A GENERALIZED EXPANSION METHOD FOR COMPUTING LAPLACEBELTRAMI EIGENFUNCTIONS ON MANIFOLDS JACKSON C. TURNER ELENA CHERKAEV AND DONG WANG

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A GENERALIZED EXPANSION METHOD FOR COMPUTING

LAPLACE–BELTRAMI EIGENFUNCTIONS ON MANIFOLDS

JACKSON C. TURNER, ELENA CHERKAEV, AND DONG WANG

Abstract. Eigendecomposition of the Laplace–Beltrami operator is instrumental for a variety of

applications from physics to data science. We develop a numerical method of computation of the

eigenvalues and eigenfunctions of the Laplace–Beltrami operator on a smooth bounded domain

based on the relaxation to the Schr¨odinger operator with ﬁnite potential on a Riemannian manifold

and projection in a special basis. We prove spectral exactness of the method and provide examples

of calculated results and applications, particularly, in quantum billiards on manifolds.

1. Introduction

The Laplace–Beltrami operator plays an important role in the diﬀerential equations that describe

many physical systems. These include, for example, vibrating membranes, ﬂuid ﬂow, heat ﬂow, and

solutions to the Schr¨odinger equation. Another example is that of spectral partitions—collections

of kpairwise disjoint open subsets such that the sum of their ﬁrst Laplace–Beltrami eigenvalues

is minimal [13,14,38,39]. This has a wide class of applications including data classiﬁcation [32],

interacting agents [18,16,19], and so on. In all the above applications, the fundamental question

is how to eﬃciently compute the eigenvalues of the Laplace–Beltrami operator in an arbitrary

domain with a proper boundary condition. Also, the Laplace–Beltrami operator is crucial to

understanding systems described by nonlinear Schr¨odinger equations, such as the propagation of

Langmuir waves in an ionized plasma [24,7], the single-particle ground-state wavefunction in a

Bose–Einstein condensate [7], the slowly-varying envelope of light waves in Kerr media [20], and

water surface wave packets [41].

The Laplace–Beltrami operator of a scalar function fon a Riemannian manifold (M, g) is deﬁned

as the surface divergence of the vector ﬁeld gradient of f,

(1.1) ∆gf=∇·∇f.

The divergence of a vector ﬁeld Xwith metric gis (in Einstein notation)

(1.2) ∇ · X=1

p|g|∂ip|g|Xi,

and the gradient of a scalar function fis

(1.3) (grad f)i=∂if=gij∂jf.

Department of Applied Physics and Applied Mathematics, Columbia University, New York City

10027

Department of Mathematics, University of Utah, Salt Lake City 84112

School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong,

518172, China, Shenzhen International Center for Industrial and Applied Mathematics, Shenzhen

Research Institute of Big Data, Guangdong, 518172, China

E-mail addresses:jackson.turner@columbia.edu, elena@math.utah.edu, wangdong@cuhk.edu.cn.

2020 Mathematics Subject Classiﬁcation. 35J05, 65N85, 47A70, 65N25,

Key words and phrases. Laplace operator, Laplace–Beltrami operator, ﬁctitious domain methods, expansion

method, quantum billiards.

arXiv:2210.10982v1 [math.NA] 20 Oct 2022

From above, we obtain the Laplace–Beltrami operator acting on functions over (M, g),

(1.4) ∆gf=1

p|g|∂ip|g|gij∂jf.

In general, the Helmholtz equation (i.e. Laplace–Beltrami eigenvalue problem) with Dirichlet

boundary conditions on Ω ⊂(M, g) is

(1.5) (−∆gu(x) = λu(x), x ∈Ω

u(x)=0, x ∈∂Ω.

In this paper, we develop a numerical method to ﬁnd the eigenvalues and eigenfunctions of the

Laplace–Beltrami operator with Dirichlet and periodic boundary conditions for arbitrary domains

on various surfaces. The idea is highly motivated by the Schr¨odinger operator and is based oﬀ

the method given in [27]. By using the Schr¨odinger operator relaxation, we relax the eigenvalue

problem on an arbitrary domain into the eigenvalue problem for a Schr¨odinger operator on a regular

domain which is convenient for the numerical discretization.

