Bloch Decomposition-Based Gaussian Beam

Method for the Schrödinger equation with Periodic

Potentials∗

Shi Jin†, Hao Wu‡, Xu Yang§and Zhongyi Huang¶

January 19, 2010

Abstract

The linear Schrödinger equation with periodic potentials is an im-portant model in solid state physics. The most efficient direct simula-tion using a Bloch decomposition based time-splitting spectral method[18] requires the mesh size to be O(ε) where ε is the scaled semiclassi-cal parameter. In this paper, we generalize the Gaussian beam methodintroduced in [23] to solve this problem asymptotically. We combinethe technique of Bloch decomposition and the Eulerian Gaussian beammethod to arrive at an Eulerian computational method that requiresmesh size of O(

√ε). The accuracy of this method is demonstrated via

several numerical examples.

Key words: Schrödinger equation, periodic potential, Bloch de-composition, Gaussian beam method, Liouville equation

1 Introduction

The linear Schrödinger equation with periodic potentials

iε∂Ψε

∂t= −ε

2

2∆Ψε + VΓ

(xε

)Ψε + U(x)Ψε, x ∈ Rn , (1.1)

∗This work was partially supported by NSF grant No. DMS-0608720, NSF FRG grantDMS-0757285, NSAF Projects 10676017, NSFC Projects 10971115, the National BasicResearch Program of China under grant 2005CB321701. SJ was also supported by a VanVleck Distinguished Research Prize from University of Wisconsin-Madison.

†Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA(jin@math.wisc.edu)

‡Department of Mathematical Sciences, Tsinghua University, Beijing, 10084, China(hwu@tsinghua.edu.cn)

§Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA. Cur-rent address: Program in Applied and Computational Mathematics, Princeton University,Princeton, NJ 08544, USA. (xuyang@math.princeton.edu)

¶Department of Mathematical Sciences, Tsinghua University, Beijing, 10084, China(zhuang@math.tsinghua.edu.cn)

1

is a simple model in solid state physics which describes the motion of elec-trons with the periodic potentials generated by the ionic cores. Here Ψε(t,x)is the wave function, ε is the re-scaled Plank constant in the semiclassicalregime, and U(x) is the smooth external potential. The oscillatory lattice-potential VΓ(z) is a periodic function in some regular lattice Γ.

We consider this model in one dimension with the two-scale WKB initialcondition:

Ψε0(x, z :=x

ε) = A0(x, z)e

iS0(x)/ε . (1.2)

Without loss of generality we assume Γ = 2πZ, i.e.

VΓ(z + 2π) = VΓ(z), ∀z ∈ R. (1.3)

We introduce several physical concepts related to (1.3) [1]:

• The fundamental domain of the lattice Γ is C = (0, 2π).

• The dual lattice Γ∗ = Z.

• The (first) Brillouin zone is B =(−12,1

2

), which is the fundamental

domain of Γ∗.

The direct numerical simulation of (1.1)-(1.2) is prohibitively expensivedue to the small parameter ε in the semiclassical regime and the highlyoscillating structure of VΓ. The standard time-splitting spectral method [3]requires the mesh size be o(ε) and the time step be o(ε). A novel time-splitting spectral method based on the Bloch decomposition was proposedrecently by Huang, Jin, Markowich and Sparber [18, 19, 20] which relaxesthe time step requirement to be O(1) with a much coarser mesh size of O(ε).However, such a mesh size is still expensive especially in high dimensionsfor a very small ε.

One efficient alternative way is to solve (1.1)-(1.2) asymptotically by theBloch band decomposition and the modifiedWKBmethod [4], which leads toeikonal and transport equations in the semi-classical regime. The problem ofthese approaches is that they do not give accurate solution around caustics.The Gaussian beam method, developed for the high frequency linear waves[34, 38, 37, 29, 30, 23], and also in the setting of quantum mechanics [14,15, 16], provides an efficient way to compute the wave amplitude aroundcaustics. The idea is to allow the phase function to be complex and choosethe imaginary part properly so that the solution has a Gaussian profile. Thedetailed construction and its validity at caustics were analyzed by Ralstonetc in [35, 7]. All these previous works gave the Gaussian beam method inthe Lagrangian framework.

In [23], we developed an Eulerian Gaussian beam method to solve thelinear Schrödinger equation asymptotically. The method consists of solv-ing n complex-valued and 1 real-valued homogeneous Liouville equations

2

for n-space dimensional problems. The solution to this method has beenshowed to have good accuracy even at caustics, with a mesh size as coarseas O(

√ε). There have also been other Eulerian Guassian beam methods [26]

that use much more complex-valued inhomogeneous Liouville equations. Inthis paper, we generalize our method in [23] for (1.1)-(1.2) with the help ofthe Bloch decomposition. The idea is to use the Eulerian Gaussian beammethod of [23] for each of the Bloch band, and then superimpose them forall the bands. (This method is restricted to adiabatic cases which do notpermit band-crossings.) Since effectively only small number of bands areneeded numerically and the Liouville equation is solved locally in the vicin-ity of a co-dimensional zero level curve [31, 33, 28], the overall cost of thismethod is much smaller than a full simulation by directly solving (1.1) whenε is small.

