Spectral operators on the Sierpinski gasket I

Abstract

We study spectral operators for the Kigami Laplacian on the Sierpinski gasket. These operators may be expressed as functions of the Laplacian (Dirichlet or Neumann), or as Fourier multipliers for the associated eigenfunction expansions. They include the heat operator, the wave propagator and spectral projections onto various families of eigenspaces. Our approach is both theoretical and computational. Our main result is a technical lemma, extending the method of spectral decimation of Fukushima and Shima to certain eigenfunctions corresponding to ‘forbidden’ eigenvalues. This enables us to compute the kernel of a spectral operator (Neumann) when one of the variables is a boundary point. We present the results of these computations in various cases, and formulate conjectures based on this experimental evidence. We also prove a new result about the trace of the heat kernel as t → 0: not only does it blow up as a power of t (known from the standard on-diagonal heat kernel estimates), but after division by this power of t it exhibits an oscillating behaviour that is asymptotically periodic in log t. Our experimental evidence suggests that the same oscillating behaviour holds for the heat kernel on the diagonal.

Keywords:

• Sierpinski gasket
• Kigami Laplacian
• spectral operators
• heat kernel
• Poisson kernel

AMS Subject Classifications::

• 28A80
• 31C99

Acknowledgements

For A. Allan and M. Barany, the research was supported by the National Science Foundation through the Research Experiences for Undergraduates (REU) Program at Cornell University. For R.S. Strichartz, the research was supported in part by the National Science Foundation, Grant DMS–0652440.