COMPUTATION OF LAPLACIAN EIGENVALUES OF TWO-DIMENSIONAL SHAPES WITH DIHEDRAL SYMMETRY DAVID BERGHAUS ROBERT STEPHEN JONES HARTMUT MONIEN

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COMPUTATION OF LAPLACIAN EIGENVALUES OF

TWO-DIMENSIONAL SHAPES WITH DIHEDRAL SYMMETRY

DAVID BERGHAUS, ROBERT STEPHEN JONES, HARTMUT MONIEN,

AND DANYLO RADCHENKO

Abstract. We numerically compute the lowest Laplacian eigenvalues of several two-

dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our ap-

proach is based on the method of particular solutions with domain decomposition. We

are particularly interested in asymptotic expansions of the eigenvalues λ(n) of shapes with

nedges that are of the form λ(n)∼xP∞

k=0

Ck(x)

nkwhere xis the limiting eigenvalue for

n→ ∞. Expansions of this form have previously only been known for regular polygons

with Dirichlet boundary condition and (quite surprisingly) involve Riemann zeta values and

single-valued multiple zeta values, which makes them interesting to study. We provide nu-

merical evidence for closed-form expressions of higher order Ck(x) and give more examples

of shapes for which such expansions are possible (including regular polygons with Neumann

boundary condition, regular star polygons and star shapes with sinusoidal boundary).

1. Introduction

Let Ψ be a function deﬁned on a domain Ω ⊂R2that satisﬁes the Laplace eigenvalue

equation

(1.1) −∆Ψ(x, y) = λ·Ψ(x, y),

where ∆ = ∂2

x+∂2

ydenotes the Laplacian in two-dimensional Euclidean space and λ∈R. We

consider the case when the two-dimensional shape Ω has the symmetry group of a regular

n-gon, and Ψ has either Dirichlet or Neumann boundary condition on ∂Ω

Ψ|∂Ω= 0 (Dirichlet) ,

∂~nΨ|∂Ω= 0 (Neumann) ,

where ∂~n denotes the normal derivative. By combining Eq. (1.1) with the boundary condi-

tion, one obtains a discrete spectrum of eigenvalues

0< λ1≤λ2≤λ3≤. . . ,

(in the case of Neumann boundary condition we additionally have λ0= 0). The eigenvalues

are invariant under translations and rotations but do depend on the area of the considered

shape. More precisely, if Ω0is obtained from Ω by a homothety, then one has

(1.2) λi(Ω)A(Ω) = λi(Ω0)A(Ω0),

where Adenotes the area of the corresponding shape. Keeping the area constant is therefore

crucial for investigating 1/n expansions. Throughout this paper, we will always normalize

the domains to have area π, which means that the shapes will approach the unit disk in the

limit as n→ ∞.

We start by considering regular polygons with Dirichlet boundary condition. Additional

examples will be introduced in Section 7. There are so far only two regular polygons whose

eigenvalues are known explicitly. These are the regular triangle where λ1= 4π/

√3 and the

square with λ1= 2π, see, e.g., P´olya and Szeg¨o [32] (one should however mention that

some eigenmodes of the regular hexagon can be obtained by piecing together eigenmodes of

regular triangles). The remaining regular polygons oﬀer challenges, both analytically and

numerically, due to the presence of non-analytic vertices (i.e. vertices that are not of the

arXiv:2210.13229v1 [math.NA] 24 Oct 2022

2 D. BERGHAUS, R. S. JONES, H. MONIEN, AND D. RADCHENKO

form π/n). It has been known from works of several authors [2,7,18,25,30] that the lowest

eigenvalue of a regular n-gon with Dirichlet boundary condition can be approximated by a

series in 1/n

(1.3) λ1(n)∼x

∞

k=0

Ck(x)

nk,

where x=j2

0,1is the liming eigenvalue (i.e., the lowest eigenvalue of the unit disk, which is

given by the square of the ﬁrst root of the Bessel function J0(x)). Since regular polygons are

approaching the circle in the limit, it is natural that C0= 1. Interestingly, the remaining

coeﬃcients seemed to be expressible in closed-form as polynomials in x=j2

0,1of degree

bi−3

2cwith coeﬃcients expressible in terms of Riemann zeta values

C0(x)=1,

C1(x)=0,

C2(x)=0,

C3(x)=4ζ(3) ,

C4(x)=0,

C5(x) = ζ(5)(−2x+ 12) ,

C6(x) = ζ(3)2(4x+ 8) ,

C7(x) = ζ(7)(−1

2x2−12x+ 36) ,

C8(x) = ζ(3)ζ(5)(2x2+ 8x+ 48) .

