ENERGY PRESERVING METHODS FOR NONLINEAR SCHR ¨ODINGER EQUATIONS

ENERGY PRESERVING METHODS FOR NONLINEAR SCHR ¨ ODINGER EQUATIONS

CHRISTOPHE BESSE, ST ´EPHANE DESCOMBES, GUILLAUME DUJARDIN, AND INGRID LACROIX-VIOLET

Abstract. This paper is concerned with the numerical integration in time of nonlinear Schr¨odinger equations using different methods preserving the energy or a discrete analog of it. The Crank-Nicolson method is a well known method of order 2 but is fully implicit and one may prefer a linearly implicit method like the relaxation method introduced in [10] for the cubic nonlinear Schr¨odinger equation. This method is also an energy preserving method and numerical simulations have shown that its order is 2. In this paper we give a rigorous proof of the order of this relaxation method and propose a generalized version that allows to deal with general power law nonlinearites. Numerical simulations for different physical models show the efficiency of these methods.

AMS Classification: 35Q41, 81Q05, 65M70

Keywords. Nonlinear Schr¨odinger equation, Gross-Pitaevskii equation, numerical methods, relax- ation methods.

1. Introduction

The nonlinear Schr¨odinger equation (NLSE) is a fairly general dispersive partial differential equation arising in many areas of physics and chemistry [1, 2, 16, 26, 12]. One of the most important application of the NLSE is for laser beam propagation in nonlinear and/or quantum optics and there it is also known as parabolic/paraxial approximation of the Helmholtz or time-independent Maxwell equations [1, 2, 16, 23, 12]. In the context of the modeling of the Bose-Einstein condensation (BEC), the nonlinear Schr¨odinger equation is known as the Gross-Pitaevskii equation (GPE), which is a widespread model that describes the averaged dynamics of the condensate [26, 6]. Depending on the physical situation that one considers, several terms in the right-hand side of the NLSE appear. Our goal is to develop numerical methods for the time integration of a fairly general NLSE including realistic physical situations, that have high-order in time and have good qualitative properties (preservation of mass, energy, etc) over finite times. We develop our analysis in spatial dimensiondP t1,2,3ubecause it fits the physical framework, even if most methods and results naturally extend to higher dimensions.

The space variablexwill sometimes lie in R^d, sometimes on thed-dimensional torusT^dδ “ pR{pδZqq^d (for someδą0). In this paper, we consider a NLSE of the form

(1) iBtϕpt, xq “ ˆ

´1

2∆`Vpxq `β|ϕ|^2σpt, xq `λ`

U˚ |ϕpt,¨q|²˘

pxq ´Ω.R

˙ ϕpt, xq,

whereϕis an unknown function from RˆR^d or RˆT^dδ to C, ∆ is the Laplace operator, V is some real-valued potential function, β P R is a parameter that measures the local nonlinearity strength, λ P R is a parameter that measures the nonlocal nonlinearity strength with convolution kernel U, ΩPR^d is a vector encoding the direction and the speed of a rotation, and R is a rotation operator that is important in the modeling of rotating BEC (for example,R “x^ p´i∇qwhen d“3). The NLSE is supplemented with an initial datumϕin. The results presented in this paper extend to more

1

general power law nonlinearities such as iBtϕpt, xq “

˜

´1

2∆`Vpxq `

K

ÿ

k“1

βk|ϕ|^2σkpt, xq `λ`

U˚ |ϕpt,¨q|²˘

pxq ´Ω.R

¸ ϕpt, xq,

but we restrict ourselves toK“1 for the sake of simplicity. Equation (1) is hamiltonian for the energy functional

(2) Epϕq “ ż

R^d

ˆ1

4}∇ϕ}²`1

2V|ϕ|²` β

2σ`2|ϕ|<sup>2σ`2</sup>`λ

4pU˚ |ϕ|²q|ϕ|²´Ω 2ϕRϕ

˙ dx,

provided U is a real-valued convolution kernel, symmetric with respect to the origin ¹. In practice the convolution kernel U may for example correspond to a Poisson equation (Upxq “ 1{p4π|x|q in dimensiond“3) or it may represent dipole-dipole interactions (see [9]).

The main goal of this paper is the analysis of numerical methods for the time integration of (1) that preserve the energy (2) or a discretized analogue of it. In particular, we are interested in the order (in time) of such methods. A well known method is the Crank-Nicolson method introduced in [14] for parabolic problems (see for example a posteriori error estimates in [3]) and applied in [17] to Schr¨o- dinger equations. For all nonlinearities, these methods are fully implicit. However, they have second order in time (see [27] for the case of the cubic NLS equation and [31] for the case of a system with possibly fractional derivatives) and preserve discrete analogues of the energy (2) as well as the total mass (squaredL²-norm) of the solutions. Unfortunately, since they are fully implicit, these methods are costly. To work around this problem, the methods introduced and analyzed in this paper belong to the family of relaxation methods.

Relaxation methods for Schr¨odinger equations were introduced in [10, 11]. They have been applied to different NLS equation for example in the context of plasma physics [25]. For cubic nonlinearities (σ“1 in (1)), they are linearly implicit hence very popular [4, 15, 6, 19, 21]. They preserve theL²-norm and a discrete analogue of (2). It is well-known that they have numerical order 2. Recently, the proof of convergence in time to second order was established in fully discrete context for nonlinear fractional Schr¨odinger equations [24] and for semilinear Heat equation [32]. However, up to our knowledge, there is no proof of order 2 in the literature when the phase space is infinite dimensionale.g. in the semi- discrete case. This paper presents two new results with respect to relaxation methods for (1)in the semi-discrete case, i.e. discrete in time and continuous in space. First, we prove rigorously that the classical relaxation method, applied to the classical cubic NLS equation (i.e. (1) with V ”0,β “1, σ“1,λ“0 and Ω“0) is of order 2. Second, we present a generalized relaxation method that allows to deal with general power law nonlinearities (σ‰1 in (1)) and with the full GPE. The generalized relaxation methods that we introduce in this context are implicit (actually explicit forσď4), have numerical order 2 and we show that they preserve an energy which is also a discretized analogue of (2).

