Figure 1.
Eigenfunctions on the cube
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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 |
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} $.
Mathematics Subject Classification:
35P05.
Citation:
Evan Greif, Daniel Kaplan, Robert S. Strichartz, Samuel C. Wiese. Spectrum of the Laplacian on regular polyhedra. Communications on Pure and Applied Analysis,
2021, 20
(1)
: 193-214.
doi: 10.3934/cpaa.2020263
![]()
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. |
| [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. |
| [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. |
| [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.
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| [5] |
P. A. Kuchment, Operator Theory: Advances and Applications, Birkhauser, 2012.
doi: 10.1007/978-3-0348-8573-7. |
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B. McCartin,
On Polygonal Domains with Trigonometric Eigenfunctions of the Laplacian under Dirichlet or Neumann Boundary Conditions, Appl. Math. Sci., 2 (2008), 2891-2901.
|
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A. N. Sengupta, Representing Finite Groups: A semisimple introduction, Springer, New York, 2012.
doi: 10.1007/978-1-4614-1231-1. |
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J. Serre, Linear Representations of Finite Groups, Springer, New York, 1977. |
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T. Shioya, Geometric Analysis on Alexandrov Spaces, Sugaku Expositions, 24 (2011), 145–167. |
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B. Simon, Representations of Finite and Compact Groups, American Mathematical Society, 1996.
doi: 10.1038/383266a0. |
| [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. |
| [12] |
R. S. Strichartz and S. C. Wiese, Spectrum of the Laplacian on regular polyhedra, http://pi.math.cornell.edu/ polyhedral. |
| [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. |
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. |
| [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. |
| [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. |
| [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.
|
| [5] |
P. A. Kuchment, Operator Theory: Advances and Applications, Birkhauser, 2012.
doi: 10.1007/978-3-0348-8573-7. |
| [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.
|
| [7] |
A. N. Sengupta, Representing Finite Groups: A semisimple introduction, Springer, New York, 2012.
doi: 10.1007/978-1-4614-1231-1. |
| [8] |
J. Serre, Linear Representations of Finite Groups, Springer, New York, 1977. |
| [9] |
T. Shioya, Geometric Analysis on Alexandrov Spaces, Sugaku Expositions, 24 (2011), 145–167. |
| [10] |
B. Simon, Representations of Finite and Compact Groups, American Mathematical Society, 1996.
doi: 10.1038/383266a0. |
| [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. |
| [12] |
R. S. Strichartz and S. C. Wiese, Spectrum of the Laplacian on regular polyhedra, http://pi.math.cornell.edu/ polyhedral. |
| [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. |
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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