SOLUTIONS OF THE TWO-DIMENSIONAL HELMHOLTZ EQUATION

SOLUTIONS OF THE TWO-DIMENSIONAL

HELMHOLTZ EQUATION

Robert S. Jones, Ph.D.

November 5, 2004

1

0 INTRODUCTION 2

0 INTRODUCTION

NOTE: This is a re-entry of my Ph.D. thesis I wrote in 1993 at The Ohio State University,

Physics Department. It is as close as practical to the original with some grammar fixes,

re-numbering, and some formatting changes.

–R.S. Jones, Autumn 2004

In this project, I examine the solutions of the two-dimensional Helmholtz equation within

several domains formed by piecing together identical 30◦-60◦-90◦ triangles. These shapes

were chosen partly because of their relation to the hexagonal region and partly because

they may be compared to the closed-form equilateral triangle eigenmodes. This study is

motivated by two important physical applications. The first is the quantum-mechanical,

three-body problem with one-dimensional infinite square wells {[4], [10]}. The second is

the electromagnetic waveguide problem [25]. Both applications involve essentially identical

mathematical formulations.

The Helmholtz equation plus the appropriate boundary conditions constitute the eigen-

value problem of the Laplacian,

∇2Ψ + λΨ = 0. (1)

The Helmholtz equation results from the Schrodinger equation for the quantum mechanical

problem {[30], for example} and from the Maxwell equations for the waveguide problems

{[11], for example}. Much physics is contained in the boundary conditions. In this project,

only Dirichlet and Neumann boundary conditions are considered. Specific boundary condi-

tions restrict the possible values (spectrum) of λ to a denumerable, infinite set (with a lower

bound).

Pedagogical as well as practical applications of the Helmholtz equation typically involve

domains for which the eigenfunctions Ψ and eigenvalues λ are well known in closed form.

The only two-dimensional domains for which complete sets of ‘exact’ or closed-form results

exist are the equilateral-triangle, the rectangle, the ellipse, and special cases of these (such as

the square, isosceles right triangle, circle, half circle, etc.). The rectangular and the elliptical

modes are unique in that they may be obtained by separation of variables. The triangular and

rectangular modes are similar in that they may be expressed as a finite sum of plane waves

0 INTRODUCTION 3

and that they may tile the plane using the Riemann-Schwarz reflection principle. There

exist no other two-dimensional domains for which complete sets of modes are expressible in

closed form, known or otherwise [35].

The closed-form equilateral triangle modes were first discovered by Lame in 1852 [16].

Curiously, these results have been left out of the modern curriculum; I have been able to

find only two textbooks which discuss these equilateral triangle modes {[22], [29]1}, despite

the fact that this problem provides a unique and elementary example of a non-separable

solution of the two-dimensional Helmholtz equation. Over the years, authors appear to

have ‘rediscovered’ these closed-form results in one form or another. Consequently, novel

derivations and interesting discussions are scattered throughout the literature. In addition

to [29] and [22], a list of modem references which refer to the closed-form modes includes

{[15], [31], [4], [5], [21], [10], [9], [26], [8], [13], [34], [20], [25], [35], [39]}.2 Appendix A of this

thesis collects important results.3

Interesting physical systems exist for which solutions within other domains are appro-

priate. For the three-body problem, the important regions include the regular hexagon and

the 60◦ rhombus, both satisfying Dirichlet boundary conditions. The application to the

waveguide problem is more practical and less pedagogical than the three-body square-well

problem. Also, it is slightly more general since more regions - with both Dirichlet and

Neumann boundary conditions - are studied. (Here, the ‘region’ or ‘domain’ refers to the

cross-section of the guide.) Indeed, five domains are considered, each of which is formed by

piecing together identical 30◦-60◦-90◦ triangles. (The hexagon also satisfies this property.)

A large part of this dissertation is concerned with the numerical solution of the eigenvalue

problem. Two different and independent methods of solutions are used, both very success-

fully.4 The first is a combination of the imaginary time-step [12] and finite-difference [1]

1However, see [25] regarding Ref. [29].2Much thanks to P. L. Overfelt for pointing out some of these to me.3I also provide my own discussion, including a simple and practical algorithm for tabulating and clasifying

the modes.4Other methods were tried and abandoned because of moderate or low success. For example, Rayleigh-

Ritz, five parameter polynomial trial-functions provided upper-bounds to only the very lowest eigenmodes

with only three or four digit accuracy in the eigenvalue.

0 INTRODUCTION 4

(ITS/FD) methods, with which 69 regular hexagon modes with Dirichlet boundary condi-

tions were computed to at least five digit accuracy in the eigenvalue. The second method

of solution is the point-matching or collocation method, whereby an exact solution of the

Helmholtz equation is made to satisfy the boundary conditions approximately.5 The TE

(Neumann) and TM (Dirichlet) modes6 of the regions of Fig. 6 (Chapter 3) are divided up

into eight symmetry classes, four of which correspond to the closed-form modes. A total

of 26 non-closed-form modes are computed to a very high (and quite possibly an unprece-

dented) precision. For example, using the point-matching method with seventy matching

points, I have determined the cutoff wavenumber of the lowest regular hexagon TM mode

(with unit-length edges) to lie between 2.674946522 and 2.674946580; whereas (to my knowl-

edge), the previous best result [4] was 2.67495 (unbounded) using Richardson extrapolation

of finite-difference results.

Perhaps the strongest advantages of both the ITS/FD and the point-matching methods

are that they are simple to implement and guaranteed to work reliably. The application

of both methods to the problem at hand was straightforward. With the ITS/FD method,

a complete FORTRAN program was developed within one week. Computation of the 69

hexagon modes to the desired precision required hundreds of CPU hours on the VAX cluster

and was completed within about two months (real time). With the point-matching method,

the symbolic/numerical programming language MAPLE was used. MAPLE programs were

developed and run within days, requiring only tens of CPU hours on the VAX cluster for

comparable results. The chosen languages were well suited to each method.7

It appears that the point-matching method is far superior to the ITS/FD method. First,

as suggested above, the amount of computer effort is significantly less for the point-matching

method. Second, the ITS/FD method required that the values at each of the 15,000 grid

5As reviewed in Chapter 3, this is done by enforcing the boundary conditions at a finite set of points on

the boundary.6The notation ‘TE’ and ‘TM’ stands form respectively transverse electric and transverse magnetic waveg-

uide modes.7It may be mentioned that MAPLE is an interpretive language which can perform arithmetic operations

to arbitrary precision, implemented in software; FORTRAN is a compiled language and numerical precision

is (essentially) limited to that of the hardware precision.

0 INTRODUCTION 5

points for each eigenfunction be stored – thus requiring a significant amount of disk space.

The point-matching method did not require any storage of the eigenfunctions. Related to this

is the disparate number of parameters used to represent each eigenfunction: ITS/FD (N ≈5, 000 corresponding to the number of grid points) vs point-matching (N < 50 corresponding

to the number of matching points) for comparable precision within the 30◦-60◦-90◦ triangle.

Third, with the ITS/FD method the precision and accuracy of each eigenmode depended

upon the precision and accuracy of the lower modes. The point-matching method was used to

determine each mode, independent of any other mode. Finally, and probably most important,

the point-matching method was used to provide very tight upper and lower bounds to each

eigenvalue. Although it is possible to estimate more accurately the eigenvalues using the

Richardson extrapolation,8 it is difficult to bound the eigenvalues using the finite-difference

method {[13], Sec. 141}. Furthermore, nothing analogous to ‘Richardson extrapolation’

exists for the eigenfunction.

Of course, using independent methods has an advantage because results can be compared.

Thus, some duplication in work paid off in that estimates of precision using the ITS/FD

method were obtained. Some other checks of precision included comparison of the numerical

results to closed-form results, and comparison of degenerate modes within separate symmetry

classes. Ten of the lowest 69 hexagon modes correspond to closed-form equilateral triangle

modes.

This thesis is logically divided into four chapters. First, I discuss the physics of the

quantum-mechanical three-body problem. It is shown that the one-dimensional three-body

problem with infinite square-well potentials is equivalent to the Dirichlet eigenvalue problem

of the hexagon or the rhombus – plus the trivial center-of-mass motion. The usual classi-

fication scheme for the eigenmodes according to permutation symmetry and parity is also

discussed. This is especially useful in understanding the relation between the hexagon and

rhombus modes and the three-body system. Appendix B collects some useful coordinate

transformation results.

Second, I describe the algorithm of the ITS/FD method and its application to the hexagon

8This is done by calculating the eigenvalues at low grid resolution and extrapolating to the limit of infinite

grid resolution.

0 INTRODUCTION 6

eigenmodes. The lowest 69 eigenfunctions results are reported using contour plots. These

contour plots are particularly interesting because they reveal both the complexity and an

inherent beauty of the problem. Because of their voluminous nature, the pictures are given

in Appendix C. The (Dirichlet) rhombus modes are not solved using this method.

Third, the waveguide problem is discussed. A relation between the modes of the various

regions shown in Fig. 6 (Chapter 3) is derived, based on the symmetry properties, and

boundary conditions of the respective eigenvalue problems. This is an interesting result based

on elementary considerations. Finally, I describe the point-matching method as a way to solve

the eigenvalue problem and how it is used to solve the three-body problem (Chapter 1) and

the waveguide problem (Chapter 3). To begin, the eigenfunction is expanded in a Fourier-

Bessel series. Next, the point-matching method provides an algorithm to determine the

expansion coefficients and the eigenvalue. I apply this method to each mode with minimal-

order computations to gain insight into how it works. Then, I apply it using high-order

computations to the non-closed-form modes. In this process, two important properties are

revealed. As mentioned above, one property is that relatively tight bounds to the eigenvalues

can be obtained. The other property is that a simple relation exists between the expansion

coefficients.

1 THE THREE-BODY PROBLEM 7

1 THE THREE-BODY PROBLEM

1.1 Preliminaries

I begin by posing the following physical problem:

Consider the quantum-mechanical three-body problem with one-dimensional in-

finite square wells. For simplicity, assume the particles are of equal mass and the

range of each square well is identical. Determine the eigenfunctions and eigen-

values of the four possible variants illustrated in Fig. 1. In systems A and C,

the bonds occur between the pairs of particles; in systems B and D, the bonds

exist between each particle and the center-of-mass location. (Systems A and B

include three bonds, while systems C and D include only two bonds.) Classify the

eigenmodes according to the symmetry group of each system and determine geo-

metrical and accidental degeneracies, if present. Also, determine any closed-form

results, if possible.

Elementary quantum texts quite often make use of the infinite square well as a model

to introduce many physical concepts such as bound states and spectra. This is primarily

because the relative simplicity of the mathematics does not obscure the physics. Curiously,

the texts rarely – if ever – go beyond the one- or two-body problem, despite the fact that

most real systems consist of several to many particles. The physical problem posed above

is perhaps the most elementary extension of introductory considerations since it contains

three particles, Despite its benign appearance, this three-body paradigm is not trivial, and

understanding this model provides a window into the very interesting world of multi-particle

systems.

This chapter is devoted to examining the solution of the systems A-D. The solution con-

sists in first separating the (trivial) center-of-mass motion from the internal dynamics. Then,

for each system, the ‘internal’ eigenstates are determined by solving the two-dimensional

Helmholtz equation - with Dirichlet boundary conditions - within either the regular hexagon

{[7], [4]} (systems A and B) or the 600 rhombus [32] (systems C and D). Invariances or

symmetries of the systems consist of spatial inversion (parity) and particle permutation.


Figure 1: The four three-body systems under study. The arrows indicate square-well inter-actions. Note that the interactions of systems B and D are between the particles and thecenter-of-mass (CM) coordinate.


Consequently, the states are classified according to the irreducible representations of these

groups. In addition, the symmetry properties directly translate into simple geometrical

operations on the hexagon or rhombus eigenmodes.

It should be emphasized that none of the systems A-D is equivalent to the system con-

sisting of three independently interacting particles, each of which interacts with a com-

mon one-dimensional infinite square well. In this system - call it E - the center-of-mass is

fixed (i.e., not free to move) and the states are equivalent to the Dirichlet modes within

a three-dimensional rectangular Parallelepiped.9 Nevertheless, contrasting and comparing

these systems is instructive.10 The central-potential problem E is the direct product of three

one-body problems; therefore, the states of E are products of closed-form single-particle

states – or linear combinations of such products. As with systems A through D, the states

of E may be classified according to the ‘three-body symmetry groups,’ i.e., permutations

(where appropriate) plus inversion.

A most interesting feature of the systems A through D is that some of the three-body

states are expressible in closed form. These closed-form states correspond to those hexagonal

or rhombic modes in which nodal lines divide the domain into equilateral triangles. This

is because the equilateral triangle modes are expressible in closed form. Physically, these

closed-form modes are shown to correspond to the fully antisymmetric states of system A,

the fully symmetric states of system B, the states antisymmetric in particles 2 and 3 of

system C, and the states antisymmetric in particle 1 with respect to the center-of-mass of

system D. Appendix A examines the closed-form equilateral triangle modes and includes

references.

9The Dirichlet problem within a rectangular parallelepiped (with various applications) is treated in many

elementary texts such as {[30], [23], [11], [28]}. if the particles are identical, the geometry is a cube.10Curiously, Ref. [8] transforms some three-dimensional cubical modes (antisymmetric states of the three-

body problem of type E) into the two-dimensional equilateral triangular modes (antisymmetric states of

the three-body of type A) simply by transforming the coordinates. Although mathematically correct, the

physical argument that systems A and E are equivalent is faulty. For example, the spectra, degeneracies,

etc., are very different.


1.2 Setting-up the Problem

In this section, the Schrodinger equation is separated according to the internal and center-of-

mass motions. The canonical form of the problem is described. It is shown that the internal

Schrodinger problem with infinite square wells is equivalent to the Dirichlet Helmholtz prob-

lem inside a regular hexagon or a 60◦ rhombus.

For equal mass particles, the Schrodinger equation is

− h2

2m

[∂2

∂r21

+∂2

∂r22

+∂2

∂r23

]Ψ(r1, r2, r3) = [ETOT − V ] Ψ(r1, r2, r3) (2)

where the translationally invariant potential V is one of

VA = v(r1 − r2) + v(r2 − r3) + v(r3 − r1); (3)

VB = v(r1 −RCM) + v(r2 −RCM ) + v(r3 − RCM); (4)

VC = v(r1 − r2) + v(r3 − r1); (5)

VD = v(r2 −RCM) + v(r3 −RCM ); (6)

and where the center-of-mass location is

RCM =r1 + r2 + r3

3. (7)

The one-dimensional infinite square well potential is defined by

v(x) =

0 if |x| < a

∞ if |x| > a, (8)

where a is the range of the force. The boundary conditions on the wavefunction are imposed

by requiring Ψ ≡ 0 unless V = 0. This occurs only if all of the arguments of the v’s have

magnitude less than a. For example, ΨA ≡ 0 unless VA = 0. Thus, if either |r1 − r2| > a,

|r2 − r3| > a, or |r3 − r1| > a, then ΨA ≡ 0.

The distance between a particle coordinate and the center-of-mass of the entire system

can be related to the distance between that particle and the center-of-mass of the remaining

pair, for example,

r1 − RCM ≡2

3

(r1 −

r2 + r3

2

). (9)


This is important because it easily generalizes problems B and D – with little or no additional

work – to those problems in which each particle interacts with the center-of-mass of the

remaining pair.

Since the center-of-mass is free to move, the eigenfunction Ψ is not normalizable. How-

ever, using the considerations reviewed in Appendix B, the center-of-mass motion is easily

isolated from the internal dynamics. The internal eigenfunction is normalizable and has a

discrete bound-state spectrum. It is this internal problem which is interesting.

Let the letters (i, j, k) represent a cyclic permutation of (1, 2, 3). Using the equal mass

condition and the transformations of Appendix B; namely11

xi =

√1

2(rj − rk) ; (10)

yi = −√

3

2(ri −RCM ) = −

√2

3

(ri −

rj + rk2

); (11)

zi =√

3RCM ; (12)

the Schrodinger equation is transformed into the two-dimensional version

−[∂2

∂x2+

∂2

∂y2

]ψ(x, y) =

2m

h2 [E − V ]ψ(x, y) (13)

where

Ψ(r1, r2, r3) = e±iκzψ(x, y) ; (14)

E = ETOT −h2κ2

2m. (15)

Note that E is the internal energy and assumes discrete positive values. Also note that if no

subscripts appear on x or y, any one of the three sets of coordinates can be used.

