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Spectrum of the Laplacian on regular polyhedra

  • * Corresponding author

    * Corresponding author

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)

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  • 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:

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  • 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
     | Show Table
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    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
     | Show Table
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    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
     | Show Table
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    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
     | Show Table
    DownLoad: CSV
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