For periodic potentials, every energy band could yield caustics in thesemiclassical regime. Since the solution is a superposition of many energybands, there could be many caustics thus significantly reduce the overallaccuracy of the semiclassical method. Thus methods accurate near causticsare highly desirable for such problems.

The paper is organized as follows. In Section 2 we give an overviewof the Bloch decomposition and the semiclassical limit of the Schrödingerequation with periodic structures. In Section 3, we formulate the Gaussianbeam method for solving (1.1)-(1.2) by combining the Bloch decompositionwith the Eulerian Gaussian beam method of [23]. We show the accuracy andefficiency of this Gaussian beam method through several numerical examplesin Section 4. In Section 4.3, we study an insulator example. Finally, we makesome conclusive remarks in Section 5.

2 Overview of the Bloch decomposition and thesemiclassical limit

2.1 The Bloch decomposition

Define Em(k) as them-th eigenvalue and χm(k, z) as the correspondingm-theigenfunction of the shifted Hamiltonian H(k, z):

H(k, z) :=1

2(−i∂z + k)2 + VΓ(z), (2.1)

H(k, z)χm(k, z) = Em(k)χm(k, z), (2.2)

χm(k, z + 2π) = χm(k, z), z ∈ R, k ∈ B. (2.3)

Em(k), k ∈ B, is called the m-th energy band, and {Em(k), χm(k, z)}mdescribe the spectral properties of the shifted Hamiltonian H(k, z). It hasbeen shown in [40] that there exists an ordered countable family of real

3

eigenvalues {Em(k)}∞m=1 such that

E1(k) ≤ E2(k) ≤ · · · ≤ Em(k) ≤ · · · , m ∈ N,

and the complete set of the eigenfunctions {χm(k, z)}∞m=1 for each k ∈ Bforms an orthonormal basis of L2(C). This allows for a decomposition of theinitial condition (1.2) in terms of Bloch waves with the help of the stationaryphase method (cf. [4, §3.2 and §4.7 of Chapter 4]):

Ψε0(x, z) =∞∑

m=1

a0m(x)χm(∂xS0, z)eiS0(x)/ε +O(ε), (2.4)

where the coefficient

a0m(x) =

∫ 2π0

A0(x, z)χm(∂xS0, z)dz. (2.5)

2.2 The semiclassical limit and its computation

Plugging the modified WKB ansatz :

Ψε(t, x) = A(t, x,

x

ε

)eiS(t,x)/ε, (2.6)

into (1.1) yields, to the leading order, the following eikonal equation for Smand transport equation for am via a separation of the slow scale x and fastscale x/ε (cf. [4]):

∂tSm + Em(∂xSm) + U(x) = 0 , (2.7)

∂tam + E′m(∂xSm)∂xam +

1

2am∂x

(E′m(∂xSm)

)+ βmam = 0, (2.8)

where βm(t, x) ∈ iR is given by

βm = ⟨∂tχm, χm⟩L2(C) −1

2∂x

(E′m(∂xSm)

)− i

2⟨(∂z + i∂xSm)∂xχm + ∂x(∂z + i∂xSm)χm, χm⟩L2(C)

(2.9)

with χm evaluated at k = ∂mSm(t, x) and ⟨·, ·⟩L2(C) defined as

⟨f, g⟩L2(C) =∫ 2π0

fgdz.

The solution to (1.1) is approximated by

Ψε(t, x) =∞∑

m=1

am(t, x)χm

(∂xSm,

x

ε

)eiSm(t,x)/ε +O(ε). (2.10)

4

Note that since βm(t, x) ∈ iR, the following conservation law holds ([4]):

∂t |am|2 + ∂x(E′m(∂xSm) |am|2) = 0.

The solution to the Hamilton-Jacobi equation (2.7) develops singulari-ties when caustic forms, and the correct semiclassical limit of the physicalobservables (density, velocity, etc.), as ε → 0, becomes multivalued beyondcaustics. To describe the dynamics beyond caustics, one can use the Wignertransform to obtain the Liouville equation along each band ([2]):

Lmwm = ∂twm + E′m(ξ)∂ywm − U ′(y)∂ξwm = 0 , (2.11)

where wm(t, x, ξ) > 0 is the density distribution of the m-th energy bandof the particle. The operator Lm is the linear Liouville operator for them-th energy band. The semiclassical limit initial data for wm, for (1.2), ismeasure-valued:

wm(0, x, ξ) =∣∣a0m∣∣2 δ(ξ − ∂xS0) , (2.12)

where a0m is given by (2.5).The (multivalued) physical observables such as ρm, um = ∂xSm etc. can

be evaluated by taking the moments of wm over ξ.An efficient numerical method to solve the Liouville equation (2.11) with

initial data (2.12) was introduced in [6, 21, 22], through a decomposition ofwm = fmδ(ϕm) where both fm and ϕm solve the same Liouville equation forthe m-th energy band (in the level set framework):

Lmϕm = 0, Lmfm = 0 .

