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Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains
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} $.
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. 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. |
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. 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. |
































# | 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 |
# | 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 |
# | 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 |
# | 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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