The parameter κ is proportional to the center-of-mass momentum PCM = p1 + p2 + p3,

since κ z = PCMRCM . Thus

κ =PCM√

3(16)

11Appendix B treats the general mass case and it is convenient to include the masses in those transfor-

mations. Here, it is more convenient to factor the mass out and not include it in the transformations. One

easy way to transcribe the results of Appendix B to this section is to first assume that m = mi = 1, and in

the end restore m 6= 1.


The internal energy is thus

E = ETOT −h2P 2

CM

2 · 3m (17)

which may have been obtained directly from elementary principles. The fact that κ can

assume arbitrary values demonstrates that the center-of-mass motion forms a doubly degen-

erate continuum of states. Indeed, the plus or minus sign on the exponential corresponds to

the fact that the bound system can be traveling as a plane wave to the right (e+iκz) or left

(e−iκz) in the one-dimensional particle configuration space. With this said, continue with

the more interesting internal problem.

Each point of the two-dimensional configuration space with coordinates (x1, y1), (x2, y2),

or (x3, y3) represents an internal configuration of the three-body system. These coordinates

can be expressed in terms of each other. Expressing the ‘2’ and ‘3’ coordinates in terms of

the ‘1’ coordinates,

x2 = −1

2x1 +

√3

2y1 (18)

y2 = −√

3

2x1 −

1

2y1 (19)

x3 = −1

2x1 −

√3

2y1 (20)

y3 = +

√3

2x1 −

1

2y1 (21)

These transformations correspond to rotations by ±2π/2 in the plane. Fig. 2 shows, these

coordinates.

Each sixty degree wedge defined by the three y-axes specifies an ordering of the particles

along the one-dimensional particle configuration space. When ‘crossing’ the yi-axis, the

particles j and k switch places. To determine the orderings, it is only necessary to use

xi ∝ rj − rk, and decide if xi > 0 or if xi < 0 for i = 1, 2, 3 within each 60◦ wedge.

These considerations are helpful in understanding the symmetry of the states. For example,

exchanging the particles at rj and rk, corresponds to the reflection (xi, yi)→ (−xi, yi). This

is described in more detail in the next section.

The potential V is translationally invariant; therefore, it does not depend on RCM . In-

deed,

VA = v(γx1) + v(γx2) + v(γx3); (22)


VB = v(γ′y1) + v(γ′y2) + v(γ′y3); (23)

VC = v(γx2) + v(γx3); (24)

VD = v(γ′y2) + v(γ′y3); (25)

where (for convenience)

γ ≡√

2 and γ′ ≡√

2

3(26)

For the infinite square well potentials, the regions in which V = 0 define polygons in the

xy-plane:

VA = 0 only if |xi| < a/γ for i = 1, 2, 3;

VB = 0 only if |yi| < a/γ′ for i = 1, 2, 3;

VC = 0 only if |xi| < a/γ for i = 2, 3;

VD = 0 only if |yi| < a/γ′ for i = 2, 3.

(27)

To draw the polygons, draw the lines xi = ±a/γ for VA and VC , and yi = ±a/γ′ for VB and

VD. These shapes are shown in Fig. 3. For VA and VB, the region is a regular hexagon, and

for VC and VD the region is a 60◦ rhombus. Note that – as described below – the edge length

of each polygon is chosen to be unity.

The internal problem is thus reduced to that of the Helmholtz equation, with Dirichlet

boundary conditions, within the regular hexagon or the 60◦ rhombus. To express the problem

in a canonical form, first let k be the eigenparameter such that Eq. (13) becomes

[∂2

∂x2+

∂2

∂y2+ k2

]ψ(k; x, y) = 0. (28)

This eigenparameter k is related to the internal-energy eigenvalue according to

E =h2k2

2m. (29)

Of course, the eigenfunction ψ is constrained to vanish on the edge of the polygonal domains

shown in Fig. 3.

Next, the domains are chosen to have unit edge-length in the xy-plane. This corresponds

to fixing the value of a according to, respectively,

system A : |rj − rk| = γ|xi| < a = γ√

32

⇒ aA ≡√

3

2; (30)


Figure 2: The internal two-dimensional configuration space of the one-dimensional three-body problem.


system B : |ri −RCM | = γ′|yi| < a = γ′√

32⇒ aB ≡

√1

2; (31)

system C : |rj − rk| = γ|xi| < a = γ√

34

⇒ aC ≡1

2

√3

2; (32)

system D : |ri −RCM | = γ′|yi| < a = γ′√

34⇒ aD ≡

1

2

√1

2. (33)

These are the conditions on the range which lead to unit-edged polygons. Note that the

canonical ranges of the systems C and D are one half those of systems A and B respectively.

Of course, it is possible to extend the results to an arbitrary ‘range’ of the potential by

using the scale invariance of the Helmholtz problem. Thus, if we transform the problem

according to (x, y)→ (βx, βy), where β is an arbitrary scale factor, then the range and the

eigenparameter change according to

a→ a′ = βa and k → k′ =k

β(34)

This corresponds to changing the edge-length of the respective polygon by a factor of a.

In the Chapters 2 and 4, the numerical solutions of these problems for 0 < k < 20 are found

to a very high precision. It is determined that there are 69 hexagon states and 21 rhombus

states in this range of k, some of which are expressible in closed form.

It may be mentioned that the present problem is equivalent to that in which a particle

is confined by the two-dimensional domains of the regular hexagon and the 60◦ rhombus.

The results may be used to examine the correspondence principle relating the classical and

quantal systems of the hexagonal and rhombical ‘billiard’ problems [35], for example.

In the next section, the symmetry of the states examined and a classification scheme is

described.

1.3 Classification of the Three-Body States

The group under which the internal three-body system is invariant consists of particle per-

mutations and spatial inversion. For systems A and B, the permutations include all three

particles, while for systems C and D, particle 1 is left out of the permutations. For two

particles, the group is S2 and consists of only the transposition of particles 2 and 3. For


Figure 3: The two-dimensional domains in which the three-body problems A-D must besolved. The size is defined by the canonical form of the equations in which the edge-lengthof each polygon is unity.


three particles, the permutation group is S3 and consists of six operations. All four systems

are invariant under inversion i.

The states are classified according to the irreducible representations of the symmetry

group. First, states are even or odd under inversion, which corresponds to the parity of the

state. A superscripted, parenthesized plus or minus sign is used to indicate the parity of a

quantity.

Regarding the two-dimensional configuration space (x, y), each axis may be either even

or odd. In general, an axis or other straight line is ‘odd’ if there is a node in the eigenfunction

along this line; while it is ‘even’ if contour lines of the eigenfunction cross this axis at right

angles.

The Riemann-Schwarz reflection principle can be used to continue a harmonic function12

through a straight even or odd edge. Indeed, if x = 0 is an even or odd line for the

eigenfunction ψ(x, y), then ψ(−x, y) = ±ψ(x, y) respectively. This process can be repeated

as long as the eigenfunction remains single valued.

Under the permutation of particles 2 and 3, systems C and D are either even or odd.

Thus, in terms of the rhombus modes, the symmetry across the y1, axis is either even or odd,

since xi ∝ r2 − r3. The closed-form states are those states of system C which are odd under

(x1, y1)→ (−x1, y1) and those states of system D which are odd under (x1, y1)→ (x1,−y1).

This is easy to visualize since in both cases the rhombus is divided into equilateral triangles.

Together with inversion, the four symmetry classes of the rhombus states are obtained by

specifying the evenness or oddness of the x and y axes.

Chapter 3 discusses the rhombus modes in more detail using waveguide language. Chap-

ter 4 examines the lowest 21 rhombus eigenmodes (with Dirichlet boundary conditions)

numerically using the point-matching method and closed-form results.

Under permutations, systems A and B are fully symmetric S, fully antisymmetric A, or

mixed M. The mixed states come in degenerate pairs that are13 respectively odd and even

under permutations of particles 2 and 3. Since x1 ∝ r2−r3 and y1 ∝ r1− (r2 +r3)/2 are also

odd and even respectively under this transformation, we can label the mixed states using the

12I.e., any solution of Eq. (28).13or ‘can be made’


notation Mx and My. Thus, if an eigenfunction transforms like x1 (y1) under permutation

of particles 2 and 3, i.e. (r1, r2, r3) → (r1, r3, r2), and is a ‘mixed’ state, then it is classified

under Mx (My).

The symmetric and antisymmetric states are easy to identify. These are the states with

respectively all even or all odd lines along each of the three y axes. Indeed, if a state

is symmetric it must not change sign under any particle permutation. The transposition

of particles j and k, of course, corresponds to the transformations (xi, yi) → (−xi, yi) for

i = 1, 2, 3; since xi ∝ rj − rk and yi ∝ ri − (rj + rk)/2.

Thus, there are eight possible types of the ‘hexagon’ states. They are S (+), S(−), A(+),

A(−), M(+)x , M(−)

x , M(+)y , and M(−)

y . The hexagon contour plots shown in Appendix C are

classified using the system A. The states of system B are trivially obtained by rotating each

hexagon by ninety degrees.

There are several characteristics common to all hexagon states of a given type or classifi-

cation. Quite simply, these can be identified at-a-glance by noting the symmetry of the state

under reflection through the xi and yi axes. For example, those states which have nodes

connecting all three pairs of opposite vertices divide the hexagon into equilateral triangles.

These hexagon modes correspond to closed-form modes (Appendix A).

To classify the states of either system A or B, use these rules. With each contour plot

listed in Appendix C,14 choose the first one occurring in this list:

• S(+): All xi and yi are even;

• S(−): All xi are even and all yi are odd;

• A(+): All xi and yi are odd;

• A(−): All xi are odd and all yi are even;

• M(+)x : Only x1 and y1 are odd;

• M(−)x : Only x1 is odd and y1 is even;

14Caution: The classification of the states listed in Appendix C is for the system A. Thus, a symmetric

odd-parity system-A state corresponds to an antisymmetric odd parity system-B state; only S (−)is given in

the label.


• M(+)y : Only x1 and y1 are even;

• M(−)y : Only x1 is even and y1 is odd.

Thus, for example, if a hexagon mode is even in every x and every y axis, it corresponds

to a symmetric, even parity state.

Only the antisymmetric states (of either parity) of system A are closed form. Only the

symmetric states (of either parity) of system B are closed form. For values of k < 20, there

is no accidental degeneracy.

In systems A and B, the mixed modes come in pairs, each member of which has the same

parity, viz., (M(+)x , M(+)

y ) and (M(−)x , M(−)

y ). In system A, the antisymmetric states come

in degenerate pairs due to a geometrical symmetry of the equilateral triangle. These pairs

of states have opposite parity, viz,, (A(+), A(−)). Likewise, the geometrically degenerate

symmetric states of system B come in pairs with opposite parity, (S (+), S(−)).

2 NUMERICAL SOLUTION OF THE HEXAGON MODES 20

2 NUMERICAL SOLUTION OF THE HEXAGON MODES

“Numerical Analysis is partially a science and partially an art, and short of writ-

ing a textbook on the subject it has been impossible to indicate where and under

what circumstances the various formulas are used or accurate, or to elucidate the

numerical difficulties to which one might be led by uncritical use.”

- P. J. Davis and L. Polonsky {[1], page 877}

2.1 Preliminaries

In this chapter, I discuss the method by which I numerically solved for the Dirichlet modes

within the regular hexagon. The method is a combination of the imaginary-time-step (ITS)

and the finite-difference (FD) [1] procedures. The ITS algorithm is an iterative, relaxation

method whereby the eigenfunctions and eigenvalues are determined. Actual numerical rep-

resentation of the eigenfunctions using two-dimensional arrays require that a FD procedure

be used. I refer to the combination of these methods as the ‘ITS/FD’ procedure.15

This document lists no FORTRAN code since such inclusion (I believe) would be of

limited utility. However, the algorithm is presented in such a way that translation into your

favorite computer language is elementary. This is more useful since many of the techniques

discussed here are applicable to a whole class of eigenvalue problems of which the bound-state

Schrodinger problem is only one.

Using the ITS/FD procedure, 69 of the hexagon modes were determined. The numerous

contour plots are given in Appendix C for easy reference. The modes are divided up into

four symmetry classes according to the symmetry along the x and y axes. The eigenvalues

were computed to more than five digits in precision. Tables ????? 2-5 collect the numerical

results. For comparison, other published results [4], the point-matching results (Chapter

IV), and closed-form results (Appendix A) are included in these tables.

Regarding the accuracy of the hexagon modes, Ref. [13] (page 178) suggests that the

15The techniques of the ITS/FD algorithm as described in this chapter were taught to me by Dave Wasson,

to whom I owe much thanks. A recent paper which discusses the imaginary-time-step procedure can be found

in [12].


non-closed-form results of [4] should be recomputed.16 My results do provide such a re-

computations and a comparison shows that the relative difference is less than 10−5. (For the

6-EE state, the relative discrepancy ∆k/k is the greatest at 8× 10−6.) In addition, the very

tight bounds obtained using the point-matching method (Chapter ??????IV) are also useful

for comparison; however, not all hexagon modes were computed this way.

2.2 The Eigenvalue Problem

The general eigenvalue problem and its desired solution are presented in the usual form. Let

H be the time-independent Hamiltonian of a system; then the orthonormalized eigenfunc-

tions ψns and the corresponding eigenvalues En of H satisfy the following relations,

Hψns = Enψns (35)

〈ψns|ψn′s′〉 = δnn′δss′ (36)

0 < E1 < E2 < · · · (37)

n = 1, 2, 3, · · · (38)

s = 1, 2, 3, · · · , gn gn is the multiplicity of En (39)

The inner product 〈f |g〉 of two well-behaved functions f and g is appropriate to the function

space under consideration. Regarding Eq. (37), it is tacitly assumed that the spectrum of H

has a lower bound. Thus, an inconsequential constant may be added to H (without changing

the physics) so that Eq. (37) is satisfied. Of course, the lower bound of the hexagon spectrum

is zero; therefore, this is not a problem.

Four symmetry classes are obtained by applying the symmetry conditions (Dirichlet

or Neumann) along the x and y axes. This division of the modes is convenient because

it separates the geometrically degenerate modes according to a simple rule.17 A useful

byproduct of doing this is that modes within separate computations may be compared. For

example, if two degenerate mixed modes agree in the eigenvalue only to d digits, we can be

confident that the eigenvalue may not be precise to more than d digits.

16This is because [4] used an incorrect Richardson extrapolation formula. Since I did not use Richardson

extrapolation, I can not make the same mistake.17Imposing the edge conditions along the x and y axes is relatively straightforward.


Figure 4: The region in which the Helmholtz equation is solved. Also shown are the gridpoints (N = 5) used in the finite-difference algorithm.


In the first quadrant of the hexagonal domain shown in Fig. ????4, the relevant equations

for the hexagon modes are

H = −[∂2

∂x2+

∂2

∂y2

](40)

〈f |g〉 =∫ √3/2

0dx

∫ 1−x/√

3

0dyf(x, y)g(x, y) (41)

There are four edges along which conditions must be imposed. The Dirichlet conditions

along an odd edge are imposed by requiring that the eigenfunction vanish along that edge.

The Neumann conditions along an even edge are imposed by requiring that the normal

derivative of the eigenfunction vanish across that edge. The actual method of imposing the

edge conditions depends upon the representation of the eigenfunction.

Let Px and Py be operators that reflect an eigenfunction through the x and y axes,

respectively. Then,

Pxψ(x, y) = ψ(−x, y) = ±ψ(x, y) ⇒

ψ|y=0 ≡ 0 (ψ odd in x)

∂ψ∂x|y=0 ≡ 0 (ψ even in x)

(42)

Pyψ(x, y) = ψ(x,−y) = ±ψ(x, y) ⇒

ψ|x=0 ≡ 0 (ψ odd in y)

∂ψ∂y|x=0 ≡ 0 (ψ even in y)

(43)

These symmetry relations are easily imposed along the x and y axes.

The inversion operator Q is obtained by successively operating with Px and Py (in either

order) on the eigenfunction. The parity of the eigenfunction is thus related to the eigenvalues

of Px and Py. Let ψ(q)(x, y) be an eigenfunction of Q = PxPy; then

Qψ(q)(x, y) = PxPyψ(q)(x, y) = ψ(q)(−x,−y) = qψ(q)(x, y) (44)

where q = ±1 is the parity.