One can compute the multivalued densities by

ρm(t, y) ∈{f(t, y, ξ)

|∂ξϕm|

∣∣∣ϕm(t, y, ξ) = 0} . (2.13)Based on this formulation, a level set method for the semiclassical limit of(1.2) was introduced in [27]. For the computations of multivalued solutionsto this problem see also [10, 11, 12, 13]. The problems with all these semi-classical methods is that ρm defined in (2.13) blows up at caustics since∂ξϕm = 0.

In Section 3, we will introduce the Bloch decomposition-based Gaussianbeam method to solve (1.1)-(1.2), which is a generalization of the EulerianGaussian beam method we developed in [23]. The key difference from (2.13)is that, one can get rid of the singularities of |∂ξϕm| by making ϕm complex.

2.3 Numerical computation of the Bloch bands

In this subsection, we briefly restate the numerical computation of the Blochbands {Em(k), χm(k, z)}m for convenience. The details are referred to [18,Section 2.2].

5

Define the Fourier transform of χm as

χ̂m(k, λ) =1

2π

∫ 2π0

χm(k, z)e−iλzdz.

By taking the Fourier transform of (2.2), one has

(λ+ k)2

2χ̂m(k, λ) +

1

2π

∫ 2π0

e−iλzVΓ(z)χm(k, z)dz = Em(k)χ̂m(k, λ) .

(2.14)The discrete formula of (2.14) for λ ∈ {−Λ, · · · ,Λ− 1} ⊂ Z reads as

H(k,Λ)

χ̂m(k,−Λ)χ̂m(k, 1− Λ)

...χ̂m(k,Λ− 1)

= Em(k)

χ̂m(k,−Λ)χ̂m(k, 1− Λ)

...χ̂m(k,Λ− 1)

, (2.15)where the 2Λ× 2Λ matrix H(k,Λ) is given by

H(k,Λ) =

(−Λ+k)2

2 + V̂Γ(0) V̂Γ(−1) · · · V̂Γ(1− 2Λ)V̂Γ(1)

(−Λ+1+k)22 + V̂Γ(0) · · · V̂Γ(2− 2Λ)

......

. . ....

V̂Γ(2Λ− 1) V̂Γ(2Λ− 2) · · · (Λ−1+k)2

2 + V̂Γ(0)

.

The eigenfunction χm(k, z) is computed by

χm(k, z) =

∫ +∞−∞

χ̂m(k, λ)eiλzdλ ≈

Λ−1∑λ=−Λ

χ̂m(k, λ)eiλz.

3 A Bloch decomposition-based Gaussian beammethod

In this section we give the Bloch decomposition-based Gaussian beammethodusing both the Lagrangian and Eulerian formulations. We first briefly intro-duce the Lagrangian formulation for solving the Schrödinger equation withperiodic potentials, then focus on the Eulerian formulation.

3.1 The Lagrangian formulation

In this subsection, we adopt the Gaussian beam approximation to the m-thenergy band of the Schrödinger equation (1.1). Denote

φε,mla (t, x, y0) = am(t, ym)χ̃m

(∂xTm,

x

ε

)ei Tm(t,x,ym)/ε, (3.1)

6

where ym = ym(t, y0), and Tm(t, x, ym) is a second order Taylor truncatedphase function

Tm(t, x, ym) = Sm(t, ym) + pm(t, ym)(x− ym) +1

2Mm(t, ym)(x− ym)2.

Note that Sm ∈ R, pm ∈ R, am ∈ C, Mm ∈ C. χ̃m(∂xTm,

x

ε

)is χm

(k,x

ε

)with real-valued k replaced by complex-valued ∂xTm and

χ̃m(k, z) = χm(k, z) for k ∈ R.

Using the Lagrangian beam summation formula (for example, [17]) and(2.10), one has the Lagrangian Gaussian beam solution to (1.1) as

Φεla(t, x) =∞∑

m=1

∫R

1√2πε

rθ(x− ym(t, y0))φε,mla (t, x, y0)dy0, (3.2)

in which rθ ∈ C∞0 (Rn), rθ ≥ 0 is a truncation function with rθ ≡ 1 in a ballof radius θ > 0 about the origin and the trajectory of the beam center ymis chosen as

dymdt

= E′m(pm), ym(0) = y0.

