the error e h,τ

τ = 1 ×

 1 10 -2 τ = 5 × 10 -3 τ = 4 × 10 -3 τ = 2.5 × 10 -3 rate δ = 0.4 4.00 × 10 -2

Figure 2 (

 2 Figure 2 (left) gives the evolution of the numerical solution with discrete ABC in the bounded domain by setting τ = 1 × 10 -3 and h = 1.25 × 10 -2 , for δ = 0.4. For completeness, we also provide

Figure 2 :

 2 Figure 2: Example 1 for δ = 0.4: left: numerical solution; right: error between the reference and numerical solutions (in log 10 scale).

21 8δ 21 /8 a 3 / 8 ,

 212138 with a ∈ [-δ, δ] and the singular kernel exponent α = 3/8. We still have [x -, x + ] = [-3, 3] and T = 2, with the same initial data as in Example 1. We choose a fixed spatial discretization step h = 5 × 10 -2 and analyze the error e h,τ ∞,T at T = 2 vs τ in Table

h = 5 ×Table 4 :

 54 10 -2 h = 2.5 × 10 -2 h = 2 × 10 -2 h = 1.25 × 10 -2 rate Example 2: ∞ J -norm error e h,τ∞,T (τ = 1 × 10 -3 ) and convergence rate vs. h for various values δ.

Figure 3 :

 3 Figure 3: Example 2 (δ = 0.4): left: numerical solution; right: error between the reference and numerical solutions (in log 10 scale).

  x j + ςξ 2 h, t) + ∂ 2 x ψ(x jςξ 2 h, t))dξ 2 dξ 1 x j + ςξ 2 , t) + ∂ 2 x ψ(x jςξ 2 , t))dξ 2 dξ 1 .(A.1)

  x ψ(x j + ξ 2 kh, t) + ∂ 2 x ψ(x jξ 2 kh, t))dξ 2 dξ 1 . (A.2)Thus, from (A.1) and (A.2), (35) can be rewritten as ς)q k (ς, x j , t)dς (A.3) x j + ςξ 2 h, t) + ∂ 2 x ψ(x jςξ 2 h, t))dξ 2 dξ 1 ,

  ς)q k (ς, x j , t)dς = + ςyh, x j , t)dy. (A.5)

1 0(ξ 1

 11 ∂ 2 x ψ(x + ξ 1 ς, t) + ξ 1 ∂ 2 x ψ(xξ 1 ς, t))dξ 1 .Now, γ δ takes its values on [-δ, δ] and has a singularity at 0 of the form |ς| -α . Thus, we can find a constant C > 0 such that:| d m dς m γ δ (ς)| ≤ C|ς -(α+m) |, for ς ∈ [-δ, δ] and m = 0, 1, 2. In addition, in (A.5), qk (ς, x, t) only takes its values ς on [(k -1)h, (k + 1)h]. Recalling that kh ≤ Kh = δ, for example the function ∂ 2x ψ(x + ξ 2 , t) in (A.7) has values for ξ 2 ∈ [-δ, δ]. Thus, for (k -1)h ≤ ς ≤ (k + 1)h, from the above expression and (38), we have|q k (ς, x, t)| 2 ≤ C(h 2 ς -2(1+α) + ς -2α + ς 2-2α )||(|∂

Table 1 :

 1 Example 1: ∞ J -norm error e h,τ ∞,T (h = 5 × 10 -2 ) and convergence rate vs. τ for various values δ. × 10 -2 h = 2.5 × 10 -2 h = 2 × 10 -2 h = 1.25 × 10 -2 rate

	δ = 0.5 δ = 0.8 δ = 1	3.16 × 10 -2 5.71 × 10 -3 2.20 × 10 -3	1.01 × 10 -2 5.06 × 10 -3 1.43 × 10 -3 5.50 × 10 -4	6.44 × 10 -3 7.79 × 10 -3 9.13 × 10 -4 3.52 × 10 -4	2.52 × 10 -3 1.98 × 10 -3 3.56 × 10 -4 1.37 × 10 -4	2.00 1.99 2.00 1.99
	e h,τ ∞,T h = 5 δ = 0.4 3.76 × 10 -2 δ = 0.5 4.74 × 10 -2 δ = 0.8 2.41 × 10 -2 δ = 1 1.39 × 10 -2	9.30 × 10 -3 1.17 × 10 -2 5.98 × 10 -3 3.45 × 10 -3	5.92 × 10 -3 7.46 × 10 -3 3.80 × 10 -3 2.19 × 10 -3	2.25 × 10 -3 2.84 × 10 -3 1.45 × 10 -3 8.36 × 10 -4	2.02 2.02 2.02 2.03

Table 2 :

 2 Example 1: ∞

J -norm error e h,τ ∞,T (τ = 1 × 10 -3 ) and convergence rate vs. h for various values δ.

