Spectral Asymptotics on Surfaces of Constant Curvature

Paper

In Progress

Regions Analyzed

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Each Page linked to below contains two items: The Predicted and Calculated Eigenvalues of the Laplacian Operator for that region, listed at the bottom of the page, and six graphs relating to the Eigenvalue Counting Function, which will be defined shortly. Additionally, for regions that do not display eigenvalues with multiplicity greater than 1, there is a scatter plot and histogram of the differences of successive eigenvalues. This is not present in other regions due to the fact that if two predicted eigenvalues are sufficiently close, it is often impossible to tell if they are one eigenvalue of multiplicity two, or two distinct eigenvalues, making it unclear whether the difference is zero and therefore should not be included in the histogram and scatter plot.

Eigenvalues for the regions we analyzed were calculated using the finite element solver built into MATLAB. Given a description of the geometry of a region, MATLAB can auto-generate a mesh over it to solve PDEs on using the finite element method. By refining the mesh, we obtained successively more accurate solutions to the eigenvalues of the Laplacian. As the finite element method always produces over-estimates of the eigenvalues, even the most accurate calculated values contain error. However, by examining the amount of error that was removed with each successive refinement, we were able to conclude that in most cases an approximately constant proportion of the remaining error was removed (usually about 3/4) with each refinement. In these cases we were able to estimate the error remaining after the greatest refinement by modeling it as a portion of the geometric series and subtracting that from most accurate calculated value. By using this method on a control group of regions where the Eigenvalues of the Laplacian are known, such as a Euclidean equilateral triangle or Euclidean disc, we were able to conclude that these predictions were extremely accurate. However, this method does not work well when the error reduction rate is not approximately constant.

The Eigenvalue Counting Function is defined as follows: N(t) = the number of eigenvalues less than t, including multiplicities. For example, if a region produces the Eigenvalues 2,3,3,5,... then N(1.9) = 0, N(2.9) = 1, and N(3.1) = 3. Weyl's Asymptotic Formula and subsequent advances gives us an approximation for N(t) on 2-Spaces of Constant Curvature given by N'(t) = (Area/4pi)t (+/-)(Perimeter/4pi)t^0.5 + c where c is a constant determined by the curvature and angles of the curve describing the region, the curvature of the space, and whether or not the region has a mixture of Neumann and Dirichlet Boundary conditions. If Dirichlet Boundary Conditions apply to the whole of the region, the second term is negative, and if Neumann Boundary Conditions apply, it is positive. If mixed conditions apply to the region, the coefficient of the second term is the sum of the negative lengths of the perimeter under Dirichlet conditions, and the positive lengths under Neumann Conditions.

Our primary motivation in doing this project is to examine the rate of decay to zero of the average value of the error E(t) = N(t) - N'(t). We conjecture that the average value of E(t) decays to zero at a rate of approximately t^(-1/4) in the Euclidean and Hyperbolic 2-spaces, and does not decay to zero in Spherical 2-space. Looking through the pages listed below, we see that the third and fourth counting graphs give a high degree of empirical support for our conjecture over a wide selection of examples in each 2-space.

The six graphs related to the Eigenvalue Counting Function are as follows:

1. N(t)
2. N(t)-N'(t)
3. Average Value of N(t)-N'(t)
4. (Average Value of N(t)-N'(t))*t^0.25
5. Graph 4 plotted against x^2 instead of x
6. Average Value of Graph 5

A zip file of all code used to compute the eigenvalues and generate our analysis of them can be downloaded by clicking here. As much of the analysis was done manually to compensate for issues in existing matlab functions, the analysis is split across several functions so the user may observe and prevent irregularities.

• Euclidean 2-Space

  • Equilateral Triangle (Dirichlet Bounds)
  • 45-45-90 Degree Triangle (Dirichlet Bounds)
  • Arbitrary Triangles (Dirichlet Bounds)
  • Arbitrary Triangles (Neumann Bounds)
  • Arbitrary Triangles (Mixed Bounds)
  • Isosceles Triangle Spectrum (Dirichlet Bounds)
  • Arrowhead (Dirichlet Bounds)
  • Triangle-in-Triangle (Dirichlet Bounds)
  • Regular Pentagon, side length 1 (Dirichlet Bounds)
  • Regular Hexagon, side length 1 (Dirichlet Bounds)
  • Regular Six-Star, side length 1 (Dirichlet Bounds)
  • Disc, radius 1 (Dirichlet Bounds)
  • Disc, radius 1 (Neumann Bounds)

• Hyperbolic 2-Space

  • Hyperbolic Equilateral Triangle Spectrum (Dirichlet Bounds)
  • Arbitrary Hyperbolic Triangles (Dirichlet Bounds)
  • Hyperbolic Discs, radius 1/2, 1, 2 (Dirichlet Bounds)

• Spherical 2-Space

  • Arbitrary Spherical Triangles (Dirichlet Bounds)
  • Equilateral Spherical Triangle, theta = pi/2 (Dirichlet Bounds)
  • Equilateral Spherical Triangle, theta = pi/2 (Neumann Bounds)
  • Spherical Disc, radius 1 (Dirichlet Bounds)

Additionally, we would like to provide you with a presentation on our research that was offered in August 2014. By looking at our goals then you can get some idea of the progress which we have made. The presentation can be found here.