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January  2021, 20(1): 193-214. doi: 10.3934/cpaa.2020263

Spectrum of the Laplacian on regular polyhedra

1. 

Harvard University, Department of Mathematics, 1 Oxford St., Cambridge, MA, USA

2. 

Imperial College London, Department of Mathematics, 180 Queens Gate, Kensington, London SW7 2RH, UK

3. 

Cornell University, Department of Mathematics, Malott Hall, Ithaca, NY 14853, USA

4. 

Universität Leipzig, Department of Mathematics, Augustusplatz 10, 04109 Leipzig, Germany

* Corresponding author

Received  September 2019 Revised  August 2020 Published  January 2021 Early access  October 2020

Fund Project: EG was supported by the National Science Foundation through the Research Experience for Undergraduates (REU) Program, Grant DMS-1156350. DK was supported by the National Science Foundation through the Research Experience for Undergraduates (REU) Program, Grant DMS-1156350. RSS was supported in part by the National Science Foundation, Grant DMS-1162045. SCW was supported by the Foundation of German Business (SDW)

We study eigenvalues and eigenfunctions of the Laplacian on the surfaces of four of the regular polyhedra: tetrahedron, octahedron, icosahedron and cube. We show two types of eigenfunctions: nonsingular ones that are smooth at vertices, lift to periodic functions on the plane and are expressible in terms of trigonometric polynomials; and singular ones that have none of these properties. We give numerical evidence for conjectured asymptotic estimates of the eigenvalue counting function. We describe an enlargement phenomenon for certain eigenfunctions on the octahedron that scales down eigenvalues by a factor of $ \frac{1}{3} $.

Citation: Evan Greif, Daniel Kaplan, Robert S. Strichartz, Samuel C. Wiese. Spectrum of the Laplacian on regular polyhedra. Communications on Pure & Applied Analysis, 2021, 20 (1) : 193-214. doi: 10.3934/cpaa.2020263
References:
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show all references

References:
[1]

L. Hillairet, Spectral theory of translation surfaces: A short introduction, Séminaire de théorie spectrale et géométrie, 28 (2009), 51-62.  doi: 10.5802/tsg.278.  Google Scholar

[2]

S. Jayakar and R. S. Strichartz, Average number of lattice points in a disk, Commun. Pure Appl. Anal., 15 (2016), 1-8.  doi: 10.3934/cpaa.2016.15.1.  Google Scholar

[3]

A. Kokotov, Polyhedral surfaces and determinant of Laplacian, Proc. Am. Math. Soc., 141 (2013), 725-735.  doi: 10.1090/S0002-9939-2012-11531-X.  Google Scholar

[4]

A. Kokotov and D. Korotkin, Tau-functions on spaces of Abelian differentials and higher genus generalization of Ray-Singer formula, J. Differ. Geom., 82 (2009), 35-100.   Google Scholar

[5]

P. A. Kuchment, Operator Theory: Advances and Applications, Birkhauser, 2012. doi: 10.1007/978-3-0348-8573-7.  Google Scholar

[6]

B. McCartin, On Polygonal Domains with Trigonometric Eigenfunctions of the Laplacian under Dirichlet or Neumann Boundary Conditions, Appl. Math. Sci., 2 (2008), 2891-2901.   Google Scholar

[7]

A. N. Sengupta, Representing Finite Groups: A semisimple introduction, Springer, New York, 2012. doi: 10.1007/978-1-4614-1231-1.  Google Scholar

[8]

J. Serre, Linear Representations of Finite Groups, Springer, New York, 1977.  Google Scholar

[9]

T. Shioya, Geometric Analysis on Alexandrov Spaces, Sugaku Expositions, 24 (2011), 145–167.  Google Scholar

[10]

B. Simon, Representations of Finite and Compact Groups, American Mathematical Society, 1996. doi: 10.1038/383266a0.  Google Scholar

[11]