By a similar derivation of the Lagrangian formulation as in [23, Section2.1], one has the set of the evolutionary ODEs (the details of the derivationare given in Appendix):

dymdt

= E′m(pm), (3.3)

dpmdt

= −U ′(ym), (3.4)

dMmdt

= −E′′m(pm)M2m − U ′′(ym), (3.5)

dSmdt

= E′m(pm)pm − Em(pm)− U(ym), (3.6)

damdt

= −12E′′mMmam + um,1(pm)U

′(ym)am , (3.7)

where um,1 is given by

um,1(k) = ⟨∂kχm, χm⟩L2(C), (3.8)

and ym = ym(t, y0), pm = pm(t, ym(t, y0)), Mm = Mm(t, ym(t, y0)), Sm =Sm(t, ym(t, y0)), am = am(t, ym(t, y0)). Note that um,1 ∈ iR, and um,1U ′(ym)is the so-called Berry phase in [5, 32, 36], which is importantly related tothe Quantum Hall effects in physics [36].

7

The equations (3.3)-(3.4) are called the ray-tracing equations; (3.5) is aRiccati equation for the Hessian Mm, which could be solved by the dynamicfirst order system of ray tracing equations:

dPmdt

= E′′m(pm)Rm,dRmdt

= −U ′′(ym)Pm, (3.9)

Mm(t, ym(t, y0)) = RmP−1m . (3.10)

According to [37, 23] we specify the initial conditions for (3.3)-(3.7) as

ym(0, y0) = y0, (3.11)

pm(0, y0) = ∂yS0(y0), (3.12)

Mm(0,y0) = ∂2yS0(y0) + i, (3.13)

Sm(0, y0) = S0(y0), (3.14)

am(0, y0) = a0m(y0), (3.15)

where a0m is given by (2.5).When one computes the Lagrangian beam summation integral using

(3.1) and (3.2), the complex-valued ∂xTm = pm + (x − ym)Mm could beapproximated by the real-valued pm with the Taylor truncated error ofO(|x− ym|), i.e.

Φεla(t, x) =∞∑

m=1

∫R

1√2πε

rθ(x−ym)am(t, ym)χ̃m(pm,x

ε)eiTm(t,x,ym)/εdy0+O(|x− ym|) .

(3.16)Since |x− ym| is of O(

√ε) (cf. [37, 23]), this approximation does not destroy

the total accuracy of the Gaussian beam method, yet it provides the benefitthat the eigenfunction χ̃m(k, z) is only evaluated for real-valued k whichimplies χ̃m(k, z) = χm(k, z).

3.2 The Eulerian formulation

In this subsection, by an application of a similar technique developed in [23]we introduce the Eulerian Gaussian beam formulation using the level setmethod to solve (1.1)-(1.2).

First, corresponding to ray tracing equations (3.3)-(3.4), an Eulerianlevel set method for computing (multivalued) velocity um = ∂xSm solves forthe zero level set of ϕm which satisfies the homogeneous Liouville equation[6, 22]:

Lmϕm = 0, (3.17)

where Lm is defined in (2.11). Next, since the Lagrangian system (3.6)-(3.7) is defined on the rays (characteristics), its Eulerian formulation can be

8

written as [26]:

LmSm = E′m(ξ)ξ − Em(ξ)− U(y), (3.18)

Lmam = −1

2E′′m(ξ)Mmam + um,1(ξ)U

′(y)am, (3.19)

and um,1 is determined by (3.8). It was observed in [23] that, if ϕm iscomplex, then Mm in (3.19) can be obtained from

Mm = −∂yϕm∂ξϕm

.

To be compatible with the initial data (3.13)-(3.15), we use the followinginitial conditions:

ϕm(0, y, ξ) = −iy + (ξ − ∂yS0) (3.20)Sm(0, y, ξ) = S0(y), am(0, y, ξ) = a

0m(y) , (3.21)

where a0m is given by (2.5).By essentially identical proofs as in [23, Theorem 3.2], one could see

that (3.20) complexifies the Liouville equation (3.17) and makes ∂ξϕm non-degenerate for all t > 0.

The multivalued velocity um is given by the zero level set of the real partof ϕm, i.e.

Re ϕm(t, y, um) = 0

Define

φε,meu (t, x, y, ξ) = am(t, y, ξ)χm(ξ,x

ε)eiTm(t,x,y,ξ)/ε, (3.22)

where

Tm(t, x, y, ξ) = Sm(t, y, ξ) + ξ(x− y) +1

2Mm(t, y, ξ)(x− y)2,

then the Eulerian beam summation formula corresponding to (3.16) is givenby (cf. [23])

Φεeu(t, x) =

∞∑m=1

∫R

∫R

1√2πε

rθ(x− y)φε,meu (t, x, y, ξ)δ(Re[ϕm])dξdy . (3.23)

Remark 3.1 Equation (3.23) could be solved by a discretized delta functionintegral method [39] or a local semi-Lagrangian method introduced in [23,Section 3.3] which is an improved version of the semi-Lagrangian methoddesigned in [26].