In [34], comparable methods on manifolds using linear and cubic FEM operators and discrete

geometric Laplacians are explored, and [17] provides a method for hyperbolic domains. There is

extensive literature on the Laplacian for planar regions [26,29,12,8,1]. In methods for solving

nonlinear Schr¨odinger equations, ﬁnite diﬀerence discretizations of the Laplace operator are often

used [5,6,16]. It is likely many of these methods above can be extended to the Laplace–Beltrami

operator on manifolds. The method we present in this paper has some immediate advantages

over the ﬁnite diﬀerence method—since the boundary of domains are characterized by a potential

function (see Theorem 7), no creation of a complicated mesh is needed, allowing for more generic

domains and producing smooth solutions. Also, the method has promise to be quite robust in

discretizing the operator on domains with corners (as in Table I), especially in applications when

computation of many eigenvalues is required, whereas the ﬁnite diﬀerence method is notoriously

ineﬃcient on such domains.

The rest of the paper is organized as follows. In Section 2, we recall the Schr¨odinger operator and

introduce the generalized expansion method. We discuss and prove the convergence and accuracy

of the relaxation and approximation in Section 3and show extensive numerical experiments in

Section 4. We investigate applications to spherical domains, periodic domains, and billiard problems

in Section 5and draw some conclusion in Section 6.

2. Generalized Expansion Method

The time-independent Schr¨odinger equation on a Riemannian manifold Mwith metric g, poten-

tial V(x), and energy levels Enis

(2.1) ˆ

Hψn(x)=[−∆g+V(x)]ψn(x) = Enψn(x),

where ∆gis the Laplace–Beltrami operator on (M, g) as in (1.4). The Schr¨odinger equation is an

eigenvalue problem for the Schr¨odinger operator ˆ

H=−∆g+V(x). When

(2.2) V(x) = 0x∈Ω

∞x6∈ Ω,

the eigenvalue problem for ˆ

His equivalent to (1.5). Eigenfunctions are normalized by setting

(2.3) ZΩ|ψn(x)|2dx = 1,

where |ψn(x)|2dx is a probability density.

In [27], a method is given to solve (1.5) with g=I2on any bounded smooth Ω ⊂R2by embedding

it in a rectangle, as in Figure 1. In order to evaluate the Laplace–Beltrami eigenvalues for Ω on

a 2-D surface, we generalize this method when considering Ω as a smooth subset of a bounded

manifold S= (M, g) using a complete set of orthonormal eigenfunctions FS

∞={φn}∞

n=1 on Swith

corresponding eigenvalues {λn(S)}∞

n=1 (with λ1< λ2≤λ3≤. . . ). Here, we assume all φn∈ FS

∞

have Dirichlet boundary conditions on ∂S, but for cases when |∂Ω∩∂S|>0, one may choose to

use other boundary conditions to obtain solutions of (1.5) with u|x∈∂Ω∩∂S 6≡ 0, as in the periodic

examples in Section 5.2.

Ω

Figure 1. With the expansion method, a bounded domain Ω is embedded onto a rectangle

Swith euclidean geometry and ˆ

His projected onto FS

In this method, we use FS

N={φn}N

n=1 as a basis on which we expand the operator ˆ

Hand seek

solutions of (1.5), or the equivalent equation involving the eigenvalue problem of the Schr¨odinger

operator ˆ

(2.4) ˆ

Hψ(x) = h−∆g+˜

V(x)iψ(x) = λψ(x),

with ˜

V(x) deﬁned as

(2.5) ˜

V(x) = 0x∈Ω

∞x∈S\Ω.

We approximate ˜

V(x) as

(2.6) V(x) = V0χS\Ω(x) = 0x∈Ω

V01x∈S\Ω.

This allows us to discretize the operator ˆ

H≈HN,

(2.7) HNnm =hφn,ˆ

Hφmi=λS

nδnm +ZS

V(x)φ∗

nφmdx,

where we truncate n, m ≤N. The eigenvalues and eigenvectors of HNapproximate those of ˆ

3. Convergence analysis

In this section, we provide a rigorous proof on the convergence of the proposed method in the

sense of V0→ ∞ and N→ ∞. To keep self-consistency of the paper, we ﬁrst recall some deﬁnitions

and preliminary results from [25,36,22].

