Journal of Computational Physics

Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime

Shingyu Leung

^a

, Jianliang Qian

^b,*

aDepartment of Mathematics, UCI, Irvine, CA 92697-3875, United States

bDepartment of Mathematics, Michigan State University, East Lansing, MI 48824, United States

a r t i c l e i n f o

Article history:

Received 22 August 2008

Received in revised form 26 December 2008 Accepted 7 January 2009

Available online 22 January 2009

a b s t r a c t

We propose Gaussian-beam based Eulerian methods to compute semi-classical solutions of the Schrödinger equation. Traditional Gaussian beam type methods for the Schrödinger equation are based on the Lagrangian ray tracing. Based on the first Eulerian Gaussian beam framework proposed in Leung et al. [S. Leung, J. Qian, R. Burridge, Eulerian Gaussian beams for high frequency wave propagation, Geophysics 72 (2007) SM61–SM76], we develop a new Eulerian Gaussian beam method which uses global Cartesian coordinates, level-set based implicit representation and Liouville equations. The resulting method gives uniformly distributed phases and amplitudes in phase space simultaneously. To obtain semi-classical solutions to the Schrödinger equation with different initial wave functions, we only need to slightly modify the summation formula. This yields a very efficient method for computing semi-classical solutions to the Schrödinger equation. For instance, in the one-dimensional case the proposed algorithm requires onlyOðsNm²Þoperations to com- putesdifferent solutions withs different initial wave functions under the influence of the same potential, whereN¼Oð1=hÞ;his the Planck constant, andmNis the number of computed beams which depends onhweakly. Numerical experiments indicate that this Eulerian Gaussian beam approach yields accurate semi-classical solutions even at caustics.

Ó2009 Elsevier Inc. All rights reserved.

1. Introduction

It is well known that classical mechanics describes the behavior of macroscopic objects, while quantum mechanics de- scribes the behavior of microscopic objects with physical characteristics comparable in values with the Planck constant, such as elementary particles, atoms, and molecules. For macroscopic objects the Planck constant is considered to be negligible so that quantum effects can be ignored and classical mechanics in this regime provides a satisfactory approximation. However, for microscopic objects it is critical to incorporate quantum effects into modeling for the best physical fidelity. There are sev- eral equivalent quantum mechanics models, such as Schrödinger’s scheme, Heisenberg’s scheme, and Feyman’s path-integral scheme.

Here we consider the Schrödinger equation for a particle with unity mass ihUtþVðxÞUh²

2DU¼0; x2R^d; t>0; ð1Þ

Uðx;0Þ ¼A0ðxÞexp i~

s

0ðxÞ h

; ð2Þ

0021-9991/$ - see front matterÓ2009 Elsevier Inc. All rights reserved.

doi:10.1016/j.jcp.2009.01.007

*Corresponding author. Tel.: +1 517 353 6334; fax: +1 517 432 1562.

E-mail addresses:syleung@math.uci.edu(S. Leung),qian@math.msu.edu(J. Qian).

Contents lists available atScienceDirect

Journal of Computational Physics

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / j c p

whereVis real and smooth,~

s

0andA0are smooth, andhh=2

p

withha small (scaled) Planck’s constant. This equation is obtained by introducing undulatory theory into classical mechanics, the so-called quantization procedure. Because the Schrödinger equation propagates oscillations of wavelengthhin space and time, resolving such oscillations by direct finite difference methods requires many grid points per wavelength ofOðhÞ, which is costly in practice. As an alternative to obtain a numerical approximation capturing quantum effects, semi-classical methods are sought to link classical and quantum mechanics. In this paper, we design an Eulerian Gaussian-beam method for the Schrödinger equation in the semi-classical regime, where the Planck constant is small.

The spirit of semi-classical approximation is expanding the quantum wave function around classical expressions in pow- ers of contributions from quantum fluctuations. In the simplest case, we consider the following leading-order WKBJ ansatz for the quantum wave function,

Uðx;tÞ Aðx;tÞexp i

s

ðx;tÞ h

; ð3Þ

whereAðx;tÞis the amplitude function and

s

ðx;tÞis the phase function. Applying this ansatz to the Schrödinger equation and considering the leading order singularities, we have the following eikonal equation for the phase function and the transport equation for the amplitude function,

s

tþVðxÞ þ1

2

s

²_x¼0; ð4Þ

Atþ

s

xAxþ1

2D

s

A¼0; ð5Þ

with the initial data,

s

ðx;0Þ ¼

s

~0ðxÞ; ð6Þ

Aðx;0Þ ¼A0ðxÞ: ð7Þ

The transport equation for the amplitude is weakly coupled to the eikonal equation in the sense that one must first solve the eikonal equation to provide related coefficients for the transport equation. Because the eikonal equation is a non-linear first- order equation, in general there exists no global smooth (classical) solution for the equation. Consequently, the WKBJ ansatz (3)might not be valid globally in terms of global smooth eikonal and amplitude. One remedy is to seek globally defined un- ique weak solutions inphysical spaceto the eikonal equation, which leads to the so-called viscosity solution[8]. Another remedy is to solve the eikonal equation inphase spacefirst using the method of characteristics and then projecting the result- ing solution intophysical space. The essential difference between these two remedies can be seen right away: the viscosity- solution based eikonal solution in physical space results in a single-valued phase function while the method-of-character- istics based eikonal solution in physical space might result in a multivalued phase function. Which remedy is more appro- priate depends on specific applications.

For linear Schrödinger equations, multivalued phases are appropriate, and the resulting amplitudes are also multivalued.

As a result, the WKBJ ansatz(3)should be modified to accommodate multivalued phases and amplitudes, which has been done systematically in terms of uniform asymptotic solutions in[20,25]. In both[20,25], one has to pay particular attention to so-called caustics, where the projections of phase-space bi-characteristics into physical space, the so-called rays, converge and the resulting amplitudes become infinite. In[20], one has to identify caustics so as to choose appropriate expanding functions, such as Airy or Pearcy functions, to treat the boundary layer effect near caustics. In[25], one has to keep track of the so-called Keller–Maslov index to compensate for the phase shift when passing through a caustic. In general, since caustics can occur anywhere along the ray, the above two approaches are not satisfactory. In contrast, Gaussian beams pro- vide a powerful framework for constructing uniform asymptotic solutions systematically even at caustics without identify- ing caustics or keeping track of Keller–Maslov index.

