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DEGENERACIES IN THE SPECTRA OF NEUMANN TRIANGLES

Pamela Overfelt

To cite this version:

Pamela Overfelt. DEGENERACIES IN THE SPECTRA OF NEUMANN TRIANGLES. 2018. �hal-

01875118�

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P. L. OVERFELT

Abstract. In the following the degeneracies of the lowest ten eigenvalue levels of the Neumann Laplacian in the space of all triangles have been determined numerically using the finite element method. A number of Neumann diabolical triangles have been found within these levels. The lowest Neumann diabolical triangle occurred for eigenvalue levels 5 and 6 of the triangle 116.714 deg, 38.994 deg, and 24.292 deg. Additionally the degeneracies of the lowest ten eigenvalue levels of the Dirichlet Laplacian have been recomputed using the above method and compared where possible with the results in [1]. In general there was good agreement between the two sets of degeneracies for the Dirichlet case.

1. Introduction

It is well known that accidental degeneracies between adjacent eigenvalue levels can occur for families of planar domains described by at least two parameters [1], [15]. In particular the lowest eigenvalues of the Laplacian over the entire space of triangles with Dirichlet boundary conditions were determined in 1984 by Berry and Wilkinson [1]. Using a Green function approach, they found certain triangles with degeneracies in their spectra. These triangles were found to be equilateral [9], [16], [17], [18], isosceles, or scalene (having no symmetry). The degeneracies of the scalene triangles in [1] were referred to as diabolical points or diabolical triangles. It was not surprising that the equilateral and certain isosceles triangles exhibited degeneracy due to their symmetry properties, but the scalene triangles with degeneracies were unusual.

In the following, the lowest eigenvalues of the Laplacian over the space of all triangles governed by Neumann boundary conditions (Neumann triangles) have been determined numerically using the finite element method (FEM) [2]. It is well known that Dirichlet Laplacians have compact resolvents and thus discrete spectra.

It is less well known that Neumann Laplacians of bounded regions may not have purely discrete spectra [3], [11], [12], [13]. In the case of the family of all triangles, the Neumann Laplacian for a piecewise smooth boundary is compact and thus the members of this family have purely discrete spectra. Also the eigenvalues vary continuously under continuous perturbation of the domain for Dirichlet boundary conditions [10], [14]. For Neumann boundary conditions the continuity still holds when a bounded domain with a smooth boundary is deformed by a ”continuously differentiable transformation” but this is not true in general [3]. The continuity of the spectrum is very important when using the FEM for eigenvalue computation where the boundary is replaced by polygonal approximations [2],[3].

Date: September 12, 2018.

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This paper accomplishes two goals. First the eigenvalues of the family of all triangles with Neumann boundary conditions have been computed for the lowest ten eigenvalue levels (see Table 1).1 Within these ten levels a number of degeneracies have been determined to within the accuracy of the FEM mesh used (for remarks on the FEM for polygonal domains, see [19] and references therein). Second the degeneracies of the family of all triangles with Dirichlet boundary conditions have been determined for the lowest ten eigenvalue levels and compared where possible with the degeneracies in [1] (see Tables 2 and 3).

For the Neumann case (where level 1 is taken to be the zero eigenvalue) there is at least one degeneracy for each set of adjacent levels excluding the lowest two levels.

Using the FEM, 42 total Neumann degeneracies have been determined within the lowest ten eigenvalue levels. Of these, 16 were Neumann diabolical triangles.

Using the FEM for the Dirichlet case, 44 total Dirichlet degeneracies have been determined within the lowest ten levels. Of these, 17 were Dirichlet diabolical triangles.

2. Calculation of Eigenvalue Curves

In the following the eigenvalues, λi, and the eigenfunctions, ψi(x, y) are com- puted numerically by solving the Helmholtz equation

(1) ∆ψi(x, y) +λiψi(x, y) = 0; (x, y)∈Ω and

(2) ∂ψi(x, y)

∂n = 0; (x, y)∈∂Ω

where (2) refers to the Neumann boundary condition and n denotes the outer normal to the boundary, ∂Ω. Ω is the interior of any triangle while ∂Ω is its boundary.

