Density of states for the Anderson model on nested fractals

We prove the existence and establish the Lifschitz singularity of the integrated density of states for certain random Hamiltonians $H^{\omega }=H_0+V^{\omega }$ on fractal spaces of infinite diameter. The kinetic term $H_0$ is given by $\phi (-\mathcal{L}),$ where $\mathcal{L}$ is the Laplacian on the fractal and $\phi$ is a completely monotone function satisfying some mild regularity conditions. The random potential $V^{\omega }$ is of alloy-type.