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Spectral operators on the Sierpinski gasket. I. (English) Zbl 1188.28006
The authors 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. Their approach is both theoretical and computational. Their 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 them to compute the kernel of a spectral operator (Neumann) when one of the variables is a boundary point. They present the results of these computations in various cases, and formulate conjectures based on this experimental evidence. They also prove a new result about the trace of the heat kernel as \(t \rightarrow 0\): 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 \(\log t\). Their experimental evidence suggests that the same oscillating behaviour holds for the heat kernel on the diagonal.

28A80 Fractals
31C99 Generalizations of potential theory
Full Text: DOI
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