Eigenvalue bounds and spectral asymptotics for fractal Laplacians

Abstract

In this work we present Lyapunov type inequalities for generalized one dimensional Laplacian operators defined by positive atomless Borel measures. As applications, we present lower bounds for the first eigenvalue when the measure is a Bernoulli convolution, with or without overlaps. Also, for symmetric Bernoulli convolutions we obtain two sided bounds for higher eigenvalues, and we recover the asymptotic growth of the spectral counting function by elementary means without using the Renewal Theorem. We also consider the Laplacian on the Sierpinsky gasket and other similar fractals, and we deduce a lower bound of their eigenvalues from a Lyapunov type inequality.