The normalized eigenvalues are given by

(3) ΛnnA

whereAis the triangle area. As in [1],Ais assumed to be fixed andα, β, andγare the three angles of a general triangle. We remove the redundancy associated with labelling the vertices by assumingα≥β ≥γ.

Specifically, by fixingA, the area, andγ, the smallest angle, normalized eigen- value curves as functions of the angle, α, were computed using the FEM imple- mented in Mathematica [2].

As close approaches of successive levels are observed, a bisect the interval tech- nique was used to determine precisely the degenerate eigenvalues to within the accuracy of the FEM mesh used.

The three triangles with analytically known eigenvalues (the isosceles right [10], the half equilateral, and the equilateral [9]) have been used to determine how well the FEM method performs given specific mesh sizes. Also the normalized eigenvalue degeneracies for the case of Dirichlet boundary conditions[1] were recomputed using this technique. This gave a comparison between the Green function formalism of Berry and Wilkinson and the FEM. In the majority of instances, the two methods were in good agreement (see Tables 2 and 3).

1Some beautiful results for the extremal problem on triangles have been produced but are beyond the scope of this paper (see [4], [5], and [6]).

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40 50 60 70 80 90 100 0.0

0.5 1.0 1.5 2.0 2.5

α(deg)

Λ

Figure 1. Normalized Eigenvalue Levels 4 (Black) and 5 (Dashed) vs. αforγ= 38 deg

Figures 1 thru 4 show representative normalized Neumann eigenvalue curves of adjacent levels as functions of the largest angle, α, where the area, A, and the smallest angle,γ, are fixed. Figure 1 shows the close approach of the 4 and 5 levels for γ = 38 deg and α near 104 deg (α is always chosen to be the largest angle).

Using further root finding to determine the precise values ofαandγ, we obtain a 4/5 level degeneracy atα= 104.046 deg, andβ =γ= 37.977 deg (see Table 1).

Figure 2 shows a close approach of the 5 and 6 levels for γ = 24.3 deg and α near 115 deg. Further precision shows that a 5/6 level degeneracy occurs for α = 116.714 deg, β = 38.994 deg and γ = 24.292 deg (see Table 1). This is the lowest degeneracy involving a Neumann diabolical triangle.

Figure 3 shows a close approach of the 6 and 7 levels for γ = 41 deg, with α near 84 deg. Again further precision shows that a 6/7 level degeneracy occurs for α= 83.805 deg,β = 55.476 deg, andγ = 40.719 deg (see Table 1). This is one of the two Neumann diabolical triangles which is an acute scalene triangle.

Finally Figure 4 shows a close approach of the 7 and 8 levels forγ= 10.2 deg and αnear 85 deg. In this case there is a 7/8 level degeneracy atα=β = 84.895 deg, andγ= 10.210 deg. This triangle is an acute isosceles triangle. When α=β such plots show only one close approach. When β = γ as in Fig. 1, the numerically smaller point of closest approach gives the value ofβ. This is also true for Figs. 2 and 3 even though in those figuresα, β, andγare all distinct.

