Spectral Asymptotics for Variational Fractals

A generalization of a classic result of H. Weyl concerning the asymptotics of the spectrum of the Laplace operator is proved for variational fractals. Physically we are studying the density of states for the diffusion through a fractal medium. A variational fractal is a couple (K,E) where K is a self-similar fractal and E is an energy form with some similarity properties connected with those of K. In this class we can find some of the most widely studied families of fractals, such as nested fractals, p.c.f. fractals, the Sierpiński carpet, etc., as well as some regular self-similar Euclidean domains. We find that if r(x) is the number of eigenvalues associated with E smaller than x, then r(x)∼xν/2, where ν is the intrinsic dimension of (K,E).