Time-dependent quantum circular billiard

The motion of a quantum particle in a time-dependent circular billiard is studied on the basis of the Schrödinger equation with time-dependent boundary conditions. The cases of monotonically expanding (contracting), non-harmonically, harmonically breathing circles the case when billiard wall suddenly disappears are explored in detail. The exact analytical solutions for monotonically expanding and contracting circles are obtained. For all cases, the time-dependence of the quantum average energy is calculated. It is found that for an harmonically breathing circle, the average energy is time-periodic in the adiabatic regime with the same period as that of the oscillation. For intermediate frequencies which are comparable with the initial frequency of the particle in unperturbed billiard, such periodicity is broken. However, for very high frequencies, the average energy once again becomes periodic. A qualitative analysis of the border between adiabatic and non-adiabatic regimes is provided.