This paper is organized as follows. In Section 2 we recall the definition of the Crank-Nicolson method and give a short explanation of its energy preserving property. Section 3 is devoted to relaxation methods applied to (1): In a first part we recall the method introduced in [10] for the cubic Schr¨odinger equation and in a second part we give a proof of the optimal order of convergence for an initial datum belonging toH^s`4pR^dq, din t1,2,3uand sąd{2. In section 4 we propose a generalized relaxation method that allows to deal with general nonlinearites and we prove that this method is also an energy preserving method. Section 5 deals with numerical results in different physical models showing the efficiency of the methods.

1For real-valued functions, symmetry with respect to the origin is equivalent to real-valued Fourier transform.

2. Preservation of energy and Crank-Nicolson scheme

A usual way to prove the conservation of the energy (2) consists in multiplying the equation (1) by Btϕpt, xq, where ¯zdenotes the conjugate value of a complexz, integrating overR^d and taking the real part of the result. This computation relies on the identity which holds for all smooth functionsϕof time with values into a space of sufficiently integrable functions

(3) Re

ż

R^d

|ϕ|^2σϕBtϕ dx“ 1 2σ`2

d dt

ż

R^d

|ϕ|^2σ`2dx, @σě0.

A possible way to derive numerical schemes that preserve an energy functional is therefore to mimic this identity at the discrete level.

In 1981, Delfour, Fortin and Payre [17], following an idea of Strauss and Vasquez [29], proposed a way to deal with the nonlinear term|ϕ|^2σϕfor the Crank-Nicolson scheme. This method generalizes the second order mid-point scheme for the linear Schr¨odinger equation

iϕn`1´ϕn

δt “ ´1

2∆ϕn`1`ϕn

2 ,

whereϕ_npxqdenotes an approximation of ϕpt_n, xq with the discrete time t_n “nδtdefined with the time step δt. Their approach can be explained as follows. If one looks for a real-valued function g:C²ÑRsuch that the scheme takes the form

(4) iϕn`1´ϕn

δt “

ˆ

´1

2∆`V `βgpϕn, ϕn`1q `λ ˆ

U˚

ˆ|ϕn`1|<sup>2</sup>` |ϕn|² 2

˙˙

´Ω.R

˙ϕn`1`ϕn

2 ,

then, multiplying this relation by iϕn`1´ϕn, integrating overs R^d and taking the real part, as we did in the time-continuous setting above, yields to 0 in the left-hand side and several terms in the right-hand side. Amongst these terms, those involvingg are equal to

β ż

R^d

gpϕ_n, ϕ<sub>n`1</sub>q`

|ϕ_n`1|²´ |ϕ_n|²˘ dx,

sinceg is real-valued. Let us denote byGthe function v ÞÑ |v|<sup>2σ`2</sup>{p2σ`2q. A sufficient condition for the method (4) to preserve an energy of the form (2) is therefore to have

gpϕn, ϕn`1q`

|ϕn`1|²´ |ϕn|²˘

“Gpϕn`1q ´Gpϕnq.

This is exactly the definition ofg chosen in [17].

In the following, the Crank-Nicolson method for the GPE (1) is therefore defined using the formula iϕ_n`1´ϕ_n

(5) δt

“ ˆ

´1

2∆`V ` β σ`1

|ϕn`1|<sup>2σ`2</sup>´ |ϕn|^2σ`2

|ϕn`1|<sup>2</sup>´ |ϕn|<sup>2</sup> `λ ˆ

U˚

ˆ|ϕn`1|<sup>2</sup>` |ϕn|² 2

˙˙

´Ω.R

˙ϕn`1`ϕn

2 .

We shall use the notation

ϕn`1“Φ^CN_δt pϕnq,

for the Crank-Nicolson method (5). In the expression above, the term corresponding to the nonlinearity should be understood as

(6) β

σ`1

|ϕ<sub>n`1</sub>|<sup>2σ`2</sup>´ |ϕ_n|^2σ`2

|ϕn`1|<sup>2</sup>´ |ϕn|<sup>2</sup> “ β σ`1

σ

ÿ

k“0

|ϕ_n`1|^2k|ϕ_n|^2pσ´kq,

so that it is indeed non-singular and it is consistent with the non-linear term β|ϕ|^2σ. The Crank- Nicolson method is fully implicit. It is known to have order two for the cubic NLS equation [27].

Moreover it preserves exactly the L²-norm of the solution as well as the following energy:

(7) ECNpϕq “Epϕq,

with Edefined by (2).

In Section 4, we shall use similar ideas to derive energy-preserving relaxation methods for general power laws nonlinearities. Before doing so, we first deal with the classical relaxation method in Section 3.

3. The classical relaxation method

3.1. An energy preserving method. In [11], Besse introduced the usual relaxation method (10) applied to the nonlinear Schr¨odinger equation (1) with V “0,λ“0, Ω“0 andσ“1 that is known as the cubic nonlinear Schr¨odigner equation

(8) iBtϕpt, xq “ ´1

2∆ϕpt, xq `β|ϕpt, xq|²ϕpt, xq,

withϕp0, xq “ϕinpxq. The idea of the relaxation method is to add to (8) a new unknown Υ“ |ϕ|² and the equation (8) is transformed in

(9)

# Υ“ |ϕpt, xq|², iB_tϕpt, xq “ ´1

2∆ϕpt, xq `βΥϕpt, xq.

The relaxation method then consists in discretizing both equations respectively at discrete timestn

andtn`1{2 and to solve iteratively

(10)

$

’&

’%

Υ<sub>n`1{2</sub>`Υ_n´1{2

2 “ |ϕ_n|², iϕn`1´ϕn

δt “

ˆ

´1

2∆`βΥ<sub>n`1{2</sub>

˙ϕn`1`ϕn

2 ,

to compute approximationsϕ_n ofϕpnδtq. This system is usually initialized with Υ_´1{2“ |ϕp´δt{2q|² or by second order approximation of|ϕp´δt{2q|². This method is linearly implicit (recall thatσ“1).

Moreover, it is known to preserve exactly theL²-norm and the discrete energy [11]:

(11) E_rlxpϕ,Υq “1

4 ż

R^d

}∇ϕ}²dx`β 2

ż

R^d

Υ|ϕ|²dx´β 4

ż

R^d

Υ²dx.