I use an ordered pair of letters out of ‘E’ for even and ‘O’ for odd to form the notation

whereby each state is classified according to the symmetry under Px and Py, respectively.

Thus the four possibilities: ‘EE’, ‘EO’, ‘OE’, and ‘OO’. Each of these classifications can easily

be related to the three-body systems A and B. The classification scheme is summarized in

Table 2. Note that the EE and OO states have even parity and the EO and the OE states

have odd parity.


Table 1: Classification scheme of the hexagon States as they relate to the three-body sys-tems A and B.

S(+) M(+)y M(+)

x A(+) A(−) M(−)x M(−)

y S(−)

EE AB AB

OO AB AB

OE A A B B

EO B B A A

This division of states separates geometrically degenerate states and is significant for two

reasons. The first is that degeneracies are removed from each computation. For example, a

tower of EE modes can be computed in which each state is not degenerate with any other

state within this EE tower.18 The second is that different computations can be compared. If

fact, it is possible to compare each of the four towers – from the lowest states to the highest –

by linking degenerate modes. The degenerate pairs must have the same eigenvalue. Thus

for system A, the (A(+), A(−)) pairs link the OO and OE towers; the (M(+)x , M(+)

y ) pairs

link the EE and OO towers; and the (M(−)x , M(−)

y ) pairs link the EO and OE towers. This

linking is shown in Fig. 5.

2.3 The Imaginary-Time-Step (ITS) Method

To introduce the ITS method, it is convenient to begin with the formal solution of the

time-dependent Schrodinger equation,

HΨ(t) = i∂

∂tΨ(t) ⇒ Ψ(t) = e−iHtΨ(0) (45)

where Ψ(0) is some ‘initial’ state. Expanding Ψ(0) in the eigenfunctions of H, given by

Eqs. (35) through (39), yields the following result,

Ψ(0) =∑

ns

cnsψns ⇒ Ψ(t) =∑

ns

cnse−iEntψns (46)

Next, replace the time t by a parameter τ using the substitutions

t→ −iτ and Ψ(t)→ Φ(τ) (47)

18See footnote 19.


where Φ(0) ≡ Ψ(0), to obtain

Φ(τ) = e−τHΦ(0) (48)

The parameter τ is an ‘imaginary time’.

I next show how to obtain the lowest eigenvalue E1 and the corresponding eigenfunc-

tion ψ1. Then the method is generalized to excited states.

2.3.1 Ground-state

As the value of τ is made to increase, the operator exp(−Hτ) projects out of Ψ(0) the

lowest-energy state (or a linear combination of such states). To see this, expand Φ(τ) in

terms of the ψns, according to Eq. (46) and observe that the coefficients multiplying ψns.

effectively become exponentially decaying functions of τ ; viz.,

Φ(τ) =∑

ns

(cnse

−Enτ)ψns (49)

Thus, it becomes evident that the term(s) with the lowest value of En will ‘decay’ at the

slowest rate. If all of the cns are non-zero, then, in the limit as τ →∞, those terms with the

smallest energy eigenvalue will survive.

If there is a degeneracy in the eigenvalue, a linear combination of the corresponding

modes will be obtained. Thus,

limτ→∞Φ(τ) = e−E1τ

∑

s

c1,sψ1,s (50)

This is inconvenient. Degeneracy is usually not of concern for the ground state, but it may

be for excited states. It is possible to separate geometrically degenerate modes according

to the symmetry group of H (as is done with the hexagon modes).19 From now on, assume

that there is no degeneracy so that we can drop the index s. Thus Eq. (50) becomes

limτ→∞Φ(τ) = e−E1τc1ψ1 (no degeneracy) (51)

19Accidental degeneracies and near degeneracies are more difficult to anticipate. Accidental degeneracies

do occur with the (closed-form) hexagonal modes; however, the computed values of E are well below the

first occurrence.


The operator exp(−Hτ) is not unitary and therefore the normalization of the approximate

eigenfunction Φ(τ) is not maintained.20 Any practical algorithm must re-normalize the

eigenfunctions as τ increases, so that the entire wavefunction does not decay away according

to Eq. (51). Thus, multiply Φ(τ) by

N(τ) =1√

〈Φ(τ)|Φ(τ)〉(normalization) (52)

to define the normalized function,

φ1(τ) ≡ N(τ)Φ(τ) (53)

satisfying

〈φ1(τ)|φ1(τ) = 1 and limτ→∞φ1(τ) = ψ1 (54)

To determine the eigenvalue E1, compute the expectation value of the Hamiltonian

〈H〉1(τ) ≡ 〈ψ1(τ)|Hψ1(τ)〉 (55)

As τ increases, 〈H〉1(τ) approaches the lowest eigenvalue,

limτ→∞〈H〉1(τ) = E1 (56)

Note that – in the spirit of the Rayleigh-Ritz method – the value of 〈H〉1(τ) approaches E1

from above since ψ1(τ) is a linear combination of ψ1, ψ2, etc.

2.3.2 Excited states

The excited states are obtained by applying the above procedure with the only additional

condition that each state must be orthogonalized with respect to the lower, previously de-

termined states. In this way a tower of orthonormalized states is determined.

Assume Eµ and ψµ for µ = 1, 2, · · · , n− 1 have been determined to some desired degree

of precision. Then, the next approximate eigenfunction ψn must be orthogonal to each of

these. Thus,

ψn(τ) = Nn(τ)Φn(τ) (57)

20Indeed, let A = exp(−Hτ), then A†A = exp(−2Hτ) 6= 1. Note that H† = H means that H is Hermitian

and its eigenvalues are real. Of course, exp(−iHt) [with t, not τ ] is unitary.


where

Nn(τ) =1√

〈Φn(τ)|Φn(τ)〉(58)

Φn(τ) = Φ(τ)−n−1∑

µ=1

ψµ〈ψµ|Φ(τ)〉 (59)

Note that we can let Φ1(τ) ≡ Φ(τ) so that the n = 1 case is recovered. Also note that the

operator

Pn ≡ 1−n−1∑

µ=1

|ψµ〉〈ψµ| (60)

is a projection operator, since it is idempotent, i.e., P 2n = Pn. In effect, Pn projects out

of Φ(τ) that part which is orthogonal to the lower states ψµ for m = 1, 2, · · · , n− 1.

For any value of τ , these orthonormalized functions φn(τ) and ψµ for µ = 1, 2, · · · , n− 1

satisfy

〈ψµ|φn(τ)〉 = 0 for µ = 1, 2, 3, · · · , n− 1 (61)

〈φn(τ)|φn(τ)〉 = 1 (62)

Using φn(τ), the expectation value of the Hamiltonian is

〈H〉n(τ) ≡ 〈ψn(τ)|Hψn(τ)〉 (63)

In the limit that τ increases to infinity, we have the following results,

limτ→∞φn(τ) = ψn (64)

limτ→∞〈H〉n(τ) = En (65)

These are valid because φn(τ) approaches the ‘lowest’ state subject to the condition φn(τ)

is orthogonal to ψµ for µ = 1, 2, · · · , n− 1. As with the n = 1 case, 〈H〉n(τ) approaches En

from above if the lower states are sufficiently exact.

2.3.3 Stepping through values of τ

The Hamiltonian requires differentiation. The inner product requires integration. Thus

‘Hψµ’ and ‘〈ψµ|ψν〉’ are relatively straightforward to implement numerically and/or symbol-

ically.


What is less straightforward is the propagation of Φ(τ) according to Eq. (48) since the

exponentiated Hamiltonian contains derivatives. We may attempt to overcome this difficulty

by first choosing a small imaginary time step τ → δτ + τ and then Taylor expanding the

exponential. To lowest order,

Φ(τ + δτ) = e−HδτΦ(τ) ≈ Φ(τ)− δτ ·HΦ(τ) (66)

This suggests an iterative method with the result that after k-steps

Φ(k · δτ) = [1− δτ ·H]k Ψ(0) (67)

Formally (i.e., in an exact treatment), since the function space is infinite and the Hamiltonian

does not have an upper bound on its spectrum, Eq. (67) does not converge, no matter how

small δτ > 0 is.21 However, in an actual computation the function space must be truncated;

therefore, there exists a maximum eigenvalue Emax and corresponding eigenfunction ψmax.

As a consequence, there exists a value of δτ > 0 (call it δτ ) below which Eq. (67) ‘converges’

to the lowest state, and above which it ‘diverges’ to the maximum state. In particular,

limτ→∞φn(τ) =

ψn if δτ > δτ > 0 (convergence)

ψmax if 0 < δτ < δτ (divergence)(68)

limτ→∞〈H〉n(τ) =

En if δτ > δτ > 0 (convergence)

Emax if 0 < δτ < δτ (divergence)(69)

Determining an optimal value of δτ that gives the fastest convergence is desirable, since

this can reduce the required computer time. In practice, the optimum value of δτ was slightly

less than than δτ . Heuristically, the value of δτ must be such that

Emax · δτ ≈ 1 (70)

Thus, the optimum value is slightly less than 1/Emax.

Calculating the optimum value of δτ required an estimate of Emax. Letting the procedure

diverge by selecting a large value of δτ was the easiest way to determine Emax.22

21The mathematical proof is analogous to the elementary proof that (1− x)k is finite as k →∞, if and

only if |x| < 1. Eigenvalues of H [do] exist such that δτ · E > 1.22If it doesn’t diverge to the maximum state, simply increase δτ until it does. Even if the initial state

is a pure eigenstate, finite numerical precision soon introduces other components. Thus, the trial function

always either converges to the lowest state ψn, or it diverges to the highest state ψmax.


One important criterion for precise results is that the computed eigenvalues must lie well

below Emax. This condition is well satisfied in the present application as described below.

2.4 Finite Differences (FD)

Numerical representation of the eigenfunction is realized with a two dimensional FORTRAN

array. Differentiating and integrating is practical and convenient with this representation

using finite differences.

A rectangular grid is laid over the first quadrant of the hexagon and numerical values of

the eigenfunction at each grid point are estimated using the ITS/FD method.23 The grid

size is chosen such that grid points lie on the hexagon boundary as well as the x and y axes.

This is so that the boundary and symmetry conditions may be easily imposed. The x and y

grid spacings are defined by

δx =

√3

2Nand δy =

1

2N(71)

where N is a positive integer. The positions of the grid points are given by

xi = i · δx where i = 0, 1, 2, · · · , N (72)

yj = j · δy where j = 0, 1, 2, · · · , 2N − i (73)

Note that yj also depends on i because the maximum value of y depends on the value of x.

Also note that the number of intervals along the x-axis (from zero to√

3/2) is N , whereas

the number of intervals in the y direction decreases from 2N (at x = 0) to N (at x =√

3/2).

The total number of grid points is (3N/2 + 1)(N + 1) ≈ 3N 2/2; however, FORTRAN must

reserve a rectangular array of size (2N + 1)(N + 1) ≈ 2N 2.

The numerical values of each eigenfunction at the grid-points are stored in an array,

ψn[i, j] ≡ ψn(xi, yj) (74)

Similarly, the numerical values of Hψn are stored in an array at the same grid points and

approximated [1] (Eq. 25.3.23), (to lowest order) according to

(Hψ)n[i, j] = − 1

δx2{ψn[i+ 1, j]− 2ψn[i, j] + ψn[i− 1, j]}

23Ref. [4] used a equilateral triangular array.


− 1

δy2{ψn[i, j + 1]− 2ψn[i, j] + ψn[i, j − 1]} (75)

Higher order approximations [1] (Eq. 25.3.24) yield better results.

According to Eq. (75), the value of (Hψ)n near the perimeter of the domain (first quadrant

of the hexagon) requires values of ψn outside of this domain. A naive approach might be

to apply the Riemann-Schwarz reflection principle [13] along all four edges. However, for

those eigenfunctions which do not have a straight nodal line connecting the vertices of the

hexagon to the center (i.e., the non-closed-form eigenmodes), this will fail near the vertex

(x, y) = (√

3/2, 1/2). This is because multiple reflection around this vertex will yield a

multiple valued function with the result that the ‘eigenfunction’ will he greatly suppressed

near this vertex. This is equivalent to the fact that the non-closed-form eigenmodes cannot

be analytically continued to tile the entire plane since nodal lines must cross at equal angles.

This problem is overcome by using one-sided differentiation along the slanted edge of the

boundary and Riemann-Schwarz along the remaining three edges.

To compute the inner product of two functions f and g represented on the grid use

〈f |g〉 = δx δyN∑

i=0

2N−1∑

j=0

w[i, j] f [i, j] g[i, j] (76)

where w[i, j] is a set of weights. Since the Dirichlet boundary conditions are imposed along

the hexagon boundary, w = 0 along these edges is sufficient. For the other grid points, the

weights are to lowest order

w[0, 0] = 1/4 (77)

w[i, 0] = 1/2 for i = 1, 2, · · · , N − 1 (78)

w[0, j] = 1/2 for j = 1, 2, · · · , 2N − 1 (79)

w[i, j] = 1 for i = 1, 2, · · · , N − 1 and j = 1, 2, · · · , 2N − i− 1 (80)

2.5 Numerical Results and Concluding Remarks

The ITS/FD method was applied to the lowest 69 states. The highest order computation

was with a grid size of N = 100, or about 15,000 grid-points. (Lower grid-size results were

used as a starting points for increasingly higher grid-size results.)


Numerical results are tabulated in Tables 2 through 5. The eigenvalue results of [4] are

listed, as are the high-order point matching results (Chapter ???IV) and the closed form

results (Appendix A), where appropriate. The numerical values of k in the ‘finite-difference’

column are rounded to seven digits. This approximation is guided by the point-matching

results, which provide bounds. Another guide is provided by comparing the degenerate modes

within separate towers. Thus, for example, the degenerate 2-EE and the 1-OO states agree

to at least seven digits.24 And still another guide is provided by comparing the numerical

results to the exact results (Appendix A) for certain modes.

In addition, contour plots of these modes are given in Appendix C. Note that the sym-

metry according to particle permutation (and parity) corresponds to that of system A.

System B is trivially obtained by rotating the hexagon by ninety degrees and determining

the symmetry according to section??????1.3.

The highest eigenvalue used to determine the optimum δτ for the final grid-size is esti-

mated to be kmax ≈ 350. This value of kmax is significantly larger than the largest eigenvalue

determined, that is 0 < k < 20.

The numerical precision of the excited states depends upon the precision of the lower

states. This can lead to some inaccurate results for the higher states. As examples, the

degenerate 19-EE and 14-OO states agree to only six digits, and the 20-EE state does

not fall within the tight bound provided by the point-matching result. Nevertheless, the

numerical results are perhaps the most precise and comprehensive to date. They differ by

at most one part per million in the worst case when compared to the exact values.

Fig. 5 displays the entire numerical results of the spectrum of the unit-edged, regular

hexagon states for k < 20. Note that they are arranged to illustrate the linking of the

degenerate modes for numerical comparison. The middle two columns contain the closed-

form states, although the eigenmodes were determined using the ITS/FD method. Note

that, for example, the tower of 20 ‘EE’ eigenstates are determined in one computation and

the separation into the respective symmetry classes (S (+) and M(+)y ) is done a posteriori,

i.e., after viewing the contour plots of the modes listed in Appendix C.25


Table 2: Even-even (EE) hexagon eigenvalues. The first column lists the symmetry accordingto permutation S3 and inversion i The finite-difference values of k are rounded to seven digits.For comparison, the Bauer-Reiss results of Ref. [4] and the bounded point-matching resultsof chapter ???IV are also listed.

