Eigenmodes of the Laplacian on some Laakso spaces

Abstract

We analyse the spectrum of a self-adjoint operator on a Laakso space using the projective limit construction originally given by Barlow and Evans. We will use the hierarchical cell structure induced by the choice of approximating quantum graphs to calculate the spectrum with multiplicities. We also extend the method for using the hierarchical cell structure to more general projective limits beyond Laakso spaces.

Keywords:

• fractal
• Laplacian
• projective limit
• eigenvalues
• eigenfunctions

AMS Subject Classifications::

• 34L40 (primary)
• 34L16
• 54B30

Acknowledgements

The authors thank Alexander Teplyaev, Daniel Ford, and Robert Strichartz for their support and feedback throughout the production of this article.