R. S. Strichartz, Average error for spectral asymptotics on surfaces, Commun. Pure Appl. Anal., 15 (2016), 9-39.  doi: 10.3934/cpaa.2016.15.9.  Google Scholar

[12]

R. S. Strichartz and S. C. Wiese, Spectrum of the Laplacian on regular polyhedra, http://pi.math.cornell.edu/ polyhedral. Google Scholar

[13]

A. Zoric, Flat Surfaces, Frontiers in Number Theory, Physics, and Geometry Ⅰ, Springer-Verlag, 2006. doi: 10.1007/978-3-540-31347-2_13.  Google Scholar

Figure 1.  Eigenfunctions on the cube
Figure 2.  Restriction of eigenfunctions on the cube to $ y = 0 $
Figure 3.  Eigenfunctions on the octahedron: (a) is obtained from (b) by rotating and dilating by $ \sqrt{3} $, which reduces the eigenvalue by a factor of $ \frac{1}{3} $
Figure 4.  Identified edges on the cube
Figure 5.  Identified edges on the tetrahedron
Figure 6.  Eigenfunctions on the tetrahedron
Figure 7.  Restriction of eigenfunctions on the tetrahedron to $ y = 0 $
Figure 8.  The torus covering
Figure 9.  Reflection symmetry
Figure 10.  Skew-symmetric (1-) reflection
Figure 11.  The hexagonal lattice
Figure 12.  A generic orbit ($ j > k > 0 $)
Figure 13.  Eigenvalues on the tetrahedron
Figure 14.  Counting function on the tetrahedron
Figure 15.  Identified edges on the octahedron
Figure 16.  Eigenfunctions on the octahedron
Figure 17.  Restriction of eigenfunctions on the octahedron to $ y = 0 $
Figure 18.  Types of reflections on the octahedron
Figure 19.  Symmetry properties of eigenfunctions on the octahedron
Figure 20.  Tetrahedron-type eigenfunctions
Figure 21.  Distribution of signs
Figure 22.  Counting function on the octahedron
Figure 23.  Eigenfunctions on the icosahedron
Figure 24.  Restriction of eigenfunctions on the icosahedron to $ y = 0 $
Figure 25.  Identified edges on the icosahedron
Figure 26.  Counting function on the icosahedron
Figure 27.  Reflections on the cube
Figure 28.  Symmetric properties of eigenfunctions on the cube
Figure 29.  Integer lattice
Figure 30.  Generic orbit
Figure 31.  Distribution of signs over the orbit on the cube
Figure 32.  Counting function on the cube
Table 1.  Normalized eigenvalues on the tetrahedron, res. 128: we can see the error (the deviation from the integer value) growing
# Eigenvalue # Eigenvalue # Eigenvalue
1 0 21 12.00181 41 21.00554
2 1.00001 22 12.00181 42 21.00554
3 1.00001 23 13.00212 43 21.00554
4 1.00001 24 13.00212 44 25.00784
5 3.00011 25 13.00212 45 25.00784
6 3.00011 26 13.00212 46 25.00784
7 3.00011 27 13.00212 47 27.00915
8 4.00020 28 13.00212 48 27.00915
9 4.00020 29 16.00321 49 27.00915
10 4.00020 30 16.00321 50 28.00984
11 7.00062 31 16.00321 51 28.00984
12 7.00062 32 19.00453 52 28.00984