Finite-difference Bauer-Reiss Point-Matching

i⊗ S3 mode k mode k mode k

S(+) 1-EE 2.674947 1 2.67495 1-(oee) 2.6749465 8022

M(+)y 2-EE 5.696653 5 5.69666

S(+) 3-EE 6.123018 6 6.12303 2-(oee) 6.1230 2111

M(+)y 4-EE 8.374984 12 8.37500

M(+)y 5-EE 9.355850 14 9.35586

S(+) 6-EE 9.489908 15 9.48994 3-(oee) 9.489 9288

S(+) 7-EE 10.99398 19 10.99401 4-(oee) 10.9939 8652

M(+)y 8-EE 12.06157

M(+)y 9-EE 12.86524

S(+) 10-EE 12.99310 5-(oee) 12.993 1206

M(+)y 11-EE 13.68729

S(+) 12-EE 14.83148 6-(oee) 14.831 5240

M(+)y 13-EE 15.35073

M(+)y 14-EE 16.23500

S(+) 15-EE 16.54452 7-(oee) 16.544 5548

M(+)y 16-EE 16.57898

M(+)y 17-EE 17.73468

S(+) 18-EE 17.97028 8-(oee) 17.970 3513

M(+)y 19-EE 18.88865

S(+) 20-EE 18.99836 9-(oee) 18.99838 5901


Table 3: Odd-Odd (OO) hexagon eigenvalues. The first column lists the symmetry accordingto permutation S3 and inversion i. The finite-difference values of k are rounded to sevendigits. For comparison, the Bauer-Reiss results of Ref [BR-78] and equilateral triangle closed-

form results (Appendix A, k = 4π√λ/3) are also listed.

Finite-difference Bauer-Reiss Closed-form

i⊗ S3 mode k mode k λ k

M(+)x 1-OO 5.696653 4 5.69666

M(+)x 2-OO 8.374984 11 8.37500

M(+)x 3-OO 9.355850 13 9.35586

A(+) 4-OO 11.08250 21 11.08252 7 11.08250

M(+)x 5-OO 12.06157

M(+)x 6-OO 12.86524

M(+)x 7-OO 13.68729

A(+) 8-OO 15.10289 13 15.10290

M(+)x 9-OO 15.35073

M(+)x 10-OO 16.23500

M(+)x 11-OO 16.57900

M(+)x 12-OO 17.73469

A(+) 13-OO 18.25849 19 18.25851

M(+)x 14-OO 18.88862

A(+) 15-OO 19.19543 21 19.19545


24Of course, they could both be wrong by the same amount.25The symmetry according to particle permutation and spatial inversion of systems A and B are listed

under the figure and may be helpful in identifying the system B symmetry of the contour plots.


Table 4: Odd-even (OE) hexagon eigenvalues. The first column lists the symmetry accordingto permutation S3 and inversion i. The finite-difference values of k are rounded to sevendigits. For comparison, the Bauer-Reiss results of Ref. [BR-78] and equilateral triangle

closed-form results (Appendix A, k = 4π√λ/3) are also listed.

Finite-difference Bauer-Reiss Closed-form

i⊗ S3 mode k mode k λ k

M(−)x 1-OE 4.258131 2 4.25814

A(−) 2-OE 7.255197 8 7.25520 3 7.255197

M(−)x 3-OE 7.752748 9 7.75276

M(−)x 4-OE 9.712134 17 9.71215

A(−) 5-OE 11.08250 20 11.08250 7 11.08250

M(−)x 6-OE 11.18237

M(−)x 7-OE 12.35573

M(−)x 8-OE 13.54386

A(−) 9-OE 14.51039 12 14.51039

M(−)x 10-OE 14.74507

A(−) 11-OE 15.10289 13 15.10290

M(−)x 12-OE 16.31809

M(−)x 13-OE 16.81988

M(−)x 14-OE 17.68348

A(−) 15-OE 18.25849 19 18.25851

M(−)x 16-OE 18.33968

A(−) 17-OE 19.19543 21 19.19545


Table 5: Even-odd (EO) hexagon eigenvalues. The first column lists the symmetry accordingto permutation S3 and inversion i. The finite-difference values of k are rounded to sevendigits. For comparison, the Bauer-Reiss results of Ref. [BR-78] and the bounded point-matching results of chapter??? IV are also listed.

Finite-difference Bauer-Reiss Point-Matching

i⊗ S3 mode k mode k mode k

M(−)y 1-EO 4.258131 3 4.25814

S(−) 2-EO 6.901403 7 6.90142 1-(ooe) 6.90140 3523

M(−)y 3-EO 7.752748 10 7.75276

M(−)y 4-EO 9.712133 16 9.71215

S(−) 5-EO 10.50498 18 10.50503 2-(ooe) 10.5049 9753

M(−)y 6-EO 11.18237

M(−)y 7-EO 12.35573

M(−)y 8-EO 13.54386

S(−) 9-EO 13.76639 3-(ooe) 13.766 4331

M(−)y 10-EO 14.74507

S(−) 11-EO 14.98915 4-(ooe) 14.9891 5845

M(−)y 12-EO 16.31809

M(−)y 13-EO 16.81987

S(−) 14-EO 17.22765 5-(ooe) 17.227 7133

M(−)y 15-EO 17.68347

M(−)y 16-EO 18.33971

S(−) 17-EO 19.06539 6-(ooe) 19.065 4433


Figure 5: The spectrum of the unit-edged, regular hexagonal region with Dirichlet boundaryconditions. The symmetry of the systems A and B is listed at the bottom. Note that thelevels are arranged so that degenerate levels line up for easy comparison. The middle twocolumns correspond to closed-form modes.

3 THE WAVEGUIDE PROBLEM 38

3 THE WAVEGUIDE PROBLEM

3.1 Preliminaries

The closed-form eigenmodes26 of the equilateral triangle were first published by Lame [16]

in 1852. Properties and derivations of these closed-form eigenmodes are discussed in the

literature in many places. Modern applications include the electromagnetic waveguide prob-

lem {[29], [21], [25], [39]} and several quantum-mechanical problems {[31], [10], [9], [8], [20],

[35]}. In addition, the eigenvalue problem is treated in Refs. {[22], [26], [13]}. Appendix A

contains a summary of the closed-form modes.

This report examines the relationship between eigenmodes of waveguides with various

cross sections which are constructed by piecing together identical 30◦-60◦-90◦ triangles, as

shown in Fig. 6 [25]. For simplicity, the guides are assumed to be uniform and perfectly

conducting. The shapes include (I) the 30◦-60◦-90◦ triangle, (II) the equilateral triangle,

(111) the kite quadrilateral, (IV) the 30◦-120◦-30◦ triangle, and (V) the rhombus. These

cross sections were chosen because many of their modes are expressible in closed form and

because this set includes all possible cross sections with a 30◦-60◦-90◦ triangular, fundamental

region with simple edge conditions.27

In the next chapter, the point-matching method is described and numerical estimates

of the eigenmodes are presented. One utility and selling point of this method is that a

relatively large number of eigenmodes are obtained with a modest amount of work. In

addition, a small number of parameters are required to represent the eigenfunctions. Using

five or six parameters, 53 distinct eigenfunctions – accurate to at least three digits in the

eigenvalue – are reported. Some additional benefits are that, with the chosen representation

of the approximate eigenfunctions, the modes may be easily compared to the modes within

a circular cross section; and unlike other methods, such as finite elements [4], (for example),

the numerical accuracy of each mode does not depend upon the accuracy of lower modes.

It was discovered that the point-matching method, when applied to the present problem,

26By ’closed-form’, it is meant that each eigenfunction may be written as a finite sum of plane waves.27The fundamental region of the regular hexagon is also a 30◦-60◦-90◦ triangle since its symmetry group

is of order twelve; however, the edge conditions are generally more complicated.


Figure 6: The waveguide cross-sections formed by piecing together identical 30◦-60◦-90◦

triangles. The dashed lines indicate the symmetry axes. These are the only shapes forwhich complete sets of eigenmodes may be obtained using the eight symmetry classes of the30◦-60◦-90◦ triangle.


may provide arbitrarily tight upper and lower bounds to each eigenvalue. Since bracketing

eigenvalues is in general rather difficult, this is a very important result. There do exist

other bracketing methods (which may use the point-matching results) {[15], [24], [14], [13]};however, they appear to provide much wider bounds for equivalent amounts of work.

3.2 The Eigenvalue Problem

The waveguide modes are determined [11] by solving the eigenvalue problem[∂2

∂x2+

∂2

∂y2+ k2

]ψ(x, y) = 0 (81)

inside the respective cross section and subject to the appropriate boundary conditions. The

eigenparameter k is the cutoff wavenumber. The Dirichlet and Neumann boundary con-

ditions divide the modes into two types – namely, the transverse magnetic (TM) and the

transverse electric (TE) modes, respectively. Let S be the boundary of the cross section;

then the boundary conditions are

TM : ψ|S = 0 (82)

TE : ∂ψ/∂n|S = 0 (83)

where ∂/∂n|S is the normal derivative on S. Note that by re-scaling the coordinates, results

may be extended; thus under the transformation (x, y)→ (ax, ay) the cutoff wavenumber k

becomes k/a.

The Riemann-Schwarz reflection principle [13] (for example) may be used to analytically

continue a harmonic function u(x, y) through a straight-line bounding edge as long as the

function remains single valued. Let x = 0 be such a line; then this line is even or odd

according to whether u(−x, y) = ±u(x, y). For any convex polygon cross section, and in

particular those of Fig. 6, the bounding edges of the TE and TM modes are respectively

even and odd.

3.3 Geometrical Classification Scheme

The geometrical symmetry of the cross section provides a classification scheme for the eigen-

modes. The cross sections of Fig. 6 have symmetry axes indicated by the dashed lines defined


by the joining of the 30◦-60◦-90◦ triangles. Every mode may be classified as either even or

odd with respect to reflection through these axes. This divides the modes into separate

orthogonal, symmetry classes, each of which contains an infinite set of modes such that

0 < λ0 ≤ λ1 ≤ λ2 ≤ · · · (84)

where λ = k2. There are no accumulations in the eigenvalue [27]. However, accidental

degeneracies of the closed-form modes may become arbitrarily large {[31], [26]}.Using the Riemann-Schwarz reflection principle, the complete sets of modes for each

symmetry class, boundary condition, and cross section of Fig. 6 may be constructed from

the complete sets of eigenmodes inside the 30◦-60◦-90◦ triangle with the eight possible edge

conditions.28 The set of cross-sections of Fig. 6 contains all of the possible domains for which

this method works. The 30◦-60◦-90◦ triangle is called the fundamental region since it is the

smallest region from which all of the modes may be obtained using reflections. The size and

orientation of the fundamental region is shown in Fig. 7.

Denote each set of edge conditions with an ordered triple of letters out of ’e’ (for even)

and ’o’ (for odd) to represent the symmetry of the edges A, B, and C respectively. For

example, if A, B, and C are even, odd, and even, respectively, then use (eoe) to denote

the edge conditions. If one or more edge conditions are arbitrary or unspecified, use an

asterisk as a place holder; thus, for example, all modes with odd edge B and even edge C

are symbolized using (*oe). The eight symmetry classes are illustrated in Fig. 8.

The classification scheme is summarized in Table 6 and makes explicit the relation be-

tween the eight symmetry classes of the fundamental region and the TE and TM modes of

the cross sections of Fig. 6. Note that the edge condition of the hypotenuse C determines

the boundary condition of the rhombus, and use, for example, ’eo-TE’, to indicate that the

shorter and longer symmetry axes are, respectively, even and odd for a TE mode.

In passing, we note that sets of modes with more complicated cross sections may be

obtained using the reflection principle. For example, each of the fundamental region eigen-

modes with an even (odd) leg A may be repeatedly reflected through legs B and C to obtain

sets of regular hexagon TE (TM) modes {[29], [4], [35]}. However, not all hexagon modes

28The 30◦-60◦-90◦ triangle TE (TM) modes correspond to all even (odd) edges.


Figure 7: Size and orientation of the 30◦-60◦-90◦ triangular, fundamental region. Both theCartesian coordinates (x, y) and Polar coordinates (r, θ) are shown. The edge lengths areLA = 1/2, LB =

√3/2, and LC = 1.

Table 6: Classification scheme of modes with the five cross sections shown in Fig. 6. Notethat the first four symmetry classes correspond to those closed form-modes of the equilateraltriangle. Thus, for example, only the even-TE and odd-TM modes of the kite quadrilateralshape (III) are expressible in closed form.

(ABC) I II III IV V

(eee) TE e-TE e-TE e-TE ee-TE

(eoe) o-TE eo-TE

(oeo) e-TM oe-TM

(ooo) TM o-TM o-TM o-TM oo-TM

(eeo) o-TE ee-TM

(oee) o-TE oe-TE

(eoo) e-TM eo-TM

(ooe) e-TM oo-TE


Figure 8: This diagram illustrates the eight symmetry classes of the 30◦-60◦-90◦ triangle.The even (Neumann) and odd (Dirichlet) edges are indicated using dashed and solid linesrespectively.


are obtained in this way. In fact only those TE (edge-A even) or TM (edge-A odd) hexagon

modes with three even (edge-C even) or three odd (edge-C odd) lines connecting opposite

pairs of vertices and three even (edge-B even) or three odd (edge-B odd) lines connecting the

centers of opposite edges are obtained using the 30◦-60◦-90◦, triangle modes with the eight

possible edge conditions. Contour plots of the lowest 69 TM hexagon modes are displayed

in appendix C.

3.4 Closed-Form and Non-Closed-Form Modes

The first four classes of Table 6, namely (eee), (eoe), (oeo), and (ooo), constitute the closed-

form modes. Appendix A lists the closed-form results; however, it is useful to digress here

to emphasize some useful and relevant properties of these modes.

First, these modes are expressible in closed form due to a reflection property whereby

they may tile the entire plane {[22], [35]}.29 Equivalently, a closed-form mode is composed of

a finite sum of plane waves [25] which interfere in such a way that there is a simple relation

between the phase of each component. The maximum number of plane waves in the sum is

six, corresponding to the six possible orientations of the reflected equilateral triangle.

Second, the closed-form modes exhibit both geometrical and accidental degeneracies {[31],

[20]}. The geometrical degeneracy arises because of an invariance of the eigenvalue problem

under the symmetric group of degree three S3 [20]. The S3 invariance is most obvious for

the equilateral triangle, but nevertheless present with the other closed-form modes. The

accidental degeneracy is due to additional symmetry properties (which do not form a group)

of the closed-form modes and is analogous to that of the well-known square {[31], [19]}.Elementary number theory, and in particular integer-domain ring theory, provides the most

convenient method to determine the eigenvalue spectrum of the equilateral triangle as well

29It is interesting to note that the only polygon cross sections with complete sets of closed-form eigenmodes

are the rectangle, the isosceles right triangle {[25], [11] (exercise 8.4)}, the equilateral triangle, and the 30◦-

60◦-90◦ triangle. These polygons are the only polygons which share this tiling property [35]. As an interesting

counter-example, the regular hexagon is the only other regular polygon, besides the square and equilateral

triangle, which may fill the plane; however, since the hexagon eigenfunction may become multi-valued upon

reflection, complete sets of hexagon modes may not be obtained in closed form [4].


as the square {[6], [31], [26], [13], [20]}.Finally, a practical result is that the first accidental degeneracy within a given symmetry

class occurs for a relatively large cutoff wavenumber. Specifically, accidental degeneracy is

absent in the lowest 19 (eee), 15 (eoe), 24 (oeo), and 19 (ooo) eigenvalues. However, the first

accidental degeneracy is between the second (eee) mode and the lowest (oeo) mode.

The last four classes of Table 6, namely (eeo), (oee), (eoo), and (ooe), constitute the

non-closed-form modes. Since these modes are not invariant under any symmetry group

with an irreducible representation of dimension greater than one, geometrical degeneracies

are impossible. Also, since these modes may not tile the entire plane, and do not have the

same high symmetry properties as the closed-form modes, any degeneracy in these modes is

very unlikely and probably absent.

In the next chapter, a great effort is made to estimate the eigenvalues and eigenfunc-

tions of the 30◦-60◦-90◦ triangular domain. This is done by expanding the eigenfunction

in a Fourier-Bessel series and using the point-matching method to estimate the expansion

coefficients and cutoff wavenumbers.

3.5 Concluding Remarks

In this chapter, the waveguide modes within guides of several cross-sectional shapes formed

by piecing together 30◦-60◦-90◦ triangles are examined. They are divided up into eight

symmetry classes, based on simple edge conditions along the edges of the 30◦-60◦-90◦ triangle.

Four of the eight symmetry classes are expressible in closed form and only these modes may

be analytically continued to tile the entire plane.

Our next step is to determine precisely the lowest eigenvalues (i.e., cutoff wavenumbers)

and eigenfunctions of the eight symmetry classes of the 30◦-60◦-90◦ triangle, at least for

those non-closed-form modes. Once this is done, practical applications which require these

lowest may be made.