9

Remark 3.2 The curve integration method for the computation of phaseS from a given multivalued velocity u = ∂xS introduced in [9, 24] cannotbe used here since the integration constant, which can not be ignored whenevaluating (3.23), is different for different bands. Therefore we use the inho-mogeneous Liouville equation (3.18) to compute the phase function directly,as in [26].

Remark 3.3 Although the Liouville equations are defined in the phase space,thus the computational dimension is doubled than a direct computation ofthe Schrödinger equation (1.1), one only needs to solve the Liouville equa-tions locally in the vicinity of a co-dimensional zero level curve of Re(ϕm)[31, 33, 28], hence with a mesh size of O(

√ε), the overall cost of this method

is much smaller than a full simulation by directly solving (1.1) when ε issmall. For cost analysis of the Eulerian Gaussian beam method see [23].

4 Numerical examples

In this section, we test the accuracy of the Bloch decomposition-based Gaus-sian beam method by several numerical examples. The ‘true’ solution of theSchrödinger equation with periodic potentials is solved by the Strang split-ting spectral method [3] using small enough mesh sizes and time steps (bothof o(ε)). In the first two subsections, we study the Mathieu’s model withthe periodic potential VΓ(z) = cos z. In the last subsection, an insulatorexample is studied. In all the numerical examples of Section 4.2 and Section4.3, the truncation parameter θ in (3.23) is chosen fairly large so that thecut-off error is almost zero.

4.1 Approximations of the Bloch decomposition

We first look at the eigenvalues {Em(k)}∞m=1 and eigenfunctions {χm(k, z)}∞m=1of the shifted Hamiltonian (2.1) for Mathieu’s model. The first eight eigen-values and modulus of eigenfunctions are shown in Figures 1-2, which arecomputed by the algorithm described in Section 2.3. We notice that in Fig-ure 1 some of the eigenvalues around k = 0, ±0.5 are very close to each otherwhich may numerically cause band crossing. The issue of band crossing isitself an interesting topic which will not be studied in this paper. To avoidunnecessary numerical complication, we do not put mesh points around thesingular points (k = 0, ±0.5).Example 1. We test the accuracy of the Bloch decomposition by the fol-lowing two initial conditions

1) A0(x, z) = e−50(x+0.5)2 , S0(x) = 0.3x+ 0.1 sinx, x ∈ [−1, 0] , (4.1)

2) A0(x, z) = e−50(x+0.5)2 cos z, S0(x) = 0.3x+ 0.1 sinx, x ∈ [−1, 0]. (4.2)

10

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−1

0

1

2

3

4

5

6

7

8

9

k

1

2

3

4

5

6

7

8

Figure 1: The eigenvalues Em(k), m = 1, · · · , 8 of the Mathieu’s model.

The l2 errors of the Bloch decomposition with different ε are given inTable 1 and Table 2. As one can see, the errors are basically independentof ε and the accuracy is good even for small number of bands.

M 6 8 10 12ε = 1/128 5.49× 10−4 9.85× 10−6 1.10× 10−7 8.37× 10−10ε = 1/512 5.49× 10−4 9.85× 10−6 1.10× 10−7 8.30× 10−10ε = 1/2048 5.48× 10−4 9.53× 10−6 1.10× 10−7 8.31× 10−10

Table 1: the l2 errors of Bloch decomposition for the initial data (4.1).

M 6 8 10 12ε = 1/128 3.83× 10−3 1.15× 10−4 1.94× 10−6 2.07× 10−8ε = 1/512 3.83× 10−3 1.15× 10−4 1.94× 10−6 2.07× 10−8ε = 1/2048 3.83× 10−3 1.15× 10−4 1.94× 10−6 2.07× 10−8

Table 2: the l2 errors of Bloch decomposition for the initial data (4.2).

4.2 The Gaussian beam approximations

In this subsection, we conduct numerical experiments to show the efficiencyand accuracy of the Bloch decomposition-based Gaussian beam method.

11

Figure 2: The modulus of the eigenfunctions |χm(k, z)|2 , m = 1, · · · , 8 ofthe Mathieu’s model.

12

We take the external potential U(x) = 0 for all the examples. This is notnecessary for the numerical method, but is convenient for us to stay awayfrom the singularity points of the Bloch eigenfunctions (k = 0, ±0.5). Thesolutions of the Liouville equations (3.17)-(3.19) can be obtained using themethod of characteristics:

ϕm(t, y, ξ) = −i(y − E′m(ξ)t) + ξ − S′0(y − E′m(ξ)t),Sm(t, y, ξ) = S0(y − E′m(ξ)t) + E′m(ξ)ξt− Em(ξ)t,

am(t, y, ξ) =a0m(y)√

1 +(i+ S′′0 (y − E′m(ξ)t)

)E′′m(ξ)t

.

We will denote the Eulerian Gaussian beam solution given by (3.23) as ΦεGB.The grids are taken as ∆y = ∆ξ = 1/Ny where Ny will be specified in eachexample.

