Anderson Transition in Quantum Chaos

We investigate the effect of classical singularities on the quantum properties of nonrandom Hamiltonians. A kicked rotator with a nonanalytical potential is discussed in detail. It is shown that classical singularities produce anomalous diffusion in the classical phase space. Quantum mechanically, the eigenstates of the evolution operator are power-law localized with an exponent given by the type of classical singularity. For logarithmic singularities, the classical motion presents 1/f noise and the quantum properties resemble those of an Anderson transition. Neither the classical nor the quantum properties depend on the details of the potential but only on the type of singularity. We thus define a new universality class in quantum chaos by the relation between classical singularities (anomalous diffusion) and quantum power-law localization.