Conformal Measure Spectra for Quasicircles

We give numerical results for 2 quasicircles (c=.2 and c =.33-.25i). This page contains eigenvalue tables and pictures of the eigenfunctions.

The highlights of our findings:

Unlike for equilibrium measure, the spectrum of our Lapcian is not a topological invariant, thus we present results for 2 quasicircles. This spectrum lacks much of the structure of the equilibrium measure. Because symmetry about the real axis makes computation simpler we have results up to level 12 for c=.2, but only up to level 9, for c=.33-.25i.

Here are pictures of the two Quasicircles we give results for:

Here is a table of eigenvalues:
- c=.2: ( unnormalized | normalized).
- c=.33-.25i: ( unnormalized | normalized).
Here are pictures of eigenfunctions on level 6-12 (6-9 for the variant). The first 256 eigenfunctions are shown 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.

c=.2:
- Level 6: PDF
- Level 7: PDF
- Level 8: PDF
- Level 9: PDF
- Level 10: PDF
- Level 11: PDF
- Level 12: PDF
c=.33-.25i:
- Level 6: PDF
- Level 7: PDF
- Level 8: PDF
- Level 9: PDF

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Last Update: 1 June 2009