NUMERICAL METHODS AND ANALYSIS OF COMPUTING QUASIPERIODIC SYSTEMS KAI JIANG SHIFENG LIAND PINGWEN ZHANG

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NUMERICAL METHODS AND ANALYSIS OF COMPUTING

QUASIPERIODIC SYSTEMS ∗

KAI JIANG†, SHIFENG LI†,AND PINGWEN ZHANG‡

Abstract. Quasiperiodic systems are important space-ﬁlling ordered structures, without decay

and translational invariance. How to solve quasiperiodic systems accurately and eﬃciently is of

great challenge. A useful approach, the projection method (PM) [J. Comput. Phys., 256: 428,

2014], has been proposed to compute quasiperiodic systems. Various studies have demonstrated that

the PM is an accurate and eﬃcient method to solve quasiperiodic systems. However, there is a

lack of theoretical analysis of PM. In this paper, we present a rigorous convergence analysis of the

PM by establishing a mathematical framework of quasiperiodic functions and their high-dimensional

periodic functions. We also give a theoretical analysis of quasiperiodic spectral method (QSM) based

on this framework. Results demonstrate that PM and QSM both have exponential decay, and the

QSM (PM) is a generalization of the periodic Fourier spectral (pseudo-spectral) method. Then we

analyze the computational complexity of PM and QSM in calculating quasiperiodic systems. The

PM can use fast Fourier transform, while the QSM cannot. Moreover, we investigate the accuracy

and eﬃciency of PM, QSM and periodic approximation method in solving the linear time-dependent

quasiperiodic Schr¨odinger equation.

Key words. Quasiperiodic systems, Quasiperiodic spectral method, Projection method, Birkhoﬀ’s

ergodic theorem, Error estimation, Time-dependent quasiperiodic Schr¨odinger equation.

AMS subject classiﬁcations. 42A75, 65T40, 68W40, 74S25

1. Introduction. Quasiperiodic systems are a natural extension of periodic sys-

tems. The earliest quasiperiodic system can trace back to the study of three-body

problem [1]. Many physical systems can fall into the set of quasiperiodicity, includ-

ing periodic systems, incommensurate structures, quasicrystals, many-body problems,

polycrystalline materials, and quasiperiodic quantum systems [1,2,3,4]. The math-

ematical study of quasiperiodic orders is a beautiful synthesis of geometry, analysis,

algebra, dynamic system, and number theory [5,6]. The theory of quasiperiodic func-

tions, even more general almost periodic functions, has been well developed to study

quasiperiodic systems in mathematics [7,8,9]. However, how to numerically solve

quasiperiodic systems in an accurate and eﬃcient way is still of great challenge.

Generally speaking, quasiperiodic systems, related to irrational numbers, are

space-ﬁlling ordered structures, without decay nor translational invariance. This

rises diﬃculty in numerically computing quasiperiodic systems. To study such im-

portant systems, several numerical methods have been developed. A widely used

approach, the periodic approximation method (PAM), employs a periodic function to

approximate the quasiperiodic function [10]. The conventional viewpoint is that the

approximation error could uniformly decay as the supercell gradually becomes large.

However, a recent theoretical analysis has demonstrated that the error of PAM may

not uniformly decrease as the calculation area increases [11]. The second method is the

quasiperiodic spectral method (QSM), which approximates quasiperiodic function by

a ﬁnite summation of trigonometric polynomials based on the continuous Fourier-Bohr

∗Submitted to ...

Funding:

†Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Lab-

oratory of Intelligent Computing and Information Processing of Ministry of Education, School of

Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan, China, 411105.

(kaijiang@xtu.edu.cn,shifengli@smail.xtu.edu.cn).

‡School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, School of Mathemat-

ical Sciences, Peking University, Beijing, 100871, China. (pzhang@pku.edu.cn).

This manuscript is for review purposes only.

arXiv:2210.04384v3 [math.NA] 17 Jan 2024

transform [10], also see Subsection 3.1. The third approach is the projection method

(PM) [12], based on the fact that the quasiperiodic system can be embedded into a

high-dimensional periodic system. Then the PM can accurately calculate the high-

dimensional periodic system over a torus in a pseudo-spectral manner. Meanwhile,

the PM is eﬃcient due to the availability of fast Fourier transform (FFT). Finally, the

PM obtains the quasiperiodic system by choosing a corresponding irrational slice of

the high-dimensional torus by the projection matrix. Extensive studies have demon-

strated that the PM can be used to compute quasiperiodic systems to high precision,

including quasicrystals [13,14], incommensurate quantum systems [15,16,17], topo-

logical insulators [18], and grain boundaries [19,20]. However, the PM still has a lack

of corresponding theoretical guarantees.

