Exact Spectral Asymptotics on the Sierpinski Gasket

TL;DR

This paper derives an exact, remainder-free formula for the spectral counting function on the Sierpinski gasket, revealing precise spectral asymptotics that surpass the typical power law behavior seen on manifolds.

Contribution

It provides the first exact formula for spectral asymptotics on a fractal, with no remainder, for almost every spectral parameter, advancing analysis on fractals beyond previous approximate results.

Findings

Exact spectral asymptotics formula for the Sierpinski gasket

No remainder term in the spectral counting function

Stronger than manifold spectral asymptotics

Abstract

One of the ways that analysis on fractals is more complicated than analysis on manifolds is that the asymptotic behavior of the spectral counting function $N(t)$ has a power law modulated by a nonconstant multiplicatively periodic function. Nevertheless, we show that for the Sierpinski gasket it is possible to write an exact formula, with no remainder term, valid for almost every $t$. This is a stronger result than is valid on manifolds.