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 \(H^\omega =H_0+V^\omega \) $H^{\omega }=H_0+V^{\omega }$ on fractal spaces of infinite diameter. The kinetic term \(H_0\) $H_0$ is given by \(\phi (-{\mathcal {L}}),\) $\varphi (-\mathcal{L}),$ where \({\mathcal {L}}\) $\mathcal{L}$ is the Laplacian on the fractal and \(\phi \) $\varphi$ is a completely monotone function satisfying some mild regularity conditions. The random potential \(V^\omega \) $V^{\omega }$ is of alloy-type.