Deﬁnition 1. Suppose Anand Aare self-adjoint operators. We say that Anconverges to Ain the

strong resolvent sense, if

(3.1) k(RAn(z)−RA(z))φk → 0,∀φ∈D(An)

for some z∈Γ = C\Σ,Σ = σ(A)∪Snσ(An)where the function RA(z)is the resolvent of A.

Deﬁnition 2. A subset D0⊆D(A)is a core of Awhen {(x, Ax) : x∈D0}is dense in {(x, Ax) :

x∈D(A)}.

Lemma 3. (6.36 of [36]): Let An, A be self-adjoint operators. Then Anconverges to Ain the

strong resolvent sense if there is a core D0of Asuch that for any ψ∈D0we have Pnψ∈D(An)

for nsuﬃciently large and AnPnψ→Aψ.

Theorem 4. (6.38 of [36]). Let Anand Abe self-adjoint operators. If Anconverges to Ain the

strong resolvent sense, we have σ(A)⊆limn→∞ σ(An).

Theorem 5. (2.2.3 of [25]). Let Lnbe a sequence of uniformly elliptic operators deﬁned on an

open set Dby

(3.2) Lnu:= −

i,j=1

∂

∂xian

ij(x)∂u

∂xj+an

0(x)u.

We assume that, for ﬁxed i, j, the sequence an

i,j is bounded in L∞and converge almost everywhere

to a function ai,j; we also assume that the sequence an

0is bounded in L∞and converges weakly-*

in L∞to a function a0. Let Lbe the (elliptic) operator deﬁned on Das in (3.2)by the functions

ai,j and a0. Then each eigenvalue of Lnconverges to the corresponding eigenvalue of L.

Theorem 6. (9.29 of [22]) Let u∈W2,p(S)∩C0(¯

S)satisfy Lu =fin S, u =ϕon ∂S where

f∈Lp(S), ϕ ∈Cβ(¯

S)for some β > 0, and suppose that ∂S satisﬁes a uniform exterior cone

condition. Then u∈Cα(¯

S)for some α > 0.

Now, we prove spectral exactness of the expansion method in V0and Nin Theorems 7and 9.

We give an example of calculating solutions and the rate of convergence in V0of eigenvalues for the

relaxed problem on an interval in Example 8. We provide intuition for eﬃcient implementation of

the expansion method in Remark 10.

Theorem 7. The eigenvalues of the Schr¨odinger operator

(3.3) ˆ

H(V0) = −∆g+V(x), V (x) = 0, x ∈Ω

V0, x 6∈ Ω

acting on a bounded Riemannan manifold (M, g), with Ωsmooth in (M, g)converge monotonically

to the eigenvalues of −∆gwith Dirichlet boundary conditions on Ωas V0→ ∞.

Proof. By considering the volume form on the manifold, we have the inner product:

(3.4) hf1, f2ig=ZM

f1(x)f2(x)(det g)1/2dx1···dxn

Now, from the Rayleigh quotient of an elliptic linear operator Lon a Riemannian manifold (M, g),

(3.5) R(L, u) = hu, Luig

hu, uig

we have

(3.6) R(ˆ

H(V0), u) = hu, −∆guig

hu, uig

+hu, V uig

hu, uig

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摘要：

AGENERALIZEDEXPANSIONMETHODFORCOMPUTINGLAPLACE{BELTRAMIEIGENFUNCTIONSONMANIFOLDSJACKSONC.TURNER,ELENACHERKAEV,ANDDONGWANGAbstract.EigendecompositionoftheLaplace{Beltramioperatorisinstrumentalforavarietyofapplicationsfromphysicstodatascience.Wedevelopanumericalmethodofcomputationoftheeigenvaluesandei...

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A GENERALIZED EXPANSION METHOD FOR COMPUTING LAPLACEBELTRAMI EIGENFUNCTIONS ON MANIFOLDS JACKSON C. TURNER ELENA CHERKAEV AND DONG WANG

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