Example 2. In this example, we take (4.1) as the initial data for theSchrödinger equation (1.1). The l2 errors between the solution of the Schrödingerequation Ψε and that of the Gaussian beam method ΦεGB are given in Table3. Here we take time t = 0.2, the number of Bloch bands M = 8, the num-ber of Gaussian beams Ny = 32 (which is enough for numerical accuracyand shows the efficiency for small values of ε). The convergence rate in ε isof order 0.6730 in the l2 norm. We plot the wave amplitudes and absoluteerrors for different ε in Figure 3.

ε 1/128 1/512 1/2048||ΦεGB −Ψε||2 6.41× 10−2 2.17× 10−2 9.92× 10−3

Table 3: the l2 errors of wave function for Example 2.

Example 3 In this example, the same experiments are carried out for initialdata (4.2). With the same numerical parameters as in Example 2, the l2

errors between the solution of the Schrödinger equation Ψε and that of theGaussian beam method ΦεGB are given in Table 4. The convergence rate in εis of order 0.7054 in the l2 norm. We plot the wave amplitudes and absoluteerrors for different ε in Figure 4.

ε 1/128 1/512 1/2048||ΦεGB −Ψε||2 4.85× 10−2 1.43× 10−2 6.86× 10−3

Table 4: the l2 errors of wave function for Example 3.

13

−1 −0.8 −0.6 −0.4 −0.2 0−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

|Ψε|

|ΦGBε |

−1 −0.8 −0.6 −0.4 −0.2 0−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|Ψε−ΦGBε |

(a) ε =1

128

−1 −0.8 −0.6 −0.4 −0.2 0−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

|Ψε|

|ΦGBε |

−1 −0.8 −0.6 −0.4 −0.2 0−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|Ψε−ΦGBε |

(b) ε =1

512

−1 −0.8 −0.6 −0.4 −0.2 0−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

|Ψε|

|ΦGBε |

−1 −0.8 −0.6 −0.4 −0.2 0−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|Ψε−ΦGBε |

(c) ε =1

2048

Figure 3: Example 2, the Schrödinger solution |Ψε| versus the Gaussianbeams solution |ΦεGB| at ε =

1128 ,

1512 ,

12048 . The left figures are the com-

parisons of the wave amplitude at t = 0.2; the right figures plot the errors|Ψε − ΦεGB|.

14

−1 −0.8 −0.6 −0.4 −0.2 0 0.2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

|Ψε|

|ΦGBε |

−1 −0.8 −0.6 −0.4 −0.2 0 0.2−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|Ψε−ΦGBε |

(a) ε =1

128

−1 −0.8 −0.6 −0.4 −0.2 0 0.2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

|Ψε|

|ΦGBε |

−1 −0.8 −0.6 −0.4 −0.2 0 0.2−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|Ψε−ΦGBε |

(b) ε =1

512

−1 −0.8 −0.6 −0.4 −0.2 0 0.2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

|Ψε|

|ΦGBε |

−1 −0.8 −0.6 −0.4 −0.2 0 0.2−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|Ψε−ΦGBε |

(c) ε =1

2048

Figure 4: Example 3, the Schrödinger solution |Ψε| versus the Gaussianbeams solution |ΦεGB| at ε =

1128 ,

1512 ,

12048 . The left figures are the com-

parisons of the wave amplitude at t = 0.2; the right figures plot the errors|Ψε − ΦεGB|.

15

4.3 An application in the insulators

In this subsection, we study an insulator example where VΓ(z) = e−20(z−π)2

([8]). This gives a band structure shown in Figure 5. Since it is an insulator,the band gap is of O(1) and does not allow the band crossing phenomenonto occur.

We use the semi-Lagrangian method introduced in [26] to compute ψε,meu .We take (3.17) for instance to illustrate the main idea. In order to get ϕmat the mesh point (yj , ξk) (or the points needed around caustic points),one traces (yj , ξk) back to the initial position (yj0, ξ

k0 ) by solving (3.3)-(3.4)

with t → −t numerically, which are the time-backward bi-characteristics of(3.17). Then one simply has ϕm(t, y

j , ξk) = ϕm(0, yj0, ξ

k0 ). (3.18)-(3.19) are

treated similarly by changing Lm → d/dt and solving them along the time-backward bi-characteristics. We comment that one can still, for example,use the finite difference method to solve the Liouville equations (3.17)-(3.19),but since we use the improved semi-Lagrangian method in Remark 3.1 totake care of the beam summation (3.23) around caustics, it will be morenatural and convenient to also use this method to compute (3.17)-(3.19).The grids are taken as ∆y = ∆ξ = 1/Ny and ∆t =

√ε/45 where Ny will be

specified in each case of ε .