The idea underlying Gaussian beams is simply to build asymptotic solutions to partial differential equations concentrated on a single curve through the domain; this single curve is nothing but a ray as shown in[33]. The existence of such solutions has been known to the pure mathematics community since sometime in the 1960s[1], and these solutions have been used to obtain results on propagation of singularities in hyperbolic PDEs[13,33]. An integral superposition of these solutions can be used to define a more general solution that is not necessarily concentrated on a single curve. Gaussian beams can be used to treat pseudo-differential equations in a natural way, including Helmholtz and Schrödinger equations.

In geophysical applications, Gaussian beam superpositions have been used for seismic wave modeling[6]and for seismic wave migration[12]. The numerical implementations in these works are based on ray-centered coordinates which prove to be computationally inefficient[6,12]. More recently, based on[33,38]a purely Eulerian computational approach was pro- posed in[17]which overcomes some of these difficulties. To the best of our knowledge, the Eulerian Gaussian beam method proposed in[17]is the first successful Eulerian Gaussian beam framework; it can be easily applied to both high frequency waves and semi-classical quantum mechanics. In[38]Lagrangian Gaussian beams are successfully constructed to simulate mountain waves, a kind of stationary gravity wave forming over mountain peaks and interfering with aviation.

In quantum mechanics, some variants of Gaussian beams, such as frozen Gaussian beams and Gaussian wave packets, have been used to construct approximate solutions to Schrödinger equations in the semi-classical regime[16,10,11]. How-

ever, these formulations were all based on the Lagrangian framework. In this paper, we propose an Eulerian formulation for Gaussian beams. We first follow the ansatz proposed in[33,38]to construct Gaussian beams to the Schrödinger equation along central rays. Mathematically, this ansatz constructs an approximate phase function with an imaginary part as a Taylor expansion around a central ray by using phase derivatives on the central ray. To have a corresponding Eulerian formulation capturing multi-valued phases and caustics, we generalize the Eulerian Gaussian beam approach first proposed in[17]to the Schrödinger equation. The Eulerian method proposed in[17]is based on level sets and paraxial Liouville equations[30,18,31]

and is designed to solve Helmholtz equations in the high frequency regime.

The advantages of this Eulerian Gaussian Beams approach over the usual Lagrangian framework are multi-fold. Unlike usual Lagrangian formulations, we obtain a uniform resolution of beam distribution so that the resulting Gaussian beam summation will have a uniform resolution as well. To obtain another asymptotic solution to the Schrödinger equation with a different initial wave function or for a different parameterh, we only slightly modify the summation formula by extracting a different level set function and corresponding computed quantities; this advantage is not shared by Lagrangian based methods. This results in a computationally very efficient algorithm when we are solving the Schrödinger equation especially for various settings of the initial wave function under the same potential. For instance, the proposed algorithm requires at mostOðsNm²Þoperations to computesdifferent solutions withsdifferent initial wave functions under the same potential, whereN¼Oð1=hÞ, andmNis the number of beams that we apply and is weakly dependent onh.

We remark that there are some other recent works for solving the Schrödinger equation in the semi-classical regime. In [22]the authors apply Wigner-transform techniques to the analysis of finite difference methods for the Schrödinger equa- tion in the case of a small Planck constant; in terms of numerical approximations of quadratic observables rather than the quantum wave function itself, they are able to obtain Wigner-measure related sharp conditions on the spatial-temporal grid which guarantee convergence for average values of observables as the Planck constant tends to zero. In[2], the authors pro- pose time-splitting spectral approximations for the linear Schrödinger equation in the semi-classical regime; in terms of numerical approximations of quadratic observables rather than the quantum wave function itself, their numerical examples and analytical considerations based on the Wigner transform show that weaker constraints on mesh sizes for the direct numerical solution of the Schrödinger equation are admissible for obtaining correct observables. In terms of computing mul- tivalued phases and amplitudes in an Eulerian framework, an approach based on level sets and Liouville equations first ini- tiated in[28]and further developed in[29,7,15], etc., has been successful; as a further development along this line, in[14]an Eulerian approach is proposed for computing multivalued physical observables for the Schrödinger equation in the semi- classical regime, but the approach unavoidably runs into difficulty at caustics, and it does not construct the quantum wave function either. Unlike[22,2]where physical observables are computed from numerical solutions obtained by direct numer- ical methods for the Schrödinger equations, the Eulerian Gaussian beam approach proposed here computes the observables from uniform asymptotic solutions of the Schrödinger equation. Since the beam-based asymptotic solution is valid even at caustics, our Eulerian Gaussian beam approach is different from the approach in[14]. See[19,34]for recent development in high frequency waves.

The rest of the paper is organized as follows. InSection 2, we summarize the Lagrangian Gaussian beam formulation and develop four different methods for initializing beam propagation. InSection 3, we develop an Eulerian Gaussian beam for- mulation and develop a semi-Lagrangian method for solving the resulting level-set equations and Liouville equations. InSec- tion 4, we analyze the complexities of the resulting algorithms. InSection 5, we show numerical examples to demonstrate the efficiency and accuracy of the algorithms.

2. Lagrangian Gaussian beams (LGB)

2.1. Construction of Gaussian beams

We would like to construct an asymptotically valid solutionWðx;t; hÞfor the Schrödinger equation, that is concentrated on a single curve

c

. Namely,jWðx;t; hÞjis small away from

c

, and it will satisfy the Schrödinger equation up toOðh^MÞfor some fixed positive numberMunder some appropriate norm. Here we are interested in constructing the lowest-order Gaussian beam so that the Schrödinger equation will be satisfied up toOð ffiffiffi

h p Þ.

To construct such Gaussian beams for the Schrödinger equation, ihUtþVðxÞUh²

2DU¼0; x2R^d; t>0; ð8Þ

we follow[33,38,24]and start with the WKBJ ansatz, Uðx;tÞ Aðx;tÞexp i

s

ðx;tÞ

h

: ð9Þ

The functionsAðx;tÞand

s

ðx;tÞare all assumed to be smooth, and these requirements are feasible because the beam solution is constructed to be concentrated on a single curve; this is the essential difference between traditional WKBJ asymptotic solutions and Gaussian beam solutions. As a result, the requirements on the phase function

s

are slightly different from those of traditional WKBJ asymptotics. We will require that

s

is real valued on

c

, but away from this curve

c

;

s

can be complex

valued with the restriction that the imaginary part of the second-order derivative

s

xxis positive definite. This will makeU look like a Gaussian distribution with variancehon planes perpendicular to

c

.