4 THE POINT-MATCHING METHOD 46

4 THE POINT-MATCHING METHOD

4.1 Preliminaries

Next, we use the point-matching or collocation method to estimate the eigenmodes within

each symmetry class. This technique is useful because it provides numerous accurate eigen-

values and eigenfunctions with a modest amount of work. Also, it provides a new technique

to tightly bounding each eigenvalue. Since bounding eigenvalues is in general rather difficult,

this is a significant result.

The point-matching method consists of approximating the eigenfunction with an exact

solution of the Eq. (81) which satisfies the edge condition along some edges (if possible) and

at a finite set of points on the remaining edges {[7], [38], [18], [2], [15], [3], [37], [13]}. If the

approximation is sufficient, this method provides a set of modes which may be associated in

a one-to-one manner with a set of exact 30◦-60◦-90◦ triangle modes. The approximation is

improved by increasing the number of points at which the edge conditions are enforced.

There are several related methods which (may) use the approximate point matching

results to bound the eigenvalues {[32], [24], [14], [13]}; however, in the present application,

the obtained bounds are not very tight. Also, the published bounding methods are designed

primarily for the modes with Dirichlet conditions on the point matching edge(s).

In the course of applying the point-matching method to the 30◦-60◦-90◦ triangular region,

an apparently new method was discovered whereby it may be possible to tightly bound any

eigenvalue. It was applied to the lowest, non-closed-form modes with the results that

k(oee) = 2.6749465 8022 N = 70 (85)

k(eeo) = 4.98984 6859 N = 50 (86)

k(ooe) = 6.90140 3523 N = 50 (87)

k(eoo) = 8.46824 7950 N = 50 (88)

using the indicated number of matching points. (See Eq. (146) and the surrounding text

for the rounding convention and the notation.) For these modes, each additional digit

required approximately an order of magnitude increase in numerical work. Only heuristic

and empirical considerations of the convergence rates are given.


To compare the above results with other results, I obtained ‘unbounded’ values of k(oee) =

2.67494652 and k(ooe) = 6.9014027, using finite elements. Ref. [4] cites k(oee) = 2.67495

and k(ooe) = 6.90142, which are obtained using Richardson extrapolation of low-order finite-

difference results. For the lowest (eeo) mode, Ref. [32] gives a bound of 4.988 < k(eeo) < 4.998,

using an affine transformation method. It may be added that the results of Eqs. (85)-(88)

are probably the world’s best for this problem to date.

The closed-form modes have a considerably faster rate of convergence – by many orders

of magnitude – than their non-closed-form cousins. This is true despite the facts that the

Bessel functions represent ‘circular waves,’ and both closed-form and non-closed-form modes

are treated equivalently.30 (See the comment made in Ref. [32] (Sec. 7) regarding ability to

tightly bound eigenvalues in the rhombus.)

The lowest closed-form (eee) mode was bounded using only two through five matching

points with the results

k(eee) =

4.188 8877 N = 2

4.188 790 204 8948 N = 3

4.188 790 204 786 39 1208 N = 4

4.188 790 204 786 390 984 616 8 7631 N = 5

(89)

(See Eq. (146) and the surrounding text for the rounding convention and the notation.)

These consistently bound the exact value of

k(eee) =4π

3= 4.188 790 204 786 390 984 616 857 844 · · · (90)

thus verifying the new bounding method. Similar incredible results were obtained for several

of the other closed-form modes; however, they are not reported here. (An obtuse method of

30May this difference in convergence rates be used as a flag to identify yet undiscovered closed-form modes

of other geometries and dimensions? For example, are certain modes of the regular tetrahedron expressible

in closed form? This shape is interesting because (analogous to the equilateral triangle in two dimensions)

its modes may fill the space using reflections. See Refs. [8], [34], for a related problem. Another interesting

geometry is the regular dodecahedron since it can be related to the one-dimensional, quantum-mechanical

four-body problem with infinite square-wells. In particular, are the four-body, antisymmetric states (Dirichlet

modes in a pyramid with a pentagon base) expressible in closed-form?


tightly bounding π is thus obtained. This is vaguely analogous to the suggestion that the

point-matching and a bounding method may be used to bracket the zeros of certain Mathieu

functions [15].)

To begin, it is necessary to consider under what conditions the point matching method

is valid and reliable {[18], [2], [3], [37]}. For this reason, all details are given in the next

section.

4.2 Details of the Point-Matching Method

With the usual polar coordinates defined by

x = r cos(θ) and y = r sin(θ) (91)

Eq. (81) becomes [∂2

∂r2+

1

r

∂

∂r+

1

r2

∂2

∂θ2+ k2

]ψ(r, θ) = 0 (92)

An arbitrary, finite solution of Eq. (92) may be expanded in a Fourier-Bessel series,

ψ(k; r, θ) =∑

m

αmφm(k; r, θ) (93)

where the sum is over all non-negative integers, the αm are coefficients, and the basis func-

tions are either

φm(k; r, θ) = Nm Jm(kr) cos(mθ) (94)

or

φm(k; r, θ) = Nm Jm(kr) sin(mθ) (95)

depending on the edge-B conditions, i.e., the symmetry of the eigenfunction under θ → −θ or

(x, y)→ (x,−y). (These results are elementary and may be found in Ref. [33], for example.)

The normalization constant Nm is given below by Eq. (115), and Jm(x) is the Bessel function

of integral order m. For any fixed value of k 6= 0, the φm(k; r, θ) form a complete set of finite

basis functions in the plane containing the origin. (If the origin is excluded, the Ym(x) must

be included to form a complete set of finite basis functions.)


The choice of basis functions φm is convenient since exact edge conditions along the longer

edges B and C may be trivially satisfied. In particular, using an obvious notation,

ψ(∗ee)(k; r, θ) =∑

m=0,6,12,···αmNm Jm(kr) cos(mθ) (96)

ψ(∗oo)(k; r, θ) =∑

m=6,12,18,···αmNm Jm(kr) sin(mθ) (97)

ψ(∗eo)(k; r, θ) =∑

m=3,9,15,···αmNm Jm(kr) cos(mθ) (98)

ψ(∗oe)(k; r, θ) =∑

m=3,9,15,···αmNm Jm(kr) sin(mθ) (99)

Note that the value of k in Eqs. (96)–(99) is not specified. To fix the value of k, the edge

condition along A must be enforced, which is done with either

ψ(o∗∗)∣∣∣A

= 0 or∂

∂xψ(e∗∗)

∣∣∣A

= 0 (100)

For a given set of edge conditions, there exists an infinite set of eigenfunctions and eigenval-

ues. Let ki for i = 1, 2, 3, · · ·; be the distinct cutoff wavenumbers where 0 < ki < ki+1. Also

let γi be the multiplicity of ki.31 Then, an exact eigenfunction may be written

ψ(s,i)(ki; r, θ) =∑

m

α(s,i)m φm(ki; r, θ) (101)

where s = 1, 2, · · · , γi labels degenerate eigenfunctions.

The point matching method provides a reliable and convenient method to determine the

numerical values of the k and α(s,i)m . In addition, the values of the coefficients permit a

simple physical interpretation of each eigenfunction by allowing a comparison between the

well-known circular modes and the 30◦-60◦-90◦ triangular modes.32 This comparison, in turn,

provides an elementary procedure to determine which terms of Eq. (101) are important. This

is good because, in practice, the sum (101) must be truncated above a certain number of

31This notation is more convenient than that of Eq. (84).32See Eq. (128) and the surrounding text for the interpretation. Briefly, α2

m is approximately equal to

the probability that the eigenfunction has a given ‘m-value’ within a hexagonal domain. Using the quantal

analog of a particle confined within a regular hexagonal domain (TM hexagon modes), this m-value is sort

of like an angular momentum (in units of h) of the particle. Of course, since the hexagon is not continuously

rotationally invariant, angular momentum is not a good quantum number and the interpretation need not

be taken literally.


terms. Before proceeding, it is necessary to examine some properties of the eigenfunctions

and basis functions.

One drawback and difficultly of the Fourier-Bessel series is that the basis functions are

orthonormalized in a circular region, not a polygonal region.33 For this reason, it is convenient

to define a 30◦ circular sector with an area equal to that of the 30◦-60◦-90◦ triangular,

fundamental region [4]. In addition, this circular region is oriented similar to the fundamental

region. More precisely, edges A, B, and C correspond respectively to the arc at r = R and

0 < θ < π/6, and the radii 0 < r < R at θ = 0 and θ = π/6. With equal-area regions (see

Eq. (113)), the radius of the sector is

R =

√3√

3

2π≈ 0.90939 (102)

Analogous to the eight symmetry classes of the 30◦-60◦-90◦ triangle, there exist eight sym-

metry classes of the 30◦ sector. The φm(k; r, θ) are normalized eigenfunctions within the

circular sector {[11], (Ex. 8.3)}. Imposing edge conditions on the arc at r = R gives the

possible values of k,

kmn ≡

jmn/R Dirichlet condition at r = R

j ′mn/R Neumann condition at r = R(103)

where jmn and j ′mn are the zeros (or roots) defined by Jm(jmn) = 0 and J′m(j ′mn) = 0, for

n = 1, 2, 3, · · · {[1], sec 9.5}. Since j ′0,1 = 0 is by convention the only vanishing zero, this

mode is trivial and may be ignored. The zeros interlace {[1], Eq. 9.5.2 and Table 9.5}according to

m ≤ j ′m,1 < jm,1 < j ′m,2 < jm,2 < j ′m,3 < · · · (104)

m < jm,1 < jm+1,1 < jm,2 < jm+1,2 < jm,3 < · · · (105)

m ≤ j ′m,1 < j ′m+1,1 < j ′m,2 < j ′m+1,2 < j ′m,3 < · · · (106)

where the equalities occur only with the unphysical j ′0,1 = 0. It is interesting to note that the

zeros of Jm and Jm+2 also interlace [36]; however, those of Jm and Jm+3 do not. In addition,

the only coincidences in the zeros occur with j ′0,n = j1,n−1 for n = 2, 3, 4, · · ·; this is because

J′0(z) = −J1(z) {[1], Eq. 9.5.1}.33Analogous to fitting a round peg (φm) into a polygonal hole (the edge conditions of ψ).


Imposing the edge conditions along the radii yield the normalized eigenfunctions in the

30◦ sector;

φ(∗ee)m (kmn; r, θ) = Nm Jm(kmnr) cos(mθ) where m = 0, 6, 12, · · · (107)

φ(∗oo)m (kkm; r, θ) = Nm Jm(kr) sin(mθ) where m = 6, 12, 18, · · · (108)

φ(∗eo)m (kmn; r, θ) = Nm Jm(kmnr) cos(mθ) where m = 3, 9, 15, · · · (109)

φ(∗oe)m (kmn; r, θ) = Nm Jm(kmnr) sin(mθ) where m = 3, 9, 15, · · · (110)

Compare these equations with Eqs. (96)–(99). The interlacing relations given by Eqs. (104)–

(106) prove that there is no degeneracy within any symmetry class of the 30◦ sector. How-

ever, a geometrical degeneracy exists between modes of different symmetry classes with

similar edge A conditions. Specifically, for m 6= 0, there exist pairs of degenerate modes

(φ(eee)m , φ(eoo)

m ); (φ(oee)m , φ(ooo)

m ); (φ(eeo)m , φ(eoe)

m ); and (φ(oeo)m , φ(ooe)

m ). These modes are obtained

from one another by rotation in the plane about the origin by π/2, which in effect trans-

forms the ‘sin’ into a ‘cos’ or visa-versa. There is no other degeneracy in these modes.34

Define the inner products of two well-behaved real functions, f and g,

(f |g) ≡ ∫ π/60 dθ∫ R0 r dr f(r, θ)g(r, θ) (30◦ sector) (111)

〈f |g〉 ≡ ∫√

3/20 dx

∫ x/√30 dy f(x, y)g(x, y) (30◦-60◦-90◦ triangle) (112)

The parentheses are used for the 30◦ circular sector (curved) while the angle brackets are

used for the 30◦-60◦-90◦ triangle (angled). The area of the sector and the fundamental region

is given by

A ≡ (1|1) =πR2

12= 〈1|1〉 =

√3

8(113)

This is used to fix the value of R given by Eq. (102).

The basis functions of a given symmetry class are partially orthonormalized in the 30◦

sector using

(φm(k)|φm′(k)) = A δmm′ (114)

34Recall that since the relevant m-values are multiples of three, the fact that j ′0,n = j1,n−1 (for n =

2, 3, 4, · · ·) does not lead to any accidental degeneracies. Of course, with a circular guide, the TE modes

J0(j′0,nr/R) are accidentally degenerate with the pairs of TM modes J1(j1,n−1r/R) sin(θ), cos(θ) {[11]

(Ex. 8.3)}


where δmm′ is the Kronecker delta function and k > 0 is (for now) arbitrary [15].35 Using

this normalization, the value of Nm depends only on the dimensionless parameter kR and is

determined using Eq. (114) with m = m′,

πR2

12= N 2

m ×[∫ π/6

0cos2(mθ) dθ

]×[k−2

∫ kR

0J2m(η) η dη

](115)

The integrals of Eq. (115) are elementary.36 Solving for Nm as a function of kR yields

Nm(kR) =

√2

σm

[J2m+1(kR) + J2

m(kR)−{

2m

kR

}Jm(kR) Jm+1(kR)

]−1/2

(116)

where σ0 = 2 and σm = 1 for m = 3, 6, 9, · · ·.The basis functions are approximately orthogonal in the fundamental region,

〈φm(k)|φm′(k)〉 = A [ δmm′ + εmm′(k)] ≈ A δmm′ (117)

where

εmm′(k) ≡[Nm(kR)Nm′(kR)

A k2

](118)

×∫ π/6

0dθ cos(mθ) cos(m′θ)

∫ k√

32 cos θ

kRη dη Jm(η) Jm′(η) (119)

is assumed to be small compared to unity. [In general, let f and g be well-behaved functions

in the plane such that (f |f) ≈ (g|g) ≈< f |f >≈< g|g >≈ A, then < f |g > −(f |g) ≡ Aε. If,

near r = R, there is either a node in f or g, or if f and g are relatively constant, then ε will

be small compared to unity. It is numerically verified that for k < 20, the |εmm′(k)| < 0.01.

For example, ε00(4.189) = −0.000218.]

With distinct value of k, the inner product of the basis functions becomes more compli-

cated. An exact treatment of the basis functions yields

(φm(kmn)|φm′(km′n′)) = Aδmm′δnn′ (120)

35The orthonormalization is partial in that only one value of k is considered. The more general expression

(φm(k)|φm′(k′)) is considered below.

36The Bessel function integral is∫ z

0 J2m(η)η dη = (z2/2)

[J′2m(z) + (1−m2/z2)J2

m(z)]{[33], Eq. 5.14.5},

where the derivative of the Bessel function is given by J′m(z) = mJm(z)/z − Jm+1(z) {[33], Eq. 5.2.6}.


where the kmn are given by Eq. (103). On the other hand, with arbitrary k 6= k′,

(φm(k)|φm′(k′)) = A δmm′Nm(kR)Nm(k′R)

R(k2 − k′2

)

(121)

× [ k′Jm(kR) J′m(k′R)− kJm(k′R) J′m(kR)] (122)

For kR and/or k′R not near a node of either Jm(η) or J′m(η), this expression is not necessarily

small. Note that in both of Eqs. (120) and (122), the δmm′ comes from the angular integration.

