Weyl’s asymptotic formula for fractal Laplacians defined by a class of self-similar measures with overlaps

We observe that some self-similar measures that we call essentially of finite type satisfy countable measure type condition. We make use of this condition to set up a framework to obtain a precise analog of Weyl’s asymptotic formula for the eigenvalue counting function of Laplacians defined by measures, emphasizing on one-dimensional self-similar measures with overlaps. As an application of our result, we obtain an analog of a semi-classical asymptotic formula for the number of negative eigenvalues of fractal Schrödinger operators as the parameter tends to infinity.