Define the following Hamiltonian for the Schrödinger equation, Hðx;pÞ ¼p²

2 þVðxÞ: ð10Þ

According to the Gaussian beam theory[33], such a curve

c

is nothing but thex-projection of bicharacteristicsðxðtÞ;pðtÞÞ satisfying the following Hamiltonian system,

x_¼dx

dt¼Hp; xj_t¼0¼x0; p_ ¼dp

dt¼ Hx; pj_t¼0¼p0; ð11Þ

wheretis time parametrizing bicharacteristics. Along bicharacteristics, the phase function satisfies

s

_ ¼d

s

dt¼1

2p²VðxÞ;

s

jt¼0¼

s

0: ð12Þ

To determine the second order derivative

s

xxalong bicharacteristics, we solve the following variational system for matrix- valued solutionsBðtÞandCðtÞ:

B_ ¼ H^T_xpBHxxC; Bj_t¼0¼B0;

C_ ¼HppBþHpxC; Cj_t¼0¼I; ð13Þ

whereIis the identity matrix, and matrixB0is chosen to take into account the initial phase function and to have an imag- inary part which is positive definite. HereB¼Bðt;x0;p₀ÞandC¼Cðt;x0;p₀Þare taken to be the variations ofp¼pðt;x0;p₀Þ andx¼xðt;x0;p₀Þalong the bi-characteristics with respect to the initial pointx0¼

a

,

Bðt;x0;p₀Þ ¼op

o

a

^{; Cðt;x0;p0Þ ¼}

ox

o

a

^{: ð14Þ}

We notice thatBC¹yields the Hessian of the phase function

s

along the bi-characteristics. Solution to the above equations exists on any intervalt2 ½0;T. Moreover, we have the following lemma on the bound of the solution; its proof can be found in[33,38].

Lemma 2.1.Under the above assumptions, CðtÞis non-singular for any t, and ImðBC¹Þis positive definite.

SincepðtÞ ¼

s

xðtÞalong bicharacteristics, we can use the following second-order Taylor expansion to define a smooth glo- bal approximate phase function:

s

ðx;t;x0;p₀Þ ¼

s

ðt;x0;p₀Þ þpðt;x0;p₀Þ ðxxðt;x0;p₀ÞÞ þ1

2ðxxðt;x0;p₀ÞÞ^TðBC¹Þðxxðt;x0;p₀ÞÞ: ð15Þ Next we need to determine the amplitude functionA. According to the beam theory, the amplitude functionAsatisfies the transport equation,

A_ ¼ 1

2traceðBC¹ÞA;Aj_t¼0¼A0ðx0;p₀Þ: ð16Þ

The initial conditionA0depends on how the initial wave function is decomposed into a summation of Gaussians centered at different locations, as we will see inSection 2.2. Solution to this equation can be found analytically and is given by the fol- lowing lemma.

Lemma 2.2.The solution for the transport equation(16)is Aðt;x0;p₀Þ ¼ A0ðx0;p0Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detðCðt;x0;p₀ÞÞ

p : ð17Þ

Lemma 2.2can be proved by applying the divergence theorem to Eq.(5)around a ray tube; this ray tube in the time-space domain is defined by intersecting two neighboring rays with two horizontal surfaces corresponding tot⁰¼t1andt⁰¼t2; see [17]for a proof of a similar lemma.

To obtain a smooth global approximate amplitude function, we use the following extension:

Aðx;t;x0;p0Þ ¼Aðt;x0;p0Þ: ð18Þ

Inserting(15) and (18)into the WKBJ ansatz yields an asymptotically valid solution:

Wðx;t;x0;p₀Þ ¼Aðx;t;x0;p₀Þexp i

s

ðx;t;x0;p0Þ h

; ð19Þ

this beam solution is concentrated on a single smooth curve

c

which is thex-projection of the bicharacteristic emanating fromðx0;p₀Þatt¼t0¼0.

The asymptotic solution is obtained by integrating all the beams parametrized by the initial pointðx₀;p₀Þ, Uðx;tÞ ¼cð

^;h;dÞ

Z

p0

Z

x0

Wðx;t;x0;p0Þdx0dp0 ð20Þ for some normalization constantcð

;h;dÞ, where

is a parameter related to initial beam width.

2.2. Initializing Gaussian beams

We will consider the following highly oscillatory initial data for the Schrödinger equation:

Uðx;0; hÞ ¼U0ðx; hÞ ¼A0ðxÞexp i

s

~0ðxÞ h

; ð21Þ

whereA0ðxÞand~

s

0ðxÞare smooth functions, andA0ðxÞhas compact support.

In practice, we may only know thatU0ðx; hÞis highly oscillatory, but we may not know the specific expressions ofA0ðxÞ and

s

~0ðxÞ. Therefore, we will initialize beam propagation by using different strategies and superpose them accordingly.

2.2.1. Initializing by asymptotic decomposition (AD)

IfA0ðxÞand~

s

0ðxÞare given in specific expressions, then we may use the following strategy to initialize beam propagation:

xj_t¼0¼x0; pj_t¼0¼p₀¼o

s

~0

oxðx0Þ;

s

j_t¼0¼~

s

ðx0Þ;

Bj_t¼0¼B0¼o²

s

~0

ox²ðx0Þ þi

I;

Cj_t¼0¼I;

Aj_t¼0¼A0ðx0Þ; ð22Þ

whereIis the identity matrix.

Consequently, the resulting beam ingredients are functions of x0 only, and the approximate functions

s

ðx;t;x0;p₀Þ ¼

s

ðx;t;x0Þand Aðx;t;x0;p₀Þ ¼Aðx;t;x0Þ. Furthermore, the beam summation formula will be modified to be the following:

Uðx;tÞ ¼cð

;h;dÞ Z

x₀

Z

p₀

Wðx;t;x0;p0Þdðp0~

s

xðx0ÞÞdp0dx0¼cð

;h;dÞ Z

x₀

W~ðx;t;x0Þdx0; ð23Þ where

W~ðx;t;x0Þ ¼Aðx;t;x0Þexp i

s

ðx;t;x0Þ h

; Aðx;t;x0Þ ¼Aðt;x0Þ;

s

ðx;t;x0Þ ¼

s

ðt;x0Þ þpðt;x0Þ ðxxðt;x0ÞÞ þ1

2ðxxðt;x0ÞÞ^TðBC¹Þðxxðt;x0ÞÞ: ð24Þ The following lemma proved in[37]holds in terms of recovering the initial data by the initial beam summation:

Lemma 2.3. Let/₀2C¹ðR^dÞbe a real-valued function and a02C¹₀ðR^dÞ. Define uðxÞ ¼a0ðxÞexp i

h/0ðxÞ

;

v

^{ðx;yÞ ¼}

2

p

h ^d₂

a0ðyÞexp i

h/0ðyÞ þ/⁰₀ðyÞðxyÞ þ 1

2h½i/⁰⁰₀ðyÞ

ðxyÞ²

: ð25Þ

Then

uðxÞ Z

R^d

v

^ðx;yÞdy

L²

6C h

1=2

ð26Þ

for some constant C.