The coefficients in the expansion (101) may be determined using

(φm(ki)

∣∣∣ψ(s,i)(ki))

= Aα(s,i)m (123)

where ki is an ‘exact’ eigenvalue of the 30◦-60◦-90◦ triangle. However, note the approximation

⟨φm(ki)

∣∣∣ψ(s,i)⟩

= A[α(s,i)m +

∑

m′εmm′(k)α

(s,i)m′

]≈ Aα(s,i)

m (124)

The eigenfunctions within a given symmetry class are orthonormalized in the fundamental

region according to

⟨ψ(s,i)(ki)

∣∣∣ψ(t,j)(kj)⟩

= A δij δst =∑

mm′α(s,i)m α

(t,j)m′

⟨φm(ki)

∣∣∣φm′(kj)⟩

(125)

On the other hand, within the 30◦ sector the same two eigenfunctions satisfy

(ψ(s,i)(ki)

∣∣∣ψ(t,j)(kj))

=∑

mm′α(s,i)m α

(t,j)m′

(φm(ki)

∣∣∣φm′(kj))

(126)

[Editor’s note (2004): Eqs. (125) and (126) differ in the type of inner product, and the latter

is not proportional to δijδst.] Setting i = j (same eigenvalue), and using the approximation

of Eq. (117), Eqs. (125) and (126) yield the approximate relation for the coefficients

δst =∑

m

α(s,i)m α(t,i)

m +∑

mm′α(s,i)m εmm′(ki)α

(t,i)m′ ≈

∑

m

α(s,i)m α(t,i)

m (127)

Eqs. (124) and (127) suggest that α(s,i)m is approximately the amount of the basis function

φm(ki; r, θ) in the exact eigenfunction ψ(s,i)(ki; r, θ). The relative importance of each term is

determined by comparing each

P (s,i)m ≡

[α(s,i)m

]2(128)

in complete analogy with the quantum mechanical interpretation of the coefficients. (The

Pm are non-negative and approximately less than unity since∑m Pm ≈ 1). Although the


regular hexagonal region – obtained by repeatedly reflecting the fundamental region through

edges B and C – is not rotationally invariant, the Pm give an approximate ‘probability’ that

the eigenfunction has a definite m-value. Thus, for example, an average m-value for each

eigenfunction may be defined,⟨m(s,i)

⟩≡∑

m

mP (s,i)m (129)

This average m-value can be used to develop a heuristic comparison between the circular

sector modes and the triangular modes.

A more useful aspect of the Pm is obtained by observing that the terms with relatively

large values of Pm are dominant or important terms in the eigenfunction. Since the eigen-

function is approximated by truncating the sum, the values of Pm allow an a posteriori

determination of the validity of the method. In particular, the (exact) Pm, for each mode,

decrease in magnitude very rapidly beyond some minimum value of m. If the sum is trun-

cated below this minimum value, there may not be a one-to-one correspondence between the

approximate and the exact modes of the 30◦-60◦-90◦ triangular region. A lack of correspon-

dence indicates the method is unreliable.37 Below, a rule-of-thumb criterion is suggested for

estimating the minimum m-value; however, an a posteriori sharp fall-off of the Pm gives a

higher level of confidence in the numerical results.

In effect, the point-matching method prescribes a definite method of approximating the

eigenfunction by truncating the sum of Eq. (101). The number of terms retained equals the

number of points at which the edge conditions are enforced on edge-A using Eq. (100). If

we match at N points,

ψ(s,i)N (kN,i; r, θ) ≡

∑

m

[N ]α

(s,i)N,m φm(kN,i; r, θ)

= α(s,i)N,m1

φm1(kN,i; r, θ) + · · ·+ α(s,i)N,mN

φmN (kN,i; r, θ) (130)

where the ‘[N ]’ on the summation indicates that only the first N terms are included, and

where mµ for µ = 1, 2, · · · , N are the possible m-values with mµ < mµ+1 = 6 + mµ. Note

that m1 = 0 for the (*ee) modes, m1 = 3 for the (*eo) and (*oe) modes, and m1 = 6 for the

37Indeed, if the domain is continuously deformed from the 30◦-60◦-90◦ triangle to the 30◦ sector, we should

expect that each mode of both shapes can be followed from one to the other. This is discussed in more detail

in a later section.


(*oo) modes. Each ψ(s,i)N thus has N+1 parameters, namely, kN,i and the N coefficients α

(s,i)N,m.

Matching at N points yields N independent, constraint equations; thus, together with the

normalization, it is possible to determine values of the N + 1 parameters for each of the

eigenfunctions. Implicit are the assumptions that the set of matching points are distinct and

exclude those points where the edge conditions are satisfied trivially – as may occur at the

end-points of A.

In the limit that the matching points become infinitely dense along the edge A, the

approximate values must approach the exact values. In particular, if δy is the maximum

separation between adjacent matching points, then

limδy→0

kN,i = ki and limδy→0

α(s,i)N,m = α(s,i)

m (131)

which imply that

limδy→0

ψ(s,i)N = ψ(s,i) (132)

These are valid because the edge conditions are satisfied exactly in the limit that δy = 0. Note

that if the matching points are (approximately) equally spaced along A, then δy ≈ 1/(2N),

and the above limits are equivalent to the N →∞ limit. For convergence to the ‘exact’

value, necessary condition is that δy → 0 [corrected 2004] which requires that N →∞.38

Next, a general rule-of-thumb criterion is given for estimating the minimum number

of terms necessary for a reliable truncation. To be definite, consider the application to

the modes with cutoff wavenumbers between zero and some maximum value, 0 < k < kmax.

These results, however elementary, appear to be new.

First consider the (o**) modes. Ignoring the possibility of cancellation between terms,

the dominant terms, i.e., those with large Pm, heuristically should have a node near r = R,

i.e. near edge A. These nodes are governed by the nodes of the Bessel functions. Keeping in

mind interlacing relation Eq. (105), it is obvious that the truncation must not occur below

order m where

kmaxR < jm,1 (odd edge A) (133)

38The N → ∞ limit does not imply that δy → 0. For example, exclusion of the points between y = 0

and y = y0, where 0 < y0 < 1/2, prevents δy from becoming smaller than y0 no matter how large N may

become.


The Bessel functions of order greater than m do not support nodes at values of r < R, and

they serve to refine the eigenfunctions near the edge A. On the other hand, the dominant

terms govern the behavior of the eigenfunctions within the interior region.

The arguments for the (e**) modes are similar. Specifically, the dominant terms should

have an node in the derivative (with respect to r) near r = R with the result that

kmaxR < j ′m,1 (even edge A) (134)

where m is again the highest order.

If the number of terms in the expansion is sufficient, a contour along which the edge A

conditions are satisfied must pass continuously from edge B, through the matching points,

to edge C. This is a more difficult criterion to enforce, yet it must be satisfied for any

approximation to be valid.

In the numerical analysis, the normalization given by Eq. (127) is inconvenient because it

involves approximations. Therefore – instead of determining the εmm′ of Eq. (119) – impose

1 ≡∑

m

[N ] [α

(s,i)N,m

]2and α

(s,i)N,m1

≥ 0 (135)

for each of the approximate eigenfunctions ψ(s,i)N given by Eq. (130). The general meaning of

the coefficients is not lost; instead, the normalization of the eigenfunction is slightly modified.

However, this modification does not affect the values of kN,i.

To illustrate the practical technique used in this project for the (o**) modes, consider an

approximate eigenfunction given by an abbreviated version of Eq. (130)

ψN(k; η) = αm1φm1(k; η) + αm2φm2(k; η) + · · ·+ αmNφmN (k; η) (136)

where η = (r, θ) and k are (for now) arbitrary. Let ηµ (for µ = 1, 2, · · · , N) denote the

N matching points on edge A. For simplicity assume that 0 ≤ yµ < yµ+1 ≤ 1/2 where

yµ = rµ sin(θµ) and xµ =√

3/2. Thus, the N independent constraints are ψN (k; ηµ) = 0.

For each non-trivial eigen-solution, the determinant must vanish,

∆N(k; {η}) ≡

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

φm1(k; η1) φm2(k; η1) · · · φmN (k; η1)

φm1(k; η2) φm2(k; η2) · · · φmN (k; η2)...

.... . .

...

φm1(k; ηN) φm2(k; ηN) · · · φmN (k; ηN)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

= 0 (137)


This function ∆N (k; {η}) depends upon the number of matching points N , the distribution

of these points symbolized by {η}, and the eigenparameter k. Restricting the search to

values of k such that 0 < k < kmax < jmN ,1/R, the zeros of ∆N (k; {η}) are determined and

identified with the kN,i. After the kN,i are found, the coefficients α(s,i)N,m are determined subject

to Eq. (135). It is estimated numerically that a small induced error k → k ± δk, changes the

coefficients by αm → αm ± δαm where |δαm| < |δk|.The procedure is similar for the (e**) modes except that the constraints are ∂ψN (k; ηµ)/∂x = 0

and the values of k are limited by 0 < k < kmax < j ′mN ,1/R. It may be added that the com-

putation of the (e**) modes is a bit more involved because of the derivatives.

To speed up numerical computations for relatively large values of N , the matrix elements

of ∆N(k; {η}) were expanded in second or third order Taylor series about an estimate of

the eigenvalue κ ≈ kN . An improved value of κ′ ≈ kN was then determined. By repeating

this procedure and expanding about κ′, it was verified that this expansion method converges

very quickly, and in particular well within the reported precision.

Also, accidental degeneracies (which are possible with the closed-form triangle modes

as pointed out in Appendix A) are not encountered. This may be anticipated since (the

arbitrary value of) kmax = 20 is considerably smaller than the lowest accidental degeneracy

at k = 4π√

49/3 ≈ 29.3215. This is fortunate since an accidental degeneracy will result

(approximately) in a double root of ∆N (k; {η}). More effort must be made in examining the

behavior of the solutions in such a case.

4.3 Low-Order Point-Matching Results

Low order computations were performed for values of k from zero to twenty for all of the

symmetry classes. Each of the eigenvalues may be compared to exact values obtained using

higher order computations (done in a later section) or other methods. However, in this

section it is demonstrated that with a modest amount of numerical work, a relatively large

number of modes may be determined with a precision of three or more digits.

For simplicity, the N point-matching coordinates were chosen such that the subtended


angle between successive points is equal; namely,

θµ =π

6N

(µ− 1

2

)(138)

rµ =

√3/2

cos(θµ)(139)

where µ = 1, 2, . . . , N . This set of matching points satisfies the necessary condition that the

maximum distance between adjacent points δy → 0 as N →∞. The endpoints of edge A

are excluded because in some cases the edge conditions are trivially satisfied at these points.

This choice of distribution is not optimal and rather arbitrary. Any attempt to bound the

eigenvalues using the present computations is not made here.

For kmax = 20, Eqs. (133) and (134) yield m(A odd) = 13.1 and m(A even) = 16.1 re-

spectively. All computations included m-values of at least m = 18 which is the smallest

multiple of three greater than 16. The results are listed in Tables 7 through 14. Selected

values of N were four or five. Since the closed-form modes appear to converge more quickly,

N = 4 seemed very sufficient. In addition, the ‘odd-edge-A’ non-closed-form modes were

slightly easier to compute than the ‘even-edge-A’ non-closed-form modes; therefore, a larger

value of N (equal to five) was used.39

The 30◦ sector modes are given by Eqs. (107)–(110). The last two columns of Tables 7

through 14 list the values of (m,n) and kmn. given by Eq. (103). These are simply ordered

according to the values of kmn. The insight gained by comparing these modes is that the

correspondence between the modes is remarkable. Indeed, at least for the lowest modes, the

eigenvalues of both shapes agree to within several percent and the m-value of the dominant

term corresponds to the m-value of the circular mode. For example, the 4-(oee) mode has an

eigenvalue k = 10.9948 and a dominant coefficient α6 = 0.9265; whereas the corresponding

circular mode (m,n) = (6, 1) has k6,1 = 10.9261. The agreement gets worse as the eigenvalue

increases in value. I return to this in the next section.

One important observation is the a posteriori fact that enough terms were included in

the truncated eigenfunction. This can be seen from the one-to-one correspondence between

the circular modes and the triangular modes. Also, the Pm = α2m (for each mode) decreases

39In a later section, high-order computations of the non-closed-form modes are made, therefore, the present

limitations are not severe.


Table 7: Point-matching results of the (eee) modes with N = 4.

ψ(i) =∑m=0,6,12,18 αmNmJm(kr) cos(mθ)

φ(m,n) = NmJm(kmnr) cos(mθ)

Point-matching results Circle modes

i k α0 α6 α12 α18 (m,n) kmn

1 4.18879 0.9985 0.0553 0.0000 0.0000 (0,1) 4.21348

2 7.25520 0.8045 -0.5939 0.0016 0.0000 (0,2) 4.21348

3 8.37758 0.6941 0.7199 0.0061 0.0000 (6,1) 8.24866

4 11.0825 0.8816 -0.4662 -0.0734 0.0002 (0,3) 11.1871

5 12.5664 0.6096 0.7669 0.2006 0.0010 (6,2) 12.9042

6 14.5105 0.5366 -0.7332 0.4177 -0.0070 (0,4) 14.6512

7 15.1027 0.7467 0.1133 -0.6545 -0.0055 (12,1) 15.2617

8 16.7545 0.4835 0.6590 0.5748 0.0399 (6,3) 16.7894

9 18.2639 0.6833 -0.7127 0.1457 0.0634 (0,5) 18.1117

10 19.1916 0.7125 0.3940 -0.5347 -0.2261 (6,4) 20.4944

rapidly above some value of m. More detailed computations verify this convergence to a very

high order. Furthermore, also verified is the fact that if the Fourier-Bessel series is truncated

prematurely, the one-to-one correspondence is lost and this shows up as ‘missing’ zeros of

∆N (k; η).

4.4 Relation Between the Triangular and Circular Modes

There exists a one-to-one correspondence between the 30◦-60◦-90◦ triangle modes and the

30◦ circular sector modes. It will be shown that this correspondence is not unique; however,

it is rather interesting for several reasons {[17], [4]}First, as suggested by Ref. [17]40, it may be possible to classify the 30◦-60◦-90◦ triangle

modes by associating each to a well known circular mode. To some extent, this has already

40This reference actually considers the mapping of a rectangle to a circle (passing through elliptical do-

mains) in order to compare and classify the modes.


Table 8: Point-matching results of the (eoe) modes with N = 4.

ψ(i) =∑m=3,9,15,21 αmNmJm(kr) sin(mθ)

φ(m,n) = NmJm(kmnr) sin(mθ)



1 4.18879 1.0000 0.0008 0.0000 0.0000 (3,1) 4.61978

2 8.37758 0.9908 0.1351 0.0002 0.0000 (3,2) 8.81384

3 11.0825 0.6795 -0.7336 0.0021 0.0000 (9,1) 11.7787

4 12.5664 0.7720 0.6354 -0.0189 0.0000 (3,3) 12.4764

5 15.1029 0.8180 -0.5542 -0.1539 0.0004 (3,4) 16.0391

6 16.7553 0.7235 0.6419 0.2540 0.0034 (9,2) 16.8099

7 18.2586 0.3172 -0.7706 0.5526 -0.0110 (15,1) 18.7162

8 19.1953 0.7663 -0.1197 -0.6308 -0.0213 (3,5) 19.5611

Table 9: Point-matching results of the (oeo) modes with N = 4.





1 7.25520 0.9988 -0.0499 0.0000 0.0000 (3,1) 7.01586

2 11.0825 0.9952 0.0974 -0.0065 0.0000 (3,2) 10.7336

3 14.5104 0.7458 -0.6611 0.0816 -0.0003 (3,3) 14.3120

4 15.1029 0.6764 0.7325 -0.0775 -0.0008 (9,1) 14.6849

5 18.2585 0.8843 -0.4587 -0.0857 0.0125 (3,4) 17.8399

6 19.1956 0.4944 0.8479 0.1897 -0.0244 (9,2) 18.9591


Table 10: Point-matching results of the (ooo) modes with N = 4.




i k α6 α12 α18 α24 (m,n) kmn

1 11.0825 0.9965 -0.0836 -0.0001 0.0000 (6,1) 10.9261

2 15.1029 0.9877 0.1558 -0.0156 -0.0000 (6,2) 14.9433

3 18.2585 0.5976 -0.7922 0.1237 -0.0004 (12,1) 18.3620

4 19.1954 0.8100 0.5833 -0.0611 -0.0027 (6,3) 18.6980

Table 11: Point-matching results of the (eeo) modes with N = 4.





1 4.99048 0.9997 0.0239 -0.0020 0.0004 (3,1) 4.61978

2 9.15640 0.9989 0.0475 0.0060 -0.0008 (3,2) 8.81384

3 11.8582 0.2970 -0.9543 -0.0322 0.0037 (9,1) 11.7787

4 13.0088 0.9483 0.3172 0.0072 0.0003 (3,3) 12.4764

5 16.3337 0.8023 -0.5860 -0.1130 -0.0117 (3,4) 16.0391

6 17.0568 0.6609 0.7279 0.1820 0.0138 (9,2) 16.8099

7 18.8026 0.0371 -0.4551 0.8893 0.0267 (15,1) 18.7162


Table 12: Point-matching results of the (oee) modes with N = 5.