The ﬁrst coeﬃcients up to third order have been discovered by Grinfeld and Strang [18]

in 2012 through deforming the circle to a regular polygon and then applying a technique

called calculus of moving surfaces. Three years later, Boady in his PhD thesis [7] managed

to compute two more terms also using methods from the calculus of moving surfaces. The

seventh and eighth coeﬃcients were recently found by the second author [25] using numerical

methods similar to the ones presented in this report.

We managed to obtain eight higher order coeﬃcients in this project

C9(x) = ζ(9)(9

4x3+ 104x2+ 438x−1020) + ζ(3)3(240x+ 96) ,

C10(x) = ζ(7)ζ(3)(x3+ 39x2−24x+ 144) + ζ(5)2(x3−6x2−12x+ 72) ,

C11(x) = ζ(11)(−5

32 x4−661

60 x3−1623

20 x2−176x+ 372)

+ζ(5)ζ(3)2(80x2+ 176x+ 96) + ζsv(3,5,3)(1

5x3+54

5x2),

C12(x) = ζ(9)ζ(3)(5

8x4+107

3x3+ 456x2−488

3x+1360

+ζ(7)ζ(5)(11

8x4+47

2x3−207x2−216x+ 432) + ζ(3)4(−16x2+272

3x+32

3),

C13(x) = ζ(13)(−7

64 x5−226501

16800 x4−1283839

8400 x3−1447393

1400 x2−618x+ 1260)

+ζ(7)ζ(3)2(x4+ 34x3+ 1236x2+ 336x+ 288)

+ζ(5)2ζ(3)(−31

10 x4+256

5x3−1128

5x2+ 336x+ 288)

+ζsv(5,3,5)(−157

1400 x4−549

350 x3−12339

175 x2)

+ζsv(3,7,3)(−5

56 x4−59

28 x3−747

14 x2).

(The expressions of the coeﬃcients C14(x), C15(x), and C16(x) are given in the Appen-

dix, the numerical expressions as well as the underlying eigenvalue data are provided as

an attachment to this paper). In order to determine these expressions we computed the

eigenvalues of 650 n-gons to at least 980 digits precision. Through this we also discovered

the appearance of single-valued multiple zeta values (MZVs) in the expansion coeﬃcients,

COMPUTING LAPLACIAN EIGENVALUES OF 2D SHAPES WITH DIHEDRAL SYMMETRY 3

∂Ω3

∂Ω2

∂Ω1

α/2β/2

∂Ω3

∂Ω2

∂Ω1

α/2

β/2·

Figure 1. Fundamental region of regular polygons for fully symmetric eigenfunctions

starting at eleventh order. In case of C16(x) the basis of single-valued multiple zeta values

was found with help of the program HyperlogProcedures developed by Oliver Schnetz.

We provide a brief introduction to MZVs and the deﬁnitions of the single-valued MZVs that

appear in our expressions in the Appendix. We should note that the approach used in this

paper does not produce explicitly proven results but rather gives conjectures with very high

numerical evidence. We did however manage to prove some of the results for regular poly-

gons with Dirichlet boundary condition in the companion paper [2]. Namely, we explicitly

derived the results up to (and including) C14 and proved that that the expansion coeﬃcients

are expressions over the space of multiple zeta values. However, the expressions for C15 and

C16 can only be considered conjectural at this point.

2. The Method of Particular Solutions

Because the eigenfunctions considered in this work are dihedrally symmetric, it is suﬃcient

to compute them inside a triangular subdomain (which we refer to as fundamental domain)

instead of working with the full shape (see Fig. 1 for an illustration of the fundamental

domain for a regular polygon). A numerical method for computing eigenvalues of triangular

shapes is given by the method of particular solutions (MPS). The MPS has been introduced

by Conway [10] in 1961 and established by a famous paper of Fox, Henrici and Moler [16]

in 1966 on computing eigenvalues of L-shape domains.

The main idea of the MPS is to expand Ψ for the spectral parameter k=√λas a

Fourier-Bessel series (in polar coordinates)

(2.1) Ψ(k, r, θ) = X

ν=1

cνψν(k, r, θ),

where

(2.2) ψν(k, r, θ) = Jmν(k·r)·(sin(mνθ)|Odd eigenfunctions ,

cos(mνθ)|Even eigenfunctions ,

and Jmdenotes the Bessel function (see for example [35]). The basis functions ψνhave the

useful property that they can serve as eigenfunctions along an unbounded wedge by a well-

made choice of mν. Consider for example a wedge with angle αwith the vertex at the origin.