Indeed

Re ˆ

Υ<sub>n`1{2</sub>ϕn`1`ϕn

2 ϕ_n`1´ϕ_n

˙

“ Υ_n`1{2 2

´

|ϕ_n`1|²´ |ϕ_n|²

¯ . But

Υ_n`1{2

´

|ϕ_n`1|²´ |ϕ_n|²

¯

“Υ_n`1{2

´

|ϕ_n`1|²´ |ϕ_n|²

¯

`Υ_n´1{2

´

|ϕ_n|²´ |ϕ_n|²

¯

“

´

Υ<sub>n`1{2</sub>|ϕn`1|²´Υ_n´1{2|ϕn|²

¯

´

´

Υ_n`1{2´Υ_n´1{2

¯

|ϕn|². Using the definition of Υ_¨`1{2in (10), a simple computation leads to

´

Υ_n`1{2´Υ_n´1{2

¯

|ϕn|²“

´ Υ_n`1{2

¯2

´

´ Υ_n´1{2

¯2

2 .

We therefore conclude that Re

ˆ

Υ<sub>n`1{2</sub>ϕn`1`ϕn

2 ϕn`1´ϕn

˙

“ Υ_{n`1{2</sub>|ϕ<sub>n`1}|²´Υ_n´1{2|ϕ_n|²

2 ´

´ Υ_n`1{2

¯2

´

´ Υ_n´1{2

¯2

4 ,

which allows to prove the conservation of the discrete energy (11).

It is interesting to note the consistency of the energy associated to relaxation scheme with the energy (2) for cubic nonlinear Schr¨odinger equation

Erlxpϕ,|ϕ|²q “Epϕq.

The relaxation method was proved to converge in [11] but consistency analysis was missing. We present it in the next subsection.

3.2. Consistency analysis for NLS equation with cubic nonlinearity. The aim of this subsec- tion is to prove that this method has temporal order 2 under fairly general assumptions. This fact is supported by numerical evidences in the literature for years. We provide the first rigorous proof below.

To this end we use ideas from [11], with slightly different notation, where the scheme was proved to be convergent, but the order was only obtained numerically.

The first equation in (10) is the discrete equivalent of the continuous constraint Υ “ |ϕ|². In particular, this constraint is notan evolution equation. Using the ideas introduced in [11], we rewrite the continuous equation (1) (recall thatV “0,λ“0 and Ω“0 andσ“1) as the system

(12)

$

’’

&

’’

%

iB_tϕ`1

2∆ϕ“βΥϕ, BtΥ“2Repvϕq, iB_tv`1

2∆v“βpB_tΥϕ`ΥB_tϕq.

The discrete system (10) has a discrete augmented equivalent (see [10, 11]). Let us denote byv_{n`</sub>1 2 “ ϕ<sub>n`1}´ϕ_n

δt the discrete time derivative of ϕ_n and define the nonlinearities as

(13)

$

’’

’’

’’

’’

’’

&

’’

’’

’’

’’

’’

% Φ_n`1

2 “Υ_n`1 2

ˆϕn`1`ϕn

2

˙ , Ξ_n`1

2 “2Re ˆ

v_n`1 2

ˆϕn`1`ϕn

2

˙˙

, V_n`¹

2 “

ˆΥ_n`3

2 `Υ_n´1 2

2

˙ ˆv_n`3

2`2v<sub>n`</sub>1

2 `v_n´1 2

4

˙

`2Re ˆ

v_n`1 2

ˆϕ<sub>n`1</sub>`ϕ_n 2

˙˙ ˆϕ<sub>n`2</sub>`ϕ<sub>n`1</sub>`ϕ_n`ϕ_n´1 4

˙ .

The augmented system writes

(14)

$

’’

’’

’’

’&

’’

’’

’’

’%

iϕn`2´ϕn`1

δt `1

2∆

ˆϕn`2`ϕn`1

2

˙

“βΦ_n`3

2, p14.bq

Υ_n`³

2´Υ_n´¹

2

2δt “Ξ_n`1

2, p14.aq

i v_n`3

2 ´v_n´1 2

2δt `1

2∆ ˆv_n`3

2 `2v<sub>n`</sub>1

2 `v_n´1 2

4

˙

“βV_n`¹

2. p14.cq This system allows to computepϕn`2,Υ<sub>n`</sub>3

2, v_n`3 2qfrom Xn :“

´ Υ_n´1

2,Υ_n`1

2, ϕn´1, ϕn, ϕn`1, v<sub>n´</sub>1 2, v<sub>n`</sub>1

2

¯ . We consider the system (14) as the mapping

XnÞÑXn`1.

The seven variables involved inX_n are not independent. If they satisfiy the five relations

$

’’

’’

’’

’’

’’

&

’’

’’

’’

’’

’’

% v_n´1

2 “ϕ_n´ϕ_n´1 δt , v_n`1

2 “ϕ_{n`1</sub>´ϕ<sub>n</sub> δt , Υ<sub>n`}¹

2 `Υ_n´¹

2 “2|ϕn|², iϕ_n´ϕ_n´1

δt “

ˆ

´1

2∆`βΥn´1{2

˙ϕ_n`ϕ_n´1

2 ,

iϕ_n`1´ϕ_n

δt “

ˆ

´1

2∆`βΥ<sub>n`1{2</sub>

˙ϕ<sub>n`1</sub>`ϕ_n

2 ,

then the seven variables inX<sub>n`1</sub>satisfy the same five relations with nreplaced byn`1. This fact is proved in [11]. We describe now how to build the seven initial data inX₀from ϕ₀“ϕ_inand Υ_´1

2 so that they satisfy the five relations above:

(15)

$

’’

’’

’’

’’

’&

’’

’’

’’

’’

’% Υ1

2 “2|ϕ₀|²´Υ_´1 2, ϕ_´1 “

ˆ 2´iδt

ˆ

´1

2∆`βΥ_´1{2

˙˙´1ˆ 2`iδt

ˆ

´1

2∆`βΥ_´1{2

˙˙

ϕ₀, ϕ₁ “

ˆ 2`iδt

ˆ

´1

2∆`βΥ_1{2

˙˙´1ˆ 2´iδt

ˆ

´1

2∆`βΥ_1{2

˙˙

ϕ₀, v_´1

2 “ pϕ0´ϕ´1q{δt, v1

2 “ pϕ1´ϕ0q{δt.

Let us define the operators

A“ pi´δt∆{4q^´1pi`δt∆{4qandB“ pi´δt∆{4q^´1, and the matrix of operators

C“

¨

˚

˚

˚

˚

˚

˚

˚

˚

˝

I 0 0 0 0 0 0

0 I 0 0 0 0 0

0 0 B 0 0 0 0

0 0 0 B 0 0 0

0 0 0 0 B 0 0

0 0 0 0 0 B 0

0 0 0 0 0 0 B

˛

‹

‹

‹

‹

‹

‹

‹

‹

‚ .

