Quasienergies, Stark Hamiltonians, and growth of energy for driven quantum rings

We study time-dependent Schrödinger operators in Aharonov-Bohm geometries where the flux threading the hole increases linearly with time. We show that the quasienergy operator has, in these cases, the same spectrum as the time-independent Stark Hamiltonian on the universal covering space. Combining known results on Stark Hamiltonians with a theorem of Bellissard, we prove that the energy of a particle on a finite ring, with smooth background potential, increases without bound as t→∞.