The parameter

in the beam decomposition controls the initial beam width since the amplitude of the beam decays away from the center in the order of

O exp

2hðxyÞ²

: ð27Þ

Theoretically, this parameter will not affect the asymptotic solution ash!0 and therefore we can arbitrarily pick this width.

One simple way is to pick unity. In this case, as seen from the estimate, the initial condition of this particular asymptotic decomposition converges to the exact initial profile in the order ofOðh¹⁼²Þ.

In numerical implementation, numerical quadrature for the beam integral will introduce errors as well. Suppose that we shoot out rays parameterized byywith a uniform spacingDx, and we approximate the integral by a simple midpoint quad- rature. Then the overall error in approximating the initial wave function will beO ^{h 1=2}þDx²

. If

¼1, then the error is Oðh¹⁼²þDx²Þ; in other words, ifhis fixed and we increase the number of beams by lettingDx!0, then the approximation error in the initial wave function is still in the order ofh¹⁼², which is undesirable.

Here we propose to balance the two errors by choosing the initial beam width

according tohandDx. Consider one Gaussian centered at zero in the form of expð

y²=2hÞ, which has the standard deviation of

r

¼ ffiffiffiffiffiffiffiffi

h=

p . Since this Gaussian decays to almost zero 3

r

away from the center, we propose to resolve this Gaussian using a fixed number of grid points.

Numerically, we use three grid points to resolve 3

r

, which gives

¼h=Dx². Another motivation of this particular choice comes from[12]in which the author makes use of a sum of shifted Gaussians to approximate unity,

1’ 1 ffiffiffiffiffiffiffi 2

p

p Dx

r

X

i

exp ðxxiÞ² 2

r

²

!

; ð28Þ

wherexi¼iDx;i¼0;1;2;. . .The conclusion leads to a restriction thatDx<2

r

. Relating this condition to

, we have

<4h=Dx². Our choice indeed satisfies this extra accuracy requirement according to[12]. Similar considerations can also be found in[10,35].

With this particular choice of the initial beam width, we have the following error estimate in the initial wave decomposition,

uðxÞ DxX

i

v

^ðx;xiÞ

L²

6CDx; ð29Þ

whereCis a constant. Givenhand

, this gives a reasonable numerical recipe for decomposing an initial wave function into Gaussians.

We remark that we may set up a mesh which is fine enough so that it will resolve Gaussians forhlarger than a certain h_min. As a result, we may not recompute necessary beam ingredients; this is essentially done in[17]for the Helmholtz equa- tion in the high frequency regime.

2.2.2. Initializing by pointwise matching decomposition (PM)

Because the amplitude functionA0has compact support, we can further improve the numerical accuracy in the initial decomposition by requiring that the summation matches with the initial wave function pointwise at the discrete level.

Mathematically, this means that we determineAðxi;0Þfor allisuch that Uðxj;0Þ ¼DxX

i

Aðxi;0Þexp i

s

0ðxj;xiÞ h

ð30Þ

for alljwith

s

0ðx;xiÞ ¼~

s

0ðxiÞ þo~

s

0ðxiÞ

ox ðxxiÞ þ1

2ðxxiÞ^To²

s

~0ðxiÞ

ox² ðxxiÞ þi1

2

ðxxiÞ²: ð31Þ

LettingAi¼Aðxi;0Þandfi¼Uðxi;0Þ, we obtainAiby solving the following system of linear equations for~A ¼ ðA1;. . .;AIÞ^T

E~A ¼F~; ð32Þ

where~F ¼ ðf1;. . .;fIÞ^T;E ¼ ½Ei;jandEi;j¼exp½i

s

0ðxj;xiÞ=hDx. Following the same argument as in the previous section, we use

¼Oðh=Dx²Þwhich resolves the Gaussian using a fixed number of grid points.

A similar decomposition has been widely used in the radial basis function method for multivariate approximation. We refer interested readers to[3]for more analysis on properties of the matrixE. In terms of a more general framework, these coefficientsA~can be found using the following projection such that fork¼1;. . .;I

Z

Uðxj;0Þexp i

s

0ðxj;xkÞ h

dxj¼ Z

DxX

i

Aðxi;0Þexp i

s

0ðxj;xiÞ h

exp i

s

0ðxj;xkÞ h

dxj: ð33Þ

2.2.3. Initializing by semi-classical Fourier transform

Applying the semi-classical Fourier transform to the initial wave functionUðx;0Þ ¼U0ðxÞ, we have

Uðp;^ 0; hÞ ¼ 1 ð2

p

hÞ^d=2

Z þ1 1

U0ðxÞexp ipx h

dx: ð34Þ

Then the following system will be solved to obtain ingredients for beam propagation:

x_¼Hp; xj_t¼0¼x0; p_ ¼ Hx; pj_t¼0¼p₀;

s

_¼1

2p²VðxÞ;

s

jt¼0¼x0p0; B_ ¼ H^T_xpBHxxC; Bj_t¼0¼i

I;

C_ ¼HppBþHpxC; Cj_t¼0¼I;

A_ ¼ 1

2traceðBC¹ÞA; Aj_t¼0¼Uðp^ ₀;0; hÞ: ð35Þ

We remark that the initial datum,

s

j_t¼0¼x0p₀, comes from the plane wave decomposition provided by the Fourier transform.

Thus, the asymptotic solution to the Schrödinger equation is obtained by integrating all the beams parametrized by the initial pointðx0;p₀Þ,

Uðx;tÞ ¼

¹⁼²

2

p

h ^dZ

p0

Z

x0

Wðx;t;x0;p₀Þdx0dp₀; ð36Þ whereWis the contribution from a single Gaussian beam defined as in(19).

It is easy to verify that att¼0 we recover the initial data in this case. Since Wðx;0;x0;p0Þ ¼Uðp^ 0;0Þexp i

s

ðx;0;x0;p₀Þ

h

; ð37Þ

where

s

ðx;0;x0;p0Þ ¼x0p0þp0 ðxx0Þ þi

2ðxx0Þ^Tðxx0Þ; ð38Þ

we have

Uðx;0Þ ¼

¹⁼²

2

p

h ^dZ

p₀

Z

x₀

Wðx;0;x0;p₀Þdx0dp₀¼

¹⁼²

2

p

h ^dZ

p₀

Z

x₀

Uðp^ ₀;0Þexp i p₀xþⁱ₂jxx0j² h

!