In this work, we present a rigorous theoretical analysis of numerical methods for

solving quasiperiodic systems. We establish the relationship between quasiperiodic

functions and their corresponding high-dimensional periodic functions based on the

idea of PM. These mathematical results provide a theoretical framework to analyze

the convergence of PM, as well as QSM. We also present another error analysis frame-

work of QSM without using high-dimensional periodic functions. These theoretical

results demonstrate that both PM and QSM have exponential convergence. Moreover,

we analyze the computational complexity of PM and QSM in solving quasiperiodic

systems. The PM can use FFT by introducing discrete Fourier-Bohr transform, see

Subsection 3.2, while the QSM cannot. Further analysis reveals that the QSM (PM)

is an extension of the periodic Fourier spectral (pseudo-spectral) method. Finally, we

investigate the accuracy and eﬃciency of PM, QSM, and PAM to solving the linear

time-dependent quasiperiodic Schr¨odinger equation (TQSE).

2. Preliminaries. Before our analysis, we give some preliminaries on quasiperi-

odic and periodic functions in this section.

2.1. Preliminaries of quasiperiodic functions. Let us recall the deﬁnition

of the quasiperiodic function [9]. Denote

Md×n={M= (m1,··· ,mn)∈Rd×n:m1,··· ,mnare Q-linearly independent},

and deﬁne P∈Md×nas the projection matrix.

Definition 2.1. Ad-dimensional function f(x)is quasiperiodic if there exists

a continuous n-dimensional periodic function F(n≥d)which satisﬁes f(x) =

F(PTx), where Pis the projection matrix.

In particular, when n=dand Pis nonsingular, f(x) is periodic. When n→ ∞,

fis almost periodic function [7]. For convenience, Fin Deﬁnition 2.1 is called the

parent function of fin the following content. QP(Rd) represents the space of all

quasiperiodic functions. In Section 4, we will show that fand Fcan be uniquely

determined by each other when the projection matrix Pis given.

Let KT={x:x∈Rd,|xj| ≤ T, j = 1,··· , d}be the cube in Rd. The mean

value M{f(x)}of f∈QP(Rd) is deﬁned as

M{f(x)}= lim

T→+∞

(2T)dZs+KT

f(x)dx:= −

Zf(x)dx,

where the limit on the right side exists uniformly for all s∈Rd. An elementary

This manuscript is for review purposes only.

calculation shows

M{eiλTxe−iβTx}=(1,λ=β,

0,λ̸=β.

(2.1)

Correspondingly, the continuous Fourier-Bohr transform of f(x) is

fλ=M{f(x)e−iλTx},(2.2)

where λ∈Rd. Denote Λ={λ:λ=P k,k∈Zn}and the Fourier series associated

with the quasiperiodic function f(x) can be written as

f(x)∼X

k∈Zn

fλkeiλT

kx,(2.3)

where λk=P k ∈Λare Fourier exponents and ˆ

fλkdeﬁned in (2.2) are Fourier

coeﬃcients. To simplify the notation, denote ˆ

fk=ˆ

fλk. Let

QP1(Rd) = nf∈QP(Rd) : X

k∈Zn|ˆ

fk|<+∞o,

with norm ∥f∥L∞(Rd)= supx∈Rd|f(x)|.

In general, the convergence of the Fourier series (2.3) is a challenging problem,

see [9] for some suﬃcient criteria. The following conclusion presents an important

convergence property of quasiperiodic function.

Theorem 2.2. ([25] Chapter 1.3) If the Fourier series of a quasiperiodic function

is uniformly convergent, then the sum of the series is the given function.

If the Fourier series of the quasiperiodic function is absolutely convergent, it is also

uniformly convergent. Therefore, for f∈QP1(Rd), we have

f(x) = X

k∈Zn

fkeiλT

kx.

As a consequence, we can obtain a subspace QP2(Rd) of QP(Rd)

QP2(Rd) = nf∈QP(Rd) : M{|f|2}<+∞o

equipped with norm

∥f∥2

L2(Rd)=M{|f|2}=X

k∈Zn|ˆ

fk|2,(2.4)

and the inner product (·,·)QP2(Rd)

(f1, f2)QP2(Rd)=−

Zf1(x)¯

f2(x)dx.

Equality (2.4) is the Parseval’s identity. Now we introduce the Hilbert space of

quasiperiodic functions. Denote |x|=Pd

j=1 |xj|with ∀x∈Rd. For any m∈N0=

{m∈Z:m > 0}, the Sobolev space Hα

QP (Rd) comprises all quasiperiodic functions

with partial derivatives order α≥1 with respect to the inner product (·,·)α

(f1, f2)α= (f1, f2)QP2(Rd)+X

|m|=α

(∂m

xf1, ∂m

xf2)QP2(Rd),

and endowed with norm ∥f∥2

α=Pk∈Zn(1 + |λk|2)α|ˆ

fk|2,and semi-norm |f|2

α=

Pk∈Zn|λk|2α|ˆ

fk|2.

This manuscript is for review purposes only.

2.2. Preliminaries of periodic functions. Let Tn= (R/2πZ)nbe the n-

dimensional torus, then the Fourier transform of F(y) deﬁned on Tn

Fk=1

|Tn|ZTn

e−ikTyF(y)dy,k∈Zn,(2.5)

and

L∞(Tn) = nF(y) : X

k∈Zn|ˆ

Fk|<+∞o.

Further, denote the Hilbert space on Tn

L2(Tn) = nF(y) : ⟨F, F ⟩<+∞o,

equipped with inner product

⟨F1, F2⟩=1

|Tn|ZTn

F1¯

F2dy.

