  We investigate the quantum properties of a non-random Hamiltonian with a
step-like singularity. It is shown that the eigenfunctions are multifractals
and, in a certain range of parameters, the level statistics is described
exactly by Semi-Poisson statistics (SP) typical of pseudo-integrable systems.
It is also shown that our results are universal; namely, they depend
exclusively on the presence of the step-like singularity and are not modified
by smooth perturbations of the potential or the addition of a magnetic flux.
Although the quantum properties of our system are similar to those of a
disordered conductor at the Anderson transition, we report important
quantitative differences in both the level statistics and the multifractal
dimensions controlling the transition. Finally the study of quantum transport
properties suggests that the classical singularity induces quantum anomalous
diffusion. We discuss how these findings may be experimentally corroborated by
using ultra cold atoms techniques.