" #

dx0dp₀

¼ 1

ð2

p

hÞ^d=2 Z

p₀

Uðp^ ₀;0Þexp ip0x h

dp₀¼U0ðxÞ:

A similar initialization has been used in modeling mountain waves in[38].

2.2.4. Initializing by the Fourier–Bros–Iagolnitzer (FBI) transform

The Fourier–Bros–Iagolnitzer (FBI) transform[23]provides another approach for decomposing the initial data into Gaus- sians. The FBI transform is defined by the following formula:

Uðx;^ p; hÞ ¼ TU0¼

a

d;hexp p² 2h

Z

y

exp ðxipyÞ² 2h

" #

U0ðyÞdy; ð39Þ

with the normalization constant given by

a

d;h¼2^d=2ð

p

hÞ^3d=4; ð40Þ

wheredis the dimensionality, andðxipyÞ²¼ ðxipyÞ ðxipyÞ.

The main advantage of the FBI transform over the Fourier transform is that this transform can provide information simul- taneously about the local behavior ofU0and that of its semi-classical Fourier transform, namely, the microlocal behavior of the functionU0[23]. Therefore, the FBI transform provides a micro-localized representation forU0in phase space so that the total number of beams in phase space can be reduced as we will see in numerical examples.

To construct beams, we solve the following system to obtain ingredients for beam propagation:

x_¼Hp; xj_t¼0¼x0; p_ ¼ Hx; pj_t¼0¼p0;

s

_¼1

2p²VðxÞ;

s

jt¼0¼0;

B_ ¼ H^T_xpBHxxC; Bj_t¼0¼iI;

C_ ¼HppBþHpxC; Cj_t¼0¼I;

A_ ¼ 1

2traceðBC¹ÞA; Ajt¼0¼Uðx^ 0;p0; hÞ: ð41Þ

The asymptotic solution to the Schrödinger equation is obtained by integrating all the beams parametrized by the initial pointðx0;p₀Þ,

Uðx;tÞ ¼

a

d;h

Z

p₀

Z

x₀

Wðx;t;x0;p0Þdx0dp0: ð42Þ It is easy to show that att¼0 we recover the initial data. Since

Wðx;0;x0;p0Þ ¼Uðx^ 0;p0; hÞexp i

s

ðx;0;x0;p₀Þ h

; ð43Þ

where

s

ðx;0;x0;p0Þ ¼p0 ðxx0Þ þi

2ðxx0Þ^Tðxx0Þ; ð44Þ

we have

Uðx;0Þ ¼

a

d;h

Z

p0

Z

x0

Wðx;0;x0;p0Þdx0dp0¼

a

d;h

Z

x0

Z

p0

Uðx^ 0;p0; hÞexp i

hðx0xÞ p0ðx0xÞ² 2h

" #

dx0dp0¼ T TU0¼U0;

whereT is the adjoint ofT. Here we have used a property of FBI transform[23]:T T ¼Iunder some appropriate condi- tions, whereIis the identity operator.

We remark that initializing beam propagation by the FBI transform is exact theoretically and is accurate computationally up to truncation error in numerical quadrature. To the best of our knowledge, initializing beam propagation by the FBI trans- form is new.

2.3. Numerical methods 2.3.1. A numerical FBI transform

For some simple cases, we can analytically compute the integral(39). However, it is generally difficult to find the closed form ofU^for an arbitraryU0. In this section, we will discuss how to numerically evaluate the integral. In fact, the FBI trans- form is related closely to the Gabor transform

GU0ðxÞðx;pÞ ¼ Z 1

1

expð

p

ðyxÞ²Þexpð2

p

ipyÞU0ðyÞdy ð45Þ

by

Uðx;^ p;hÞ ¼

a

d;h

ffiffiffiffiffiffiffiffiffi 2

p

h

p exp ixp h

G_U

0ðx=^pffiffiffiffiffiffi_2ph

Þ

ffiffiffiffiffiffiffiffiffi 2

p

h p x; ffiffiffiffiffiffiffiffiffi

2

p

h

p p

: ð46Þ

It is still an active research area to develop a fast discrete Gabor transform[32,39]. Therefore in this paper, instead of apply- ing these newly developed methods, we simply implement the followingOðn³Þalgorithm to embed the one-dimensional ini- tial wave function in the phase space using the FBI transform, wherenis the number of grid points in each direction in phase space.

We assume thatU0has compact support and we denoteUi¼U0ðxiÞat the grid pointsxifori¼1;. . .;I. Consider an equiv- alent form of the FBI transform,

U^¼ TU0¼

a

d;h

Z

y

exp ðxyÞ²

2h þp ðxyÞi h

" #

U0ðyÞdy: ð47Þ

One way to determineUðx^ _i⁰;p_j⁰; hÞis to approximate the above integral using the midpoint quadrature,

Uðx^ i⁰;pj⁰; hÞ ¼

a

d;hDxX^I

j¼1

exp ðx_i⁰xjÞ²

2h þp_j⁰ ðx_i⁰xjÞi h

" #

Uj ð48Þ

for each individualðx_i⁰;p_j⁰Þ. InMATLAB, we do not implement this in a point-by-point fashion. For a one dimensional numer- ical FBI transform, we first construct two matricesAandBwith each entry given byðx_i⁰xjÞ²=2handiðx_i⁰xjÞ=h, respec- tively. These two matrices are independent ofp_j0and are stored separately from the integration routine. Then, for eachp_j0

we construct the matrix expðAþp_j⁰BÞ and multiply it by the vector containing U0ðx_i⁰Þ. This gives Uðx^ _i⁰;p_j⁰; hÞ for all i⁰¼1;. . .;Ifor a fixedp_j⁰.

The above approximation converges to the exact solution asDx!0. However, in practice, we have to avoid the aliasing error as in the numerical Fourier transform. For simplicity, we consider the one-dimensional case whereU0ðyÞis real. We first rewrite(39)into

TU0¼

a

d;hexp ixp h Z

y

exp ðxyÞ² 2h

" #

exp iyp h

U0ðyÞdy: ð49Þ

To well-sample the oscillations from expðiyp=hÞ, we require maxðp_min;p_maxÞDx

h <

p

2; ð50Þ

which impliesp2 ðh

p

=2Dx;h

p

=2DxÞ.