The mappingX_nÞÑX_n`1 reads

(16)

¨

˚

˚

˚

˚

˚

˚

˚

˚

˚

˝ Υ_n`1

2

Υ_n`3 2

ϕn

ϕn`1

ϕn`2

v_n`¹

2

v_n`3 2

˛

‹

‹

‹

‹

‹

‹

‹

‹

‹

‚

“

¨

˚

˚

˚

˚

˚

˚

˚

˚

˝

0 I 0 0 0 0 0

I 0 0 0 0 0 0

0 0 0 I 0 0 0

0 0 0 0 I 0 0

0 0 0 0 A 0 0

0 0 0 0 0 0 I

0 0 0 0 0 A A´I

˛

‹

‹

‹

‹

‹

‹

‹

‹

‚

¨

˚

˚

˚

˚

˚

˚

˚

˚

˚

˝ Υ_n´1

2

Υ_n`1 2

ϕn´1

ϕn

ϕn`1

v_n´¹

2

v_n`1 2

˛

‹

‹

‹

‹

‹

‹

‹

‹

‹

‚

`δtC

¨

˚

˚

˚

˚

˚

˚

˚

˚

˝ 0 2Ξ_n`1

2

0 0 βΦ_n`3

2

0 2βV_n`1

2

˛

‹

‹

‹

‹

‹

‹

‹

‹

‚ .

In a more compact form, we defineB andMso that the mapping (16) reads (17) X<sub>n`1</sub>“BXn`δtCMpXn, X_n`1q.

We introduce the hypotheses that will allow us to prove our consistency and convergence result for the classical relaxation method (10) in Theorem 8.

Remark 1. The results below extend to more general cases. In particular, one may treat the case whereV is non zero smooth autonomous potential such that the multiplication byV is a bounded linear operator between Sobolev spaces and the case whereV is a nonautonomous such operator with sufficient regularity with respect to time.

Hypotheses 2. We fixdP t1,2,3uandsąd{2². We assumeϕ₀PH^{s`4</sup>pR<sup>d</sup>qis given. We denote by T<sup>˚</sup> ą0 the existence time of the maximal solution ϕof the Cauchy problem (1) (with V “0,λ“0, Ω “0 and σ“1) in H<sup>s`4}pR^dq. We assume there existsδt₀ ą0 such that τ ÞÑ ϕpτ,¨q is a smooth map from p´δt₀, T^˚q to H^{s`4</sup>pR<sup>d</sup>q. Moreover, we assume that there exists R<sub>1</sub>, R<sub>2</sub> ą0 such that for all Υ<sub>´1{2</sub> PH<sup>s`4}pR^dq with }Υ_´1{2}H^{s`4</sup> ďR<sub>1</sub>, the numerical solution X<sub>n</sub> (with initial datum (15)) is uniquely determined by (16) for all δtP p0, δt<sub>0</sub>q and all nP N such that nhď T, and it satisfies }X<sub>n</sub>}pH<sup>s`2}pR^dqq⁷ ďR₂ for all suchn.

Remark 3. The hypotheses above on the exact solution are fullfilled in several cases. For example, for the exact solution ϕ, it is well known (see [20]) thatT^˚“ `8in at least two cases:

‚ if ϕ_in has small H^s`4-norm andβă0

‚ if βą0and ϕ_inPH^s`4pR^dq.

For the numerical solution, they are fullfilled provided T^˚ ă `8(see Theorem 1.1 in [11]) and also when T<sup>˚</sup> “ `8andβą0 (see[10]).

Lettn“nδtdenote the discrete times andtÞÑXptqthe vector

(18) Xptq “

¨

˚

˚

˚

˚

˚

˚

˚

˚

˝

|ϕpt´δt{2,¨q|²

|ϕpt`δt{2,¨q|² ϕpt´δt,¨q

ϕpt,¨q ϕpt`δt,¨q Btϕpt´δt{2,¨q Btϕpt`δt{2,¨q

˛

‹

‹

‹

‹

‹

‹

‹

‹

‚ .

Using the definition ofM(see (13) and (17)), the fact that H^spR^dq, is an algebra since sąd{2, and the fact that the exact and numerical solutions stay in a bounded set of H^spR^dq, it is easy to prove the following lemma.

Lemma 4. Assume Hypotheses 2 is satisfied. There existsC ą0 such that for all δtP p0, δt₀q, all nPNsuch thatpn`1qδtďT

}MpXn, Xn`1q ´MpXptnq, Xptn`1qq}pH^spR^dqq⁷

ď C`

}X_n´Xpt_nq}pH^spR^dqq⁷` }X<sub>n`1</sub>´Xpt_n`1q}pH^spR^dqq⁷

˘. Note that the constantC depends only on the initial data.

Before starting the proof of our main result of this section (see Theorem 8), we state and prove another lemma.

Lemma 5. There exists a constantbą0 such that the operatorB defined in (16)and (17)satisfies for allδtą0, andnPN,

(19) ~BⁿC~ ďb,

where~ ¨ ~is the norm of linear continuous operators frompH^spR^dqq⁷ to itself.

Proof. The operatorB is defined by three diagonal blocks B1“

ˆ0 I I 0

˙

, B2“

¨

˝

0 I 0 0 0 I

0 0 A

˛

‚, and B3“

ˆ0 A A A´I

˙ .

2The hypothesis są d{2 can be relaxed for dP t1,2,3u sinceH^{s`2</sup> and H<sup>s`4} are algebras in that case.

However it is more natural in higher dimensions.

The first blockB1is an isometry frompH^spR^dqq²to itself and so are all its powers. The powers of the second blockB2 read for allně2

B₂ⁿ“

¨

˝

0 0 A^n´2 0 0 A^n´1 0 0 Aⁿ

˛

‚.

Since all the powers of A are of norm less than one, we infer that the norm of Bⁿ₂ is less than ? 3.

Since the norm ofB is less than one, we infer that the norm of Bⁿ₂

¨

˝

B 0 0

0 B 0

0 0 B

˛

‚,

is less than? 3.

