Equilibrium Measure Spectra

We have numerical results for graph levels 2-13. This page contains eigenvalue tables and pictures of the eigenfunctions for the level 6-11 approximations.

The highlights of our findings:

There are two different ways of dividing the eigenfunctions into two families. First we can devide the eigenfunctions into vertical and horizontal families based on their support. Secondly we can devide the eigenfunctions into symmetric and antisymmetric families with respect to the reflection z->-z. Notably given any member of the antisymmetric family, u with eigenvalue l, u(P) (u pre-composed with z^2-1) is an eigenfunction with 2*sqrt(2)*l. Each of the horizontal-symmetric eigenfunctions can be derived from a member horizontal-antisymmetric family by such a method. For the vertical-antisymmetric eigenvalues this process produces and eigenfunction supported on a union of disjoint sets. In this case we find that we have an eigenvalue with multiplicity equal to the number of disjoint sets on which the derived function is supported.

Based on these findings horizontal antisymmetric eigenfunctions are labeled as H and an increasing number, Vertical V. Derived eigenfunctions are labeled as D and number specifying the order of the derivation from the base antisymmetric. For example D3(H5) is the 3 derived function of the fifth horizontal function.

• Here is a table of eigenvalues ( unnormalized | normalized). Note: the little question marks are an artifact of transferring my Excel file to HTML, not a comment on the data.
• Using the calculated eigenvalues on level 2-13 we can predict the eigenvalues in the limit that the level goes to infinity. Taking the differences between successive levels and then the ratios of these differences, we find that the ratios alternate. Using this pattern we can predict the final values of the first 512 eigenvalues. Using these predictions we get pictures of the eigenvalue counting function and Weyl ratio.
• Here are pictures of eigenfunctions on level 6-11. The HTML version contains the first 52 at each level, the PDF contains the first 256 eigenfunctions, where available. Each row of pictures corresponds to a single eigenfunction, the first column is the ray angle description of the function, the second is the support of function colored by the sign of function, the third shows the restriction of the function to the inner circle.
• Level 6: PDF
• Level 7: PDF
• Level 8: PDF
• Level 9: PDF
• Level 10: PDF
• Level 11: PDF

Last Update: 27 May 2009