One possible improvement to the current approach is to truncate the Gaussians in the kernel and limit the evaluation of the integral in a small neighborhood of eachx. For instance, the matrix expðAÞcan be approximated by a sparse matrix by ignoring those off-diagonal entries whenðx_i⁰xjÞis significantly large. Presumably, this will reduce the computational com- plexity fromOðn³ÞtoOðn²lognÞ; however, this is left for future work.

2.3.2. Algorithm: LGB-AD

Here we summarize the Lagrangian Gaussian Beam summation algorithm for solving the one-dimensional Schrodinger equation by initializing beams based on the asymptotic decomposition (AD). It is relatively straight-forward to generalize the algorithm to higher dimensions or use different initialization for beam propagation.

Algorithm 1. LGB-AD

1. Discretize the computational domain xi¼xminþ ði1ÞDx; i¼1;2;. . .;I;

p_i¼pðxiÞ ¼

s

⁰₀ðxiÞ; i¼1;2;. . .;I;

tk¼t0þkDt; k¼1;2;. . .;K; ð51Þ

whereDx¼ ðxmaxxminÞ=ðI1ÞandDt¼ ðtft0Þ=K. Initialize

. 2. At a fixedt¼t_k, for eachi⁰¼1;. . .;I,

(a) solve the ray tracing system(11)–(16)ifA0ðx_i⁰Þ–0 using the initial conditions

xðt¼t0Þ ¼xi⁰; pðt¼t0Þ ¼pi⁰;

s

ðt¼t0Þ ¼~

s

⁰₀ðxi⁰Þ;

Aðt¼t0Þ ¼A0ðxi⁰Þ;

Bðt¼t0Þ ¼o²

s

0ðx_i⁰Þ ox² þi

I;

Cðt¼t0Þ ¼I: ð52Þ

(b) for each~i¼1;. . .;~I, compute

Wðx~i;tk;x_i⁰;p_i⁰Þ ¼AðtkÞexp i

s

ðx~i;tkÞ h

; ð53Þ

where

s

ðx~i;tkÞ ¼

s

ðtkÞ þpðtkÞðx~ixðtkÞÞ þ1

2ðx~ixðtkÞÞ^TðBðtkÞCðtkÞ¹Þðx~ixðtkÞÞ: ð54Þ 3. Sum up the individual wave function. For each~i¼1;. . .;~I,

Uðx~i;tkÞ ¼cð

;h;nÞDxX^I

i⁰¼1

Wðx~i;tk;xi⁰;pi⁰Þ: ð55Þ Theoretically,(17)is the exact solution to the transport equation for the amplitude. However, sinceCðt;x0;p₀Þis complex, we have to choose the correct branch for the square root in(17)so thatAðt;x0;p₀Þis continuous along the characteristics.

Therefore, it is easier to directly solve(16)which will automatically determine the continuous solution in time.

As described above, we approximate the integral in the Gaussian beam summation using the midpoint quadrature. We discretize thex-direction of the phase space using a mesh with grid sizeDx. This resolution is chosen solely according to the accuracy in the solution. The finer the discretization, the better the approximation will be in calculating the Gaussian summation integral using the midpoint rule. Further, in the cases when we pick

¼h=Dx², this mesh size also contributes an error ofOðDxÞto the asymptotic decomposition of the initial wave function. This accuracy requirement is different from the resolution requirement. To visualize the solution, we have to sample the continuous solutionUðx;tÞ,(20), defined for all x2R^d. These sampling locations might be different from those grid locations we use to discretize thex-direction in phase space. In general, both the number and the location of these sampling points can be arbitrary. However, to resolve fine oscil- lations in the solution, we have to sample the solution on a mesh which is finer than the scale of those oscillations. This is a requirement imposed only on the visualization step. To distinguish these sampling points from the discretization pointsxi, we have denoted them byx~iin the above algorithm.

When we initialize beam propagation in phase space using one of the transforms, we only modify the above algorithm by discretizing thep-direction usingp_j¼p_minþ ðj1ÞDp, shoot more rays, and sum more beams with a summation ofj⁰from 1 toJin the last step.

In terms of numerical implementation, the initial beams in the Lagrangian formulation are uniformly distributed over phase space, while the beam locations at the final time are not uniformly distributed. Since we have no control of these beam locations, it is generally difficult to determine the size of the computational domain and the total number of beams used.

3. Eulerian Gaussian beams (EGB)

3.1. Formulations

By the level set methodology we embed the ray tracing system into the Liouville equation in phase space. Let /ðx;p;tÞ 2R^d,wðx;p;tÞ 2R^d, andTðx;p;tÞ 2R¹. We have the following level set equations and the phase equation,

/tþHp/xHx/p¼0; /ðx;p;0Þ ¼x;

w_tþHpw_xHxw_p¼0; wðx;p;0Þ ¼p;

TtþHpTxHxTp¼1

2p²VðxÞ; Tðx;p;0Þ ¼

s

0ðxÞ: ð56Þ

Similarly we have the equations forA;BandC.

AtþHpAxHxAp¼ 1

2traceðBC¹ÞA; Aj_t¼0¼A0ðx;pÞ;

BtþHpBxHxBp¼ H^T_xpBHxxC; Bj_t¼0¼B0ðx;pÞ;

CtþHpCxHxCp¼HppBþHpxC; Cj_t¼0¼I: ð57Þ

The initial conditions for functionsAandBare specified according to how the initial wave function is decomposed into a summation of Gaussians. If we initialize beam propagation by the FBI transform, then we set B0ðx;pÞ ¼iI and A0ðx;pÞ ¼Uðx;^ p; hÞ. If we initialize beam propagation by the asymptotic decomposition or the pointwise matching decom- position, then we setB0ðx;pÞ ¼^o_ox^2~s2⁰ðxÞ þi

IandA0ðx;pÞ ¼A0ðxÞ.