13 7.00062 33 19.00453 53 28.00984
14 7.00062 34 19.00453 54 28.00984
15 7.00062 35 19.00453 55 28.00984
16 7.00062 36 19.00453 56 31.01206
17 9.00102 37 19.00453 57 31.01206
18 9.00102 38 21.00554 58 31.01206
19 9.00102 39 21.00554 59 31.01206
20 12.00181 40 21.00554 60 31.01206
# Eigenvalue # Eigenvalue # Eigenvalue
1 0 21 12.00181 41 21.00554
2 1.00001 22 12.00181 42 21.00554
3 1.00001 23 13.00212 43 21.00554
4 1.00001 24 13.00212 44 25.00784
5 3.00011 25 13.00212 45 25.00784
6 3.00011 26 13.00212 46 25.00784
7 3.00011 27 13.00212 47 27.00915
8 4.00020 28 13.00212 48 27.00915
9 4.00020 29 16.00321 49 27.00915
10 4.00020 30 16.00321 50 28.00984
11 7.00062 31 16.00321 51 28.00984
12 7.00062 32 19.00453 52 28.00984
13 7.00062 33 19.00453 53 28.00984
14 7.00062 34 19.00453 54 28.00984
15 7.00062 35 19.00453 55 28.00984
16 7.00062 36 19.00453 56 31.01206
17 9.00102 37 19.00453 57 31.01206
18 9.00102 38 21.00554 58 31.01206
19 9.00102 39 21.00554 59 31.01206
20 12.00181 40 21.00554 60 31.01206
Table 2.  Normalized eigenvalues on the octahedron, Res. 128
# Eigenvalue # Eigenvalue # Eigenvalue
1 0 21 5.45089 41 10.67867
2 0.54376 22 5.45089 42 12.00723
3 0.54376 23 6.37226 43 12.00723
4 0.54376 24 6.37226 44 12.83710
5 1.33342 25 6.37226 45 12.83710
6 1.33342 26 6.84597 46 12.83710
7 1.89224 27 6.84597 47 12.86814
8 1.89224 28 6.84597 48 12.86814
9 2.84941 29 8.38948 49 12.86814
10 2.84941 30 8.38948 50 12.90939
11 2.84941 31 8.38948 51 12.90939
12 3.62006 32 9.18907 52 12.90939
13 3.62006 33 9.18907 53 14.41173
14 3.62006 34 9.18907 54 14.41173
15 4.00080 35 9.33771 55 14.41173
16 4.00080 36 9.33771 56 16.01286
17 5.33476 37 9.33771 57 16.01286
18 5.33476 38 9.33771 58 16.72998
19 5.45089 39 10.67867 59 16.72998
20 5.45089 40 10.67867 60 16.72998
# Eigenvalue # Eigenvalue # Eigenvalue
1 0 21 5.45089 41 10.67867
2 0.54376 22 5.45089 42 12.00723
3 0.54376 23 6.37226 43 12.00723
4 0.54376 24 6.37226 44 12.83710
5 1.33342 25 6.37226 45 12.83710
6 1.33342 26 6.84597 46 12.83710
7 1.89224 27 6.84597 47 12.86814
8 1.89224 28 6.84597 48 12.86814
9 2.84941 29 8.38948 49 12.86814
10 2.84941 30 8.38948 50 12.90939
11 2.84941 31 8.38948 51 12.90939
12 3.62006 32 9.18907 52 12.90939
13 3.62006 33 9.18907 53 14.41173
14 3.62006 34 9.18907 54 14.41173
15 4.00080 35 9.33771 55 14.41173
16 4.00080 36 9.33771 56 16.01286
17 5.33476 37 9.33771 57 16.01286
18 5.33476 38 9.33771 58 16.72998
19 5.45089 39 10.67867 59 16.72998
20 5.45089 40 10.67867 60 16.72998
Table 3.  Normalized eigenvalues on the icosahedron, Res. 128
# Eigenvalue # Eigenvalue # Eigenvalue
1 0 21 2.11277 41 4.64893
2 0.22032 22 2.32749 42 4.64893
3 0.22032 23 2.32749 43 4.64893
4 0.22032 24 2.32750 44 4.64894