Table 3 :

 3 Example 2: ∞

J -norm error e h,τ ∞,T (h = 5 × 10 -2 ) and convergence rate vs. τ for various values δ.

  ςh -|y|) 2 |q k (kh + y, x, t)| 2 dy/h 2 + ξ 2 , t) + ∂ 2 x ψ(xξ 2 , t))dξ 2 dξ 1 (A.7) ∂ 2 x ψ(x + ξ 1 ς, t) + ξ 1 ∂ 2 x ψ(xξ 1 ς, t))dξ 1

	Consequently, we deduce
								K-1	1	ςh
	|Q(x, t)| 2 ≤ C	k=1	0	dς	-ςh
			1				0		h
	+C	0	dς	-ςh	| qK (Kh + y, x, t)| 2 dy/h + C	-h	|∂ 2 x ψ(x + y, t)| 2 dy/h 2α-1 .	(A.6)
	From (38), the terms qk (ς, x, t) and qk (ς, x, t) satisfy
	qk (ς, x, t) = x ψ(x -2 d 2 dς 2 (ςγ δ (ς)) 1 0 khξ 1 ςξ 1 (∂ 2 1 d dς (ςγ δ (ς)) 0 1 (ξ 1 -ςγ δ (ς) 0 (ξ 2 1 ∂ 3 x ψ(x + ξ 1 ς, t) -ξ 2 1 ∂ 3 x ψ(x -ξ 1 ς, t))dξ 1
	and
	qK (ς, x, t) =	d dς	(ςγ δ (ς))	0	1	ςξ 1 Khξ 1	(∂ 2
									1	ξ 1
									0
	≤	C h α	h -h	|∂ 2 x ψ(x + y, t)|dy.

0 (∂ 2 x ψ(x + ςξ 2 h, t) + ∂ 2 x ψ(xςξ 2 h, t))dξ 2 dξ 1 | (x ψ(x + ξ 2 , t) + ∂ 2 x ψ(xξ 2 , t))dξ 2 dξ 1 -(ςγ δ (ς))

  2 x ψ(y, t)| + |∂ 3 x ψ(y, t)|) 2 || L ∞ x,δ , | qK (ς, x, t)| 2 ≤ C(h 2 ς -2α + ς 2-2α ) |∂ 2 x ψ(y, t)| 2 || L ∞ x,δ .The above inequality gives after some computations ςhy)2 |q k (kh + y, x, t)| 2 dy/h 2 (A.8) ≤ C (|∂ 2 x ψ(y, t)| + |∂ 3 x ψ(y, t)|) 2 || L ∞On the other hand, there existsx δ ∈ [xδ, x + δ] such that |∂ 2 x ψ(y, t)| 2 || L ∞ x,δ = |∂ 2 x ψ(x δ , t)| 2 ≤ C|∂ 2 x ψ(x, t)| 2 + C

							δ
							-δ	|∂ 3 x ψ(x + y, t)| 2 dy.	(A.11)
	Combining (A.10) and (A.11), we have
					∞		4
		h	j=-∞	|Q(x j , t)| 2 ≤ C	R	p=2	|∂ p x ψ(y, 0)| 2 dy
	from Lemma 5. Thus, this ends the proof of (36).
	1	ςh
	dς
	0	-ςh
							x,δ	(k+1)h (k-1)h	1 |y| 2α dy + Ch (|∂ 2 x ψ(y, t)| + |∂ 3 x ψ(y, t)|) 2 || L ∞ x,δ ,
	and
		1			0
		0	dς	-ςh	| qk (Kh + y, x, t)| 2 dy/h ≤ C |∂ 2 x ψ(y, t)| 2 || L ∞ x,δ .	(A.9)

(Taking (A.8), (A.9) into (A.6), for α < 1/2 we have

|Q(x, t)| 2 ≤ C (|∂ 2 x ψ(y, t)| + |∂ 3 x ψ(y, t)|) 2 || L ∞ x,δ . (A

.10)

Acknowledgments. This research is partially supported by NSFC under grant Nos. 11502028. Research conducted within the context of the Sino-French International Associated Laboratory for Applied Mathematics-LIASFMA. X. ANTOINE thanks the LIASFMA funding support of the Université de Lorraine.