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40 60 80 100 120 1.6

1.8 2.0 2.2 2.4 2.6 2.8 3.0

α(deg)

Λ

Figure 2. Normalized Eigenvalue Levels 5 (Black) and 6 (Dashed) vs. αforγ= 24.3 deg

40 50 60 70 80 90 100

2.8 3.0 3.2 3.4

α(deg)

Λ

Figure 3. Normalized Eigenvalue Levels 6 (Black) and 7 (Dashed) vs. αforγ= 41 deg

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0 50 100 150 0.0

0.5 1.0 1.5 2.0 2.5 3.0

α(deg)

Λ

Figure 4. Normalized Eigenvalue Levels 7 (Black) and 8 (Dashed) vs. αforγ= 10.2 deg

3. Discussion of Degenerate Neumann Eigenvalues

The degeneracies in the eigenvalues of the family of all triangles governed by Neu- mann boundary conditions are given in Table 1. The results for the Neumann case will now be considered in detail. The lowest degeneracy is the 2/3 level degeneracy found in the equilateral triangle only. The next lowest degeneracy occurs at the 3/4 level for an acute isosceles triangle withα=β = 74.436 deg,γ= 31.128 deg. This is the only 3/4 level degeneracy. There are two 4/5 level degeneracies, one of which occurs for an acute isosceles triangle, α = β = 79.6355 deg, γ = 20.729 deg; the other occurs for an obtuse isosceles triangle,α= 104.046 deg,β =γ= 37.977 deg.

So far all degeneracies have involved the equilateral triangle or some kind of isosceles triangle with enough symmetry to explain the degeneracy [1].

But (just as in the Dirichlet case) at the 5/6 eigenvalue levels, there are five de- generacies. Of these five triangles, one is equilateral, three are isosceles, and one is diabolical. This triangle, withα= 116.714 deg,β = 38.994 deg, andγ= 24.292 deg has a 5/6 level degeneracy and appears to be the lowest Neumann diabolical trian- gle. The lowest Dirichlet diabolical triangle occurred between levels 5 and 6 also [1].

For the lowest ten eigenvalue levels of the Neumann case, there are 16 diabolical triangles showing degeneracies. Their parameters and normalized eigenvalues are given in Table 1 and are denoted by the bold type in the table. In terms of distri- bution the 2/3, 3/4, and 4/5 levels have no diabolical triangles; the 5/6 level has one, the 6/7 level has two, the 7/8 level has three, the 8/9 level has three, and the 9/10 level has seven. Thus the number of diabolical triangles (as well as the total number of degeneracies) increases with level number.

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In both the Neumann and Dirichlet cases, the number of degeneracies at each level is finite and each degeneracy is an isolated point.

In Table 1 it is interesting to note also that of the 16 diabolical triangles, 14 are obtuse and only 2 are acute. For the lowest 10 normalized eigenvalue levels of triangles with Neumann boundary conditions, 42 total degeneracies have been determined.