Now we have all the ingredients for constructing Eulerian Gaussian beams. If the beam propagation is initialized by the Fourier transform or the FBI transform, the asymptotic solution is given by

Uðx;tÞ ¼cð

^;h;dÞ Z

p⁰

Z

x⁰

Wðx;t;x⁰;p⁰ÞDðx0;p₀;x⁰;p⁰;tÞdx⁰dp⁰; ð58Þ where

Wðx;t;x⁰;p⁰Þ ¼Aðx⁰;p⁰;tÞexp i1

h

s

ðx;t;x⁰;p⁰Þ

; ð59Þ

s

ðx;t;x⁰;p⁰Þ ¼Tðx⁰;p⁰;tÞ þp⁰ ðxx⁰Þ þ1

2ðxx⁰Þ^TðBC¹Þðxx⁰Þ ð60Þ

andDðx0;p₀;x⁰;p⁰;tÞis the Jacobian of the mapðx⁰;p⁰Þ ! ðx0;p₀Þ, whereðx0;p₀Þrefers to the initial values of the level sets/ andwarriving atðx⁰;p⁰Þat timet. Moreover, the Jacobian of the mapðx0;p₀Þ ! ðx⁰;p⁰Þsatisfies the conservation law

Dtþ ðHpDÞx ðHxDÞp¼0; Dt¼0¼1 ð61Þ and it follows thatDðx0;p₀;x⁰;p⁰;tÞ 1. This implies that the asymptotic solution is simply given by

Uðx;tÞ ¼cð

;h;nÞ Z

p⁰

Z

x⁰

Wðx;t;x⁰;p⁰Þdx⁰dp⁰: ð62Þ On the other hand, if the beam propagation is initialized by the asymptotic decomposition or the pointwise matching decom- position, we extract the necessary information by looking into the zero level set defined by

fðx;pÞ:wðx;p;tÞ o~

s

0½/ðx;p;tÞ=ox¼0g: ð63Þ For simplicity of notations, we will below use~

s

⁰₀to denote^o~_ox^s0. Then the wave function can be computed by

Uðx;tÞ ¼cð

;h;dÞ Z

p⁰

Z

x⁰

Wðx;t;x⁰;p⁰Þd½wðx⁰;p⁰;tÞ ~

s

⁰₀ð/ðx⁰;p⁰;tÞÞdx⁰dp⁰: ð64Þ To avoid discretizing thed-function, we will use the properties of thed-function to simplify the above integral; such a tech- nique has been used in[4]. We will first consider the case ofd¼1, and we will comment on the case ofdP2. Ifd¼1, we first explicitly determine the set of points where the zero level set intersects with the grid lines, i.e.

C¼ ðx; pÞ:wðxi;pÞ ¼

s

~⁰₀½/ðxi;pÞorwðx;p_jÞ ¼~

s

⁰₀½/ðx;p_jÞ fori¼1;. . .;I; j¼1;. . .;J

; ð65Þ

then we interpolate all necessary ingredients at these locations and the wave function can be integrated by Uðx;tÞ ¼cð

^;h;dÞX

C

D/ðCÞWðx;t;CÞ; ð66Þ

where the weightD/ðCÞ(the Jacobian) is computed by taking the difference in the take-off value ofxbetween two adjacent beams, as shown inFig. 1. A similar approach has been developed in[17]in the context of solving the Helmholtz equation in the high frequency regime. Such an approach can be generalized todP2 in the following way. Assuming thatx2R^d, we first determine a set of pointsCin phase spaceR^2d which samples the zero level set. For example, this can be done by the isosurfacing algorithm presented in[27]. In higher dimensions, since the initial implicit surfacefðx;pÞ:p¼r

s

^~0ðxÞg can be parameterized byx2R^d, we can apply the summation formula(66)accordingly by computing the necessary Jacobian;

this will be reported in a forthcoming paper.

We remark that even though the Lagrangian formulation(20)and the Eulerian formulation(62)are theoretically equiv- alent to each other, there are multiple advantages of the above Eulerian formulation. The first advantage is that we have uni- form resolution of beam distribution, so that the Gaussian beam summation will have uniform resolution as well. The second advantage is that we can generate wave functions under the same external potential with different initial profiles by slightly modifying the solution from the amplitude equation in(57). The idea is to useLemma 2.2which essentially means that the amplitude att>0 is proportional to the initial conditionA₀. Using this relation, we have

A^newðxi;pj;tkÞ ¼A^oldðxi;pj;tkÞA^new₀ ð/ðxi;p_j;tkÞ;wðxi;p_j;tkÞÞ

A^old₀ ð/ðxi;p_j;tkÞ;wðxi;p_j;tkÞÞ; ð67Þ whereA^oldis the solution obtained by using the initial conditionA^old₀ from the previous initial wave function, andA^newis the new amplitude solution with a different initial wave function.

3.2. A preliminary numerical method

We give the Eulerian Gaussian Beam summation algorithm for constructing wave functions when the beam propagation is initialized by either the Fourier transform or the FBI transform. It is straight-forward to generalize this algorithm to other initializations or problems in higher dimensions.

Fig. 1.Eulerian Gaussian beam with the initial wave function decomposed using the asymptotic decomposition or the pointwise matching decomposition.

The set of sampling pointsCare shown in blue dots. The useful level set is plotted using a black solid line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Algorithm 2. EGB-Tran-PDE:

1. Discretize the computational domain xi¼xminþ ði1ÞDx; Dx¼xmaxxmin

I1 ; i¼1;2;. . .;I p_j¼p_minþ ðj1ÞDp; Dp¼pmaxpmin

J1 ; j¼1;2;. . .;J tk¼t0þkDt; Dt¼tft0

K ; k¼1;2;. . .;K; ð68Þ

and initialize

and all functions/_i;j;0,w_i;j;0;Ti;j;0;Ai;j;0;Bi;j;0andCi;j;0according to(56) and (57).

2. Solve the Liouville equations according to(56) and (57). For eachi¼1;. . .;I,j¼1;. . .;Jandk¼2;. . .;K, determine /ðxi;p_j;tkÞ;wðxi;p_j;tkÞ;Tðxi;p_j;tkÞ;Aðxi;p_j;tkÞ;Bðxi;p_j;tkÞ;Cðxi;p_j;tkÞ: ð69Þ 3. Construct the individual wave function. At a fixedt¼tk, for eachi⁰¼1;. . .;Iandj⁰¼1;. . .;J, compute

Wðx~i;tk;x_i⁰;p_j⁰Þ ¼Aðx_i⁰;p_j⁰;tkÞexp i

s

ðx~i;tk;xi⁰;pj⁰Þ h

; ð70Þ

where

s

ðx~i;tk;xi⁰;pj⁰Þ ¼Tðxi⁰;pj⁰;tkÞ þpj⁰ ðx~ixi⁰Þ þ1

2ðx~ixi⁰Þ^TðBðxi⁰;pj⁰;tkÞCðxi⁰;pj⁰;tkÞ¹Þðx~ixi⁰Þ: ð71Þ 4. Sum up the individual wave function. For each~i¼1;. . .;~I,

Uðx~i;tkÞ ¼cð

^;h;nÞDxDpX^I

i⁰¼1

X^J

j⁰¼1

Wðx~i;tk;x_i⁰;p_j⁰Þ: ð72Þ

Unfortunately, this algorithm has two main drawbacks. One is related to the accuracy. To obtain accurate solution of the Liouville equations(56) and (57), we have to use a very fine computational grid even though those PDEs are solved using a high-order numerical scheme such as WENO5-TVDRK3[36]. Since the number of computational grids is essentially the same as the number of beams used in sampling the integral, this underlying grid refinement might introduce unnecessary beams to over-sample the solution.

