We study properties of the eigenvalues of the operator \(\frac{d}{d\mu }\frac{d}{dx}\) with homogeneous Neumann and Dirichlet boundary conditions. Our aim is to determine the asymptotic growth behaviour of the eigenvalue counting function if the measure \(\mu \) has an irregular recursive structure described by a so called environment sequence. This construction also allows an application to random homogeneous Cantor-like measures.