The block B₃ can be diagonalized by blocks as B3“

ˆ I I

´I A

˙ ˆ´I 0

0 A

˙ ˆ I I

´I A

˙´1

, so that for allnPN,

(20) Bⁿ₃ “

ˆ I I

´I A

˙ ˆp´Iqⁿ 0

0 Aⁿ

˙ ˆ I I

´I A

˙´1

. Therefore, the last 2ˆ2 block ofBⁿC reads

Bⁿ₃

ˆB 0

0 B

˙

“

ˆ pp´IqⁿA`A<sup>n</sup>qBpI`Aq^´1 pAⁿ´ p´IqⁿqBpI`Aq<sup>´1</sup> pp´Iq<sup>n`1</sup>A`A<sup>n`1</sup>qBpI`Aq<sup>´1</sup> pp´Iq<sup>n</sup>`A<sup>n`1</sup>qBpI`Aq^´1

˙ . SinceBpI`Aq^´1“ p1{2qI, we infer

Bⁿ₃

ˆB 0

0 B

˙

“ 1 2

ˆ pp´IqⁿA`A<sup>n</sup>q pA<sup>n</sup>´ p´Iq<sup>n</sup>q pp´Iq<sup>n`1</sup>A`A<sup>n`1</sup>q pp´Iqⁿ`A<sup>n`1</sup>q

˙ . Since all the powers of Aare of norm less than 1, we infer that

@nPN,

Bⁿ₃

ˆB 0

0 B

˙ ď4.

This proves the result.

Remark 6. The powers of the operator B are not uniformly bounded. However, the powers of B mutiplied byC are uniformly bounded as shown above. The main reason is that the third matrix in the right hand side of (20)becomes singular at the end of the spectrum of∆.

Lemma 7. Assume d, s, R1, δt0 and ϕin are given as in Hypotheses 2. There exists cą0 such that for allΥ_´1{2PH^{s`4</sup> with}Υ<sub>´1{2</sub>}H<sup>s`4} ďR₁ and allδtP p0, δt₀q,

(21) }X0´Xp0q}_pHs`2pR<sup>d</sup>qq<sup>7</sup> ďc`

}Υ_´1{2´ |ϕp´δt{2q|²}_Hs`2pR<sup>d</sup>q`δt²˘ .

Proof. TheH^s`2-norm of each of the seven components of the vectorX0´Xp0qis estimated separately.

TheH^{s`2</sup>-norm of the first component Υ<sub>´1{2</sub>´ |ϕp´δt{2q|<sup>2</sup>is bounded byitself. For the H<sup>s`2}-norm of the second component, we define fptq “ |ϕptq|², which is a smooth function from p´δt0, δt0q to H^s`4, thanks to Hypotheses 2. We may write using a Taylor formula at 0

f ˆ

´δt 2

˙

“ |ϕ0|²´2Repϕ0Btϕp0qqδt 2 `

ż´δt{2 0

ˆ´δt 2 ´σ

˙

f²pσqdσ, and similarly

f ˆδt

2

˙

“ |ϕ0|²`2Repϕ0Btϕp0qqδt 2 `

żδt{2 0

ˆδt 2 ´σ

˙

f²pσqdσ.

Therefore, one has Υ_1{2´f

ˆδt 2

˙

“ 2|ϕ₀|²´Υ_´1{2´f ˆδt

2

˙

“ f ˆ

´δt 2

˙

´ ż´δt{2

0

ˆ´δt 2 ´σ

˙

f²pσqdσ´ żδt{2

0

ˆδt 2 ´σ

˙

f²pσqdσ´Υ_´1{2. By triangle inequality, we infer that

}Υ_1{2´ |ϕpδt{2q|²}H<sup>s`2</sup>ďc`

}Υ_´1{2´ |ϕp´δt{2q|²}H<sup>s`2</sup>`δt²˘ , (22)

}Υ_1{2´ |ϕpδt{2q|²}_Hs`4 ďc`

}Υ_´1{2´ |ϕp´δt{2q|²}_Hs`4`δt²˘ , (23)

wherec“maxp1,sup_{σPp´δt0,δt0q}}f²pσq}_Hs`4qonly depends on the exact solution of (1). The inequality (22) provides us with a bound for the second component ofX0´Xp0q consistent with (21). The inequality (23) shows that theH<sup>s`4</sup>-norm of Υ_1{2is bounded provided that of Υ_´1{2 is bounded and δtis small. This will be useful later in the proof (see (27)). We now estimate theH^{s`2</sup>-norm of the fifth component ofX0´Xp0q. Note that theH<sup>s`2}-norm of the third component can be estimated the very same way and that theH^s`2-norm of the fourth component is zero. We start with the identity

ϕ₁“ϕ₀´iδt ˆ

´1

2∆`βΥ_1{2

˙ϕ₁`ϕ₀

2 ,

and we denote byrpδtqthe consistency error defined by rpδtq “ϕpδtq ´ϕ0`iδt

ˆ

´1

2∆`βΥ_1{2

˙ϕpδtq `ϕ0

2 .

Using the fact thattÞÑϕpt,¨q is a smooth function fromp´δt₀, δt₀qtoH^s`4, we may write another Taylor expansion to obtain

