Restrictions of Laplacian eigenfunctions to edges in the Sierpinski gasket

TL;DR

This paper investigates how harmonic and eigenfunctions of the Laplacian behave when restricted to edges in the Sierpinski gasket, providing criteria for their extrema and confirming a conjecture about eigenfunction restrictions.

Contribution

It introduces general criteria for the extrema of restricted eigenfunctions and harmonic functions, and uses spectral decimation to precisely count local extrema, confirming a prior conjecture.

Findings

Harmonic functions restricted to edges are monotone or have a single extremum.

Eigenfunctions can have multiple local extrema along edges.

Spectral decimation allows exact counting of extrema in eigenfunctions.

Abstract

In this paper, we study the restrictions of both the harmonic functions and the eigenfunctions of the symmetric Laplacian to edges of pre-gaskets contained in the Sierpinski gasket. For a harmonic function, its restriction to any edge is either monotone or having a single extremum. For an eigenfunction, it may have several local extrema along edges. We prove general criteria, involving the values of any given function at the endpoints and midpoint of any edge, to determine which case it should be, as well as the asymptotic behavior of the restriction near the endpoints. Moreover, for eigenfunctions, we use spectral decimation to calculate the exact number of the local extrema along any edge. This confirm, in a more general situation, a conjecture of K. Dalrymple, R.S. Strichartz and J.P. Vinson \cite{DSV} on the behavior of the restrictions to edges of the basis Dirichlet eigenfunctions, suggested by the numerical data.