ψ(i) =∑m=0,6,12,18,24 αmNmJm(kr) cos(mθ)



i k α0 α6 α12 α18 α24 (m,n) kmn

1 2.67501 0.9994 -0.0342 0.0016 -0.0002 0.0000 (0,1) 2.64443

2 6.12323 0.9958 0.0916 -0.0074 0.0008 -0.0001 (0,2) 6.07008

3 9.49068 0.9602 -0.2770 0.0357 -0.0027 0.0004 (0,3) 9.51595

4 10.9948 0.3649 0.9265 -0.0918 0.0030 -0.0005 (6,1) 10.9261

5 12.9942 0.9231 -0.3825 -0.0389 0.0089 -0.0009 (0,4) 12.9664

6 14.8342 0.4995 0.8489 0.1710 -0.0247 0.0017 (6,2) 14.9433

7 16.5460 0.8379 -0.5436 0.0441 -0.0193 0.0023 (0,5) 16.4186

8 17.9763 0.5007 0.5962 -0.6170 0.1146 -0.0048 (12,1) 18.3620

9 18.9987 0.4017 0.4999 0.7630 -0.0814 -0.0048 (6,3) 18.6980

Table 13: Point-matching results of the (eoo) modes with N = 4.




i k α6 α12 α18 α24 (m,n) kmn

1 8.47039 0.9995 0.0307 -0.0029 0.0005 (6,1) 8.24866

2 13.0323 0.9924 0.1229 0.0117 -0.0011 (6,2) 12.9042

3 15.3093 0.3567 -0.9337 -0.0298 0.0029 (12,1) 15.2617

4 17.1089 0.9356 0.3529 0.0111 0.0017 (6,3) 16.7894


Table 14: Point-matching results of the (ooe) modes with N = 5.

ψ(i) =∑m=3,9,15,21,27 αmNmJm(kr) sin(mθ)



i k α3 α9 α15 α21 α27 (m,n) kmn

1 6.90190 0.9975 -0.0705 0.0034 -0.0006 0.0001 (3,1) 7.01586

2 10.5062 0.9908 0.1342 -0.0160 0.0016 -0.0001 (3,2) 10.7336

3 13.7697 0.9082 -0.4215 0.0704 -0.0047 0.0006 (3,3) 14.3120

4 14.9895 0.4625 0.8811 -0.0988 0.0002 -0.0002 (9,1) 14.6849

5 17.2330 0.9058 -0.4163 -0.0774 0.0182 -0.0014 (3,4) 17.8399

6 19.0686 0.5391 0.8164 0.2045 -0.0319 0.0010 (9,2) 18.9591

been done by dividing the 30◦ triangle modes up according to the edge conditions and

ordering them according to the cutoff wave number. The circular modes Jm(kmnr) cos(mθ)

and Jm(kmnr) sin(mθ) have m diametrical nodes and n − 1 interior circular nodes. The

diametrical nodes (antinodes) translate directly into Dirichlet (Neumann) conditions along

edges B and C of the 30◦-60◦-90◦ triangle. However, the circular nodes do not have such a

simple interpretation. Although the triangular modes do not possess circular nodes, a loose

connection may be drawn if we compare the typical wavelengths (which depend upon the

eigenvalue) in the different shapes.

Second, inscribed and circumscribed circles provide, respectively, upper and lower bounds

of the hexagon eigenvalues ([4], for example). These bounds are certainly not very tight and

are not used in this project. A failing of this bounding method is that it is not a priori

evident which eigenvalue is bounded, since the mapping is not unique.

Finally, the comparison between the modes can provide great insight into devising accu-

rate methods of solution. The point-matching method provides an appropriate example.

How may the one-to-one correspondence be shown? And how may the uniqueness fail?

First, imagine that the boundary of each region is continuously deformed from the 30◦-60◦-90◦

triangle to the 30◦ circular sector. Each intermediate shape (with appropriate edge condi-


tions) supports an infinite set of modes which may be associated in a one-to-one manner

with the modes of each nearby, nearly identical shape. Thus, given a definite method of de-

formation, each mode may be unambiguously and continuously followed (i.e. mapped) from

beginning to end and the one-to-one correspondence is established. This also demonstrates

that there is no creation or ‘destruction’ of modes in the mapping process. To be unique, the

one-to-one correspondence must be independent of the method of deformation. Uniqueness

may be proved if it can be shown that there are never any degenerate modes anywhere in the

mapping process. For, if this is true, every mode may be ordered according to the eigenvalue

and this ordering remains intact and the mapping is unique. The negative of the converse

is also valid; namely, if there may ever be any degenerate modes, then the mapping is not

unique. This is because the mapping depends upon the linear combination of degenerate

modes. Consider, for example, a definite method of deformation which supports two inter-

mediately degenerate modes, ψ and φ. Working from this intermediate shape, mapping the

linear combination aψ + bφ back to the circular and triangular modes depends upon the

(arbitrary) values of a and b. In effect, the levels may either actually cross or simply come

together and touch without crossing.41 Thus, the mapping in this case is ambiguous.

It is easy to see that the mapping is not unique. This is especially true for the closed-form

triangle modes since they may support arbitrarily high degeneracies. However, for any set

of modes, by simply choosing an isosceles right triangle (with appropriate edge conditions)

as an intermediate shape, every eigenvalue (with only one exception) may be brought into

coincidence with another. Thus, the mapping cannot be unique. The only exception occurs

with the lowest (ooo) mode, since this mode may never be degenerate for any domain.

For several modes, it was numerically verified that if the boundary is minimally deformed

from the 30◦-60◦-90◦ triangle to the 30◦ sector, the ordering of the eigenvalues is maintained.

The minimal deformation – of the edge connecting edges B and C – was given by

x2 + εy2 = εR2 + (1− ε) · 3

4(140)

where ε varies continuously from zero (triangle, x =√

3/2) to unity (sector, x2 + y2 = R2).

The ‘radial’ edges B and C remained intact and the area of the intermediate regions did not

41A ‘crossing’ indicates that the ordering of the eigenvalues changes in the mapping process.


vary by more than a few percent.

The coefficients αm(ε) and the eigenparameter k(ε) can be considered functions of the

parameter ε. When ε = 0, the values of the αm(ε = 0) and k(ε = 0) correspond to those

of those of the 30◦-60◦-90◦ triangle mode. At the other extreme, when ε = 1, the values

correspond to those of the 30◦ sector mode. And, since the basis functions are eigenfunctions

of the sector, only one of the coefficients is non-zero. Specifically, for each mode,

αm(ε = 1) = ±δmm′ (141)

where m′ is the m-value of the circular sector eigenmode to which the triangle mode is

mapped. The sign is determined using the normalization of Eq. (135).

Fig. 9 plots the approximate cutoff wavenumber and the values of the leading coefficients

as the eighth and ninth (oee) modes are mapped from the triangle to the sector. Observe that,

for example, the ninth (oee) mode is mapped into the ninth circular mode (m,n) = (6, 3)

despite the fact that α212(0) ≈ 0.58 is significantly greater than the other dominant terms

with α26(0) ≈ 0.25 and α2

0(0) ≈ 0.16. This provides an example of the fact that the largest

Pm = α2m for a given triangle mode does not necessarily determine the m-value of the asso-

ciated circular mode. Note that the computations were performed with N = 5, equiangular

matching points; and only the first four coefficients are plotted, since the fifth coefficient (in

both cases) is always less than 0.01 in magnitude.

4.5 High-Order Point-Matching Results

High order computations were performed for the non-closed form modes using the point-

matching method. In addition to the properties mentioned in the previous sections, two

important discoveries were made. One was that very tight bounds to the eigenvalues may

be obtained. The other was that, above a given order, the coefficients of the Fourier-Bessel

series follow a simple rule.

First, observe that the numerical values of the parameters (and their accura- cies) depend

upon the distribution as well as the number of matching points. For each approximate

eigenfunction ψN , with N fixed, there exists an optimal distribution of matching points.

This distribution is such that the maximum value of |ψN | or |∂ψN/∂x| along edge A is a


minimum if A is odd or even, respectively. Refs. ([15], [24] and [14]) show how to bracket

the odd-A eigenvalues using the maximum value of |ψN | along A. In the present application,

it was numerically discovered that the bounds using this method were not very tight. Also,

determining an ‘optimal distribution’ proved very difficult.

An apparently new method of bounding any eigenvalue is discussed next. Rather than

strive to optimize the distribution of matching points, it was discovered that by using two

distributions of matching points, the exact eigenvalues can be approached from above and

from below by increasing N . Dropping the suffix i, and using ‘[a]’ and ‘[b]’ to indicate

the distinct distributions, it was discovered that either of the following conditions may be

satisfied. First, the k[a]N and k

[b]N may be absolutely convergent, thus providing lower and

upper bounds respectively,

k[a]N−1 < k

[a]N < k(exact) < k

[b]N < k

[b]N−1 (142)

for any N above some minimum value. Second, the k[a]N and k

[b]N may alternately provide

upper and lower bounds according to

k[b]N−1 < k

[a]N < k(exact) < k

[b]N < k

[a]N−1 (143)

where N is (exclusively) either even or odd, as well as above some minimum value.

For the non-closed-form modes, Eq. (142) was satisfied, i.e. the approximate eigenvalues

were absolutely convergent from above or from below. This was verified for values of N from

three or four to up to seventy. No attempt was made to theoretically prove convergence. Also,

no attempt was made to optimize the actual distributions of points which gave this result.

Nevertheless, the results are numerically verified by comparing them to other independent

results.

For the clored-form modes, Eq. (143) seemed to be the rule. However, since these modes

are known exactly anyway, no further studies were made.

To report the bounds, the following round-off procedure is used. First, the upper and

lower bounds are compared. Then, the two digits following the matching digits are rounded

up for the upper bound and down for the lower bound. (This is a bit conservative, but helps

avoid the reporting of excessive digits.) These two digits are appended to the matching


Figure 9: Plots of the eighth and ninth (oee) eigenmodes as the cross section is deformedaccording to Eq. (140). The mapping is made as ε varies from zero (30◦-60◦-90◦ triangle) tounity (30◦ circular sector). Top: The eigenvalue mapping. Bottom: The coefficient mapping.


digits as shown in the following example. For N = 70, the approximate wavenumbers for

the lowest (oee) mode were determined to be

k[a]70 = 2.674946522660... (lower bound) (144)

k[b]70 = 2.674946579843... (upper bound) (145)

The first eight digits agree; therefore, use

k70 = 2.674 946 5 8022 (146)

to report the bounded eigenvalue.

The relative separation in the bounds is determined using the more precise values {e.g.

Eqs. (144) and (145)} and rounded up using two significant figures. Thus, for the lowest

(oee) mode in the above example,

∆k70/k70 < 2.2× 10−8 (147)

where ∆k ≡∣∣∣k[b] − k[a]

∣∣∣.

The convergence of the approximate eigenvalue as the number of matching points was

increased was very smooth. An empirical functional dependence may be discovered and used

to extrapolate the eigenvalue. However, only moderate success was made in this direction.

A plot of the lowest (oee) approximate wavenumber as a function of number of matching

points is made in Fig. 10. Note that the convergence rates for the two distributions are not

the same. In fact, the actual eigenvalue is probably closer to the upper bound than the lower

bound.

Next, to consider empirically the convergence of the Fourier-Bessel series of Eq. (101),

examine how the coefficients behave for large orders. Let the ratio of successive coefficients

be defined by

βm ≡αmαm+6

(148)

for each eigenfunction. It is observed that for terms of order m not less than the highest

dominant order – given by m of Eq. (133) or (134) – βm behaves very smoothly. This suggests

a simple functional dependence of αm upon m. Also, for modes within a given symmetry

class, it was observed that this functional dependence was nearly identical for each of the


Figure 10: Plot of the approximate wavenumber as a function of the number of matchingpoints for the lowest (oee) mode. This illustrates the convergence from above and frombelow using two different distributions of matching points. (Convergence for this mode wasactually verified up to N = 70.)


modes. The difference between the modes is of course indicated by different wavenumbers

and different leading order terms (i.e. the αm for m < m). Also, the larger the |βm| as m

increases, the more rapidly the series converges.

In Fig. 11, the ratio of coefficients βm is plotted for m-values from 6 to 394 for the

lowest (eoo) mode. This includes 49 terms. The number of matching points used in this

computation is N = 50. For simplicity, I use the notation mµ, where µ = 1, 2, · · · , N , in

which case, mµ = 6µ. Note that above µ = 40 (m = 240), the βm assume apparently

random values. This is probably a byproduct of the point-matching method, not numerical

‘round-off’.42 In fact, this random behavior of several of the highest order terms is present

in all computations, no matter what the value of N . More important is the behavior of the

first few terms (in this case, one) and the intermediate terms before the apparently erratic

behavior. Note that the values of βm for m > 6 are negative, indicating that the coefficients

alternate in sign. Also, with the normalization of Eq. (135), the value α6 ≈ 0.99954. Some

unsuccessful attempts were made to fit the smooth curve to elementary functions of m.43

The distributions of points which successfully bounded the eigenvalues are described

next. One set, which consistently gives lower bounds for each non-closed-form symmetry

class, were equally spaced points along edge A,

xµ =

√3

2and yµ =

1

2N

[µ− 1

2

](149)

for µ = 1, 2, · · · , N . These points exclude the end points and so may be used for each

symmetry class without modification. The other set of points, which consistently give com-

plementary upper bounds, are also equally spaced along edge A; however, they include the

end points of A only where appropriate,

xµ =

√3

2and yµ =

(µ−1)2(N−1)

for (*ee) modes

mu2N

for (*oe) modes

(µ−1)2N

for (*eo) modes

µ2(N−1)

for (*oo) modes

(150)

42The program MAPLE was used in the calculations. Setting the ‘Digits’ variable to increasingly larger

values from 20 up to 50 did not reduce the erratic behavior of the last few coefficients. Round-off error is

thus excluded.43For example, βm = A+B/m+ C/m2 + · · · did not work very well.


Figure 11: Plot of βm = αm/αm+1 for the lowest (eoo) mode using the approximation withN = 50. Fore convenience, on the abscissa, i = 1, 2, · · · , 49 is used where mi = 6i. Note thatthis mode has k ≈ 8.46825 and α6 ≈ 0.99954. Also, β6 ≈ 32.9871 and β12 ≈ −11.0693 areoff the vertical scale.

for µ = 1, 2, · · · , N . The excluded end points are locations where the edge conditions axe

satisfied trivially. (If they are included, the determinant ∆(k, {η}) vanishes identically.)

Note that the points of each distribution (149) and (150) are interleaved.

Although no mathematical proof is available, a heuristic argument is offered which ex-

plains why these distributions may give upper and lower bounds. The explanation is based

on the fact that increasing (decreasing) the area of the 30◦-60◦-90◦ decreases (increases) the

eigenvalue. Consider the slightly curved contour passing through the matching points where

the edge A conditions are enforced. Thus, if A is odd (or even), a node (or antinode) must

pass through the matching points and intersect with edges B and C. These ‘curved’ con-

tours, together with the radial edges, define two regions of slightly different shapes and sizes.

Although the eigenvalue depends upon the shape as well as the area, I conjecture that the

region with a slightly larger (smaller) area gives a slightly smaller (larger) eigenvalue. The

eigenfunctions appear to be most sensitive to the matching points near the intersections of

edges A and C. I also conjecture that the distribution given by Eq. (150) gives a contour

which bounds a slightly smaller region than that given by Eq. (149). Both regions approach


the exact region of the 30◦-60◦-90◦ triangle as the number of matching points increase, and

in particular, δy → 0 as N →∞; thus, the convergence toward the exact eigenvalue for both

distributions.

Table 15 summarizes the high-order point-matching results for the non-closed-form modes

with cutoff wavenumber below kmax = 20. Where available, results obtained using the imag-

inary time-step/finite-difference (ITS/FD) method are listed in the last two columns. Note

the remarkable agreement; except for the 20-EE mode, the ITS/FD estimates fall within the

indicated bounds.

The eigenvalues of both the closed-form and the non-closed-form modes are plotted in

Fig. 12. Note the regularity of the closed-form modes compared to the non-closed-form

modes. The curious near accidental degeneracy between the 4-(eeo), 5-(oee), and 2-(eoo)

may have been misinterpreted as an actual accidental degeneracy without such high-order

computations.