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Level α β γ Λ

2, 3 60 60 60 0.605

3, 4 74.436 74.436 31.128 1.063

4, 5 79.6355 79.6355 20.729 1.489

4, 5 104.046 37.977 37.977 1.653

5, 6 82.2665 82.2665 15.467 1.906

5, 6 116.714 38.994 24.292 2.114

5, 6 71.1885 71.1885 37.623 2.648

5, 6 64.410 64.410 51.180 2.497

5, 6 60 60 60 2.418

6, 7 83.8455 83.8455 12.309 2.319

6, 7 123.240 39.106 17.655 2.549

6, 7 77.8645 77.8645 24.271 2.986

6, 7 131.040 24.480 24.480 2.558

6, 7 83.805 55.476 40.719 3.166

7, 8 84.895 84.895 10.210 2.728

7, 8 127.171 39.023 13.806 2.975

7, 8 137.927 24.380 17.693 2.984

7, 8 81.083 81.083 17.834 3.358

7, 8 102.900 49.803 27.297 3.684

7, 8 60 60 60 4.232

8, 9 85.642 85.642 8.716 3.135

8, 9 129.797 38.894 11.309 3.397

8, 9 141.947 24.246 13.807 3.404

8, 9 82.984 82.984 14.032 3.749

8, 9 144.799 17.6005 17.6005 3.404

8, 9 113.301 47.233 19.466 4.036

8, 9 121.184 29.408 29.408 4.180

8, 9 107.491 36.2545 36.2545 4.302

8, 9 70.864 70.864 38.272 4.730

9, 10 86.200 86.200 7.600 3.541

9, 10 131.678 38.758 9.564 3.815

9, 10 144.583 24.117 11.300 3.821

9, 10 148.765 17.500 13.735 3.820

9, 10 119.119 45.821 15.060 4.410

9, 10 80.580 80.580 18.840 5.022

9, 10 131.109 29.010 19.881 4.533

9, 10 78.2265 78.2265 23.547 4.847

9, 10 77.4135 77.4135 25.173 4.812

9, 10 110.575 41.816 27.609 5.019

9, 10 79.682 57.266 43.052 5.616

9, 10 64.029 64.029 51.942 5.538

9, 10 60 60 60 5.441

Table 1. Degeneracies in Triangle Spectra - Neumann Boundary Conditions. Note: Angles are in degrees. Bold entries denote diabolical triangles.

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4. Discussion of Degenerate Dirichlet Eigenvalues

Tables 2 and 3 give the parameters and normalized eigenvalues of the family of triangles governed by Dirichlet boundary conditions. This table gives a comparison between the Green function technique used by Berry and Wilkinson [1] and the Mathematica [2] finite element method used in this paper. In Tables 2 and 3 for each set of adjacent levels, the first line gives the results from this paper (PO) and the second line gives the Berry and Wilkinson result (BW). Generally there is good agreement between the two methods. One difference between the methodologies is that Berry and Wilkinson used the energy (normalized eigenvalue) region 0< En<

18, while this paper concentrated on degeneracies in the eigenvalues of the lowest ten levels. Thus Berry and Wilkinson have degeneracies that this paper does not cover and vice versa. And of course the FEM (like most numerical methods) loses accuracy for triangles with very small angles (less than 7 deg).

With these caveats in mind, the degeneracies for the Dirichlet case will be con- sidered in detail. The 2/3, 3/4, and 4/5 levels are very similar when compar- ing the two techniques. At the 5/6 level there is a slight discrepancy in the pa- rameters for the lowest Dirichlet diabolical triangle. Berry and Wilkinson values are α = 130.57 deg, β = 30.73 deg, and γ = 18.70 deg [1] while the FEM gives α= 130.483 deg,β = 30.803 deg, andγ= 18.714 deg. This discrepancy is not large and is only mentioned because this is the lowest level diabolical triangle for the Dirichlet case [1]. The Dirichlet case has five degeneracies at the 5/6 levels - one equilateral, three isosceles, and one diabolical.

The first major discrepancy between the two methods occurs at the 6/7 lev- els. The FEM gives two diabolical triangles at the 6/7 levels while [1] only found one. From the FEM there is another diabolical triangle at α = 136.390 deg, β = 30.365 deg and γ = 13.245 deg with Λ = 12.213 (see Table 2). With the addition of a second diabolical triangle, the total number of Dirichlet degeneracies at the 6/7 levels is five (Berry and Wilkinson had four).

At the 7/8 levels, using the FEM, there are two diabolical triangles not found in [1]. One is the triangle given by α = 119.470 deg, β = 38.315 deg, and γ = 22.215 deg with Λ = 12.193. The other is the triangle given by α= 148.837 deg, β= 18.126 deg, andγ= 13.037 deg with Λ = 14.424.

On the other hand, Berry and Wilkinson find a degenerate 7/8 level isosceles triangle at α= 149.80 deg,β =γ= 15.10 deg with E= 14.47[1]. Using the FEM, the residual seemed too large in comparison to other degeneracies to indicate a true degeneracy there. Thus the FEM gives six 7/8 level total degeneracies with three of them being diabolical. Berry and Wilkinson found five total 7/8 level degeneracies with only one diabolical triangle.

The 8/9 levels show only one discrepancy between the two methods. The FEM determines a diabolical triangle for α = 142.024 deg, β = 29.699 deg, and γ = 8.277 deg with λ = 16.680 which does not appear in [1]. There are 11 total degeneracies at these levels of which three are diabolical.