For any integer α≥0, the α-derivative Sobolev space on Tnis

Hα(Tn) = {F∈L2(Tn) : ∥F∥α<∞},

where ∥F∥α=Pk∈Zn(1+∥k∥2α

2)|ˆ

Fk|21/2,with ∥k∥2

2=Pn

j=1 |kj|2. The semi-norm

of Hα(Tn) can be deﬁned as |F|α=Pk∈Zn∥k∥2α

2|ˆ

Fk|21/2.

3. Algorithms. In this paper, our purpose is to establish the theoretical analysis

of QSM and PM. In this section, we introduce these algorithms before delving into

the numerical analysis. Moreover, we present the implementation framework of PM

by deﬁning the discrete Fourier-Bohr transform of quasiperiodic functions.

For an integer N∈N0and a given projection matrix P∈Md×n, denote

N={k= (kj)n

j=1 ∈Zn:−N≤kj< N},

and

Λd

N={λ=P k :k∈Kn

N} ⊂ Λ.(3.1)

Obviously, the order of the set Λd

Nis #(Λd

N) = (2N)n. The ﬁnite dimensional linear

subspace of QP(Rd) is

SN= span{eiλTx,x∈Rd,λ∈Λd

N}.

We denote PN: QP(Rd)7→ SNthe projection operator. For a quasiperiodic function

f(x)∈QP1(Rd) and its Fourier exponent λk∈Λ, we can split it into two parts

f(x) = X

k∈Kn

fkeiλT

kx+X

k∈Zn/Kn

fkeiλT

kx=PNf+ (f− PNf).(3.2)

Next, we present QSM and PM, respectively.

This manuscript is for review purposes only.

3.1. Quasiperiodic spectral method (QSM). The QSM directly approxi-

mates quasiperiodic function fby PNf,

f(x)≈ PNf(x) = X

k∈Kn

fkeiλT

kx,x∈Rd,

where the quasiperiodic Fourier coeﬃcient ˆ

fkis obtained by the continuous Fourier-

Bohr transform (2.2). We will give the error analysis of QSM in Subsection 5.1, and

describe the numerical implementation of solving quasiperiodic system in Subsec-

tion 6.1.1. Note that quasiperiodic Fourier coeﬃcients in QSM are obtained through

the continuous Fourier-Bohr transform (2.2), resulting in the QSM cannot use FFT.

A further computational complexity analysis will be presented in Subsection 6.1.1.

3.2. Projection method (PM). The PM embeds the quasiperiodic function

f(x) into a high-dimensional parent function F(y), then directly replace the discrete

quasiperiodic Fourier coeﬃcients by the discrete parent Fourier coeﬃcients [10,12].

We can use the periodic Fourier spectral method to obtain the parent Fourier coef-

ﬁcients. Concretely, we ﬁrst discretize the tours Tn. Without loss of generality, we

consider a fundamental domain [0,2π)nand assume the discrete nodes in each dimen-

sion are the same, i.e.,N1=N2=··· =Nn= 2N,N∈N0. The spatial discrete

size h=π/N. The spatial variables are evaluated on the standard numerical grid Tn

with grid points yj= (y1,j1, y2,j2, . . . , yn,jn), y1,j1=j1h,y2,j2=j2h, . . . , yn,jn=jnh,

0≤j1, j2, . . . , jn<2N. We deﬁne the grid function space

GN:= {F:Zn7→ C:Fis Tn

N-periodic}.

Given any periodic grid functions F, G ∈ GN, the ℓ2-inner product is deﬁned as

⟨F, G⟩N=1

(4πN)nX

yj∈Tn

F(yj)G(yj).

For k,ℓ∈Zn, we have the discrete orthogonality condition

⟨eikTyj, eiℓTyj⟩N=(1,k=ℓ+ 2Nm,m∈Zn,

0,otherwise.

(3.3)

The discrete Fourier coeﬃcient of F∈ GNis

Fk=⟨F, eikTyj⟩N,k∈Kn

N.(3.4)

The PM directly takes ˜

fk=˜

Fk. We deﬁne the discrete Fourier-Bohr transform of

quasiperiodic function f(x) as

f(xj) = X

λk∈Λd

fkeiλT

kxj,(3.5)

where collocation points xj=P yj,yj∈Tn

N. The trigonometric interpolation of

quasiperiodic function is

INf(x) = X

λk∈Λd

fkeiλT

kx.(3.6)

This manuscript is for review purposes only.

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NUMERICALMETHODSANDANALYSISOFCOMPUTINGQUASIPERIODICSYSTEMS∗KAIJIANG†,SHIFENGLI†,ANDPINGWENZHANG‡Abstract.Quasiperiodicsystemsareimportantspace-fillingorderedstructures,withoutdecayandtranslationalinvariance.Howtosolvequasiperiodicsystemsaccuratelyandefficientlyisofgreatchallenge.Ausefulapproach,thep...

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NUMERICAL METHODS AND ANALYSIS OF COMPUTING QUASIPERIODIC SYSTEMS KAI JIANG SHIFENG LIAND PINGWEN ZHANG

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