rpδtq

“ ϕ₀`δtB<sub>t</sub>ϕp0q `δt²

2 B²_tϕp0q ` żδt

0

pδt´σq²

2 B³_tϕpσqdσ´ϕ₀

`iδt ˆ

´1

2∆`βΥ_1{2

˙˜ ϕ₀`δt

2Btϕp0q `1 2

żδt 0

pδt´σqB²_tϕpσqdσ

¸

“ ´iδt²βRe ˆi

2ϕ0∆ϕ0

˙

ϕ0`iδtβ`

|ϕ0|²´Υ´1{2

˘ ˆ

ϕ0´iδt 2

ˆ

´1

2∆`β|ϕ0|²

˙ ϕ0

˙

`iδt 2

ˆ

´1

2∆`βΥ_1{2

˙ żδt 0

pδt´σqB_t²ϕpσqdσ` żδt

0

pδt´σq²

2 B_t³ϕpσqdσ

“ ´iδt²βRe ˆi

2ϕ₀∆ϕ₀

˙

ϕ₀`iδtβ`

|ϕ₀|²´ |ϕp´δt{2q |²˘ ˆ

ϕ₀´iδt 2

ˆ

´1

2∆`β|ϕ₀|²

˙ ϕ₀

˙

`iδtβ`

|ϕp´δt{2q |²´Υ_´1{2˘ ˆ

ϕ₀´iδt 2

ˆ

´1

2∆`β|ϕ₀|²

˙ ϕ₀

˙

`iδt 2

ˆ

´1

2∆`βΥ_1{2

˙ żδt 0

pδt´σqB_t²ϕpσqdσ` żδt

0

pδt´σq²

2 B_t³ϕpσqdσ.

Using ˇ ˇ ˇ ˇ ϕ

ˆ

´δt 2

˙ˇ ˇ ˇ ˇ

2

“ |ϕp0q|²`δtRe ˆ

iϕ0

ˆ

´1

2∆ϕ0`β|ϕ0|²ϕ0

˙˙

` ż´δt{2

0

p´δt{2´σqB²_tp|ϕ|²qpσqdσ,

we obtain rpδtq

“ iδtβ`

|ϕp´δt{2q |²´Υ_´1{2˘ ˆ

ϕ₀´iδt 2

ˆ

´1

2∆`β|ϕ₀|²

˙ ϕ₀

˙

`δt³ 2 βRe

ˆi 2ϕ0∆ϕ0

˙ ˆ

´1

2∆`β|ϕ0|²

˙ ϕ0

´iδtβ ˆ

ϕ0´iδt 2

ˆ

´1

2∆`β|ϕ0|²

˙ ϕ0

˙ ż´δt{2 0

p´δt{2´σqB²_tp|ϕ|²qpσqdσ

`iδt 2

ˆ

´1

2∆`βΥ1{2

˙ żδt 0

pδt´σqB²_tϕpσqdσ` żδt

0

pδt´σq²

2 B³_tϕpσqdσ.

Note that we have

›

›

›

›

› żδt

0

pδt´σq²

2 B³_tϕpσqdσ

›

›

›

›

›H^s`2

ďcδt³,

›

›

›

›

› iδt

2 ˆ

´1

2∆`βΥ_1{2

˙ żδt 0

pδt´σqB²_tϕpσqdσ

›

›

›

›

›H^s`2

ďcδt³,

›

›

›

›

› iδtβ

ˆ

ϕ₀´iδt 2

ˆ

´1

2∆`β|ϕ₀|²

˙ ϕ₀

˙ ż´δt{2 0

p´δt{2´σqB²_tp|ϕ|²qpσqdσ

›

›

›

›

›H^s`2

ďcδt³, and

›

›

›

› δt³

2 βRe ˆi

2ϕ0∆ϕ0

˙ ˆ

´1

2∆`β|ϕ0|²

˙ ϕ0

›

›

›

›H^s`2

ďcδt³, wherecdoesn’t depend onδt. Then we infer from the estimates above that (24) }rpδtq}_Hs`2ďcδt`

}Υ_´1{2´ϕp´δt{2q}_Hs`2`δt²˘ .

Now, we denote by epδtqthe fifth componentϕ₁´ϕpδtqof the vectorX₀´Xp0qand we have epδtq “ ´iδt

2 ˆ

´1

2∆`βΥ_1{2

˙

epδtq ´rpδtq.

We want to estimate theH<sup>s`2</sup>-norm ofepδtq using this relation and the estimate (24). To this aim, we takeαPN<sup>d</sup> with|α| “α1` ¨ ¨ ¨ `αdďs`2 and differentiate the relation above to obtain that

B^α_xepδtq “iδt1

4∆B^α_xepδtq ´iδt

2βB^α_xpΥ_1{2epδtqq ´ B_x^αrpδtq.

Multiplying this relation by B^α_xepδtqintegrating overR^d, and taking the real part, we obtain (25) }B^α_xepδtq}²₂“ ´βδt

2Re ˆ

i ż

R^d

B_x^αepδtqB^α_xpΥ_1{2epδtqq

˙

´Re ˆż

R^d

B^α_xepδtqB_x^αrpδtq

˙ . Whenα“0_Nd, the first term on the right hand side in the equation above vanishes and

}epδtq}²₂ď }epδtq}₂}rpδtq}₂, using Cauchy-Schwartz inequality. We infer,

(26) }epδtq}₂ď }rpδtq}₂.

Now, whenα‰0_Nd, the Leibniz’ rule applied in (25) provides us with:

}B_x^αepδtq}²₂“ ´βδt 2

ÿ

tk:kďαu

cpk, αqRe ˆ

i ż

R^d

B_x^αepδtqB_x^kΥ_1{2B_x^α´kepδtq

˙

´Re ˆż

R^d

B_x^αepδtqB_x^αrpδtq

˙ ,

wherecpk, αqare integers. Note that since Υ_1{2 is real valued, the term corresponding tok“0_Nd in the sum vanishes. Then, with Cauchy-Schwarz inequality, we infer

}B^α_xepδtq}²₂ ď βδt 2

ÿ

tk:kďα,|k|ě1u

cpk, αq}B_x^αepδtq}₂}B^k_xΥ_1{2B_x^α´kepδtqq}₂` }B^α_xepδtq}₂}B_x^αrpδtq}₂.

ď βδt 2

ÿ

tk:kďα,|k|ě1u

cpk, αq}B_x^αepδtq}2}B^k_xΥ_1{2}8}B^α´k_x epδtqq}2` }B_x^αepδtq}2}B^α_xrpδtq}2. Sincekďαin the sum above, we have|k| ď |α|. For such k, one has

(27) }B_x^kΥ_1{2}₈ďc}B^k_xΥ_1{2}_Hpd`1q{2 ďc}Υ<sub>1{2</sub>}<sub>H</sub>pd`1q{2`|k| ďc}Υ<sub>1{2</sub>}<sub>H</sub>pd`1q{2`s`2ďc}Υ_1{2}H^{s`4</sup>, wherecis the Sobolev constant of the injection fromH<sup>pd`1q{2}pR^dqtoL⁸pR^dqand where we have used the fact thatdP t1,2,3u. Since theH^s`4-norm of Υ_1{2 is controlled by (23), we have

}B^α_xepδtq}2ďcα}epδtq}_H|α|´1` }B^α_xrpδtq}2,

where cα does not depend on δtP p0, δt0q. Then, by induction on |α| P t0, . . . , s`2ustarting with (26) forα“0<sub>N</sub>d, we have for some positive constantcs`2:

}B^α_xepδtq}2ďcs`2}rpδtq}<sub>H</sub>s`2,

for allδtP p0, δt0qand allαPN^d with|α| ďs`2. Therefore, we have}epδtq}<sub>H</sub>s`2 is controlled by }rpδtq}_Hs`2 and the conclusion for the fifth term follows using (24):

(28) }ϕ1´ϕpδtq}_Hs`2 ďcδt`

}Υ_´1{2´ϕp´δt{2q}_Hs`2`δt²˘ , wherecdoes not depend onδtP p0, δt₀q.