At the 9/10 levels, regardless of the magnitude of the normalized eigenvalue (or energy), the FEM gives 13 total degeneracies of which eight are diabolical. Within the same eigenvalue (energy) range used in [1], from the FEM, we find two 9/10 level diabolical triangles that do not appear in [1]. One isα= 84.167 deg,β= 73.190 deg, andγ= 22.643 deg with Λ = 13.857. The other isα= 118.360 deg,β= 37.863 deg, and γ = 23.777 deg with Λ = 14.777. Finally at the 9/10 levels, there is an

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isosceles triangle with angles given byα=β= 62.642 deg andγ= 54.716 deg with Λ = 12.831 that does not appear in [1]. This degeneracy is close to the degeneracy at the 9/10 levels of the equilateral triangle.

For the lowest 10 normalized eigenvalue levels of triangles with Dirichlet bound- ary conditions, 44 total degeneracies of which 17 are diabolical have been deter- mined.

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Level α β γ Λ Source

2, 3 60.000 60.000 60.000 4.232 PO

60.00 60.00 60.00 4.23 BW

3, 4 75.967 75.967 28.066 6.072 PO

75.97 75.97 28.06 6.07 BW

4, 5 81.137 81.137 17.726 8.175 PO

81.14 81.14 17.72 8.17 BW

4, 5 117.636 31.182 31.182 7.845 PO

117.64 31.18 31.18 7.85 BW

5, 6 83.5975 83.5975 12.805 10.346 PO

83.59 83.59 12.82 10.34 BW

5, 6 130.483 30.803 18.714 10.003 PO

130.57 30.73 18.70 10.02 BW

5, 6 74.0385 74.0385 31.923 8.866 PO

74.03 74.03 31.94 8.87 BW

5, 6 61.157 61.157 57.686 7.880 PO

61.15 61.15 57.70 7.88 BW

5, 6 60.000 60.000 60.000 7.860 PO

60.00 60.00 60.00 7.86 BW

6, 7 85.0165 85.0165 9.967 12.546 PO

85.02 85.02 9.96 12.55 BW

6, 7 136.390 30.365 13.245 12.213 PO

- - - - BW

6, 7 143.202 18.399 18.399 12.196 PO

143.32 18.34 18.34 2.15 BW

6, 7 79.9375 79.9375 20.125 10.645 PO

79.95 79.95 20.10 10.65 BW

6, 7 84.949 55.332 39.719 9.793 PO

84.98 55.30 39.72 9.79 BW

7, 8 85.934 85.934 8.132 14.765 PO

85.93 85.93 8.14 14.76 BW

7, 8 139.797 29.999 10.204 14.441 PO

139.88 29.94 10.18 14.44 BW

7, 8 148.837 18.126 13.037 14.424 PO

- - - - BW

7, 8 82.774 82.774 14.452 12.632 PO

82.78 82.78 14.44 12.63 BW

7, 8 - - - - PO

149.80 15.10 15.10 14.47 BW

7, 8 119.470 38.315 22.215 12.193 PO

- - - - BW

7, 8 60.000 60.000 60.000 11.487 PO

60.00 60.00 60.00 11.49 BW

Table 2. Part A of Degeneracies in Triangle Spectra - Dirichlet Boundary Conditions. Note: Angles are in degrees. Boldentries denote diabolical triangles.