Example 4 We consider the external harmonic potential U(x) = 12x2 and

the initial conditions

A0(x, z) = e−50(x+π)2 cos z, S0(x) = 0.3− 0.3 sinx. (4.3)

In this example, one caustic forms at time t = 0.5 which correspondingto the peaks around x = −0.75 in Figure 6. We choose [−3π, π] as thecomputational domain which is large enough so that the zero boundarycondition could be applied. We take the number of Bloch bands M = 8 andthe number of Gaussian beams Ny ∼ ε−

12 as discussed in [23]. The l2 errors

between the ‘exact’ solution Ψε and the Gaussian beam solution ΦεGB aregiven in Table 5, and plotted in Figure 6. The convergence rate in ε is oforder 1.17 in the l2 norm.

(ε,Ny) (1/128, 24) (1/256, 32) (1/512, 48) (1/1024, 64)||ΦεGB −Ψε||2 8.34× 10−2 4.27× 10−2 1.71× 10−2 7.25× 10−3

Table 5: The l2 errors of wave function for Example 4.

5 Conclusion

In this paper, we developed an efficient Eulerian computational method forthe linear Schrödinger equation with periodic potentials. Using the Bloch

16

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

0

1

2

3

4

5

6

7

8

123

4

5

6

7

8

k

Bloch Eigenvalues for first eight bands.

Figure 5: The eigenvalues Em(k), m = 1, · · · , 8 of the insulator examplewith VΓ(z) = e

−20(z−π)2 .

−1.5 −1 −0.5 0 0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

x/2π

ε=1/256

|Ψε|

−1.5 −1 −0.5 0 0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

x/2π

ε=1/512

|Ψε|

−1.5 −1 −0.5 0 0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

x/2π

ε=1/1024

|Ψε|

−1.5 −1 −0.5 0 0.5

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

x/2π

ε=1/256

|Ψε−ΦGBε |

−1.5 −1 −0.5 0 0.5

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

x/2π

ε=1/512

|Ψε−ΦGBε |

−1.5 −1 −0.5 0 0.5

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

x/2π

ε=1/1024

|Ψε−ΦGBε |

Figure 6: Example 4, at time t = 0.5. Top: The wave amplitude for the‘exact’ solution Ψε with different ε. Bottom: The errors between the ‘exact’solution Ψε and the Gaussian beam solution ΦεGB.

17

decomposition, we generalize the Gaussian beam method introduced in [23]to solve the problem with periodic potentials asymptotically with an errorof O(

√ε), where ε is the small semiclassical parameter. While the classi-

cal numerical method, such as the recently developed Bloch-decompositionbased time-splitting spectral method, for the original Schrödinger equationrequires the mesh size to be of O(ε), this new method requires the mesh sizeto be merely of O(

√ε). Several numerical examples are given to demonstrate

the accuracy and effectiveness of this Bloch decomposition-based Gaussianbeam method.

Acknowledgement

The authors would like to thank Prof. Weinan E for suggesting the insulatorexample used in Section 4.3.

Appendix

In this appendix, we give the detailed derivation of the Lagrangian formu-lation for the Bloch decomposition-based Gaussian beam method.

For convenience we drop the index m and denote the modified WKBansatz as

Ψε(t, x, y) = a(t, y)χ̃(Tx,x

ε)eiT/ε, (A.1)

where y = y(t, y0), χ̃(Tx, z :=x

ε) is χ(k, z :=

x

ε) with the real-valued k

replaced by the complex-valued Tx and T = T (t, x, y) is given by

T (t, x, y) = S(t, y) + p(t, y)(x− y) + 12M(t, y)(x− y)2. (A.2)

Note that when x = y, Tx = p(t, y) ∈ R, which implies χ̃(Tx, z) = χ(Tx, z).The properties of the eigenfunction χ̃(k, z) for complex k could be found inan early work of W. Kohn [25].

Since in the two scale exapasion dx → ∂x +1

ε∂z, one has

Ψεt =

(da

dtχ̃+ aχ̃k

dTxdt

+i

εaχ̃

dT

dt

)eiT/ε

dxΨε =

(aχ̃kTxx +

1

εaχ̃z +

i

εaχ̃Tx

)eiT/ε,

d2xΨε =

(aχ̃kkT

2xx +

2

εaχ̃kzTxx + aχ̃kTxxx +

2i

εaχ̃kTxxTx

)eiT/ε

+

(1

ε2aχ̃zz +

2i

ε2aχ̃zTx −

1

ε2aχ̃T 2x +

i

εaχ̃Txx

)eiT/ε.

18

Plugging them into (1.1) and matching the leading order asymptotic coeffi-cient give

(Tt + ytTy)χ̃−1

2χ̃zz − iTxχ̃z +

1

2T 2x χ̃+ VΓχ̃+ Uχ̃ = 0,

which can be written as

[Tt + ytTy + U(x)]χ̃ =1

2(∂z + iTx)

2χ̃− VΓ(z)χ̃. (A.3)

Evaluating (A.3) at x = y gives

[St + yt(Sy − p) + U(y)]χ =1

2(∂z + ip)

2χ− VΓ(z)χ,

where we have used the fact that when x = y, Tx = p(t, y) ∈ R, whichimplies χ̃(Tx, z) = χ(Tx, z). This fact will be used again later.