The TM rhombus modes correspond to the three-body systems C and D as discussed

in chapter I of this thesis. According to Table 6, these correspond to those 30◦-60◦-90◦

triangle modes with an odd edge C, i.e., the (**o) modes. The lowest 21 states of these

three-body problems are thus obtained. There are seven (eeo) modes, four (eoo), six (oeo),

and four (ooo) states with k < 20. The closed-form equilateral triangle results are reviewed

in Appendix A. It is interesting to note that the degeneracies of the closed-form modes in

this lower range of k-values is due to a geometrical symmetry of the equilateral triangle

modes, not a three-body symmetry.


Table 15: Summary of the bounded eigenvalues obtained using the point-matching method.Only the non-closed-form modes with k < 20 are given here. Where available, the lastcolumns list the mode and cutoff wavenumber (rounded to at least seven digits), obtainedwith the finite-difference method (Chapter II).

Point-matching Finite-difference

mode N kN ∆kN/kN mode k

1-(oee) 70 2.674 946 5 8022 2.2× 10−8 1-EE 2.674 946 5

2-(oee) 20 6.123 0 2111 1.5× 10−6 3-EE 6.123 018

3-(oee) 20 9.489 9288 3.4× 10−6 6-EE 9.489 908

4-(oee) 20 10.993 9 8652 3.1× 10−6 7-EE 10.993 98

5-(oee) 20 12.993 1206 3.4× 10−6 10-EE 12.993 10

6-(oee) 20 14.831 5240 7.0× 10−6 12-EE 14.831 48

7-(oee) 20 16.544 5548 3.4× 10−6 15-EE 16.544 52

8-(oee) 20 17.970 3513 1.2× 10−5 18-EE 17.970 28

9-(oee) 25 18.998 38 5901 2.6× 10−7 20-EE 18.998 36

1-(eeo) 50 4.989 84 6859 1.5× 10−7

2-(eeo) 20 9.155 8 1201 1.2× 10−6

3-(eeo) 20 11.851 2210 1.1× 10−5

4-(eeo) 20 13.008 789 6250 8.9× 10−9

5-(eeo) 20 16.327 2616 5.4× 10−6

6-(eeo) 20 17.050 5646 5.2× 10−6

7-(eeo) 20 18.777 8855 1.8× 10−5

1-(ooe) 50 6.901 40 3523 1.7× 10−7 2-EO 6.901 403

2-(ooe) 20 10.504 9 9753 4.1× 10−6 5-EO 10.504 98

3-(ooe) 20 13.766 4331 8.2× 10−6 9-EO 13.76639

4-(ooe) 20 14.989 1 5845 7.9× 10−7 11-EO 14.989 15

5-(ooe) 20 17.227 7133 1.0× 10−5 14-EO 17.227 65

6-(ooe) 20 19.065 4433 5.2× 10−5 17-EO 19.065 39

1-(eoo) 50 8.468 24 7950 3.4× 10−7

2-(eoo) 20 13.029 6357 3.9× 10−6

3-(eoo) 20 15.298 9675 1.3× 10−5

4-(eoo) 20 17.108 6 1106 2.4× 10−7


Figure 12: The spectrum of the 30◦-60◦-90◦ trianglular region with the eight possible edgeconditions. The closed-form and non-closed-form modes are listed in the left and right halvesrespectivley. Note the regular behavior of the closed-form spectrum. The only degeneraciesoccur in the closed-form eigenvalues; however, note the near degeneracy of the 4-(eeo),5-(oee), and 2-(eoo) modes.

5 CLOSED-FORM RESULTS 75

5 Closed-form Results

The following is a summary of the closed-form, equilateral triangle results. The form of the

equations shown below appears to be the most popular and convenient for the waveguide

problem. An algorithm for determining the spectrum is described and illustrated with a

tabulation and classification of the lowest 35 distinct eigenvalues (69 TE modes and 51 TM

modes). The derivation and mathematical proofs are omitted for brevity.

Using the coordinate system of Fig. 7, the eigenfunctions of Eq. (81) are

ψ(eee) = cos(lπx) cos[(m− n)πy/√

(3)]

+ cos(mπx) cos[(n− l)πy/√

(3)]

+ cos(nπx) cos[(l −m)πy/√

(3)] (151)

ψ(eee) = cos(lπx) sin[(m− n)πy/√

(3)]

+ cos(mπx) sin[(n− l)πy/√

(3)]

+ cos(nπx) sin[(l −m)πy/√

(3)] (152)

ψ(eee) = sin(lπx) cos[(m− n)πy/√

(3)]

+ sin(mπx) cos[(n− l)πy/√

(3)]

+ sin(nπx) cos[(l −m)πy/√

(3)] (153)

ψ(eee) = sin(lπx) sin[(m− n)πy/√

(3)]

+ sin(mπx) sin[(n− l)πy/√

(3)]

+ sin(nπx) sin[(l −m)πy/√

(3)] (154)

where l, m, and n are integers such that l + m + n = 0 and ψ 6= 0. The corresponding

eigenvalue is

λ = k2 =(

4π

3

)2

λ (155)

where the integer44

λ ≡ m2 +ml + l2 (156)

44Eq. (156) is a Diophantine (or Diofantine) equation.


is the eigenvalue in units of (4π/3)2 ≈ 4.188792. Modern derivations of these results using

novel methods based on the superposition of plane waves and/or the tiling property, may

be found in Refs. {[22], [21], [10], [9], [26], [8], [20], [25], [35]}. Note that Riemann-Schwarz

reflection of the closed-form modes is effected simply by extending the domain of x and y to

include adjacent equilateral triangles.

Each of Eqs. (151)–(154) and (156) is fully symmetric under permutations of the indices

l, m, and n. Therefore, we assume that

0 6= m ≥ l ≥ 0 (157)

because this excludes multiple counting of eigenmodes and trivial results.45 The index

n = −(l +m) is thus fixed by the values of l and m.

With a given pair of l and m, certain eigenfunctions may vanish identically. To determine

these uninteresting modes, observe that

ψ(eoe) ≡ 0 if l = m (158)

ψ(oeo) ≡ 0 if l = 0 (159)

ψ(ooo) ≡ 0 if l = 0 or l = m (160)

For example, to obtain Eq. (159), substitution of n = −m into Eq. (153) causes the first

term to vanish and the second two terms to cancel. Eqs. (158)–(160) give the only vanishing

modes; the remaining modes are non-vanishing.

The eigenfunctions of Eqs. (151)–(154) are normalized in the 30◦ − 60◦ − 90◦ triangle of

Fig. 2 using

N ≡ 〈ψ|ψ〉 =∫ √3/2

0dx∫ x/

√3

0dy ψ(x, y)2 (161)

The closed-form expressions for N are quite cumbersome46 except for the special cases l = 0

and l = m, where they reduce to

N (eee)(l = 0) =

√3

96(mπ)2

[18(mπ)2 + 8

√3(mπ) sin(

√3mπ)

+ 8 + 72 cos(mπ/√

3)− 63 cos(2mπ/√

3)

− 8 cos(√

3mπ)− 9 cos(4mπ/√

3)]

(162)

45If m = l = n = 0, then ψ(ooo) = 3 and the other functions vanish.46Hundreds of terms and several pages long


N (eoe)(l = 0) =

√3

96(mπ)2

[18(mπ)2 + 8

√3(mπ) sin(

√3mπ)

+ 8− 72 cos(mπ/√

3) + 63 cos(2mπ/√

3)

− 8 cos(√

3mπ) + 9 cos(4mπ/√

3)]

(163)

N (eee)(l = m) =

√3

96(mπ)2

[18(mπ)2 + 2

√3(mπ)

{sin(2

√3mπ)

+ 4 sin(√

3mπ)}

+ 9− 9 cos(2√

3mπ)]

(164)

N (oeo)(l = m) =

√3

96(mπ)2

[18(mπ)2 − 2

√3(mπ)

{sin(2

√3mπ)

+ 4 sin(√

3mπ)}− 9 + 16 cos(

√3mπ)

− 7 cos(2√

3mπ)]

(165)

The asymptotic value of Eqs. (162)–(163) is N = 3√

3/16 +O(1/m). If m > l > 0, then the

asymptotic value is N = 3√

3/32 +O(1/m) +O(1/l). For small values of m, the values of

N do not vary by much more than ten percent of these asymptotic values. These results do

not appear in the published literature.

There are two types of accidental degeneracy. The first type occurs if the Diophantine

problem, namely Eq. (156) subject to Eq. (157), admits multiple solutions. The second type

occurs if two or more distinct modes with the same l and m do not belong to the mixed

representation of S3. To determine the exact nature of the spectrum and the classification

of the closed-form modes, we need some results from elementary number theory.

Every integer λ of the form m2 +ml + l2 has the unique Gaussian factorization

λ =

[3α ·

r∏

i=1

pγii

]· 2β0 ·

k∏

j=1

qβjj

2

(166)

where the p and q are rational primes congruent to respectively 1 and 5, modulo 6.47 (The

converse is true, any number given by Eq. (166) is an eigenvalue λ of the form m2 +ml + l2.)

The lowest values of these primes are

p = 7, 13, 19, 31, 37, 43, 61, ... (167)

q = 5, 11, 17, 23, 29, 41, 47, ... (168)

47In other words, p and q are primes of the form p = 6µ+ 1 and q = 6µ− 1 where µ = 1, 2, 3, ....


Table 16: The accidental degeneracy of an eigenvalue within each symmetry class.

α γ g(eee) g(eoe) g(oeo) g(ooo)

even even (g + 2)/4 (g + 2)/4 (g − 2)/4 (g − 2)/4

odd even (g + 2)/4 (g − 2)/4 (g + 2)/4 (g − 2)/4

odd g/4 g/4 g/4 g/4

The exponents α (of 3) and the r non-zero γ ≡ {γi|i = 1, 2, 3, ..., r} (of the pi) play an

essential role in this formalism. We use the notation that if all of the γi are even, then γ

is said to be ’even’; while if some or all of the γi are odd, then γ is said to be ’odd’. To

illustrate, if α and γ are both even, then λ is the square of an integer.

A number-theoretic result using integer-domain ring theory is that the total number of

distinct TE and TM modes belonging to a given eigenvalue λ is

g = 2r∏

i=1

[γi + 1] (169)

Furthermore, the number of distinct solutions of the Diophantine problem, Eqs. (156) and

(157), is (g + 2)/4 is γ is even and g/4 if γ is odd. Table 16 lists the accidental degeneracy

of an eigenvalue within each symmetry class. Discussion and proof of these results may be

found in Refs. {[6], [31], [26]}.Multiple solutions of the Diophantine problem yield linearly independent solutions within

each symmetry class; however, degenerate eigenfunctions are not necessarily orthogonal.48

The Gram-Schmidt orthogonalization procedure {[22], for example} may be used to obtain

orthogonal functions.

The equilateral triangle eigenvalue problem is invariant under the geometrical point

group S3, i.e., the symmetry group of degree three [20]. Thus it is helpful to further classify

the closed-form modes according to the irreducible representations of S3 – namely, the one-

dimensional symmetric S, the one-dimensional antisymmetric A, and the two-dimensional

mixed M representations. According to S3 invariance, the only geometrical degeneracy

48The TE (TM) modes within separate symmetry classes are orthogonal by construction.


occurs in the pairs of mixed modes; any other degeneracy is accidental.

The symmetric and antisymmetric eigenfunctions have respectively even and odd lines

connecting each of the three vertices to the center of the opposite edge of the equilateral

triangle. A judicious linear combination of the degenerate mixed modes is made such that

each pair is split into separate symmetry classes [25]. Specifically, these mixed modes,Meand

Mo, have exactly one even and one odd line, respectively, connecting a vertex to the center

of the opposite edge.

If α > 0 for a given eigenvalue, then the modes belong to either the symmetric or

antisymmetric representations. But, if α = 0, then the modes come in pairs of mixed modes.

The classification scheme is summarized in Table 17. The only geometrical degeneracy due

to S3 invariance occurs with pairs of either TE (Me-TE and Mo-TE) or TM (Me-TM and

Mo-TM) modes. Every other degeneracy is accidental, including multiple sets of mixed

modes.

The reason why modes with α = 0 belong to the M representation, while those with

α > 0 belong to either the Aor Srepresentations, is related to a reflect-and-shrink property.

Indeed, any closed form 30◦ − 60◦ − 90◦ triangle mode ψ with eigenvalue λ may be reflected

three times to form another larger 30◦ − 60◦ − 90◦ triangle mode. If this is reduced to the

size of the original triangle (using (x, y)→ (x/3, y/3)), we obtain an Aor Smode ψ ′ with

eigenvalue 3λ. Every Sand Amode may be obtained this way.

The edge conditions of ψ determine the S3 symmetry and boundary conditions of ψ′.

Thus, if the edge conditions of ψ are (eee), (eoe), (oeo), or (ooo), then, the edge conditions

of ψ′ are (eee)=S-TE, (oeo)=S-TM, (eoe)=A-TE, or (ooo)=A-TM, respectively. If α = 0,

then this mode cannot be obtained using the reflect-and-shrink property of a lower mode

and must consequently be an Mmode, since it is neither and Snor an Amode. Because not

all inverses exist, the reflect-and-shrink operations do not form a group.

Tabulation of the eigenmodes is obtained using the following procedure: First, select

a number λmax > λ and fill in the triangular array as shown in Fig. 13 keeping in mind

Eq. (157). In this example, λmax = 100. Note that λ = 49 and 91 appear twice.

Second, arrange the possible l, m, and λ as shown in the first three columns of Table 18.

In the first column, prime factor each λ according to Eq. 166.


Table 17: Classification of the closed-form modes according to S3 based on the value of α.

eee eoe oeo ooo

α = 0 Me-TE Mo-TE Me-TM Mo-TM

α > 0 S-TE A-TE S-TM A-TM

9 81 91

8 64 73 84 97

7 49 57 67 79 93

6 36 43 52 63 76 91

5 25 31 39 49 61 75

4 16 21 28 37 48 –

3 9 13 19 27 – –

2 4 7 12 – – –

1 1 3 – – – –

m/l 0 1 2 3 4 5

Figure 13: Tabulation of the possible λ < 100.


Third, for each pair l and m, determine the modes of Eqs. (151)–(154) which vanish

identically by applying the rules of Eqs. (158)–(160). Mark a ’–’ in columns 4-6 of Table 18

for each vanishing mode.

Finally, classify each non-vanishing mode according to S3, using the value of α and

Table 17. From Table 18, we observe that there are five possibilities for a given pair of l and

m. They are illustrated with (l, m) = (0, 1), (1, 1), (1, 2), (0, 3), and (1, 4).


Table 18: Tabulation and classification of all closed-form eigenmodes with λ < 100. (Thefirst row indicates the boundary conditions of the equilateral triangle.)

TE modes TM modes

λ l m (eee) (eoe) (oeo) (ooo)

1 0 1 Me Mo – –3 1 1 S – S –4 = 22 0 2 Me Mo – –7 1 2 Me Mo Me Mo

9 = 32 0 3 S A – –12 = 22 · 3 2 2 S – S –13 1 3 Me Mo Me Mo

16 = 24 0 4 Me Mo – –19 2 3 Me Mo Me Mo

21 = 3 · 7 1 4 S A S A25 = 52 0 5 Me Mo – –27 = 33 3 3 S – S –28 = 22 ∗ 7 2 4 Me Mo Me Mo

31 1 5 Me Mo Me Mo

36 = 22 · 32 0 6 S A – –37 3 4 Me Mo Me Mo

39 = 3 · 13 2 5 S A S A43 1 6 Me Mo Me Mo

48 = 24 · 3 4 4 S – S –49 = 72 0 7 Me Mo – –49 = 72 3 5 Me Mo Me Mo

52 = 22 · 13 2 6 Me Mo Me Mo

57 = 3 · 19 1 7 S A S A61 4 5 Me Mo Me Mo

63 = 32 · 7 3 6 S A S A64 = 26 0 8 Me Mo – –67 2 7 Me Mo Me Mo

73 1 8 Me Mo Me Mo

75 = 3 · 52 5 5 S – S –76 = 22 · 19 4 6 Me Mo Me Mo

79 3 7 Me Mo Me Mo

81 = 34 0 9 S A – –84 = 22 · 3 · 7 2 8 S A S A91 = 7 · 13 1 9 Me Mo Me Mo

91 = 7 · 13 5 6 Me Mo Me Mo

93 = 3 · 31 4 7 S A S A97 3 8 Me Mo Me Mo

REFERENCES 83

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REFERENCES 84

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SOLUTIONS OF THE TWO-DIMENSIONAL HELMHOLTZ EQUATION

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