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Level α β γ Λ Source

8, 9 86.5725 86.5725 6.855 16.994 PO

86.58 86.58 6.84 16.92 BW

8, 9 142.024 29.699 8.277 16.680 PO

- - - - BW

8, 9 152.032 17.907 10.061 16.666 PO

152.06 17.90 10.04 16.67 BW

8, 9 84.411 84.411 11.178 14.709 PO

84.42 84.42 11.16 14.76 BW

8, 9 154.273 12.8635 12.8635 16.664 PO

154.34 12.83 12.83 16.67 BW

8, 9 128.154 36.539 15.307 14.268 PO

128.16 36.55 15.29 14.27 BW

8, 9 79.1765 79.1765 21.647 13.593 PO

79.18 79.18 21.64 13.59 BW

8, 9 136.022 21.989 21.989 14.226 PO

136.00 22.00 22.00 14.22 BW

8, 9 78.0325 78.0325 23.935 13.209 PO

78.03 78.03 23.94 13.21 BW

8, 9 73.013 73.013 33.974 12.311 PO

73.01 73.01 33.98 12.31 BW

8, 9 111.045 34.4775 34.4775 12.862 PO

111.04 34.48 34.48 12.86 BW

9, 10 87.042 87.042 5.916 19.231 PO

- - - - BW

9, 10 143.595 29.455 6.950 18.925 PO

- - - - BW

9, 10 154.096 17.731 8.173 18.913 PO

- - - - BW

9, 10 85.4665 85.4665 9.067 16.833 PO

85.47 85.47 9.06 16.83 BW

9, 10 157.334 12.724 9.942 18.912 PO

- - - - BW

9, 10 133.054 35.329 11.617 16.403 PO

133.10 35.30 11.60 16.42 BW

9, 10 143.368 21.493 15.139 16.354 PO

143.35 21.53 15.12 16.35 BW

9, 10 82.2305 82.2305 15.539 15.319 PO

82.23 82.23 15.54 15.32 BW

9, 10 84.167 73.190 22.643 13.857 PO

- - - - BW

9, 10 118.360 37.863 23.777 14.777 PO

- - - - BW

9, 10 83.107 56.286 40.607 13.739 PO

83.10 56.29 40.61 13.74 BW

9, 10 62.642 62.642 54.716 12.831 PO

- - - - BW

9, 10 60.000 60.000 60.000 12.697 PO

60.00 60.00 60.00 12.70 BW

Table 3. Part B of Degeneracies in Triangle Spectra - Dirichlet Boundary Conditions. Note: Angles are in degrees. Boldentries denote diabolical triangles.

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5. Comparison between Degeneracies of Dirichlet and Neumann Triangles

Within the first 10 eigenvalue levels, there are 44 total degeneracies for the Dirichlet boundary condition cases determined using the FEM. Of these, 17 are diabolical. Of the 17 Dirichlet diabolical triangles, 14 are obtuse and 3 are acute.

The lowest diabolical Dirichlet triangle occurs at the 5/6 levels. By level, there is one diabolical Dirichlet triangle at the 5/6 levels, two at the 6/7 levels, three at the 7/8 levels, three at the 8/9 levels, and eight at the 9/10 levels, To compare these results with those in [1], (see Tables 2 and 3), Berry and Wilkinson have eight total diabolical Dirichlet triangles for the first 10 levels (where their normalized eigenvalue or energy is restricted to be less than 18). Discounting any triangles at the 9/10 levels in Table 3 which have normalized eigenvalues greater than 18, Table 3 shows 14 total Dirichlet diabolical triangles in the first 10 levels with normalized eigenvalue magnitudes less than 18.

For the Neumann case, within the first 10 eigenvalue levels, there are 42 total degeneracies. Of these 16 are diabolical. Of the 16 Neumann diabolical triangles, 14 are obtuse and 2 are acute. The lowest diabolical Neumann triangle occurs at the 5/6 levels. By level, there is one diabolical Neumann triangle at the 5/6 levels, two at the 6/7 levels, three at the 7/8 levels, three at the 8/9 levels, and 7 at the 9/10 levels.

The number of total degeneracies between Dirichlet and Neumann cases is very similar, the level distribution for total degeneracies is the same except for the 8/9 levels, and the diabolical triangles for each case are similar in both total number and in level distribution.