Making use of the Bloch eigenvalue problem (2.1)-(2.3), one has

[St + yt(Sy − p) + U(y)]χ = −H(p, z)χ = −E(p)χ,

which is equivalent to

St + ytSy − pyt + E(p) + U(y) = 0. (A.4)

Taking derivative with respect to x of (A.3) gives

(Txt + ytTxy + Ux)χ̃+ (Tt + ytTy + U)χ̃kTxx

= iTxx(∂z + iTx)χ̃+

(1

2(∂z + iTx)

2 − VΓ(z))χ̃kTxx. (A.5)

Evaluating (A.5) at x = y yields

(pt + yt(py −M) + Uy)χ+ (St + yt(Sy − p) + U)Mχk

= iM(∂z + ip)χ+

(1

2(∂z + ip)

2 − VΓ(z))Mχk.

After simplification and taking inner product with χ of the above equation,

pt + ytpy + Uy = [yt − p+ i⟨χz, χ⟩]M + ⟨(E −H)χk, χ⟩M. (A.6)

We introduce a theorem below which helps our further derivation.

Theorem A.1 The derivatives of the Bloch eigenfunction E(k) satisfy thefollowing relations

E′(k) = k − iu3,E′′(k) = 1 + 2iu2 + 2iu1u3,

19

where

u1(k) = ⟨χk, χ⟩,u2(k) = ⟨χk, χz⟩ = −⟨χkz, χ⟩,u3(k) = ⟨χz, χ⟩.

Moreover, we have the equalities

⟨(E −H)χk, χ⟩ = 0,⟨(E −H)χkk, χ⟩ = 0,u2 + u2 + 2u1u3 = 0.

Proof: By taking derivatives of (2.2) with respect to k, we have

Hkχ+Hχk = E′χ+ Eχk.

Taking inner product with χ, one gets

E′ = ⟨Hkχ, χ⟩+ ⟨(H − E)χk, χ⟩

The first term of the right-hand side above gives k − iu3 because

Hk = −i∂z + k,

and the second term is zero since H is self-adjoint,

⟨(H − E)χk, χ⟩ = ⟨χk, (H − E)χ⟩ = 0.

Hence we haveE′(k) = k − iu3.

The other equalities could be easily proved similarly. �

Using these equalities, (A.6) becomes

pt + ytpy + Uy = (yt − E′(p))M. (A.7)

Taking derivative with respect to x of (A.5), we have

(Txxt + ytTxxy + Uxx)χ̃+ 2(Txt + ytTxy + Ux)χ̃kTxx

+(Tt + ytTy + U)(χ̃kkT2xx + χ̃kTxxx)

= iTxxx(∂z + iTx)χ̃− T 2xxχ̃+ 2iTxx(∂z + iTx)χ̃kTxx

+

(1

2(∂z + iTx)

2 − VΓ(z))(χ̃kkT

2xx + χ̃kTxxx).

Evaluating the last equation at x = y produces

(Mt + ytMy + Uyy)χ+ 2(yt − E′(p))χkM2

= (2ytχk − χ+ 2iχkz − 2pχk)M2 + (E −H)χkkM2.

20

Taking inner product with χ and simplifying it lead to

(Mt+ytMy+Uyy)+2(yt−E′(p))M2u1 = (2ytu1−1−2iu2−2pu1)M2. (A.8)

By matching the next order in the asymptotic expansion, one has,

(at + ytay)χ̃+ aχ̃k(Txt + ytTxy)− iaχ̃kzTxx + aχ̃kTxxTx +1

2aχ̃Txx = 0.

Evaluating it at x = y gives

(at+ytay)χ−aχkUy+aχk(yt−E′(p))M+(−χkyt− iχkz+χkp+1

2)aM = 0.

By taking the inner product with χ and simplifying it, one has

(at+ytay)−au1Uy+au1(yt−E′(p))M+(−u1yt+iu2+u1p+1

2)aM = 0. (A.9)

Considering the y−trajectory defined by

dy

dt= E′(p),

and using the equalities

2ytu1 − 1− 2iu2 − 2pu1 = 2E′u1 − 1− 2iu2 − 2pu1= 2u1(E

′ − p)− 1− 2iu2= −2u1u3 − 1− 2iu2 = −E′′,

(A.4), (A.7)-(A.9) can be written as a set of ODEs:

dy

dt= E′(p), (A.10)

dp

dt= −Uy, (A.11)

dS

dt= pE′(p)− E(p)− U, (A.12)

dM

dt= −E′′M2 − Uyy, (A.13)

da

dt= au1Uy −

1

2E′′aM. (A.14)

21

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24