Another drawback is related to the computational efficiency. Since the Liouville equations are hyperbolic type, the march- ing step size is restricted by the CFL condition. The complexity of the whole algorithm isOðn^2dþ1Þ, wherenis the number of grid points in each direction in phase space anddis the dimension of physical space.

3.3. Semi-Lagrangian methods (SL)

To improve accuracy and efficiency of the above algorithm, we follow[18,17]which solve Liouville Eqs.(56) and (57) using a semi-Lagrangian method. This approach can be easily generalized to higher dimensions. We apply the method of characteristics to the level set equations and the phase equation, giving

D/

Dt ¼0;

Dw Dt ¼0;

DT Dt¼1

2p²VðxÞ; ð73Þ

whereD=Dtis the material derivative defined by D

Dt¼o otþHp

o oxHx

o

op: ð74Þ

At each grid pointðx_i;p_j;t_kÞfori¼1;. . .;I,j¼1;. . .;Jandk¼2;. . .;Kin the phase space, one tracesbackwardfromt¼t_kto t¼t0along the characteristic by integrating^dx_dt¼Hpand^dp_dt¼ Hxto obtainðxðt0Þ;pðt0ÞÞ. For the level set equations, we assign /ðxi;p_j;tkÞ ¼xðt0Þandwðxi;p_j;tkÞ ¼pðt0Þ. For the phase equation, we use the reciprocal principle and integrate the source term½pðtÞ²=2VðxðtÞÞalong the characteristic to obtainTðxi;p_j;tkÞ.

As forA;BandC, we apply the method of characteristics and obtain DA

Dt¼ 1

2traceðBC¹ÞA; Aj_t¼0¼A0ð/ðxi;p_j;tkÞ;wðxi;p_j;tkÞÞ;

DB

Dt¼ H^T_xpBHxxC; Bj_t¼0¼B0ð/ðxi;pj;tkÞ;wðxi;pj;tkÞÞ;

DC

Dt ¼HppBþHxpC; Cjt¼0¼C0ð/ðxi;pj;tkÞ;wðxi;pj;tkÞÞ; ð75Þ

with the initial conditions imposed on the levelt¼t0. In this case, we do not have the reciprocal principle as for the phase equation anymore; we need to use theforwardray tracing to solve these quantities along the same characteristic provided by the backward ray tracing. This means that we first compute the ray trajectory by integrating^dx_dt¼Hpand^dp_dt¼ Hxbackward int, and then we integrate(75)forward intalong the same characteristic, as shown inFig. 2.

In terms of numerical integration of the Hamiltonian system, we adopt a symplectic scheme. Specifically, to preserve the symplectic structure of the Hamiltonian system,

q_ ¼fðq;

v

Þ;

v

_ ¼gðq;

v

Þ; ð76Þ

we adopt the following fourth order implicit Runge–Kutta method[5,9]:

ki¼f q_nþhX^s

j¼1

aijkj;

v

ⁿþhX^s

j¼1

^aijlj

!

;

li¼g q_nþhX^s

j¼1

aijkj;

v

nþhX^s

j¼1

a^ijlj

!

;

qnþ1¼qnþhX^s

i¼1

biki;

v

nþ1¼

v

nþhX^s

i¼1

bili; ð77Þ

wheres¼2 and

½aij ¼ ½^aij ¼ 1=4 1=4 ffiffiffi p3

=6 1=4þ ffiffiffi

p3

=6 1=4

!

: ð78Þ

andbi¼1=2 (i¼1;2). The equations forkiandli(i¼1;. . .;s) are nonlinear and have to be solved by fixed-point iteration, provided that the step sizehis sufficiently small.

Therefore, the accuracy in the solution now depends solely on the size of the time stepDtbut independent of the number of grid points in discretizing the phase space, since we treat each of these grid points independently. This allows us to use relatively small number of beams to sample the solution, which in most cases are sufficient to give reasonable qualitative solution.

Fig. 2.Eulerian Gaussian beam computed using the semi-Lagrangian method.

Table 1

Complexity of various Eulerian Gaussian beam algorithms.

AD PM FBI

Wave function decomposition n n³ b

Evolution n³ n³ n³

Extraction nlogn nlogn –

Summation Nnlogn Nnlogn Nn²

Total n³þNnlogn n³þNnlogn n³þbþNn²

Each extra wave function Nnlogn n³þNnlogn bþNn²

Algorithm 3. EGB-Tran-SL:

1. Discretize the computational domain xi¼xminþ ði1ÞDx; Dx¼xmaxxmin

I1 ; i¼1;2;. . .;I;

pj¼pminþ ðj1ÞDp; Dp¼p_maxp_min

J1 ; j¼1;2;. . .;J;

tk¼t0þkDt; Dt¼tft0

K ; k¼1;2;. . .;K ð79Þ

and initialize

. Table 2

Complexity of various algorithms for solvingsdifferent initial wave functions or differenth.

FDCN sN²

SP2 sNlogN

LGB-AD sðn²þNnÞ

LGB-PM sðn³þNnÞ

LGB-FBI sðn³þbþNn²Þ

EGB-AD n³þsNnlogn

EGB-PM sn³þsNnlogn

EGB-FBI n³þsbþsNn²

−1 −0.5 0 0.5 1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x

t

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

x

p

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

x

p

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

x

p

a b

d c

Fig. 3.(Example 5.1) The initial wave function is decomposed using the asymptotic decompositions or the pointwise matching decomposition. (a) Rays in thextspace; (b) the terminal locations of bicharacteristics in thexpspace att¼0:5; (c) the terminal locations of bicharacteristics in thexpspace at t¼1:0; (d) the terminal locations of bicharacteristics in thexpspace att¼2:0.

x

p

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

1 2 3 4 5 6 7 8 9 10 11

Fig. 4.(Example 5.1) Contour plot ofjA0jobtained by embedding the initial wave function in phase space using the FBI transform.