It remains to estimate theH^s`2-norm of the sixth and seventh components of the vectorX₀´Xp0q.

We only give the details for the seventh term since the computation is similar and even simpler for the sixth term. Let us denote byppδtqthe consistency error defined as

ppδtq “ ϕpδtq ´ϕ0

δt ´ Btϕ ˆδt

2

˙ .

Using a Taylor expansion, since the exact solution is a smooth function from p´δt0, δt0qto H^s`2, we have

(29) }ppδtq}_Hs`2ďcδt².

Let us denote byqpδtq “v_1{2´ Btϕ ˆδt

2

˙

the seventh component ofX0´Xp0q. We have qpδtq ´ppδtq “ϕ1´ϕpδtq

δt . Using estimates (28) and (29) we have by triangle inequality,

}qpδtq}_Hs`2 ď }ppδtq}<sub>H</sub>s`2` 1

δt}ϕ₁´ϕpδtq}_Hs`2 ďcδt<sup>2</sup>`c`

}Υ_´1{2´ϕp´δt{2q}_Hs`2`δt²˘ .

This concludes the proof of the lemma.

We prove below that the relaxation method (10) is of order 2.

Theorem 8.Assumed,s,R1ą0,ϕinPH^{s`4</sup>pR<sup>d</sup>q,T ăT<sup>‹</sup>are given and satisfy Hypotheses 2. There existsC ą0andδt0ą0 (smaller than the one in the hypotheses) such that for allΥ<sub>´1{2</sub>PH<sup>s`4}pR^dq with}Υ_´1{2}H^s`4pR^dqďR1, allnPNand all δtP p0, δt0qwithnδtďT,

(30) }Xn´Xptnq}pH^spR^dqq⁷ďC`

}Υ_´1{2´ |ϕp´δt{2q|²}H<sup>s`2</sup>pR<sup>d</sup>q`δt²˘ .

Proof. First, the initial datumX₀is computed fromϕ₀“ϕ_in and Υ_´1{2 using (15). Therefore, using Lemma 7, there exists a constantcą0 such that for allδtP p0, δt0qand all }Υ´1{2}_Hs`4pR^dqďR1, we have the estimate (21)

}X₀´Xp0q}_pHs`2pR<sup>d</sup>qq<sup>7</sup> ďc`

}Υ_´1{2´ |ϕp´δt{2q|²}H<sup>s`2</sup>pR<sup>d</sup>q`δt²˘ .

Second, we define the consistency error of the relaxation scheme at time t_n“nδtďT by the formula Rnpδtq “BXptnq `δtCMpXptnq, Xptn`1qq ´Xptn`1q.

Substracting this definition from (17), we obtain

(31) Xn`1´Xptn`1q “BpXn´Xptnqq `δtCpMpXn, Xn`1q ´MpXptnq, Xptn`1qqq `Rnpδtq.

From now on, we set for allnPNsuch thatnδtďT,e_n “X_n´Xpt_nq. Iterating the relation above, we obtain, as long asnδtďT,

en “Bⁿe0`δt

n´1

ÿ

k“0

B^n´k´1CpMpXk, Xk`1q ´MpXptkq, Xptk`1qqq `

n´1

ÿ

k“0

B^n´k´1Rkpδtq.

This implies

}e_n}pH^spR^dqq⁷ ď }BⁿCC^´1e₀}pH^spR^dqq⁷

`δt

n´1

ÿ

k“0

}B^n´k´1CpMpXk, Xk`1q ´MpXptkq, Xptk`1qqq }pH^spR^dqq⁷

`

n´1

ÿ

k“0

}B^n´k´1CC^´1Rkpδtq}_pHspR^dqq⁷

ď b}C^´1e0}_pHspR^dqq⁷

`δtb

n´1

ÿ

k“0

} pMpXk, X_{k`1</sub>q ´MpXptkq, Xpt<sub>k`1}qqq }_pHspR^dqq⁷

`b

n´1

ÿ

k“0

}C^´1R_kpδtq}_pHspR^dqq⁷

ď b}e0}_pHs`2pR^dqq⁷

`Cδtb

n´1

ÿ

k“0

`}e<sub>k</sub>}<sub>pH</sub>spR<sup>d</sup>qq<sup>7</sup>` }e_k`1}_pHspR^dqq⁷

˘

`b

n´1

ÿ

k“0

}R_kpδtq}_pHs`2pR^dqq⁷, using Lemmas 4 and 5. Then

p1´Cδtbq}en}pH^spR^dqq⁷ ďb}e0}pH<sup>s`2</sup>pR<sup>d</sup>qq<sup>7</sup>`2Cδtb

n´1

ÿ

k“0

}ek}pH^spR^dqq⁷`b

n´1

ÿ

k“0

}Rkpδtq}_pHs`2pR^dqq⁷.

Letδtbe small enough to ensure that 1´Cδtbě1

2. This implies }e_n}pH^spR^dqq⁷ ď2b}e₀}pH<sup>s`2</sup>pR<sup>d</sup>qq<sup>7</sup>`4Cδtb

n´1

ÿ

k“0

}e_k}pH^spR^dqq⁷`2b

n´1

ÿ

k“0

}R_kpδtq}_pHs`2pR^dqq⁷. Taylor expansions and the fact that one can differentiate the last two lines of (12) with respect to time show that there exists a constantQsuch that for allnandδtwithnδtďT

}Rkpδtq}_pHs`2pR^dqq⁷ ďQδt³.

This implies that 2břn´1

k“0}R_kpδtq}_pHs`2pR<sup>d</sup>qq<sup>7</sup> ď2bQT δt<sup>2</sup>. Then we obtain }en}<sub>pH</sub>spR<sup>d</sup>qq<sup>7</sup> ď2b}e0}<sub>pH</sub>s`2pR^dqq⁷`2bQT δt<sup>2</sup>`4Cδtb

n´1

ÿ

k“0

}ek}_pHspR^dqq⁷. Using a discrete Gronwall Lemma (see section 5 in [22]), we get

}en}_pHspR^dqq⁷ ď p2b}e0}_pHs`2pR<sup>d</sup>qq<sup>7</sup>`2bQT δt²qexpp4CT bq.