Dirichlet boundary conditions impose that the function has to vanish along the boundary. If

4 D. BERGHAUS, R. S. JONES, H. MONIEN, AND D. RADCHENKO

one chooses mν=νπ/α for the odd eigenfunction basis, one obtains functions that trivially

fulﬁll the boundary condition along both edges of the wedge. This property (combined with

the exponential decay of the basis functions) makes the MPS very useful when computing

eigenvalues of triangular shapes: the boundary condition on two of the three edges can be

trivially fulﬁlled by the construction of ψν. To obtain the spectral parameter kone truncates

the expansion of Ψ to some ﬁnite order Nand searches for the parameter kfor which the

expansion vanishes on a discrete set of points on the remaining edge up to the desired

numerical precision (this approach is often referred to as point-matching). This results in a

linear system of equations





ψ1(k, r1, θ1). . . ψN(k, r1, θ1)

.....

ψ1(k, rN, θN). . . ψN(k, rN, θN)

·





=



0

,

which is square if the number of point-matching points is also chosen to be N. The parame-

ter know corresponds to an approximation of the true spectral parameter if the coeﬃcients

are essentially invariant under the choice of matching points. In view of this, one computes

the value of kfor which the determinant becomes zero (i.e., when the matrix becomes sin-

gular) using root-ﬁnding algorithms (we used the secant method) [16]. By increasing N, one

can obtain better approximations and hence better precision for the eigenvalue.

This approach has been applied by the second author [24, 25] who expanded from the

vertex with angle β/2 (see Fig. 1) and applied point-matching to the edge ∂Ω2to compute

eigenvalues of polygons with n≤150 to 50 digits of precision. We remark that the MPS

has the limitation that the point-matching matrix becomes ill-conditioned for large N. This

has been further analyzed by Betcke and Trefethen [4,5] in 2005 who proposed an improved

approach that is referred to as modiﬁed MPS. Their main idea has been to compute an

additional matrix that consists of (randomly chosen) points in the interior of the considered

region. This matrix is designed to ensure that the eigenfunction is non-zero in the interior.

By minimizing the lowest generalized singular value of these two matrices, one can reliably

locate the eigenvalues of a given shape because spurious solutions are excluded and Ncan be

chosen as required, without conditioning the matrix too poorly. The modiﬁed MPS might be

regarded as a successor of the MPS and is superior in most applications. For the computation

of eigenvalues of relatively simple shapes at arbitrary precision arithmetic, the original MPS

can however still prove itself to be very competitive because the ill-conditioning can be

overcome by increasing the working precision and because the generalized singular value

decomposition is very computationally expensive compared to the LU-decomposition that

is required for the determinant computation. Additionally, the function of the determinant

w.r.t. kbecomes almost linear close to the eigenvalues which makes the location of the roots

very eﬃcient [24].

We further remark that recent progress has been achieved in making the results of the

MPS rigorous (i.e., with certiﬁed error bounds) [11, 20]. For our application, certifying the

eigenvalues yields no beneﬁt because the derivation of the closed-form expressions using the

LLL algorithm is non-rigorous (unless one knows a bound of the height of the coeﬃcients

in advance, which we are unaware of). Additionally, speciﬁcally certifying the eigenvalues

results in a loss of precision.

3. Domain Decomposition

Despite the absolute convergence of the MPS (i.e., one always gets better estimates by

increasing Nas long as the working precision is suﬃciently increased) the algorithm which

has been applied by the second author in [24,25] has the disadvantage that the convergence

rate (which is the eigenvalue precision per amount of matching points) drastically decreases

for polygons with more edges (and hence “thinner” fundamental regions). To overcome

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COMPUTATIONOFLAPLACIANEIGENVALUESOFTWO-DIMENSIONALSHAPESWITHDIHEDRALSYMMETRYDAVIDBERGHAUS,ROBERTSTEPHENJONES,HARTMUTMONIEN,ANDDANYLORADCHENKOAbstract.WenumericallycomputethelowestLaplacianeigenvaluesofseveraltwo-dimensionalshapeswithdihedralsymmetryatarbitraryprecisionarithmetic.Ourap-proachisbasedo...

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COMPUTATION OF LAPLACIAN EIGENVALUES OF TWO-DIMENSIONAL SHAPES WITH DIHEDRAL SYMMETRY DAVID BERGHAUS ROBERT STEPHEN JONES HARTMUT MONIEN

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