5 0.65895 25 2.32750 45 4.64894
6 0.65895 26 3.05440 46 4.83217
7 0.65896 27 3.05441 47 4.83219
8 0.65896 28 3.05442 48 4.83221
9 0.65895 29 3.40530 49 4.83222
10 1.22415 30 3.40530 50 5.69309
11 1.22415 31 3.40530 51 5.69313
12 1.22415 32 3.40727 52 5.69313
13 1.39760 33 3.40728 53 6.19595
14 1.39761 34 3.40730 54 6.19595
15 1.39761 35 3.40732 55 6.19595
16 1.39762 36 3.40732 56 6.27054
17 2.11275 37 4.00080 57 6.27057
18 2.11276 38 4.56435 58 6.27060
19 2.11277 39 4.56436 59 6.27062
20 2.11277 40 4.56439 60 6.27062
# Eigenvalue # Eigenvalue # Eigenvalue
1 0 21 2.11277 41 4.64893
2 0.22032 22 2.32749 42 4.64893
3 0.22032 23 2.32749 43 4.64893
4 0.22032 24 2.32750 44 4.64894
5 0.65895 25 2.32750 45 4.64894
6 0.65895 26 3.05440 46 4.83217
7 0.65896 27 3.05441 47 4.83219
8 0.65896 28 3.05442 48 4.83221
9 0.65895 29 3.40530 49 4.83222
10 1.22415 30 3.40530 50 5.69309
11 1.22415 31 3.40530 51 5.69313
12 1.22415 32 3.40727 52 5.69313
13 1.39760 33 3.40728 53 6.19595
14 1.39761 34 3.40730 54 6.19595
15 1.39761 35 3.40732 55 6.19595
16 1.39762 36 3.40732 56 6.27054
17 2.11275 37 4.00080 57 6.27057
18 2.11276 38 4.56435 58 6.27060
19 2.11277 39 4.56436 59 6.27062
20 2.11277 40 4.56439 60 6.27062
Table 4.  Normalized eigenvalues on the cube
# Eigenvalue # Eigenvalue # Eigenvalue
1 0 21 4.52692 41 8.70184
2 0.42105 22 4.52697 42 8.70209
3 0.42171 23 4.54599 43 8.71324
4 0.42197 24 4.61381 44 9.41359
5 1.16475 25 4.61602 45 9.44725
6 1.16502 26 5.65888 46 9.70349
7 1.16512 27 5.66338 47 9.1256
8 1.42522 28 5.66512 48 9.71602
9 1.43001 29 6.13609 49 9.96909
10 2.00027 30 6.15305 50 10.00591
11 2.59432 31 6.63945 51 11.02694
12 2.60125 32 6.65077 52 11.03827
13 2.60384 33 6.65518 53 11.04246
14 2.67862 34 7.00648 54 11.39163
15 2.67925 35 7.02039 55 11.39266
16 2.68175 36 7.02786 56 11.42616
17 3.81367 37 8.00428 57 11.95575
18 3.81781 38 8.05707 58 11.96738
19 3.81940 39 8.07340 59 12.69329
20 4.00067 40 8.07945 60 12.72420
# Eigenvalue # Eigenvalue # Eigenvalue
1 0 21 4.52692 41 8.70184
2 0.42105 22 4.52697 42 8.70209
3 0.42171 23 4.54599 43 8.71324
4 0.42197 24 4.61381 44 9.41359
5 1.16475 25 4.61602 45 9.44725
6 1.16502 26 5.65888 46 9.70349
7 1.16512 27 5.66338 47 9.1256
8 1.42522 28 5.66512 48 9.71602
9 1.43001 29 6.13609 49 9.96909
10 2.00027 30 6.15305 50 10.00591
11 2.59432 31 6.63945 51 11.02694
12 2.60125 32 6.65077 52 11.03827
13 2.60384 33 6.65518 53 11.04246
14 2.67862 34 7.00648 54 11.39163
15 2.67925 35 7.02039 55 11.39266
16 2.68175 36 7.02786 56 11.42616
17 3.81367 37 8.00428 57 11.95575
18 3.81781 38 8.05707 58 11.96738
19 3.81940 39 8.07340 59 12.69329
20 4.00067 40 8.07945 60 12.72420
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