6. Conclusions

The normalized eigenvalues for the family of triangles with both Neumann and Dirichlet boundary conditions have been considered and a number of degeneracies have been determined numerically for the lowest 10 eigenvalue levels using a finite element method [2]. Where possible the degeneracies of the Dirichlet case have been compared with the results in [1]. In general there is good agreement between the Green function formalism and the FEM. The degeneracies found in both cases were expected to occur when solving the Helmholtz equation for two-parameter planar domains.

It has been conjectured [8] and subsequently proved [7] that the Laplacian with Dirichlet boundary conditions on a generic triangle has a simple spectrum. Al- though the generic Neumann case has yet to be proved, the numerical degeneracies in this paper and in [1] for both types of boundary conditions are at least an indi- cation that a number of triangles do have degenerate spectra.

References

[1] M. V. Berry and M. Wilkinson, Diabolical points in the spectra of triangles, Proc. R. Soc.

Lond. A392, 15-43 (1984).

[2] Wolfram Research, Inc.,Mathematica, Version 11.0, Champaigne, IL (2017).

[3] D. S. Grebenkov and B. - T. Nguyen, Geometrical structure of Laplacian eigenfunctions, arXiv:1206.1278v2(math.AP) 8 Feb (2013).

[4] A. Henrot, Minimization problems for eigenvalues of the Laplacian, Journal of Evolution Equations3, 443-461 (2003)

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[5] R. S. Laugesin and B. A. Siudeja,Triangles and Other Special Domains, Shape Optimization and Spectral TheoryChapter 6, 149-200, De Gruyter Open, Antoine Henrot, Editor (2017) [6] P. R. S. Antunes and P. Freitas, On the inverse spectral problem for Euclidean triangles,

Proc. R. Soc. A,DOI:10.1098/rspa.2010.0540(2010).

[7] L. Hillairet and C. Judge,Spectral simplicity and asymptotic separation of variables, arXiv:

1001.0235 V1(math.SP) 4 Jan (2010).

[8] L. Hillairet and C. Judge,Generic spectral simplicity of polygons, Proc. Amer. Math. Soc.

137, 2139-2145 (2009): arXiv: math/0703616 V3(math.SP).

[9] M. A. Pinsky,The eigenvalues of an equilateral triangle, SIAM J. Math. Anal.11, 819-827 (1980).

[10] R. Courant and D. Hilbert,Methods of Mathematical PhysicsVol. 1(Wiley Classics, 1989).

[11] R. Hempel, L. Seco, and B. Simon,The essential spectrum of Neumann Laplacians on some bounded singular domains, J. Funct. Anal.102, 448-483 (1991).

[12] J. N. V. Gomes and M. A. M. Marrocos,On eigenvalue generic properties of the Laplace- Neumann operator, arXiv: 1510.07067V3(math.DG) 24 Apr. (2018)

[13] M. A. M. Marrocos and A. L. Pereira,Eigenvalues of the Neumann Laplacian in symmetric regions, J. Math. Phys.56, 111502 (2015).

[14] V. Kozloz,Domain dependence of eigenvalues of elliptic type operators, arXiv:1203.2093V1 (math.SP) 9 March (2012).

[15] T. Betcke and L. N. Trefethen,Computations of eigenvalue avoidance in planar domains, Proc. Appl. Math. Mech.4, 634-635 (2004).

[16] M. Prager,Eigenvalues and eigenfunctions of the Laplace operator on an equilateral triangle, Appl. Math.43, 311-320 (1998).

[17] M. Prager,Eigenvalues and eigenfunctions of the Laplace operator on an equilateral triangle for the discrete case, Applications of Mathematics46, 231-239 (2001).

[18] B. J. McCartin,Laplacian Eigenstructure of the Equilateral Triangle(Hikari Ltd, 2011).

[19] X. Liu and S. Oishi,Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape, SIAM J. Numer. Anal.51, 1634-1654 (2013).

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