This estimate and (21) prove the result.

Remark 9. The Theorem 8 is written forxPR^d. The result remains true forxPT^dδ. It is also valid for a smooth bounded domain O ĂR^d with homogeneous Dirichlet boundary conditions. The main ingredient is that the functional space at hand (H^spR^dq, H^spT^dδq, etc) is an algebra.

4. The generalized relaxation method for general nonlinearities

4.1. Generalized relaxation method for NLS equation. We start by considering the simplified equation (1) with zero potential (V “ 0), no convolution operator (U “ 0) and without rotation (Ω“0). AssumingσPN^‹, we are therefore dealing with the classical nonlinear Schr¨odinger equation

(32) iB_tϕpt, xq “ ´1

2∆ϕpt, xq `β|ϕpt, xq|^2σϕpt, xq, with initial datumϕ_in.

The original relaxation method applied to (32) would consists in adding the variable Υ to (32) and discretizing the following continuous system as discrete times t_n andt_n`1{2

(33)

# Υpt, xq “ |ϕpt, xq|^2σ, iBtϕpt, xq “ ´1

2∆ϕpt, xq `βΥϕpt, xq.

It is however known to not conserve energy functional.

As a generalization of the classical relaxation method (10), which is designed for the special cubic case (σ“1), we propose the following method, which allows for a general nonlinearity exponentσPN^‹. We propose to substitute system (33) by

(34)

# γ^σpt, xq “ |ϕpt, xq|^2σ, iBtϕpt, xq “ ´1

2∆ϕpt, xq `βγ^σϕpt, xq.

The modification seems ligth but allows to build an energy preserving scheme. The second equation is approximated to second order at timet_n`1{2by

iϕ_n`1´ϕ_n

δt “

ˆ

´1

2∆`βγ<sub>n`1{2</sub>^σ

˙ϕ<sub>n`1</sub>`ϕ_n

2 .

We now want to find an approximation of γ^σ “ |ϕ|^2σ “ γ^σ´1|ϕ|² that allow energy conservation following the ideas that were presented in section 2. As for the classical relaxation method, we note that

Re ˆ

γ<sub>n`1{2</sub><sup>σ</sup> ϕn`1`ϕn

2 ϕ_n`1´ϕ_n

˙

“ γ_n`1{2^σ

2

´

|ϕ_n`1|²´ |ϕ_n|²

¯ . The last term also reads

γ^σ_n`1{2´

|ϕ_n`1|²´ |ϕ_n|²

¯

“γ_n`1{2^σ ´

|ϕ_n`1|²´ |ϕ_n|²

¯

`γ_n´1{2^σ ´

|ϕ_n|²´ |ϕ_n|²

¯

“

´

γ_{n`1{2</sub><sup>σ</sup> |ϕ<sub>n`1}|²´γ_n´1{2^σ |ϕ_n|²

¯

´

´

γ_n`1{2^σ ´γ_n´1{2^σ ¯

|ϕ_n|².

The only choice that allow to preserve energy is to choose

´

γ_n`1{2^σ ´γ_n´1{2^σ

¯

|ϕn|²“ σ σ`1

´

γ<sub>n`1{2</sub><sup>σ`1</sup> ´γ_n´1{2^σ`1

¯ . Moreover, we remark that

1 σ`1

γ<sub>n`1{2</sub><sup>σ`1</sup> ´γ_n´1{2^σ`1

δt “ 1

σ

γ_n`1{2^σ ´γ_n´1{2^σ δt |ϕ_n|² is a second order approximation ofγ^σ“γ^σ´1|ϕ|² at timet“t_n.

This method is therefore designed so that it preserves exactly the following energy E_rlxpϕ, γq “ 1

4 ż

R^d

}∇ϕ}²dx`β 2

ż

R^dγ^σ|ϕ|²dx´β σ 2pσ`1q

ż

R^dγ^σ`1dx.

Since at continuous levelγ^σ“ |ϕ|^2σ,Erlxpϕ, γqreduces to the true energy 1

4 ż

R^d

}∇ϕ}²dx` β 2σ`2

ż

R^d

|ϕ|^2σ`2dx.

Moreover, the generalized relaxation method preserves theL²-norm of the solution. We present in Theorem 10 a more general method, which includes possibly non zero other terms in the equation (see Section 4.2).

Starting with ϕ₀“ϕ_in andγ_´1{2 approximating|ϕp´δt{2q|², the generalized relaxation method is

(35)

$

’’

’’

&

’’

’’

%

γ<sub>n`1{2</sub><sup>σ`1</sup> ´γ^σ`1_n´1{2

γ<sub>n`1{2</sub><sup>σ</sup> ´γ<sup>σ</sup><sub>n´1{2</sub> “σ`1 σ |ϕ_n|², iϕ_n`1´ϕ_n

h “

ˆ

´1

2∆`βγ<sub>n`1{2</sub>^σ

˙ϕ<sub>n`1</sub>`ϕ_n

2 .

Like for the Crank-Nicolson scheme and equation (6), the first equation also reads γ_n`1{2^σ “

ˆσ`1

σ |ϕn|²´γ_n´1{2

˙˜_σ´1 ÿ

k“0

γ_n`1{2^k γ_n´1{2^σ´1´k

¸ . so we also have the other version of generalized relaxation method

(36)

$

’’

’’

&

’’

’’

%

γ_n`1{2^σ “

ˆσ`1

σ |ϕn|²´γn´1{2

˙˜_σ´1 ÿ

k“0

γ_n`1{2^k γ^σ´1´k_n´1{2

¸ ,

iϕn`1´ϕn

δt “

ˆ

´1

2∆`βγ<sub>n`1{2</sub>^σ

˙ϕn`1`ϕn

2 .

Note that, when σ “ 1, the generalized relaxation method (36) reduces to the classical relaxation method (10). In contrast to the classical relaxation method (10), whenσě2, the generalized relaxation method (36) is fully implicit on its first stage, and linearly implicit in its second stage. For small values of σ (σ “ 1,2,3,4) the first stage of (36) is polynomial of degree σ and hence we can use explicit formulas for the computation of the solutionγ_n`1{2. For example, for quintic nonlinearity andσ“2, we have explicit solutions to the quadratic equation

γ_{n`1{2</sub><sup>2</sup> ´k1γ<sub>n`1{2}´k1γ_n´1{2“0, where k1“ p3{2q|ϕn|²´γ_n´1{2.